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[47] 20-inch Astrograph.

Photograph courtesy of Lick Observatory, University of California, Mt. Hamilton, California.

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[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

(28)

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

(29)

where r_O is the radius of the aperture. The intensityat a point off the optical axis, by an angle , is given by

(30)

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 :

(31)

whereis the mean intensity atand 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.....

Figure 14. Schematic representation of the variation in light intensity,, in a clear-aperture telescope as a function of distance from the optical axis(= 0). Off-axis distance is expressed in terms of Airy radii; the light is assumed to be monochromatic.

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(32)

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 atarcsec.

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

(33)

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 is10^-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.

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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.

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Figure 19. Photograph of a model planetary system using a clear aperture (no apodizer); = 10^-3 for this system.

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Figure 21. Photograph of a model planetary system ( = 10^-4) using a mask that is the complement of the mask used in figure 20 (see text for discussion). One of the bright spikes obscures the companion.

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Figure 22. Photograph of a model planetary system (= 10^-4) using a mask that is the complement of the mask used in figure 20 (see text for discussion). The mask has been rotated to align a dark interspike region with the location of the companion.

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Figure 23. Photograph of a model planetary system (= 10^-6) using a mask that is the complement of the mask used in figure 20 (see text for discussion). The mask has been rotated to align a dark interspike region with the location of the companion.

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[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

(34)

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

(35)

where u is a coordinate on the exit aperture in an appropriate direction. Summing these amplitudes gives

(36)

The fringe brightness will be at a maximum whenever The distance between adjacent bright fringes is given by

(37)

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,

(38)

It follows that all wave fronts incident upon the interferometer at, so thatwhere

(39)

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,

(40)

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:

(41)

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

(42)

[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

(43)

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 resolutionof an interferometer of baseline S is given by

(44)

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

(45)

where here is the angle of the optical axis of the interferometer and is the angle between successive maxima (or minima) in thepattern 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

(46)

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

(47)

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

(48)

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]

(49)

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 (B_p = 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 (A_p - B_p)/(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.

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 4component, 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 2component and the 4component will be only 6 percent as large Thus a considerable benefit derives from working with the 4component.

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/Hz^1/2) 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/Hz^1/2). 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/Hz^1/2.

[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(T_p) is the brightness of the planet, and T_p is the temperature:

For the SPS, observed with a telescope having A = 1 m², B = 0.05 Hz, T_p = 128 K, and v = 7.5 x 10¹² Hz (= 40 ~m), the NEP is about10^-26 W/Hz^1/2, 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/Hz^1/2) 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/Hz^1/2.

[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:

(52)

	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 thatfor 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.

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