Dear Mr. Bode, dear Dr. Dunham, a few days ago Mr. Bode asked me to determine an orbit for Ceres with my orbit determination and improvement software and to calculate the path of the June 10 occultation of PPM 231555 by Ceres. A. Orbit determination ---------------------- Ceres is regularly obseverd with the Carlsberg Meridian Circle on La Palma. These positions are very accurate and periodically published in the Carlsberg Meridian Catalogue (CAMC). I choose 50 positions from the CAMC between 1984 and 1992 to determine an orbit of Ceres. Unfortunately the CAMC gives *apparent* positions for Ceres; but my software has problems in handling more than 50 *apparent* positions, whereas up to 300 *astrometric* positions are dealt with correctly. The following table gives the Julian Date (TDT), apparent geocentric right ascension, the sign of declination, declination and station code 500 for Ceres as extracted from the CAMC: 2448916.313212 20 16 32.203 -1 29 46 31.37 500 2448891.372449 20 3 30.534 -1 31 12 29.28 500 2448876.412902 20 2 46.998 -1 31 41 14.04 500 2448840.525492 20 23 25.467 -1 30 58 9.79 500 2448814.612446 20 46 28.444 -1 28 35 36.57 500 2448799.659929 20 55 53.851 -1 26 56 54.86 500 2448773.733483 20 59 35.907 -1 24 38 32.02 500 2448418.394264 13 30 10.669 +1 0 34 11.92 500 2448392.470578 13 37 51.777 +1 2 27 6.09 500 2448362.569094 14 1 50.331 +1 2 18 54.67 500 2448313.717169 14 23 54.708 -1 0 29 39.08 500 2448287.780745 14 11 45.641 -1 0 50 16.42 500 2447953.333840 5 29 38.716 +1 28 29 55.73 500 2447929.393175 5 20 42.069 +1 27 50 46.46 500 2447904.467976 5 30 8.725 +1 26 59 39.74 500 2447892.508012 5 40 38.569 +1 26 25 52.62 500 2447876.563178 5 57 13.048 +1 25 27 50.71 500 2447849.651908 6 18 53.300 +1 23 35 6.30 500 2447817.739554 6 19 16.938 +1 21 55 4.85 500 2447500.331336 23 40 2.430 -1 14 17 47.79 500 2447457.448534 23 39 44.187 -1 17 53 20.10 500 2447439.505585 23 51 8.984 -1 17 50 33.00 500 2447419.571574 0 7 34.988 -1 16 34 31.47 500 2447400.632781 0 21 3.261 -1 14 38 36.02 500 2447385.677959 0 27 8.985 -1 13 6 12.03 500 2447374.708887 0 28 26.415 -1 12 9 44.23 500 2447028.362483 17 24 7.171 -1 27 47 19.41 500 2447001.437841 17 26 28.916 -1 27 9 17.41 500 2446984.491903 17 37 31.168 -1 26 36 1.34 500 2446971.535613 17 49 22.793 -1 26 1 12.84 500 2446962.566289 17 58 11.455 -1 25 31 57.11 500 2446951.603354 18 8 20.462 -1 24 51 57.08 500 2446939.642314 18 17 17.154 -1 24 6 32.75 500 2446909.728954 18 24 6.747 -1 22 30 3.79 500 2446560.366179 10 44 19.399 +1 21 52 20.33 500 2446540.417362 10 39 22.627 +1 24 6 22.29 500 2446527.454443 10 41 40.059 +1 24 59 30.72 500 2446510.507401 10 51 6.746 +1 25 12 48.32 500 2446497.550282 11 1 46.658 +1 24 33 39.13 500 2446477.616354 11 18 19.771 +1 22 23 38.89 500 2446446.707963 11 28 23.331 +1 18 31 22.43 500 2446408.796777 11 6 48.645 +1 16 45 58.76 500 2446079.348878 2 42 57.852 +1 11 25 54.06 500 2446041.457735 2 50 19.646 +1 8 58 32.60 500 2446030.493605 2 58 45.229 +1 8 45 38.19 500 2446016.540750 3 11 37.940 +1 8 47 27.62 500 2446000.594441 3 26 4.625 +1 9 6 38.33 500 2445987.636176 3 35 5.276 +1 9 27 29.39 500 2445972.680986 3 40 39.150 +1 9 48 36.16 500 2445942.759148 3 35 14.311 +1 9 57 56.30 500 The orbit is calculated with full perturbations from Mercury to Pluto (LE200/DE200) and gives the following elements for the osculation epoch 1996 June 10 (JD 2450244.5 TT): M omega Omega i e mu a (deg) (deg) (deg) (deg) (deg/day) (AU) 89.77889 71.91589 80.66058 10.59982 .0760455 .21398452 2.7683084 The accuracy is quite satisfactory: the rms values for fitting the above positions are 0.19" in right ascension and 0.17" in declination. The rms values are of the same order as the errors for a position measurement of the Carlsberg Meridian Circle. B. Orbits for Ceres in the literature ------------------------------------- The above determined orbit can be compared to highly accurate orbits found in the literature. The Ephemerides for Minor Planets 1996 (EMP 1996) have copied an orbit by Bowell (MPC 24219) which is based on positions measured between 1839 and 1994. From the EMP 1996 elements I calculated osculating elements for 1996 June 10 TT: M omega Omega i e mu a (deg) (deg) (deg) (deg) (deg/day) (AU) 89.77897 71.91584 80.66051 10.59981 .0760455 .21398452 2.7683084 The above determined orbit is in excellent agreement with the EMP 1996 orbit. Bec-Borsenberger determined an orbit for Ceres for the HIPPARCOS project using positions between 1950 and 1990 (Astronomy&Astrophysics Suppl. 98, 77 [1993]). His orbit gives the following osculating elements for 1996 June 10: M omega Omega i e mu a (deg) (deg) (deg) (deg) (deg/day) (AU) 89.77893 71.91592 80.66044 10.59981 .0760453 .21398452 2.7683085 The agreement between the three orbits is not perfect, but very good. C. The occultation of PPM 231555 by Ceres on 1996 June 10 --------------------------------------------------------- 1. Astrometric position of the star: In this case there are two main sources for a good position of the star: - the PPM catalogue, - the CAMC 1986 (this position can either be used with PPM proper motions or with CAMC proper motions). astrometric J2000 (FK5) position of PPM 321555 for 1996.45: h m s o o " 16 20 29.408 -18 41 25.00 (PPM) 16 20 29.414 -18 41 24.97 (CAMC position+PPM proper motions) 16 20 29.416 -18 41 24.95 (CAMC positions and proper motions) The accuracy of the star's position should be around 0.1" or better. An error in declination shifts the occultation path north or south, whereas an error in right ascension translates into a time error (about 10s in the case). 2. Moment of minimum geocentric separation between star and Ceres Using the PPM position of the star the following times and minimum separations for the different orbits have been calculated: 1996 June 10 2h 45h 00s TT 1.86" (orbit Federspiel) 1996 June 10 2 44 33 1.83 (EMP 1996) 1996 June 10 2 44 02 1.76 (Bec-Borsenberger) I assume that the EMP 1996 orbit is the most accurate one. However, the uncertainty for the occultation due to errors in the orbit is about 30s in time and about 0.07" in position of the path. With the CAMC star position and PPM proper motions we obtain the following respective numbers: 1996 June 10 2h 44h 50s TT 1.88" (orbit Federspiel) 1996 June 10 2 44 23 1.85 (orbit EMP 1996) 1996 June 10 2 43 52 1.78 (orbit Bec-Borsenberger) 3. The central line on the Earth Now the central lines for the different combinations of orbits and stellar positions can be calculated. The following tables give geographic latitude, longitude and time (TT) for several points on the central line. For these calculations the rotation of the Earth has been taken into account. The times are followed by the polar coordinates of this point of the central line, i.e. the distance in Earth radii and the direction in degrees from the apparent center of the Earth as seen from Ceres at that particular moment. The northern and southern limits for the occultation may be calculated from the apparent radius of Ceres of 0.40" which corresponds to 0.08 Earth radii in projection for this occultation. The maximum duration of the occultation for an observer on the central line will be 88s. Orbit EMP 1996, star position PPM latitude longitude time TT radius direction -18.50 27.49 2:36:25. ! .966 104.86 -22.30 17.06 2:37: 2. ! .905 106.49 -25.04 9.44 2:37:38. ! .845 108.34 -27.30 2.93 2:38:15. ! .785 110.48 -29.26 -2.97 2:38:51. ! .727 112.97 -30.99 -8.50 2:39:28. ! .671 115.88 -32.56 -13.79 2:40: 4. ! .617 119.32 -33.98 -18.92 2:40:41. ! .565 123.40 -35.26 -23.95 2:41:17. ! .517 128.27 -36.43 -28.92 2:41:54. ! .473 134.10 -37.49 -33.87 2:42:30. ! .435 141.02 -38.44 -38.83 2:43: 7. ! .404 149.13 -39.28 -43.81 2:43:43. ! .383 158.34 -40.01 -48.85 2:44:20. ! .373 168.34 -40.64 -53.94 2:44:56. ! .374 178.58 -41.16 -59.12 2:45:33. ! .387 188.43 -41.56 -64.39 2:46: 9. ! .411 197.39 -41.84 -69.78 2:46:46. ! .443 205.22 -41.99 -75.31 2:47:22. ! .483 211.87 -41.99 -80.99 2:47:59. ! .528 217.45 -41.85 -86.85 2:48:36. ! .577 222.11 -41.52 -92.95 2:49:12. ! .630 226.02 -41.01 -99.32 2:49:49. ! .684 229.32 -40.25 -106.07 2:50:25. ! .741 232.12 -39.20 -113.33 2:51: 2. ! .800 234.52 -37.74 -121.41 2:51:38. ! .859 236.59 -35.63 -130.96 2:52:15. ! .920 238.38 -31.83 -144.86 2:52:51. ! .981 239.96 Orbit Federspiel, star position PPM latitude longitude time TT radius direction -18.75 27.84 2:36:51. ! .968 105.23 -22.60 17.28 2:37:28. ! .907 106.87 -25.36 9.61 2:38: 4. ! .847 108.75 -27.63 3.07 2:38:41. ! .788 110.92 -29.60 -2.86 2:39:17. ! .730 113.43 -31.34 -8.41 2:39:54. ! .674 116.37 -32.91 -13.71 2:40:31. ! .620 119.83 -34.34 -18.86 2:41: 7. ! .568 123.93 -35.63 -23.91 2:41:44. ! .520 128.82 -36.81 -28.90 2:42:20. ! .477 134.65 -37.87 -33.88 2:42:57. ! .440 141.55 -38.82 -38.86 2:43:33. ! .410 149.59 -39.67 -43.87 2:44:10. ! .389 158.69 -40.40 -48.93 2:44:46. ! .379 168.54 -41.03 -54.05 2:45:23. ! .380 178.61 -41.55 -59.26 2:45:59. ! .393 188.30 -41.95 -64.57 2:46:36. ! .417 197.14 -42.23 -69.99 2:47:12. ! .449 204.87 -42.38 -75.55 2:47:49. ! .488 211.48 -42.38 -81.27 2:48:25. ! .533 217.03 -42.23 -87.18 2:49: 2. ! .582 221.70 -41.90 -93.31 2:49:38. ! .634 225.61 -41.37 -99.74 2:50:15. ! .689 228.92 -40.60 -106.54 2:50:51. ! .745 231.74 -39.53 -113.87 2:51:28. ! .803 234.15 -38.04 -122.02 2:52: 5. ! .863 236.23 -35.87 -131.72 2:52:41. ! .923 238.05 -31.86 -146.20 2:53:18. ! .984 239.64 Orbit Bec-Borsenberger, star position PPM latitude longitude time TT radius direction -18.11 26.15 2:35:53. ! .960 104.09 -21.74 16.18 2:36:30. ! .899 105.67 -24.41 8.71 2:37: 6. ! .838 107.49 -26.63 2.29 2:37:43. ! .778 109.58 -28.55 -3.55 2:38:19. ! .720 112.02 -30.27 -9.02 2:38:56. ! .663 114.88 -31.81 -14.25 2:39:33. ! .608 118.28 -33.21 -19.34 2:40: 9. ! .555 122.33 -34.48 -24.32 2:40:46. ! .506 127.19 -35.63 -29.25 2:41:22. ! .461 133.05 -36.67 -34.15 2:41:59. ! .422 140.08 -37.60 -39.06 2:42:35. ! .391 148.38 -38.43 -43.99 2:43:12. ! .369 157.89 -39.16 -48.96 2:43:48. ! .358 168.27 -39.78 -54.00 2:44:25. ! .360 178.91 -40.29 -59.11 2:45: 1. ! .374 189.11 -40.69 -64.32 2:45:38. ! .398 198.33 -40.96 -69.64 2:46:14. ! .432 206.30 -41.11 -75.09 2:46:51. ! .472 213.01 -41.12 -80.69 2:47:27. ! .518 218.60 -40.98 -86.48 2:48: 4. ! .568 223.24 -40.67 -92.48 2:48:40. ! .622 227.11 -40.17 -98.76 2:49:17. ! .677 230.36 -39.43 -105.41 2:49:53. ! .735 233.10 -38.41 -112.56 2:50:30. ! .794 235.44 -37.00 -120.48 2:51: 7. ! .854 237.46 -34.96 -129.81 2:51:43. ! .915 239.21 -31.43 -142.97 2:52:20. ! .976 240.74 Orbit EMP 1996, star position CAMC+PPM proper motions latitude longitude time TT radius direction -18.56 28.01 2:36:15. ! .968 105.05 -22.43 17.42 2:36:52. ! .907 106.68 -25.18 9.74 2:37:28. ! .847 108.55 -27.46 3.20 2:38: 5. ! .788 110.70 -29.42 -2.72 2:38:41. ! .730 113.19 -31.17 -8.27 2:39:18. ! .673 116.11 -32.74 -13.57 2:39:54. ! .619 119.55 -34.16 -18.71 2:40:31. ! .568 123.64 -35.46 -23.75 2:41: 7. ! .519 128.51 -36.63 -28.74 2:41:44. ! .476 134.32 -37.69 -33.70 2:42:20. ! .438 141.22 -38.64 -38.68 2:42:57. ! .408 149.28 -39.49 -43.67 2:43:33. ! .387 158.41 -40.23 -48.72 2:44:10. ! .376 168.32 -40.86 -53.83 2:44:46. ! .378 178.46 -41.38 -59.03 2:45:23. ! .391 188.23 -41.78 -64.32 2:45:59. ! .414 197.14 -42.06 -69.73 2:46:36. ! .446 204.93 -42.21 -75.27 2:47:13. ! .485 211.56 -42.21 -80.97 2:47:49. ! .530 217.15 -42.06 -86.86 2:48:26. ! .579 221.82 -41.74 -92.98 2:49: 2. ! .631 225.75 -41.22 -99.38 2:49:39. ! .686 229.06 -40.46 -106.15 2:50:15. ! .743 231.87 -39.40 -113.45 2:50:52. ! .801 234.28 -37.93 -121.56 2:51:28. ! .861 236.36 -35.80 -131.17 2:52: 5. ! .921 238.17 -31.92 -145.27 2:52:41. ! .982 239.76 Orbit Federspiel, star position CAMC+PPM proper motions latitude longitude time TT radius direction -18.81 28.36 2:36:41. ! .970 105.41 -22.72 17.64 2:37:18. ! .909 107.06 -25.50 9.91 2:37:54. ! .849 108.96 -27.79 3.34 2:38:31. ! .790 111.13 -29.76 -2.61 2:39: 8. ! .732 113.65 -31.52 -8.17 2:39:44. ! .676 116.60 -33.10 -13.49 2:40:21. ! .622 120.07 -34.53 -18.66 2:40:57. ! .571 124.18 -35.83 -23.72 2:41:34. ! .523 129.06 -37.01 -28.73 2:42:10. ! .480 134.87 -38.07 -33.72 2:42:47. ! .443 141.75 -39.03 -38.71 2:43:23. ! .413 149.74 -39.88 -43.73 2:43:60. ! .392 158.77 -40.62 -48.81 2:44:36. ! .382 168.53 -41.25 -53.95 2:45:13. ! .384 178.51 -41.77 -59.17 2:45:49. ! .397 188.12 -42.17 -64.50 2:46:26. ! .420 196.90 -42.45 -69.94 2:47: 2. ! .452 204.60 -42.60 -75.53 2:47:39. ! .491 211.19 -42.60 -81.27 2:48:16. ! .535 216.75 -42.45 -87.20 2:48:52. ! .584 221.42 -42.11 -93.36 2:49:29. ! .636 225.34 -41.58 -99.81 2:50: 5. ! .690 228.67 -40.80 -106.63 2:50:42. ! .747 231.50 -39.72 -114.00 2:51:18. ! .805 233.92 -38.22 -122.20 2:51:55. ! .864 236.02 -36.03 -131.96 2:52:31. ! .924 237.84 -31.93 -146.70 2:53: 8. ! .986 239.44 Orbit Bec-Borsenberger, star position CAMC+PPM proper motions latitude longitude time TT radius direction -18.19 26.61 2:35:43. ! .962 104.28 -21.87 16.51 2:36:20. ! .901 105.87 -24.56 8.99 2:36:56. ! .840 107.70 -26.79 2.55 2:37:33. ! .780 109.80 -28.72 -3.31 2:38:10. ! .722 112.25 -30.44 -8.80 2:38:46. ! .665 115.12 -31.99 -14.05 2:39:23. ! .610 118.53 -33.40 -19.14 2:39:59. ! .558 122.58 -34.67 -24.14 2:40:36. ! .509 127.45 -35.83 -29.08 2:41:12. ! .464 133.30 -36.87 -33.99 2:41:49. ! .426 140.30 -37.81 -38.91 2:42:25. ! .394 148.55 -38.65 -43.86 2:43: 2. ! .373 157.98 -39.37 -48.85 2:43:38. ! .362 168.26 -40.00 -53.90 2:44:15. ! .364 178.80 -40.51 -59.03 2:44:51. ! .377 188.92 -40.90 -64.26 2:45:28. ! .401 198.07 -41.18 -69.59 2:46: 4. ! .435 206.01 -41.33 -75.06 2:46:41. ! .475 212.71 -41.34 -80.68 2:47:18. ! .521 218.30 -41.20 -86.49 2:47:54. ! .570 222.95 -40.88 -92.52 2:48:31. ! .624 226.83 -40.38 -98.82 2:49: 7. ! .679 230.09 -39.64 -105.49 2:49:44. ! .736 232.85 -38.61 -112.67 2:50:20. ! .795 235.21 -37.19 -120.63 2:50:57. ! .855 237.24 -35.13 -130.02 2:51:33. ! .916 239.00 -31.54 -143.35 2:52:10. ! .977 240.54 D. Conclusion ------------- The occultation of PPM 231555 by Ceres on 1996 June 10 will be favourable for observers in southwestern Africa and the southern part of South America. The expected errors in the position of the star and in the ephemeris of Ceres is about 10% of Ceres' projected shadow on the Earth. I guess that the most accurate orbit is EMP 1996, because it is based on accurate positions that cover a very wide range in time, and that the most reliable position of the star comes from CAMC with PPM proper motions. However, orbits determined by Bec-Borsenberger and Federspiel in combination with PPM or CAMC star positions give quite similar results which help to estimate probable errors. Good luck to all observers Martin Federspiel ------------------------------------------------------------------------- Martin Federspiel ***E-mail: federspielm@ubaclu.unibas.ch Astronomisches Institut ******** oder: federspi@astro.unibas.ch der Universitaet Basel *** oder: martin@sirius.astro.unibas.ch Venusstr. 7 Telefon: (CH)-61/2055413 CH-4102 Binningen (Schweiz) Fax: (CH)-61/2055455 -------------------------------------------------------------------------