Since the announcement of the discovery of sources of gamma ray radiation in 1973, many more reports of such bursts have been published. Numerous artificial satellites have been equipped with gamma ray detectors including GRO. Unfortunately, almost no progress has been made in identifying the sources of this high energy radiation. Only one visible counterpart is known. It is suspected that this is a consequence of the methods currently used to define gamma ray burst source 'error boxes'. An alternative procedure was proposed in 1988 by Taff. Herein, Monte Carlo simulations are reported of the efficacy of this technique using realistic burst timing uncertainties and satellite location errors as well as a variety of satellite constellations. Since these are controlled numerical experiments, the dependence is examined of the statistics of the errors in the deduced burst wavefront normal as a function of the timing inconsistencies, detector location standard deviations, and especially the number and distribution of the detectors. The results clearly show that an arc minute prediction of a unique burst location is routinely obtainable once there are at least two interplanetary detectors

Holfeltz, S. T.

Taff, L. G.

English

NASA Technical Reports Server

N92-2i913
GAMMA-RAY BURST ASTROMETRY II: NUMERICAL TESTS
L. G. TAFF and S. T. HOLFELTZ
SPACE TELESCOPE SCIENCE INSTITUTE
3700 San Martin Drive
Baltimore, MD 21218
Since the announcement of the discovery of sources of bursts of gamma-ray radiation in
1973, many more reports of such bursts have been published. Numerous artificial satellites
have been equipped with gamma-ray detectors including GRO. Unfortunately, we have made
almost no progress in identifying the sources of this high energy radiation. Only one visible
counterpart is known. We suspect that this is a consequence of the methods currently used
to define gamma-ray burst source 'error boxes'. An alternative procedure was proposed in
1988 by Taft. In the current paper we rcport on Monte Carlo simulations of the efficacy of
this technique using realistic burst timing uncertainties and satellite location errors as well
as a variety of satellite constellations. Since these are controlled numerical experiments, we
can examine the dependence of the statistics of the errors in the deduced burst wavefront
normal as a function of the timing inconsistencies, detector location standard deviations,
and especially the number and distribution of the detectors. The results clearly show that
an arc minute prediction of a unique burst location is routinely obtainable once there are at
least two interplanetary detectors.
I. INTRODUCTION
An alternative to the customary "time difference of arrival" method of gamma-ray burst
source location was presented by Taft in 1988. This technique predicts a unique location
for the source of the burst independent of the number of different gamma-ray detectors
registering the burst. All the detector locations and times of observation are folded into a
single, simple computation. In contrast, the standard method only defines a (circular) locus
of points on the celestial sphere on which the burst source location should reside. When there
are multiple detections of the same burst, a finitc area on the celestial sphere is, in practice,
delineated via a pair-wise analysis of the location and timing data. With real data--and with
a very difficult problem of time registration of bursts observed with detectors of different
responsivities and sensitivities, spacecraft clock errors, timing errors arising from binning
the recorded photons, differing thresholds before recording is initiated, and so forth--the
geometrically pure problem is degraded into one whose best possible outcome is that all the
intersection points lie near each other. This small area is used to define an 'error-box' in
which the burst source is thought to lie (see Fig. 1).
The location deduced in this fashion will, in general, not be the statistically most likely
position for the source of the burst. The hope that the circle drawn for each pair of detecting
sensors is centered in a region of high source location probability will not be consistently
realized in practice; indeed, the circle must lie completely outside the one-sigma error region
fairly often. Furthermore, the supposition of a probabilistically uniform region surrounding
301
https://ntrs.nasa.gov/search.jsp?R=19920012670 2020-03-17T13:15:47+00:00Z
___True Circle #2
Error Contaminated
Circle #2 Plus 1_ Band __
Error Contaminated
Circle #1 Plus lc Band
_ "Error
Box"
\
\
\
\
\
\
\
`%
Fig. 1. General intersection area for "time difference of arrival" gamma-ray burst source
location determination. Note error bands need not be symmetrically placed relative to
circular locus nor include the true circle.
the most probable source location circle may be too simplistic--it is not clear that the
underlying probability function is that well-behaved.
Finally, random errors in detector location became systematic errors in the "time differ-
ence of arrival" method circle (see Fig. 2; timing errors change the radius of the circle). In
Fig. 2 the random error in the location of the spacecraft at P causes us to place it at pt
instead. This alters the axis of the cone from OP to OP I thereby systematically shifting the
locus of possible source locations. Similarly, a random error in the relative timing between
the two spacecraft will cause 0 to be over-estimated or under-estimated thereby systemati-
cally enlarging or decreasing the locus of possible source locations. One could compute the
one-sigma regions in an attempt to produce reliable error estimates. However, none have
been published and the amount of computation necessary for the customary method must
exceed that necessary to similarly characterize the results of Taft's method by the ratio of
(number of points in a circle):one since this is the ratio of their prediction volumes. Hence,
such a calculation would be unwieldy and extremely expensive computationally. A more
complete comparison between the different aspects of the two methods is given in Table 1.
302
Q! .. .... ..
.i
O P
,4-QO P=D ",4-Q' 0 P'=e
P P'=Ar
Fig. 2. Locus of potential burst source locations systematically shifted by the random
spacecraft positional error ppI.
TABLE 1. COMPARISON OF THE TWO METHODS
TAFF'S CUSTOMARY
ASPECT METHOD METHOD
PREDICTS A UNIQUE LOCATION
SIMULTANEOUSLY USES ALL THE
OBSERVATIONAL DATA
SUBSUMES THE OTIIER METHOD AS A
SPECIAL CASE
IS EASILY AMENABLE TO MONTE CARLO
SIMULATION
CAN EASILY ASSESS THE QUALITY
OF THE PREDICTION
YES
YES
YES
YES
YES
NO
NO
NO
NO
NO
303
The techniquedevelopedin Taft (1988)is easilyamenableto numericalsimulation. This
paper reports on extensiveMonte Carlo computationsof its predictions for the location of
gamma-ray bursts. Thesecalculations explore rangesof numbersof potential spacecraft-
carrying burst detectors,in cislunar (2, 3, or 4) and interplanetary (1, 2, 3, or 4) space,and
all 47rsteradiansof potential burst sourcelocations. In addition, becausethesearecontrolled
numerical experiments,wecanstatistically characterizeboth the accuracyand the precision
of the results. In sum, they conclusivelyshowthat a minute of arc is routinely availableonce
there are two interplanetary spacecraftin the burst detectionnetwork.
In the next sectionwe briefly review the fundamental ideasbehind this method. The
third sectionof the paper describesthe Monte Carlo simulations wehaveperformed. Com-
mentson our oral presentationhavedefinedadditional areasof research;theseareoutlined
in the fourth section. Becauseof the closenessof the Huntsville meeting, we shall report
thereon theseadditional topics. Thesewill include morerealistic simulations and analytical
progresson solving our form of the problem. Usingour software,within a year an entirely
new catalog of burst sourcelocations, for everygamma-rayburst multiply detected,could
be computed. Moreover, eachnew sourcelocation would be describedby a reliable error
estimate (onceaccessto the data is obtained).
II. CONCEPT
The essentialconceptbehind the techniqueis to usethe one pieceof information about
the gamma-rayburst that weindisputedly know; to wit, that the phaseof the burst (whether
planar or spherical)is an invariant for all the detectors. Could the detectorson the inter-
planetary burst network measurethe phase¢ of the burst wavefront, they would all obtain
the samevalue (absentobservationalerrorsof course),namely
¢ = k. r- wt (1)
where r is the solar system barycentric location of the spacecraft, t is the time of arrival of
the burst at that spacecraft (t is assumed to be a shared inertial timescale), k is the wave
vector of the burst wavefront, and w is the angular frequency of the burst (= 27ru where
v is the frequency of the photon; v = c/,_ where c is the speed of light in vacuo and
is the wavelength of the burst). Rewriting k as ku, where u is the wavefront normal, the
pseudo-invariant _ can be defined, viz.
¢b = u . r - ct. (2)
Although neither ¢ nor q5 can be directly measured, they do include all the observational
data at our disposal and the quantity we want to determine -- u. Taft's (1988) concept
was, that especially in the presence of unknown systematics and the very difficult time
registration problem we have, enforcing the constraint that each sensor's (albeit unknown and
unmeasurable) value of dp be the same would lead to a mathematically well-posed problem
for the computation of u. The explicit method Taft (1988) proposed for doing so was to
minimize the quantity
1 N
T=_ _ (¢n-¢m) 2 (3)
304
subject to the constraint that u • u = 1. Taft suggested that the constraint be incorporated
via a Lagrange multiplier.
By explicit computation, Taft (1988) further showed that the customary time of arrival
analysis was contained in this principle as a special case of minimizing Eq. (3) (i.e., they are
represented by the cases of N = 2 and N = 3). Taft also explicitly computed the coplanar
solution for u when N was equal to 4.
HI. MONTE CARLO SIMULATIONS
The only drawback of the first paper was that it did not contain any numerical testing
of this novel form of statistical estimation. This defect will be repaired immediately.
Since many of the gamma-ray bursts detected in the past have involved at least two
near-Earth (i.e., within cislunar space) detectors and at least one interplanetary detector,
this formed the minimum configuration for our simulations. All configurations of Ndose = 2,
3, or 4 and Nfa r = 1, 2, 3, or 4 have been considered. Not surprisingly, the algorithm works
much better once gfa r is at least two.
In addition to considering two groups of spacecraft, we considered a bifurcated distribu-
tion of spacecraft location uncertainties. For those in near-Earth orbit location uncertainties
of 100 or 200 km were allowed (that is to the actual location of the spacecraft we added
a vector for which each component was normally distributed about zero with a standard
deviation of 100 or 200 km in length). For those spacecraft in interplanetary space we used
the larger values of 1000 and 2000 km to span likely uncertainties. Finally, we also had
to assign timing errors. These represent a convolution of spacecraft clock errors, detection
timing (e.g., binning), sensor-to-sensor correlation, and so on. We used standard deviations
of 25 and 50 milli-seconds coupled with the zero mean normal distribution to represent
these errors. Note that 25 milli-seconds represents 7500 km at the speed of light. Hence,
since spacecraft locations arc bound to improve, timing errors will continue to dominate the
problem.
The spacecraft were always coplanar, in the plane of the ecliptic. This is a very good
(numerical) approximation to the real case and significantly simplifies the (analytic aspects
of the) computations. Spacecraft in cislunar space where deposited uniformly in azimuth
relative to the Earth-Sun line and uniformly in distance between geosynchronous distance
(42,000 km) and the mean lunar distance (400,000 km). Similarly, the interplanetary space-
craft were uniformly distributed in azimuth relative to the Earth-Sun line and uniformly in
heliocentric distance between 0.5 and 4 A. U.
Once the standard deviations of the locations and timing were set, and the numbers of
spacecraft fixed, the spacecraft were strewn across the solar system. For this constellation
of spacecraft we thcn chose 100 different unit wavefront normal vectors to represent the
gamma-ray burst (n). These were uniformly distributed over the celestial sphere. We next
computed the relative timings, corrupted the spacecraft locations and timings as described
above, and solved for u. The angle 0, given by the scalar product between n and u,
cos0 = n. u (4)
tells us the angular mis-distance between the actual direction of gamma-ray burst source
and the calculated one (note that the direction of the gamma-ray burst source is -n and the
computed direction is -u). Another way to think of 0 is that it is the 3a radius of our error
305
TABLE 2. RESULTS OF MONTE CARLO SIMULATIONS
(km) (km) (msec) ( ) (') (km) (km) () (')
2
2
2
2
3
3
3
3
4
4
4
4
2
2
2
2
3
3
3
3
4
4
4
4
2
2
2
2
3
3
3
3
4
4
4
4
2
2
2
2
3
1 10 100 25
2 10 100 25
3 10 100 25
4 10 100 25
1 10 100 25
2 10 100 25
3 10 100 25
4 10 100 25
1 10 100 25
2 10 100 25
3 10 100 25
4 10 100 25
1 20 100 25
2 20 100 25
3 20 100 25
4 20 100 25
1 20 100 25
2 20 100 25
3 20 100 25
4 20 100 25
1 20 100 25
2 20 100 25
3 20 lO0 25
4 20 lO0 25
1 10 200 25
2 lO 200 25
3 10 2O0 25
4 10 200 25
1 10 200 25
2 10 200 25
3 10 200 25
4 10 200 25
1 10 200 25
2 10 2O0 25
3 10 200 25
4 10 200 25
1 20 200 25
2 20 200 25
3 20 200 25
4 20 200 25
1 20 2OO 25
452.60 380.60 100 1000 359.20 333.80
1.26 2.45 100 1000 2.50 4.20
0.88 1.51 100 1000 0.87 1.50
0.84 1.34 100 1000 0.81 1.32
107.60 123.30 100 1000 112.10 130.20
1.60 2.88 100 1000 4.14 6.31
0.81 1.31 100 1000
0.79 1.20 lO0 1000
81.12 lO0.10 lO0 1000
1.87 3.29 lO0 1000
0.83 1.39 lO0 1000
0.73 1.16 lO0 lO00
0.80 1.33
0.78 1.16
75.77 91.00
4.63 7.10
0.68 1.10
0.77 1.19
542.30 460.40 200
1.66 2.59 200
0.90 1.53 200
0.76 1.15 200
119.60 138.40 200
2.36 4.03 200
0.77 1.33 2O0
0.73 1.16 200
72.86 90.62 200
1.30 2.48 200
0.76 1.24 200
0.76 1.23 200
1000 467.80 427.80
1000 11.70 22.05
1000 0.83 1.44
1000 0.70 1.13
1000 121.20 137.60
1000 2.06 3.94
1000 0.80 1.39
1000 0.73 1.16
1000 71.43 93.99
1000 1.25 2.25
lO00 0.75 1.32
lO00 0.72 1.15
370.20 324.70 lO0 2000 397.40 370.80
2.37 4.07 lO0 2000 1.45 2.66
0.75 1.24 100 2000 0.84 1.49
0.73 1.13 100 2000 0.74 1.22
123.80 141.10 100 2000 115.70 133.10
2.72 5.21 100 2000 1.15 2.12
0.81 1.39 lO0 2000 0.80 1.27
0.75 1.22 lO0 2000 0.76 1.29
72.13 93.13 100 2000 82.14 105.50
1.85 3.41 lO0 2000 1.64 2.95
0.79 1.32 lO0 2000 0.76 1.37
0.71 1.16 lO0 2000 0.77 1.17
491.50 423.50 200 2000 442.70 401.00
2.98 4.31 200 2000 3.29 4.67
0.77 1.29 200 2000 0.81 1.34
0.77 1.19 200 2000 0.72 1.16
116.60 126.70 200 2000 124.30 143.50
306
Table 2. Continued.
Nclose N far O'reloze (TrJar Grt(km) (o) () (,)(t) (t) (km) _3 (0) a 0
3 2 20 200 25
3 3 20 200 25
3 4 20 200 25
4 1 20 200 25
4 2 20 200 25
4 3 20 200 25
4 4 20 200 25
2 1 10 100 50
2 2 10 100 50
2 3 10 100 50
2 4 10 100 50
3 1 10 100 50
3 2 10 100 50
3 3 10 100 50
3 4 10 100 50
4 1 10 100 50
4 2 10 100 5O
4 3 10 100 50
4 4 10 100 50
2 1 20 100 50
2 2 20 100 50
2 3 20 100 50
2 4 20 100 50
3 1 20 100 50
3 2 20 100 50
3 3 2O 100 50
3 4 2O 100 50
4 1 20 100 50
4 2 20 100 50
4 3 20 100 50
4 4 20 100 50
2 1 10 200 50
2 2 10 20O 50
2 3 10 200 50
2 4 10 200 50
3 1 10 200 50
3 2 l0 200 50
3 3 l0 200 50
3 4 10 2O0 50
4 1 10 200 50
4 2 10 200 50
2.68 4.20 200 2000 2.07 4.15
0.75 1.22 200 2000 0.87 1.42
0.72 1.14 200 2000 0.84 1.30
72.60 92.00 200 2000 76.02 92.77
1.66 2.99 200 2000 1.34 2.77
0.80 1.32 200 2000 0.81 1.35
0.83 1.30 200 2000 0.78 1.21
685.60 575.30 100 1000 718.70 615.70
2.27 4.89 100 1000 6.35 9.42
1.01 1.76 100 1000 1.14 1.99
0.86 1.38 100 1000 0.92 1.49
227.80 219.50 100 1000 197.90 195.20
2.84 4.77 100 1000 3.02 4.81
1.07 1.86 100 1000 1.02 1.78
0.89 1.39 100 1000 0.78 1.27
137.20 147.90 100 1000 128.90 138.20
2.10 3.95 100 1000 2.44 4.03
1.00 1.61 100 1000 0.97 1.78
0.84 1.30 100 1000 0.80 1.37
729.50 605.20 200 1000 616.50 536.50
1.93 3.67 200 1000 3.89 5.74
0.99 1.55 200
0.97 1.50 200
208.10 197.20 200
1.95 3.39 200
0.96 1.62 200
0.93 1.43 200
152.10 157.70 200
3.50 5.O9 20O
0.93 1.61 200
0.94 1.44 200
662.60 560.00 100
3.73 5.66 100
0.99 1.70 100
0.98 1.57 100
235.40 224.40 100
2.66 4.03
1.05 1.75
0.90 1.53
116.00 125.50
3.32 5.96
1000 1.10 1.93
1000 0.88 1.37
1000 206.10 199.50
1000 2.43 4.67
1000 0.95 1.67
1000 0.80 1.23
1000 148.10 154.90
1000 2.15 3.81
1000 1.00 1.63
1000 0.82 1.31
2000 617.80 534.40
2000 2.31 4.11
2000 1.06 1.76
2000 0.91 1.41
2000 194.50 192.70
100 2000 2.47 4.40
100 2000 0.94 1.52
100 2000 0.82 1.26
100 2000 164.40 173.10
100 2000 3.30 4.93
307
Table 2. Continued.
Nclose N far ff reloje _7relo.e fir,tar
(km) _rl_ 3 at IO) a 0 (Oil a 0(msec) (t) (t) (km) (km) () (')
4 3 10 200 50 0.91 1.64 100 2000 0.97 1.62
4 4 10 200 50 0.87 1.50 100 2000 0.85 1.44
2 1 20 200 50 653.10 550.60 200 2000 530.40 472.00
2 2 20 200 50 12.94 13.02 200 2000 2.44 4.55
2 3 20 200 50 1.10 1.86 200 2000 1.13 2.05
2 4 20 200 50 0.87 1.35 200 2000 0.90 1.46
3 1 20 200 50 220.30 209.80 200 2000 215.00 216.00
3 2 20 200 50 4.96 6.59 200 2000 2.92 5.19
3 3 20 200 50 1.03 1.90 200 2000 0.96 1.76
3 4 20 200 50 0.89 1.46 200 2000 0.80 1.25
4 1 20 200 50 144.50 150.40 200 2000 138.30 151.40
4 2 20 200 50 1.96 3.45 200 2000 1.71 3.03
4 3 20 200 50 0.92 1.55 200 2000 1.06 1.90
4 4 20 200 50 0.82 1.34 200 2000 0.81 1.36
circle. By averaging over the 100 randomly chosen wavefront normals, and then again over
100 different constellations of the same set of sensors, we can unambiguously demonstrate
the power of the technique. These results are given in the right-hand portion of Table 2 for
the larger location uncertainties.
Table 2's mid-section has the identical format to its right-hand portion except that all
the location uncertainties have been reduced by an order of magnitude (i.e., 10 and 20 km for
spacecraft in cislunar space and 100 and 200 km for those in interplanetary space). Clearly,
once there are at least two detectors located in interplanetary space an arc minute is the
routine performance of the technique. Finally, since we do know the true location of the burst
source, we can also compute the standard deviation about the mean [which is necessarily
non-zero owing to the form in Eq. (4)]. These values are also in Table 2 (i.e., aO).
IV. ADDITIONAL AREAS OF RESEARCH
One consequence of the oral presentation of these results were the following suggestions
for future work: (1) Simulations tailored to the Ulysses, Pioneer-Venus Orbiter, GRO, and
GRANAT configuration; (2) More analytical work in describing the error distribution of this
method, and (3) Simulations of the BATSE instrument alone as a gamma-ray burst source
location detector. These additional computations will be reported on in paper III to be given
at the Huntsville Gamma-Ray Burst Workshop.
Reference
Taft, L. G., 1988, Ap J 326, 1032
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