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 1 9 7 2 ) and classical data on bright star

counts (see especially Seares -- et al.

Soneira 1 9 8 0 ) . 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 1 9 8 1 on

1 9 2 5 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 con-

tained 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 1 9 7 8 ) .

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

constant between 4660 8 to 7 0 0 0 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 c3 2 a a a K

W z J e a

J

0 0 LL

a

2

t o t h e Visua l band, b u t has a factor of 2 . 7 g r e a t e r energy

throughput and 2.8 g r e a t e r p h o t o n throughput f o r t h e c o l o r d i s -

t r i b u t i o n of stars b r i g h t e r t han V = 1 4 . 0 ; see Bahcal l and

Sonei ra 1981, § 111. For V = 15, t h e energy and photon

throughputs are 2 .7 and 2 . 9 , r e s p e c t i v e l y . )

3 .

stars (i = 1 , 2 ) i s 9 < m < m where 1 4 5 ml 5 1 5 and

(m2 - ml) could i n some des igns be as l a r g e as 2 magnitudes.

4 .

have a magnitude s e p a r a t i o n g r e a t e r t han Am = 0 . 5 mag.

5 .

B S = 1 3 . 5 arcmin.

The magnitude range from t h e primary and secondary guide

- V i'

S t a r s closer than B e = 1 . 0 arcmin t o a guide s tar must

A p a i r of gu ide stars must be s e p a r a t e d by a t least

b ) A n a l y t i c Calcu la t ion

An a n a l y t i c s t a t i s t i ca l treatment of t h e selection c r i te r ia

i s u s e f u l f o r g a i n i n g an unders tanding of t h e GS p r o b a b i l i t i e s . .

( C a l c u l a t i o n s u s i n g Monte Carlo methods are more accurate and,

i n some ways, more conven ien t . )

The p r o b a b i l i t y P = P ( R , b , m ) of f i n d i n g a t least one

s u i t a b l e G S pa i r i n t h e d i r e c t i o n (Il,b) w i t h appa ren t magni-

t u d e s b r i g h t e r t h a n rn = ml = m2 i s g iven t o h igh accuracy by:

3 2 P(R,b,m) = P1 + (2Q + l ) P 1 ( l - P1) I

w i t h :

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 / ; t ia ,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, require-

ment 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 > e s .

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 . 8 7 3 ' . 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 ap-

proximation.)

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 . 8 2 .

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 . 0 2 0 ( k ) e ~ 0.5 e

For mv > 14, the influence of Be and Am on P decreases monotonical-

ly with increasing mv. 6

The above value for P is calculated for the Weistrop field

centered at R = 65.5', b = 85.5' (SA 5 7 ) .

star density is expected to be at a minimum slightlyawayfrom

At mv = 14 mag, the

the Galactic pole in the direction of the galactic anti-

center R = 1 8 0 ° , 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 obscura-

tion 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 = 8 0 ° 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 4 2 % ) . 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 dis-

7

" 0 0" T? /r,

G A L A C T I C C E N T E R

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 representa-

tive of the star density at its galactic latitude and longi-

tude).

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 presum-

9

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, ( 1 4 ~ 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 al-

lowing 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

~

P -

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

Note that for d2 >> dl P - 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 to-

gether 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. P - Approximate 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 = 8 0 ° ) 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 proba-

bility 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 improve-

ment 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

Bahca l l , J . N . , and Sone i r a , R. M. 1 9 8 0 , Ap.J. Suppl . ,

i n p r e s s (Paper I ) .

. 1 9 8 1 , submitted to Ap.J.

Groth, E . J. 1 9 7 8 , " P r o b a b i l i t i e s of Finding S u i t a b l e Guide

S ta rs , " i n " U s e of t h e WFC as a 'F inder C a m e r a ' f o r t h e

Small Aper ture SI'S," NASA r e p o r t .

S e a r e s , F. H . , van Rh i jn , P. J . , Joyner , M. C . , and Richmond,

M. L. 1925, Ap.J . , 6 2 , 320.

Weistrop, D . 1 9 7 2 , A . J . , - 77, 366.

. 1 9 8 0 , p r i v a t e communication.

13

N w f - w w w Q m m I - * m O m w m m f - m f - m l v w * m w * w m w w m w m m l v m ( u ( u w o o o o o o o o o o o o o o o o o o o l v 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . . . .

m f - * r l 4 m r l m o r l m w l v r l r l w m m m m r l f - w N o o o o 0 0 r l ~ 4 r l r l 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . .

t - m m w m r l o r l w r l r - m m r l m f - m r l o o o r l ( v l v r l o o o o o 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . .

w m m m I - o l v m w m o m r l o ( v m ( v r l o o 0 0 0 0 0 0 0 0 . . . . . . .

w I - r l m m o m f - f - o r l m o ~ w m o m m l n m m m r - b w w o m m d m w m l v 4 N r l r l O o o o o o o o o o o o o o o o o o o o r l 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . . . .

N l - c v d m O c o m d d d d L n m r l r l f - r l m N d o O o o r l l v ( u r l r l o o o o o o 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . .

r l w o o o o o m r l m m f - I - c o m 4 o o r l ( u ~ ~ 0 0 0 0 0

0 0 0 0 0 0 0 0 0 . . . . . . . . .

w m m m b o c v m w m o m r 1 0 ( U m l v 4 0 0 0

0 0 0 0 0 0 0 . . . . . . .

L n m m o ( u b ( v m m m b m m m l v r l f - w w m m r l m m m m w * m l v l v r l r l ~ r l r l o o o l v r l r l r l o o o o o o o o o o o o o o o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . . . .

m w l v w r l l v o f - N r l 4 m w m r l m l v o o o r l ( u ( u r l ~ 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 . . . . . . . . . .

w a m m r - o m m w m o m r l o ( u m l v r l o o o 0 0 0'0 0 0 0 . . . . . . .

* m m m b o ( u m w m o m r l o l v m l v r l o o o 0 0 0 0 0 0 0 . . . . . . .

14

h a a, 7 c -4 4J c 0 W * m c 0 4 4J 7 R -4 k 4J rn -4 cl h 4 .A rl -4

J2 0 k PI

k

4J v)

al a -4 3 c!J

Y

9

m

rl

w l=l

3

It I I

E c N

m w m ~ r l m N w m m N m r l o W w m w r - 0 N r l m m w m w m s m s m w m m m n l m N w O O O O O O O O O O O O O O O O O O O m

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . . . .

r - r - N r - m r l w m d m m w N r l r l w m r - w w ~ r - w r l o o o o 0 0 r l r l r l ~ r l 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . .

r - w m w w r l o r l w r l r - m m r l m r - m r l o o 0 r l N N r l 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 . . . . . . . . . .

r - w m w w r l o r l w r l r - m m r l m r - m l - i o o 0 r l N N ~ 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 . . . . . . . . . .

w w r l m N w w o N w w ~ N N m m w m m w N r l w m m m m w m w w a w m m m N N N w O O O O O O O O O O O O O O O O O O O N

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . . . .

m r - w m m m w 0 0 0 m w N r l r - w u 3 r - w o r - w N o o o 0 0 r l r l r l ~ r l 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . .

r - w m w W ~ 0 r l w l - l I - m m r l m r - m r l o o 0 r l N N r l 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 . . . . . . . . . .

r l w o o o o o m r l m a r - r - w m r l o o r l N N r l 0 0 0 0 0

0 0 0 0 0 0 0 0 0 . . . . . . . . .

0 N m r l m N w w W w w " N m m m w r - m W m r - r - r - m W w m w w m w m N N N r l d N o o o o o o o o o o o o o o o o o o o r l 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . . . .

r l r - r - o w m r l o N w N r l m m O r l r - ~ w m ~ 0 0 0 0 r l N N r l ~ 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . .

r l w o o o o o m r l m w r - r - w m r l o o r l N N r l 0 0 0 0 0

0 0 0 0 0 0 0 0 0 . . . . . . . . .

r l w o 0 0 0 0 m r l m w r - r - w m r l o o r l N N ~ 0 0 0 0 0

0 0 0 0 0 0 0 0 0 . . . . . . . . .

. rl

d w w c o w c n n l m r l o m w w m * ~ ~ m m r - m d r - r - c o m w m m l n m m m m m N m r l d r - d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . . . . . . . .

c n m o r - d ~ o r l m O o m 4 r l d w d ~ c o m d r - m N d 0 0 0 0 0 d d d d d 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . . .

F 03 m w a3 rl 0 rl w rl r - m m r l m r - m r l o o O r l c v N ~ 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 . . . . . . . . . .

m m c o r - m r l O d d N 0 0 m m r l o o o 0 0 0 0 0 0 . . . . . .

m c 0 -4 c, -d c -4 4-1 a, a k O w 0 .rl

0) F m a al al CA c

16

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

m = 1 5 . 0

n = 1 .89 n = 2.56 0

m = 1 4 . 0 m = 14 .5

n = 1.37 0

m = 1 3 . 5

n = 0.980 0

m = 13.0

n = 0.684 0 0

1 1 1 1 n

1

0

1

2

3

4 ..

5

6

7

8

9

1 0

11

1 2

1 3

1 4

1 5

1 6

17

1 8

0 .128

0.264

0.270

0.185

0 .095

0 .039

0 .013

0 .004

0 . 0 0 1

0 .053

0.155

0.228

0 .224

0 .165

0 .097

0.047

0.020

0.007

0 .002

0 . 0 0 1

0.016

0.067

0.139

0.190

0.195

0.160

0.110

0.064

0.033

0.015

0.006

0 .002

0 . 0 0 1

0 .003

0 .020

0 .055

0 .105

0 .148

0 .168

0 .159

0 .129

0 . 0 9 1

0 .058

0.033

0 .017

0.008

0.003

0 . 0 0 1

0 .001

0.000

0.004

0 .014

0.035

0.067

0.103

0 .132

0 .144

0.139

0 .118

0 . 0 9 1

0 .063

0 . 0 4 1

0.024

0 .013

0 .007

0 .003

0 . 0 0 1

0 .001

17

-Al-

APPENDIX A

11. BRIGHT-STAR DENSITY

A. The Best Available Data

The principal source of data that we used for the bright

( ~ 1 6 ~ ~ ~ ) 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 magni-

tude 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 esti-

mating 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) con-

version 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

-AS-

using 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

M > 3.7 , (1 I -

which a c c u r a t e l y f i t s t h e data of Sandage (1957), Schmidt

(1959) and McCuskey (1966) . A scale h e i g h t of 250 p c is

adopted for t h e g i a n t s (see paper I ) .

-A7- REFERENCES

Allen, C. W. 1 9 6 3 , Astrophysical Quantities, second edition

(London: Athlone Press).

. 1 9 7 3 , Astrophysical Quantities, third edition

(London: Athlone Press).

~

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

(Paper I)

Brown, G. S. 1 9 7 9 , A.J., - 8 4 , 1 6 4 7 .

Faber, S. M., Burstein, D., Tinsley, B., and King, I. R. 1 9 7 6 ,

A.J., 8 1 , 4 5 . - Groth, E. J. 1 9 7 8 , "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. 1 9 8 0 , private communication.

Kron, R. G. 1 9 7 8 , Ph.D. Thesis, Univ. of California, Berkeley.

McCuskey, S. W. 1 9 6 6 , Vistas in Astronomy, 7 , 1 4 1 .

Peterson, B. A., E l l i s , R. S. Kibblewhite, E. J., Bridgeland,

M. T., Hooley, T., and Horne, D. 1 9 7 9 , Ap.J. (Letters),

233 , L109.

Rhijn, P. J. van 1 9 2 9 , Pub. Kapteyn Astro. Lab. Groningen, N o . , 4 3 .

Sandage, A. 1 9 5 7 , Ap.J., 1 2 5 , 422 .

Scheffler, H., and Elsasser, H. 1 9 6 5 , Landolt-Bornstein Tables,

Group VI, 1, p. 6 0 1 .

Schmidt, M. 1 9 5 9 , Ap.J., 1 2 9 , 2 4 3 .

Seares,.F. H. and Joyner, M. C., 1 9 2 8 , Ap.J., 6 7 , 2 4 .

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

M. L. 1 9 2 5 , Ap.J., 6 2 , 3 2 0 .

Stebbins, J., Whitford, A. E., and Johnson, H. L. 1 9 5 0 , Ap.J.,

1 1 2 , 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 1 2 1 3 1 4 1 5 1 6

W e i s t r o p 1 . 9 3 . 7 7 .6 1 8 3 7 74 135 224

a Seares e t a l . -- 1 . 2 3 . 3 8.0 1 8 3 6 7 1 134 216

B a h c a l l - S o n e i r a 1 . 5 3 .5 7 . 7 1 6 3 4 70 1 3 8 254 Mode 1

A l l e n 1 . 4 3 . 6 9 . 1 2 1 49 1 0 2 1 8 6 347

a R e d u c e d t o t h e V i s u a l band i n 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 pur- pose 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 con- venient 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 func- tion 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 ac- curacy 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.

g-

1

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. T I T L E AN0 S U B T I T L E

Guide Star Probabilities

7. AUTHOR(5)

Raymond M. Soneira and John N. Bahcall 9. PERFORMING ORGANIZATION N A M E AND ADDRESS

Institute for Advanced Study Princeton, New Jersey 08540

12. SPONSORING AGENCY N A M E AND ADDRESS

National Aeronautics and Space Administration Washington, D. C. 20546

3. R E C I P I E N T ' S CATALOG NO.

5 . REPORT DATE January 1981

6. PERFORMING ORGANIZATION CODE

8 . PERFORMING ORGANIZATION REPDR r $t

10. WORK U N I T NO.

M-321 1 1 . CONTRACT OR GRANT NO.

NAS8 -32902 13, TYPE OF REPOR-,' PERIOD COVEREC

Contractor Report

1.2. SPONSORING AGENCY CODE

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

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 s t a r densities are summarized and a previous e r r o r 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

2 2 . P R I C E

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 C L A S S I F . (of thin r e p d l 20. SECURITY CLA!

18. D I S T R I B U T I O N S T A T E M E N T

Unclassified - Unlimited

NASA-Langley, 1981