NASA Contractor Report 3374 

Guide Star Probabilities 

Raymond M. Soneira and John N. Bahcall 

Institute for Advanced Study 

Princeton, New Jersey 

Prepared for 

Marshall Space Flight Center 

under Contract NAS8-32902 

National Aeronautics 

and Space Administration 

Scientific and Technical 

Information Branch 

1981 
INTRODUCTION 

We have calculated the probabilities for finding a suitable 

pair of Guide Stars (GS) in the Fine Guidance System (FGS) of 

the Space Telescope using the Weistrop north galactic pole 

star catalog (Weistrop 1972) and classical data on bright star 

counts (see especially Seares -- et al. 

Soneira 1980). Many of the considerations and techniques 

discussed here are relevant for other space astronomy missions. 

The data are discussed fully in Appendix A of this report 

(which is a section of a paper by Bahcall and Soneira 1981 on 

1925 and Bahcall and 

I 

the distribution of bright stars). The available observational 

data are rather limited but are in good agreement with each 

other. The definition of our task in this report is contained in Appendix B. 

The most important conclusion from this report is that 

the acquisition probability for guide stars can be greatly 

increased if the FGS will allow the second member of a guide 

star pair to be appreciably fainter than the primary (bright) 

guide star. 

a) GS Selection Requirements 

The requirements for GS selection that we have used are: 

1. Each FGS detector sees an area of sky L? = 69.2 square 

arcmin. The geometry for the 3 detectors is shown in figure 

1 (which is taken from Groth 1978). 

2. The sensitivity of the GS detector may be modeled as a 

constant between 4660 8 to 7000 8, and zero (1; ... tside this 

range. (This color sensitivity of the GS detector is similar 
W 

0 

z 

a 

n 

0 

3 

(3 

W 

z 

Lr, 

- 

k 

2 

w 

= 

W 

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2 

a 

a 

a 

K 

W 

z 

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e 

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to the Visual band, but has a factor of 2.7 greater energy 

throughput and 2.8 greaterphoton throughput for the color distribution of stars brighter than V = 14.0; see Bahcall and 

Soneira 1981, § 111. For V = 15, the energy and photon 

throughputs are 2.7 and 2.9, respectively.) 

3. 

stars (i = 1,2) is 9 < m < m where 14 5 ml 5 15 and 

(m2 - ml) could in some designs be as large as 2 magnitudes. 

4. 

have a magnitude separation greater than Am = 0.5 mag. 

5. 

BS = 13.5 arcmin. 

The magnitude range from the primary and secondary guide 

- V i' 

Stars closer than Be = 1.0 arcmin to a guide star must 

A pair of guide stars must be separated by at least 

b) Analytic Calculation 

An analytic statistical treatment of the selection criteria 

is useful for gaining an understanding of the GS probabilities.. 

(Calculations using Monte Carlo methods are more accurate and, 

in some ways, more convenient.) 

The probability P = P(R,b,m) of finding at least one 

suitable GS pair in the direction (Il,b) with apparent magnitudes brighter than rn = ml = m2 is given to high accuracy by: 

3 2 P(R,b,m) = P1 + (2Q + l)P1(l - P1) I 

with: 

3 
e 

R aAm - B = 1 + (10 1) n , assuming , 

&= CIOam , a and C constants , 

-aAm) - 'e 

y = $ (10a*m - 10 R 

(4) 

(5) 

e/z/ = c/;tia ,b,m) is the number of stars per solid angle brighter 

than magnitude rn satisfying requirement 3 above; 

&' is Jz/ after the application of requirement 4; 

R is the solid angle seen by one FGS detector, (R = 69.2 

square arcmin, requirement 1); 

0 is the minimum angular distance of any star to a GS with a 

e 

magnitude difference less than Am (ee = 1.0 arcmin, requirement 4) ; 

R~ = ree is the area enclosed by Be (Re = 3.1 square arcmin) ; 

Am is the minimum magnitude difference allowed for a star 

closer than Be to a GS. (Am = 0.5 mag, requirement 4) ; 

0 is the minimum angular separation for a pair of GSs 

S 

(es = 13.5 arcmin, requirement 5); 

~ 4 
Q1 

Q1 

01 

6 

= Q1(Bs) is the (ensemble average) probability that a single 

pair of GSs, each star having a randomly chosen position 

within its GS field, are separated by an angular distance >es. 

for adjacent detector fields is given by the following expression: 

90°-Ae 

de jR2 dr2 lR2 rlr2 f ( 8 , rl , r2 ) drl 

, (9) - /A8 R1 R1 

R2 R2 Q1 - 90°-Ae 

de dr2 I rlr2drl 

/A@ R1 R1 

180°-A8-$-8 f= I 900-2~e 

-1 $ = cos I 

L '-1 

where: R1 = 10.22, R2 = 14.09, A6 = 2.873'. R1, R2 and 

A8 define the non-vignetted detector area. A numerical 

integration gives Q = 0.735. For non-adjacent GS fields, 

Q, = 1.0; 

is the logarithmic (base 10) slope of the star counts near 

1 

m. (a = 0.32 from the Weistrop data); 

is the correction factor expressing the probability that 

a GS is rejected due to proximity of another star less than 

Be and Am away [B = 1.0201 ; 

P1 is the probability of finding at least one GS in one FGS 

field satisfying requirement 4; 

Pi is the Poisson probability distribution; 

5 
The above formulae are approximate for the following 

reasons : 

1. They assume that stars are randomly distributed and therefore 

ignore star clustering. (Our results suggest that this 

assumption introduces a - 1% overestimate in P); 

2. &is assumed to be exactly an exponential. (This is also 

an excellent approximation introducing a negligble error); 

3. Correction factors due to finite boundaries are ignored in 

the calculation of B and y. (This is also an excellent approximation.) 

Formulae (1) - (11) were used to calculate the probability 

of acquiring a GS pair in the memorandum in Appendix B, 

which describes our definition of the task. 

From the Weistrop (1972, 1980) data, the surface density 

of stars with 9.0 - < mv < 14.0 is 71.5 stars per square degree. 

The mean number of stars per detector is 1.37. Then P1 = 0.74 

and Q : 0.954 so that P = 0.41 + 0.41 = 0.82. 

Note that although the single pair form factor, 

Q1 0.75, the actual form factor Q =: 0.95 because there 

are many possible combinations of GS pairs when there is 

more than one GS per field. 

The €le/ Am constraint has only a minor effect on P. 

For mv = 14.0 mag, 

p - 0.84 - 0.020(k)e~ 0.5 e 

For mv > 14, the influence of Be and Am on P decreases monotonically with increasing mv. 

6 
The above value for P is calculated for the Weistrop field 

centered at R = 65.5', b = 85.5' (SA 57). 

star density is expected to be at a minimum slightlyawayfrom 

At mv = 14 mag, the 

the Galactic pole in the direction of the galactic anticenter R = 180°, b = 80'. (This is based on the galactic 

model of Bahcall and Soneira 1980; there is no direct data 

presently available to confirmthis point.) 

not expected to be at the pole because of the galactic spheroidal 

The minimum is 

bulge contribution to the star counts. For R = 180°, b < 80' 

the increasing star density from the disk overcomes the decrease 

from the spheroid. Figure 2 is an isodensity contour plot of 

the integrated star counts to mv = 14 mag for b > 20' calculated 

from the Bahcall and Soneira model (assuming a Sandage obscuration model, see paper I). 

- 

(For inv > 14 the counts increase like 

with only a slight dependence of the slope on latitude 

and longitude.) The model predicts that the integrated star 

density to mv = 14 in the direction R = 180°, b = 80° is 37% 

smaller than in the direction R = 65.S0, b = 85.5O. (For 

rn 

stars per detector to 1.32 and P = 0.80. Near m = 14, 

P =: 0.5(1+0.45n), where n is the mean number of stars per detector. 

1 

1 = 15, the value is 42%). This reduces the mean number of 

V 

V 

This approximation is accurate to a few percent for n between 

1.0 and 1.8 corresponding to the magnitude interval 

13.65 - - < m < 14.35. 

c) Monte Carlo Calculation 

The GS probabilities have also been calculated using a 

Monte Carlo simulation. The stars in each FGS field were dis7 
"0 0" T? 

/r, 

GALACTIC 

CENTER 

Fig. 2 

8 
tributed randomly; the number of stars in each field was chosen 

from a Poisson distribution with a specified mean number of 

stars per field and magnitudes were assigned randomly according 

to a distribution function that results in a number-magnitude 

relation that increases as loam. 

through an algorithm that implements the GS selection requirements 

(section A, above). 

The stars were then processed 

This procedure simulates an ST pointing. 

In order to accumulate good statistics on the distribution of 

GS pairs, the procedure was repeated several thousand times. 

The results of this method agree with the analytic calculation 

tG within the statistical uncertainties. 

are almost identical to the analysis on the real Weistrop data, 

we defer discussion to the next section. 

Because the results 

d) GS Selection with the Weistrop Data 

The Weistrop survey area is - 700 times larger than the 

area of one FGS field and since there are 984 stars brighter 

than mv = 14 in the catalog, there is a sufficient data base 

for accurately determining GS statistics empirically near the 

pole (assuming, of course, the Weistrop field is representative of the star density at its galactic latitude and longitude). 

The only significant difference between the results for 

the Monte Carlo calculation (5 c) and the Weistrop data is 

the GS rejection due to the proximity of another star: 

for the random case at V = 14m it is 2% and, for the Weistrop 

data it is 3%. The larger rejection factor in the Weistrop 

data is due to star clustering. The Weistrop catalog presum9 
ably identifies all stars that are separated by more than the 

seeing disk, < 5 2arcsec (Weistrop 1980); hence only very compact 

binaries (which may be a problem for the FGS) are missed. 

(The anticorrelation length for the measured stellar positions 

is - 6 arcsec. The absolute positions of the stars are accurate 

to - 1 arcmin, Weistrop 1980.) 

We present in Table 1, GS probak lities for the following 

combinations of (ml,m2) : (14,141, (14~4.5)~ (14,15), (14.5,14.5), 

(14.5, 15.0), (15.0, 15.01, and (13,151. For each of these 

pairs of magnitudes we list ni = JZ/(mi)51, 

stars per detector, and several GS probability distributions: 

PI (n) and P2 (n) are the probabilities for finding n primary or 

secondary stars per FGS detector before application of any 

selection constraints; Ps(n) is the probability for finding 

n suitable secondary guide stars for a primary guide star; 

the mean number of 

l and P (n) is the probability for finding n suitable guide P 

star pairs per ST pointing. 

For the "standard" case of m = m2 = 14, Pp (21) ; i.e., 

the probability of finding at least one suitable GS pair is 

0.80, and P (>2) = 0.55. (This latter value is important for 

considering backup GS pairs if close binaries are a problem.) 

For (14,14.5), PP()l) = 0.91 and P (>lo) = 0.35. Hence allowing the secondary GSs to be 1/2 maq dimmer than the pri- 

- maries brinqs about a dramatic increase in the probability of 

acquirinq a suitable GS pair. 

and P (>1) = 0.87 for (13,lS). 

P 

of 5(2Ql+l)n2 and P (>1) - 1 - e 

10 

1 

P 

~ 

PFor (14,151, Pp(~l) = 0.97 

PNote that for d2 >> dl 

follows closely the Poisson distribution for a mean density S 2 -3nl 

P- 
For >> d1, the number distribution of GS pairs, 

can be significantly reduced if any of the primary Guide 

Stars is a close binary system. The probability of finding 

pP 

n candidate primary GSs in the three FGS fields taken together is then the statistic of most practical interest 

because we are virtually guaranteed of finding a secondary 

GS for each candidate primary GS since >> dl. This 

statistic (a Poisson distribution for a mean density of 3nl) 

is presented in Table 2 for each of the limiting magnitudes 

considered in this paper. 

As in section b), the "worst case" probabilities need to 

be corrected from the direction R = 65.5', b = 85.5' to the 

density minimum at R = 180°, b = 80'. For the "standard" 

case of ml = m2 = 14, P (>1) is then 0.79. PApproximate estimates for the GS probabilities for 

directions other than the galactic pole can be obtained from 

figure 2 by matching the surface densities &for a given 

direction (R,b) as closely as possible to the entries in Table 1. 

For Jz/> ., 100 stars per deg2, coresponding to mv 

the galactic pole, the probability of acquiring a suitable 

pair of Guide Stars is > .., 95%, sufficiently high to be of little 

concern. Appropriate Guide Stars will nearly always be available 

(P (21) - > 0.95) for all galactic latitudes and longitudes below 

b - 40' if ml = m2 - > 14m (see Figure 2 and Table 1) . 

14.5mag at 

P 

e) Conclusion 

The probability for acquiring a suitable pair of Guide 

Stars in the direction of minimum star density (k = 180°, 
b = 80°) is estimated in this report to be 0.79 when both Guide 

Stars are required to be brighter than mv = 14m, somewhat less 

than the design specification of 0.85. (The acquisition probability will be further reduced if there are a significant number 

of close binaries with angular separation 5 2 arcsec.) In 

order to meet the design specification with the same brightness 

limit for both the primary and secondary Guide Stars, one would 

require m = m2 = 14.25 Visual magnitudes. 1 

If, however, the secondary Guide Star is allowed to be 

dimmer than the primary, then there can be a significant improvement in the acquisition probability. For ml = 14.0 and m 

= 14.5m, the Guide Star acquisition probability is 0.90. 

If we use the maximum allowed brightness difference between 

"2 

the primary and secondary Guide Stars, 2 magnitudes, then 

the primary star need only be brighter than 13th magnitude 

(Visual) in order to meet the design specification in the 

direction of the density minimum near the galactic pole. 

12 
REFERENCES 

Bahcall, J. N., and Soneira, R. M. 1980, Ap.J. Suppl., 

in press (Paper I). 

. 1981, submitted to Ap.J. 

Groth, E. J. 1978, "Probabilities of Finding Suitable Guide 

Stars," in "Use of the WFC as a 'Finder Camera' for the 

Small Aperture SI'S," NASA report. 

Seares, F. H., van Rhijn, P. J., Joyner, M. C., and Richmond, 

M. L. 1925, Ap.J., 62, 320. 

Weistrop, D. 1972, A.J., - 77, 366. 

. 1980, private communication. 

13 
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16 
TABLE 2. Number Distribution of Primary Guide Stars For All Three 

Dectectors. 

m = 15.0 

n = 1.89 n = 2.56 

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n = 1.37 

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m = 13.5 

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5 

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7 

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0.270 

0.185 

0.095 

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0.013 

0.004 

0.001 

0.053 

0.155 

0.228 

0.224 

0.165 

0.097 

0.047 

0.020 

0.007 

0.002 

0.001 

0.016 

0.067 

0.139 

0.190 

0.195 

0.160 

0.110 

0.064 

0.033 

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0.006 

0.002 

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0.105 

0.148 

0.168 

0.159 

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0.017 

0.008 

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0.144 

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0.118 

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0.063 

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0.013 

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0.001 

17 
-AlAPPENDIX A 

11. BRIGHT-STAR DENSITY 

A. The Best Available Data 

The principal source of data that we used for the bright 

(~16~~~) star densities in paper I was Seares et al. (1925), 

who list star counts in the direction of the north Galactic 

pole for magnitudes in the range 4 - < mv - 18. The original 

data from some of the star catalogs used by Seares et al. had 

large systematic errors (up to 2 magnitudes) that Seares and 

his collaborators tried to remove. However, a scale error 

in the "corrected" magnitudes was discovered by Stebbins, 

Whitford and Johnson (1950). The corrections of Stebbins 

-- et al. were applied in paper I, but it was conceivable that 

some systematic errors remained. 

- -- 

-7 

Modern counts of bright star densities have been rare. 

In paper I, the brightest recent star counts considered were 

those of Brown (1979). Brown measured integrated star counts 

in several fields near the north Galactic pole in the magnitude range 17 - < mv - < 20.5. 

(1979) extend to 13th magnitude: 'however, stellar images 

brighter than 17th magnitude were saturated and hence required 

(The counts of Tyson and Jarvis 

an indirect magnitude 

In this paper we 

gth to 16th magnitude 

1980) that was kindly 

measurement.) 

consider additionally star counts from 

obtained from the data of Weistrop (1972, 

made available to us in the form of 
-A2- 

the original computer tape and printout. The Weistrop catalog 

comes from a field of area 13.5 square degrees, centered on 

SA57 in the direction of the north Galactic pole. There are 

2989 stars in her catalog brighter than V = 16 (244 brighter 

than V = 12). The catalog extends to V = 20, but magnitude 

uncertainties appear to increase significantly dimward of 

16L" magnitude (Faber -- et al. 1976; Weistrop 1980; King 1980). 

According to Weistrop (1980) , the magnitude errors for 

V - < 16 are 2 0.12m (one standard deviation). 

Faber -- et al. 1976 of the Weistrop photographic magnitudes, 

A comparison by 

to photoelectric magnitudes, VpE, for 23 stars with 

16 gives <Vw - VpE> = - 0.02 t 0.09. 

vW 

V 

resulting from minor differences between the photographic and 

photoelectric pass bands are of concern here only when estimating B-V colors, see below.) King (1980) has compared his 

own photographic photometry to that of Weistrop. His results 

for 16 stars with V - < 16 is <Vw - VK> = + 0.01 2 0.09. These 

estimates all yield a magnitude uncertainty - 2 0.1 . The 

(Systematic errors 

- 

m 

zero point and magnitude scale appear to be well established 

for V - < 16. 

Figure 1 displays a compilation of all the available 

differential star densities (number of stars per magnitude 

per square degree) in the direction of the north Galactic 

pole (see paper I). The solid curve is the density predicted 

by our standard Galaxy model of paper I, including a separate 

contribution from giants (see below). In Table 1 we list 

the total integrated star counts (stars brighter than the 
-A3- 

specified magnitudes limit) for the Weistrop data, the Seares 

et al. data as reduced to the Visual band in paper I, the 

calculated densities from our standard Galaxy model (see 

below), and the densities tabulated in Allen (1973). 

B. Allen's Recommended Densities 

The star densities listed in Allen (1963, 1973) are 

typically more than 40% larger than the densities listed above 

and in paper I for m 2 13. Allen's results have been used 

as the basis for considering the performance of the Guide 

Star System for the Space Telescope (see, e.g. , Groth 1978). 

Because of the importance of the Guide Star System for the 

entire Space Telescope Observatory, it is necessary to examine 

the origin of the discrepancies between Allen's numbers and 

the other results given in Table 1. (Allen 1973 notes that 

his star densities are 25% larger than the Landolt-B6rnstein 

Tables [Scheffler and Elsasser 19651, his reference number 6, 

although the same sources were used.) The sources listed 

by Allen are Seares -- et al. (1925), Seares and Joyner (1928) I 

and van Rhijn (1929). The Seares and Joyner paper examines 

star counts for b - < 70°, and thus is not relevant for the 

present discussion. The van Rhijn paper combines star count 

data of Seares -- et al. and the Harvard-Durchusterung survey. 

A comparison of the Seares -- et al. and van Rhijn counts 

in the international photographic band with corresponding 

counts listed in Allen (1973), shows that Allen must have 

re-reduced the data introducing small changes, -lo%, (generally) 

upward in the bright star densities. 
-A4- 

Counts in the Visual band are desired for many purposes. 

In order to transform the photographic band counts into the 

Visual band,Bahcall and Soneira (1980) used the color equation 

in Seares et al. Allen used the following (incorrect) conversion from m to mv: 

Pg 

This conversion is more incorrect at higher galactic latitudes, 

since most of the bright stars are counted near the plane of 

the disk and stellar type (i.e., - color) changes systematically 

with Galactic latitude. 

The net discrepancy is unexpectedly large (>40%) between, 

on the one hand, either the modern star counts in V (Weistrop 

1972, 1980) or the older Seares -- et al. (1925) data as reduced 

in paper I, and, on the other hand, the star densities tabulated 

by.Allen (1973). In order to illustrate where in the analysis 

of Allen the errors appear to have originated, we divide the 

transformation into several steps. For specificity, we 

consider the star density in the Visual band at 14th magnitude 

for which Allen cites 102 stars deg-2. 

to result from the different manner in which Allen combined each 

of the three (tabulated) quantities needed in equation 2 from 

his two sources. If we apply Allen's transformation (equation 1) 

to just the Seares -- et al. data alone, we obtain 94 stars deg-2. 

(Van Rhijn tabulates only 2 of the 3 quantities, so it is not 

possible to perform the Allen transformation solely with 

Part of the error appears 

van Rhijn's data.) If the band transformation is calculated 
-ASusing the correct color equation described in Seares -- et al., 

the density further decreases to 82 stars deg-2. 

if the magnitude corrections of Stebbins, Whitford, and 

Finally, 

Johnson (1950) are applied to the Seares -- et al. data (as in 

paper I) , we obtain 71 stars deg 

magnitude in the direction of the north Galactic pole. 

Weistrop counted 73.5 stars deg-2; the Bahcall-Soneira 

Galaxy model predicts 70.4 stars deg-2 brighter than V = 14. 

-2 brighter than 14th Visual 

Hence, it appears that Allen's result can be accounted for 

by 3 systematic errors, each of order lo%, that add "fortuitously" 

in the same direction. 

C. Theoretical Model 

In calculating the star counts for the bright apparent 

magnitudes considered here using the Bahcall-Soneira Galaxy 

model, it is necessary to separate the contribution of the 

stars on the giant branch from the more common main sequence 

stars which completely dominate the counts at faint magnitudes 

(see § IIa of paper I for a detailed discussion of this point). 

The giants were not included separately in paper I. 

The differential star counts for the giants peak at 

llth mag, where they contribute about 20% to the total counts; 

by 13th mag their contribution has fallen to 3%. We take the 

fraction of stars, f, on the main sequence in the plane of 

the disk to be 

[Io.0001S(M+8)3~3, M < 3.7 , f= exp 

(1 M > 3.7 , I - 
which accurately fits the data of Sandage (1957), Schmidt 

(1959) and McCuskey (1966). A scale height of 250 pc is 

adopted for the giants (see paper I). 
-A7- 

REFERENCES 

Allen, C. W. 1963, Astrophysical Quantities, second edition 

(London: Athlone Press). 

. 1973, Astrophysical Quantities, third edition 

(London: Athlone Press). 

~ 

Bahcall, J. N., and Soneira, R. M. 1980a, Ap.J. Suppl., 44, 

(Paper I) 

Brown, G. S. 1979, A.J., - 84, 1647. 

Faber, S. M., Burstein, D., Tinsley, B., and King, I. R. 1976, 

A.J., - 81, 45. 

Groth, E. J. 1978, "Probabilities of Finding Suitable Guide 

Stars,'' in "Use of the WFC as a 'Finder Camera' for the 

Small Aperture SI'S," NASA report. 

King, I. R. 1980, private communication. 

Kron, R. G. 1978, Ph.D. Thesis, Univ. of California, Berkeley. 

McCuskey, S. W. 1966, Vistas in Astronomy, 7, 141. 

Peterson, B. A., Ellis, R. S. Kibblewhite, E. J., Bridgeland, 

M. T., Hooley, T., and Horne, D. 1979, Ap.J. (Letters), 

233, L109. 

Rhijn, P. J. van 1929, Pub. Kapteyn Astro. Lab. Groningen, No.,43. 

Sandage, A. 1957, Ap.J., 125, 422. 

Scheffler, H., and Elsasser, H. 1965, Landolt-Bornstein Tables, 

Group VI, 1, p. 601. 

Schmidt, M. 1959, Ap.J., 129, 243. 

Seares,.F. H. and Joyner, M. C., 1928, Ap.J., 67, 24. 

Seares, F. H., van Rhijn, P. J., Joyner, M. C., and Richmond, 

M. L. 1925, Ap.J., 62, 320. 

Stebbins, J., Whitford, A. E., and Johnson, H. L. 1950, Ap.J., 

112, 469. 
-A8- 

Tyson, J. A. and Jarvis, J. F. 1979, Ap.J. (Letters), 230, L153. 

Weistrop, D. 1972, A.J. - 77, 366. 

. 1980, private communciation. 
-A9- 

TABLE 1. TOTAL INTEGRATED STAR DENSITIES 

PER SQUARE DEGREE 

V 9 10 11 12 13 14 15 16 

Wei s trop 1.9 3.7 7.6 18 37 74 135 224 

a Seares -- et al. 1.2 3.3 8.0 18 36 71 134 216 

Bahcall-Soneira 1.5 3.5 7.7 16 34 70 138 254 

Mode 1 

Allen 1.4 3.6 9.1 21 49 102 186 347 

a Reduced to the Visual band in paper I. 
-A10- 

FIGURE CAPTIONS 

FIGURE 1: Differential star densities d per magnitude per 

square degree for the Galactic pole. The solid 

curve is the density predicted by the standard 

Galaxy model of paper I assuming no obscuration 

at the pole and including a separate contribution 

from giants (see § I1 c). Data from Seares et al. 

(1925) as reduced to the Visual-band in paper I 

are plotted as filled-circles, data from Weistrop 

(1972, 1980) are plotted as filled-triangles, data 

from Brown (1979) corrected to the pole (see paper I) 

are plotted as filled-squares, data from Kron (1978) 

are plotted as open triangles, data from Tyson and 

Jarvis (1979) are plotted as open circles, and 

data from Peterson et al. (1979) as open squares. 

For conversions between magnitude systems, see 

paper I (and § 111). 
Fig. 1 
-B1- 

APPENDIX B 

THE INSTITUTE FOR ADVANCED STUDY 

PRINCETON, NEW JERSEY 08540 

Telephone-609-924-4400 

SCHOOL OF NATURAL SCIENCES 

ibEMORANDUM 

February 13, 1980 

To : C. R. O'Dell and W. Jeffreys 

From: John N. Bahcall and Raymond Soneira 

Subject: Guide-star probabilities. 

We are undertaking to supply more accurate estimates for 

the probabilities of finding guide stars in the FGS. The purpose ofthis preliminary note is to inform you of the scope of 

our activity and to present some general numbers. We also wish 

to indicate our summary of the problem so that you can correct 

any misunderstandings we may have before we comslete a lot of 

numerical work. 

Definition of the Problem 

1. Each FGS detector has an area of 69.2 arcmin squared. 

2. The magnitude limit is presently specified as 95m;X 

where X is in the range 14 to 14.5. 

will not be important numerically for our purposes.) 

13.5 arcmin. 

separation in magnitude must be at least 0.5m. For a guide 

star of magnitude X,.the exclusion applies to magnitude 

X 5 0.5m. 

5. The sensitivity of the detector may be modeled as constant 

between 4660 2 to 7000 8, and zero outside this range. 

detector really requires a fixed number of photons per square 

centimeter per second in this band. 

(The lower limit, mv=9, 

3. The separation of the two required guide stars must exceed 

4. For nearby stars within 1 arcmin of a guide star, the 

The 

Scope of Our Activity 

A. Probabilities. We will calculate probabilities for a convenient grid in latitude and longitude of finding two guide 

stars in the FGS within the limiting magnitude range of 2 

(above) for X = 14.0 to 14.5, in steps of 0.1. For these 

calculations, we shall use the model star densities in our 

Paper 1'. We will calculate the corrections to these proba- 
-B2- 

C. R. O'Dell & W. Jeffreys -2- February 13, 1980 

bilities due to constraints (3) and (4) above. We will 

make similar calculations for mg. 

B. Color Correction. The detector sensitivity is not that 

of any standard astronomical band. Stars of the same 

visual magnitude will have different photon fluxes in 

this band. We will normalize our results to an A0 V star, 

assuming it has the limiting visual magnitude X of consraint 

2 above. Then we will reevaluate the photon fluxes for all 

stars close to magnitude X using their known intrinsic 

spectra and our estimate of the relative probabilities of 

stars of different types from Paper I. This will lead to 

revised color-corrected estimates of the probabilities of 

finding guide stars in the FGS. 

C. Star Clumpinq. The bright star N-point correlation function is not known, but it may not be accurate to assume, 

as in all previous calculations we have seen, that bright 

stars are Poisson distributed on the sky. We will measure 

the departures from a Poisson distribution using the star 

data for 13.5 square degrees near SA57 by Dr. D. Weistrop2. 

We will correct all the above probabilities for the effect 

of star clumping, if appreciable. 

I 

Initial Results 

We have completed an initial set of calculations for part A 

I above. We find the following. 

a. The minimum star density is slightly displaced from the 

b. At the minimum, we find from Paper I: 

pole. It actually occurs at R = 180°, b = 80'. 

N(mv 1. 14.0) = 65 per square degree. 

Subsequent to the writing of Paper I, we have verified 

our overall normalization (originally based on Seares et a1.3 

and recalibrated in 1950 using photoelectric standards by 

Stebbins et a1.4) for the star densities by inde endent 

comparisons with data of D. Weistrop and I. King' for SA57 

(latitude 86'). Our results agree with theirs to an accuracy of about 10 percent in the range mv = 14 to 15. 

c. The probabilities given in the DRM and by Ed Groth in his 

recent report are based on Allen's Astrophysical Quantities6. 

Allen uses an ad hoc and incorrect conversion from m to mv, 

i.e., 

T 

P9 

/total number of stars brighter than m-. from b=Ooto 90°\ V 

)\ corresponding number for m Pg )- N(zmv) = N(<m 

- Pg __ 

I This conversion is more incorrect the closer one approaches the Galactic poles (where Guide star probabilities are mainly 
-B3- 

C.R. O'Dell f W. Jeffreys February 13, 1980 

d. 

e. 

f. 

g1 

2 

3 

4 

5 

6 

of interest) since most of the bright stars are counted 

near the plane of the disk and stellar type (i.e., color) 

changes systematically with Galactic latitude. Comparable 

errors, fortuitously all in the same direction, arise from 

a magnitude error (uncorrected for by Allen) in the Seares 

et al. counts and by Allen's curious reanalysis of the 

primary star count data. 

or a factor of 1.6 more than our estimate of 66.5 at the 

pole. 

Two other numbers of interest (at the minimum) are: 

At the pole, Allen gives 

N(mv) = 102 per degree squared , 

Nv'mv - < 14.5) = 93.5 per square degree and 

Nvhv - -c 15.0) = 132 per square degree. 

Criterion 3 above reduces the probability of finding 

Guide stars by about four percent in adjacent detectors 

(zero otherwise) and criterion 4 makes a further reduction 

of about two percent near the poles (3 percent at lower 

latitudes). The net reduction of the overall probability 

near the Galactic poles from these criteria is about three 

percect. 

The probability of successful Guide star acquisition at 

the pole is (ignoring color corrections, star clumping 

and close binaries), 

0.75 at mv = 14.0 

0.90 at mv = 14.5 

0.95 at mv = 15.0 . At R = Oo, b = 20°, mv = 14.0, the corresponding probability 

is 0.995. 

Bahcall, J.N., and Soneira, R.M. 1980, "The Universe at Faint 

Weistrop, D. 1972, A.J., 77, 366. 

Seares, F.H., van Rhijn, P.J., Joyner, MC., and Richmond, M.L. 

Stebbins, J., Whitford, A.E., and Johnson, H.L. 1950, Ap.J., 

King, I. 1980, private communication. 

Allen, C.W. 1973, Astrophysical Quantities, The Athlone Press 

Magnitudes: I. Models for the Galaxy and the Predicted Star 

Counts,'' Ap.J. (Supplement), in press. (Paper I). 

. 1980, private communication. 

1925, Ap.J., 62, 320. 

112, 469. 

(University of London), 3rd edition. Also see 1st and 2nd 

editions, 1955 and 1963, respectively. 
1. REPORT NO. 2. GOVERNMENT ACCESSION NO. 

NASA CR-3374 

4. TITLE AN0 SUBTITLE 

Guide Star Probabilities 

7. AUTHOR(5) 

Raymond M. Soneira and John N. Bahcall 

9. PERFORMING ORGANIZATION NAME AND ADDRESS 

Institute for Advanced Study 

Princeton, New Jersey 08540 

12. SPONSORING AGENCY NAME AND ADDRESS 

National Aeronautics and Space Administration 

Washington, D. C. 20546 

3. RECIPIENT'S CATALOG NO. 

5. REPORT DATE 

January 1981 

6. PERFORMING ORGANIZATION CODE 

8. PERFORMING ORGANIZATION REPDR r $t 

10. WORK UNIT NO. 

M-321 

11. CONTRACT OR GRANT NO. 

NAS8 -32902 

13, TYPE OF REPOR-,' PERIOD COVEREC 

Contractor Report 

1.2. SPONSORING AGENCY CODE 

IF. (of this page) 21. NO. OF PAGES 

15. SUPPLEMENTARY NOTES 

Marshall Technical Monitor: Dr. C. R. O'Dell 

Final Report 

16. ABSTRACT 

Probabilities are calculated for acquiring suitable Guide Stars (GS) with the Fine 

Guidance System (FGS) of the Space Telescope. A number of the considerations and techniques 

described here are also relevant for other space astronomy missions. 

FGS are reviewed. The available data on bright star densities are summarized and a previous 

error in the literature is corrected. 

probabilities are described. A simulation of Space Telescope pointing is carried out using the 

Weistrop north Galactic pole catalog of bright stars. Sufficient information is presented so that 

the probabilities of acquisition can be estimated as a function of position in the sky. 

The constraints of the 

Separate analytic and Monte Carlo calculations of the 

22. PRICE 

The probability of acquiring suitable guide stars is greatly increased if the FGS can allow 

an appreciable difference between the (bright) primary GS limiting magnitude and the (fainter) 

secondary GS limiting magnitude. 

Unclassified 

7. KEY WORDS 

I 32 A02 Unclassified 

Guide Star Probabilities 

Monte Carlo Calculations 

Weistrop Data 

19. SECURITY CLASSIF. (of thin repdl 20. SECURITY CLA! 

18. DISTRIBUTION STATEMENT 

Unclassified - Unlimited 

NASA-Langley, 1981