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Journal of Space Safety Engineering 7 (2020) 318–324

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Journal of Space Safety Engineering

journal homepage: www.elsevier.com/locate/jsse

Quantitative assessment of a threshold for risk mitigation actions

Theodore H. Sweetser a , ∗ , Barbara M. Braun b , Michael Acocella c , Mark A. Vincent d

a Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr. Pasadena, CA, 91109 United States b The Aerospace Corporation, 2155 Louisiana Blvd. NE, Suite 5100, Albuquerque, NM, 87110 United States c Northrop Grumman Space Systems, 45101 Warp Dr., Dulles, VA, 20166 United States d Raytheon Intelligence & Space, 300 N. Lake Ave., Suite 1120, Pasadena, CA 91101 United States

a r t i c l e i n f o

Article history:

Received 8 June 2020

Accepted 5 July 2020

Available online 9 August 2020

a b s t r a c t

Many low-Earth-orbit missions have a policy that if a future conjunction with a secondary object such as a piece

of orbital debris is detected, a go/no-go meeting will be held to decide about a risk mitigation action before

the time of closest approach. Commonly, the policy is that a probability of collision (Pc) above a predetermined

action threshold at the time of the meeting means the mission will take action to reduce the risk. The value to

which the action threshold is set is a compromise —if it is higher, then there is a higher probability that a collision

might occur when action is not taken; if it is lower, then more actions will be taken, increasing the cumulative

costs and risks of the actions themselves. This paper shows how a policy using an action threshold affects the

overall mission risk of a collision with a large object. We augment this with estimates of action success, expected

hard-body radius, and expected covariance to obtain an algorithm for estimating the risk reduction associated

with an action threshold policy. We apply this algorithm to the OCO-2 and CloudSat missions as examples, using

historical conjunction data for these two missions, and show how this algorithm can guide developing missions

in setting an action threshold.

1

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. Introduction

The risk that any Earth-orbiting spacecraft will collide with an ob- ect in space is low —NASA requires that every space mission show that here is less than one in a thousand chance of a collision over the entire ime in orbit before it will be allowed to launch. But the risk is always ncreasing, and with proposals being pursued to launch thousands of pacecrafts for some missions the concern about the risk is increasing ven faster.

One approach to reducing risk is simply to think small —a smaller ross-section has a lower chance of being hit. But this is not often a easible approach. Many Earth-orbiting missions have adopted a more ctive strategy and have established a process to dodge away from any rbiting object (the secondary) that is predicted to come too close to the ission spacecraft (the primary). This generally involves performing a aneuver to change the orbit of the primary, but can also be done by

hanging or delaying a maneuver that was planned for other purposes, .g., an inclination adjustment or for drag make-up, but that puts the rimary in harm’s way.

Either way, the necessary first step is detecting the threat. This in- olves not only noticing that an object is going to pass nearby, i.e., will e in conjunction, but also evaluating the risk posed to see if it is high nough to be worth taking action, which is operationally expensive and oses risks of its own. Almost always this evaluation is done by mea- uring the estimated probability of a collision, traditionally denoted Pc,

∗ Corresponding author.

E-mail address: [email protected] (T.H. Sweetser).

ttps://doi.org/10.1016/j.jsse.2020.07.009

468-8967/© 2020 Published by Elsevier Ltd on behalf of International Association

gainst an action threshold, T a —if Pc > T a then action will be taken, ither a maneuver will be performed or an otherwise already-planned aneuver will be modified. The setting of the action threshold is a policy ecision on the part of a mission project. This paper presents a method or quantifying the effect of using the action threshold to help the mis- ion set the value of T a .

This study is not the first to address the issue of setting the action hreshold. Several studies have been done using a variety of approaches. ften these have attempted to quantify the overall risk and the mitigated

isk, which requires a model of the population of objects in orbit. Some tudies have assumed that a conjunction is happening and focused on ethods to measure the risk of that conjunction given just an estimated iss vector. Many studies have turned the problem around and focused

n quantifying the chance that a collision will not be detected, i.e., that c will be less than T a but that nevertheless a collision happens —this s called the missed-detection problem. Our study concentrates on the isk reduction aspect; we estimate the fraction of overall risk that is itigated as a function of T a , independent of particular conjunctions or hat the actual risk is.

. Missed detection state of the art

Foster and Estes’s paper in 1992 [1] was an early and seminal study o address this issue. They take the approach of calculating the actual

for the Advancement of Space Safety.

T.H. Sweetser, B.M. Braun, M. Acocella et al. Journal of Space Safety Engineering 7 (2020) 318–324

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isk to the mission and the actual risk removed by mitigation actions; hey use flux models to quantify the relationships in a triad of maneuver hresholds, residual risk and maneuver rates.

The analysis of Foster and Stansbery [2] looked at a different re- ion of space for the International Space Station (ISS) and Space Shuttle STS) than the analysis presented here for satellites in the A-Train Con- tellation. They binned secondary objects according to the uncertainty n their orbits, similar to the binning shown in Section 7 below. Thus, heir 15 bins and our 19 bins correspond to a different-sized ellipse in he conjunction plane for a fixed threshold (see Section 5 ). Their over- ll results were also similar to ours: a threshold of 1 × 10 − 4 produced residual risk of about 20% with a few maneuvers needed per year. here was one exception for the STS when a threshold of 3 × 10 − 5 was ore appropriate, but of course doing an avoidance maneuver with the

huttle is in some ways simpler than moving a satellite in the A-Train. Sanchez-Ortiz et al . [3] looked at risk thresholds and the resulting

isk reductions and number of maneuvers for a low-Earth-orbit satellite Envisat), the ISS, and geosynchronous and geosynchronous-transfer or- its. The results for the first is most applicable for comparisons to the esults in this paper, though there has to be some translation since they lot number of maneuvers vs. risk threshold and then risk reduction s. number of maneuvers. They also produced numerical results using a uropean flux model named MASTER.

Chan’s textbook [4] contains explanations of several of the formulas sed here. In particular, in Chapter 4 he introduces the simple formula iven in Eq. 5 below. He also introduces the Rician integral [5] with a implified solution in the form of a truncated series expansion to calcu- ate the 2-D Pc. He notes that if the miss distance is zero then the Rician ntegral becomes the Rayleigh integral which has a closed-form solu- ion. Plus, in Chapter 12 he creates an annual probability of collision or a reference satellite at 618 km altitude which can be compared to he flux models used by others.

Carpenter et al. [6] used statistical methods, in particular the Wald equential Probability Ratio test, with both a frequentist and Bayesian pproach to test the two hypotheses: whether there will be a collision r whether there will not be collision. Notable is that the test, which s based on the entire observation history for a future conjunction, can upport either hypothesis or neither, and in the latter case suggest more bservations are needed. In this strategy, the thresholds for doing a ma- euver (when the test supports a collision hypothesis) or dismissing the onjunction without a maneuver (when the test supports a non-collision ypothesis) depend on a priori values considered acceptable for the prob- bilities of a missed detection and of a false alarm. Their assertion is that his is a better strategy than the action threshold strategy being exam- ned here.

Alfano and Oltrogge [7] present relationships between Mahalanobis pace and Pc. These relationships are then used to derive bounding val- es for false alarms (Type I errors) and missed alarms that result in a ollision (Type II errors), the latter of which is the subject of this paper. owever, they use a somewhat more complicated formula to relate a Pc

hreshold to a Mahalanobis distance threshold (the latter is called dm alert n their paper, but r M in this paper) than is used by us. The 17

th figure n their paper is the probability of being outside a given Mahalanobis creening threshold, which can be compared to Eq. 4 below.

The origin of the analysis presented in Vincent and Sweetser 8] came from the question: “how much credit should be given to doing isk mitigation maneuvers when calculating the mission lifetime prob- bility of being hit by large objects for End of Mission requirements. ” he analysis then focused on the probability of detecting and mitigating conjunction that was actually going to be a collision. That analysis is resented here in Sections 4 and 5 . A side benefit of that analysis was its pplication to a previously-developed tool named PC4 [ 9 , 10 ]. It uses the urrent probability density function (PDF) along with a model of what he covariance (at Time of Closest Approach – TCA) would be from a uture observation of the secondary object to calculate the likelihood hat the future Pc will cross a chosen threshold. As it turns out, if the

319

iss distance is chosen to be zero and the future covariance is modelled o be the same as the current one, this likelihood turns out to be the robability of detection as derived in Section 5 .

. Some basic statistical concepts

These basic concepts are given to establish our notation and recall hose concepts that are used in the development. We use boldface to rep- esent multidimensional variables, such as vectors and matrices; capital etters are used for random variables and for matrices.

.1. Multivariate normal distributions

For our purposes, we consider multivariate normal distributions that re full-rank affine transformations of a standard normal random vector f the same dimension. Thus X = ( X i ), i = 1 to n , is a multivariate normal istribution of dimension n here if and only if X = x 0 + TS , where T is matrix with non-zero determinant and S is the vector ( S i ) of the same imension as X , where each S i is a random variable that is normally istributed with mean 0 and variance 1. The matrix Z = TT T is the co- ariance matrix of X and is positive definite here by definition, where he superscript operator T stands for “transpose ” (and -T means “inverse ranspose ”). The probability density function (PDF) f for a point x is

( 𝒙 ) = 1∕ √

( 2 𝜋) 𝑛 |𝒁 |𝑒 ( 𝒙 − 𝒙 0 ) T 𝒁 −1 ( 𝒙 − 𝒙 0 ) (1) here x 0 is the mean of the n -dimensional random vector X . The square

oot of the determinant of Z is just the determinant of T , which is the caling factor for the change in volume that happens when the trans- ormation T is applied; it is this scaling factor division which keeps the ntegral of the PDF over the space equal to 1 and also preserves the umulative distribution over a volume, i.e., the cumulative distribution ver a region in the space of the standard normal random vectors is the ame as the cumulative distribution over the image of that region by in the multivariate normal vector space. There is no closed-form gen- ral formula for calculating a cumulative distribution value for different olumes within the distribution space, but such a formula does exist for olumes within any fixed Mahalanobis distance of the mean vector.

.2. Mahalanobis distance

In our space navigation work we typically characterize multivariate istributions by considering ellipsoids (ellipses in two dimensions) cen- ered at the mean point and representing one sigma, two sigma, etc., istances from the mean. Fig. 1 shows this for a typical conjunction. hese ellipsoids are the images under the mapping T of the integer ra- ius spheres (circles) in the standard normal random vector space. On hese ellipsoids, as on their preimages, the value of the PDF is constant, s is their distance from the origin in the preimage. This distance in the tandard normal vector space gives what is called the Mahalanobis dis- ance [11] in the multivariate vector space, which is defined this way or any point in the multivariate vector space. More explicitly the Maha- anobis distance r M ( x ) of a point x from the mean x 0 in the multivariate ector space is given by

𝑀 ( 𝒙 ) = ‖‖‖𝑻 −1 (𝒙 − 𝒙 0 )‖‖‖ =

√ ( 𝒙 − 𝒙 0

)T 𝒁

−1 ( 𝒙 − 𝒙 0

) . (2)

Note that this lets us write the PDF for x (see Eq. 1 ) as

( 𝒙 ) = 1∕ √

( 2 𝜋) 𝑛 |𝒁 |𝑒 𝑟 𝑀 ( 𝒙 ) 2 . (3) One characteristic of the Mahalanobis distance is that the distribu-

ion of the Mahalanobis distance from the mean is a chi distribution hose number of degrees of freedom is the dimension of X , since the ahalanobis distance is the distance in the standard normal random

istribution. In two dimensions this means that the cumulative distribu- ion inside the distribution ellipse for any r M is given by the Rayleigh istribution (which is a chi distribution with two degrees of freedom)

T.H. Sweetser, B.M. Braun, M. Acocella et al. Journal of Space Safety Engineering 7 (2020) 318–324

Fig. 1. The distribution of errors in the estimate of the miss vector, relative to

the estimate, is the same as the distribution of possible true miss vectors given

this estimated miss.

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𝐹

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t v u i f a

t w

m a m

a t t o v

1 c t

P

s m s w i b s

5

t w s a t

i o e t a t m

t

Fig. 2. This distribution of possible OD solutions assumes that the true miss

distance is zero.

ith unit scale parameter. Thus the cumulative distribution function CDF) F ( r M ) relative to the mean, i.e., the probability of being within an M -ellipse in Fig. 1 , is just

( 𝑟 𝑀

) = 1 − 𝑒 −( 𝑟 𝑀

2 ∕2) . (4)

. Miss distance and probability of collision

We call the vector from the primary body (our spacecraft) to the econdary body (an object passing nearby) at the closest approach of he secondary to the primary the miss vector . We have a collision if the olume of the secondary body intersects the volume of the primary body long the relative trajectory, either at or just before or after the closest pproach. To simplify our analysis, we approximate this situation by ombining the two body volumes into a sphere centered at the primary ody; then if the miss vector is inside the combined volume we take that o be a collision, and Pc is the integral of the PDF of the miss vector over he combined volume.

To better visualize and to simplify this situation, we map it into the onjunction plane, which has just two dimensions. The conjunction plane s a plane perpendicular to the relative velocity vector at the miss vector; ince the miss vector is at the point of closest approach, the conjunction lane contains both the secondary and the primary body. We coordi- atize the conjunction plane by placing the origin at the primary body nd choosing perpendicular axes in the plane, usually by making the vertical ” axis (when drawing the plane) as the most radial axis in the lane, so that the “horizontal ” axis is truly horizontal with respect to he Earth. Alternative coordinatizations choose the x-axis to include the iss vector, or align the x-axis with the semi-major axis of the error

llipse. Now we have the situation shown in Fig. 1 . We approximate the

ovariance of the estimated miss vector by taking the sum of the covari- nces of the estimated positions of the primary and secondary bodies nd then projecting that onto the conjunction plane. The region of in- erest is the circle centered at the origin and with radius equal to the ombined hard-body radius r HB —if the actual miss vector is within this ircle then we consider that to be a collision. As above, Pc is equal to he integral of the PDF over the area of this circle.

Foster and Estes’s seminal paper in 1992 [1] is widely attributed to be he first to describe the above method for reducing the calculation of Pc alues to 2-D. For the actual calculation they gave two algorithms —one ses the Monte Carlo method and the other simply calls for numerically ntegrating the PDF function (though the formula they give for the PDF unction depends on using coordinates in the conjunction plane that are ligned with the principal axes of the error ellipse). The boundary of

320

he area of the combined hard body is described in polar coordinates, hich are used for the numerical integration.

Subsequent authors [ 12 , 13 ] have implemented various quadrature ethods to do the same numerical integration. Chan [4] has proposed

nalytical techniques and Alfano [12] compared them against the nu- erical methods for computational speed and accuracy.

In particular, Carpenter [6] (following Chan [4] ) has proposed using n ellipse instead of a circle to represent the combined hard body, where he ellipse is similar in shape and orientation to the error ellipse but has he same area as the circular representation; the cumulative distribution ver this ellipse is given by the Rice distribution [5] , which gives the alue of the CDF as an infinite sum involving modified Bessel functions.

In this study we use a much simpler approximation (called Method in [8] ): we simply multiply the value of the PDF at the center of the ircle (i.e., the origin in Fig. 1 ) by the area of the circle, which is 𝜋r HB

2 , hus

c = 𝜋𝑟 𝐻𝐵 2 ∕ √

( 2 𝜋) 2 |𝒁 |𝑒 𝑟 𝑀 ( 𝒙 ) 2 . (5) The compelling advantage of this approximation is shown in the next

ection. Chan (Chapter 4, [4] ) warns of the limitations of this approxi- ation, though in general it is a reasonable approximation if the r HB is

mall compared to the covariance. For example, in the isotropic case, here | Z | = 𝜎2 , Eq. 5 gives the highest overestimate of Pc when x

s the origin because the curvature of the surface is maximum there, ut this maximum overestimate is less than 1% when r HB < 0.2 𝜎. Two- ignificant-digit accuracy is sufficient in the following analysis.

. Probability of detection

In Section 4 we considered the probability of collision given an es- imated miss vector, i.e., the probability that the true miss vector is ithin the combined hard-body radius. Now we turn that problem up-

ide down: given that we are on a collision course, what is the prob- bility that the estimated miss vector will have a Pc above the action hreshold? This situation is illustrated in Fig. 2 .

Note that the covariance of the distribution of possible miss vectors n Fig. 2 is derived from the covariances of the primary and secondary rbit distributions, and is the same as the distribution of errors in the stimate of a miss vector —indeed this is the source of the error distribu- ion. This symmetry is illustrated in Fig. 3 . Thus the value of the Pc for ny particular estimated miss vector is equal to the cumulative distribu- ion over the area of the combined hard body centered on the estimated iss vector, which is shown in blue in Fig. 2 .

Now consider the region on the conjunction plane where Pc is greater han the action threshold value. The cumulative distribution of possible

T.H. Sweetser, B.M. Braun, M. Acocella et al. Journal of Space Safety Engineering 7 (2020) 318–324

Fig. 3. There is a symmetry between possible miss vectors for a given estimate

and possible estimates for a given miss vector —the distributions are the same.

Fig. 4. If the latest estimate of the miss vector is within the mission’s action

threshold ellipse (in blue), then some action will be taken to mitigate the risk.

m e t l “ t

b t t c t t E

l e a t c r

m s s t

6

t r w l m t

a o p c S a t T

t p d i g

w t a

𝑟

b

𝑃

w

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iss vectors over this region gives the probability that the Pc for the stimated miss vector for this conjunction will be equal to or greater han the action threshold, given that the true relative trajectory is a col- ision. This is traditionally called the probability of detection, although probability of threat recognition ” would be better since it depends on he action threshold chosen.

Now the significance of our choice of method for approximating Pc ecomes apparent. In the conjunction plane, and given a particular dis- ribution covariance, if Pc depends only on the PDF value at the cen- er of the combined hard body, then contours of constant PDF are also ontours of constant Pc, which are contours of constant r M . This is illus- rated in Fig. 4 . And we know how to calculate the cumulative distribu- ion within an r M -ellipse such as the blue ellipse in Fig. 4 —it is given in q. 4 .

We have assumed here that the miss vector is zero, i.e., that the col- ision is dead center. What if the collision is off-center, or even at the dge of the hard-body radius? Then the ellipse is shifted and a slice of higher PDF region on one side is replaced by a lower PDF region on he other side. We have used numerical approximations of the CDFs for entered and shifted ellipses for T a = 0.0001 and for various values of HB . From these results we estimate that the centered ellipse overesti- ates the CDF by much less than 1% when r HB < 0.2 𝜎b , where 𝜎b is the

emi-minor axis of the error ellipse in the conjunction plane; this is the ame limit that bounds the error in the approximation of Pc to be less han 1%, as shown in Section 4 .

321

. Quantifying the risk mitigated

We are now prepared to find out how much a strategy of risk mitiga- ion using an action threshold actually reduces risk. The fraction of risk emoved is the product of four factors: the probability that the threat ill be noticed and analyzed; the probability of detection given an ana-

yzed threat and a predetermined action threshold; the probability of a itigating action being successfully taken; and the fraction of risk that

he action risk removes. Given current orbital tracking capabilities and the large volume ex-

mined for conjunctions around each satellite, a conservative estimate f the probability that a threat will be noticed is 99%. Operational ex- erience with satellites that do risk mitigation indicates that 90% is a onservative estimate that a mitigation action will be successful. (See ections 7 & 8 below for specific examples.) And if we require that an ction be designed to reduce the risk by at least two orders of magni- ude, then the fraction of risk removed by the action is at least 99%. his leaves the probability of detection to be determined.

Suppose now that a project has determined or chosen values for r HB , he combined hard-body radius, T a , the action threshold, and Z E , the ex- ected value of | Z |, the determinant of the covariance, which for orbit etermination depends on the partial of the state with respect to nav- gation data measurement error and the size of those errors expected iven the equipment making the navigation data measurements. Then

henever 𝑇 𝑎 < 𝜋𝑟 𝐻𝐵 2 ∕ √

( 2 𝜋) 2 |𝒁 |, we can use Eq. 5 to solve for the r M hat gives the boundary of the region in which the Pc is greater than T a s follows:

𝑀 =

√ √ √ √ − ln ( ( 2 𝜋) 2 |𝒁 |( 𝑇 𝑎 𝜋𝑟 𝐻𝐵

2

) 2 ) . (6)

Now we can use that expression for r M in Eq. 4 to get P D , the proba- ility of detection:

𝐷 = max ⎧ ⎪ ⎨ ⎪ ⎩ 1 − 𝑒

− 1 2

( − ln

( ( 2 𝜋) 2 |𝒁 |( 𝑇 𝑎

𝜋𝑟 𝐻𝐵 2

) 2 ) ) , 0

⎫ ⎪ ⎬ ⎪ ⎭ ,

hich is

𝐷 = max ⎧ ⎪ ⎨ ⎪ ⎩ 1 −

√ √ √ √ ( ( 2 𝜋) 2 |𝒁 |( 𝑇 𝑎 𝜋𝑟 𝐻𝐵

2

) 2 ) , 0

⎫ ⎪ ⎬ ⎪ ⎭ . (7)

Note that if | Z | is too large, the right-hand formula for P D becomes egative, which is an impossible value for P D ; this just means that r M n Eq. 6 was imaginary, which happens if there is no r M at which Pc s equal to or greater than T a , even if the object is on a direct collision ourse with the satellite. In such a situation P D is actually zero.

Eq 7 can be simplified further. If 𝜎a and 𝜎b are the semi-major and emi-minor axes of the error ellipse for the covariance, then | Z | = 𝜎a 2 𝜎b 2 nd we have the following surprisingly simple formula:

𝐷 = max { 1 − 2 𝑇 𝑎

𝜎𝑎 𝜎𝑏

𝑟 𝐻𝐵 2 , 0

} . (8)

Unfortunately, there is no single covariance that applies for all con- unctions —instead there is a wide range of covariances. In the cases we xamine below, the average value of | Z | is large enough that the P D is nrealistically small. So instead we take a divide-and-conquer approach.

To calculate the overall risk reduction, we assume a distribution of he values of | Z | for each satellite and represent that as a histogram by inning the values of | Z | into logarithmic bins from 10 1 to 10 19 and stimating a frequency of occurrence of values of | Z | in each bin. Then, e used the upper bound of each bin in Eq. 8 to compute the P D for

hat bin, which is zero when | Z | is so large that P D is zero. Finally, e multiplied the risk capture of each bin by the percentage of total

vents in each bin, and summed the results over all bins to determine he overall risk capture.

T.H. Sweetser, B.M. Braun, M. Acocella et al. Journal of Space Safety Engineering 7 (2020) 318–324

Fig. 5. Histogram of the determinants (in m 4 ) of the 2-D covariance matrix for 2323 conjunctions experienced by CloudSat from 2015-04 through 2019-09.

t m a t l c p b

7

e o a s l a E T

u t r n

v t t t a t E T p i t f

o b f d j o n o i o 7

8

b T s C

a t v 1 T t

w u

Table 1

Comparing OCO-2 Probabilities of detection for different thresholds.

Threshold 1 × 10 − 4 4.4 × 10 − 4 1.0 × 10 − 3

Probability of Missed Detection 15.1% 29.5% 40.1%

Probability of Detection 84.9% 70.5% 59.9%

If the | Z | associated with an individual bin is large enough, the P D for hat bin is zero, which is equivalent to the probability of the associated issed detection being unity. In other words, in these cases the covari-

nce is large enough that the probability of collision never rises above he action threshold T a , even if the miss distance is zero. Note that this ower limit on P D (i.e., it is never less than 0) for an individual bin has onsequences for the overall combined probability P D , in particular the robability of a missed detection is not linear with respect to T a , as can e seen in Section 8 below.

. Risk mitigation for CloudSat

To estimate the risk reduction over time for the space environment xperienced by CloudSat, we began by computing the value | Z | based n the uncertainty reported in the last report for each conjunction that rrived at least 24 hours before TCA. From this report, we used the ize of semi-major and semi-minor axes of the one-sigma uncertainty el- ipse, squared, which is the variance in each principal axis of the prob- bility distribution. The product of these two values is the value | Z | in qs. 1 & 3 . Given a hard body radius r HB = 3.5 m and an action threshold a = 0.0001 for CloudSat, the Mahalanobis radius r M can be calculated sing Eq. 3 . The risk captured can then be calculated from the action hreshold Mahalanobis radius by using Eq. 4 . However, this gives the isk captured only for conjunctions which have that value for | Z |, and ot for the overall risk reduction inherent in the maneuver approach.

To calculate the overall risk reduction, we created a histogram of the alues of | Z | for CloudSat over the period of time examined. We binned he values of | Z | into logarithmic bins from 10 1 to 10 19 and calculated he frequency of occurrence of values of | Z | in each bin. Then, we used he upper bound of each bin in Eq. 3 to compute the Mahalanobis radius nd the risk capture for that bin. As the value of | Z | increases (i.e., as he combined uncertainty grows larger), the value in the radicand of q. 3 becomes negative, and no real solution for the risk capture exists. his represents the situation where the uncertainty is so large that the robability of collision will never reach the action threshold value, even f the object is on a direct collision course with CloudSat. In this case, he risk capture is zero, and a zero percent risk reduction was entered

or that bin.

322

Finally, we multiplied the risk capture of each bin by the percentage f total events that fell into each bin, and summed the results over all ins to determine the overall risk capture. The data for this calculation or CloudSat are given in Fig. 5 , and the resulting percent likelihood of etection is 75.2%. We then consider the likelihoods of noticing the con- unction and performing the mitigation action, and factor in the amount f risk reduction from the mitigation. CloudSat has had six failed ma- euver attempts out of 230 attempts, giving a 95% confidence interval f 0.95 to 0.99 for the probability of success, using the normal approx- mation method of finding the confidence interval. Using the lower end f the confidence interval, the result is a total risk reduction estimate of 0%.

. Risk mitigation for OCO-2 and other A-Train satellites

The same methodology was applied to OCO-2, which has a hard- ody radius of 6 m and the same action threshold as CloudSat, 0.0001. he data for this calculation for OCO-2 are given in Fig. 6 , and the re- ulting percent likelihood of detection is 84.7%, which is larger than loudSat’s because the combined hard-body radius is larger.

As mentioned previously, to compare different threshold values the nalysis has to be redone for the distribution of | Z | values. In Table 1 , he OCO-2 operational value of 1 × 10 − 4 is compared to a 4.4 × 10 − 4 alue, which is the current red threshold used in CARA reports, and .0 × 10 − 3 , a high value that has been proposed for certain situations. he probability of missed detections is not linear with respect to the hreshold values, as can be seen in the table.

Table 2 gives the likelihood of detection using this method for a hole suite of A-Train and C-Train satellites. The second major contrib- tor to the estimate of risk reduction is the likelihood of succeeding to

T.H. Sweetser, B.M. Braun, M. Acocella et al. Journal of Space Safety Engineering 7 (2020) 318–324

Fig. 6. Histogram of the determinants (in m 4 ) of the 2-D covariance matrix for 2402 conjunctions experienced by OCO-2 from 2015-01 through 2019-09.

Table 2

Probabilities of detection for five A-Train and C-Train sun-synchronous orbiters at about 700 km altitude.

Satellite Hard Body Radius Threshold Estimated Detection Rate P(Pc > T) Data Span Number of Reports Number of Days

CloudSat 3.5 1.00E-04 75.2% Apr 2015 - Sep 2019 2323 1632

CALIPSO 14.8 1.00E-04 92.5% Jan 2015 - Apr 2019 2261 1586

Aqua 20 1.00E-04 96.3% Jun 2016 - Apr 2019 1560 1047

Aura 20 1.00E-04 96.6% Jun 2016 - Apr 2019 1772 1045

OCO-2 6 1.00E-04 84.7% Jan 2015 - Sep 2019 2402 1720

p 5 m g s d

9

t p r f

w f i l

o t d a i s T d

c f

i N o l l t v h

o e t m o p t r s l l t f a m

erform the planned mitigation action. OCO-2 has attempted to make 7 maneuvers in orbit with no failures; the mission deferred two of the aneuvers to avoid conjunctions, which counts as successful risk miti-

ation actions, so the 95% confidence interval for the probability of a uccessful maneuver is from 0.98 to 1.0 by the Clopper-Pearson interval efinition. For OCO-2 this means a total risk reduction estimate of 81%.

. Setting an action threshold for new missions

No organizational-level standards currently exist for setting opera- ional action thresholds for on-orbit collision risk mitigation. Lifetime robability of collision limits, however, do exist in a number of agency egulations; for example, NASA Technical Standard 8719.14B, Process or Limiting Orbital Debris , states the probability of accidental collision ith space objects larger than 10 cm in diameter may not exceed 0.001

or satellites passing through low earth orbit. The techniques described n this paper can help new missions set action thresholds based on these ifetime probability of collision requirements.

To accomplish this, a new mission would need an estimate of not nly the debris flux expected for its orbital environment, but also the ypical distribution of uncertainties. As an example, Fig. 7 shows that istribution for five satellites in the A-Train and C-Train orbits, which re sun-synchronous at approximately 700 km altitude; compare this nclusive plot to Figs. 5 & 6 to see how distributions for individual mis- ions can vary. These individual distributions give the results shown in able 2 , which depend almost entirely on the combined hard-body ra- ius that applies to each mission. We expect this distribution pattern of

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ovariances to apply for any sun-synchronous orbiter, but the pattern or spacecraft at different inclinations could be different.

Missions would accomplish an initial calculation of lifetime probabil- ty of collision based on the overall debris flux using current tools (i.e., ASA’s Debris Analysis Software and similar tools). Should the results f this analysis (which takes into account the different stages of a satel- ite’s lifetime, from launch through operations and disposal) indicate a ifetime probability of collision that meets organizational requirements, he satellite would theoretically be done; no collision-avoidance maneu- ers would be required by this regulation. As discussed elsewhere [8] , owever, in practical applications the situation is more complicated.

Should this calculation result in a violation of the lifetime limit – r should the mission’s program management or organizational lead- rship desire a stricter limit – the mission could determine an action hreshold using the techniques described in this paper. With an esti- ate of the uncertainties of the debris crossing through its operational

rbit, the mission could calculate, based on its hard-body radius and pro- osed action threshold, the percentage of the risk that would be reduced hrough mitigation actions during its operational lifetime. The residual isk would be added to the probability of collision calculated during the atellite’s non-operational lifetime (the period of time, for example, fol- owing passivation when no maneuvers are possible) to provide a new ifetime probability of collision number that meets the standard. This ac- ion threshold would become the on-orbit operational action threshold or risk mitigation. To ensure a conservative analysis, missions might lso want to consider factoring in an estimate of the likelihood of a risk itigation maneuver being performed successfully.

T.H. Sweetser, B.M. Braun, M. Acocella et al. Journal of Space Safety Engineering 7 (2020) 318–324

Fig. 7. Histogram of the determinants (in m 4 )

of the 2-D covariance matrix for 10,316 con-

junctions experienced by Aqua, Aura, Cloud-

Sat, CALIPSO, and OCO-2 between 2015-01

and 2019-09.

1

m a p c d t e t o

o r t m

D

i t

A

P

t t m

R

[

[

[

[

0. Suggestions for future work

Eq. 8 depends on two significant approximations: that we can esti- ate probability of collision by taking the PDF at the center to be the

verage of the PDF over the combined hard-body radius, and that the robability of detection for any point in the hard-body radius is very lose to the value for the center of the hard body. We have some evi- ence that when the combined hard-body radius is less than 0.2 times he smaller sigma in the covariance (the semi-minor axis of the error llipse), then these assumptions give over-estimates that are much less han 1%. The relationship between the approximation error and the size f the combined hard body deserves a closer look.

Another area of study is the further application of the above method- logy to data from operational missions. Estimates of the actual risk and isk mitigated for the missions considered above can be made and dis- ributions of the covariances for missions in other orbits could be deter- ined.

eclaration of Competing Interest

The authors declare that they have no known competing financial nterests or personal relationships that could have appeared to influence he work reported in this paper.

cknowledgements

The work described in this paper was carried out in part at the Jet ropulsion Laboratory, California Institute of Technology, under a con-

324

ract with the National Aeronautics and Space Administration. The au- hors thank NASA Goddard Space Flight Center’s Conjunction Assess- ent Risk Analysis (CARA) team for providing covariance data.

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