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Pole and Shape Determination for 12 Asteroids
ICARUS
123, 456–462 (1996) 0171
ARTICLE NO.
Pole and Shape Determination for 12 Asteroids TADEUSZ MICHAŁOWSKI Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, California 91109; and Astronomical Observatory, Adam Mickiewicz University, ul. Słoneczna 36, 60–286 Poznan´, Poland E–mail: [email protected] Received May 1, 1995; revised January 26, 1996
The results of pole and shape determination for 12 asteroids are presented. They have been obtained by using the method described in Michałowski (1993; Icarus 106, 563–572). These parameters have been determined for the first time for 4 objects and they have been improved or confirmed for the rest. Because of the small number of lightcurves available in the present study, future observations, taking into account the results from this paper, are proposed. 1996 Academic Press, Inc.
INTRODUCTION
Physical properties of asteroids, such as their shapes, spin periods, and spin axes, can improve our knowledge of their collisional evolution. It has been suggested that the outcome of collisions is strongly size dependent (Davis et al. 1989). According to this, the large asteroids, more than 100 km in diameter, are most likely to be gravitationally bound ‘‘rubble piles’’ and have quasi–equilibrium shapes due to their rapid rotations. The existence of such shapes would permit determination of the bulk densities of such asteroids. On the other hand, objects smaller than about 100 km in diameter may be fragments, whose shapes are very irregular. The determinations performed by many authors (Drummond et al. 1988, 1991; Magnusson 1990; Michałowski 1993; De Angelis 1995) showed that hydrostatic equilibrium shapes, in general, did not describe asteroids. Only a few asteroids, out of more than 50 objects, are the best candidates for quasi–equilibrium ‘‘rubble piles.’’ A distribution of spin axes seems to be quite isotropic in sin bp (bp is the ecliptic latitude of an asteroid pole), but with a slight majority of prograde rotating asteroids (Magnusson 1986, 1990, 1992; Drummond et al. 1988, 1991; Michałowski 1993; De Angelis 1995). All these authors confirmed the conclusion of Barucci et al. (1986) concerning the apparent lack of poles close to the ecliptic plane. There are three proposed explanations of this bimodality in pole distribution. First, this is a selection effect in the choosing asteroids for the study, and connected with observational data available. Second, the observed distribution 456 0019-1035/96 $18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
may simply be a statistical fluke. Third, the observed bimodality may be real, perhaps reflecting some primordial distribution. The sample of the asteroid with known poles and senses of rotation is too small and does not give any support for one of these possibilities. This paper presents results obtained for 12 asteroids, mainly because the new observations have been accessible in the literature. The main goal of the present work was to enlarge the sample of asteroids with known rotational and shape parameters which should allow a more reliable analysis in the future. RESULTS
The various methods, as described in Magnusson et al. (1989), have been used to obtain the spin vectors, sidereal periods, and triaxial ellipsoid models of asteroids. In the present paper the method described in Michałowski (1993) has been used. It combines three well-known methods (epoch, amplitude, and magnitude) in one set of nonlinear equations, which is solved by an iterative least-square fitting. The observed amplitudes and magnitudes of the brightness maxima have been reduced to zero phase angle by using the m parameter (Zappala` et al. 1990) and the HG–magnitude system (Bowell et al. 1989), respectively. The basic parameters of the asteroids are summarized in Table I. Their IRAS diameters and albedos are taken from Tedesco et al. (1992), while the taxonomic types are from Tholen (1989). The next two columns display the m and G values obtained during reduction the amplitudes and magnitudes, respectively, to zero phase angle. If the existing data were insufficient for such reduction, the average value for a given taxonomic type is taken from Zappala` et al. (1990) for the m parameter and from Harris (1989) for G. Assumed values are given in parentheses. The maximum brightness of the asteroid obtained for aspect 908 and zero solar angle H(90, 0) is shown in the last column of Table I. The source of the lightcurves used in calculation was primarily the Asteroid Photometric Catalogue by La-
POLES AND SHAPES OF 12 ASTEROIDS
TABLE I Asteroid Parameters
457
should be only 0.24–0.27 mag assuming values from other papers. In 1983 Lutetia was observed close to equatorial aspect and the amplitude was only 0.24 mag. In other words, the a/b obtained by De Angelis is overestimated. For a possible explanation, see the description of asteroid 201 Penelope below. 30 Urania
gerkvist et al. (1987, 1989, 1992, 1993) (hereafter APC). It contains all published lightcurves of asteroids up to 1992 and is more accessible than the many separate papers. Only papers with photometric data published after 1992 are referenced in present work. The results for 12 asteroids are presented in Table II. This table contains the sidereal periods, the senses of rotation (P, prograde; R, retrograde), the ecliptic coordinates for the poles, the a/b and b/c parameters of the triaxial ellipsoid models, and the method used in calculation (E, epoch; A, amplitude; M, magnitude). If any previous results exist, they are also included in Table II with appropriate references. This is not a complete list of such results. The full data can be found in Magnusson (1989). If no previous results are listed in Table II, it means the results obtained in this work are the first published for the given asteroids. 21 Lutetia This asteroid was observed during four oppositions: 1962, 1981, 1983, 1985 (APC). There are a few papers (see Table II) containing determined parameters for this asteroid. All of them are in quite good agreement with the exception of sidereal periods, and a/b as determined by De Angelis (1995). On two nights in March 1995, Denchev (personal communication) observed this asteroid again. These new data have been included in the present study. The new results (Table II) confirm the previous, however the sidereal period is longer. Lutetia was observed in longitude from 08 to 1808 and probably because of this one pole solution was obtained. As mentioned above, the a/b 5 1.41 as from De Angelis (1995) is higher than from other papers (1.25–1.29). The a/b parameter is connected with the maximum observed amplitude (in equatorial aspect) by the relation A max 5 2.5 log(a/b). According to a value presented by De Angelis (1995) such amplitude should be about 0.37 mag, while it
APC reports the observations from two oppositions: 1958 and 1978. There are now available data from 1987 (Wisniewski et al. 1995). This asteroid showed a large amplitude of about 0.53 mag in 1978 and only 0.13 mag in 1987, within the synodic period of 13.686 hr. These data do not allow a determination of the sidereal period and sense of rotation. Presented here (Table II) a pole and triaxial ellipsoid model are the first published parameters for this asteroid. 47 Aglaja There are lightcurves from four oppositions: 1978, 1979, 1984, and 1989 (APC). The synodic period is 13.178 hr and observed amplitude changed from 0.04 mag in 1979 to 0.20 mag in 1978. The sense of rotation seems to be prograde (see Table II), but the shorter sidereal period of 0.549477 days is also possible. New observations are needed to improve the results presented here. 55 Pandora APC gives the lightcurves of this asteroid obtained in four apparitions: 1977, 1982–1983, 1984, 1988–1989. Drummond et al. (1988) determined the retrograde sense of rotation for Pandora, but in Drummond et al. (1991) they reported the prograde one. Michałowski (1993) also confirmed the prograde one. In contrast, De Angelis (1995) obtained the retrograde sense of rotation (see Table II). Recently, Shevchenko et al. (1993) reported their observations from the 1989 and 1991 oppositions. These new data have been included for the calculation and the prograde sense of rotation is still confirmed (Table II). The difference in senses of rotation as reported by Drummond et al. (1988, 1991) was probably caused by not sufficient data in their earlier study. It is not clear why De Angelis (1995) determined the retrograde rotation of this asteroid. 63 Ausonia This asteroid was observed during five oppositions: 1976, 1980, 1981, 1983, and 1985 (APC). Because of these available data, Ausonia has been included in the present work. There is a quite good agreement between all existing results (see Table II).
458
TADEUSZ MICHAŁOWSKI
TABLE II Results
79 Eurynome There are lightcurves from three apparitions (1974, 1983, 1989) in APC. Using these data Michałowski and Velichko (1990) and De Angelis (1993) determined the pole and shape of this asteroid—see Table II. Velichko (personal communication) observed this asteroid in October and November 1993. The lightcurve amplitude was less than 0.1 mag. New pole and shape determination confirms the previous results but the shorter sidereal period of 0.2490087
days is still possible. The new observations should resolve this ambiguity in the period. 110 Lydia This asteroid was observed during only three apparitions: 1958–1959, 1969, and 1990 (see APC). Observed amplitudes changed between 0.15 and 0.19 mag and the synodic period is 10.927 hr. This could suggest the pole of Lydia far from the ecliptic plane.
POLES AND SHAPES OF 12 ASTEROIDS
459
TABLE II—Continued
It has not been possible to determine sidereal period and sense of rotation (see Table II). A few solutions with b/c varying from 1.6 to 2.0 with a rather large error of 0.4 have been obtained. This would be possible for an asteroid with a small number of observations and its pole far from ecliptic plane. New observations (see Table III and Fig. 1) are strongly required. 135 Hertha There are lightcurves from the 1978, 1980, 1981, 1984, and 1985 apparitions in APC. Some of them from 1985 were unknown during previous study (Michałowski 1993) and this is a reason for including this asteroid in the present paper. All obtained results are presented in Table II. The sidereal period and sense of rotation from Michałowski (1993) have not been confirmed in the present study. New observations, especially carried out in ecliptic longitudes of 458
and 2258, are required to resolve this problem. Table III and Fig. 1 show that there will not be such oppositions before 2000. 201 Penelope The lightcurves from eight oppositions, 1980, 1983, 1984, 1985–1986, 1987, 1988, 1989, and 1993 (APC; Busarev and Krugly 1995), are available for pole and shape determination of this asteroid. The results presented here agree with those previous published (see Table II). The EA method described in De Angelis (1993) and also used in De Angelis (1995) was derived from the paper by Michałowski and Velichko (1990), where a linear relation was used for a reduction of the observed amplitudes to zero phase angle. In these papers, they did this by assuming the linear relation with the same slope for all oppositions for a given asteroid. Such linear relations were confirmed by Zappala` et al. (1990). However, the slopes of
TABLE III Future Oppositions a
a
Each opposition is described by: date a,d( J2000) l,b( J2000) aspect. 460
461
POLES AND SHAPES OF 12 ASTEROIDS
is minimal, larger a/b and lower pole latitude are obtained. Probably, this is an explanation for the existing differences between results presented by De Angelis (1995) and other authors (see Table II). 505 Cava According to APC, this asteroid was observed during three apparitions: 1975, 1979, and 1982. However, the 1975 lightcurve is of very poor quality and it has not been possible to use it in calculation. Luckily, Velichko (personal communication) observed 505 Cava on May 10, 1993 and the amplitude was 0.20 mag, versus 0.11 and 0.23 mag in 1979 and 1982, respectively. The synodic period is 8.180 hr. The data from three apparitions (1979, 1982, and 1993) have not allowed a determination of the sidereal period and sense of rotation. Results for pole and shape of this asteroid are presented in Table II. 516 Amherstia
FIG. 1. Previous observed and expected amplitudes in the 1996–2000 oppositions versus the ecliptic longitude. The points for each asteroid are arbitrary shifted in amplitude scale for better display. The tickmarks on the ordinate show steps of 0.2 mag.
these relationships appear to depend linearly on the corresponding amplitudes reduced to zero phase angle. This means that the slopes of these relations depend on the aspect angles and may be different for different oppositions for a given asteroid. Zappala` et al. (1990) gave the new formula for reducing the observed amplitude to zero phase angle. This new approach has already been used for pole and shape determination (Dotto et al. 1992; Erikson and Magnusson, 1993; Michałowski 1993). If the same slope is used for all apparitions (De Angelis 1993, 1995) then it may overestimate the reduced amplitudes when they are large, and underestimate them when they are small. This effect will be very serious for asteroids which have rather large changes in their aspects and show big differences in observed amplitudes. As a consequence of using a larger amplitude when it is maximal and a smaller one while it
This asteroid was observed during only three oppositions: 1978, 1985, and 1989 (APC). The synodic period is 7.49 hr and the observed amplitude changed from 0.15 mag in 1978 to 0.48 mag in 1985. These data have not been sufficient to determine sidereal period and sense of rotation. Parameters obtained here (Table II) are in good agreement only in longitude of the pole presented by De Angelis (1995). According to the a/b from De Angelis (1995) maximum amplitude for this asteroid should be 0.64–0.67 mag. In 1985, Amherstia was close to its equatorial aspect and the amplitude was 0.48 mag, lower than predicted using the a/b determined by De Angelis. The value presented here for a/b is consistent with this observed amplitude. There are also rather large differences between b/c and the ecliptic latitude of the pole from De Angelis (1995) and presented here. 584 Semiramis There are lightcurves from five oppositions (1981, 1982– 1983, 1984, 1985, and 1987) in APC. Semiramis has been included in the present study because the data from the 1981 apparition was unknown during previous works. Results obtained here are in good agreement with those previously published (see Table II). FUTURE WORK
Results presented here do not greatly change our knowledge about the asteroid population as described in the Introduction. However, obtained values support the bimodality in pole distribution, because all studied asteroids have their poles far from the ecliptic plane. Because of a small number of existing lightcurves for the
462
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asteroids presented here, future observations are needed. Using the results from this paper, it is possible to indicate when such observations should be carried out to improve our understanding of these asteroids. Table III lists the oppositions of studied asteroids in the period 1996–2000. Each opposition is identified by its date in the first row. The next two rows show the equatorial (a, d ) and ecliptic (l, b) coordinates, respectively. The last row gives the aspect angle calculated for the first solution of the pole taken from Table II. The aspect for the second result (if it exists) is changed symmetrically according to aspect 908. The results obtained (Table II) also allow one to calculate the amplitudes for the future oppositions. These expected amplitudes together with those observed during previous apparitions are presented in Fig. 1 versus the ecliptic longitude of the asteroids. It is visible that some future oppositions will be at such geometry where the asteroids have not been observed. These apparitions are also indicated by bold letters in Table III. The new photometric data from such oppositions will be the most helpful in improving the results for the objects studied in this paper. ACKNOWLEDGMENTS This work was performed at the Jet Propulsion Laboratory while the author held a Fulbright Scholarship. F. P. Velichko and P. Devchev kindly allowed the use of their unpublished data for asteroids 21, 79, and 505. A. Kryszczyn´ska calculated ephemeries for Table III. W. D. Sears read and improved the text of manuscript.
REFERENCES BARUCCI, M. A., D. BOCKELEE-MORVAN, A. BRAHIC, S. CLAIREMIDI, J. LECACHEUX, AND F. ROQUES 1986. Asteroid spin axes: Two additional pole determination and theoretical implications. Astron. Astrophys. 163, 261–268. BOWELL, E., B. HAPKE, D. DOMINGUE, K. LUMME, J. PELTONIEMI, AND A. W. HARRIS 1989. Application of photometric models to asteroids. In Asteroid II (R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 524–556. Univ. of Arizona Press, Tucson. BUSAREV, V. V., AND YU.N. KRUGLY 1995. A spot of hydrated silicates on the M asteroid 201 Penelope? In Lunar and Planetary Science XXVI, pp. 197–198. Lunar and Planetary Institute, Houston. DAVIS, D. R., S. J. WEIDENSCHILLING, P. FARINELLA, P. PAOLICHI, AND R. P. BINZEL 1989. Asteroid collisional history: Effects on sizes and spins. In Asteroid II (R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 805–826. Univ. of Arizona Press, Tucson. DE ANGELIS, G. 1993. Three asteroid pole determinations. Planet. Space Sci. 41, 285–290. DE ANGELIS, G. 1995. Asteroid spin, pole and shape determinations. Planet. Space Sci. 43, 649–682. DOTTO, E., M. A. BARUCCI, M. FULCHIGNONI, M. DI MARTINO, A. ROTUNDI, R. BURCHI, AND A. DI PAOLANTONIO 1992. M-type asteroids: rotational properties of 16 objects. Astron. Astrophys. Suppl. Ser. 95, 195–211. DRUMMOND, J. D., S. J. WEIDENSCHILLING, C. R. CHAPMAN, AND D. R. DAVIS 1988. Photometric geodesy of main-belt asteroids. II. Analysis of lightcurves for poles, periods, and shapes. Icarus 76, 19–77.
DRUMMOND, J. D., S. J. WEIDENSCHILLING, C. R. CHAPMAN, AND D. R. DAVIS 1991. Photometric geodesy of main-belt asteroids. IV. An updated analysis of lightcurves for poles, periods, and shapes. Icarus 89, 44–64. ERIKSON, A., AND P. MAGNUSSON 1993. Pole determination of asteroids. Icarus 103, 62–66. HARRIS, A. W. 1989. The H 2 G asteroid magnitude system: mean slope parameters. In Lunar and Planetary Science XX, pp. 375–376. Lunar and Planetary Institute, Houston. LAGERKVIST, C.-I., M. A. BARUCCI, M. T. CAPRIA, M. FULCHIGNONI, P. MAGNUSSON, AND V. ZAPPALA` 1989. Asteroid Photometric Catalogue, first update. Consiglio Nazionale Delle Ricerche, Rome. LAGERKVIST, C.-I., P. MAGNUSSON, I. BELSKAYA, A. ERIKSON, M. DAHLGREN, AND M. A. BARUCCI 1993. Asteroid Photometric Catalogue, third update. Uppsala Astronomical Observatory. LAGERKVIST, C.-I., M. A. BARUCCI, M. T. CAPRIA, M. DAHLGREN, A. ERIKSON, M. FULCHIGNONI, AND P. MAGNUSSON 1992. Asteroid Photometric Catalogue, second update. Uppsala Astronomical Observatory. LAGERKVIST, C.-I., M. A. BARUCCI, M. T. CAPRIA, M. FULCHIGNONI, L. GUERRIERO, E. PEROZZI, AND V. ZAPPALA` 1987. Asteroid Photometric Catalogue. Consiglio Nazionale Delle Ricerche, Rome. LUPISHKO, D. F., AND F. P. VELICHKO 1987. Sense of rotation of asteroid 21, 63, 216 and 349. Kinem. Phys. Celest. Bodies 3, 57–65. LUPISHKO, D. F., F. P. VELICHKO, I. N. BELSKAYA, AND V. G. SHEVCHENKO 1987. Pole coordinates and phase dependence of brightness of the asteroid 21 Lutetia. Kinem. Phys. Celest. Bodies 3, 36–38. MAGNUSSON, P. 1986. Distribution of spin axes and senses of rotation for 20 large asteroids. Icarus 68, 1–39. MAGNUSSON, P. 1989. Pole determination of asteroids. In Asteroid II (R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 1180–1190. Univ. of Arizona Press, Tucson. MAGNUSSON, P. 1990. Spin vectors of 22 large asteroids. Icarus 85, 229–240. MAGNUSSON, P. 1992. Spin vectors of asteroids: Determination and cosmogonic significance. In Observations and Physical Properties of Small Solar System Bodies (A. Brahic, J.-C. Gerard, J. Surdej, Eds.), pp. 163–179. Univ. de Liege. MAGNUSSON, P., M. A. BARUCCI, J. D. DRUMMOND, K. LUMME, S. J. OSTRO, J. SURDEJ, R. C. TAYLOR, AND V. ZAPPALA` 1989. Determination of pole orientations and shapes of asteroids. In Asteroids II (R. P. Binzel, T.Gehrels, and M. S. Matthews, Eds.), pp. 66–97. Univ. of Arizona Press, Tucson. MICHAŁOWSKI, T. 1993. Poles, shapes, senses of rotation, and sidereal periods of asteroids. Icarus 106, 563–572. MICHAŁOWSKI, T., AND F. P. VELICHKO 1990. Photoelectric photometry, parameters of rotation and shapes of asteroids 22 Kalliope and 79 Eurynome. Acta Astron. 40, 321–332. SHEVCHENKO, V. G., YU.N. KRUGLY, D. F. LUPISHKO, A. W. HARRIS, AND G. P. CHERNOVA 1993. Lightcurves and phase relations of the asteroid 55 Pandora. Astron. Vestnik 27, 75–80. TEDESCO, E. F., G. J. VEEDER, J. W. FOWLER, AND J. R. CHILLEMI 1992. The IRAS Minor Planet Survey, Phillips Laboratory, Report PL-TR92-2049. THOLEN, D. J. 1989. Asteroid taxonomic classifications. In Asteroids II (R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 1139–1150. Univ. of Arizona Press, Tucson. WISNIEWSKI, W. Z., T. MICHAŁOWSKI, A. W. HARRIS, AND R. S. MCMILLAN 1996. Photoelectric observations of 125 asteroids. Icarus, submitted. ZAPPALA` , V., A. CELLINO, M. A. BARUCCI, M. FULCHIGNONI, AND D. F. LUPISHKO 1990. An analysis of the amplitude–phase relationship among asteroids. Astron. Astrophys. 231, 548–560.
123, 456–462 (1996) 0171
ARTICLE NO.
Pole and Shape Determination for 12 Asteroids TADEUSZ MICHAŁOWSKI Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, California 91109; and Astronomical Observatory, Adam Mickiewicz University, ul. Słoneczna 36, 60–286 Poznan´, Poland E–mail: [email protected] Received May 1, 1995; revised January 26, 1996
The results of pole and shape determination for 12 asteroids are presented. They have been obtained by using the method described in Michałowski (1993; Icarus 106, 563–572). These parameters have been determined for the first time for 4 objects and they have been improved or confirmed for the rest. Because of the small number of lightcurves available in the present study, future observations, taking into account the results from this paper, are proposed. 1996 Academic Press, Inc.
INTRODUCTION
Physical properties of asteroids, such as their shapes, spin periods, and spin axes, can improve our knowledge of their collisional evolution. It has been suggested that the outcome of collisions is strongly size dependent (Davis et al. 1989). According to this, the large asteroids, more than 100 km in diameter, are most likely to be gravitationally bound ‘‘rubble piles’’ and have quasi–equilibrium shapes due to their rapid rotations. The existence of such shapes would permit determination of the bulk densities of such asteroids. On the other hand, objects smaller than about 100 km in diameter may be fragments, whose shapes are very irregular. The determinations performed by many authors (Drummond et al. 1988, 1991; Magnusson 1990; Michałowski 1993; De Angelis 1995) showed that hydrostatic equilibrium shapes, in general, did not describe asteroids. Only a few asteroids, out of more than 50 objects, are the best candidates for quasi–equilibrium ‘‘rubble piles.’’ A distribution of spin axes seems to be quite isotropic in sin bp (bp is the ecliptic latitude of an asteroid pole), but with a slight majority of prograde rotating asteroids (Magnusson 1986, 1990, 1992; Drummond et al. 1988, 1991; Michałowski 1993; De Angelis 1995). All these authors confirmed the conclusion of Barucci et al. (1986) concerning the apparent lack of poles close to the ecliptic plane. There are three proposed explanations of this bimodality in pole distribution. First, this is a selection effect in the choosing asteroids for the study, and connected with observational data available. Second, the observed distribution 456 0019-1035/96 $18.00 Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
may simply be a statistical fluke. Third, the observed bimodality may be real, perhaps reflecting some primordial distribution. The sample of the asteroid with known poles and senses of rotation is too small and does not give any support for one of these possibilities. This paper presents results obtained for 12 asteroids, mainly because the new observations have been accessible in the literature. The main goal of the present work was to enlarge the sample of asteroids with known rotational and shape parameters which should allow a more reliable analysis in the future. RESULTS
The various methods, as described in Magnusson et al. (1989), have been used to obtain the spin vectors, sidereal periods, and triaxial ellipsoid models of asteroids. In the present paper the method described in Michałowski (1993) has been used. It combines three well-known methods (epoch, amplitude, and magnitude) in one set of nonlinear equations, which is solved by an iterative least-square fitting. The observed amplitudes and magnitudes of the brightness maxima have been reduced to zero phase angle by using the m parameter (Zappala` et al. 1990) and the HG–magnitude system (Bowell et al. 1989), respectively. The basic parameters of the asteroids are summarized in Table I. Their IRAS diameters and albedos are taken from Tedesco et al. (1992), while the taxonomic types are from Tholen (1989). The next two columns display the m and G values obtained during reduction the amplitudes and magnitudes, respectively, to zero phase angle. If the existing data were insufficient for such reduction, the average value for a given taxonomic type is taken from Zappala` et al. (1990) for the m parameter and from Harris (1989) for G. Assumed values are given in parentheses. The maximum brightness of the asteroid obtained for aspect 908 and zero solar angle H(90, 0) is shown in the last column of Table I. The source of the lightcurves used in calculation was primarily the Asteroid Photometric Catalogue by La-
POLES AND SHAPES OF 12 ASTEROIDS
TABLE I Asteroid Parameters
457
should be only 0.24–0.27 mag assuming values from other papers. In 1983 Lutetia was observed close to equatorial aspect and the amplitude was only 0.24 mag. In other words, the a/b obtained by De Angelis is overestimated. For a possible explanation, see the description of asteroid 201 Penelope below. 30 Urania
gerkvist et al. (1987, 1989, 1992, 1993) (hereafter APC). It contains all published lightcurves of asteroids up to 1992 and is more accessible than the many separate papers. Only papers with photometric data published after 1992 are referenced in present work. The results for 12 asteroids are presented in Table II. This table contains the sidereal periods, the senses of rotation (P, prograde; R, retrograde), the ecliptic coordinates for the poles, the a/b and b/c parameters of the triaxial ellipsoid models, and the method used in calculation (E, epoch; A, amplitude; M, magnitude). If any previous results exist, they are also included in Table II with appropriate references. This is not a complete list of such results. The full data can be found in Magnusson (1989). If no previous results are listed in Table II, it means the results obtained in this work are the first published for the given asteroids. 21 Lutetia This asteroid was observed during four oppositions: 1962, 1981, 1983, 1985 (APC). There are a few papers (see Table II) containing determined parameters for this asteroid. All of them are in quite good agreement with the exception of sidereal periods, and a/b as determined by De Angelis (1995). On two nights in March 1995, Denchev (personal communication) observed this asteroid again. These new data have been included in the present study. The new results (Table II) confirm the previous, however the sidereal period is longer. Lutetia was observed in longitude from 08 to 1808 and probably because of this one pole solution was obtained. As mentioned above, the a/b 5 1.41 as from De Angelis (1995) is higher than from other papers (1.25–1.29). The a/b parameter is connected with the maximum observed amplitude (in equatorial aspect) by the relation A max 5 2.5 log(a/b). According to a value presented by De Angelis (1995) such amplitude should be about 0.37 mag, while it
APC reports the observations from two oppositions: 1958 and 1978. There are now available data from 1987 (Wisniewski et al. 1995). This asteroid showed a large amplitude of about 0.53 mag in 1978 and only 0.13 mag in 1987, within the synodic period of 13.686 hr. These data do not allow a determination of the sidereal period and sense of rotation. Presented here (Table II) a pole and triaxial ellipsoid model are the first published parameters for this asteroid. 47 Aglaja There are lightcurves from four oppositions: 1978, 1979, 1984, and 1989 (APC). The synodic period is 13.178 hr and observed amplitude changed from 0.04 mag in 1979 to 0.20 mag in 1978. The sense of rotation seems to be prograde (see Table II), but the shorter sidereal period of 0.549477 days is also possible. New observations are needed to improve the results presented here. 55 Pandora APC gives the lightcurves of this asteroid obtained in four apparitions: 1977, 1982–1983, 1984, 1988–1989. Drummond et al. (1988) determined the retrograde sense of rotation for Pandora, but in Drummond et al. (1991) they reported the prograde one. Michałowski (1993) also confirmed the prograde one. In contrast, De Angelis (1995) obtained the retrograde sense of rotation (see Table II). Recently, Shevchenko et al. (1993) reported their observations from the 1989 and 1991 oppositions. These new data have been included for the calculation and the prograde sense of rotation is still confirmed (Table II). The difference in senses of rotation as reported by Drummond et al. (1988, 1991) was probably caused by not sufficient data in their earlier study. It is not clear why De Angelis (1995) determined the retrograde rotation of this asteroid. 63 Ausonia This asteroid was observed during five oppositions: 1976, 1980, 1981, 1983, and 1985 (APC). Because of these available data, Ausonia has been included in the present work. There is a quite good agreement between all existing results (see Table II).
458
TADEUSZ MICHAŁOWSKI
TABLE II Results
79 Eurynome There are lightcurves from three apparitions (1974, 1983, 1989) in APC. Using these data Michałowski and Velichko (1990) and De Angelis (1993) determined the pole and shape of this asteroid—see Table II. Velichko (personal communication) observed this asteroid in October and November 1993. The lightcurve amplitude was less than 0.1 mag. New pole and shape determination confirms the previous results but the shorter sidereal period of 0.2490087
days is still possible. The new observations should resolve this ambiguity in the period. 110 Lydia This asteroid was observed during only three apparitions: 1958–1959, 1969, and 1990 (see APC). Observed amplitudes changed between 0.15 and 0.19 mag and the synodic period is 10.927 hr. This could suggest the pole of Lydia far from the ecliptic plane.
POLES AND SHAPES OF 12 ASTEROIDS
459
TABLE II—Continued
It has not been possible to determine sidereal period and sense of rotation (see Table II). A few solutions with b/c varying from 1.6 to 2.0 with a rather large error of 0.4 have been obtained. This would be possible for an asteroid with a small number of observations and its pole far from ecliptic plane. New observations (see Table III and Fig. 1) are strongly required. 135 Hertha There are lightcurves from the 1978, 1980, 1981, 1984, and 1985 apparitions in APC. Some of them from 1985 were unknown during previous study (Michałowski 1993) and this is a reason for including this asteroid in the present paper. All obtained results are presented in Table II. The sidereal period and sense of rotation from Michałowski (1993) have not been confirmed in the present study. New observations, especially carried out in ecliptic longitudes of 458
and 2258, are required to resolve this problem. Table III and Fig. 1 show that there will not be such oppositions before 2000. 201 Penelope The lightcurves from eight oppositions, 1980, 1983, 1984, 1985–1986, 1987, 1988, 1989, and 1993 (APC; Busarev and Krugly 1995), are available for pole and shape determination of this asteroid. The results presented here agree with those previous published (see Table II). The EA method described in De Angelis (1993) and also used in De Angelis (1995) was derived from the paper by Michałowski and Velichko (1990), where a linear relation was used for a reduction of the observed amplitudes to zero phase angle. In these papers, they did this by assuming the linear relation with the same slope for all oppositions for a given asteroid. Such linear relations were confirmed by Zappala` et al. (1990). However, the slopes of
TABLE III Future Oppositions a
a
Each opposition is described by: date a,d( J2000) l,b( J2000) aspect. 460
461
POLES AND SHAPES OF 12 ASTEROIDS
is minimal, larger a/b and lower pole latitude are obtained. Probably, this is an explanation for the existing differences between results presented by De Angelis (1995) and other authors (see Table II). 505 Cava According to APC, this asteroid was observed during three apparitions: 1975, 1979, and 1982. However, the 1975 lightcurve is of very poor quality and it has not been possible to use it in calculation. Luckily, Velichko (personal communication) observed 505 Cava on May 10, 1993 and the amplitude was 0.20 mag, versus 0.11 and 0.23 mag in 1979 and 1982, respectively. The synodic period is 8.180 hr. The data from three apparitions (1979, 1982, and 1993) have not allowed a determination of the sidereal period and sense of rotation. Results for pole and shape of this asteroid are presented in Table II. 516 Amherstia
FIG. 1. Previous observed and expected amplitudes in the 1996–2000 oppositions versus the ecliptic longitude. The points for each asteroid are arbitrary shifted in amplitude scale for better display. The tickmarks on the ordinate show steps of 0.2 mag.
these relationships appear to depend linearly on the corresponding amplitudes reduced to zero phase angle. This means that the slopes of these relations depend on the aspect angles and may be different for different oppositions for a given asteroid. Zappala` et al. (1990) gave the new formula for reducing the observed amplitude to zero phase angle. This new approach has already been used for pole and shape determination (Dotto et al. 1992; Erikson and Magnusson, 1993; Michałowski 1993). If the same slope is used for all apparitions (De Angelis 1993, 1995) then it may overestimate the reduced amplitudes when they are large, and underestimate them when they are small. This effect will be very serious for asteroids which have rather large changes in their aspects and show big differences in observed amplitudes. As a consequence of using a larger amplitude when it is maximal and a smaller one while it
This asteroid was observed during only three oppositions: 1978, 1985, and 1989 (APC). The synodic period is 7.49 hr and the observed amplitude changed from 0.15 mag in 1978 to 0.48 mag in 1985. These data have not been sufficient to determine sidereal period and sense of rotation. Parameters obtained here (Table II) are in good agreement only in longitude of the pole presented by De Angelis (1995). According to the a/b from De Angelis (1995) maximum amplitude for this asteroid should be 0.64–0.67 mag. In 1985, Amherstia was close to its equatorial aspect and the amplitude was 0.48 mag, lower than predicted using the a/b determined by De Angelis. The value presented here for a/b is consistent with this observed amplitude. There are also rather large differences between b/c and the ecliptic latitude of the pole from De Angelis (1995) and presented here. 584 Semiramis There are lightcurves from five oppositions (1981, 1982– 1983, 1984, 1985, and 1987) in APC. Semiramis has been included in the present study because the data from the 1981 apparition was unknown during previous works. Results obtained here are in good agreement with those previously published (see Table II). FUTURE WORK
Results presented here do not greatly change our knowledge about the asteroid population as described in the Introduction. However, obtained values support the bimodality in pole distribution, because all studied asteroids have their poles far from the ecliptic plane. Because of a small number of existing lightcurves for the
462
TADEUSZ MICHAŁOWSKI
asteroids presented here, future observations are needed. Using the results from this paper, it is possible to indicate when such observations should be carried out to improve our understanding of these asteroids. Table III lists the oppositions of studied asteroids in the period 1996–2000. Each opposition is identified by its date in the first row. The next two rows show the equatorial (a, d ) and ecliptic (l, b) coordinates, respectively. The last row gives the aspect angle calculated for the first solution of the pole taken from Table II. The aspect for the second result (if it exists) is changed symmetrically according to aspect 908. The results obtained (Table II) also allow one to calculate the amplitudes for the future oppositions. These expected amplitudes together with those observed during previous apparitions are presented in Fig. 1 versus the ecliptic longitude of the asteroids. It is visible that some future oppositions will be at such geometry where the asteroids have not been observed. These apparitions are also indicated by bold letters in Table III. The new photometric data from such oppositions will be the most helpful in improving the results for the objects studied in this paper. ACKNOWLEDGMENTS This work was performed at the Jet Propulsion Laboratory while the author held a Fulbright Scholarship. F. P. Velichko and P. Devchev kindly allowed the use of their unpublished data for asteroids 21, 79, and 505. A. Kryszczyn´ska calculated ephemeries for Table III. W. D. Sears read and improved the text of manuscript.
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