Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.gao.spb.ru/english/personal/malkin/publ/zm09f.pdf Äàòà èçìåíåíèÿ: Tue May 11 13:11:03 2010 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 03:10:57 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: jupiter

ISSN 0038-0946, Solar System Research, 2009, Vol. 43, No. 4, pp. 313­318. © Pleiades Publishing, Inc., 2009. Original Russian Text © Z.M. Malkin, V.N. L'vov, S.D. Tsekmeister, 2009, published in Astronomicheskii Vestnik, 2009, Vol. 43, No. 4, pp. 327­332.

Forthcoming Close Angular Approaches of Planets to Radio Sources and Possibilities to Use Them as GR Tests
Z. M. Malkin, V. N. L'vov, and S. D. Tsekmeister
Center Astromical Observatory at Pulkovo of the Russian Academy of Scienced, Pulkovskoe sh. 65, St. Petersburg, 196140 Russia
Received October 30, 2008

Abstract--During close angular approaches of solar system planets to astrometric radio sources, the apparent positions of these sources shift due to relativistic effects and, thus, these events may be used for testing the theory of general relativity; this fact was successfully demonstrated in the experiments on the measurements of radio source position shifts during the approaches of Jupiter carried out in 1988 and 2002. An analysis, performed within the frames of the present work, showed that when a source is observed near a planet's disk edge, i.e., practically in the case of occultation, the current experimental accuracy makes it possible to measure the relativistic effects for all planets. However, radio occultations are fairly rare events. At the same time, only Jupiter and Saturn provide noticeable relativistic effects approaching the radio sources at angular distances of about a few planet radii. Our analysis resulted in the creation of a catalog of forthcoming occultations and approaches of planets to astrometric radio sources for the time period of 2008­2050, which can be used for planning experiments on testing gravity theories and other purposes. For all events included in the catalog, the main relativistic effects are calculated both for ground-based and space (Earth­Moon) interferometer baselines. PACS numbers: 04.80.Cc, 95.10.Gi, 95.75.Kk DOI: 10.1134/S0038094609040054

INTRODUCTION Apart from the theory of general relativity (GR), generally accepted among physicists and astronomers, alternative theories of gravitation have been suggested. The most serious of them do not contradict the available observational data, but predict deviations from GR for circumstances that have not been observed so far or for accuracies higher than those that have been already achieved. Therefore, the testing of gravity theories on the basis of different methods and more and more precise observations is an urgent astronomical and physical problem. Among the proposed GR tests are the Very Long Baseline Interferometry (VLBI) observations of radio sources at the time instants when solar system planets closely approach them (Treuhaft and Lowe, 1991; Kopeikin, 2001; Fomalont and Kopeikin, 2003, 2008). In particular, these papers present the observations of relativistic time delays of radio source signals at the instances of close approaches of Jupiter in 1988 and 2002. Similar observations of Jupiter and Saturn are planned at O.A. Titov's proposal by the International VLBI Service for Geodesy and Astrometry (SchlÝter and Behrend, 2007) in 2008­2009 (http://ivscc.gsfs.nasa.gov/program/opc.html). However, the reduction of already available experimental data showed that some formulated problems still have no unambiguous answer, in particular, because the achieved accuracy proved to be insufficient, mostly as a result of the fairly large angular distances between the radio sources

and Jupiter. Therefore, further observations are necessary, which should be performed in the most favorable conditions, first of all, during the closest approaches (further, we imply apparent approaches) and occultations. Note that so far the observations have been carried out in an irregular way, i.e., an experimenter used the closest in time approach of a planet to a known astrometric radio source. Another researcher who decided to perform a similar experiment has to search for forthcoming approaches himself. In this situation, the most interesting phenomena, i.e., the closest angular approaches, may be missed. In addition, the observations of relativistic effects caused by light deflections and gravitational time delays near a planet require significant, usually international, resources and should be planned well in advance. Therefore, the possibility of early planning of the experiments that can be performed in the most favorable conditions is highly desirable. In the present work, we tried to create a catalog of approaches of solar system planets to astrometric radio sources for the convenient and reliable planning of observations. During the work, we found that two characteristics of such experiments, which are currently treated as obvious, are wrong. First, we found that such events occur much more often than it had been considered before. Second, one of the most interesting relativistic effects related to the speed of gravity propagation proved to be observable not only in the case of giant planets, but in the case of all other planets, including

313


314

MALKIN et al.

Table 1. The maximum values of relativistic effects for the case of observations at the edges of a planet's disks, ns Baseline, thousand km 8 Effect P 400 P Mercury 0.01 0.15 0.5 7.3 Venus 0.06 0.32 3.2 16 Mars 0.02 0.13 0.75 6.7 Jupiter 2.1 1.2 110 62 Saturn 0.77 0.68 39 34 Uranus 0.27 0.73 14 37 Neptune 0.33 1.14 16 57 Pluto 0.00 0.09 0.05 4.3

Table 2. The values of relativistic effects for a 30-arcsec angular distance between the planet center and the source, ns Baseline, thousand km 8 400 Effect P P Mercury 0.00 0.00 0.04 0.05 Venus 0.01 0.01 0.51 0.41 Mars 0.00 0.00 0.05 0.03 Jupiter 1.1 0.33 56 17 Saturn 0.20 0.04 9.8 2.2 Uranus 0.02 0.00 0.78 0.12 Neptune 0.01 0.00 0.60 0.08 Pluto 0.00 0.00 0.00 0.00

Pluto. In addition, the development of space technology makes it possible to perform VLBI experiments with baselines of several hundred thousand kilometers; with these baselines, the main relativistic effects can be observed in the case of all planets, including the largest among the minor planets. OBSERVABLE EFFECTS When a planet approaches a radio source, one can observe two relativistic effects which contribute into the measured interferometric delay, defined as the difference between the instants of electromagnetic wave front arrivals to two VLBI antennas. These effects are the Shapiro time delay and the delay of gravity propagation P caused by the ray propagation near a moving body, i.e., near a planet in our case. Kopeikin (2001) showed that their measurement makes it possible to determine two fundamental relativistic parameters: , which is equal to unity in GR, and the parameter of gravity propagation , which is equal to zero in GR, i.e., the gravity speed is considered to be equal to the speed of light. The values of these effects can be estimated from the formulas derived by Kopeikin (2001; Eq. 13), which after the obvious transformations take the form 2 ( 1 + ) GMrB ------------------------------------ , 3 c Rd v P ( 1 + ) ------- , cd (1)

this case, Eq. (1) within the frames of GR can be rewritten in the form 4 GMB ---------------- , 3 cR v P ------- . cr (2)

Equation (2) shows that at the edge of the disk, does not depend on the apparent size of the disk, i.e., on the planet distance, and P reaches its maximum when the distance between the Earth and the planet becomes the largest. The maximum values of the observable relativistic effects are present in Table 1 for ground-based and space-based (Earth­Moon) interferometer baselines. Taking into account the linear dependence of the considered effects on B, these values can be easily recalculated for any baseline. Note that the relativistic effects that are of the order of P can be measured using the simplest interferometric technique of the group delay measurements. At the same time, the effects of the order of P require differential phase measurements in carefully planned observations. One can reasonably expect a significant increase of the accuracy of relativistic effect measurements with setting in operation the new VLBI 2010 stations (Behrend et al., 2008). Calculations show that nearly all planets during their closest approaches to radio sources provide relativistic effects, especially for P, that can be observed even with ground-based interferometers. However, with an increase of the angular distances between the planets' centers and radio sources to 30 arcsec, the relativistic effects become so weak that ground-based interferometers can produce sufficient accuracy only for Jupiter and Saturn (Table 2).
SOLAR SYSTEM RESEARCH Vol. 43 No. 4 2009

where GM is the planetocentric gravitational constant, B is the interferometer baseline, r is the apparent angular radius of the planet, d is the angular distance between the source and the planet center for a station, R is the planet radius, v is its orbital speed, and ð is the speed of light. One can see that and P achieve their maxima at the edge of the planet's disk, i.e., when d = r. In


FORTHCOMING CLOSE ANGULAR APPROACHES OF PLANETS Table 3. Occultations of astrometric radio sources by planets Planet Venus Mars Venus Venus Venus Venus Jupiter Saturn Jupiter Venus Venus Jupiter Venus Venus Jupiter Venus Venus Date, y m d 2011 2011 2012 2015 2020 2020 2025 2028 2033 2035 2037 2043 2043 2043 2045 2049 2049 02 05 12 08 01 07 09 10 02 07 01 02 02 02 09 01 11 26.6 03.8 24.4 06.8 16.7 17.7 18.6 24.8 04.2 03.3 03.8 01.1 15.6 17.7 24.4 13.5 02.2 Source 1946­200 0127+084 1631­208 0947+064 2220­119 0446+178 0725+219 0223+113 2104­173 0558+234 1734­228 1734­228 1858­212 1908­211 2221­116 2243­081 1333­082 19 1 16 9 22 4 7 2 21 6 17 17 19 19 22 22 13 Source and , J2000 h min s 49 30 34 50 22 49 28 25 07 01 37 37 01 11 24 45 36 53 28 30 03 56 13 21 42 27 47 02 02 04 54 08 49 08 deg arcmin arcsec ­19 +8 ­20 +6 ­11 +17 +21 +11 ­17 +23 ­22 ­22 ­21 ­21 ­11 ­7 ­8 57 42 58 15 44 54 53 34 08 24 51 51 12 02 26 55 29 13 46 26 04 26 32 06 25 10 53 55 55 01 44 21 19 52 S. America, Australia, Antarctica America Africa, S. America, Antarctica America Europe, Africa, S. America America America, Antarctica Annular occultation, Asia, Africa, Europe S. America, Australia, Antarctica Europe, Asia, Africa S. America, Australia, Antarctica Antarctica, Asia, Africa, Antarctica America America, Australia S. America, Australia, Antarctica Asia, Europe, Africa Africa, Antarctica Notes, seeing

315

PRECOMPUTATION OF OCCULTATIONS AND APPROACHES On the basis of the results presented in the previous section, one can conclude that the instants of occultations should be precomputated for all planets from Venus to Neptune, and for Jupiter and Saturn the circumstances of the approaches should be precomputated in more detail. In this case, all of the approaches of planets to radio sources that provide the most noticeable relativistic effects of signal propagation will be taken into account. Their amount makes it possible to plan experiments not involving Mercury, Mars, and minor planets, for these planets' relativistic effects can be measured, but with fairly large relative errors. However, the data omitted in this paper can be easily supplied by the authors at the request of interested parties. Most computations of the circumstances of planet approaches to radio sources were performed using the codes APPROACH and OCCULT, which utilize the Ephemeride Package for Objects of the Solar System (EPOS; L'vov et al., 2001) data and environment. Source coordinates were taken from the Goddard center of space flight's catalog (Petrov, 2008), adding sources from the ICRF-2 catalog (Fey et al., 2004). The total number of sources proved to be 3958; their list and optical characteristics are available at the website http://www. gao.spb.ru/english/as/ac_vlbi/sou_car.dat. The list of source occultations by planets is presented in Table 3, and the circumstances of Jupiter and
SOLAR SYSTEM RESEARCH Vol. 43 No. 4 2009

Saturn approaches to radio sources are given in Tables 4 and 5, respectively. The tables exhibit the circumstances of all occultations and approaches closer than 10 over the time interval from September 2008 (the time of writing this paper) to 2050. There is no occultation for Uranus and Neptune over this time interval. An interesting feature of the list is the presence of multiple approaches due to the retrograde apparent motion of the planets. In this case, a planet approaches a radio source from different directions, which may have a particular interest for studying the influence of a moving planet on the signal delay (P term). The apparent angular diameters of planets in Tables 4 and 5 are calculated from their mean radii. The and P values are calculated for a 8000-km long baseline. In this case, we used an equation which obviously follows from Eq. 1 (GR case) 4 GMB = ---------------- , 3 c Dd (3)

where D is the distance between the Earth and the planet. Data for other baselines, including space-based ones, can be easily obtained by proportionally recalculating the present values. The results of the experiment performed in 2002 (Fomalont and Kopeikin, 2003) showed that the relativistic time delay P 6 ps (more precisely, an equivalent light deflection of 51 µs) proved to be measurable with an accuracy of 20% using the VLBA interferometer supplemented by the 100-


316

MALKIN et al.

Table 4. Apparent approaches of Jupiter to astrometric radio sources Date, y m d 2008 2009 2011 2011 2012 2012 2013 2013 2013 2013 2017 2019 2020 2020 2021 2022 2022 2023 2023 2024 2025 2025 2025 2029 2029 2031 2031 2031 2033 2034 2035 2035 2037 2037 2037 2041 2045 2045 2045 2045 2045 2046 2047 2047 2049 2049 11 03 07 09 02 02 02 10 11 11 10 10 08 10 02 11 12 06 11 01 09 10 11 03 09 02 06 10 02 01 05 05 05 08 09 09 01 02 05 09 12 01 04 05 05 08 19.0 08.6 03.6 13.1 04.0 20.3 28.1 23.0 07.0 22.1 13.7 28.4 02.0 24.2 19.9 13.8 04.1 11.1 05.4 02.1 15.4 25.0 29.1 15.3 28.5 23.2 07.1 05.6 27.2 28.9 14.0 24.1 28.4 27.9 19.0 11.6 20.1 12.0 29.4 20.3 04.5 10.7 28.4 08.3 11.4 29.5 Source 1922 2104 0210 0229 0201 0210 0420 0723 0725 0723 1352 1723 1922 1922 2104 2354 2354 0210 0229 0210 0723 0741 0741 1333 1352 1734 1734 1723 2126 2245 0201 0210 0558 0725 0741 1352 2104 2126 2245 2223 2223 2245 0201 0210 0558 0741 ­ ­ + + + + + + + + ­ ­ ­ ­ ­ ­ ­ + + + + + + ­ ­ ­ ­ ­ ­ ­ + + + + + ­ ­ ­ ­ ­ ­ ­ + + + + 224 173 119 131 113 119 210 219 219 219 104 229 224 224 173 021 021 119 131 119 219 214 214 082 104 228 228 229 158 091 113 119 234 219 214 104 173 158 091 114 114 091 113 119 234 214 Source and , J2000 h min s 19 21 2 2 2 2 4 7 7 7 13 17 19 19 21 23 23 2 2 2 7 7 7 13 13 17 17 17 21 22 2 2 6 7 7 13 21 21 22 22 22 22 2 2 6 7 25 07 13 31 03 13 23 26 28 26 54 26 25 25 07 57 57 13 31 13 26 44 44 36 54 37 37 26 29 47 03 13 01 28 44 54 07 29 47 25 25 47 03 13 01 44 40 27 05 46 47 05 02 14 21 14 47 59 40 40 27 25 25 05 46 05 14 47 47 08 47 02 02 59 12 52 47 05 47 21 47 47 27 12 52 44 44 52 47 05 47 47 deg arcmin arcsec ­22 ­17 +12 +13 +11 +12 +21 +21 +21 +21 ­10 ­22 ­22 ­22 ­17 ­1 ­1 +12 +13 +12 +21 +21 +21 ­8 ­10 ­22 ­22 ­22 ­15 ­8 +11 +12 +23 +21 +21 ­10 ­17 ­15 ­8 ­11 ­11 ­8 +11 +12 +23 +21 19 08 13 22 34 13 08 53 53 53 41 58 19 19 08 52 52 13 22 13 53 20 20 29 41 51 51 58 38 50 34 13 24 53 20 41 08 38 50 13 13 50 34 13 24 20 35 10 11 55 45 11 02 20 06 20 03 02 35 35 10 16 16 11 55 11 20 00 00 52 03 55 55 02 41 22 45 11 53 06 00 03 10 41 22 41 41 22 45 11 53 00 d , arcsec r, arcsec 83 277 341 149 490 342 216 123 388 351 69 184 79 355 149 159 177 28 199 396 215 30 274 432 47 261 55 312 417 342 433 173 306 159 29 74 192 283 459 228 466 83 294 308 129 179 17 16 18 23 19 18 19 20 21 21 15 16 23 18 16 22 21 17 24 21 17 19 22 21 15 17 23 18 16 17 16 17 16 16 17 16 16 16 19 24 19 17 16 16 16 16
Vol. 43

, ps 443 127 116 328 83 114 191 343 114 132 471 192 631 112 232 304 257 1321 261 117 174 1374 169 105 704 142 877 121 83 105 81 206 112 222 1271 455 180 121 91 224 89 449 118 113 272 196
No. 4

P, ps 48 4.1 3.0 20 1.5 3.0 8.0 25 2.6 3.4 62 9.4 72 2.8 14 17 13 426 12 2.6 7.3 406 5.5 2.2 136 4.9 143 3.5 1.8 2.8 1.7 11 3.3 13 391 55 8.4 3.8 1.8 8.9 1.7 48 3.6 3.3 19 9.9
2009

SOLAR SYSTEM RESEARCH


FORTHCOMING CLOSE ANGULAR APPROACHES OF PLANETS Table 5. Apparent approaches of Saturn to astrometric radio sources Source and , J2000 Date, y m d Source h min s 2009 02 10.2 2009 06 26.0 2015 06 19.1 2015 11 19.1 2016 11 22.9 2017 12 13.3 2021 08 10.8 2021 12 08.1 2023 04 13.3 2023 04 18.2 2024 01 04.6 2024 03 18.5 2026 04 01.5 2026 10 19.0 2028 05 20.6 2030 11 31.0 2032 04 03.5 2033 05 24.2 2034 06 15.7 2034 07 16.2 2037 01 16.1 2037 07 24.1 2043 10 18.4 2044 02 27.6 2045 09 20.4 2046 09 17.5 2047 10 17.1 2048 11 28.4 1125 + 062 1109 + 076 1548 ­ 177 1614 ­ 195 1658 ­ 217 1752 ­ 225 2044 ­ 188 2044 ­ 188 2221 ­ 116 2223 ­ 114 0220 ­ 119 2252 ­ 090 0019 ­ 001 0037 + 011 0208 + 106 0409 + 188 0503 + 216 0620 + 227 0725 + 219 0741 + 214 1013 + 127 1013 + 127 1459 ­ 149 1548 ­ 177 1614 ­ 195 1658 ­ 217 1752 ­ 225 1853 ­ 226 11 27 37 11 12 10 15 51 15 16 17 27 17 02 10 17 55 26 20 47 38 20 47 38 22 24 08 22 25 44 2 13 05 22 55 04 0 22 25 0 40 14 2 11 13 4 12 46 5 06 34 6 23 18 7 28 21 7 44 47 10 15 44 10 15 44 15 02 25 15 51 15 16 17 27 17 02 10 17 55 26 18 56 36
Vol. 43 No. 4 2009

317

d, arcsec r, arcsec deg arcmin arcsec +5 55 32 + 7 24 49 ­17 55 02 ­19 41 32 ­21 30 03 ­22 32 11 ­18 41 41 ­18 41 41 ­11 26 21 ­11 13 41 +12 13 11 ­08 44 04 + 0 14 56 + 1 25 46 +10 51 35 +18 56 37 +21 41 00 +22 41 36 +21 53 06 +21 20 00 +12 27 07 +12 27 07 ­15 08 53 ­17 55 02 ­19 41 32 ­21 30 03 ­22 32 11 ­22 36 17 80 146 156 64 194 73 20 114 33 276 370 158 472 145 79 306 71 206 38 157 72 233 220 33 46 51 367 321 9 8 9 7 7 7 9 8 8 8 8 8 8 9 8 10 9 8 8 8 10 8 7 8 8 8 8 7

, ps

P, ps

92 44 44 88 29 78 347 52 183 22 16 37 13 51 77 25 93 31 163 39 103 26 26 193 132 120 16 18

7.7 2.0 1.9 9.1 1.0 7.1 115 3 37 0.5 0.3 1.5 0.2 2.3 6.5 0.5 8.8 1.0 28 1.7 9.6 0.8 0.8 39 19 16 0.3 0.4

SOLAR SYSTEM RESEARCH


318

MALKIN et al.

meter antenna at Effelsberg, Germany. Less close approaches should be observed with space interferometers for which the effect is larger proportionally to the ratio of space/ground-based interferometer baseline lengths. Note that in the present work, the approach circumstances are calculated for the geocenter. For a real observer, the angular distances will differ from the values presented in Tables 4 and 5 up to the value of the R0/D ratio, where R0 is the distance from the geocenter to the mid-baseline, depending on the position angle and baseline orientation. Clearly, this difference reaches its maximum at the epochs of oppositions, and for ground-based interferometers may be as large as 3 for Jupiter and 1 for Saturn. The data presented in Table 3 were calculated for ground-based observations; in regard to space interferometers, the relevant calculations should be performed for their specific configurations. CONCLUSIONS We have calculated the circumstances of the approaches of solar system planets to astrometric radio sources for the time interval of 2008­2050. Especially interesting are the radio occultations for which the relativistic effects reach their maximum values. These occultations make it possible to measure the effects with minimum relative errors, which is a question of the utmost importance for testing gravity theories. One can efficiently use all of the planets from Venus to Saturn. The present work demonstrates that the apparent approaches of planets to radio sources and even the radio occultations are not as rare of events as it is generally considered. The number of events in consider-

ation grows with the expansion of the list of ecliptic radio sources, thus increasing the possibilities to perform relevant experiments. REFERENCES
Behrend, D, Boehm, J, Charlot, P, et al., Proc. 2007 IAG General Assembly. Observing our Changing Earth. Perugia, Italy, July 2­13, 2007, pp. 833­840. Fey, A.L., Ma C., Arias E.F., et al. The second extension of the International Celestial Reference Frame: ICRFExt.2, Astron. J., vol. 127, pp. 3587­3608. Fomalont, E.B. and Kopeikin, S.M., The Measurement of the Light Deflection from Jupiter: Experimental Results, Astrophys. J., 2003, vol. 598, pp. 704­711. Fomalont, E.B and Kopeikin, S.M, Radio interferometric tests of general relativity, Proc. IAU Symposium No. 248A "Giant Step: from Milli- to Micro-arcsecond Astrometry", 2008, pp. 383­386. Kopeikin, S.M., Testing the Relativistic Effect of the Propagation of Gravity by Very Long Baseline Interferometry, Astrophys. J., 2001, vol. 556, pp. L1­L5. L'vov, V.N., Smekhacheva, R.I., and Tsekmeister, S.D., EPOS - programmnaya sistema dlya podderzhki issledovanii ob"ektov Solnechnoi sistemy, Trudy Konf. "Okolozemnaya Astronomiya" (Proc. Conf. "Near-Earth Astronomy"), Zvenigorod, 2001, pp. 21­25. Petrov, L., Goddard VLBI astrometric catalogue, 2008b. http://vlbi.gsfc.nasa.gov/solutions/2008b_astro. SchlÝter, W. and Behrend, D., The International VLBI Service for Geodesy and Astrometry (IVS): Current Capabilities and Future Prospects, J. Geodesy, 2007, vol. 81, pp. 379­387. Treuhaft, R.N. and Lowe, S.T., A Measurement of Planetary Relativistic Deflection, Astron. J., 1991, vol. 102, pp. 1879­1888.

SOLAR SYSTEM RESEARCH

Vol. 43

No. 4

2009