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Pole and shape determinaton of asteroids. II
Planetary and Space Science 48 (2000) 973–982
www.elsevier.nl/locate/planspasci
Pole and shape determinaton of asteroids. II C. Blanco, M. Cigna, D. Riccioli Physics and Astronomy Department of Catania University, Via S. Soÿa 78, 95125 Catania, Italy Received 4 October 1999; accepted 11 October 1999
Abstract To update the research work on asteroids to obtain their rotation axis orientation and axes ratios, we went on implementing the collection of lightcurves. By means of new observations carried out and new lightcurves published, we were able to apply the computation methods of the rotation parameters to 18 minor planets. The found values of the pole coordinates and of the axes ratios are reported. For 14 objects, this is the ÿrst determination of these elements which are necessary to study the collisional evolution of c 2000 Elsevier Science Ltd. All rights reserved. the asteroids.
1. Introduction The implementation of asteroid lightcurves devoted to the determination of their rotational period, pole coordinates, shape and B and V magnitudes was recently increased with the conclusion of the observations and the data reduction of 39 asteroids. The results of 18 objects were recently published (Blanco et al., 2000) and the other 21 were submitted for publication (Riccioli et al., 1999). All the observations were performed at the M.G. Fracastoro Station of Catania Astrophysical Observatory. By adding these new data to the ones already collected by us (Riccioli and Blanco, 1995) and to those existing in the literature, we were able to apply the amplitude– magnitude (AM) method (ZappalÂa et al., 1983) to 5 Astrea, 6 Hebe, 13 Egeria, 26 Proserpina, 34 Circe, 63 Ausonia, 66 Maya, 137 Meliboea, 176 Iduna, 250 Bettina, 258 Tyche, 335 Roberta, 352 Gisela, 419 Aurelia, 432 Pythia, 471 Papagena, 537 Pauly, 984 Gretia. For 14 objects this is the ÿrst determination of the rotation axis orientation and shape. For the others for which we found previous determinations in the literature, even if for the same asteroid, signiÿcant di erences between di erent models can be found, the values found by us substantially agree with the published ones. Together with a comparison between these results, we report the observed and theoretical amplitude–longitude (A–) plots and the values of the pole coordinates and of the axes ratios found by us. 2. Pole and shape determination As reported in a previous paper (Blanco and Riccioli, 1998) and according to ZappalÂa and Knezevic (1984), the
(AM) method for spin-vector determination allows us to obtain only a preliminary indication of the rotational properties of the asteroid. But for its simple and fast application it is particularly suitable to cases, as the present, of a very large number of rotation axis direction and shape determination. The availability of a consistent set of new lightcurves, mostly obtained by means of our photoelectric observations (Blanco et al., 2000, Riccioli et al., 1999), allows us to have the minimum conditions to apply the (AM) method to a congruous number of asteroids, mainly without previous determinations of parameters of rotation. The method, based on the assumed ellipsoidal shape of the asteroid (with semiaxes a ¿ b ¿ c) and on the relationships between the aspect angle, the lightcurve amplitude (A) and the magnitude V of the asteroid lightcurve maxi mum, requires the knowledge of these parameters at least in three oppositions well distributed in longitude (). If we assume the smaller axis c to be the asteroid rotation axis, the ratio between the other two axes and subsequently their single values, can be obtained from the amplitude– longitude (A–) plot. The modelling curves were obtained using the least-squares method. The absence of observed lightcurves at the longitudes of the maximum or minimum may imply that the extrema of the theoretical curves seem to be overestimated with respect to the observed values. This fact depends on the computing program that in these circumstances takes into account the slope of the ascending or descending branches. At every given orbital position of the asteroid, from the axes ratios it is possible to obtain the value of the aspect angle (with an uncertain deÿnition of the north and south poles) and hence the pole longitude. The lightcurve amplitude was corrected for its dependence on the phase angle by means of the
c 2000 Elsevier Science Ltd. All rights reserved. 0032-0633/00/$ - see front matter PII: S 0 0 3 2 - 0 6 3 3 ( 0 0 ) 0 0 0 6 5 - 9
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C. Blanco et al. / Planetary and Space Science 48 (2000) 973–982
Fig. 1.
Fig. 2.
◦
relationship A(0 ) = A( )=(1 + m ) (ZappalÂa et al., 1990), where A( ) is the observed lightcurve amplitude, is the solar phase angle and m is a coecient depending on the asteroid taxonomic class. The V magnitude at 1 a.u. and at a given phase angle of each asteroid, V0 (1; ), was computed adopting the value m , the arithmetic average of all the phase angles. Due to the available lightcurves (only lightcurves covered at least 90% were utilized), their minimum number (at least three) necessary to apply the (AM) method and to their distribution in longitude, it was possible to compute the pole coordinates and the axes ratios for 18 asteroids. In Figs. 1–18, using di erent symbols for the di erent references from which the values were taken, the (A–) plots of these asteroids are reported. The values adopted are the mean values computed over the duration of each lightcurve. The ÿlled symbols indicate the observed values of the amplitude A, the empty ones
◦
the corresponding values at longitudes (+180 ), the continuous and dashed (in the case of two solutions) lines are the theoretical curves.
3. Results Table 1 lists the asteroids to which it was possible to apply the (AM) method and the relative results obtained. In the columns from left to right, the name of the asteroids, the average solar phase angle, the adopted maximum amplitude, the coordinates of the pole and the axes ratios are reported. When two pairs of solutions were found, since it is pratically impossible to distinguish between them and to reject one pair in favour of the other, the choice may take into account one’s stand on the smaller error or on the better ÿt with the theoretical plot. In
C. Blanco et al. / Planetary and Space Science 48 (2000) 973–982
Fig. 3.
Fig. 4.
Fig. 5.
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Fig. 6.
Fig. 7.
Fig. 8.
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Fig. 9.
Fig. 10.
Fig. 11.
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Fig. 12.
Fig. 13.
Fig. 14.
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Fig. 15.
Fig. 16.
Fig. 17.
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Fig. 18.
our case the P1 solution normally has a smaller error than the P2 one. When this occurs, the two solutions di er ◦ by about 180 in eclipting longitude. Since the applied method does not allow us to distinguish between prograde and retrograde rotation about the same axis, every tabulated solution has a symmetric one with equal probability, which is not reported in the table. The choice between the prograde or the retrograde reported solutions was made according to the solution given by the computation program. A comparison between the values found by us and the ones by other authors, mainly computed by di erent methods, shows a certain agreement.
5 Astrea: The rotation parameters of 5 Astrea were published by Taylor (1978) who gives only one value of 0 ; ÿ0 , by ZappalÂa and Di Martino (1986) who give four solutions of 0 ; ÿ0 and the value of a=b and b=c axes ratios, by Erikson and Magnusson (1993) who give two solutions of 0 ; ÿ0 and the value of a=b ratio and by De Angelis (1995) who gives one value of 0 ; ÿ0 and the values of both the axes ratios. Taking into account the symmetric solution, the value of 0 of our solution agrees with the ones given by the above-mentioned authors, while the ÿ0 value is in agreement only with P3 and P4 solutions by ZappalÂa and Di Martino (1986). The values of the axes
Table 1 The results of asteroid pole and shape determinations Asteroid Astreaa
5 6 Hebea 13 Egeria 26 Proserpina 34 Circe 63 Ausoniaa 66 Maja 137 Meliboea 176 Iduna 250 Bettinaa 258 Tyche 335 Roberta 352 Gisela 419 Aurelia 432 Pythia 471 Papagena 537 Pauly 984 Gretia a Marks
m
Amax
0 (P1 )
ÿ0 (P1 )
11.366 11.700 10.403 9.480 8.190 11.835 7.940 6.887 12.179 7.981 15.200 8.723 8.683 6.805 10.825 8.135 7.218 4.922
0.40 0.30 0.39 0.16 0.30 0.95 0.55 0.18 0.36 0.60 0.45 0.80 0.42 0.27 0.34 0.24 0.24 0.88
132 ± 6 128 ± 2 103 ± 4 47 ± 1 113 ± 5 125 ± 2 156 ± 6 149 ± 3 85 ± 1 95 ± 5 252 ± 15 258 ± 4 213 ± 5 13 ± 2 121 ± 9 21 ± 3 290 ± 31 228 ± 9
−58 ± 3 30 ± 3 13 ± 10 −4 ± 7 17 ± 22 −36 ± 3 62 ± 1 8±3 36 ± 1 −1 ± 30 −20 ± 15 25 ± 9 53 ± 5 −34 ± 2 65 ± 6 31 ± 3 40 ± 31 −12 ± 9
0 (P2 )
ÿ0 (P2 )
227 ± 13
0 ± 13
42 ± 24 80 ± 7
−40 ± 30 15 ± 15
192 ± 6
34 ± 6
199 ± 10
−29 ± 10
46 ± 2
47 ± 1
the asteroids for which previous pole and shape determinations have been found in the literature.
a=b
b=c
1.44 1.32 1.43 1.16 1.32 2.39 1.66 1.18 1.39 1.74 1.51 2.09 1.47 1.28 1.37 1.25 1.25 2.25
1.30 1.11 1.26 1.40 1.00 1.00 1.40 1.11 1.28 1.58 1.25 1.14 1.38 1.16 1.27 1.38 1.88 1.00
C. Blanco et al. / Planetary and Space Science 48 (2000) 973–982
ratios given by us are mainly greater by one-tenth than the already published values. 6 Hebe: This asteroid was studied by Gehrels and Owings (1962), Gehrels and Taylor (1977), ZappalÂa and Knezevic (1984), Magnusson (1986), Michalowski (1988), De Angelis (1995) and Dotto et al. (1995). Gehrels and Owings (1962) give one value of 0 ; ÿ0 while Gehrels and Taylor (1977) give one value of 0 ; ÿ0 and one value of the axes ratios. They are quite in agreement with our values. ZappalÂa and Knezevic (1984) give four solutions of 0 ; ÿ0 and one solution of the axes ratios. Taking into account the symmetric solution, the longitude value is in agreement with our value while only two ÿ0 values coincide with the one given by us, the other two are opposite. The axes ratios substantially agree with our values. The remaining authors give one solution of 0 ; ÿ0 and one value of the axes ratio, except Michalosky (1988) who gives only the pole coordinates. All the 0 values di er ◦ by about 220 from our values, while the ÿ0 and the axes ratio values agree with the ones given by us. 63 Ausonia: In the literature we found rotation parameters and shape elements of 63 Ausonia by ZappalÂa et al. (1983), ZappalÂa and Knezevic (1984), Magnusson (1986) and De Angelis (1995). The value of 0 and those of the axes ratios, given only by ZappalÂa et al. (1983), agree very well with those given by us while, among those given by ZappalÂa and Knezevic (1984) (four values of 0 ; ÿ0 and one value of the axes ratios), the ÿ0 value of the P1 and P2 solutions is in disagreement, being opposite to the value found by us. Well in agreement are the data published by Magnusson (1986), who gives a pair of solutions 0 ; ÿ0 and one value of a=b and b=c and those published by De Angelis (1995) who only gives a value of the pole longitude and latitude and of the axes ratios. 250 Bettina: The data of this asteroid found in the literature are on an average in disagreement with those reported by us. Among the values of 0 ; ÿ0 and of the axes ratios published by Drummond et al. (1991) only the pole coordinates agree while the shape elements di er by some tenth. The values by Dotto et al. (1992), that present four solutions of 0 ; ÿ0 and one value of a=b and b=c, are those that show the best agreement with our values. The other values found in the literature by Michalowski (1992,1993) Erikson and Magnusson (1993) and De Angelis (1995), all present two solutions of 0 ; ÿ0 and one value of the axes ratios, present an agreement only with the value of the longitude and of some value of the axes ratios. Acknowledgements The authors would like to give their thanks to M. Di Martino, G. De Sanctis and P. Tanga for the collaboration to the observations and to D. Recupero for the help in editing this note. The work has been supported by grant ASI ARS-98-87 from the Italian Space Agency.
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References Ahmad, I.I., 1954. Photometric studies of asteroids. IV. The lightcurves of Ceres, Hebe, Flora, and Kalliope. Astrophys. J. 120, 551–560. Barucci, M.A., Di Martino, M., Fulchignoni, M., 1992. Rotational properties of small asteroids: photoelectric observations. Astron. J. 103, 1679–1686. Binzel, R.P., 1987. A Photoelectric Survey of 130 Asteroids. Icarus 72, 135–208. Blanco, C., Di Martino, M., Riccioli, D., 2000. New rotational period of 18 asteroids. Planetary and Space Science, 48, 271–284. Blanco, C., Riccioli, D., 1998. Pole coordinates and shape of 30 asteroids. Astron. Astrophys. Suppl. Ser. 131, 385–394. Chang, Y.C., Chang, C.S., 1962. Photometric investigation of seven variable asteroids. Acta Astron. Sinica 10, 101–110. Chang, Y.C., Chang, C.S., 1963. Photometric observations of variable asteroids II. Acta Astron. Sinica 11, 139–148. De Angelis, G., 1995. Asteroid spin, pole and shape determinations. Planet. Space Science 43, 649–682. Di Martino, M., Blanco, C., Riccioli, D., De Sanctis, G., 1994. Lightcurves and rotational periods of nine main belt asteroids. Icarus 107, 269–275. Di Martino, M., Cacciatori, S., 1984. Photoelectric photometry of 14 asteroids. Icarus 60, 75–82. Di Martino, M., Ferreri, W., Fulchignoni, M., De Angelis, G., Barucci, M.A., Lecacheux, J., Burchi, R., Di Paolantonio, A., 1990. 66 Maja and 951 Gaspra: Possible Flyby Targets for Cassini and Galileo Missioms. Icarus 87, 372–376. Dotto, E., Barucci, M.A., Fulchignoni, M., Di Martino, M., Rotundi, A., Burchi, R., Di Paolantonio, A., 1992. M-type asteroids: rotational properties of 16 objects. Astron. Astrophys. Suppl. Ser. 95, 195–211. Dotto, E., De Angelis, G., Di Martino, M., Barucci, M.A., Fulchignoni, M., De Sanctis, G., Burchi, R., 1995. Pole orientation and shape of 12 asteroids. Icarus 117, 313–327. Drummond, J.D., Weidenschilling, S.J., Chapman, C.R., Davis, D.R., 1991. Photometric geodesy of main belt asteroids. Icarus 89, 44–64. Erikson, A., Magnusson, P., 1993. Pole determinations of asteroids. Icarus 103, 62–66. Gehrels, T., Owings, D., 1962. Photometric studies of asteroids. IX. Additional lightcurves. Astrophys. J. 135, 906–924. Gehrels, T., Taylor, R.C., 1977. Minor planets and related objects. XXII. Phase function for (6) Hebe. Astron. J. 82, 229–237. Gil Hutton, R., Campos, J., 1991. Rotation period and lightcurve of asteroid 137 Meliboea. Minor Planet Bull 18, 17–18. Hainaut-Rouelle, M.-C., Hainaut, O.R., Detal, A., 1995. Lightcurves of selected minor planets. Astron. Astrophys. Suppl. Ser. 112, 125–142. Harris, A.W., Young, J.W., 1989. Asteroids lightcurve observations from 1979–1981. Icarus 81, 314–364. Harris, A.W., Young, J.W., Dockweiler, T., Gibson, J., 1992. Asteroid lightcurve observations from 1981. Icarus 95, 115–147. Lagerkvist, C.-I., 1978. Photographic photometry of 110 main-belt asteroids. Astron. Astrophys. Suppl. 31, 361–381. Lagerkvist, C.-I., 1981. Physical studies of asteroids II: photoelectric observations of the asteroids 63, 93, 135 and 409. Astron. Astrophys. Suppl. Ser. 44, 345–347. Lagerkvist, C.-I., Erikson, A., Debehogne, H., Festin, L., Magnusson, P., Mottola, S., Oja, T., De Angelis, G., Belskaja, I.N., Dahlgren, M., Gonano-Beurer, M., Lagerros, J., Lumme, K., Pohjolanen, S., 1995. Physical studies of asteroids. XXIX. Photometry and anlysis of 27 asteroids. Astron. Astrophys. Suppl. Ser. 113, 115–129. Lagerkvist, C.-I., Hahn, G., Magnusson, P., Rickman, H., 1987. Physical studies of asteroids XVI: photoelectric photometry of 17 asteroids. Astron. Astrophys. Suppl. Ser. 70, 21–32.
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Lagerkvist, C.-I., Magnusson, P., Debehogne, H., Ho mann, M., Erikson, A., De Campos, A., Cutispoto, G., 1992. Physical studies of asteroids. XXV: photoelectric photometry of asteroids obtained at ESO and Hoher List Observatory. Astron. Astrophys. Suppl. Ser. 95, 461–470. Lagerkvist, C.-I., Magnusson, P., Williams, I.P., Buontempo, M.E., Gibbs, P., Morrison, L.V., 1989. Physical studies of asteroids. XIX. Phase relation and composite lightcurves obtained with the Carlsberg Meridian Circle. Astron. Astrophys. Suppl. Ser. 78, 519–532. Lagerkvist, C.-I., Rickman, H., 1981a. The spin periods of M asteroids. Moon and Planets 24, 437–440. Lagerkvist, C.-I., Rickman, H., 1981b. Physical studies of asteroids V: photoelectric observations of the asteroids 70, 101, 369 and 432. Astron. Astrophys. Suppl. Ser. 45, 177–179. Lagerkvist, C.-I., Williams, I.P., 1987. Physical studies of asteroids XV. Determinations of slope parameters and absolute magnitudes for 51 asteroids. Astron. Astrophys. Suppl. Ser. 68, 295–315. Licandro, J., Gillardo, T., Tancredi, G., 1990. Photometric observations of asteroids 31, 118, 13, 196 and 471. Rev. Mex. Astron. Astroÿs. 21, 590–592. Lustig, G., 1977. The period of rotation and the photoelectric lightcurve of the minor planet 471 Papagena. Astron. Astrophys. Suppl. 30, 117–119. Magnusson, P., 1986. Distribution of Spin Axes and Senses of Rotation for 20 Large Asteroids. Icarus 68, 1–39. Magnusson, P., Lagerkvist, C.-I., 1991. Physical studies of asteroids XXII. Photoelectric photometry of the asteroids 34, 98, 115, 174, 270, 389, 419 and 804. Astron. Astrophys. Suppl. Ser. 87, 269–275. Michalowski, T., 1988. Photometric astrometry applied to the asteroids: 6, 15, 43 and 624. Acta Astronomica 38, 455–468. Michalowski, T., 1992. Spin vector of asteroids 21 Laetitia, 196 Philomela, 250 Bettina, 337 Devosa and 804 Hispania. In: Harris, A.W., Bowell, E. (Eds.), Asteroids, Comets, Meteors 1991. Lunar and Planetary Institute, Houston, pp. 417–419. Michalowski, T., 1993. Poles shapes, senses of rotation, and sidereal periods of asteroids. Icarus 106, 563–572. Pavlovski, K., Knezevic, Z., Muminovic, M., 1981. Photometry of the asteroid (5) Astrea. Astron. Obs. Sarajevo 1, 225–230. Piironen, J., Bowell, E., Erikson, A., Magnusson, P., 1994. Photometry of eleven asteroids at small phase angles. Astron. Astrophys Suppl. Ser. 106, 587–595.
Riccioli, D., Blanco, C., 1995. Amplitude–Longitude (A–) plot catalogue of asteroids. Catania Astrophysical Observatory Special Publication 1995, 143. Riccioli, D., Blanco, C., Cigna, M., 1999. New rotational period of Asteroids. II. Planetary and Space Science (submitted). Scaltriti, F., ZappalÂa, V., 1977. A photometric study of the minor planet 63 Ausonia. Icarus 31, 498–502. Scaltriti, F., ZappalÂa, V., 1978. Photoelectric photometry of asteroids 37, 80, 97, 216, 270, 313, and 471. Icarus 34, 428–435. Scaltriti, F., ZappalÂa, V., 1979. Photoelectric photometry and rotation periods of the asteroids 26 Proserpina, 194 Prokne, 287 Nephthys, and 554 Peraga. Icarus 39, 124–130. Schober, H.J., Erikson, A., Hahn, G., Lagerkvist, C.-I., Albrecht, R., Orning, W., Schroll, A., Stadler, M., 1994. Physical studies of asteroids. XXVIII. Lightcurves and photoelectric photometry of asteroids 2, 14, 51, 105, 181, 238, 258, 369, 377, 416, 487, 626, 679, 1048 and 2183. Astron. Astrophys. Suppl. Ser. 105, 281–300. Surdej, A., Surdej, J., 1977. Rotation period and photoelectric lightcurves of asteroid 471 Papagena. Astron. Astrophys. Suppl. 30, 121–124. Taylor, R.C., 1978. Minor planets and related objects. XXIV. Photometric observations for (5) Astrea. Astron. J. 83, 201–204. Weidenschilling, S.J., Chapman, C.R., Davis, D.R., Greenberg, R., Levy, D.H., Binzel, R.P., Vail, S.M., Magee, M., Spaute, D., 1990. Photometric Geodesy of main-belt Asteroids III. Additional lightcurves. Icarus 86, 402–447. Yang, X.-y., Zhang, Y.-y., Li, X.-q., 1965. Photometric observations of variable asteroids. III. Acta Astron. Sinica 13, 66–73. ZappalÂa, V., Cellino, A., Barucci, A.M., Fulchignoni, M., Lupishko, D.F., 1990. An analysis of the amplitude-phase relationship among asteroids. Astron. Astrophys. 231, 548–560. ZappalÂa, V., Di Martino, M., 1986. Rotation axis of asteroids via amplitude-magnitude method: results for 10 objects. Icarus 68, 40– 54. ZappalÂa, V., Knezevic, Z., 1984. Rotation Axes of Asteroids: Results for 14 Objects. Icarus 59, 436–455. ZappalÂa, V., Scaltriti, F., Di Martino, M., 1983. Photoelectric Photometry of 21 Asteroids. Icarus 56, 325–344. Zhou, X.-h., Yang, X.-y., Wu, Z.-x., 1982. Lightcurves of variable asteroids (V). Acta Astron. Sinica 23, 349–352.
www.elsevier.nl/locate/planspasci
Pole and shape determinaton of asteroids. II C. Blanco, M. Cigna, D. Riccioli Physics and Astronomy Department of Catania University, Via S. Soÿa 78, 95125 Catania, Italy Received 4 October 1999; accepted 11 October 1999
Abstract To update the research work on asteroids to obtain their rotation axis orientation and axes ratios, we went on implementing the collection of lightcurves. By means of new observations carried out and new lightcurves published, we were able to apply the computation methods of the rotation parameters to 18 minor planets. The found values of the pole coordinates and of the axes ratios are reported. For 14 objects, this is the ÿrst determination of these elements which are necessary to study the collisional evolution of c 2000 Elsevier Science Ltd. All rights reserved. the asteroids.
1. Introduction The implementation of asteroid lightcurves devoted to the determination of their rotational period, pole coordinates, shape and B and V magnitudes was recently increased with the conclusion of the observations and the data reduction of 39 asteroids. The results of 18 objects were recently published (Blanco et al., 2000) and the other 21 were submitted for publication (Riccioli et al., 1999). All the observations were performed at the M.G. Fracastoro Station of Catania Astrophysical Observatory. By adding these new data to the ones already collected by us (Riccioli and Blanco, 1995) and to those existing in the literature, we were able to apply the amplitude– magnitude (AM) method (ZappalÂa et al., 1983) to 5 Astrea, 6 Hebe, 13 Egeria, 26 Proserpina, 34 Circe, 63 Ausonia, 66 Maya, 137 Meliboea, 176 Iduna, 250 Bettina, 258 Tyche, 335 Roberta, 352 Gisela, 419 Aurelia, 432 Pythia, 471 Papagena, 537 Pauly, 984 Gretia. For 14 objects this is the ÿrst determination of the rotation axis orientation and shape. For the others for which we found previous determinations in the literature, even if for the same asteroid, signiÿcant di erences between di erent models can be found, the values found by us substantially agree with the published ones. Together with a comparison between these results, we report the observed and theoretical amplitude–longitude (A–) plots and the values of the pole coordinates and of the axes ratios found by us. 2. Pole and shape determination As reported in a previous paper (Blanco and Riccioli, 1998) and according to ZappalÂa and Knezevic (1984), the
(AM) method for spin-vector determination allows us to obtain only a preliminary indication of the rotational properties of the asteroid. But for its simple and fast application it is particularly suitable to cases, as the present, of a very large number of rotation axis direction and shape determination. The availability of a consistent set of new lightcurves, mostly obtained by means of our photoelectric observations (Blanco et al., 2000, Riccioli et al., 1999), allows us to have the minimum conditions to apply the (AM) method to a congruous number of asteroids, mainly without previous determinations of parameters of rotation. The method, based on the assumed ellipsoidal shape of the asteroid (with semiaxes a ¿ b ¿ c) and on the relationships between the aspect angle, the lightcurve amplitude (A) and the magnitude V of the asteroid lightcurve maxi mum, requires the knowledge of these parameters at least in three oppositions well distributed in longitude (). If we assume the smaller axis c to be the asteroid rotation axis, the ratio between the other two axes and subsequently their single values, can be obtained from the amplitude– longitude (A–) plot. The modelling curves were obtained using the least-squares method. The absence of observed lightcurves at the longitudes of the maximum or minimum may imply that the extrema of the theoretical curves seem to be overestimated with respect to the observed values. This fact depends on the computing program that in these circumstances takes into account the slope of the ascending or descending branches. At every given orbital position of the asteroid, from the axes ratios it is possible to obtain the value of the aspect angle (with an uncertain deÿnition of the north and south poles) and hence the pole longitude. The lightcurve amplitude was corrected for its dependence on the phase angle by means of the
c 2000 Elsevier Science Ltd. All rights reserved. 0032-0633/00/$ - see front matter PII: S 0 0 3 2 - 0 6 3 3 ( 0 0 ) 0 0 0 6 5 - 9
974
C. Blanco et al. / Planetary and Space Science 48 (2000) 973–982
Fig. 1.
Fig. 2.
◦
relationship A(0 ) = A( )=(1 + m ) (ZappalÂa et al., 1990), where A( ) is the observed lightcurve amplitude, is the solar phase angle and m is a coecient depending on the asteroid taxonomic class. The V magnitude at 1 a.u. and at a given phase angle of each asteroid, V0 (1; ), was computed adopting the value m , the arithmetic average of all the phase angles. Due to the available lightcurves (only lightcurves covered at least 90% were utilized), their minimum number (at least three) necessary to apply the (AM) method and to their distribution in longitude, it was possible to compute the pole coordinates and the axes ratios for 18 asteroids. In Figs. 1–18, using di erent symbols for the di erent references from which the values were taken, the (A–) plots of these asteroids are reported. The values adopted are the mean values computed over the duration of each lightcurve. The ÿlled symbols indicate the observed values of the amplitude A, the empty ones
◦
the corresponding values at longitudes (+180 ), the continuous and dashed (in the case of two solutions) lines are the theoretical curves.
3. Results Table 1 lists the asteroids to which it was possible to apply the (AM) method and the relative results obtained. In the columns from left to right, the name of the asteroids, the average solar phase angle, the adopted maximum amplitude, the coordinates of the pole and the axes ratios are reported. When two pairs of solutions were found, since it is pratically impossible to distinguish between them and to reject one pair in favour of the other, the choice may take into account one’s stand on the smaller error or on the better ÿt with the theoretical plot. In
C. Blanco et al. / Planetary and Space Science 48 (2000) 973–982
Fig. 3.
Fig. 4.
Fig. 5.
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C. Blanco et al. / Planetary and Space Science 48 (2000) 973–982
Fig. 6.
Fig. 7.
Fig. 8.
C. Blanco et al. / Planetary and Space Science 48 (2000) 973–982
Fig. 9.
Fig. 10.
Fig. 11.
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C. Blanco et al. / Planetary and Space Science 48 (2000) 973–982
Fig. 12.
Fig. 13.
Fig. 14.
C. Blanco et al. / Planetary and Space Science 48 (2000) 973–982
Fig. 15.
Fig. 16.
Fig. 17.
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C. Blanco et al. / Planetary and Space Science 48 (2000) 973–982
Fig. 18.
our case the P1 solution normally has a smaller error than the P2 one. When this occurs, the two solutions di er ◦ by about 180 in eclipting longitude. Since the applied method does not allow us to distinguish between prograde and retrograde rotation about the same axis, every tabulated solution has a symmetric one with equal probability, which is not reported in the table. The choice between the prograde or the retrograde reported solutions was made according to the solution given by the computation program. A comparison between the values found by us and the ones by other authors, mainly computed by di erent methods, shows a certain agreement.
5 Astrea: The rotation parameters of 5 Astrea were published by Taylor (1978) who gives only one value of 0 ; ÿ0 , by ZappalÂa and Di Martino (1986) who give four solutions of 0 ; ÿ0 and the value of a=b and b=c axes ratios, by Erikson and Magnusson (1993) who give two solutions of 0 ; ÿ0 and the value of a=b ratio and by De Angelis (1995) who gives one value of 0 ; ÿ0 and the values of both the axes ratios. Taking into account the symmetric solution, the value of 0 of our solution agrees with the ones given by the above-mentioned authors, while the ÿ0 value is in agreement only with P3 and P4 solutions by ZappalÂa and Di Martino (1986). The values of the axes
Table 1 The results of asteroid pole and shape determinations Asteroid Astreaa
5 6 Hebea 13 Egeria 26 Proserpina 34 Circe 63 Ausoniaa 66 Maja 137 Meliboea 176 Iduna 250 Bettinaa 258 Tyche 335 Roberta 352 Gisela 419 Aurelia 432 Pythia 471 Papagena 537 Pauly 984 Gretia a Marks
m
Amax
0 (P1 )
ÿ0 (P1 )
11.366 11.700 10.403 9.480 8.190 11.835 7.940 6.887 12.179 7.981 15.200 8.723 8.683 6.805 10.825 8.135 7.218 4.922
0.40 0.30 0.39 0.16 0.30 0.95 0.55 0.18 0.36 0.60 0.45 0.80 0.42 0.27 0.34 0.24 0.24 0.88
132 ± 6 128 ± 2 103 ± 4 47 ± 1 113 ± 5 125 ± 2 156 ± 6 149 ± 3 85 ± 1 95 ± 5 252 ± 15 258 ± 4 213 ± 5 13 ± 2 121 ± 9 21 ± 3 290 ± 31 228 ± 9
−58 ± 3 30 ± 3 13 ± 10 −4 ± 7 17 ± 22 −36 ± 3 62 ± 1 8±3 36 ± 1 −1 ± 30 −20 ± 15 25 ± 9 53 ± 5 −34 ± 2 65 ± 6 31 ± 3 40 ± 31 −12 ± 9
0 (P2 )
ÿ0 (P2 )
227 ± 13
0 ± 13
42 ± 24 80 ± 7
−40 ± 30 15 ± 15
192 ± 6
34 ± 6
199 ± 10
−29 ± 10
46 ± 2
47 ± 1
the asteroids for which previous pole and shape determinations have been found in the literature.
a=b
b=c
1.44 1.32 1.43 1.16 1.32 2.39 1.66 1.18 1.39 1.74 1.51 2.09 1.47 1.28 1.37 1.25 1.25 2.25
1.30 1.11 1.26 1.40 1.00 1.00 1.40 1.11 1.28 1.58 1.25 1.14 1.38 1.16 1.27 1.38 1.88 1.00
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ratios given by us are mainly greater by one-tenth than the already published values. 6 Hebe: This asteroid was studied by Gehrels and Owings (1962), Gehrels and Taylor (1977), ZappalÂa and Knezevic (1984), Magnusson (1986), Michalowski (1988), De Angelis (1995) and Dotto et al. (1995). Gehrels and Owings (1962) give one value of 0 ; ÿ0 while Gehrels and Taylor (1977) give one value of 0 ; ÿ0 and one value of the axes ratios. They are quite in agreement with our values. ZappalÂa and Knezevic (1984) give four solutions of 0 ; ÿ0 and one solution of the axes ratios. Taking into account the symmetric solution, the longitude value is in agreement with our value while only two ÿ0 values coincide with the one given by us, the other two are opposite. The axes ratios substantially agree with our values. The remaining authors give one solution of 0 ; ÿ0 and one value of the axes ratio, except Michalosky (1988) who gives only the pole coordinates. All the 0 values di er ◦ by about 220 from our values, while the ÿ0 and the axes ratio values agree with the ones given by us. 63 Ausonia: In the literature we found rotation parameters and shape elements of 63 Ausonia by ZappalÂa et al. (1983), ZappalÂa and Knezevic (1984), Magnusson (1986) and De Angelis (1995). The value of 0 and those of the axes ratios, given only by ZappalÂa et al. (1983), agree very well with those given by us while, among those given by ZappalÂa and Knezevic (1984) (four values of 0 ; ÿ0 and one value of the axes ratios), the ÿ0 value of the P1 and P2 solutions is in disagreement, being opposite to the value found by us. Well in agreement are the data published by Magnusson (1986), who gives a pair of solutions 0 ; ÿ0 and one value of a=b and b=c and those published by De Angelis (1995) who only gives a value of the pole longitude and latitude and of the axes ratios. 250 Bettina: The data of this asteroid found in the literature are on an average in disagreement with those reported by us. Among the values of 0 ; ÿ0 and of the axes ratios published by Drummond et al. (1991) only the pole coordinates agree while the shape elements di er by some tenth. The values by Dotto et al. (1992), that present four solutions of 0 ; ÿ0 and one value of a=b and b=c, are those that show the best agreement with our values. The other values found in the literature by Michalowski (1992,1993) Erikson and Magnusson (1993) and De Angelis (1995), all present two solutions of 0 ; ÿ0 and one value of the axes ratios, present an agreement only with the value of the longitude and of some value of the axes ratios. Acknowledgements The authors would like to give their thanks to M. Di Martino, G. De Sanctis and P. Tanga for the collaboration to the observations and to D. Recupero for the help in editing this note. The work has been supported by grant ASI ARS-98-87 from the Italian Space Agency.
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References Ahmad, I.I., 1954. Photometric studies of asteroids. IV. The lightcurves of Ceres, Hebe, Flora, and Kalliope. Astrophys. J. 120, 551–560. Barucci, M.A., Di Martino, M., Fulchignoni, M., 1992. Rotational properties of small asteroids: photoelectric observations. Astron. J. 103, 1679–1686. Binzel, R.P., 1987. A Photoelectric Survey of 130 Asteroids. Icarus 72, 135–208. Blanco, C., Di Martino, M., Riccioli, D., 2000. New rotational period of 18 asteroids. Planetary and Space Science, 48, 271–284. Blanco, C., Riccioli, D., 1998. Pole coordinates and shape of 30 asteroids. Astron. Astrophys. Suppl. Ser. 131, 385–394. Chang, Y.C., Chang, C.S., 1962. Photometric investigation of seven variable asteroids. Acta Astron. Sinica 10, 101–110. Chang, Y.C., Chang, C.S., 1963. Photometric observations of variable asteroids II. Acta Astron. Sinica 11, 139–148. De Angelis, G., 1995. Asteroid spin, pole and shape determinations. Planet. Space Science 43, 649–682. Di Martino, M., Blanco, C., Riccioli, D., De Sanctis, G., 1994. Lightcurves and rotational periods of nine main belt asteroids. Icarus 107, 269–275. Di Martino, M., Cacciatori, S., 1984. Photoelectric photometry of 14 asteroids. Icarus 60, 75–82. Di Martino, M., Ferreri, W., Fulchignoni, M., De Angelis, G., Barucci, M.A., Lecacheux, J., Burchi, R., Di Paolantonio, A., 1990. 66 Maja and 951 Gaspra: Possible Flyby Targets for Cassini and Galileo Missioms. Icarus 87, 372–376. Dotto, E., Barucci, M.A., Fulchignoni, M., Di Martino, M., Rotundi, A., Burchi, R., Di Paolantonio, A., 1992. M-type asteroids: rotational properties of 16 objects. Astron. Astrophys. Suppl. Ser. 95, 195–211. Dotto, E., De Angelis, G., Di Martino, M., Barucci, M.A., Fulchignoni, M., De Sanctis, G., Burchi, R., 1995. Pole orientation and shape of 12 asteroids. Icarus 117, 313–327. Drummond, J.D., Weidenschilling, S.J., Chapman, C.R., Davis, D.R., 1991. Photometric geodesy of main belt asteroids. Icarus 89, 44–64. Erikson, A., Magnusson, P., 1993. Pole determinations of asteroids. Icarus 103, 62–66. Gehrels, T., Owings, D., 1962. Photometric studies of asteroids. IX. Additional lightcurves. Astrophys. J. 135, 906–924. Gehrels, T., Taylor, R.C., 1977. Minor planets and related objects. XXII. Phase function for (6) Hebe. Astron. J. 82, 229–237. Gil Hutton, R., Campos, J., 1991. Rotation period and lightcurve of asteroid 137 Meliboea. Minor Planet Bull 18, 17–18. Hainaut-Rouelle, M.-C., Hainaut, O.R., Detal, A., 1995. Lightcurves of selected minor planets. Astron. Astrophys. Suppl. Ser. 112, 125–142. Harris, A.W., Young, J.W., 1989. Asteroids lightcurve observations from 1979–1981. Icarus 81, 314–364. Harris, A.W., Young, J.W., Dockweiler, T., Gibson, J., 1992. Asteroid lightcurve observations from 1981. Icarus 95, 115–147. Lagerkvist, C.-I., 1978. Photographic photometry of 110 main-belt asteroids. Astron. Astrophys. Suppl. 31, 361–381. Lagerkvist, C.-I., 1981. Physical studies of asteroids II: photoelectric observations of the asteroids 63, 93, 135 and 409. Astron. Astrophys. Suppl. Ser. 44, 345–347. Lagerkvist, C.-I., Erikson, A., Debehogne, H., Festin, L., Magnusson, P., Mottola, S., Oja, T., De Angelis, G., Belskaja, I.N., Dahlgren, M., Gonano-Beurer, M., Lagerros, J., Lumme, K., Pohjolanen, S., 1995. Physical studies of asteroids. XXIX. Photometry and anlysis of 27 asteroids. Astron. Astrophys. Suppl. Ser. 113, 115–129. Lagerkvist, C.-I., Hahn, G., Magnusson, P., Rickman, H., 1987. Physical studies of asteroids XVI: photoelectric photometry of 17 asteroids. Astron. Astrophys. Suppl. Ser. 70, 21–32.
982
C. Blanco et al. / Planetary and Space Science 48 (2000) 973–982
Lagerkvist, C.-I., Magnusson, P., Debehogne, H., Ho mann, M., Erikson, A., De Campos, A., Cutispoto, G., 1992. Physical studies of asteroids. XXV: photoelectric photometry of asteroids obtained at ESO and Hoher List Observatory. Astron. Astrophys. Suppl. Ser. 95, 461–470. Lagerkvist, C.-I., Magnusson, P., Williams, I.P., Buontempo, M.E., Gibbs, P., Morrison, L.V., 1989. Physical studies of asteroids. XIX. Phase relation and composite lightcurves obtained with the Carlsberg Meridian Circle. Astron. Astrophys. Suppl. Ser. 78, 519–532. Lagerkvist, C.-I., Rickman, H., 1981a. The spin periods of M asteroids. Moon and Planets 24, 437–440. Lagerkvist, C.-I., Rickman, H., 1981b. Physical studies of asteroids V: photoelectric observations of the asteroids 70, 101, 369 and 432. Astron. Astrophys. Suppl. Ser. 45, 177–179. Lagerkvist, C.-I., Williams, I.P., 1987. Physical studies of asteroids XV. Determinations of slope parameters and absolute magnitudes for 51 asteroids. Astron. Astrophys. Suppl. Ser. 68, 295–315. Licandro, J., Gillardo, T., Tancredi, G., 1990. Photometric observations of asteroids 31, 118, 13, 196 and 471. Rev. Mex. Astron. Astroÿs. 21, 590–592. Lustig, G., 1977. The period of rotation and the photoelectric lightcurve of the minor planet 471 Papagena. Astron. Astrophys. Suppl. 30, 117–119. Magnusson, P., 1986. Distribution of Spin Axes and Senses of Rotation for 20 Large Asteroids. Icarus 68, 1–39. Magnusson, P., Lagerkvist, C.-I., 1991. Physical studies of asteroids XXII. Photoelectric photometry of the asteroids 34, 98, 115, 174, 270, 389, 419 and 804. Astron. Astrophys. Suppl. Ser. 87, 269–275. Michalowski, T., 1988. Photometric astrometry applied to the asteroids: 6, 15, 43 and 624. Acta Astronomica 38, 455–468. Michalowski, T., 1992. Spin vector of asteroids 21 Laetitia, 196 Philomela, 250 Bettina, 337 Devosa and 804 Hispania. In: Harris, A.W., Bowell, E. (Eds.), Asteroids, Comets, Meteors 1991. Lunar and Planetary Institute, Houston, pp. 417–419. Michalowski, T., 1993. Poles shapes, senses of rotation, and sidereal periods of asteroids. Icarus 106, 563–572. Pavlovski, K., Knezevic, Z., Muminovic, M., 1981. Photometry of the asteroid (5) Astrea. Astron. Obs. Sarajevo 1, 225–230. Piironen, J., Bowell, E., Erikson, A., Magnusson, P., 1994. Photometry of eleven asteroids at small phase angles. Astron. Astrophys Suppl. Ser. 106, 587–595.
Riccioli, D., Blanco, C., 1995. Amplitude–Longitude (A–) plot catalogue of asteroids. Catania Astrophysical Observatory Special Publication 1995, 143. Riccioli, D., Blanco, C., Cigna, M., 1999. New rotational period of Asteroids. II. Planetary and Space Science (submitted). Scaltriti, F., ZappalÂa, V., 1977. A photometric study of the minor planet 63 Ausonia. Icarus 31, 498–502. Scaltriti, F., ZappalÂa, V., 1978. Photoelectric photometry of asteroids 37, 80, 97, 216, 270, 313, and 471. Icarus 34, 428–435. Scaltriti, F., ZappalÂa, V., 1979. Photoelectric photometry and rotation periods of the asteroids 26 Proserpina, 194 Prokne, 287 Nephthys, and 554 Peraga. Icarus 39, 124–130. Schober, H.J., Erikson, A., Hahn, G., Lagerkvist, C.-I., Albrecht, R., Orning, W., Schroll, A., Stadler, M., 1994. Physical studies of asteroids. XXVIII. Lightcurves and photoelectric photometry of asteroids 2, 14, 51, 105, 181, 238, 258, 369, 377, 416, 487, 626, 679, 1048 and 2183. Astron. Astrophys. Suppl. Ser. 105, 281–300. Surdej, A., Surdej, J., 1977. Rotation period and photoelectric lightcurves of asteroid 471 Papagena. Astron. Astrophys. Suppl. 30, 121–124. Taylor, R.C., 1978. Minor planets and related objects. XXIV. Photometric observations for (5) Astrea. Astron. J. 83, 201–204. Weidenschilling, S.J., Chapman, C.R., Davis, D.R., Greenberg, R., Levy, D.H., Binzel, R.P., Vail, S.M., Magee, M., Spaute, D., 1990. Photometric Geodesy of main-belt Asteroids III. Additional lightcurves. Icarus 86, 402–447. Yang, X.-y., Zhang, Y.-y., Li, X.-q., 1965. Photometric observations of variable asteroids. III. Acta Astron. Sinica 13, 66–73. ZappalÂa, V., Cellino, A., Barucci, A.M., Fulchignoni, M., Lupishko, D.F., 1990. An analysis of the amplitude-phase relationship among asteroids. Astron. Astrophys. 231, 548–560. ZappalÂa, V., Di Martino, M., 1986. Rotation axis of asteroids via amplitude-magnitude method: results for 10 objects. Icarus 68, 40– 54. ZappalÂa, V., Knezevic, Z., 1984. Rotation Axes of Asteroids: Results for 14 Objects. Icarus 59, 436–455. ZappalÂa, V., Scaltriti, F., Di Martino, M., 1983. Photoelectric Photometry of 21 Asteroids. Icarus 56, 325–344. Zhou, X.-h., Yang, X.-y., Wu, Z.-x., 1982. Lightcurves of variable asteroids (V). Acta Astron. Sinica 23, 349–352.