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Concept of a space telescope able to see the planets and even the satellites around the nearest stars
Acta Astronautica Vol. 12. No. 3, pp. 195-201. 1985 Printed in Great Britain.
0094 5765/85 $ 3 . 0 0 + .DO Pergamon Press Ltd.
CONCEPT OF A SPACE TELESCOPE ABLE TO SEE THE PLANETS AND EVEN THE SATELLITES AROUND THE NEAREST STARSt C. MARCHAL~ Office National d'Etudes et de Recherches A~rospatiales (ONERA), 92320 Chatillon, France (Received 19 January 1984; in revised form 13 June 1984)
Abstract--The Space Telescope will be launched in 1986 and will improve considerably our observational possibilities, but not the visibility of outer planetary systems. The solar system as seen from Alpha Centauri is presented, as well as the nearest stars and the main cameras of the Space Telescope. As a possible improvement the action of a distant and star-shaped screen is described; that screen is 100 to 800 meters and placed 105 to 106 km in front of the telescope; it allows one to avoid the dazzling effect of the stars and to look for planets such as Jupiter and Saturn up to 20 to 40 light-years. Such planets as Earth and Venus are a little less visible. The visibility of satellites such as the Moon is discussed; it remains at the limit of our technical possibilities. This conceptual paper does not consider in detail the technical difficulties involved.
1. INTRODUCTION
Let us note the following drawbacks. 1. The gap between the Sun and the planets is very large: 21.6 magnitudes between the Sun and Jupiter. The magnitude scale being a logarithmic scale (ratio 10 for a magnitude difference of 2.5), it means that the Sun is 4.4 × 108 times brighter than Jupiter at its full phase (and 2.8 billion times brighter than Earth). 2. A planet is at its full phase only when it is behind its star, i.e. at the worst position, and the maximum angular distance from the star corresponds to the quarters. The quarters are of course less bright than the full phase and the magnitude difference is about 2.7 (ratio 12) for a celestial body without atmosphere (Mercury, Moon); it is smaller but already 1.5 (ratio 4) for a planet with atmosphere such as Venus. It seems that the position half-way between the quarter and the full phase corresponds to the greatest visibility. 3. Alpha Centauri is almost the nearest star and at twice its distance, the distance of Sirius, we must add 1.5 to the magnitude of Table 1 and the Sun-planet angular distances are twice as small.
After some delays the Space Telescope will be launched in 1986 by the shuttle on a circular orbit at about 500 km altitude. That revolutionary new telescope will avoid the atmospheric screen and will represent a major step in many fields with respect to existing telescopes. It contains of course many instruments (wide field camera, planetary camera, faint object camera, faint object spectrograph, e t c . . . ) and will certainly be the source of many great discoveries. The astronomers have already prepared many series of experiments and observations in all fields of astronomy. However, and this is a surprise, the space telescope will be unable to see directly the planets around the nearest stars. It has already the power necessary to see them but, in spite of an excellent coronograph, the light of the star has a much too large dazzling effect and the astronomers must look for planets through the old indirect ways (irregularities in the motion of the star, etc.), The simplest pictures of an outer planetary system would give us much more information, but how to get them? A coronograph fights the dazzling effect of a powerful source of light and its efficiency increases with its distance from the telescope; we will analyse coronographs at thousands of kilometers in front of the space telescope.
3. THE 13 STARS AT LESS THAN
10 LIGHT-YEARS[I]
That small set of stars shows a wide variety: five single stars and four binaries, four visible stars (Sun, Alpha Centauri, Alpha Centauri B, Sirius) and nine invisible stars, many different spectral types, etc. (Table 2). The number of nearby stars increases rapidly with the distance: 79 stars are at less than 18 light-years (31 single stars, 15 binaries and six triple systems) and we have thus many possible targets.
2. THE SOLAR SYSTEM AS SEEN FROM THE DISTANCE OF ALPHA CENTAURI
The solar system is of course a good example of what we are looking for and the Table 1 shows how small are the possibilities of seeing something.
4. ON THE INSTRUMENTS OF THE SPACE TELESCOPE tPaper presented at the 34th Congress of the International Astronautical Federation, Budapest, Hungary, 10-15 October 1983. :[:Member A.I.A.A.
The Space Telescope (Fig. 1 and Ref, [:2]) will weigh 10 tons, be 15 meters long, with 2.4-m m i r r o r . . , and have an "astronomical" price of one billion dollars. 195
C. MARCHAL
196
Table 1. The solar system as seen from the distance of Alpha Centauri Magnitude at the full phase
Maximum angular
distance the Sun
from
.................................................. SUN 0.3 MERCURY 24.9 03| " ................................................. VENUS 22.4 0.54" EARTH 23.9 O. 75" MOON 27.5 -_-~ .............................................. MARS 26.8 1.14" JUPITER 21.9 3.9" SATURN 23.4 7.2" .................................................. URANUS NEPTUNE PLUTO
26.8 27.5 35 to 36
from the visibility of ordinary planets, and Fig. 2 shows the dispersion of light due to unavoidable diffraction effects. The astronomers still hope to detect many planets around the nearest stars[3], but they will use the old indirect ways (irregularities in the motion and in the radial velocity of the stars). These methods were used for the first time by Bessel in 1844 for the detection of the stellar companions of Sirius and Procyon and they show the existence of at least two very large planets around Barnard's star[4]. However, of course, the simplest picture of an outer planetary system will give us much more information.
14.4" 22.6" 37"(at aphelion)
With respect to terrestrial telescopes it will allow one to see 50 times fainter objects with 10 times better resolution. Its main instruments have the characteristics given in Table 3. The Space Telescope will also contain various kinds of spectrographs and many other smaller instruments. Moreover, the faint object camera can, under favourable conditions, be used in a f/288 mode that gives an angular resolution of 0.007", that is the root mean square of guiding error of the Space Telescope. The faint object camera has a coronograph allowing one to detect at 1" from a bright point object a small faint object 16 to 17 magnitudes fainter (in the best cases). These numbers should be compared to those of Table 1: even for the nearest stars we will be very far
5. A DISTANT SCREEN AS CORONOGRAPH The efficiency of a coronograph increases with its distance to the image and since, in space, very large distances are possible, we can look for the action of a distant screen. Let us consider the usual electro-magnetic equations of Maxwell (in empty space). curlE +
0--~- =
=0By. 0 divE = 0;divB c 2 curl B
t = 0
~-t
with: E = electric field (in volt/m), B = magnetic induction (in Tesla), t = time, c = velocity of light.
Table 2. The 13 nearest stars Star
Distance (light-years)
Sun
0
Proxlma Centaurl
4.22
Right
Declination
Ascension
14 h 26 m
-62 ° 28'
Alpha Centaurl Alpha Centaurl
4.35
14 h 36 m
star Wolf 359
21185
Sirius Sirius
Ross 154
M 5
ll
G 2
O
K 0
1.3
4 ° 33'
M 5
9.5
7.75
I0 h 54 m
7 ° 19'
M 8
13.3
M 2
7.5
8.22
11 h 01 m
36 = 1 8 ' M
t
B
- 26.9
17 h 55 m
B
Luyten 726-8 Luyten 726-8 B
G 2
5.98
Lalande 21185
Lalande
[
Magnitude (as seen from Earth)
-60 = 38'
B
Bar'nard
Spectral type
8.43
1 h 36 m
?
M 5
12.4
M 5
12.9
-18 ° 13'
6 h 43 m
-16 = 39'
A 1 DA
-I.5
8.65
9.45
Ig h 47 m
-23 = 53'
M 4
10.6
8.7
(1)
Space telescope concept
197
Fig. 1. The Space Telescope and the Shuttle.
A usual solution of (1) is the polarized and plane electromagnetic wave (with a usual 0xyz set of reference): E = (0, E>., O)
E,. = cB_ = f ( c t -
x).
(2)
B = (0,0, B:) The function f ( c t - x) can be arbitrary; it is usually decomposed into sinusoidal components: E~ = cB: = Eo e x p [ j k ( c t -
x)],
(3)
with:
j = square root of minus one, k = 2n/2; 2 = wavelength.
5.1 Diffraction given by a hole
Let us look for the propagation of the plane polarized and sinusoidal electromagnetic wave given by (2)(3)(4) after its passage through a hole in the Oyz plane (Fig. 3). With an excellent approximation and when x(P) is large with respect to ,~ the electromagnetic field in P is
Table 3. Characteristics of the main cameras of the Space Telescope Instrument
Field of view
Wide field
Planetary
camera
camera
2'40"x2'40"
Angular resolution (one pixel) Wavelengh range
(quantum efficiency~lg~a) Photometric accuracy Dynamic
0.I"
I'9"xi'9"
0.043"
0.115 to
0.Ii to
i.I~
1.1~
1%
~ 1% 28
8.5(mv~28
Faint object cameral(f/48)
22"x22"
O. 045"
0.12 to 0.6U
at least 2%
21.< mv~ 28
Fain object camera2(f/96)
ll"xll"
0.022"
0.12 to 0.6
at least 2%
21~ mv~ 28
range
'm = v i s u a l ~
m~gnit~de]
AA 12:3-E
signal/nolse ~ 3
(4)
c u m u l a t i v e ten hours o b s e r vation) signal/noise ~ 4
198 Z
C: MARCHAL ~3
1
g ~ , 0 -q . re"
N qO-z"
:~ ~0 - ~ . I-r ,,q
7.. ~o-4 m
¢~ I 0 "su. ,,vI0 - 6 . l,u > ,~10 -r. _1 t,.u r,'-
>. or r
f
if)
/
tangular three-hedron of the wave:
! l/
v /
(7)
B ~ = (Eolc) • "y exp { j k ( c t -
(8)
z,,, O
#
-
0.s
et • OM)}.
O "' to
i
Eu = Eo • 13 exp { j k ( c t - et • OM)},
LI.I
/. " ,
Then (5) and (6) give:
lit
j E o ~ dy • dz Ep = ~ j~ r
• exp { j k ( c t - ot • O M - r)} • (u x ~ + ' , / )
05
t0 .t5 .20 .25 .30 IMAGE RADIUS (ARC-SECONDS)
Fig. 2. The expected optical performance (2 = 6328 ,/k) is illustrated in terms of the image surface brightness distribution and the encircled energy. The central obscuration and deviations from perfect mirror figure and alignment have been incorporated in these predictions.
given by the following equations[5]:
• exp { j k ( c t - a • O M - r)} • u X ( ~ + ~/ X u).
(5)
(10)
The integrations (9) and (10) are not easy; however we can obtain the main part of Ep and Bp in the following way: A. The unit vectors u and ot are very close to each other, their angle is less than 10" and we can put: u --- ,a + c,
U
(9)
j E o ~ dy • dz Bp = ~ c .~ r
0
x - exp { - j k r } ,
x u,
I1,~11< 5 x 10%
(11)
hence:
r
(u x 13 + ' , t )
x u=
., (6)
• exp { - j k r } ,
where S represent the hole and its surface, M an arbitrary point of the hole, y and z its coordinates, r the distance M P and u the unit vector of MP. These expressions remain valid even if the wave has not the direction of Ox and we will look for the result of the integrations (5) and (6) when both the wave and M P are only slightly inclined on Ox. Let us call tx, I~, 'f the unit vectors of the right rec-
l
[(a +,~)
= 2',t x
However, with t • u (~
x
13)
u
x
=
=
u +
x I] + " t ] (t
x
x u
I~) x
u.
(12)
g2/2:
I~(~ " u) -
~(13" u)
= [3 ~ - e(l~ " e),
(13)
(u X I~ + ~/) X u = 2'y X u + O(e2),
(14)
u x ( ~ + ',/ x u) = 2 u x [3 + O(e2).
(15)
and thus:
Hence, if we put:
z
u
i
V =
2 .Is d y ' d z ' - r
(16)
• exp { j k ( c t - ot • O M - r)}, /'/
r
P (xp, yp, zp)
we shall get: x
Ep = ~1 x V + O{EoSe2/xp2},
(17)
Bp -
(18)
V
x______~ +
O{EoSe2/%2c} '
C
Fig. 3. Analysis of the diffraction given by a hole.
[with 2 = 2 n / k = wavelength; xp = x(P)].
.k99
Space telescope concept In the integration (16), we can substitute to u / r its average value u0/r0 or even a/ro with only an error of order e 2 on the radial component of V and an error of order e on its circumferential components (all this implying only an error of order e-' on the received power). We thus arrive to: V -~ V0 = Eoa exp {jk(ct - tx • OP)} • J,
(19)
with
that is, with u = p2/2 and ~(p) = ~u(u):
I -j
= I _Jk--f"/du ro Jo • ~(u).exp
~'-Jk---Y-u}.
k
A first choice can be ~,(u) = 1 (circular screen of radius p:), it leads to: 1 - J = exp {-jku//ro}.
dy • dz" exp {jk(a • M P - r)}.
J = ~ro
-2r--'~o(MMa)2 "
d y " d z " exp
~u2(u) = 1 - -
U
u:
implies:
(22)
Note that if J = 1 the eqns (17), (18), (19) give E n and Bp as are EM and E z in (7) and (8): the hole has no effect. The factor J is then really the amplifying factor of the hole for the point P the direction r, and the wavelength 2. 5.2 Diffraction given by a screen The effect of a hole and the effect of a screen of the same place and shape are complementary, their sum correspond to the undisturbed propagation of the electromagnetic wave, and hence the amplifying factor of the screen is 1 - J with the same expression of J given in (22), S being now the surface of the screen. 5.3 The choice of the distant coronograph We want to switch off the light of the star for all interesting wavelengths but we also want to keep the light of its planets. Hence we need an amplifying factor 1 - J as near to zero as possible when M, is at the center of the screen or very near to it, but we also need 1 - J of the order of I when M~ is at about 1" or more from the center of the screen. We must integrate (22) for a suitable shape of screen; for instance let us use the polar coordinate p: p = V'y 2 + z 2,
1 - J is not in the vicinity of zero and the choice is not good. Let us consider some other cases leading to the starshaped screen of Fig. 4:
(21)
We thus arrive to the "amplifying factor" J:
J = ~ro
(26)
(20)
(at • MP - r) is zero at the point Ma where M~P is parallel to ot (Fig. 3), and in its vicinity we can write, with again the same order of error: a • M P - r = -(MMa)2/2ro .
(25)
ro J
(27) ro
1 - J2 = j'k-uuy(1 - e x p { - j k u / / r o } ) .
.2
¥3(u) = 1 - 3 u } + 2 u } implies: 6r.....~o 2 1 - J3 = k2u} [ - 1 - exp {-jku//r0}]
(28)
12jr~
+ k3u---~ [exp { - j k u : / r o } -
(a)
1].
(b)
(23)
and let us assume that at the distance p from its center the screen intercepts the proportion tp(p) of the light. We will obtain:
-J=
l - ~ r ~ fop/27zp
dp
(c)
ox,{
(24)
(d)
Fig. 4. The shape of some possible screens for various ~, functions. (a) ~ = ~¢2. (b) ~ = ~ . (c) ~g = ~u4. (d) q/ = ¢q.
200
C. MARCHAL
These choices are good if for the wavelength of interest (2 = 2~/k <~ 1.1 p) the ratios ro/ku~ are small. For instance with r0 = 105 km, pf = 100 m and uf = 5000 m 2 we get ro/ku s <<- 1/285, that is I1 - J_,l ~< 1/142 for the ~2 screen and ]1 - J31 ~< 1 / 13,600 for the q/3 screen. The powers are proportional to the square of the amplifying factors; hence the first screen adds at least 10.7 to the magnitude of the star and the second screen adds at least 20.6 (with 20.6 = 2.5 log (13,600)z); it becomes useless to use the inner coronograph of the Space Telescope! (But the central zone of darkening is very small.) Let us note the following points: A. At 105 km an angle of 1" corresponds to 500 m, hence the light of a planet at 1" from its star will be almost not disturbed by our screens with only p~ = 100 m. B. If we choose r0 = 106 km and Pl = 100 m • ~ / ~ , we obtain the same amplifying factors and the same effects on the light of the star, but we can look for planets at % / ~ times smaller angles. We found again that the furthest coronographs are the most efficient (we can also prefer to choose p~ = 1000 m and to obtain a better darkening of the star). C. If we take account of the approximations presented in (14), (15) and in the second half of section 5.1, we arrive to an almost negligible correction of functions ~'2(u) and q/3(u). D. The main drawback of the functions ~Uz(U) and q/3(u) is that they are less than one as soon as u is positive and the corresponding screens are discontinuous at the center (Fig. 4). Fortunately if we choose: ~'4(u) = ~us(u) = 1 q/4(u)
~,/,/2(U
~,~(u)
~u~(u
~t
when0<~u<~u~,
u 0 ~ when u, ~< u ~< ut + uj,
(29)
u,)j
We found the following amplifying factors: exp {-jkuflro},
(30)
(1 - Js) = (1 - J3) " exp {-jkuJro},
(31)
(1 - J4) = (1 - J2)
"
and the efficiency of the two last screens (Fig. 4) are exactly that of the corresponding first screens. Forr0 = 1 0 5 k m a g o o d c h o i c e c a n b e u ~ = 2 0 0 m 2 (i.e. p~ = 20 m), hence ui + u~ = 5200 m 2 and the corresponding p is 102 m. This enlargement has another interest: it enlarges the central zone of darkening. E. We can look for functions ~u(u) better than q/2 to ~us, for instance ~J6(U) = 1 - lOu3/u} + 1 5 u 4 / u } 6uS~u}; however it would require a more accurate positioning of the screen and it is necessary to study accurately the width of the central zone of darkening and to look for large zones of this type.
6. TECHNICAL CONSIDERATIONS
The theoretical results of the previous section lead to many constraints. A. The screen being at 105 to 106 km in front of the
telescope, we cannot use that screen if the Space Telescope is on a low circular orbit at an altitude of 500 km; we need a Space Telescope at at least five or six times the distance of the Moon (Fig. 5). B. It is necessary to fight the dazzling effect of the reflection of the solar light by the screen. We can for instance 1. use a black screen, 2. use only the face of the screen in the shade, 3. put the screen in the shadow of a secondary screen. C. With a diameter of 200 to 800 meters the screen will have the size of the sails projected for the years 1990; its fine attitude control is given by the solar magnetic field (2 to 7 gammas) and some large electric loops[6]. D. For long time exposures, we need a very small relative motion of the Space Telescope and its screen. The Sun attraction and the corresponding accelerations require that the observed star be at about 90 ° from the direction of the Sun. Each star can thus be observed with a long exposure only twice a year. E. Let us only mention the remaining technical difficulties: fine position control of the telescope and the screen (about one meter), telescope operations and displacements of the screen(s), etc. However, these difficulties seem to be within the reach of tomorrow's technology.
7. VISIBILITY OF THE SATELLITES The Space Telescope will have the power necessary to see the full Moon from the distance of Alpha Centauri; however we meet the following difficulties. From Alpha Centauri the Earth-Moon angular distance is only 0'.'002 and the best angular separation of the Space Telescope is 0':007. Hence we must still gain a factor 4 to 10 on both the pixel size and the guiding error of the Space Telescope. Note that a secondary annular screen in the direction of the planet, with inner and outer radius p, and P2, will give the following amplifying factor: (1 - J) = 1 - exp (-jkp]/2ro) + exp (-jkpZ/2ro).
10 s t o 106 km
t5m 4 -
~ /
F////~-~
lO"
(32)
1 1
. /
A Screen Star
_
4 - -
Space Telescope 100 to 800 m
For long t i m e exposures the Sun must be
] I
I
~- -....of kilometers
at about 9 0 ° f r o m the observed star
I I
"~ ~.
~. The Earth at millions
" " ~ ~-~
~
Fig. 5. The Space Telescope and its screen in space.
Space telescope concept
201
That factor can be 3 for the best wavelengths; it means a multiplication by 9 of the received power and the research of satellites becomes much easier (the secondary screen has a radius of 10 to 30 m).
The observation of the satellites of outer planetary systems remains at the limit of our present technical possibilities but at least, within 10 years, we should have taken pictures of several outer planetary systems.
8. CONCLUSION
REFERENCES
The method of the distant and star-shaped screen improves very much our observational possibilities and will allow us to take pictures of outer planetary systems around the nearest stars. Its main drawback is the necessity to put the Space Telescope on a solar orbit at millions of kilometers from Earth, and there are of course many other technical difficulties that have not been considered in this conceptual paper. It seems that single stars represent our best hopes for both cosmogonic and celestial mechanics reasons; it is also simpler to use only one large screen.
1. D. Benest, Les 6toiles h port6e de la main. L'Astronomie 96, 501-507 (1982). 2. D. N. B. Hall (editor), The Space Telescope Observatory. NASA C.P. 2244 (1982). 3. W. A. Baum, The ability of the space telescope to detect extra-solar planetary systems. Celestial Mechanics 22, 183190 (1980). 4. P. Van de Kamp, A la recherche des plan~tes en dehors du syst~me solaire. L'Astronomie 94, 207-228 (1980). 5. G. Bruhat, Cours de Physique g6n6rale Optique, 6th 6dition. MASSON (1965). 6. C. Marchal, Minimum horizontal rotations. Optimal control, Applications and Methods 4, 357-363 (1983).
0094 5765/85 $ 3 . 0 0 + .DO Pergamon Press Ltd.
CONCEPT OF A SPACE TELESCOPE ABLE TO SEE THE PLANETS AND EVEN THE SATELLITES AROUND THE NEAREST STARSt C. MARCHAL~ Office National d'Etudes et de Recherches A~rospatiales (ONERA), 92320 Chatillon, France (Received 19 January 1984; in revised form 13 June 1984)
Abstract--The Space Telescope will be launched in 1986 and will improve considerably our observational possibilities, but not the visibility of outer planetary systems. The solar system as seen from Alpha Centauri is presented, as well as the nearest stars and the main cameras of the Space Telescope. As a possible improvement the action of a distant and star-shaped screen is described; that screen is 100 to 800 meters and placed 105 to 106 km in front of the telescope; it allows one to avoid the dazzling effect of the stars and to look for planets such as Jupiter and Saturn up to 20 to 40 light-years. Such planets as Earth and Venus are a little less visible. The visibility of satellites such as the Moon is discussed; it remains at the limit of our technical possibilities. This conceptual paper does not consider in detail the technical difficulties involved.
1. INTRODUCTION
Let us note the following drawbacks. 1. The gap between the Sun and the planets is very large: 21.6 magnitudes between the Sun and Jupiter. The magnitude scale being a logarithmic scale (ratio 10 for a magnitude difference of 2.5), it means that the Sun is 4.4 × 108 times brighter than Jupiter at its full phase (and 2.8 billion times brighter than Earth). 2. A planet is at its full phase only when it is behind its star, i.e. at the worst position, and the maximum angular distance from the star corresponds to the quarters. The quarters are of course less bright than the full phase and the magnitude difference is about 2.7 (ratio 12) for a celestial body without atmosphere (Mercury, Moon); it is smaller but already 1.5 (ratio 4) for a planet with atmosphere such as Venus. It seems that the position half-way between the quarter and the full phase corresponds to the greatest visibility. 3. Alpha Centauri is almost the nearest star and at twice its distance, the distance of Sirius, we must add 1.5 to the magnitude of Table 1 and the Sun-planet angular distances are twice as small.
After some delays the Space Telescope will be launched in 1986 by the shuttle on a circular orbit at about 500 km altitude. That revolutionary new telescope will avoid the atmospheric screen and will represent a major step in many fields with respect to existing telescopes. It contains of course many instruments (wide field camera, planetary camera, faint object camera, faint object spectrograph, e t c . . . ) and will certainly be the source of many great discoveries. The astronomers have already prepared many series of experiments and observations in all fields of astronomy. However, and this is a surprise, the space telescope will be unable to see directly the planets around the nearest stars. It has already the power necessary to see them but, in spite of an excellent coronograph, the light of the star has a much too large dazzling effect and the astronomers must look for planets through the old indirect ways (irregularities in the motion of the star, etc.), The simplest pictures of an outer planetary system would give us much more information, but how to get them? A coronograph fights the dazzling effect of a powerful source of light and its efficiency increases with its distance from the telescope; we will analyse coronographs at thousands of kilometers in front of the space telescope.
3. THE 13 STARS AT LESS THAN
10 LIGHT-YEARS[I]
That small set of stars shows a wide variety: five single stars and four binaries, four visible stars (Sun, Alpha Centauri, Alpha Centauri B, Sirius) and nine invisible stars, many different spectral types, etc. (Table 2). The number of nearby stars increases rapidly with the distance: 79 stars are at less than 18 light-years (31 single stars, 15 binaries and six triple systems) and we have thus many possible targets.
2. THE SOLAR SYSTEM AS SEEN FROM THE DISTANCE OF ALPHA CENTAURI
The solar system is of course a good example of what we are looking for and the Table 1 shows how small are the possibilities of seeing something.
4. ON THE INSTRUMENTS OF THE SPACE TELESCOPE tPaper presented at the 34th Congress of the International Astronautical Federation, Budapest, Hungary, 10-15 October 1983. :[:Member A.I.A.A.
The Space Telescope (Fig. 1 and Ref, [:2]) will weigh 10 tons, be 15 meters long, with 2.4-m m i r r o r . . , and have an "astronomical" price of one billion dollars. 195
C. MARCHAL
196
Table 1. The solar system as seen from the distance of Alpha Centauri Magnitude at the full phase
Maximum angular
distance the Sun
from
.................................................. SUN 0.3 MERCURY 24.9 03| " ................................................. VENUS 22.4 0.54" EARTH 23.9 O. 75" MOON 27.5 -_-~ .............................................. MARS 26.8 1.14" JUPITER 21.9 3.9" SATURN 23.4 7.2" .................................................. URANUS NEPTUNE PLUTO
26.8 27.5 35 to 36
from the visibility of ordinary planets, and Fig. 2 shows the dispersion of light due to unavoidable diffraction effects. The astronomers still hope to detect many planets around the nearest stars[3], but they will use the old indirect ways (irregularities in the motion and in the radial velocity of the stars). These methods were used for the first time by Bessel in 1844 for the detection of the stellar companions of Sirius and Procyon and they show the existence of at least two very large planets around Barnard's star[4]. However, of course, the simplest picture of an outer planetary system will give us much more information.
14.4" 22.6" 37"(at aphelion)
With respect to terrestrial telescopes it will allow one to see 50 times fainter objects with 10 times better resolution. Its main instruments have the characteristics given in Table 3. The Space Telescope will also contain various kinds of spectrographs and many other smaller instruments. Moreover, the faint object camera can, under favourable conditions, be used in a f/288 mode that gives an angular resolution of 0.007", that is the root mean square of guiding error of the Space Telescope. The faint object camera has a coronograph allowing one to detect at 1" from a bright point object a small faint object 16 to 17 magnitudes fainter (in the best cases). These numbers should be compared to those of Table 1: even for the nearest stars we will be very far
5. A DISTANT SCREEN AS CORONOGRAPH The efficiency of a coronograph increases with its distance to the image and since, in space, very large distances are possible, we can look for the action of a distant screen. Let us consider the usual electro-magnetic equations of Maxwell (in empty space). curlE +
0--~- =
=0By. 0 divE = 0;divB c 2 curl B
t = 0
~-t
with: E = electric field (in volt/m), B = magnetic induction (in Tesla), t = time, c = velocity of light.
Table 2. The 13 nearest stars Star
Distance (light-years)
Sun
0
Proxlma Centaurl
4.22
Right
Declination
Ascension
14 h 26 m
-62 ° 28'
Alpha Centaurl Alpha Centaurl
4.35
14 h 36 m
star Wolf 359
21185
Sirius Sirius
Ross 154
M 5
ll
G 2
O
K 0
1.3
4 ° 33'
M 5
9.5
7.75
I0 h 54 m
7 ° 19'
M 8
13.3
M 2
7.5
8.22
11 h 01 m
36 = 1 8 ' M
t
B
- 26.9
17 h 55 m
B
Luyten 726-8 Luyten 726-8 B
G 2
5.98
Lalande 21185
Lalande
[
Magnitude (as seen from Earth)
-60 = 38'
B
Bar'nard
Spectral type
8.43
1 h 36 m
?
M 5
12.4
M 5
12.9
-18 ° 13'
6 h 43 m
-16 = 39'
A 1 DA
-I.5
8.65
9.45
Ig h 47 m
-23 = 53'
M 4
10.6
8.7
(1)
Space telescope concept
197
Fig. 1. The Space Telescope and the Shuttle.
A usual solution of (1) is the polarized and plane electromagnetic wave (with a usual 0xyz set of reference): E = (0, E>., O)
E,. = cB_ = f ( c t -
x).
(2)
B = (0,0, B:) The function f ( c t - x) can be arbitrary; it is usually decomposed into sinusoidal components: E~ = cB: = Eo e x p [ j k ( c t -
x)],
(3)
with:
j = square root of minus one, k = 2n/2; 2 = wavelength.
5.1 Diffraction given by a hole
Let us look for the propagation of the plane polarized and sinusoidal electromagnetic wave given by (2)(3)(4) after its passage through a hole in the Oyz plane (Fig. 3). With an excellent approximation and when x(P) is large with respect to ,~ the electromagnetic field in P is
Table 3. Characteristics of the main cameras of the Space Telescope Instrument
Field of view
Wide field
Planetary
camera
camera
2'40"x2'40"
Angular resolution (one pixel) Wavelengh range
(quantum efficiency~lg~a) Photometric accuracy Dynamic
0.I"
I'9"xi'9"
0.043"
0.115 to
0.Ii to
i.I~
1.1~
1%
~ 1% 28
8.5(mv~28
Faint object cameral(f/48)
22"x22"
O. 045"
0.12 to 0.6U
at least 2%
21.< mv~ 28
Fain object camera2(f/96)
ll"xll"
0.022"
0.12 to 0.6
at least 2%
21~ mv~ 28
range
'm = v i s u a l ~
m~gnit~de]
AA 12:3-E
signal/nolse ~ 3
(4)
c u m u l a t i v e ten hours o b s e r vation) signal/noise ~ 4
198 Z
C: MARCHAL ~3
1
g ~ , 0 -q . re"
N qO-z"
:~ ~0 - ~ . I-r ,,q
7.. ~o-4 m
¢~ I 0 "su. ,,vI0 - 6 . l,u > ,~10 -r. _1 t,.u r,'-
>. or r
f
if)
/
tangular three-hedron of the wave:
! l/
v /
(7)
B ~ = (Eolc) • "y exp { j k ( c t -
(8)
z,,, O
#
-
0.s
et • OM)}.
O "' to
i
Eu = Eo • 13 exp { j k ( c t - et • OM)},
LI.I
/. " ,
Then (5) and (6) give:
lit
j E o ~ dy • dz Ep = ~ j~ r
• exp { j k ( c t - ot • O M - r)} • (u x ~ + ' , / )
05
t0 .t5 .20 .25 .30 IMAGE RADIUS (ARC-SECONDS)
Fig. 2. The expected optical performance (2 = 6328 ,/k) is illustrated in terms of the image surface brightness distribution and the encircled energy. The central obscuration and deviations from perfect mirror figure and alignment have been incorporated in these predictions.
given by the following equations[5]:
• exp { j k ( c t - a • O M - r)} • u X ( ~ + ~/ X u).
(5)
(10)
The integrations (9) and (10) are not easy; however we can obtain the main part of Ep and Bp in the following way: A. The unit vectors u and ot are very close to each other, their angle is less than 10" and we can put: u --- ,a + c,
U
(9)
j E o ~ dy • dz Bp = ~ c .~ r
0
x - exp { - j k r } ,
x u,
I1,~11< 5 x 10%
(11)
hence:
r
(u x 13 + ' , t )
x u=
., (6)
• exp { - j k r } ,
where S represent the hole and its surface, M an arbitrary point of the hole, y and z its coordinates, r the distance M P and u the unit vector of MP. These expressions remain valid even if the wave has not the direction of Ox and we will look for the result of the integrations (5) and (6) when both the wave and M P are only slightly inclined on Ox. Let us call tx, I~, 'f the unit vectors of the right rec-
l
[(a +,~)
= 2',t x
However, with t • u (~
x
13)
u
x
=
=
u +
x I] + " t ] (t
x
x u
I~) x
u.
(12)
g2/2:
I~(~ " u) -
~(13" u)
= [3 ~ - e(l~ " e),
(13)
(u X I~ + ~/) X u = 2'y X u + O(e2),
(14)
u x ( ~ + ',/ x u) = 2 u x [3 + O(e2).
(15)
and thus:
Hence, if we put:
z
u
i
V =
2 .Is d y ' d z ' - r
(16)
• exp { j k ( c t - ot • O M - r)}, /'/
r
P (xp, yp, zp)
we shall get: x
Ep = ~1 x V + O{EoSe2/xp2},
(17)
Bp -
(18)
V
x______~ +
O{EoSe2/%2c} '
C
Fig. 3. Analysis of the diffraction given by a hole.
[with 2 = 2 n / k = wavelength; xp = x(P)].
.k99
Space telescope concept In the integration (16), we can substitute to u / r its average value u0/r0 or even a/ro with only an error of order e 2 on the radial component of V and an error of order e on its circumferential components (all this implying only an error of order e-' on the received power). We thus arrive to: V -~ V0 = Eoa exp {jk(ct - tx • OP)} • J,
(19)
with
that is, with u = p2/2 and ~(p) = ~u(u):
I -j
= I _Jk--f"/du ro Jo • ~(u).exp
~'-Jk---Y-u}.
k
A first choice can be ~,(u) = 1 (circular screen of radius p:), it leads to: 1 - J = exp {-jku//ro}.
dy • dz" exp {jk(a • M P - r)}.
J = ~ro
-2r--'~o(MMa)2 "
d y " d z " exp
~u2(u) = 1 - -
U
u:
implies:
(22)
Note that if J = 1 the eqns (17), (18), (19) give E n and Bp as are EM and E z in (7) and (8): the hole has no effect. The factor J is then really the amplifying factor of the hole for the point P the direction r, and the wavelength 2. 5.2 Diffraction given by a screen The effect of a hole and the effect of a screen of the same place and shape are complementary, their sum correspond to the undisturbed propagation of the electromagnetic wave, and hence the amplifying factor of the screen is 1 - J with the same expression of J given in (22), S being now the surface of the screen. 5.3 The choice of the distant coronograph We want to switch off the light of the star for all interesting wavelengths but we also want to keep the light of its planets. Hence we need an amplifying factor 1 - J as near to zero as possible when M, is at the center of the screen or very near to it, but we also need 1 - J of the order of I when M~ is at about 1" or more from the center of the screen. We must integrate (22) for a suitable shape of screen; for instance let us use the polar coordinate p: p = V'y 2 + z 2,
1 - J is not in the vicinity of zero and the choice is not good. Let us consider some other cases leading to the starshaped screen of Fig. 4:
(21)
We thus arrive to the "amplifying factor" J:
J = ~ro
(26)
(20)
(at • MP - r) is zero at the point Ma where M~P is parallel to ot (Fig. 3), and in its vicinity we can write, with again the same order of error: a • M P - r = -(MMa)2/2ro .
(25)
ro J
(27) ro
1 - J2 = j'k-uuy(1 - e x p { - j k u / / r o } ) .
.2
¥3(u) = 1 - 3 u } + 2 u } implies: 6r.....~o 2 1 - J3 = k2u} [ - 1 - exp {-jku//r0}]
(28)
12jr~
+ k3u---~ [exp { - j k u : / r o } -
(a)
1].
(b)
(23)
and let us assume that at the distance p from its center the screen intercepts the proportion tp(p) of the light. We will obtain:
-J=
l - ~ r ~ fop/27zp
dp
(c)
ox,{
(24)
(d)
Fig. 4. The shape of some possible screens for various ~, functions. (a) ~ = ~¢2. (b) ~ = ~ . (c) ~g = ~u4. (d) q/ = ¢q.
200
C. MARCHAL
These choices are good if for the wavelength of interest (2 = 2~/k <~ 1.1 p) the ratios ro/ku~ are small. For instance with r0 = 105 km, pf = 100 m and uf = 5000 m 2 we get ro/ku s <<- 1/285, that is I1 - J_,l ~< 1/142 for the ~2 screen and ]1 - J31 ~< 1 / 13,600 for the q/3 screen. The powers are proportional to the square of the amplifying factors; hence the first screen adds at least 10.7 to the magnitude of the star and the second screen adds at least 20.6 (with 20.6 = 2.5 log (13,600)z); it becomes useless to use the inner coronograph of the Space Telescope! (But the central zone of darkening is very small.) Let us note the following points: A. At 105 km an angle of 1" corresponds to 500 m, hence the light of a planet at 1" from its star will be almost not disturbed by our screens with only p~ = 100 m. B. If we choose r0 = 106 km and Pl = 100 m • ~ / ~ , we obtain the same amplifying factors and the same effects on the light of the star, but we can look for planets at % / ~ times smaller angles. We found again that the furthest coronographs are the most efficient (we can also prefer to choose p~ = 1000 m and to obtain a better darkening of the star). C. If we take account of the approximations presented in (14), (15) and in the second half of section 5.1, we arrive to an almost negligible correction of functions ~'2(u) and q/3(u). D. The main drawback of the functions ~Uz(U) and q/3(u) is that they are less than one as soon as u is positive and the corresponding screens are discontinuous at the center (Fig. 4). Fortunately if we choose: ~'4(u) = ~us(u) = 1 q/4(u)
~,/,/2(U
~,~(u)
~u~(u
~t
when0<~u<~u~,
u 0 ~ when u, ~< u ~< ut + uj,
(29)
u,)j
We found the following amplifying factors: exp {-jkuflro},
(30)
(1 - Js) = (1 - J3) " exp {-jkuJro},
(31)
(1 - J4) = (1 - J2)
"
and the efficiency of the two last screens (Fig. 4) are exactly that of the corresponding first screens. Forr0 = 1 0 5 k m a g o o d c h o i c e c a n b e u ~ = 2 0 0 m 2 (i.e. p~ = 20 m), hence ui + u~ = 5200 m 2 and the corresponding p is 102 m. This enlargement has another interest: it enlarges the central zone of darkening. E. We can look for functions ~u(u) better than q/2 to ~us, for instance ~J6(U) = 1 - lOu3/u} + 1 5 u 4 / u } 6uS~u}; however it would require a more accurate positioning of the screen and it is necessary to study accurately the width of the central zone of darkening and to look for large zones of this type.
6. TECHNICAL CONSIDERATIONS
The theoretical results of the previous section lead to many constraints. A. The screen being at 105 to 106 km in front of the
telescope, we cannot use that screen if the Space Telescope is on a low circular orbit at an altitude of 500 km; we need a Space Telescope at at least five or six times the distance of the Moon (Fig. 5). B. It is necessary to fight the dazzling effect of the reflection of the solar light by the screen. We can for instance 1. use a black screen, 2. use only the face of the screen in the shade, 3. put the screen in the shadow of a secondary screen. C. With a diameter of 200 to 800 meters the screen will have the size of the sails projected for the years 1990; its fine attitude control is given by the solar magnetic field (2 to 7 gammas) and some large electric loops[6]. D. For long time exposures, we need a very small relative motion of the Space Telescope and its screen. The Sun attraction and the corresponding accelerations require that the observed star be at about 90 ° from the direction of the Sun. Each star can thus be observed with a long exposure only twice a year. E. Let us only mention the remaining technical difficulties: fine position control of the telescope and the screen (about one meter), telescope operations and displacements of the screen(s), etc. However, these difficulties seem to be within the reach of tomorrow's technology.
7. VISIBILITY OF THE SATELLITES The Space Telescope will have the power necessary to see the full Moon from the distance of Alpha Centauri; however we meet the following difficulties. From Alpha Centauri the Earth-Moon angular distance is only 0'.'002 and the best angular separation of the Space Telescope is 0':007. Hence we must still gain a factor 4 to 10 on both the pixel size and the guiding error of the Space Telescope. Note that a secondary annular screen in the direction of the planet, with inner and outer radius p, and P2, will give the following amplifying factor: (1 - J) = 1 - exp (-jkp]/2ro) + exp (-jkpZ/2ro).
10 s t o 106 km
t5m 4 -
~ /
F////~-~
lO"
(32)
1 1
. /
A Screen Star
_
4 - -
Space Telescope 100 to 800 m
For long t i m e exposures the Sun must be
] I
I
~- -....of kilometers
at about 9 0 ° f r o m the observed star
I I
"~ ~.
~. The Earth at millions
" " ~ ~-~
~
Fig. 5. The Space Telescope and its screen in space.
Space telescope concept
201
That factor can be 3 for the best wavelengths; it means a multiplication by 9 of the received power and the research of satellites becomes much easier (the secondary screen has a radius of 10 to 30 m).
The observation of the satellites of outer planetary systems remains at the limit of our present technical possibilities but at least, within 10 years, we should have taken pictures of several outer planetary systems.
8. CONCLUSION
REFERENCES
The method of the distant and star-shaped screen improves very much our observational possibilities and will allow us to take pictures of outer planetary systems around the nearest stars. Its main drawback is the necessity to put the Space Telescope on a solar orbit at millions of kilometers from Earth, and there are of course many other technical difficulties that have not been considered in this conceptual paper. It seems that single stars represent our best hopes for both cosmogonic and celestial mechanics reasons; it is also simpler to use only one large screen.
1. D. Benest, Les 6toiles h port6e de la main. L'Astronomie 96, 501-507 (1982). 2. D. N. B. Hall (editor), The Space Telescope Observatory. NASA C.P. 2244 (1982). 3. W. A. Baum, The ability of the space telescope to detect extra-solar planetary systems. Celestial Mechanics 22, 183190 (1980). 4. P. Van de Kamp, A la recherche des plan~tes en dehors du syst~me solaire. L'Astronomie 94, 207-228 (1980). 5. G. Bruhat, Cours de Physique g6n6rale Optique, 6th 6dition. MASSON (1965). 6. C. Marchal, Minimum horizontal rotations. Optimal control, Applications and Methods 4, 357-363 (1983).