Abstract

The accuracy of locating weak photographic star images is described from a theoretical viewpoint. The objective is to determine the accuracy limitations imposed by the granular nature of the photographic image, by background radiation, and by image size and shape. After selecting models for both saturated and unsaturated images, lower bounds are derived for the rms location errors. These relationships are based on results developed for photoelectric images. The bounds apply to every method of interrogating the photographic images, and thus represent intrinsic limitations. For unsaturated images, the bound is a monotone function of the image “spread” σ; it is approximately proportional to σ(c-lnσ)-12 where c is a constant. For saturated images, the bound is not necessarily a monotone function of the image spread. The bound may decrease as the image spread increases. The error bounds are compared to experimental errors. For an eighteenth-magnitude star, the bound is 70% of the experimental errors observed with plates from the 48-in. Schmidt telescope on Palomar.

© 1966 Optical Society of America

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References

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  1. W. A. Baum, [Chapter 1] in Astronomical Techniques, edited by W. A. Hiltner (University of Chicago Press, Chicago, Illinois, 1962); J. C. Marchant and A. G. Millikan, J. Opt. Soc. Am. 55, 907 (1965); R. C. Jones, J. Opt. Soc. Am. 51, 1159 (1961).
    [CrossRef]
  2. The effect of scanning a gaussian-shaped image with a rectangular aperture is described by E. J. Farrell and C. D. Zimmerman, in Optical and Electro-Optical Information Processing, edited by J. T. Tippett and et al. (MIT Press, Cambridge, Massachusetts, 1965).
  3. W. A. Baum, Trans. Int. Astr. Un. 9, 681 (1955).
  4. E. J. Farrell, J. Opt. Soc. Am. 56, 578 (1966).
    [CrossRef]
  5. Derivation given in reference cited in Ref. 4.
  6. See reference cited in Ref. 1.
  7. N. M. Evensen and R. E. Mohr investigated the characteristics of star images as a complement to a proper-motion survey being conducted by Professor Willem Luyten at the University of Minnesota.
  8. This relationship is developed by B. E. Bayer, J. Opt. Soc. Am. 54, 1485 (1964).
    [CrossRef]
  9. For small values of r, I0(r)≈r/8.

1966 (1)

1964 (1)

1955 (1)

W. A. Baum, Trans. Int. Astr. Un. 9, 681 (1955).

Baum, W. A.

W. A. Baum, Trans. Int. Astr. Un. 9, 681 (1955).

W. A. Baum, [Chapter 1] in Astronomical Techniques, edited by W. A. Hiltner (University of Chicago Press, Chicago, Illinois, 1962); J. C. Marchant and A. G. Millikan, J. Opt. Soc. Am. 55, 907 (1965); R. C. Jones, J. Opt. Soc. Am. 51, 1159 (1961).
[CrossRef]

Bayer, B. E.

Evensen, N. M.

N. M. Evensen and R. E. Mohr investigated the characteristics of star images as a complement to a proper-motion survey being conducted by Professor Willem Luyten at the University of Minnesota.

Farrell, E. J.

E. J. Farrell, J. Opt. Soc. Am. 56, 578 (1966).
[CrossRef]

The effect of scanning a gaussian-shaped image with a rectangular aperture is described by E. J. Farrell and C. D. Zimmerman, in Optical and Electro-Optical Information Processing, edited by J. T. Tippett and et al. (MIT Press, Cambridge, Massachusetts, 1965).

Mohr, R. E.

N. M. Evensen and R. E. Mohr investigated the characteristics of star images as a complement to a proper-motion survey being conducted by Professor Willem Luyten at the University of Minnesota.

Zimmerman, C. D.

The effect of scanning a gaussian-shaped image with a rectangular aperture is described by E. J. Farrell and C. D. Zimmerman, in Optical and Electro-Optical Information Processing, edited by J. T. Tippett and et al. (MIT Press, Cambridge, Massachusetts, 1965).

J. Opt. Soc. Am. (2)

Trans. Int. Astr. Un. (1)

W. A. Baum, Trans. Int. Astr. Un. 9, 681 (1955).

Other (6)

For small values of r, I0(r)≈r/8.

W. A. Baum, [Chapter 1] in Astronomical Techniques, edited by W. A. Hiltner (University of Chicago Press, Chicago, Illinois, 1962); J. C. Marchant and A. G. Millikan, J. Opt. Soc. Am. 55, 907 (1965); R. C. Jones, J. Opt. Soc. Am. 51, 1159 (1961).
[CrossRef]

The effect of scanning a gaussian-shaped image with a rectangular aperture is described by E. J. Farrell and C. D. Zimmerman, in Optical and Electro-Optical Information Processing, edited by J. T. Tippett and et al. (MIT Press, Cambridge, Massachusetts, 1965).

Derivation given in reference cited in Ref. 4.

See reference cited in Ref. 1.

N. M. Evensen and R. E. Mohr investigated the characteristics of star images as a complement to a proper-motion survey being conducted by Professor Willem Luyten at the University of Minnesota.

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Figures (9)

Fig. 1
Fig. 1

Photographic image of a weak star.

Fig. 2
Fig. 2

Densitometer traces of original plates.

Fig. 3
Fig. 3

Illustration of image elongation from error in right-ascension tracking.

Fig. 4
Fig. 4

Densitometer traces of duplicated plates.

Fig. 5
Fig. 5

Models of grain distribution. US, unsaturated image; S, saturated image; P, photographic plate.

Fig. 6
Fig. 6

Bounds on Var x ˆ 0, Var ŷ0, and Var x ˆ 0 Var ŷ0.

Fig. 7
Fig. 7

Graph of IR0(r).

Fig. 8
Fig. 8

Transmittance profile as an example.

Fig. 9
Fig. 9

Location error. A, observed error with a single scan; B, estimated error with multiple scans; C, minimum achievable error.

Tables (1)

Tables Icon

Table I Calculation of minimum location error for images on plates from the 48-in. Schmidt telescope.

Equations (18)

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G 1 ( x , y ) = N s / 2 π σ x σ y ( 1 - ρ 2 ) 1 2 × exp { - [ 1 / 2 ( 1 - ρ 2 ) ] × [ x 2 / σ x 2 + y 2 / σ y 2 - 2 ρ ( x / σ x ) ( y / σ y ) ] } ,
G 2 ( x , y ) = { g 1             for             ( x / σ x ) 2 + ( y / σ y ) 2 - 2 ρ ( x / σ x ) ( y / σ y ) ( 1 - ρ 2 ) R 0 g 2 G 1 ( x , y )             otherwise ,
- G 2 ( x , y ) d x d y = N s .
g 1 = [ N s / 2 π σ x σ y ( 1 - ρ 2 ) 1 2 ] [ 1 / ( R 0 / 2 + 1 ) ] g 2 = e R 0 / 2 / ( R 0 / 2 + 1 ) .
R G i ( x , y ) d x d y             i = 1 , 2.
Var x ˆ 0 B x - 1 ,             Var y ˆ 0 B y - 1 , Var x ˆ 0 Var y ˆ 0 ( B x B y - B x y 2 ) - 1 ,
B x = - [ x G i ( x , y ) ] 2 G i ( x , y ) + G b d x d y B y = - [ y G i ( x , y ) ] 2 G i ( x , y ) + G b d x d y B x y = - x G i ( x , y ) y G i ( x , y ) G i ( x , y ) + G b d x d y ,
x G i ( x , y ) = ( / x ) G i ( x , y ) y G i ( x , y ) = ( / y ) G i ( x , y ) i = 1 , 2.
B x = [ N s / σ x 2 ( 1 - ρ 2 ) ] I 0 ( r ) B y = [ N s / σ y 2 ( 1 - ρ 2 ) ] I 0 ( r ) B x B y - B x y 2 = [ N s I 0 ( r ) ] 2 / σ x 2 σ y 2 ( 1 - ρ 2 ) ,
I 0 ( r ) = - x 2 2 π exp ( - x 2 - y 2 ) exp ( - x 2 / 2 - y 2 / 2 ) + 2 / r d x d y ,
r = N s / π σ x σ y ( 1 - ρ 2 ) 1 2 G b .
R e = { [ 1 / ( 1 - ρ 2 ) ] [ ( x / σ x ) 2 + ( y / σ y ) 2 - 2 ρ ( x / σ x ) ( y / σ y ) ] 1 } .
σ x / [ N s I 0 ( r ) ] 1 2
I 0 ( r ) = 0.174 ln ( r ) + 0.06.
B x = [ N s / σ x 2 ( 1 - ρ 2 ) ] I R 0 ( r ) B y = [ N s / σ y 2 ( 1 - ρ 2 ) ] I R 0 ( r ) B x B y - B x y 2 = [ N s I R 0 ( r ) ] 2 / σ x 2 σ y 2 ( 1 - ρ 2 ) ,
I R 0 ( r ) = x 2 + y 2 R 0 x 2 2 π exp ( - x 2 - y 2 ) exp ( - x 2 / 2 - y 2 / 2 ) + 2 / r d x d y = 1 4 R 0 v e - v e v / 2 + 2 / r d v .
G = ( 1 / a ) ln ( 1 / T ) ,
N s = 2 π σ x 2 G 1 ( 0 , 0 ) 1.3 × 10 5 .