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Appendix A The Large Star Sample Required for A Photometric Planetary Search Consider a planet In a circular orbit, of radius a, around a main- sequence star of mass Ma and radius R*. An empirical relation between stellar size and mass is given approximately by Rut ~ M*0 78. The amplitude of an occultation is just the ratio of the cross-sectional areas, or A = 1.3p-2/3Mp 213 M -~.s6 where Mp and M. are in solar units and p is the mean density of the planet, which could be somewhere in the range O.S < p < 8 g cm~3. A minimal detection of the occultation requires that A 3~, where ~ is the photometric fractional uncertainty in the measurements. In the case of the Sun, this detection would require ~ = 3 x 1O-s for an Earth occultation up = s.s Mp = 3 X 10-6) and a = 3 x 10-3 for a Jupiter occultation (p1.3, Mp = 10-3~. In the fo310~g, p = Z1 is used, which yields the geometric mean value for crtp); the above range of values for p would yield values for ~ within a factor of 2.6. The corresponding required photometric uncertainty is ~ ~ 0.26Mp 213M* _ i.56 If Poisson statistics apply, recording the event requires that 1/~2 pho- toelectrons be counted. Assuming measurements in a 1oo~A visible pass- band with a 1-m telescope operating at 5 percent quantum efficiency, the on-target integration time required per star is 83
84 t = 3 x 10~9~~210° Seconds where ma is the apparent visual magnitude of the star. For main-sequence stars, the apparent magnitude can be reckoned from the mass and distance: me ~ - 8.4 log M* + 5 log r + 1, and so t = 10~7r2M ~0 24Mp ~4/3seconds For occultations of a 0.3-M star by the Earth and Jupiter viewed from 10 parsecs, t = 300 s and 0.13 s, respectively. The maximum duration of an occultation is 0.54 ai/2 M*0 28 days, where a is in astronomical units, and it occurs once in an orbital period P = a3/2 M*-~/2 yr. In order not to miss an event, each program star must be measured about four times dunag fiche minimum event duration sought. For estimation purposes, the limiting planet path is chosen to be offset by 0.5 stellar radii from the starts center. Then all the program stars must be measured In time: T = 0.25~/~) 0.54 amin i/2M0 28 days _ 1O4amin i/2M* 0 28 seconds, where awn is the inner boundary of the search. For amen = 1 AU and M* = 0.3,T=7.1 x 103sor2h. The occultation visibility zone decreases with increasing a. The prob- abilibr of the planet passing within 0.5 stellar radius of the star's center, as seen by an observer in a random direction, is Rota/2a or 2.33 x 10-3 M*0 78 am. Ib be confident that N. stars have been sampled for planetary occultations, that number times the inverse of the probability of visibility must be observed in the monitoring program, or No = N`, x 43Oamal`M* 0 78, where amen is the outer boundary of me search. For arm1 AU and M* = 0.3, No = 1.1 x 103 Nil. The survey program must therefore measure No stars, each for an integration time t, in a total Cycle time T. Assuming an observational efflcienc y of 0.25, this requires 4Not < T. For a Apical star (M* = 0.3) and a program duration of 10 yr (P < 10, am = 3.1) this requires
85 amin > 7.3 x 10~10Mp~~/3r2 . For the extreme case one takes Ash = aim,` and Mp = Mp(~max) = 2 x 10-3. For a minimum N. of 10, this requires r < 13 parsecs and No=34X 103N, =3.4x 104 stars. However, there are only about 1000 stars within 13 parsecs, so this method could not give an accurate estimate of the fraction of even Jupiter-size planets occurring around nearby stars.