1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
�
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
a)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
16
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
b)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives orbits within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
c)
isochrone: This potential has the interesting property that the orbital period is a function
only of the energy of the orbit. Recall that the Kepler potential has a similar degeneracy – the
energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
d)
modified Hubble:
The modified Hubble law starts with a surface brightness profile that at
one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
e)
power-law density: These models have simple power-law expressions for the density profile
with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
f)
Dehnen (with special cases Jaffe and Hernquist): These have “cuspy” density profiles for
small r and a power law of the form δ r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
g)
King
models: Starting with the isothermal sphere, King introduced an energy cutoff to the
distribution function that limits the size of the system. The associated truncation radius r
T
is
often suggestively called a tidal radius.
h)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in gravita-
tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
i)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical coordinates (R, z, θ)
it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
j)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer model
that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
k)
“cored” logarithmic: A modified logarithmic potential that avoids a singularity at
R =
0
and scales the z axis by the factor q so as to give a spheroidal potential, with q
<
1 in the
oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
l)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
m)
self-similar logarithmic: As its name implies, the both the equipotential surfaces and the
equidensity contours all have the same shape. It is separable in r and β, with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
n)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner and outer
surfaces are similar spheroids.
A thin homoeoid is the limiting case where the inner and
outer shells approach each other. Thin homoeoids have the remarkable property their external
equipotentials are spheroids that are confocal with the thin homoeoid, while their internal
potentials are constant. This can be used to generate the potentials of arbitrary spheroidal
mass distributions. Many of the results for spheroidal systems are most simply expressed in
spheroidal coordinates. If one draws nested confocal spheroids, there is a corresponding family
of confocal hyperboloids of revolution. One coordinate is constant on the ellipses (playing the
role of the radius in spherical coordinates) and the other is constant on the hyperboloids of
revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations
because of the compu- tationally efficient methods for solving the Laplacian
with spherical harmonics.
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
o)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
p)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives orbits within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
q)
isochrone: This potential has the interesting property that the orbital period is a function
only of the energy of the orbit. Recall that the Kepler potential has a similar degeneracy – the
energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
r)
modified Hubble:
The modified Hubble law starts with a surface brightness profile that at
one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
s)
power-law density: These models have simple power-law expressions for the density profile
with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
19
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
t)
Dehnen (with special cases Jaffe and Hernquist): These have “cuspy” density profiles for
small r and a power law of the form δ r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
u)
King
models: Starting with the isothermal sphere, King introduced an energy cutoff to the
distribution function that limits the size of the system. The associated truncation radius r
T
is
often suggestively called a tidal radius.
v)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in gravita-
tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
w)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical coordinates (R, z, θ)
it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
x)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer model
that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
y)
“cored” logarithmic: A modified logarithmic potential that avoids a singularity at
R =
0
and scales the z axis by the factor q so as to give a spheroidal potential, with q
<
1 in the
oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
z)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
aa)
self-similar logarithmic: As its name implies, the both the equipotential surfaces and the
equidensity contours all have the same shape. It is separable in r and β, with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
bb)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner and outer
surfaces are similar spheroids.
A thin homoeoid is the limiting case where the inner and
outer shells approach each other. Thin homoeoids have the remarkable property their external
equipotentials are spheroids that are confocal with the thin homoeoid, while their internal
potentials are constant. This can be used to generate the potentials of arbitrary spheroidal
mass distributions. Many of the results for spheroidal systems are most simply expressed in
spheroidal coordinates. If one draws nested confocal spheroids, there is a corresponding family
of confocal hyperboloids of revolution. One coordinate is constant on the ellipses (playing the
role of the radius in spherical coordinates) and the other is constant on the hyperboloids of
revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations because of the compu-
tationally efficient methods for solving the Laplacian with spherical harmonics.
16
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
�
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
cc)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
dd)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives orbits within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
ee)
isochrone: This potential has the interesting property that the orbital period is a function
only of the energy of the orbit. Recall that the Kepler potential has a similar degeneracy – the
energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
ff)
modified Hubble:
The modified Hubble law starts with a surface brightness profile that at
one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
gg)
power-law density: These models have simple power-law expressions for the density profile
with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
18
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
hh)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
ii)
King
models: Starting with the isothermal sphere, King introduced an energy cutoff to the
distribution function that limits the size of the system. The associated truncation radius r
T
is
often suggestively called a tidal radius.
jj)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in gravita-
tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
kk)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical coordinates (R, z, θ)
it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
ll)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer model
that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
mm)
“cored” logarithmic: A modified logarithmic potential that avoids a singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal potential, with q
<
1
in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
nn)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
oo)
self-similar logarithmic: As its name implies, the both the equipotential surfaces and the
equidensity contours all have the same shape. It is separable in r and β, with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
pp)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner and outer
surfaces are similar spheroids.
A thin homoeoid is the limiting case where the inner and
outer shells approach each other. Thin homoeoids have the remarkable property their external
equipotentials are spheroids that are confocal with the thin homoeoid, while their internal
potentials are constant. This can be used to generate the potentials of arbitrary spheroidal
mass distributions. Many of the results for spheroidal systems are most simply expressed in
spheroidal coordinates. If one draws nested confocal spheroids, there is a corresponding family
of confocal hyperboloids of revolution. One coordinate is constant on the ellipses (playing the
role of the radius in spherical coordinates) and the other is constant on the hyperboloids of
revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations
because of the compu- tationally efficient methods for solving the Laplacian
with spherical harmonics.
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
qq)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
rr)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives orbits within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
ss)
isochrone: This potential has the interesting property that the orbital period is a function
only of the energy of the orbit. Recall that the Kepler potential has a similar degeneracy – the
energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
tt)
modified Hubble:
The modified Hubble law starts with a surface brightness profile that at
one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
uu)
power-law density: These models have simple power-law expressions for the density profile
with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
19
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
vv)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
ww)
King
models: Starting with the isothermal sphere, King introduced an energy cutoff to
the distribution function that limits the size of the system. The associated truncation radius r
T
is often suggestively called a tidal radius.
xx)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in gravita-
tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
yy)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical coordinates (R, z, θ)
it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
zz)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer model
that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
aaa)
“cored” logarithmic: A modified logarithmic potential that avoids a singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal potential, with q
<
1
in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
bbb)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
ccc)
self-similar logarithmic: As its name implies, the both the equipotential surfaces
and the equidensity contours all have the same shape. It is separable in r and β, with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
ddd)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner and
outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the inner
and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations because of the compu-
tationally efficient methods for solving the Laplacian with spherical harmonics.
16
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
�
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
eee)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
fff)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives orbits within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
ggg)
isochrone: This potential has the interesting property that the orbital period is a
function only of the energy of the orbit. Recall that the Kepler potential has a similar degeneracy –
the energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
hhh)
modified Hubble:
The modified Hubble law starts with a surface brightness profile
that at one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
iii)
power-law density: These models have simple power-law expressions for the density profile
with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
18
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
jjj)
Dehnen (with special cases Jaffe and Hernquist): These have “cuspy” density profiles for
small r and a power law of the form δ r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
kkk)
King
models: Starting with the isothermal sphere, King introduced an energy cutoff to
the distribution function that limits the size of the system. The associated truncation radius r
T
is often suggestively called a tidal radius.
lll)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in gravita-
tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
mmm)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical coordinates
(R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
nnn)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer
model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
ooo)
“cored” logarithmic: A modified logarithmic potential that avoids a singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal potential, with q
<
1
in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
ppp)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
qqq)
self-similar logarithmic: As its name implies, the both the equipotential surfaces
and the equidensity contours all have the same shape. It is separable in r and β, with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
rrr)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner and outer
surfaces are similar spheroids.
A thin homoeoid is the limiting case where the inner and
outer shells approach each other. Thin homoeoids have the remarkable property their external
equipotentials are spheroids that are confocal with the thin homoeoid, while their internal
potentials are constant. This can be used to generate the potentials of arbitrary spheroidal
mass distributions. Many of the results for spheroidal systems are most simply expressed in
spheroidal coordinates. If one draws nested confocal spheroids, there is a corresponding family
of confocal hyperboloids of revolution. One coordinate is constant on the ellipses (playing the
role of the radius in spherical coordinates) and the other is constant on the hyperboloids of
revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations
because of the compu- tationally efficient methods for solving the Laplacian
with spherical harmonics.
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
sss)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
ttt)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives orbits within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
uuu)
isochrone: This potential has the interesting property that the orbital period is a
function only of the energy of the orbit. Recall that the Kepler potential has a similar degeneracy –
the energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
vvv)
modified Hubble:
The modified Hubble law starts with a surface brightness profile
that at one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
www)
power-law density: These models have simple power-law expressions for the density
profile with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
19
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
xxx)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
yyy)
King
models: Starting with the isothermal sphere, King introduced an energy cutoff to
the distribution function that limits the size of the system. The associated truncation radius r
T
is often suggestively called a tidal radius.
zzz)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in gravita-
tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
aaaa)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical coordinates
(R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
bbbb)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer
model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
cccc)
“cored” logarithmic: A modified logarithmic potential that avoids a singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal potential, with q
<
1
in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
dddd)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
eeee)
self-similar logarithmic: As its name implies, the both the equipotential surfaces
and the equidensity contours all have the same shape. It is separable in r and β, with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
ffff)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner and
outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the inner
and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations because of the compu-
tationally efficient methods for solving the Laplacian with spherical harmonics.
16
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
�
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
gggg)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
hhhh)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives orbits
within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
iiii)
isochrone: This potential has the interesting property that the orbital period is a function
only of the energy of the orbit. Recall that the Kepler potential has a similar degeneracy – the
energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
jjjj)
modified Hubble:
The modified Hubble law starts with a surface brightness profile that at
one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
kkkk)
power-law density: These models have simple power-law expressions for the density
profile with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
18
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
llll)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
mmmm)
King
models: Starting with the isothermal sphere, King introduced an energy cutoff to
the distribution function that limits the size of the system. The associated truncation radius r
T
is often suggestively called a tidal radius.
nnnn)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in
gravita- tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
oooo)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical coordinates
(R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
pppp)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer
model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
qqqq)
“cored” logarithmic: A modified logarithmic potential that avoids a singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal potential, with q
<
1
in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
rrrr)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
ssss)
self-similar logarithmic: As its name implies, the both the equipotential surfaces
and the equidensity contours all have the same shape. It is separable in r and β, with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
tttt)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner and
outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the inner
and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations
because of the compu- tationally efficient methods for solving the Laplacian
with spherical harmonics.
�
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
uuuu)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
17
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
vvvv)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives orbits
within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
wwww)
isochrone: This potential has the interesting property that the orbital period is a
function only of the energy of the orbit. Recall that the Kepler potential has a similar degeneracy –
the energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
xxxx)
modified Hubble:
The modified Hubble law starts with a surface brightness profile
that at one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
yyyy)
power-law density: These models have simple power-law expressions for the density
profile with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
zzzz)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
aaaaa)
King
models: Starting with the isothermal sphere, King introduced an energy cutoff to
the distribution function that limits the size of the system. The associated truncation radius r
T
is often suggestively called a tidal radius.
bbbbb)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in
gravita- tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
ccccc)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical coordinates
(R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
ddddd)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer
model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
eeeee)
“cored” logarithmic: A modified logarithmic potential that avoids a singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal potential, with q
<
1
in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
fffff)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
ggggg)
self-similar logarithmic: As its name implies, the both the equipotential surfaces
and the equidensity contours all have the same shape. It is separable in r and β, with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
hhhhh)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner and
outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the inner
and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations because of the compu-
tationally efficient methods for solving the Laplacian with spherical harmonics.
16
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
�
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
iiiii)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
jjjjj)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives orbits within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
kkkkk)
isochrone: This potential has the interesting property that the orbital period is a
function only of the energy of the orbit. Recall that the Kepler potential has a similar degeneracy –
the energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
lllll)
modified Hubble:
The modified Hubble law starts with a surface brightness profile that
at one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
mmmmm)
power-law density: These models have simple power-law expressions for the
density profile with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
18
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
nnnnn)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
ooooo)
King
models: Starting with the isothermal sphere, King introduced an energy cutoff to
the distribution function that limits the size of the system. The associated truncation radius r
T
is often suggestively called a tidal radius.
ppppp)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in
gravita- tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
qqqqq)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical coordinates
(R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
rrrrr)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer
model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
sssss)
“cored” logarithmic: A modified logarithmic potential that avoids a singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal potential, with q
<
1
in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
ttttt)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
uuuuu)
self-similar logarithmic: As its name implies, the both the equipotential surfaces
and the equidensity contours all have the same shape. It is separable in r and β, with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
vvvvv)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner and
outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the inner
and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations
because of the compu- tationally efficient methods for solving the Laplacian
with spherical harmonics.
�
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
wwwww)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
xxxxx)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives orbits
within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
yyyyy)
isochrone: This potential has the interesting property that the orbital period is a
function only of the energy of the orbit. Recall that the Kepler potential has a similar degeneracy –
the energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
zzzzz)
modified Hubble:
The modified Hubble law starts with a surface brightness profile
that at one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
aaaaaa)
power-law density: These models have simple power-law expressions for the density
profile with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
bbbbbb)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
cccccc)
King
models: Starting with the isothermal sphere, King introduced an energy cutoff to
the distribution function that limits the size of the system. The associated truncation radius r
T
is often suggestively called a tidal radius.
dddddd)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in
gravita- tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
eeeeee)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical coordinates
(R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
ffffff)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer
model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
gggggg)
“cored” logarithmic: A modified logarithmic potential that avoids a singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal potential, with q
<
1
in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
hhhhhh)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
iiiiii)
self-similar logarithmic: As its name implies, the both the equipotential surfaces
and the equidensity contours all have the same shape. It is separable in r and β, with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
jjjjjj)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner and
outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the inner
and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations because of the compu-
tationally efficient methods for solving the Laplacian with spherical harmonics.
16
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
�
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
kkkkkk)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
llllll)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives orbits
within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
mmmmmm)
isochrone: This potential has the interesting property that the orbital period is
a function only of the energy of the orbit. Recall that the Kepler potential has a similar
degeneracy – the energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
nnnnnn)
modified Hubble:
The modified Hubble law starts with a surface brightness profile
that at one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
oooooo)
power-law density: These models have simple power-law expressions for the density
profile with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
18
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
pppppp)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
qqqqqq)
King
models: Starting with the isothermal sphere, King introduced an energy cutoff to
the distribution function that limits the size of the system. The associated truncation radius r
T
is often suggestively called a tidal radius.
rrrrrr)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in
gravita- tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
ssssss)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical coordinates
(R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
tttttt)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer
model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
uuuuuu)
“cored” logarithmic: A modified logarithmic potential that avoids a singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal potential, with q
<
1
in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
vvvvvv)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
wwwwww)
self-similar logarithmic: As its name implies, the both the equipotential
surfaces and the equidensity contours all have the same shape. It is separable in r and β,
with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
xxxxxx)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner and
outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the inner
and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations
because of the compu- tationally efficient methods for solving the Laplacian
�
�
with spherical harmonics.
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
yyyyyy)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
zzzzzz)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives orbits
within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
aaaaaaa)
isochrone: This potential has the interesting property that the orbital period is a
function only of the energy of the orbit. Recall that the Kepler potential has a similar degeneracy –
the energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
bbbbbbb)
modified Hubble:
The modified Hubble law starts with a surface brightness profile
that at one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
ccccccc)
power-law density: These models have simple power-law expressions for the density
profile with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
ddddddd)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
eeeeeee)
King
models: Starting with the isothermal sphere, King introduced an energy cutoff to
the distribution function that limits the size of the system. The associated truncation radius r
T
is often suggestively called a tidal radius.
fffffff)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in
gravita- tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
ggggggg)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical coordinates
(R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
hhhhhhh)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer
model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
iiiiiii)
“cored” logarithmic: A modified logarithmic potential that avoids a singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal potential, with q
<
1
in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
jjjjjjj)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
kkkkkkk)
self-similar logarithmic: As its name implies, the both the equipotential surfaces
and the equidensity contours all have the same shape. It is separable in r and β, with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
lllllll)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner and
outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the inner
and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations because of the compu-
tationally efficient methods for solving the Laplacian with spherical harmonics.
16
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
�
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
mmmmmmm)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
nnnnnnn)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives orbits
within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
ooooooo)
isochrone: This potential has the interesting property that the orbital period is a
function only of the energy of the orbit. Recall that the Kepler potential has a similar degeneracy –
the energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
ppppppp)
modified Hubble:
The modified Hubble law starts with a surface brightness profile
that at one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
qqqqqqq)
power-law density: These models have simple power-law expressions for the density
profile with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
18
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
rrrrrrr)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
sssssss)
King
models: Starting with the isothermal sphere, King introduced an energy cutoff to
the distribution function that limits the size of the system. The associated truncation radius r
T
is often suggestively called a tidal radius.
ttttttt)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in
gravita- tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
uuuuuuu)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical coordinates
(R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
vvvvvvv)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer
model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
wwwwwww)
“cored” logarithmic: A modified logarithmic potential that avoids a
singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal
potential, with q
<
1 in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
xxxxxxx)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
yyyyyyy)
self-similar logarithmic: As its name implies, the both the equipotential surfaces
and the equidensity contours all have the same shape. It is separable in r and β, with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
zzzzzzz)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner and
outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the inner
and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations
because of the compu- tationally efficient methods for solving the Laplacian
with spherical harmonics.
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
aaaaaaaa)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
bbbbbbbb)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives
orbits within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
cccccccc)
isochrone: This potential has the interesting property that the orbital period is a
function only of the energy of the orbit. Recall that the Kepler potential has a similar degeneracy –
the energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
dddddddd)
modified Hubble:
The modified Hubble law starts with a surface brightness
profile that at one time was thought to be a close fit to those observed in for elliptical
galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
eeeeeeee)
power-law density: These models have simple power-law expressions for the
density profile with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
19
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
ffffffff)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
gggggggg)
King
models: Starting with the isothermal sphere, King introduced an energy cutoff to
the distribution function that limits the size of the system. The associated truncation radius r
T
is often suggestively called a tidal radius.
hhhhhhhh)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles
in gravita- tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
iiiiiiii)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical coordinates
(R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
jjjjjjjj)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer
model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
kkkkkkkk)
“cored” logarithmic: A modified logarithmic potential that avoids a singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal potential, with q
<
1
in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
llllllll)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
mmmmmmmm)
self-similar logarithmic: As its name implies, the both the equipotential
surfaces and the equidensity contours all have the same shape. It is separable in r and β,
with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
nnnnnnnn)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner
and outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the
inner and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations because of the compu-
tationally efficient methods for solving the Laplacian with spherical harmonics.
16
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
�
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
oooooooo)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
pppppppp)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives
orbits within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
qqqqqqqq)
isochrone: This potential has the interesting property that the orbital period is
a function only of the energy of the orbit. Recall that the Kepler potential has a similar
degeneracy – the energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
rrrrrrrr)
modified Hubble:
The modified Hubble law starts with a surface brightness profile
that at one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
ssssssss)
power-law density: These models have simple power-law expressions for the density
profile with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
18
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
tttttttt)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
uuuuuuuu)
King
models: Starting with the isothermal sphere, King introduced an energy
cutoff to the distribution function that limits the size of the system. The associated truncation
radius r
T
is often suggestively called a tidal radius.
vvvvvvvv)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in
gravita- tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
wwwwwwww)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical
coordinates (R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
xxxxxxxx)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer
model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
yyyyyyyy)
“cored” logarithmic: A modified logarithmic potential that avoids a singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal potential, with q
<
1
in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
zzzzzzzz)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
aaaaaaaaa)
self-similar logarithmic: As its name implies, the both the equipotential
surfaces and the equidensity contours all have the same shape. It is separable in r and β,
with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
bbbbbbbbb)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner
and outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the
inner and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations
because of the compu- tationally efficient methods for solving the Laplacian
with spherical harmonics.
Density-potential pairs & orbits
17
�
�
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
ccccccccc)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
ddddddddd)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives
orbits within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
eeeeeeeee)
isochrone: This potential has the interesting property that the orbital period is
a function only of the energy of the orbit. Recall that the Kepler potential has a similar
degeneracy – the energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
fffffffff)
modified Hubble:
The modified Hubble law starts with a surface brightness profile
that at one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
ggggggggg)
power-law density: These models have simple power-law expressions for the
density profile with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
19
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
hhhhhhhhh)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
iiiiiiiii)
King
models: Starting with the isothermal sphere, King introduced an energy cutoff to
the distribution function that limits the size of the system. The associated truncation radius r
T
is often suggestively called a tidal radius.
jjjjjjjjj)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in
gravita- tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
kkkkkkkkk)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical
coordinates (R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
lllllllll)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer
model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
mmmmmmmmm)
“cored” logarithmic: A modified logarithmic potential that avoids a
singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal
potential, with q
<
1 in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
nnnnnnnnn)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
ooooooooo)
self-similar logarithmic: As its name implies, the both the equipotential
surfaces and the equidensity contours all have the same shape. It is separable in r and β,
with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
ppppppppp)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner
and outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the
inner and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations because of the compu-
tationally efficient methods for solving the Laplacian with spherical harmonics.
16
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
�
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
qqqqqqqqq)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
rrrrrrrrr)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives orbits
within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
sssssssss)
isochrone: This potential has the interesting property that the orbital period is a
function only of the energy of the orbit. Recall that the Kepler potential has a similar degeneracy –
the energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
ttttttttt)
modified Hubble:
The modified Hubble law starts with a surface brightness profile
that at one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
uuuuuuuuu)
power-law density: These models have simple power-law expressions for the
density profile with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
18
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
vvvvvvvvv)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
wwwwwwwww)
King
models: Starting with the isothermal sphere, King introduced an energy
cutoff to the distribution function that limits the size of the system. The associated truncation
radius r
T
is often suggestively called a tidal radius.
xxxxxxxxx)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in
gravita- tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
yyyyyyyyy)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical
coordinates (R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
zzzzzzzzz)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical Plummer
model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
aaaaaaaaaa)
“cored” logarithmic: A modified logarithmic potential that avoids a
singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal
potential, with q
<
1 in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
bbbbbbbbbb)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
cccccccccc)
self-similar logarithmic: As its name implies, the both the equipotential
surfaces and the equidensity contours all have the same shape. It is separable in r and β,
with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
dddddddddd)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner
and outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the
inner and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations
because of the compu- tationally efficient methods for solving the Laplacian
with spherical harmonics.
Density-potential pairs & orbits
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
�
�
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
eeeeeeeeee)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
ffffffffff)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives orbits
within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
gggggggggg)
isochrone: This potential has the interesting property that the orbital period is
a function only of the energy of the orbit. Recall that the Kepler potential has a similar
degeneracy – the energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
hhhhhhhhhh)
modified Hubble:
The modified Hubble law starts with a surface brightness
profile that at one time was thought to be a close fit to those observed in for elliptical
galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
iiiiiiiiii)
power-law density: These models have simple power-law expressions for the density
profile with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
19
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
jjjjjjjjjj)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
kkkkkkkkkk)
King
models: Starting with the isothermal sphere, King introduced an energy
cutoff to the distribution function that limits the size of the system. The associated truncation
radius r
T
is often suggestively called a tidal radius.
llllllllll)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in
gravita- tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
mmmmmmmmmm)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical
coordinates (R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
nnnnnnnnnn)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical
Plummer model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
oooooooooo)
“cored” logarithmic: A modified logarithmic potential that avoids a
singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal
potential, with q
<
1 in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
pppppppppp)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
qqqqqqqqqq)
self-similar logarithmic: As its name implies, the both the equipotential
surfaces and the equidensity contours all have the same shape. It is separable in r and β,
with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
rrrrrrrrrr)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner and
outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the inner
and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations because of the compu-
tationally efficient methods for solving the Laplacian with spherical harmonics.
16
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
�
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
ssssssssss)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
tttttttttt)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives orbits
within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
uuuuuuuuuu)
isochrone: This potential has the interesting property that the orbital period is
a function only of the energy of the orbit. Recall that the Kepler potential has a similar
degeneracy – the energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
vvvvvvvvvv)
modified Hubble:
The modified Hubble law starts with a surface brightness
profile that at one time was thought to be a close fit to those observed in for elliptical
galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
wwwwwwwwww)
power-law density: These models have simple power-law expressions
for the density profile with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
18
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
xxxxxxxxxx)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
yyyyyyyyyy)
King
models: Starting with the isothermal sphere, King introduced an energy
cutoff to the distribution function that limits the size of the system. The associated truncation
radius r
T
is often suggestively called a tidal radius.
zzzzzzzzzz)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles in
gravita- tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
aaaaaaaaaaa)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical
coordinates (R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
bbbbbbbbbbb)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical
Plummer model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
ccccccccccc)
“cored” logarithmic: A modified logarithmic potential that avoids a
singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal
potential, with q
<
1 in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
ddddddddddd)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
eeeeeeeeeee)
self-similar logarithmic: As its name implies, the both the equipotential
surfaces and the equidensity contours all have the same shape. It is separable in r and β,
with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
fffffffffff)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner and
outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the inner
and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations
because of the compu- tationally efficient methods for solving the Laplacian
with spherical harmonics.
Density-potential pairs & orbits
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
�
�
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
ggggggggggg)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
hhhhhhhhhhh)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives
orbits within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
iiiiiiiiiii)
isochrone: This potential has the interesting property that the orbital period is a
function only of the energy of the orbit. Recall that the Kepler potential has a similar degeneracy –
the energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
jjjjjjjjjjj)
modified Hubble:
The modified Hubble law starts with a surface brightness profile
that at one time was thought to be a close fit to those observed in for elliptical galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
kkkkkkkkkkk)
power-law density: These models have simple power-law expressions for the
density profile with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
19
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
lllllllllll)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
mmmmmmmmmmm)
King
models: Starting with the isothermal sphere, King introduced an
energy cutoff to the distribution function that limits the size of the system. The associated
truncation radius r
T
is often suggestively called a tidal radius.
nnnnnnnnnnn)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles
in gravita- tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
ooooooooooo)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical
coordinates (R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
ppppppppppp)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical
Plummer model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
qqqqqqqqqqq)
“cored” logarithmic: A modified logarithmic potential that avoids a
singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal
potential, with q
<
1 in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
rrrrrrrrrrr)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
sssssssssss)
self-similar logarithmic: As its name implies, the both the equipotential
surfaces and the equidensity contours all have the same shape. It is separable in r and β,
with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
ttttttttttt)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner and
outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the inner
and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations because of the compu-
tationally efficient methods for solving the Laplacian with spherical harmonics.
16
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
�
�
Density-potential pairs & orbits
Recall that the first step in constructing a galaxy is to pick a gravitational potential. Since differen-
tiation is generally much more manageable than integration, it should be no surprise that we rely on
Poisson’s Equation to get the density distribution from the potential (Λ δ via differentiation rather
than δ Λ via integration). We shall loosely speak of a density-potential pair being a “model” for a
galaxy, but such models are incomplete. As we saw in Figure 1.3 there are many ways to populate a
given density with different orbits. A complete description requires not just a physical density
δ
(
ξ
x
),
but a phase space density
f
(
ξ
x
,
ξ
p
)
(#
stars/unit volume/unit momentum).
Different density-potential pairs are good for different purposes. Some have the virtue of simplicity
or some special symmetry. Others permit one to write the phase space density in closed form, often
a function of only a limited number of variables:
f = f
(E), or
f = f
(E, L
2
) or
f = f
(E, L
z
), where
E,
L
and
L
z
are respectively the total energy of the orbit, its angular momentum and its angular
momentum around the z-axis. Yet others have the property that orbits are self-similar, meaning that
an orbit at any radius can be scaled by an arbitrary factor producing another legitimate orbit. This
is especially useful when generating orbit libraries.
Before proceeding we will want to tighten up our language a bit. We shall use ellipsoid to mean
something whose equi-something contours are elliptical in cross-section but which generically have
three unequal axes. A spheroid has also elliptical cross-sections but has two equal axes. Oblate
spheroids have their long axes equal while prolate spheroids have their short axes equal.
We present the following list of density-potential pairs so that our reader is not taken by surprise
when he encounters one in a talk or in the literature. He might even ask “Please remind me what
special property of the something-something potential makes it appropriate for the problem at hand”
and then say “ah yes” upon hearing the answer.
spherical potentials:
uuuuuuuuuuu)
point mass: The Kepler potential with
GM
Λ(r) =
−
r
.
(1.41)
The energy of an object with mass µ is a function only of the elliptical orbit’s semi-major axis
1.5. DENSITY-POTENTIAL PAIRS & ORBITS
17
3
�
2
�
r
2
3
a:
E(a) =
−
GMµ
2a
.
(1.42)
vvvvvvvvvvv)
homogeneous sphere: A constant density sphere of mass M =
4
ψδR
3
gives
orbits within the
sphere with constant periods
T
=
3ψ
.
(1.43)
Gδ
Formulæ like this are common throoughout astrophysics. The typical dynamical time scale
T
dyn
must vary as (Gδ)
−
1/2
. The potential outside a homogeneous sphere is the same as that
of the point mass Λ(r)
=
−
GM/r while for r
<
R,
Λ(r)
=
−
2ψGδ
�
R
−
r
.
(1.44)
wwwwwwwwwww)
isochrone: This potential has the interesting property that the orbital
period is a function only of the energy of the orbit. Recall that the Kepler potential has a similar
degeneracy – the energy is a function only of the semi-major axis.
GM
Λ(r)
=
−
b
+
∞
b
2
+
r
2
,
(1.45)
giving a central density of
3M
δ
o
=
16ψb
3
(1.46)
xxxxxxxxxxx)
modified Hubble:
The modified Hubble law starts with a surface brightness
profile that at one time was thought to be a close fit to those observed in for elliptical
galaxies,
2j
0
a
I(r) =
1
+
(R/a)
2
,
(1.47)
where j
0
is a central surface brightness and a is a scale length. It is a variant of a similar one
called the Hubble-Reynolds Law. To first order it matches the profile for a non-singular isother-
mal sphere. The scale length plays the role of the core radius of the non-singular isothermal
sphere. The associated analytic potential is somewhat cumbersome, but it is still of some use
(as we shall see below) for clusters of galaxies.
yyyyyyyyyyy)
power-law density: These models have simple power-law expressions for the
density profile with
δ(r)
=
δ
o
⎞
r
o
⎢
γ
.
(1.48)
A special case of the power-law density potential is the singular isothermal sphere.
It has
a constant velocity dispersion and a density profile δ
→
r
−
2
.
This gives our old friend, the
logarithmic potential described in Sec. (1.2). It has a flat rotation curve and infinite mass and
radius. The logarithmic potentials produce self-similar orbits. There are therefore theoretical
as well as practical reasons for choosing them.
18
CHAPTER 1. GALAXIES: DYNAMICS, POTENTIAL THEORY, AND EQUILIBRIA
→ � ∼
s
s
�
�
s
s
Λ(R, z)
=
2 v
o
ln R
c
+
R
+
q
2
.
(1.54)
zzzzzzzzzzz)
Dehnen (with special cases Jaffe and
Hernquist): These have “cuspy” density profiles for small r and a power law of the form δ
r
−
4
as r
. The phase-space distribution function
can be written in closed form as a function of energy and angular momentum,
f
(E, L
2
).
aaaaaaaaaaaa)
King
models: Starting with the isothermal sphere, King introduced an energy
cutoff to the distribution function that limits the size of the system. The associated truncation
radius r
T
is often suggestively called a tidal radius.
bbbbbbbbbbbb)
Navarro-Frenk-White: This is an empirical fit to the observed density profiles
in gravita- tional N-body experiments,
4δ
s
δ(r) = (r/r )(1 + r/r )
2
,
(1.49)
where r
s
is a scale length. It is a member of the larger family
2
3
−
ρ
δ
s
δ(r)
=
(r/r )
ρ
(1
+
r/r )
(
3
−
ρ)
,
(1.50)
of which the ρ
=
1.5 case is called the Moore profile. Astronomy is not immune to fashion and
these days the NFW and Moore profiles are quite fashionable, both as models for galaxy halos
and for clusters of galaxies.
axisymmetric potentials
cccccccccccc)
Plummer-Kuzmin: also known as “Toomre’s model 1.” In cylindrical
coordinates (R, z, θ) it has
Λ(R, z)
=
−
GM
,
(1.51)
R
2
+
(a
+
|
z
|
)
2
giving a infinitesimally thin sheet of mass density at z
=
0 with mass surface density
aM
Ψ(R)
=
2ψ(R
2
+
a
2
)
3/2
.
(1.52)
dddddddddddd)
Miyamoto: A combination of the Plummer-Kuzmin model and the spherical
Plummer model that is often used for globular clusters,
Λ(R, z)
=
−
GM
.
(1.53)
R
2
+ (a +
∞
b
2
+ z
2
)
2
eeeeeeeeeeee)
“cored” logarithmic: A modified logarithmic potential that avoids a
singularity at
R =
0 and scales the z axis by the factor q so as to give a spheroidal
potential, with q
<
1 in the oblate case and q
>
1 in the prolate case:
1
2
�
2 2 z2 �
In the limit of vanishing core radius Rc, the equipotentials all have the same spheroidal shape
and the equidensity contours likewise all have the same shape (though not spheroidal). The
orbits are therefore self-similar.
15
o
r
2l
+
1
r
l+1
0
lm
r
a
l
−
1
ffffffffffff)
Mestel disk: As in Sec. (1.2), we write this in terms of spherical coordinates,
Λ(r, β)
=
v
2
�
ln r
+
ln 1
+
|
cos
β
|
�
.
(1.55)
c
R
c
2
gggggggggggg)
self-similar logarithmic: As its name implies, the both the equipotential
surfaces and the equidensity contours all have the same shape. It is separable in r and β,
with
Λ(r, β)
=
v
2
ln[(r
·
Q(β)]
and
(1.56)
δ(r, β)
=
δ
o
⎞
r
o
⎢
2
S(β),
(1.57)
where Q(β) and S(β) are arbitrary functions for the potential and density (but of course related
to each other via Poisson’s equation). The spheroidal logarithmic potentials describe above are
special cases of this.
hhhhhhhhhhhh)
homoeoids: A homoeoid is a uniform density shell of finite thickness whose inner
and outer surfaces are similar spheroids.
A thin homoeoid is the limiting case where the
inner and outer shells approach each other. Thin homoeoids have the remarkable property their
external equipotentials are spheroids that are confocal with the thin homoeoid, while their
internal potentials are constant. This can be used to generate the potentials of arbitrary
spheroidal mass distributions. Many of the results for spheroidal systems are most simply
expressed in spheroidal coordinates. If one draws nested confocal spheroids, there is a
corresponding family of confocal hyperboloids of revolution. One coordinate is constant on the
ellipses (playing the role of the radius in spherical coordinates) and the other is constant on the
hyperboloids of revolution, playing the role of the polar angle.
non-axisymmetric potentials and the multipole expansion
The logarithmic potential and homoeoid above are straightforwardly generalized to non-axisymmetric
cases. In the logarithmic case of the potential, the density and the functions Q and S are then
functions of r, β and θ. The homoeoids are by extension ellipsoidal rather than spheroidal.
For more general three-dimensional systems, last resort is to expand the potential in spherical
harmonics:
Λ(r, β, θ)
=
−
4ψG
Σ
Y
lm
(β, θ)
�
1
�
r
δ
(a)a
l+2
da
+ r
l
�
≈
δ
lm
(a) da
�
,
(1.58)
where Ylm are the spherical Legendre functions and
δ
lm
(a)
=
�
Y
l
*
m
(β, θ)δ(a, β, θ)dΛ.
(1.59)
The multipole expansion method is frequently used in N-body simulations because of the
compu- tationally efficient methods for solving the Laplacian with spherical harmonics.
