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[9] 61-cm Sproul Refractor.

Photograph courtesy of Sproul Observatory, Swarthmore College, Swarthmore, Pennsylvania.

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[11] It is useful to examine the magnitude of the planetary detection problem and the fundamental limitations of either a physical or technological nature, particularly as such considerations significantly affect systems design concepts. This chapter outlines some of the key factors involved in Project Orion as it progressed from a definition of the problem toward the formulation of the design concepts discussed later.

The definition of "planet" used by ancient astronomers centered on the erratic apparent motion of certain celestial bodies. It is perhaps surprising that there is no generally accepted modern definition of "planet." Questions arise as to whether the definition should be only in terms of intrinsic properties of a body (e.g., mass) or whether it should also include properties related to orbital motion (e.g., that the orbit be nearly circular). The view taken here is that the term "planet" refers to any object whose mass is comparable to or less than the mass of Jupiter (~1.9 x 10²⁷ kg). Adoption of this operational definition does not imply that bodies which are more massive than Jupiter are not planets; the upper limit of the mass of a planet is set by the lowest mass consistent with the definition of a star(subscriptdenotes the Sun, so that= mass of the Sun, or 1.998 x 10³⁰ kg). The choice of this definition of planes arises directly from the challenge addressed in this study, namely, to design instrumentation that would permit unambiguous detection of objects with the mass of Jupiter. Such instrumentation would detect more massive objects with relative ease.

[12] The three techniques for detecting extrasolar planets considered here are: (1) astrometric searches for perturbations in stellar motion, (2) infrared searches for intrinsic thermal extrasolar planetary radiation, and (3) optical searches for reflected visual stellar radiation. For reasons indicated later, technique 1 is referred to as an indirect detection technique, whereas techniques 2 and 3 are referred to as direct detection techniques. It will be useful to briefly discuss the physical basis for effects arising from extrasolar planets which give rise to phenomena that may be studied by one or more of the techniques listed above. This discussion will also serve to indicate both the informational content (i.e., what might be learned concerning a detected planet) of each technique and the order of magnitude of the detection problem.

What are the observables related to each of the three techniques considered here, and what are the relationships between those observables and characteristics of a potential extrasolar planet? Imagine a rather simple planetary system consisting of a star and a single planetary companion (fig. 3). The mass, diameter, and effective temperature of the star are denoted by M_*, d_*, and T_*, respectively, whereas the corresponding parameters for the planetary companion are denoted by M_p, d_p, and T_p. The barycenter (center of mass) of this ersatz planetary system is located at distances R_* and R_p from the centers of mass of the star and planet, respectively, where M_pR_p = M_*R_* with

(1)

and

(2)

The quantity R = R_* + R_p is the semimajor axis of the planet's orbit about the star, taken here to be circular to simplify the discussion. The three techniques are discussed in the order listed above and in the context of detecting this simple planetary system.

A detailed discussion of astrometry will not be given here, but interested readers can consult any number of excellent books on the subject (e.g., ref. 5). If the system under study is an isolated (no companions) star, the barycenter lies at the center of mass of the star. If, however, the star has a companion (planet?), the system barycenter is displaced from the star's center of mass by a distance R_* (eq (1)). In this latter case, the star and companion will revolve about the barycenter with a period determined by M_*, M_p, and R. Precise observations of the star could, in principle, reveal that its motion departed from rectilinear motion.

[14] The amplitude of this nonlinear motion is R_*. However, R_* is not the directly observed quantity. The observed quantity is essentially an angle , where

(3)

where D is the distance from Earth to the star under observation. In general, M_pR_p << M_*D, so that

(4)

where M_p and M_* are, respectively, in units of Jovian and solar masses, C₁ R_p/M_*, R_p is in astronomical units (AU) (1 AU = 1.5 x 10¹¹ m, the mean distance from Earth to the Sun), and D is in parsecs (1 pc = 3 x 10¹⁶ m). Representative values of as a function of D are shown in figures 4 and 5, respectively, for M_p = 1 and 0.003 (an Earth mass planet). The technique of detecting the presence of extrasolar planets by astrometric observations is referred to as indirect detection because the presence of a planet is deduced from observations of a star, not of the planet.

Planetary companions to stars are "sources" of electromagnetic radiation. This radiation can be characterized as either thermal or reflected. A planet will radiate at some temperature which is determined by a balance between the rate at which the planet receives energy from both internal (e.g., radioactive) and external (e.g., its parent star) sources of energy, and the rate at which the planet loses energy by radiation. This thermal radiation is most pronounced in the infrared region of the spectrum because planets are relatively cool objects. A planet can also reflect radiation from its parent star, the amount of reflected radiation depending both on the size of the planet and the nature of the reflecting medium. This reflected component of a planet's radiation spectrum is generally strongest at those wavelengths where the central star emits most of its radiation, namely, the visual portion of the spectrum. The total energy flux.....

Figure 4. Maximum angular displacement () of a star due to a Jovian mass companion shown as a function of its distance (D) from an observer. Present level of observational accuracy indicated by dashed line; various slanting lines refer to differing values of R_p/M_*.

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.....(per unit frequency interval) from a planet is the sum of its intrinsic thermal flux,, and reflected stellar flux,:

(5)

If represents the radiant energy flux leaving the surface of the star depicted in figure 3, only a fraction f_p of that flux will fall on the planet, with

(6)

Figure 5. Maximum angular displacement () of a star due to an Earth mass dark companion shown as a function of its distance (D) from an observer. Present level of observational accuracy indicated by dashed line; various slanting lines refer to differing values of R_p/M_*.

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In general, only a fraction of the radiation incident upon a planet will be reflected (that fraction denoted by ):

(7)

The parameter is usually a function of frequency, but that complication is ignored here. Assuming that the intrinsic thermal flux from the planet is that of a blackbody of temperature T_p gives

(8)

[17] where c, h, and k are, respectively, the speed of light in vacuum, Planck's constant, and Boltzmann's constant. Combining equations (5), (7), and (8) gives

(9)

where the constant (in mks units) and

The number of photons emitted per second per unit frequency interval N_p(v), is obtained by dividing by the energy per photon, hv. Equation (9) gives the flux at the planet's surface. Of more interest from the detection standpoint is the energy E_DET(v) which arrives at a detector located on or near Earth. If one ignores effects such as limb darkening, E_DET(V) is given approximately by

(10)

where A and B are, respectively, the area and the frequency bandwidth of the detector, and D is the distance defined previously. The number of photons, N_DET(V), is given by

(11)

for B << v. Figure 6 shows for the Sun and for Jupiter, assuming that

One other quantity of interest is the ratio H(v) of planer radiation to stellar radiation of frequency v, namely,

(12)

whereis given by equation (8) with d_* and T_* substituted, respectively, for d_p and T_p. At high frequencies (hv >> kT_*), [19] H(v) approaches the limiting value, which is independent of stellar properties and of v. At low frequencies (hv << kT_p), H(v) is independent of v, but depends on both stellar and planetary properties, namely,

where and The techniques of detecting the presence of extrasolar planets by observations of planetary radiation, either intrinsic thermal or reflected visual radiation, are referred to as direct detection techniques because detection arises from a direct observation of the planet.

Equations (4) and (10)-(12) provide the analytical basis for estimating the order of magnitude of observable parameters for assumed extrasolar planetary systems and for determining what each of the three techniques considered here might provide by way of information concerning these planetary systems.

In order to make subsequent discussion and numerical examples specific, it is useful to define a "standard planetary system" (SPS). The basic free parameters that define an SPS are those depicted in connection with figure 3, as well as the number of planets. The standard chosen here is that of a spectral type G2 main sequence star (e.g., the Sun) around which revolves a single planet. The mass, dimensions, and temperature of that planet are taken to be identical to those of Jupiter. The planet's orbit is taken to be circular with a radius of 5 AU. The principal characteristics of the SPS are summarized in table 2. The equations given earlier indicate how observable parameters scale with different assumptions concerning the characteristics of the star, planet, and orbit.

The order of magnitude for observables relating to the SPS are obtained here for two assumed values of the distance parameter D, namely 0 = 10 and 30 pc. Equation (4) indicates that the magnitude of the angular perturbation at these distances is

(D = 10 pc) = 9.8 x 10^-4 x 0.5 = 4.9 x 10^-4 arcsec

and

(D = 30 pc) = 9.8 x 10^-4 x 0.16 = 1.6 x 10^-4 arcsec

By comparison the highest accuracy obtained with current astrometric observations is about 3 x 10^-3 arcsec and typical performance is more like 5 x 10^-3 arcsec. These accuracy figures pertain to so-called yearly mean normal points; that is. they represent the results of many observations per year with the accuracy of the mean increased over the accuracy of individual measurements in proportion to the square root of the number of observations per year. Note that the performance necessary to detect the SPS at D = 10 pc exceeds that of present astrometric facilities by a factor of 6 and by a factor of about 20 if it is required that the formal error be no greater than 1/3 the maximum values of .

Equation (10) gives the fraction of energy flux from an extrasolar planet which arrives at Earth. It is convenient to estimate the photon flux rather than the energy flux, as photon noise is an important parameter in the detection of radiation. This noise varies inversely as the square root of the number of photons. The frequency vm of maximum photon emission per unit wavelength interval from a blackbody at temperature T is given by v_m = 8.175 x10¹⁰ T (Hz). Thus, v_m for the star in the SPS is v_m* = 4 .74 x 10¹⁴ Hz (= 6.33 x 10^-7 m) whereas v_m for the planet in the SPS is v_m = 1.05 x 10¹³ Hz (= 2.87 x 10^-5 m). Evaluating the photon flux equivalent of equation (10) at these two frequencies and at distances of D = 10 and 30 pc yields the results given in table 3. Note that the fluxes given in table 3 are per unit wavelength interval, [21] not per unit frequency interval The detector bandwidth and telescope aperture used to determine the values given in table 3 are respectively B = 0.1 v (Hz) and A = 1 m².

The values of N_IR and N_V given in table 3 indicate that the standard planet would not be readily detectable in the visible portion of the spectrum beyond distances of about 4 pc (N_V ~ 1/sec) With A = 1 m² and B = 0.1 v, but could be detected at even greater distances if either more collecting area or more light (larger B) were used in the observation The important point is not that a specific planetary system is or is not detectable using a specific set of parameters but that both visible and especially infrared radiation from extrasolar planets revolving around stars located at reasonable distances (~10-30 pc) from Earth gives rise to a detectable number of photons The question of detectors and noise sources for these measurements is discussed elsewhere.

The essential aspects of information content for direct detection are contained in equation (10) The net energy flux on a detector E_DET(V), is essentially the measured quantity The frequency v and fractional bandpass are also known quantities. Assuming that distance D to the star under observation is known independently, equation (10) may be recast in the form:

(13)

where

[22] and the variablehas been used rather than

In the infrared portion of the spectrum,

(14)

so that

If observations are made at two infrared wavelengths, the temperature T_p can be determined, thereby making it possible to determine the size of the planet, d_p, from the measured parameter . Further, the orbital period can also be determined from the observations. If the mass of the star and the orbital period are known, the value of R_p can be calculated from Kepler's third law. The usefulness of knowledge concerning R_p is that a comparison of T_p with temperature T_BS of a perfectly conducting black sphere (T_BS = 0.354 T_* (d_*/R_p)^1/2) could provide an indication of atmospheric effects on the planet.

In the visual portion of the spectrum, the inequality given equation (14) is reversed so that

Values for T_* and d_* are known approximately from stellar evolution theory and spectral observations of stars. As in the infrared, the orbital period and hence R_p can be determined. However, this [23] knowledge only places constraints on the product ; there is no way to determine independently either or d_p. It may be possible to make multicolor observations that could provide useful clues regarding the nature of the reflecting medium. For example, the Earth appears blue relative to solar light because of the selective reflection of shorter wavelengths by Earth's atmosphere.

The defining equation for astrometric studies is equation (4), with the measured quantity being directly related to . The mass of the star (M_*) under study can be estimated independently, and the distance to the star (D) can be measured. Thus, the product M_pR_p can be related to measured or known quantities. The orbital period of the planet yields the value of R_p (given M_*), so that the mass of the planet can be determined. Although it is not clear from the somewhat simplified equations presented here, astrometric studies would also reveal much concerning the nature of the planet's orbit (e.g., its eccentricity and relative orientation to the line of sight).

Clearly, comprehensive studies to search for extrasolar planetary systems could, if successful, reveal something more about our neighbors in space other than their existence. The various kinds of information that can, in principle, be obtained are listed in table 4.

[24] Having outlined the physical basis underlying the detection techniques considered here, it is useful to inquire as to limitations placed on the detection problem, and thereby on possible design concepts, by Earth's atmosphere.

Earth's atmosphere presents many difficulties for any method of extrasolar planetary detection. The atmosphere is not quiescent, rather it is often highly turbulent, and it is generally disturbed by pressure waves. Light from a point source in space, such as a star, enters the atmosphere as a bundle of nearly parallel rays. However, due to atmospheric effects, these rays do not remain parallel. The principal atmospheric effect arises because all light rays do not traverse identical portions of the atmosphere. The net result is that the image of the star as seen at the bottom of the atmosphere is "smeared out" -- part of the reason why stars do not appear as true points of light. This smearing out process is not uniform. As pressure waves propagate through the atmosphere, they may affect a given light ray differently than they affect neighboring light rays. This causes the apparent image to flicker or, in more familiar language, twinkle. The twinkling of stars is a consequence of their point source nature. A source sufficiently close so as not to be a point, such as a planet in the solar system, does not appear to twinkle because the twinkle arising from a particular portion of the object tends to cancel the twinkle from another portion of the object, so the eye perceives a fairly constant pattern.

The atmospheric effect discussed above is known as "seeing" and it is generally expressed in terms of the minimum size of a stellar image as seen through a telescope. Seeing at ground-based observatories typically ranges from 1.5 to 2.5 arcsec, but may occasionally be as good as ~0.5 arcsec at exceptional sites. Since the maximum angular separation in the standard system is only 0.5 arcsec, it is clear [25] that any direct imaging cannot be clone from the ground; it must be done in space.¹

Astrometric techniques can be divided into those of absolute and relative astrometry. In absolute astrometry, stellar positions on the sky are measured relative to a system of coordinates defined by the direction of Earth's axis of rotation and the vertical at the place of observation. As in most other fields, absolute measurements are generally much less accurate than are relative measurements. In relative astrometry, the position of a star is measured relative to positions of other stars situated in angular proximity to it on the sky. Relative astrometry, with its inherently higher precision, is a more suitable approach than absolute astrometry for detecting planetary induced wobble in the motion of a star.

The errors of measurements in relative astrometry can be traced to the following four sources:

1. Inaccuracy of devices used to measure the relative positions of star images formed by a telescope

2. Changes in relative positions of star images caused by instabilities of the optical components of a telescope

3. Effects of Earth's atmosphere

4. Effects intrinsic to the observed stars, such as variable color or brightness of reference stars

(Items 1 and 2 are discussed later; item 4 involves topics that fall outside the scope of this report, but are currently under investigation by some of the Project Orion personnel; and item 3 is discussed next.)

[26] The angular separation between two stars can be split into two perpendicular components: a vertical component directed toward the zenith and a horizontal component. Measuring the vertical component requires knowledge of refraction in the atmosphere, which depends on the color of a star and on atmospheric pressure and temperature. The horizontal component, which is on the average less dependent on these factors, can be measured with considerably higher accuracy than the vertical component. Differences in both right ascension and declination of two stars can be obtained from measurements of the horizontal component of their angular separation if the observations are made at two azimuths, for example, at A = ±30°. The following discussion will therefore be limited to measurements of the horizontal component.

The analysis given here is based on Reiger's model (ref. 6) of atmospheric turbulence. This model successfully explains various observed properties of scintillation of starlight (ref. 7) and the average size of excursions of star images (appendix II in ref. 6). Reiger's theory is based on the Kolmogorov spectrum of isotropic turbulence, and characterizes the turbulence by a one-dimensional (radial) power spectrum of the form

(15)

where C₁ is a constant, k is the wave number, and is the so-called "outer scale" of the turbulence (i.e., the size scale at which energy is input to drive the turbulence). It should be stressed that the turbulence spectrum represented in equation (15) is based only on thermally generated atmospheric fluctuations and does not contain contributions at very long wavelengths (small k) due to mechanically generated fluctuations (e.g., by pressure waves, mountains).

A good measure of the astrometric error introduced by atmospheric turbulence can be had by examining the variance in optical path for light from a star as detected at two apertures separated by a distance B. The difference in optical path length, L₂, at two points separated by distance B is

(16)

[27] where the integration is over a vertical path extending from the ground to an altitude h in the tropopause, and is the fluctuation in the refractive index of the atmosphere along vector as a function of altitude z. The mean value of L₂(B) vanishes as an ensemble average (taken here as a long-term average). However, the variance in L₂ is nonzero in general; this variance is the quantity that sets the limit on the precision of relative astrometry measurements. A rigorous discussion of the determination of the variance in L₂(B), although informative, is mathematically complex and so will not be given here. A rather detailed discussion is given in appendix A. The salient aspects of that detailed discussion are summarized here.

The limiting rms error in the measurement of angle between two stars is related to the variance in optical path of the aperture by

(17)

where B is the telescope aperture (baseline) and <L₂²(B)> is the variance. The theoretical analysis given in appendix A, as well as the behavior of measured variance, indicates how path-length variance depends on baseline (fig. 7). The variations of(is the wavelength of observed light) versus log B are shown for a range of assumed values of the outer scale L₀ of the turbulence and for excellent seeing conditions. For baselines that are small compared to L₀,

(18)

Combining equations (17) and (18) yields

(19)

Thus, the limiting error varies linearly with the wavelength of observed light and it varies slowly (B^1/6) with aperture. Taking....

.... L₀ = 50 m, B = 1 m, and = 5 x 10^-7 m gives rise to ~ 0.08 arcsec, and = 0 04 arcsec for B = 50 m. These values Of pertain to the error expected for measurements taken over a time scale that is comparable to the time scale for wind-driven eddies to sweep across the field of view. For 5-10 m/s winds and B ~ 1 m, these transit times are < 0.2 sec. Additional empirical evidence for this behavior is shown in figure 8, where positional errors for observations on the ~0.8-m Thaw telescope at Allegheny Observatory are shown as a function of observing time. Note that the errors at the short time limit are consistent with the value inferred from the analysis given here. Also evident from figure 8 is that the photographic plate, used as a detector, leads to poorer long-time accuracy than if data were taken in short bursts where the inherently....

.....less accurate but statistically independent atmosphere effects can be used to define a more precise, long-term position. If measurements were taken every seconds over a nightly observing time T_obs, the nightly rms error would be

(20)

The theoretical arguments used to derive equation (20) indicate that if = 0.2 sec, the expected precision using a 1-m-aperture telescope with a perfect detector is more than an order of magnitude better than for that telescope with a photographic plate as a detector.

The principal conclusion to be drawn here is that Earth's atmosphere places rather fundamental limits on the precision with which [30] astrometric measurements can be made. The best nightly precision attainable with a 1-m-aperture astrometric telescope and a perfect detector (i.e., one that introduces no astrometric errors) is ~0.0006 arcsec (= 0.2 sec and T_obs= l hr= 3600 sec), and is only ~0.0003 arcsec for a 50-m-baseline telescope. Additional theoretical and experimental studies of the effects of the atmosphere on astrometric precision are clearly needed, but the results given here are unlikely to be much in error (an independent estimate of this limiting precision has been carried out by P. Connes; his findings are consistent with those of Project Orion). Whether very long (~l km) baselines can be used to increase precision, as suggested by Currie and co-workers, is an open question. The principal uncertainty of long baselines is the effect of mechanically induced atmosphere disturbances.

Comparison between obtainable astrometric accuracy, usually assigned to yearly mean relative positions of stars, and the accuracy required to detect the SPS requires that the nightly precision given above be increased approximately in proportion to the square root of the number of nightly observations. Assuming an average of 36 nights/yr of observing, the limiting accuracy is ~10^-4 arcsec.

Having discussed the effects of Earth's atmosphere that give rise to errors in relative astrometry observations, we will now discuss errors due to (1) instabilities in the optical components of a telescope and (2) measurement devices. Because the main focus of Project Orion was on astrometric detection of extrasolar planets, the discussion here is limited to astrometric telescopes.

If a refracting telescope is used for astrometry, very small lateral shifts of lenses and changes in their separation may cause appreciable changes in lateral chromatic aberration. The resulting shifts of star images depend on the color of a star. Effects of this kind, amounting [31] to 0.03 arcsec (ref. 8), led to spurious data concerning the motion of Barnard's star.

The color effects mentioned above are absent in reflecting telescopes. Among the aberrations of reflecting telescopes, coma is most harmful for precise astrometry. Born and Wolf (ref. 9) state, in their classical treatise on optics: "Because of the asymmetrical appearance of an image in the presence of coma, this aberration must always be suppressed in telescopes, as it would make precise positional measurements impracticable." If a nonlinear detector, such as a photographic plate, is used, coma causes shifts of star images, depending on the brightness of a star and its color (since transmittance of Earth's atmosphere is color-dependent) and on the characteristics of the photographic plate. The relative positions of stars can appear to change slowly with time when measured on photographs taken year after year. These apparent changes could be caused by advances in photographic technology, by increasing air pollution, and by climatic changes, particularly those with periods of 1 year and 11 years (e.g., sunspot cycle).

Fortunately, coma of reflecting telescopes can be completely eliminated by using a secondary mirror of suitable shape. However, secondary mirrors are a source of astrometric or positional errors of another kind. Every star in the observed field in the sky illuminates a different area on a secondary mirror. Lateral shifts in position of a secondary mirror and slight changes in the shape of its reflecting surface, occurring over a time scale of years, cause astrometric errors. Such changes in shape may be caused by varying temperature, by relief of stresses in the glass and sagging of the mirror, or as a result of re-aluminizing the surface.

Suppose that a reflecting surface (e.g., a secondary mirror) which is approximately flat is placed between the entrance aperture of a telescope and the image plane, at a distance (1 - K)F from the former and KF from the latter. Here F is the focal length of a mirror or lens placed at the entrance aperture. It is assumed that temporal changes in shape of the reflecting surface remain correlated for points on this surface separated by a distance smaller than s, the correlation distance for that surface. Also assume that the rms relative shift perpendicular to the surface for two points separated by a distance larger than s is /100, where 0.5µm is the wavelength of visible light. The value /100 is the precision with which surface [32] deformations can be measured and hence calibrated out at the present state of the art.

On these assumptions, the tilt of a surface element of diameter s fluctuates with an rms deviation of /(100s) radians. If an angular separation between the star under study (the program star) and a reference star is radians, the centers of areas on the approximately flat reflecting surface which are illuminated by these two stars are separated byand the diameters of these areas are d = KD, where D is the diameter of entrance aperture.

Consider two cases: The first case (ds) occurs when the reflecting surface (secondary mirror) is very close to the image plane. The rms tilt of the reflecting surface situated at distance KF from the image of a star shifts this image as if the star had moved in the sky by The change of angular distance (in radians) between two stars, that is, the astrometric error caused by tilts of corresponding elements of mirror surface, is

(21)

where

(22)

In the second case, d > s, so that the average number of randomly tilted areas on the secondary mirror illuminated by each star is d²/s² Therefore, the astrometric error is smaller than that in the first case, by a factor of (d²/s²)^1/2 = KD/s. This error equals

(23)

If s is smaller than both r and d, the astrometric error caused by surface instability is given (in radians) by

(24)

depending only on the diameter D of the telescope. As an example, for D = 1.5 m (Flagstaff astrometric telescope), an error [33]arcsec is produced. This error can be considered as a systematic error slowly changing with time.

If astrometry is based on direct imaging of stars, without interferometry, the position of a star is defined by the position of a centroid of its seeing or diffraction disk. The position of a star becomes independent of the location of this centroid if a Michelson stellar interferometer is used. In the latter case, the position of a star is defined by the location of a white fringe produced by the interferometer, as shown schematically in figure 9. An aberration such as coma affects the white fringe location, but can be calibrated in an interferometer.

A telescope becomes a Michelson stellar interferometer if, instead of its entire entrance aperture, only two portions of this aperture are utilized, each situated a distance B/2 from the center of aperture. The distance B is the baseline of the interferometer. Light from each of the small apertures will combine on the image plane to form an image of each star in the field of view. As a consequence of the nature of light, these images will contain interference patterns....

[34] ....comprised of colored, rainbow-like fringes. Only the central bright fringe is white, because it is common to light of all wavelengths. This white fringe is formed by those stellar light rays for which the optical paths through the two arms of the interferometer are exactly equal.

Assume that a secondary mirror is located a distance KF from an image plane. Each star illuminates an area of diameter d on the secondary mirror, and the centers of these areas are separated by a distance r. The rms relative shift perpendicular to the surface is again assumed to be /100 for two points on the mirror separated by a distance larger than s. Let s be smaller than r and d. If the path difference L₁-L₂ between the two arms of the interferometer changes randomly by for the one star and for another star, the astrometric error of the angular distance between the two stars is

(25)

On the assumptions that the rms of, or is, the astrometric error (in radians) is

(26)

For a given baseline B, the error is largest if s = d. In this case, for a baseline B = 50 m (adopted in ch. 4), the astrometric error caused by slow changes in mirror shape is

(27)

If a secondary mirror is cut into two halves, each corresponding to an arm of the interferometer, small relative tilts or shifts of these halves are inconsequential, causing only changes in scale of an image of the star field; the tilts are considered small if star images remain strongly overlapping. Equation (26) applies also to any tracking flat mirrors placed in front of an entrance aperture. In this case, d >> s, and therefore the astrometric error is very small.

[35] This discussion, and equations (24) and (26), indicate that an interferometer of long baseline B, as compared to a telescope of diameter D, is less sensitive, by a factor of 2^-1/2 B/D, to slow changes in the shape of its optical surfaces.

Astrometric errors can arise not only in the transmission of light by the optical system of a telescope, but also in the measurement of the parameters of interest. Several components of hardware fall under the rubric of "measuring devices," including the detection system and any postdetection, telescope-independent measuring machines. It will be useful to begin this section with some brief comments on detection systems, followed by comments on techniques 'or obtaining high precision in measuring angles.

The principal requirements of a detection system for high-precision astrometric observations are: sensitivity, dynamic range, geometric fidelity, and simultaneous detection of the star under study and its reference stars. The level of sensitivity required for planetary detection is not overly demanding in terms of limiting magnitude (m_v ~ +15). Photographic plates and photoelectric detectors can easily reach 15th magnitude, given sufficient integration time. However, as remarked in the section on atmospheric limitations to precision, it is desirable to observe in short exposures (<0.1 sec). Because of this short exposure time, the detector must be characterized by both high quantum efficiency and low noise. To obtain astrometric precision at the level required (<10^-4 arcsec) for a significant search for extrasolar planets, it is necessary that a reference system be used which contains many (10-20) stars (the requirement for many reference stars is discussed in detail in appendix B), and these stars must lie reasonably close (within ~1°) to the star under study. Allowing for the possibility that there may be reference stars as bright as m_v ~ +5 for a 15th-magnitude program star requires a detector with a dynamic range ~10⁴. The characteristic of "geometric fidelity" for a detector means simply that the detector does not introduce any shift in the apparent relative position of the stars (target plus reference).

[36] The tried-and-true detector of astronomy is the photographic plate, which provides for increased integration times over its forerunner, the human eye, which is capable only of integration times of a fraction of a second. However, photographic plates have a very low quantum efficiency (i.e., the fraction of incident photons that is detected), typically less than 1 percent. Photographic plates do not have a large dynamic range (about 100), and are highly nonlinear detectors over that dynamic range. Finally, the geometric fidelity of photographic plates is relatively poor in at least two crucial aspects. First, if a telescope has asymmetrical optical aberrations (e.g., coma), the images of each star in the field of view are distorted differently. As positional measurements from photographic plates are made by locating the centroid of the stellar images as they appear on the plate, this distortion leads to a positional error. The most precise machine for measuring positions on a photographic plate (ref. 10) is capable of locating centroid positions to within ~0.7 µm, which corresponds to an angular error of about 0.007 arcsec for a focal length of 20 m, a factor of about 70 times larger than the effect expected for the SPS. The second aspect of photographic plates that leads to positional error arises from distortion of the plate, both during the interval of time from detection to measurement and during storage subsequent to any remeasurement of a plate. It is very difficult to quantitatively assess the positional error attendant upon this type of geometric distortion. In light of the inherent limitations of the photographic plate (as applied to this problem), it is clear that a more modern, photoelectric detection system must be used.

Advances in technology related to detection of visual light have been significant over the past decade, leading to a plethora of devices such as charge coupled devices (CCD's) and charge injection crevices (CID's). No attempt is made here to provide a review of this new technology. Suffice it to say that these new detectors have low noise, high quantum efficiency (>80%), good dynamic range (although quoted dynamic ranges for CCD devices, for example, are not as large as 104 at present), and excellent geometric fidelity. The geometric fidelity of these detectors, which is crucial to a search for extrasolar planets, is aided by the fact that positional measurements can be made in situ and essentially in real time. A major drawback to these detectors relative to photographic plates is that they have limited archival ability: photographic plates properly cared for may be [37] re-examined decades after they have been exposed. However, this archival attribute is judged to be of secondary importance for the task considered by Project Orion.

In the final analysis, any attempt to detect and study extrasolar planetary systems by astrometric observations depends on the precision with which small angles can be measured. It is of little use to circumvent or minimize errors or uncertainties in determining the relative positions of stars due to the combined effects of Earth's atmosphere and telescope optics if the final step in the process introduces significant errors (as is the case with photographic plates and plate-measuring machines). Perhaps the most precise way to measure the relative position of two sources of light is by means of interferometry, where light from the objects under study is modulated by some type of obstructing edge. In concluding this discussion of the factors that led toward the design concept advanced in detail in chapter 4, it is useful to remark on three possible alternative techniques by which the interferometric properties of light might be used to provide high-precision positional data. These techniques involve (1) sequences of gratings, (2) gratings with lenses, and (3) delay lines.

Sequences of gratings- A familiar- device that is useful to define angles is a pair of picket fences, that is, two surfaces separated by distance a and obstructing lines in each surface separated by the semiregular distance d. Like a rule inscribed with periodic marks for convenience in length measures, a pair of picket fences is a device that defines periodic directions separated by angle d/a. Such a device is experimentally attractive because of nonsensitivity to misalignments. Nominally, the fence surfaces should be parallel, the laths in different surfaces should be parallel, and the viewing direction should be along the coincident normals to each fence. Let the corresponding tilt, skew, and slant error angles be t, s, and w, respectively. Then if attention is given to the local value of the distance a between the fences along the viewing direction, the periodic angle becomes (d/a)cos t ^. cos s ^. cos w. Moreover, when the concept is extended to the use of ruled transmission gratings in parallel light, we note that d can be an extremely precise length and that none of the aberrations apply that trouble angle measures made with the aid of a lens: coma, astigmatism, spherical aberration, or distortion.

[38] Most applications of this technique to date have involved monochromatic light (see refs. 11-17). It seems worthwhile, therefore, to note that broadband light can be used for angle measures if it is made periodic in wavelength by passing it through either another grating or a Fabry-Perot etalon. The resulting interferences are comparable to the white-light effects, called Brewster's fringes, that are used to adjust Fabry-Perot etalons to integer length ratios in measures of the meter in terms of light wavelengths.

The grating arrangement depicted in figure 10 leads to light of wavelength being diffracted by angle , where and j is an integer. The deflection of light at distance a is . As the wavelength is changed, the light energy that passes grating G2 tends to alternate between orders with change in deflection by distance d. Therefore, the light needs to be periodic with interval . Such a relation arises in a Fabry-Perot etalon of length L, where L is determined by

giving rise to

[39] The periodic energy distribution along grating G2 that is thus selected constitutes a self-imaging of grating G1, known as the Talbot effect (refs. 18 and 19), that persists even if the incident light is not collimated (ref. 20) because the reconstruction is holographic. Note thatfor first-order diffraction and that, if is small, then L can be so small that the Fabry-Perot can be constructed by thin-film techniques in many cases of interest.

A new technology relevant to these remarks is the advent of holographic phase gratings whose surface amplitude is chosen so that the energy in the undiffracted zeroth-order beam is vanishingly small tor a small range of visible wavelengths. The energy from one diffraction is mostly in orders of ±1. After light passes through two such gratings, the energy alternates mostly between orders zero and ±2. The latter orders are easy to eliminate and the zeroth order can be fully modulated. Furthermore, the gratings can be physically large (~0.2 m).

Gratings used with lenses- Ronchi (ref. 21) has reviewed the theoretical development and use of a single small grating near the focus of a lens, as depicted in figure 11. The plane grating at distance q from a focal plane of the lens in the light from a point source causes light to pass through points P₁ and P_-1 in addition to P₀ in the focal plane. At an observation screen at distance r beyond the focal plane, there are intensity oscillations (fringes) in the overlap region BE. The sources P_O and P₁ are separated by approximately v = q(/d). Such sources are coherent since they are derived from the same source. Therefore, the light fields reinforce at separations z on the observing screen, where Note that if r = q, the separation distance z equals the grating spacing, independent of light wavelength. The display seems to be a rectilinear projection of the grating onto the screen. But a wave-optical formulation is essential, as shown by a fringe spacing of z/2 in the region CD illuminated by P₁ and P_-1 because the interfering sources are separated by 2y. More generally, many spacings of the form z, z/2, z/3, etc., may be present if r/q is large and many orders interfere. Notably, the fringe spacing for all orders of interference becomes infinite as the grating moves into the focal plane. If the grating is behind the focal plane so that q in effect changes sign, sources P₁ and P_-1 become virtual.

Malacara and Cornejo (ref. 20) showed that, to obtain constructive interference for all orders on the observing screen, distance q ....

....should be an integer multiple of the Rayleigh distance Such a length emerges in Talbot self-imaging when the diffraction involves mainly zeroth-order transmission and weak ±1 orders.

The two methods discussed above provide alternative means by which light from a source can be manipulated so as to produce a well-defined interference pattern. The next step involves manipulation of such a fringe pattern to allow measurement of the separation between two light sources.

[41] A plane grating near the focal plane of a telescope can be used as a modulator of star images to measure distances in the focal plane. If two star images are separated by a distance (i + g)d, where i is an integer and 0g < 1, then a grating modulator produces two wave-forms that are similar, but one of which is delayed in time by the fraction g of a modulation cycle (see fig. 12). The grating motion can be Iinear or vibratory, depending on whether the grating perfection is locally accurate enough for the measurement purpose. Linear motion permits averaging over many grooves. The second coordinate can be obtained by rotating the grating by a known angle or using a grating with grooves in two directions. Note that when a position accuracy of radians is desired, the rotation angle between the modulation directions must be known to radians. If a mechanical rotation were planned, the observer could provide for repeatability of the angles by arranging to look through a two-direction grating at the rotated grating. The moiré fringes of nearly aligned grating grooves permit highly accurate rotation settings.

[42] Two principal approaches to the grating modulator emerge. A grating of good local perfection can be placed in the telescope focal plane, or two gratings can be used to synthesize a grating of good local perfection in the focal plane. The former might be a Ronchi ruling or holographic amplitude grating. If the grating spacing is less than the apparent star image diameter, there must be several detectors within the pupil. The synthesized grating could be obtained by placing two gratings symmetrically about the focal plane. Then the following pupil contains many fringes because of interferences between sources P_O, P₂, and (shown in fig. 11). If the many detectors, which are thus required in the pupil, are objectionable, one can form an image of these sources, block all but P_O, and put one or more detectors in the pupil that follows. Clearly, the designer of a grating modulator who uses the principles of Abbe imaging will have much flexibility in that design.

Delay lines- Miller (ref. 22) suggested in a context of small-angle measures that the angles might be measured with high precision in the geometry of figure 13. If mirrors of fixed separation are used....

Figure 13. Schematic representation of a stellar interferometer. Light collected by two apertures separated by a distance B has a time delay S/c, where S = B sin). This time delay due to longer path must be compensated by a delay in the detector.

[43] ....to collect light, then angles can be measured by introducing changes in a delay line. As usual, angles are inferred from ratios of lengths in a geometric construction. In recent years, the technology of infrared spectroscopy has developed precise and reliable delay lines.

It is important to realize that the geometry in figure 13 is completely equivalent to imaging with a lens. The delays appropriate to a lens arise from suitably placed reflecting or refracting bodies and tree-space propagation to the image point corresponding to each object point. But when the lens is uncomfortably large so that only small elements of the lens can be furnished, the necessary delays can be furnished by various means. The path lengths can be monitored by laser interferometers so that high accuracy is attained. The use of delay lines for high-precision relative astrometric angle measurement is currently under consideration by Currie and co-workers.

The use of any optical system to precisely measure angles between light sources involves the interference properties of light; the number of such systems that are adequately accurate is large. The essential factors controlling interferometer design are nicely stated by Ronchi (ref. 21): "Finally, fringes given by the grating interferometer bring one to the same conclusions that are reached with the use of any interferometer whatsoever: the difference between one interferometer and another in the testing of optical systems is reduced to a question of practicality, rapidity, and economy."

The fundamental aspects of the planetary detection problem, namely, the physical basis of the observable phenomena and the effects of Earth's atmosphere and telescope hardware on observations, have been discussed. The primary emphasis of that discussion was to focus on those factors which set the constraints that influence the formulation of design concepts. Primary consideration was given to those aspects of the detection problem that might affect the design of a ground-based astrometric telescope. The major conclusions, as they relate to defining the foundation of a design concept for such a telescope, are summarized below.

[44] 1. Astrophysical aspects

Expected angular displacement is small, beyond the measurement capability of the best current astrometric telescope

Anticipated observing time comparable to planetary orbital periods (long-term stability required)

2. Terrestrial aspects

Atmospheric thermal turbulence limits ground-based astrometry to a precision of10^-4 arcsec

Highest accuracy obtained from taking many independent, short (< 0.2 see) observations/night rather than con-tinuously integrating

Important length scale for thermal turbulence is about 50 - 100 m; therefore an aperture or baseline of at least this size is desirable

Highest accuracy is obtained from measurements that minimize refractive effects of the atmosphere (i.e., mea-surement in azimuth)

3. Hardware aspects

Optics

Reflecting optics preferred over refracting optics because of absence of chromatic effects

Reflecting optics subject to coma; however, coma can be eliminated by suitably configured secondary mirrors

Changes in position or figure of secondary optical elements give rise to astrometric errors; sensitivity of telescope to these errors varies inversely with aperture (or interferometer baseline)

Measuring devices

Requirements of low noise, high sensitivity, dynamic range, and geometric fidelity rule out photographic plates as detectors

Precise measurements of angle between two stars best obtained from measurements of the relative linear positions of white fringes produced by interfering light from each star

The findings listed here indicate that a promising design concept for a high-precision, ground-based astrometric telescope is one which [45] has (1) reflecting optics, (2) large aperture (to minimize sensitivity to astrometric errors arising from secondary mirrors and to minimize atmospheric effects from turbulence), and (3) "electric" detection of interference fringes from stars. One very important additional characteristic is that the telescope simultaneously image many stars ( 10 to 20). (This characteristic has not been discussed in detail in this chapter, but it is discussed later.) Briefly, a large number of stars is essential to obtaining enough conditional equations to determine parameters that permit modeling of both short- and long-term drifts in the data. Simultaneous imaging is essential to minimize variance of those parameters to the extent that they do not limit the attainable accuracy of the system.

A design concept which embodies the above elements is that of a long-baseline interferometer that simultaneously "images" the white light fringes of many stars. Such a concept is the one conceived and developed during Project Orion. Details of the Orion imaging stellar interferometer are given in chapter 4.

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