Abstract

This paper presents a theoretical study of the diffraction pattern which will be obtained with the telescope launched aboard the HIPPARCOS satellite. The output signal is studied by simulating observations of various stars of different spectral types and classes. In the case of very faint stars, the signal will consist of only a few photons during a sampling interval, so that a statistical recovery of the signal will be needed. A simulation of such a signal is described. The last part of this paper deals with the data processing.

© 1981 Optical Society of America

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References

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  1. “Space Astrometry, HIPPARCOS, Report on the phase A study,” Eur. Space A. Sci. ( 79) 10 (1979).
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).
  3. N. Mauron, M. Froeschle, J. Kovalevsky, “Etude de la tache de diffraction de HIPPARCOS,” unpublished paper (1979).
  4. M. Golay, Introduction to Astronomical Photometry (Reidel, Dortrecht, 1974).
    [CrossRef]
  5. C. W. Allen, Astrophysical Quantities (Athlone, London, 1973).
  6. W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, 1970, Vol. 1, Chap. 12).
  7. J. Kovalevsky, Celestial Mech. 22, 153 (1980).
    [CrossRef]
  8. H. J. Fogh Olsen, L. Helmer, “The Carlsberg Automatic Meridian,” in International Astronomical Union Colloquium 48, F. V. Prochazka, R. H. Tucker, Eds. (Institute of Astronomy, Vienna, 1979), p. 219.
  9. C. Barbieri, P. L. Bernacca, Eds. European Satellite Astrometry (Padova, 1979).
  10. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965).

1980 (1)

J. Kovalevsky, Celestial Mech. 22, 153 (1980).
[CrossRef]

1979 (1)

“Space Astrometry, HIPPARCOS, Report on the phase A study,” Eur. Space A. Sci. ( 79) 10 (1979).

Allen, C. W.

C. W. Allen, Astrophysical Quantities (Athlone, London, 1973).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Feller, W.

W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, 1970, Vol. 1, Chap. 12).

Fogh Olsen, H. J.

H. J. Fogh Olsen, L. Helmer, “The Carlsberg Automatic Meridian,” in International Astronomical Union Colloquium 48, F. V. Prochazka, R. H. Tucker, Eds. (Institute of Astronomy, Vienna, 1979), p. 219.

Froeschle, M.

N. Mauron, M. Froeschle, J. Kovalevsky, “Etude de la tache de diffraction de HIPPARCOS,” unpublished paper (1979).

Golay, M.

M. Golay, Introduction to Astronomical Photometry (Reidel, Dortrecht, 1974).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965).

Helmer, L.

H. J. Fogh Olsen, L. Helmer, “The Carlsberg Automatic Meridian,” in International Astronomical Union Colloquium 48, F. V. Prochazka, R. H. Tucker, Eds. (Institute of Astronomy, Vienna, 1979), p. 219.

Kovalevsky, J.

J. Kovalevsky, Celestial Mech. 22, 153 (1980).
[CrossRef]

N. Mauron, M. Froeschle, J. Kovalevsky, “Etude de la tache de diffraction de HIPPARCOS,” unpublished paper (1979).

Mauron, N.

N. Mauron, M. Froeschle, J. Kovalevsky, “Etude de la tache de diffraction de HIPPARCOS,” unpublished paper (1979).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Celestial Mech. (1)

J. Kovalevsky, Celestial Mech. 22, 153 (1980).
[CrossRef]

Eur. Space A. Sci. (1)

“Space Astrometry, HIPPARCOS, Report on the phase A study,” Eur. Space A. Sci. ( 79) 10 (1979).

Other (8)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

N. Mauron, M. Froeschle, J. Kovalevsky, “Etude de la tache de diffraction de HIPPARCOS,” unpublished paper (1979).

M. Golay, Introduction to Astronomical Photometry (Reidel, Dortrecht, 1974).
[CrossRef]

C. W. Allen, Astrophysical Quantities (Athlone, London, 1973).

W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, 1970, Vol. 1, Chap. 12).

H. J. Fogh Olsen, L. Helmer, “The Carlsberg Automatic Meridian,” in International Astronomical Union Colloquium 48, F. V. Prochazka, R. H. Tucker, Eds. (Institute of Astronomy, Vienna, 1979), p. 219.

C. Barbieri, P. L. Bernacca, Eds. European Satellite Astrometry (Padova, 1979).

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1965).

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Figures (10)

Fig. 1
Fig. 1

Optical system of the satellite HIPPARCOS.

Fig. 2
Fig. 2

Entrance pupil. ➀, ➁ collecting surface of the complex mirror; ➂ Secondary Baker Schmidt mirror obscuration.

Fig. 3
Fig. 3

Diffraction pattern. Every closed curve is a solution of the equation 10 + 2 log[I(u,v)/I(0,0) = C. 0.56 cm = 1 sec of arc.

Fig. 4
Fig. 4

Comparison between the computed and observed diffraction pattern.

Fig. 5
Fig. 5

Intensity distribution of the light along the u axis. λ = 400 nm. Curve represents the function g(u) = 5 + log[F(u)/F(u0)].

Fig. 6
Fig. 6

Intensity distribution of the light along the u axis for the whole spectra. Curves give the number of photons per unit of time. U is in sec of arc, and the distance is 10 parsec. 1, TE = 15,000 K; 2, TE = 9700 K; 3, TE = 6000 K; and 4, TE = 3900 K.

Fig. 7
Fig. 7

Normed deterministic signal for a whole slit.

Fig. 8
Fig. 8

Numerical simulation of the output for two typical stars. Upper and lower diagrams are obtained, respectively, for 100 and 3 photons at the maximum of the modulation.

Fig. 9
Fig. 9

Precision (2σ) in the measurement of the angular distance of two similar stars in terms of the duration of one observation, and of the average number of photoelectrons at the maximum of the modulation during 1 msec. This set of curves refers to the first method described in the text.

Fig. 10
Fig. 10

Precision (2σ) in the measurement of the angular distance of two similar stars, in terms of the duration of one observation, and of the average number of photoelectrons at the maximum of the modulation during 1 msec. This set of curves refers to the second method described in the text.

Tables (3)

Tables Icon

Table I Comparison between Visible, Blue, and HIPPARCOS Magnitude

Tables Icon

Table II Number of Expected Photoelectrons per Millisecond a,b

Tables Icon

Table III Example of a Deterministic Signal (Line I) for a Star of mH = 11.5

Equations (24)

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A ( u , v ) = i λ f ( x , y ) exp [ 2 i π λ ( u x + v y ) ] d x d y ,
A ( u , v ) = H i λ [ a c exp ( π i v c / λ ) sinc ( u a λ ) sinc ( v c λ ) - b d exp ( π i v d / λ ) sinc ( u b λ ) sinc ( v c λ ) ] ,
I ( u , v ) = H 2 λ 2 { a 2 c 2 sinc 2 u a λ sinc 2 v c λ + b 2 d 2 sinc 2 u b λ sinc 2 v d λ - 2 a b c d sinc u a λ sinc v c λ sinc u c λ sinc v d λ cos [ π v λ ( c - d ) ] } .
F ( u ) = - + I ( u , v ) d v .
F ( u ) = H 2 λ ( a 2 c sinc u a λ + b 2 d sinc u b λ - 2 a b d sinc u a λ sinc u b λ ) .
B ( ν ) = 2 h ν c 2 / [ exp ( h ν / k T ) - 1 ] .
B ( ν ) = 2 h ν 3 c 2 exp ( - h ν / k T ) .
Q = 0             for λ < 150 nm Q = 0.2             for 150 nm < λ < 400 nm Q = [ - 0.56 × 10 6 λ + 0.43 or [ - 1.7 × 10 14 ν + 0.43 Q = 0             for λ > 750 nm }             for 400 nm < λ < 750 nm .
N ( u ) d u F ( u ) Q ( ν ) h ν T ( ν ) d ν d u .
H 2 ( ν ) = π B ( ν ) ( R D ) 2 ,
m H = - 2.5 log E ( ν ) Q ( ν ) h ν T ( ν ) d ν + C ,
m H - m B = - 2.5 log E ( ν ) Q ( ν ) h ν T ( ν ) d ν E ( ν ) B 0 ( ν ) d ν + C ,
C ( u 0 ) = G ( u ) N ( u - u 0 ) d u .
ρ = C max - C min C max + C min ,
P 1 ( u 0 ) = G ( u ) F ( u - u 0 ) d u / F ( u - u 0 ) d u .
P ( N = N 0 ) = K N 0 exp ( - K ) N 0 ! .
P ( N p = n ) = N 0 P ( N = N 0 ) · P [ ( N p = n ) / N 0 ] .
P ( N p = n ) ( P 1 P 2 K ) n n ! exp - P 1 P 2 K .
P ( N p = n ) = q n n ! exp ( - q ) ,
S 2 ( a , b , x ) = exp ( a x ) sin 2 b x d x = exp ( a x ) 2 a - exp ( a x ) a 2 + 4 b 2 ( a 2 cos 2 b x + b sin 2 b x ) , C 1 ( a , b , x ) = exp ( a x ) cos b x d x = exp ( a x ) a 2 + b 2 ( a cos b x + b sin b x ) , C 1 ( a , b , x ) = x exp ( a x ) cos b x d x = exp ( a x ) a 2 + b 2 × [ ( a x - a 2 - b 2 a 2 + b 2 ) cos b x + ( b x - 2 a b a 2 + b 2 ) sin b x ] , S 2 ( a , b , x ) = x exp ( a x ) sin 2 b x d x = exp ( a x ) 2 ( x a - 1 a 2 ) - 1 2 · C 1 ( a , 2 b , x ) .
N ( u ) = 2 h c 2 ν 3 exp ( - h ν k T ) · Q ν h ν { a 2 c sin 2 [ ( π u a ν ) / c ] [ ( π u a ν ) / c ] 2 + b 2 d sin 2 [ ( π u b ν ) / c ] [ ( π u b ν ) / c ] 2 - 2 a b d sin [ ( π u a ν ) / c ] / sin [ ( π u b ν ) / c ] [ ( π u a ν ) / c ] · [ ( π u b ν ) / c ] } · H 2 ν c d ν .
N ( u ) = 2 H 2 c + 1 π 2 u 2 ν exp ( - h ν k T ) Q ν × [ c sin 2 π u a ν c + d sin 2 π u b ν c - d cos π u ν c ( a - b ) + d cos π u ν c ( a + b ) ] d ν .
N ( u ) = Q 0 [ S 0 ν ( u ) ] ν 2 ν 3 + Q 1 [ S 1 ν ( u ) ] ν 1 ν 2 + Q 2 [ S 0 ν ( u ) ] ν 1 ν 2
S 0 ν ( u ) = 2 H 2 c · 1 π 2 u 2 · { C · S 2 ( - h k T , π u a c , ν ) + d · S 2 ( - h k T , π u b c , ν ) - d S 1 [ - h k T , π u c ( a - b ) , ν ] + d S 1 [ - h k T , π u c ( a + b ) , ν ] } , S 1 ν ( u ) = 2 H 2 c · 1 π 2 u 2 · { C · S 2 ( - h k T , π u a c , ν ) + d · S 2 ( - h k T , π u b c , ν ) - d C 1 [ - h k T , π u c ( a - b ) , ν ] + d C 1 [ - h k T , π u c ( a + b ) , ν ] } ,

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