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CH6telescopes.pdf
CH. 6 Telescopes
Background and Directions
One of the main themes of this course is for you to learn that Mathematics, at its root, is a powerful tool
for understanding a variety of natural phenomena. Math is the language of nature. In this problem set, we use a
few simple equations to demonstrate and understand the limits of what telescopes can do.
In the last part of this activity, we retrace an elementary calculation that Edwin Hubble conducted from
spectra collected from a galaxy. He used this approach to determine the speed at which galaxies are receding
from us which became the Hubble Expansion of the Universe Constant. There is much ongoing research into the
precise value and interpretation of this constant.
The relevant equations are:
1) Angular Separation = Physical separation (distance between objects) x 360o___ 2Π x DISTANCE (from you)
Equation 1 is Mathematical Insight 6.1 in your textbook. For a detailed review of the origin of Equation 1, refer to
Mathematical Insight 2.1.
This will give an answer in units of degrees. To get an answer in arcseconds, note that 360o = 3600 arcseconds (“). The right side of equation 2 is simplified to:
A.S = 206,265”* P.S distance
2) Diffraction Limit~ 2.5 x 105 x wavelength of Light (arcseconds) Diameter of telescope
Equation 2 is found in Mathematical Insight 6.2 in your textbook. Note that the units of the diffraction limit come
out to be in arcseconds (“). Look at figure 2.8 in textbook and figure below for a review of arcseconds.
Note-to successfully apply these equations make sure that the metric units you use are the same in numerator and
denominator. For example, in equation 2, if wavelength is in meters, the diameter must also be in meters.
Useful conversion factor:
1 light year = 9.46 x 1015m 1 angstrom = 10-10 m
1. Given two sets of far- away binary stars, one set hot, one set cold, which set does equation 2 predict will be
easier to resolve (see two separate stars as opposed to a big blur)? Explain your answer in terms of equation 2.
2. You are in charge of designing a new telescope. Your goal is to design a telescope that has an angular
resolution of .1”(arcseconds) for light at a wavelength of 2 μm (micrometers). How large must the diameter of the primary mirror be to achieve this goal? Hint: read section 6.2 from textbook.
3. A) The angular resolution/separation of a 30 meter diameter radio telescope observing a radio signal of
21 centimeters would be how many arcseconds”?
B) What is the angular resolution/separation of the fully dilated human eye, with diameter of about 5 mm,
gathering light at an average of 6x10-7 meters in wavelength?
4. How much more light does the Mt Palomar telescope gather with a primary mirror diameter of 5 m
compared to the human eye with D= 5 mm?
5. A) How many times fuzzier will the view be using a 1m diameter telescope in the blue light region
compared to the infrared region? Use blue light = 400 nm and Infrared = 900 nm
B) How many times fuzzier (less resolved) will the view be using a 1m diameter
telescope in the blue light region compared to a radiotelescope (use appropriate
wavelength)?
6. When looking at the moon, Hubble has an Angular Resolution (A.R.) of .018”. Can Hubble see the Apollo
landers that are about 50 meters? Use the equation:
7. A binary star system is 15 light years away and its two stars are separated by 250 million kilometers. Can
Hubble resolve the two stars separately with an Angular Resolution of 0.05 arcseconds? Show work.
8. Measuring Redshifts (textbook section 5.5)
1) find the spectrum of something (usually a galaxy) that shows spectral lines
2) from the pattern of lines, identify which line was created by which atom, ion, or molecule
3) measure the shift of any one of those lines with respect to its expected wavelength, as measured in a
laboratory on Earth
4) use a formula that relates the observed shift to the object's velocity
Since hydrogen is the most common element in the universe, it is often seen in galaxies. Shown below is the
spectrum from galaxy 587731512071880746. We will use only the Hydrogen α β, γ,δ, wavelengths to estimate the
speed of galaxy.
The rest (laboratory) wavelengths (λ) for Hydrogen are given in the following table.
Name Color λ (angstroms =10-10 meters)
Alpha (α) RED 6562.8
Beta (β) Blue-Green 4861.3
Gamma (γ) Violet 4340.5
Delta (δ) Deep Violet 4101.7
A) Carefully read off the observed wavelengths from the spectrum below and fill in the wavelengths in the table
below. Pay no attention to the units of the Y axis. You do not need to use those units for this problem. All you
need to do is read off the wavelengths and fill in the table.
Name Color λ (angstroms) Ζ
Alpha (α) RED
Beta (β) Blue-Green
Gamma (γ) Violet
Delta (δ) Deep Violet
The definition of the redshift (z) is given by:
1 + z = λ observed from spectrum / λ measured in laboratory.
And does not depend on which line you choose. Solve each wavelength for Z.
B) Take the average of z for all wavelengths. Z avg =____________
C) Is the galaxy moving towards or away from us? Justify your answer.
D)To convert from redshift to velocity use the formula:
V = C*Z where C is the speed of light in m/s.
How fast is the galaxy moving? GALAXY SPEED= _____________
E) Discuss your answer in D). Does your answer surprise you? How do you interpret this number?
SPECTRUM FOR GALAXY 587731512071880746
