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Comparison of geometrical effects in radio and stellar occultations
ICAFWJS26,175-177
(1975)
Comparison of Geometrical Effects in Radio and Stellar Occultations W. B. HUBBARD Department of Planetary Sciencea, Lunar and Planetary Labwatory, University of Arizona, Tucson, Atizona 85721 Received April 24, 1975
An important difference between the radio and steller occultation techniques for probing planetary etmospheres is that EXcrucial equation relating refraction angle with ray impact parameter is solved in opposite directions in the two techniques. The solution is tolerant of small geometrical errors for stellar occultations but, unfortunately, not for radio occultations.
It has been claimed in the literature (Wasserman and Veverka, 1973) that the method used for inverting radio ocoultation data for planetary atmospheres is basically the same as that which is applied to stellar occultation data. While it is true that the two techniques have certain mathematical manipulations in common, in particular the use of the Abel integral transform, there is a crucial difference which must be noted in order to understand why radio occultation inversions can be highly sensitive to small changes in the geometry of the occultation while stellar occultation inversions are not. This paper explains the difference.
is significantly attenuated by scattering and absorption, whereas d(t) can only be inverted if the attenuation is produced solely by refractive defocusing of the rays. As is well known (Hubbard et al., 1972), we compute the change in impact parameter of the ray in the planet’s atmosphere from 6h = up
Copyright 0 1975 by Academic Press, Inc. AI1 rights of reproduction io any form reserved. Printed in Great Britain
I+,, $(t’) dt’,
(1)
where 6h is the change in impact parameter during time interval at, and vP is the velocity perpendicular to the limb. If we make some small error of d% in oomputing v,, it is clear that 6h will contain the same error. Then, we compute the change in refraction angle using the equation
FLUX INVERSIONS Stellar occultations give as data the function #(t), where + is the received stellar flux as a function of time, in units of the unocculted flux. In a radio occultation (Kliore, 1972), we have available not only +(t), but also f(t), where f is the frequency residual produced by refraction in the planet’s atmosphere. In contrast to stellar occultation, however, the radio occultation technique does not use d(t) as a principal source of data for inversion to find the atmospheric structure. This is because radio occultations of spacecraft normally probe atmospheric layers where the signal
s
68 = D-‘(v,
6t - ah),
(2)
where 68 is the change in angle of refraction during at, and D is the distance from the limb to the observer. Since 6h a vD, any error A in v, will also propagate into 68 by the same amount. The final step involves computing the refractivity v, which is proportional to vUD3/* (Hubbard et aE., 1972). Thus a small error in v,, will show up in v increased by only about 50%. In summary, the technique for inverting b(t) is very tolerant of small errors in the geometry of the occultation. 175
176
W. B. HUBBARD
in the deeper Jovian atmosphere by several hundred percent! In radio occultation inversions, the On the other hand, the geometry of the refractive angle 8 is computed flrst. The Beta Scorpii occultations by Jupiter inversion procedure is somewhat more (Hubbard et al., 1972); Veverka et al, 1974) oomplicated thsn in the stellar occultation is also subject to revision. The perpendicucase, but the essentials can be illustrated lar velocities used by Hubbard et a.!. were by using the result for an atmosphere with obtained from an astrometric analysis by oonstant scale height, Hubbard and van Flendern (1972). The latter analysis included as a free parameter 68 = ~~-1 hsf, (3) the oblateness of Jupiter, which was calcuwhere X is the radio wavelength and 68 and lated to be 0.060. Subsequent direct Sfare the increments in refraction angle and measurement of the oblateness of Jupiter frequency which occur during time interval by Pioneer 10 (Anderson et al., 1974) gave 8t (Kliore, 1972). Clearly, if v, is too large a value of 0.0647 at the 1bar level. If we by d%, 68 will be too small by d%. now fix the oblateness at the latter value We then calculs,teM by solving (2) in the and repeat Hubbard and van Flandern’s opposite direction : analysis, we find that the best-fit center of 6h=v,&DM Jupiter moves slightly south, such that V~ = vp 6t - Dv,-’ XSf, (4) is increased very slightly for all of the occultation events. The corrections are where D is now the distance from the limb summarized as follows. Immersion and to the spacecraft. emersion of /3ScoA at Johsnnesburg have Equation (4), unlike (2), can be highly V~ increased by 0.8% ; immersion and unforgiving of errors in vD. The reason is emersion of /3ScoA at N&i Tal have V~ that once 4 + 1, A changes very slowly increased by 0.2%. The immersion and with t, as can be seen from (1), i.e., emersion velocities of /3ScoC are changed &i/&t < vp. It follows that in this case, the by a negligible amount. We also checked second term in the right-hand side of (4) the revision to vp for /3ScoA at Boyden, nearly cancels the first term. An error A in using the published values of vp and time vp will act in one direction on the first term of half-intensity (Veverka et al., 1974). The and in the other direction on the second new lip is revised upward from the pubterm. Depending on the sign of A, 6h crtn lished value of 7.72kmlsec by 0.8% which be calculated to be many times larger than suggests a close relationship between the the correct value or even become reversed astronomic analysis of Veverka et al. and in sign. that of Hubbard and van Flendern. In summary, the technique for inverting It is clear from the discussion in this f(t) is highly intolerant of small errors in paper that such revisions in vp will affect either vupor f, once the true value of dhldt the temperatures in Jupiter’s atmosphere becomes much less than v~.This sensitivity derived from the ~SCO occultations by was apparently first noted by Hunten only a few degrees Kelvin. Unfortunately, (1975), and independently by Eshlemrtn similar revisions applied to the Pioneer (1975). lo/l1 occultations would be entirely nonnegligible. FREQUENCY INVERSIONS
CONCLUDING REMARKS
We can illustrate the point of the above discussion with two examples. The Pioneer
10111radio occultations at Jupiter (Kliore et d., 1975) have been reduced with correot values of v,, taking into account the oblateness of the planet (Hubbard et al., 1975). Typictally, a 5% reduction in v, has the effeot of reducing the inverted scale height
ACIINO’WLEDOMENT This work was supported by NASA Grant NSG-7045. REFERENCES ANDERSON,J. D., NULL, G. W., m WON& S. K. (1974). Gravity results from pioneer 10 Doppler data. J. Ueu@y~~Rea. 79,3661-3664.
RADIO mD
STELLAR
OCCULTATIONS
177
ESHLEMAN, V. R. (1975). Unpublishedmemoran- KLIORE, A. (1972). Current methods of radio dum to MJS and Pioneerprojects, April 10. occultation data inversion. In Mathematics of Profile Invertion (L. Colin, Ed.), p. 150, HUBBARD,W. B., HUNTEN,D. M., ANDKLIORE, A. (1975). Effect of the Jovian oblateness on Sections 3-2-3-16. NASA TMX-62. Pioneer 10/l 1 radio occultations. Geqphg8. KLIORE, A., FJELDBO, G., SEIDEL, B. L., Res. Letter8 2, 265-268. SESPLAUKIS, T. T., SWEETNAM,D. W., AND HUBBARD,W. B., NATHER,R. E., EVANS,D. S., WOICESHYN,P. M. (1975). Preliminaryresults TULL, R. G., WELLS, D. C., VAN CITTERS, on the atmosphereof Jupiter from the Pioneer 11 S-band occultation experiment. Science G. W., WARNER, B., AND VANDENBOUT, P. 188,474-476. (1972). The occultation of Beta Scorpii by Jupiter and 10. I. Jupiter. Astron.J. 77,41-59. VEVERKA,J., WASSERMAN,L. H., ELLZOT,J., SAGAN, C., AND LILLER, W. (1974). The HUBBARD,W. B., ANDVANFLANDERN, T. (1972). occultation of Beta Soorpiiby Jupiter. I. The The occultationof Beta Scorpiiby Jupiter and structure of the Jovian upper atmosphere. IO. III. Astrometry. Astron. J. 77, 65-74. fistron. J. 79, 73-84. HUNTEN,D. M. (1975). Remarks on the radiooccultation measurement of Jupiter. Collo- WASSER~, L., ANDVEVERKA,J. (1973). On the reduction of occultation light cxuves. Ic~ross quium presented at University of Arizona, 29, 322-345. March 18.
(1975)
Comparison of Geometrical Effects in Radio and Stellar Occultations W. B. HUBBARD Department of Planetary Sciencea, Lunar and Planetary Labwatory, University of Arizona, Tucson, Atizona 85721 Received April 24, 1975
An important difference between the radio and steller occultation techniques for probing planetary etmospheres is that EXcrucial equation relating refraction angle with ray impact parameter is solved in opposite directions in the two techniques. The solution is tolerant of small geometrical errors for stellar occultations but, unfortunately, not for radio occultations.
It has been claimed in the literature (Wasserman and Veverka, 1973) that the method used for inverting radio ocoultation data for planetary atmospheres is basically the same as that which is applied to stellar occultation data. While it is true that the two techniques have certain mathematical manipulations in common, in particular the use of the Abel integral transform, there is a crucial difference which must be noted in order to understand why radio occultation inversions can be highly sensitive to small changes in the geometry of the occultation while stellar occultation inversions are not. This paper explains the difference.
is significantly attenuated by scattering and absorption, whereas d(t) can only be inverted if the attenuation is produced solely by refractive defocusing of the rays. As is well known (Hubbard et al., 1972), we compute the change in impact parameter of the ray in the planet’s atmosphere from 6h = up
Copyright 0 1975 by Academic Press, Inc. AI1 rights of reproduction io any form reserved. Printed in Great Britain
I+,, $(t’) dt’,
(1)
where 6h is the change in impact parameter during time interval at, and vP is the velocity perpendicular to the limb. If we make some small error of d% in oomputing v,, it is clear that 6h will contain the same error. Then, we compute the change in refraction angle using the equation
FLUX INVERSIONS Stellar occultations give as data the function #(t), where + is the received stellar flux as a function of time, in units of the unocculted flux. In a radio occultation (Kliore, 1972), we have available not only +(t), but also f(t), where f is the frequency residual produced by refraction in the planet’s atmosphere. In contrast to stellar occultation, however, the radio occultation technique does not use d(t) as a principal source of data for inversion to find the atmospheric structure. This is because radio occultations of spacecraft normally probe atmospheric layers where the signal
s
68 = D-‘(v,
6t - ah),
(2)
where 68 is the change in angle of refraction during at, and D is the distance from the limb to the observer. Since 6h a vD, any error A in v, will also propagate into 68 by the same amount. The final step involves computing the refractivity v, which is proportional to vUD3/* (Hubbard et aE., 1972). Thus a small error in v,, will show up in v increased by only about 50%. In summary, the technique for inverting b(t) is very tolerant of small errors in the geometry of the occultation. 175
176
W. B. HUBBARD
in the deeper Jovian atmosphere by several hundred percent! In radio occultation inversions, the On the other hand, the geometry of the refractive angle 8 is computed flrst. The Beta Scorpii occultations by Jupiter inversion procedure is somewhat more (Hubbard et al., 1972); Veverka et al, 1974) oomplicated thsn in the stellar occultation is also subject to revision. The perpendicucase, but the essentials can be illustrated lar velocities used by Hubbard et a.!. were by using the result for an atmosphere with obtained from an astrometric analysis by oonstant scale height, Hubbard and van Flendern (1972). The latter analysis included as a free parameter 68 = ~~-1 hsf, (3) the oblateness of Jupiter, which was calcuwhere X is the radio wavelength and 68 and lated to be 0.060. Subsequent direct Sfare the increments in refraction angle and measurement of the oblateness of Jupiter frequency which occur during time interval by Pioneer 10 (Anderson et al., 1974) gave 8t (Kliore, 1972). Clearly, if v, is too large a value of 0.0647 at the 1bar level. If we by d%, 68 will be too small by d%. now fix the oblateness at the latter value We then calculs,teM by solving (2) in the and repeat Hubbard and van Flandern’s opposite direction : analysis, we find that the best-fit center of 6h=v,&DM Jupiter moves slightly south, such that V~ = vp 6t - Dv,-’ XSf, (4) is increased very slightly for all of the occultation events. The corrections are where D is now the distance from the limb summarized as follows. Immersion and to the spacecraft. emersion of /3ScoA at Johsnnesburg have Equation (4), unlike (2), can be highly V~ increased by 0.8% ; immersion and unforgiving of errors in vD. The reason is emersion of /3ScoA at N&i Tal have V~ that once 4 + 1, A changes very slowly increased by 0.2%. The immersion and with t, as can be seen from (1), i.e., emersion velocities of /3ScoC are changed &i/&t < vp. It follows that in this case, the by a negligible amount. We also checked second term in the right-hand side of (4) the revision to vp for /3ScoA at Boyden, nearly cancels the first term. An error A in using the published values of vp and time vp will act in one direction on the first term of half-intensity (Veverka et al., 1974). The and in the other direction on the second new lip is revised upward from the pubterm. Depending on the sign of A, 6h crtn lished value of 7.72kmlsec by 0.8% which be calculated to be many times larger than suggests a close relationship between the the correct value or even become reversed astronomic analysis of Veverka et al. and in sign. that of Hubbard and van Flendern. In summary, the technique for inverting It is clear from the discussion in this f(t) is highly intolerant of small errors in paper that such revisions in vp will affect either vupor f, once the true value of dhldt the temperatures in Jupiter’s atmosphere becomes much less than v~.This sensitivity derived from the ~SCO occultations by was apparently first noted by Hunten only a few degrees Kelvin. Unfortunately, (1975), and independently by Eshlemrtn similar revisions applied to the Pioneer (1975). lo/l1 occultations would be entirely nonnegligible. FREQUENCY INVERSIONS
CONCLUDING REMARKS
We can illustrate the point of the above discussion with two examples. The Pioneer
10111radio occultations at Jupiter (Kliore et d., 1975) have been reduced with correot values of v,, taking into account the oblateness of the planet (Hubbard et al., 1975). Typictally, a 5% reduction in v, has the effeot of reducing the inverted scale height
ACIINO’WLEDOMENT This work was supported by NASA Grant NSG-7045. REFERENCES ANDERSON,J. D., NULL, G. W., m WON& S. K. (1974). Gravity results from pioneer 10 Doppler data. J. Ueu@y~~Rea. 79,3661-3664.
RADIO mD
STELLAR
OCCULTATIONS
177
ESHLEMAN, V. R. (1975). Unpublishedmemoran- KLIORE, A. (1972). Current methods of radio dum to MJS and Pioneerprojects, April 10. occultation data inversion. In Mathematics of Profile Invertion (L. Colin, Ed.), p. 150, HUBBARD,W. B., HUNTEN,D. M., ANDKLIORE, A. (1975). Effect of the Jovian oblateness on Sections 3-2-3-16. NASA TMX-62. Pioneer 10/l 1 radio occultations. Geqphg8. KLIORE, A., FJELDBO, G., SEIDEL, B. L., Res. Letter8 2, 265-268. SESPLAUKIS, T. T., SWEETNAM,D. W., AND HUBBARD,W. B., NATHER,R. E., EVANS,D. S., WOICESHYN,P. M. (1975). Preliminaryresults TULL, R. G., WELLS, D. C., VAN CITTERS, on the atmosphereof Jupiter from the Pioneer 11 S-band occultation experiment. Science G. W., WARNER, B., AND VANDENBOUT, P. 188,474-476. (1972). The occultation of Beta Scorpii by Jupiter and 10. I. Jupiter. Astron.J. 77,41-59. VEVERKA,J., WASSERMAN,L. H., ELLZOT,J., SAGAN, C., AND LILLER, W. (1974). The HUBBARD,W. B., ANDVANFLANDERN, T. (1972). occultation of Beta Soorpiiby Jupiter. I. The The occultationof Beta Scorpiiby Jupiter and structure of the Jovian upper atmosphere. IO. III. Astrometry. Astron. J. 77, 65-74. fistron. J. 79, 73-84. HUNTEN,D. M. (1975). Remarks on the radiooccultation measurement of Jupiter. Collo- WASSER~, L., ANDVEVERKA,J. (1973). On the reduction of occultation light cxuves. Ic~ross quium presented at University of Arizona, 29, 322-345. March 18.