ICARUS 54, 13-22 (1983) Pole Orientation of Asteroid 44 Nysa via Photometric Astrometry, Including a Discussion of the Method's Application and Its Limitations R. C. TAYLOR* AND EDWARD F. TEDESCOt 'j *Lunar and Planetary Laboratory, The University of Arizona, Tucson, Arizona 85721, and t Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109 Received April 19, 1982; revised December 16, 1982 The results of photometric astrometry, a method of determining the orientation of a rotation axis, as applied to asteroid 44 Nysa are presented. The pole orientation of Nysa was found to be h0 = 100 °,/30 = +60 ° with an uncertainty of 10°. The sidereal period is 0a.26755902 -+ 0.00000006, and the rotation prograde. Refinements to, and limitations of, the application of the method of photometric astrometry are discussed. In light of the results presented herein, we believe that all photometric astrometry pole determinations of the past should be redone. INTRODUCTION shifts (see Section IV of Taylor, 1979) do not need to be applied because the transit Photometric astrometry is a method for times of the zero meridian have already determining the pole orientation, sidereal been corrected in a similar fashion. period, and sense of rotation of solar sys- tem objects from measurements of intervals I. OBSERVATIONS between epochs over long time scales. De- Figure I presents five lightcurves of tails of the method are given in Taylor Nysa. The observers and telescopes used (1979) and summarized briefly in this paper. are given in the figure caption. Two light- We present results of applying this method curves totaling less than 2 hr (from April 12, to asteroid 44 Nysa. Our computer code 1970 and March 30, 1977) are not included. was checked using Mars as a test case since Table I gives the aspect data and photomet- its sidereal period, sense of rotation, and ric parameters for the asteroid, k and B are pole orientation are well known. The dis- the geocentric longitude and latitude, A and cussions on Mars within this paper are r the geocentric and heliocentric distances, given in order to explicate the method of and o~ the solar phase angle. V0(l,o0 is the photometric astrometry, its efficacy, and its magnitude of the primary maximum at limitations. phase angle a corrected to unit distance The fundamental input to photometric as- from the Sun and Earth. B-V and U-B are trometry is the time that a surface feature the observed colors. Table I! gives posi- on an asteroid, which is normally assumed tions and photometry data for the compari- to be represented by a lightcurve extre- son stars. Table III presents the lightcurves mum, is observed from the Earth. In the used in applying photometric astrometry case of Mars, equivalent times are the Uni- along with an identification label and ref- versal times of transit of the zero meridian erence. across the center of the geometric disk. These times are tabulated in the American II. ESTIMATING THE SIDEREAL PERIOD Ephemeris and Nautical Almanac. Time The average of the synodic periods of an i Currently a National Research Council Resident asteroid over its entire orbit is called the Research Associate at the Jet Propulsion Laboratory. mean synodic period and defines the num- 13 0019-1035/83 $3.00 Copyright © 1983 by Academic Press, Inc. All rights of reproduction in any form reserved. 14 TAYLOR AND TEDESCO ' I ' I ' I ' 1 mOO 0c #0o • e~, @ : • ..-":" • ® IC 0 ° 2C % 3c ; • 3C II dUN 1970 • .4C 4O • 0 ° B A • I n L ~ ] , ] I I i I , l L 4:00 5:00 6:00 7:00 O:O0 UT oo 6 oo ?oo OT I I ' I ' I ~ ] f m l ' I ~o O0 o• @ ql• •ee f•o 0% .IO ," o I0 MAR 1977 .20 I , I J I 5:00 6:00 7:00 UT ' I ' I ' I ' I • ° o m ' I oO O0 o~eOo 22 MAR 19'77 "-o ,., • o~ .10 % ..,, .,, cP \ .20 ® o • o 22 May 1981 • 30 D E I ~ I , I , I ~ i , I l , t , 1 , I i 300 4:00 5:o0 6:oo 7:00 8:00 UT 12 O0 t3 O0 14 O0 15 O0 UT FIG. 1. (A, B) Lightcurves of Nysa obtained by R. E. Sather at the Kitt Peak National Observatory No. 3 41-cm telescope. (C) Lightcurve of Nysa obtained by J. Degewij at the University of Arizona's Catalina Station 102-cm telescope. (D) Lightcurve of Nysa obtained by J. Degewij at the University of Arizona's Catalina Station 155-cm telescope. (E) Lightcurve of Nysa obtained by P. V. Birch at the Perth Observatory 61-cm telescope. ber of synodic rotation cycles, N, over long synodic period by only a few seconds and intervals. Taylor (1979, p. 488) noted that can be found by estimating the asteroid's "The most pressing problem in photomet- sidereal rotation period. Each possible ric astrometry is... the determination of mean synodic period generates a different a unique mean synodic period..." since N, each of which yields its own pole and this must be known in order to determine sidereal period. The true mean synodic pe- N. riod is the one which most closely agrees We can now unambiguously determine with the estimated sidereal period and the mean synodic period. The mean syn- sense of rotation. odic period will differ from the observed The estimated sidereal period and sense PHOTOMETRIC ASTROMETRY OF 44 NYSA 15 of rotation are found via linear least- squares fit to the object’s angular velocity (A$JAt) on the sky versus synodic periods derived from timings of lightcurve extrema within a single opposition. The intersection at A$lAt = 0 is the synodic period at the stationary point (i.e., - the sidereal pe- riod). A positive slope indicates prograde (and a negative slope retrograde) rotation. The absolute value of the slope is s(P,,,J2/ 360 which for Nysa is 0.00019 day2/deg. Using lightcurve observations from the 1979 apparition (Birch et al., 1982), we computed the 52 independent synodic pe- riod-angular velocity pairs given in Table IV and plotted in Fig. 2. Synodic periods derived from superpositions of two light- curves each having two or more extrema in common were given double weight. A weighted linear least-squares fit to these data gives an estimated sidereal period of Od267581 L 0d000002 (1~). The slope, + 0.00012 + 0.00002 day2/deg, implies pro- grade rotation. III. MEAN SYNODIC PERIOD The mean synodic period is obtained by dividing each time interval between pairs of epochs by trial periods until a period is found which gives quotients within kO.1 of a whole number. There are probably a limit- less number of such “mean synodic peri- ods” that could be generated; in this case there were four within a 13-min span. For example, for the Mars data used in our check, the true mean synodic period was ld027497 + OrlOOOOO9since it alone gave the estimated sidereal period (ld02596) and sense of rotation obtained via the analysis discussed in Section II (see Fig. 3) using the techniques reviewed in Section V. About 2 min from the correct mean synodic period lies another at ld028845. This one, how- ever, gives a sidereal period of ld027305 (the open circle in Fig. 3) and therefore is rejected. For Mars, we found that for every interval where the difference in solar phase angle was greater than 20” the quotient of the time interval divided by the adopted 16 TAYLOR AND TEDESCO TABLE II IDENTIFICATION AND PHOTOMETRY OF THE COMPARISON STARS AND THE QUALITY OF THE NIGHTS Date RA Declination V B-V Scatter of Comments (1950) Comparison (mag) 1970 Jun 10 14h4m.7 -6054 ' 11.91 -+ 0.01 +0.96 -+ 0.02 0.018 The same star used both nights Jun 11 14 4.7 -6 54 11.92 - 0.03 +0.94 -+ 0.02 0.018 Windy both nights 1977 Mar 10 Not recorded No photometry done 0.003 Mar 22 Not recorded No photometry done 0.007 Four comparison stars used inadvertently mean synodic period differed from a whole TABLE Ill number by more than 0.1 and was therefore rejected. This difficulty, although not yet NYSA LIGHTCURVES USED IN physically understood, is easily avoided by PHOTOMETRIC ASTROMETRY never using intervals involving such large Date ID Reference differences in phase angle. From Table III we selected the Nysa 1949 Nov 6 1 Shatzel (1954) lightcurves (ID = 1-5, 7-11, 16-20, 22, 23, Nov 7 2 Shatzel (1954) 26) that were used to find its mean synodic 1954 Jan 6 3 Groeneveld and Kuiper (1954) period. Lightcurve extrema were used only Jan 7 4 Groeneveld and Kuiper if their times could be determined to within (1954) +-5 min. Intervals were formed from tim- Jan 11 5 Groeneveld and Kuiper ings between extrema from different oppo- (1954) sitions where the solar phase angles dif- 1958 Jan 13 6 Gehrels and Owings (1962) 1962 Mar 2 7 Chang and Chang (1962) fered by less than 20 ° . 1964 Oct 8 8 Yang et al. (1965) By examining a few test cases it was Oct 29 9 Yang et al. (1965) found that the difference between a mean 1970 Jun 10 10 This paper synodic period and the sidereal period it Jun 11 11 This paper generated was -0q00005. Since the esti- 1974 May 16-17 12 Zappal/~ and van Houten Groeneveld (1979) mated sidereal period was 0.d26758 (see Sec- 1977 Mar 22 13 This paper tion II), it follows that the expected mean 1979 Jul 30 14 Birch et al.