[49] The information content intrinsic to indirect (astrometric) detection techniques and to direct (at visual or infrared wavelengths) detection techniques was discussed in chapter 2. An important aspect of that discussion is that indirect and direct detection techniques are complementary in terms of information content. In addition, direct and indirect techniques are also complementary with regard to the types of planetary systems that the techniques are best suited to discover. Direct techniques are best suited for detecting bright (or hot) planets, which would be found around more massive bright stars. Indirect techniques are best suited for detecting planets that revolve around low-mass stars. In view of the possible role direct detection techniques could, and should, play in a comprehensive program to search for extrasolar planetary systems, a small effort was made during Project Orion to consider direct detection systems. The results of that effort are presented in this chapter.
It was pointed out in chapter 2 that direct detection almost certainly must be done from space. No consideration was given during Project Orion to design aspects of potential spacecraft that might house a telescope that could undertake a search for visual light reflected by extrasolar planets. Rather, consideration was confined to defining a system concept that might serve as a baseline for more detailed future studies.
The resolving power of a telescope, as a function of
aperture D and wavelength
of the observed light, is given
by
This relation is generally attributed to
Rayleigh and hence is known as the Rayleigh criterion. Equation (28)
shows that a very modest telescope in space, D0.25 m, would have an angular
resolution at
[50] = 5 x 10-7 m of 0.5 arcsec,
adequate to resolve the planet-star pair in the SPS. However, the
Rayleigh criterion is applicable only to resolving two sources of
equal
intensity. It does not adequately describe the relation
for cases when
the intensity ratio,
,
between two sources is very small, as is the case in the SPS.
The transmission of light in a normal clear-aperture telescope is constant over the aperture and goes abruptly to zero at the edge of the aperture. This produces a classical intensity diffraction pattern of the form (2J1 (R)/R)2, where J1 is the first-order Bessel function and R is the distance from the optical axis. If the light is monochromatic, all of the higher order maxima would be present; however, in most practical situations a range of wavelengths is involved, causing all but the first 10 maxima to be smeared out. The maxima outside the central peak are often referred to as the rings of the diffraction pattern. The intensity produced by a clear aperture at the center of the image, I(0), is given by
where rO is the radius of the
aperture. The intensityat a point off the optical axis, by an angle
, is given by
where is in radians. The quantity
for
monochromatic light is shown in figure 14 as a function of Airy
radii. For values of
such that
<< I(0), an asymptotic expansion of
J12 (X) may be used to
give
:
whereis the mean intensity at
and D0 =
2r0.
Equation (31) may be inverted to solve for the product
(in marcsec
(for
, in
m)). As
, the aperture size D0 is given
by.....
To resolve the SPS ( = 2X 10-9,
= 0.5) with
= 5 x 10-7 m, a telescope must
have an aperture of 56 m! Obviously, a clear-aperture telescope, even
though located in space, is not the instrument to use in a search for
extrasolar planets. Other approaches must be used to provide the
necessary
-value at
arcsec.
One approach is to reduce or eliminate the
ring maxima that occur outside the central, bright region of the
diffraction pattern. A device that accomplishes this is called an
apodizer. The first thorough examination of the problem of
resolution in the limit of very small -value was by Jacquinot (ref. 23),
who first coined the phrase "apodization." (Jacquinot referred to the
rings in a classic diffraction pattern as feet, and the process of
minimizing or removing the light contained in the rings ("feet") as
apodizing, or removing the feet.) As the rings arise due to the
abrupt termination in the transmission at the edge of an aperture,
the concept of apodization centers on [52] techniques that
produce a more gradual transition to zero transmission at the edge.
Two types of apodizer seem most feasible. One is a diaphragm that
alters the shape of the pupil; the second is a filter whose
transmission varies with distance from the optical axis. Either type
must be located at a pupil plane (i.e., any real plane conjugate to
the telescope aperture) in order that light rays from every object
point will be distributed over the apodization device in exactly the
same way as at the center of the distribution.
A second approach toward a direct imaging system involves reducing the bright, central Airy disk. The simplest means, conceptually, to accomplish this is to place an obstruction at the focal plane, or at some plane within the optics which is conjugate to the focal plane. There are, however, objections to this approach. The presence of such an obstruction makes both alignment and pointing of a telescope difficult. When a program or target star is lined up perfectly, it can no longer be seen. Also, the central obscuration would scatter light into the region where a potential planet's image would be located. This scattering can be reduced somewhat by apodization, but it raises the noise level with which apodization must contend.
Another method of reducing the effect of stellar radiation without affecting the planetary light is by means of an interferometer. Planetary visible light is reflected stellar light. Light from the planet has traveled a greater distance to arrive at the observer than has light coming directly from the star. The important point is that the extra length in optical path is large compared to a coherence length,1 so that stellar light and reflected planetary light will not interfere. It is possible to place an interferometer in the optical system of a telescope so that it would destructively interfere the [53] stellar light with itself, but not interfere with the planetary light. This operation should be carried out before the light reaches the image plane of the telescope.
A telescope containing these optical elements has special requirements. If the telescope is to be carried into orbit on a shuttle, it must be light and compact. Moreover, it must have a real pupil plane, conjugate to the dominant aperture, where the light-reducing elements may be located. These requirements strongly indicate that such a telescope would be a variant of a folded Gregorian. Aspects of one such system concept are discussed later in this chapter.
There are many possible transmission functions that can, in principle, be used to apodize a telescope. One family of functions, which has been studied by Oliver (unpublished data, 1976) in the context of detecting extrasolar planetary systems, is the so-called Sonine family. These functions lead to transmission T(r) over an aperture of the form
where a is the aperture size and µ is a
positive semidefinite (0) index. The aperture form
characterized by equation (33) is rotationally symmetric about the
optical axis. The tests performed during Project Orion were designed
to test the apodization concept by means of aperture masks. The form
for such masks is discussed in appendix C.
A variation on the concept of a tapered transmission function is that of an opaque mask at the aperture. Although such a mask does not generally give rise to a circularly symmetric tapered transmission, it does lead to the transmission being reduced smoothly in an average sense in certain preferred directions. The "directionality" of a given mask depends on the geometry of that mask.
A well-known application of the mask technique
was the study of the Sirius A-Sirius B system by Lindenblad (ref. 24)
and van [54] Albada (ref. 25). Both workers used a regular
hexagonal mask on a clear circular aperture. The diffraction pattern
produced by the brighter member (Sirius A) of the pair is like a
snowflake (cf. fig. 18). A clear aperture produces a white-light
diffraction pattern where the average intensity decreases as the
inverse third power of the distance from the optical axis (cf. eq.
(31)). The hexagonal mask gave rise to an intensity distribution that
decreases more slowly (as the inverse second power) in the spikes of
the pattern, but which decreases more rapidly (as the inverse fourth
power) in the interspike regions of the pattern. By rotating the
principal axes of the hexagonal mask, Lindenblad was able to place
one of the relatively dark, interspike regions in register with the
location of the image of Sirius B, thereby recording the first direct
detection of the B-component. It should be noted that the angular
separation of Sirius A from Sirius B is about 10 arcsec and the
brightness ratio
is
10-4 .
Detecting Sirius B is many orders of magnitude (6 - 7) easier than detecting the SPS. Further, the hexagonal mask or any aperture edge obstruction is unlikely to give rise to a diffraction pattern with the required spatial-intensity resolution. Masks that vary smoothly are more likely to provide the required diffraction pattern. In this connection, masks predicated on Sonine functions are potential candidates for apodizing to achieve an acceptable diffraction pattern. Although not referred to as "Sonine masks," such masks were studied theoretically by Tuvikene (ref. 26), and further analyzed and utilized by van Albada (ref. 25). In view of the potential of such mask geometries for apodization, an empirical investigation was conducted during Project Orion to ascertain the performance of some rather simple masks.
The time available for Project Orion precluded an in-depth experimental investigation of the problem of apodization. Consequently, it was decided to utilize the facilities available at Ames Research Center, specifically a 150-m-long dark tunnel, to test a limited number of apodizing masks on a 0.076-m-aperture telescope loaned to Project Orion by Stanford University. Figure 15 is a block diagram of the experimental system. At one end of the 150-m tunnel....
....was a model planetary system, comprised of a projection lamp in a box with two pinholes (fig. 16). The pinholes were identical in size, and the intensity of light transmitted through one of the pinholes (the "planet") was controlled by neutral density filters. The telescope (fig. 17) was mounted 150 m away from the model planetary system. Masks made from heavy construction paper were placed over the objective lens of the telescope. A camera was mounted at the eyepiece and time exposures were taken using Kodak Plus-X pan film.
The first test involved a model of the Sirius A-Sirius B system. As remarked above, this system is not a severe test of apodization, but does afford a test of the modeling scheme. The results of the Sirius test are shown in figure 18. The model Sirius B is clearly visible at ten o'clock in a null in the diffraction pattern. The mask used for this test was a regular hexagonal mask of the type used by Lindenblad. The effective "resolution" of the model telescope system is far more than adequate to resolve the model binary system, but the characteristic six-spike diffraction pattern is clearly evident.
Having demonstrated that the test system yielded qualitatively correct behavior, tests were then run on the scale model of the SPS. As the telescope concept developed during Project Orion for direct visual detection of extrasolar planets has a 1.5-m aperture (see discussion below), the tests were run to simulate the resolution of such a telescope in space. The 0.5-arcsec angular separation that characterizes the SPS corresponds to a linear separation of 3.6 x 10-4 m at a distance of 150 m. The resolution of a diffraction-limited telescope varies inversely as the aperture of the telescope (cf. eq. (28)); thus a 1.5-m aperture would have about 20 times the resolution of a 0.076-in. aperture, such as used for these tests. Including this factor in the scaling of the model SPS leads to a linear separation of 0.007 m at a distance of 150 m.
Tests were run with various aperture
configurations, namely, a clear aperture, an aperture with a Sonine
mask (µ = 7), and an aperture with the complement to the µ
= 7 Sonine mask. Figures 19 through 23 are photographs taken of the
model planetary system [58] using various
aperture configurations (fig. 19, clear aperture; fig. 20, Sonine
mask; figs. 21 - 23, Sonine complement) and various -values
(10-3, 10-3,
10-4, 10-4, and
10-6, respectively). The general result is that the Sonine
complement aperture mask gave better results than the Sonine mask
which, in turn, gave better results than the clear aperture. The
Sonine aperture was unable to resolve the model planet at an
-value of
10-4. The results with the Sonine complement mask were as
expected in that the distribution of light on the image plane was
complementary to the distribution from the Sonine aperture. However,
there was a quantitative distinction in the sense that relatively
more light was concentrated in the bright areas and relatively less
light fell in the null regions of the diffraction pattern. As a
consequence, proper orientation of the complementary occulting mask
permitted visual detection of the planetary companion at an
-value of
10-6. Unfortunately, the object is sufficiently faint that
reproduction in this report washes out the planet; it is, however,
clearly visible on the film negative.
[61] These tests are not,
nor were they intended to be, a definitive and exhaustive exploration
of the apodizing problem. They do, however, clearly indicate that
relatively unsophisticated masks provide significant off-axis light
suppression. It is not unreasonable to suspect that more
sophisticated masks, optics, and detectors could reach -values of
10-7
to 10-8. Although these
-values are higher than that of the
SPS, it seems possible to obtain the additional two orders 0f
magnitude by, as mentioned previously, combining an apodizer with a
system that can reduce the intensity of the stellar Airy disk. (It is
tempting to carry the physiological analogy of the diffraction
structure to an extreme by associating the Airy disk with the "head"
and denote a device that removes the head as an acephalizer, that
temptation will, however, be overcome.)
As discussed above, a combination of an apodizer and a device to cancel or minimize the light from the star that might reach the final image plane appears capable, in principle, of obtaining spatial-intensity resolution of the type needed to detect certain extrasolar planetary systems. Perhaps the simplest and most direct method for canceling light from a star is to place an obstruction in the focal plane of the telescope, or at any plane in the optical train which is conjugate to the focal plane. This approach is used successfully, for example, in Lyot's coronagraph (ref. 27) for obscuring the solar disk. However, the Sun is an extended object - a star image is not. The very nature of a star image makes this procedure an engineering challenge. Most stars that would be studied in a search for extrasolar planets are so distant that their images will be unresolved. The object to be occulted therefore is the central region of the diffraction pattern, the so-called Airy disk. The size of the Airy disk depends on the aperture of the telescope and its focal length. Generally, the bigger the telescope, the higher its resolving power, the smaller the Airy disk and, therefore, the more difficult it is to make a device small enough and to locate it with sufficient precision to obscure it. It is conceivable that a telescope design could include an intermediate focal plane highly magnified so that obscuration could be accomplished there and this obscured image could be relayed to the system's final focal plane where it is detected.
[62] Light that is
blocked in this way is backscattered with much of it leaving the
optical system. However, a small fraction of the backscattered light
will "rattle" around through the optical system, ultimately arriving
at the final image plane where it will either enhance the light in
the Airy disk or, worse yet, appear as a spurious ghost image. In
current applications utilizing central obscuration of this type,
scattered light is not as severe a problem as it is for the low -value regime
encountered in the planetary detection problem.
An alternative to the occulting disk approach is suggested here. This alternative makes use of the fact that light from a planetary companion to a star would not interfere with light coming directly from the star. The method is to line up the star with the axis of an interferometer so that the light in the two interferometer beams is recombined at the exit beamsplitter so that total destructive interference of starlight occurs. That is to say, the angle between the emerging wave fronts is so small that the distance between fringes exceeds the diameter of the exit aperture. Then an adjustment is made in the optical path length of one of the beams so that a dark fringe is centered on the exit aperture. Light from potential planetary companions would, if bright enough, be visible against this dark background. The basic concepts of such a dark-field device are discussed by Ken Knight (ref. 28).
In order not to cancel out the planetary light while destructively interfering the two beams of stellar light, the interferometer is so constructed as to rotate one of the beams 180° relative to the other. One approach is as follows. The basic design (see fig. 24) is that of a Mach-Zehnder interferometer (ref. 29, pp. 312-315). Light enters the device at A (fig. 24), where it falls in a beamsplitter that divides the light into two approximately equal beams. One beam is reflected to mirror B where it is again reflected to the mixer located at D. The second beam passes through the beamsplitter to mirror C where it is also reflected into the mixer at D. Here the two beams are recombined. The mixer is structurally identical to a beamsplitter.
If all four elements are exactly parallel, the wave fronts incident in the mixer will be exactly parallel. However, if one of the mirrors, say B, is tilted slightly with respect to D, then the two wave fronts will not merge at the mixer but will be slightly inclined to one another. If the optical path lengths of two beams are equal, interference will occur and fringes will appear on the mixer. The direction of....
....the fringes indicates the direction of the tilt; the separation of the fringes is inversely related to the magnitude of the tilt. This useful characteristic of the Mach-Zehnder is discussed later.
If the optical path length of one of the beams is increased, the fringes will move across the surface of the mixer, becoming fainter and fainter as the optical path difference approaches the coherence length of the radiation. If there is only a slight tilt between the two mirrors, the distance between the fringes will be large, perhaps larger than the width of the mixer plate. In that case, as the optical path [64] length varies, the entire mixer plate will vary in brightness from a maximum to a minimum. It is this principle that is applied to cancel the starlight without affecting the planetary light. A perfectly aligned Mach-Zehnder interferometer, with proper adjustment of the relative optical path length between the two beams, will null any entering wave front.
To null a wave front only when it is lined up
with the axis requires rotating one beam 180° with respect to
the other beam. The desired effect is accomplished by flipping one
beam top to bottom and reversing the other beam left to right. This
is done by replacing two of the plane mirrors in figure 24 with
suitably oriented prisms (fig. 25). A wave front entering so that its
normal makes an angle with the axis of the interferometer will result in two
wave fronts at the mixing plane making an angle of 2
with each
other. With a perfectly aligned system, only a wave front with
= 0 will null
itself. All.....
[65] ....others will
produce fringes whose separations are a function of 2. As mentioned
previously, the orientation of the fringes shows the direction of the
tilt.
To evaluate the efficiency of the
interferometer, it is useful to briefly review how interference
occurs. The notation used is that of Ditchburn (ref. 30). The
amplitude across the wave frontis given by
where t represents time and x is a coordinate of the wave front. Interference occurs at the exit beamsplitter which makes an angle of 45° with the interferometer axis. To simplify calculations, it can be assumed that interference takes place on the exit aperture. The tangential components of the wave-front amplitudes on the exit aperture are
where u is a coordinate on the exit aperture in an appropriate direction. Summing these amplitudes gives
The fringe brightness will be at a maximum
whenever The
distance between adjacent bright fringes is given by
Asincreases, the fringe spacing
decreases to the extent that the fringes are unresolvable. On the
other hand, as
->
0, the fringes move farther apart until finally the distance between
fringes exceeds the diameter of the exit aperture, and starlight
cancellation becomes possible. If there is a fringe whose width is
twice that of the aperture, then the entire aperture will appear
bright with a 50-percent decrease in intensity at the edge. If the
optical path length in one of the [66] beams is adjusted so
that a dark fringe appears at the center of the aperture, then the
entire field will appear dark with a 50-percent brightness at the
edge.
Let d be the diameter of the exit aperture of the interferometer. Then this dark-field effect occurs when the distance between fringes exceeds 2d, that is,
It follows that all wave fronts incident upon
the interferometer at, so that
where
will be interfered. Equation (39) may be
inverted to determine d so that all wave fronts entering the aperture
at the interferometer at angles less than are to be annulled, namely,
A telescope design that might be well suited to incorporating the apodizing and dark-field devices is a folded Gregorian, with several modifications. In the usual Gregorian, the primary is a paraboloid and the secondary, located beyond the primary focus, is an ellipsoid. In this design (see fig. 26), a tilted secondary convex spherical mirror is placed at or near the focus of the paraboloid directing the rays laterally toward an ellipsoidal tertiary mirror. This, in turn, reflects the beam of light to a focus. The light is intercepted before this focus by a diverging lens that acts as a collimator. Up to this negative lens, the telescope is an afocal system. The angular magnification M relates ray angles in object space and ray angles following the collimator as follows:
Here represents the angle away from a
star and
represents the off-axis angle of the corresponding ray
in the region following the collimating lens.
Equations (40) and (41) may be used to obtain
an estimate of the exit aperture of the interferometer. The Airy
disk, which is the portion of the diffraction pattern that the
interferometer is designed to annul, has an angular extent given by
, where D is
the aperture of the telescope. The post-collimator angular size
of
the Airy disk is given by equation (41) as
However, and
are very small angles so that tan
~
, or
[68] where, as discussed
above, M is the angular magnification of the afocal part of the
optics. To obtain cancellation over an angular subtent comparable to
, the exit aperture must be
Note that d is independent of (so long as
the star is not resolved). For D = 1.5 m and a magnification M = 50,
d = 6.14 x10-3 m. An objective lens at or near the exit aperture of
the interferometer would then image a dark field with doubled images
of any off-axis objects (planets?).
A basic requirement for the system to achieve
cancellation is that the interferometer optical axis be aligned
rather precisely with the star under observation. This alignment must
be achieved with a precision , namely, 4.07
x 10-7 M (in rad) (
has been taken to be 5 x
10-7
m). Assuming that M = 50,
arcsec. That level of alignment
precision is not overly demanding. Further, the system has a built-in
alignment indicator. One attractive aspect of a Mach-Zehnder type of
interferometer is that it can have two outputs. One may be used to
obtain the image that will be analyzed for the presence of planetary
companions. Output from a second beam is bright, with light from the
star, and may be used to indicate the amplitude and direction of
misalignment by noting, respectively, the separation of fringes and
the direction of fringe tilting. This latter output may be
automatically monitored and used to activate a servosystem that would
align the optics to within the required tolerance.
The present study and analysis has not been
carried to the point where it is possible to make a quantitative
assessment of the extent to which the interferometer will provide a
dark field. If it were capable of a reduction factor of100,
the combination of the apodizer and interferometer would allow
detection of systems with
-values of ~l0-9 to
10-10.
It is useful to make a few additional detailed remarks about the Gregorian design concept developed during Project Orion. The telescope system resembles a Gregorian reflector (see fig. 26). The primary mirror is a paraboloid with a 1.5-m diameter and with a focal length of 4.0 m. The diameter of the field at prime focus is therefore 35 mm. Located at the prime focus is a spherical mirror that operates a field lens. Its center of curvature lies on a line inclined 45° to the [69] axis of the paraboloid; it has a radius of curvature of 10 cm and a focal length of 0.05 m; and its diameter is 50 mm. Its axis is determined by the intersection of the axis of the paraboloid with its surface, and there are provisions to translate this mirror in two orthogonal directions for fine tracking.
The spherical mirror will produce a virtual image of the primary tilted approximately 24°. Since the dominant aperture of the system must be the primary, this image will be a pupil plane.
The third element is an ellipsoidal mirror. Its axis is 45° to the nominal axis of the spherical mirror and 90° to that of the parabolic primary. Its first (short) focus lies on the prime focus, a distance of 0.8 m from the ellipsoid. Its second (long) focus is at a distance of 2.4 m. Thus a beam of light is directed across the diameter of the primary. The ellipsoid's focal length is 0.6 m. The diameter of the field at the second focus is 3.1 m!
The ellipsoid will produce a pupil plane whose center lies at a distance of 1.936 m from the ellipsoid. This is an image of the primary mirror through the sphere and the ellipsoid. The pupil is tilted approximately 25.5° to the axis of the ellipsoid, and has a diameter of about 5.9 cm.
The next element in the optical train is a removable plane mirror tilted 45° to the axis of the ellipsoid. When it is in place, it diverts the full field to an objective lens where the light could be focused on a detector array. The full 0.25° field will cover the detector. Reference crosshairs will enable an operator to study the entire star field, to select a program star, and to point the telescope toward it. The removable flat mirror will be located 1.6 m from the ellipsoidal mirror; its dimensions will be 5 by 7 cm.
A lens at this point will form an image of the full 0.25° star field at its focus. Its field should be equivalent to the dimensions of the detector, that is, 3.6 m. These constraints are rather severe and, in the configuration described, the lens would resemble an inverted microscope.
Located at or near the pupil plane will be an achromatic collimating lens. If located 1.936 m from the ellipsoid, the required focal length of this lens will be -0.464 m. It will be tilted approximately 65.3° to the axis of the ellipsoid so that the pupil plane will be perpendicular to the axial ray of the telescope. The full aperture of this lens will be about 5.9 cm. The image of the pupil at this point is [70] reduced in size to about 9 mm. However, the semiangular field from the collimating lens is 65.3°. In practice, it would be better to shift the collimating lens slightly toward the ellipsoidal mirror, thus producing a real image of the pupil plane. It is at that location that the proposed apodizing element would be located.
The next item in the design configuration is the dark-field interferometer. Its function is twofold: (1) to cancel the starlight without affecting the light from the planet and (2) to detect how far and in what direction the star has drifted from the telescope axis and to provide a correction signal to adjust the position of the spherical mirror (which lies between the primary parabola and the ellipsoid).
Although the full-field output of the collimating lens is to be ±65.3°, only a portion of this need enter the interferometer. The spatial resolution required to detect a planet in the SPS (0.25 arcsec) translates into 0.07° in this space. The entrance aperture must be sufficiently large to accept a spread of only, say, 5° half-angle. The size of this pupil, as well as the detailed structure of the interferometer, were discussed above.
A lens at this point focused at infinity will produce two images of the planet in its focal plane. If the angular field of the rays emerging from the interferometer is, say, 5° half-angle, in which we wish to resolve an angle of 0.07° half-angle, a lens with a focal length of 41.1 mm is required. Its diameter will be kept small if it is moved up near the interferometer exit aperture. The field will now cover a radius of 3.6 mm to cover the 4- by 6-mm area of the detection device. At the other exit aperture of the interferometer, one will see a bright field when the star is properly aligned. If the star drifts off axis, fringes will appear. As noted earlier, because of the reversal and inversion of the wave fronts in the interferometer, the fringes will line up in the direction of the pointing error. These fringes are localized on the exit beamsplitter itself. A lens can be used to image these fringes onto the 4- by 6-mm field of the detection device.
It is not the intent of this report to suggest that this telescope design concept is by any means the optimal one for visual detection of extrasolar planets, let alone even a feasible concept in practice. To be sure, some of the quantitative aspects of the telescope appear to be absurd on the face of it. However, these absurdities can be greatly mitigated, if not altogether removed, by minor alterations in the design parameters.
[71] In retrospect, several alternative procedures should be entertained. In this design, the tilt in the pupil plane, induced by the introduction of the spherical mirror, is corrected by tilting the collimating lens. It may be more expedient to tilt the ellipsoid instead. Here the advantage would be that the axis of the light emerging from the ellipsoid would be tilted so that the central obscuration caused by the spherical mirror and its accoutrements could be avoided.
The portion of the electromagnetic spectrum
that is loosely defined as the "infrared" lies in the wavelength
interval 0.75µm << 1000 µm (1 mm). As with
observations at visual wavelengths, turbulence in Earth's atmosphere
effectively makes it impossible to search for extrasolar planets at
infrared wavelengths from the ground. The atmosphere also presents
problems in that molecules (e.g., H2O and
CO2)
in the atmosphere are very efficient absorbers of radiation over many
regions in the infrared. These absorption effects can be minimized by
taking observations from high-flying aircraft, such as the NASA C-141
Kuiper Astronomical Observatory (fig. 27), and they can be eliminated
entirely by observing from space.
As noted in chapter 2, there are many appealing aspects of the infrared as a wavelength regime in which to search for extrasolar planets. Successful detection provides data concerning a number of significant parameters of the detected planet, most notably, temperature and diameter. In addition, the brightness ratio between star and planet (cf. fig. 6) is lower in the infrared than it is in the visual portion of the spectrum. However, detection of extrasolar planets by means of infrared (IR) observations is not without problems. This section discusses those problems and possible solutions.
It might be supposed that an IR telescope could rather easily have sufficient spatial resolution to detect the SPS. However,....
....although the IR -value (~10-4) is indeed much
more favorable than the visual ~-value (~2 x 10-9), it is still low
enough to require a very large, clear-aperture telescope. Use of
equation (32) for
= 40 µm and
= 10-4 shows that the
clear aperture required is ~120 m. A further complication arises from
the fact that, relatively speaking, apodization is less helpful
at
= 40 µm
than at
= 0.5
µm (because of the respective
values). Use of the Sonine
functions as apodizing functions leads to a reduction in aperture
size by about a factor of 40 at
~ 2 x 10-9, but only by about
a factor of 3 at
~10-4. Thus, an apodized IR telescope must still be
40 m. The approach taken in Project
Orion to circumvent this problem was to consider an interferometer
rather than a filled-aperture telescope. The angular resolution
of an
interferometer of baseline S is given by
Equation (44) describes the situation where
radiation from a given source (a star) travels slightly farther, a
distance of , to reach one [73] of the two apertures
separated by distance S. The light amplitudes received at the
detector must be added vectorially, giving rise to a null signal as
the radiation at one aperture is 180° out of phase with
radiation at the other aperture. Requiring that
~ 0.5 arcsec
at
= 40 µm
gives S ~ 8 m, a reasonable size.
An interferometer has the added advantage of simultaneously providing angular and intensity resolution. If the star under study were a true point source, and if a space-based interferometer could be pointed with infinite precision and would remain absolutely stable, the stellar signal could be nulled out and the power from the two apertures balanced. Each of the "ifs" mentioned above is critical, and it is worthwhile to examine each in more detail.
Considering the SPS as a specific example, it
may be seen that the star is not a point source. Its angular extent
is ~0.001 arcsec. I he intensity pattern produced by the
interferometer is of the form
where here is the angle of the optical
axis of the interferometer and
is the angle between successive maxima (or minima) in
the
pattern For S ~
8 m,
1 arcsec. The
choice of
= 1 arcsec
places the SPS planet at the maximum of the intensity pattern and the
star at the minimum (see fig. 28). The effective brightness ratio
between planet and star is
where the interferometer intensity pattern is
integrated over the two bodies, K is a constant, and is the normalized intensity of the
star as a line source. Taking the star image to be a uniformly bright
disk of radius b, the integral over the star becomes
The corresponding integral over the much smaller planet is simply the solid angle of the planet as seen from Earth. Thus,
which gives~ 60 for the
SPS; that is, the interferometer can, in principle, null out the
stellar signal to the extent that the planetary signal is about 60
times stronger than the stellar signal.
Prior to discussion of the effects of pointing
errors on the ability of an IR interferometer to detect extrasolar
planets, it is necessary to make a few remarks concerning the
operational aspects of such a device. In situations where small
signals have to be detected in the presence of unwanted signals that
might cause confusion, it is a well-established practice to modulate
the desired signal. Consider the [75] previously described
interferometer to be spinning with angular frequency about an axis
passing through the star. Then the signal from the star would not
vary in strength, but the signal from the planet would rise and fall
with a fundamental frequency 2
(fig. 29). The waveform would not
be strictly sinusoidal, but of a characteristic flat-topped form
containing a noticeable amount of the 4
-harmonic
(later shown to be 6 percent).
Very faint signals can be recovered, if they
are modulated at a known frequency 2, by synchronous detection that
filters out that frequency. Alternatively, where the modulation is
substantially nonsinusoidal, the received signal can be broken into
segments of duration (1/2)
and averaged. The nonsinusoidal
form may turn out to play a significant role. With this operational
concept in mind, the effects of pointing error can be
analyzed.
An element of the stellar disk at a distance e
from the center of rotation and in position angle (fig. 30) will
produce an output fluctuation proportional to
, and the total
output clue to the star will be obtained by integrating such
contributions over all the elements of the disk.
Instead of determining the waveform in its full detail, attention will be given to the peak value A* and the trough value B*. The configurations corresponding to peak and trough are shown in figure 31. Let E be the displacement of the center of the star from the rotation axis and let the radius of the star be R; then
The sin2 factors may be
replaced by the squares of their arguments provided E and R are small
compared with . The peak-to-trough amplitude is [77]
The peak-to-trough amplitude of the variation
due to the planet is calculated by giving the planet a radius R/10
and assuming that the received power falls to zero as the planet
crosses the interference null (Bp = 0). if the planet
had the same brightness as the star, the peak-to-trough amplitude
would be, but, of course, the brightness is less by a factor of
about 250. Hence we have as an approximate value for the amplitude
ratio, when E <<
.
Numerical examples of the ratio
(Ap
- Bp)/(A* - B*) as a function of pointing error E are given in table
5. These results show that the spin axis of the interferometer may
fall 4 stellar radii away from the center of the star before the
unwanted amplitude of the 2 variation due to the star builds
up to equal or exceeds that due to the planet.
Pointing error E (arcsec) |
|
. | |
0.0005 |
|
.001 |
|
.002 |
|
.003 |
|
.005 |
|
It would not be surprising, although it would be technically demanding, if a passive object spinning in space could be guided to milli-arcsecond accuracy on a star.
If the planetary signal could be distinguished
from the stellar signal, then there would be no need for the
amplitude ratio tabulated above to fall below unity. It would only be
necessary for the received planetary signal to exceed some noise
level, the nature of which will depend on instrumental design choices
yet to be discussed. One possible design strategy (considered below)
is to make me planetary modulation markedly different from the
stellar modulation (in the presence of pointing error) by compressing
the interference fringe pattern so that the planet is fringe
spacings away from the star instead of half a spacing. This
compression can be obtained by lengthening the interferometer
baseline or by shortening the wavelength.
Under the arrangement previously discussed,
the signal from the planet was of the form but will now
become
. The second
harmonic content of this waveform is distinctly enhanced; in fact, it
is possible to choose
so as to suppress the fundamental component at 2
completely. On
the other hand, the signal due to the star, which is closer to the
rotation axis, will be much more nearly sinusoidal with a fundamental
frequency 2
.
Hence, by singling out the 4
component, a frequency that is very
precisely known, it should be possible to gain an advantage. One way
of using this advantage would be to relax the pointing accuracy
requirement on me spinning interferometer. Figure 32 shows that
situation with
= 0.82. As the fringe pattern rotates about
the....
....center C, the planet P (shown for t = 0)
moves relative to the fringes around the dotted circle. The received
power rises and falls as shown. This is me case mentioned above where
there is no 2component at all. Meanwhile, if the star is off axis at
a position corresponding to
= 0.5, an angular pointing error
of about 0.16 arcsec, it will deliver mainly a 2
component and the 4
component will
be only 6 percent as large Thus a considerable benefit derives from
working with the 4
component.
Before giving the basis on which the magnitude
of the benefit may be calculated, it should be pointed out that
another effect could contribute to distinguishing the planet from the
star. There are even higher harmonics present in the waveforms, as is
particularly obvious from inspection of figure 32, but the mix of
harmonics is different for planet and star. Thus it will be possible
to do better than would appear from comparing the 4components
alone. All that can be done now is to demonstrate that an interesting
possibility exists for relaxing pointing accuracy, but a careful
quantitative study will require more effort.
The determination of harmonic content of the interferometer output waveforms is done as follows. To Fourier analyze
note that [80]
which leads to me results indicated in table 6.
These results indicate that for = 0.5,
a4/a2 = 6 percent, as previously mentioned. The waveform
, that is,
= 1.64,
roughly maximizes the amount of 4
component and minimizes the amount
of 2
.
There are three principal categories of potential noise sources for an IR planetary detection system: instrument-related noise, natural noise, and spacecraft-related noise.
Instrument-related noise can arise from both the detector and amplifier. The latter contributes Johnson noise, flicker noise, and shot noise. The former suffers from Johnson noise, current noise, temperature noise, and generation-recombination noise (ref. 31). The standard way to express noise level in IR astronomy is in terms of "noise equivalent power" (NEP). NEP (in units of W/Hz1/2) is the minimum power that can be detected in 1 sec of integration time. Typically, good amplifiers have extremely low noise levels (NEP about 10-21 W/Hz1/2). The noise level from a detector depends strongly on the type of detector, but IR detectors have been constructed with NEP's of 3 x 10-17 W/Hz1/2.
[81] Among possible natural sources of noise are radiation noise from the source (planet), noise due to residual constituents of Earth's atmosphere, and noise due to zodiacal and background starlight. The NEP due to IR photon fluctuation from the planet is given by
where h is Planck's constant, v is the
frequency (in Hz), is the emissivity of the emitting surface, A is the
telescope aperture (in m),
is the solid
angle subtended by the planet (in rad2), B is the
instrument bandwidth (in Hz), B(Tp) is the brightness
of the planet, and Tp is the
temperature:
For the SPS, observed with a telescope having
A = 1 m2, B = 0.05 Hz, Tp = 128 K, and v = 7.5
x 1012 Hz (= 40 ~m), the NEP is about10-26
W/Hz1/2, a negligible effect.
Even at altitudes above 250 km, there are
residual components of Earth's atmosphere, notably O,
N2,
O2,
Ar, H, He, NO, N, CO, and the ionic forms of these species. The atoms
and molecules that could affect an IR observation are O, NO, CO, and
NO+
(see fig. 33). The noise contribution from these species decreases
rapidly with increasing altitude. An interesting feature of figure 33
is the absence of significant (10-17
W/Hz1/2) IR radiation from the residual atmosphere between= 6 and 40
µm, the wavelength region of interest to a search for extrasolar
planets.
Noise contributions from zodiacal light at T =
304 K and star light at T = 5500 K are also shown in figure 33.
Zodiacal light is caused by interplanetary dust particles. From
figure 33, it appears that zodiacal light produces noise dominating
that due to the residual atmosphere. However, the zodiacal light
curve shown in this figure obtains for the ecliptic plane, and the
maximum intensity is at wavelengths below 20 µm. At a latitude
±10° off the ecliptic plane, the intensity decreases by a
factor of 3. At a latitude of ±10° and at= 40 µm ,
the zodiacal light has a noise level of 3 x 10-17
W/Hz1/2.
[83] If the IR interferometer view is farther away from the ecliptic plane, then the zodiacal light should not be an important noise contribution if the wavelength is above 30µm. However, in the range 5 - 30µm, the zodiacal background could be a fundamental limitation of the IR space system.
The starlight shown in figure 33 is typical of
a 5500 K source. The straight line is only the long wavelength
portion of the complete curve which is similar to the blackbody
radiation curve. Starlight is a troublesome noise contribution for
< 10
µm. Only a small number of stars have significant influence on
the background radiation above
= 10µm.
Moonlight, earthshine, and other planets in the solar system are other natural sources that increase the background radiation. Moonlight and earthshine are the two strongest and an IR system should be shielded from these sources to reduce background radiation.
One other potential source of noise for IR observations is that arising from effluent contaminants from a space shuttle or satellite. Two of the most important contaminants are H2O and CO2 (ref. 32). The high noise levels of both H2O and CO2 cover a wide range of wavelength spectrum. Even at an altitude of 400 km, the decrease of noise power as a function of altitude is rather slow compared to that of natural H2O and CO2.
Deposition of condensable gases (both natural and contaminants) on the surface of an IR telescope causes undesired absorption and scattering that degrades the performance of the IR space system. Because some of the contaminants generated by a space system have unusually high noise radiation, they should be minimized by: (1) choosing an orbit that requires a minimal number of convection maneuvers, (2) reducing the payload of the space system, (3) stabilizing the telescope and the interferometer rather than the shuttle or orbiter, (4) using low outgassing materials, and (5) storing waste water and venting it when not observing or when at poor observing locations. Items (2) and (3) are listed to minimize the required fuel.
Although a detailed analysis of the various noise sources discussed above has not been carried out, it appears that zodiacal light may be the principal contributor. It would indeed be ironic if the particulate debris within our own planetary system prevented us from discovering other planetary systems by means of IR observations.
[84] Development of technology in the area of IR detectors has been remarkably rapid. Any statement made here concerning the state of the art will certainly be passé at the time this report is printed. However, it is useful to delineate a few general aspects of IR detectors as applied to the planetary detection problem.
With the exception of the Golay cell, all of the effective IR detectors are solid-state devices. Any choice of an IR detector should involve the following factors:
1. Range of spectral wavelength
2. Required detectivity or sensitivity
3. Response time
4. Operating conditions (e.g., temperature and stability) IR detectors can be divided into two main types:
1. Thermal detectors-the IR radiation is detected by measuring the change of properties or characteristics due to thermal effect. Some of the thermal detectors are thermocouples, thermopiles, Golay cells, bolometers, and pyroelectric detectors. Thermal detectors have a long response time, on the order of 1 msec.
2. Photoconductive detectors (or photon detectors) - these detectors utilize various internal photoeffects in semiconducting materials, such as a change in electrical conductivity or photoconductivity due to IR radiation. This type of detector has an extremely short response time, usually of the order of 1 µsec or less. This type of detector includes HgCdTe and PbSnTe detectors (intrinsic photon detectors), Ge and Si extrinsic detectors, MOSFET, and the Josephson detector.
The performance of an IR detector is based on the detector NEP (or the specific detectivity D*= NEP-1 ). An ideal IR detector (refs. 33 and 34) has the following NEP:
|
Stefan's constant |
k |
Boltzmann's constant |
T |
absolute temperature, in K |
[85] A |
detector effective area |
|
amplifier bandwidth |
|
emissivity of detector |
In order to keep NEP low, the detector must operate at a low temperature, and the emissivity should be large (~1). Theoretically, the detector area should be small, but there are two limiting factors:
1. The practical feasibility of the detector - the detector cannot be very much smaller than the operating wavelength
2. The detector must be matched to the collecting optics, which requires that
for the diffraction limited case
The bandwidth should be kept low, but not too low, to avoid losing the important signal. The broader the bandwidth, the lower the operating temperature must be for compensation.
The problems attendant upon direct detection of extrasolar planets at either visual or infrared wavelengths are formidable. However, the potential gains are great, and the analyses carried out during Project Orion indicate that it may be possible to overcome these problems. Much more must be done, in far greater depth and detail, than was possible during Project Orion. More detailed studies on direct detection systems are being conducted by Stanford University, Hewlett-Packard, and Lockheed under the direction of D. C. Black of Ames Research Center. Hopefully, the preliminary efforts outlined here will provide a useful baseline for such future studies.
1 The idea of
coherence length can be explained in terms of the Michelson
interferometer in which a beam of light is divided into two by a
beamsplitter. Each beam is directed down an arm of the interferometer
to a plane mirror, then it is reflected back to the beamsplitter
where the two beams are recombined. There interference fringes are
formed by introducing a slight tilt in one of the mirrors. When the
lengths of the two arms are equal, the contrast of the fringes is
greatest. If one of the arms is lengthened, fringe contrast drops off
until at some point they disappear completely. This displacement, the
coherence length is a property of the light source. A well-filtered
thermal source barely exhibits interference and therefore has a
coherence length near zero. A mercury discharge lamp will exhibit a
coherence length of several centimeters while a helium neon laser has
produced interference over several hundred meters.