2 N AT IONAL A E RO N A UT I CS AN D SPACE ADM I N ISTRAT IO N Technical Report 32-7382 The Motion of (48) Doris and the Mass of Jupiter J. William Zielenbach JET PROPULSION LABORATORY CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA April 15, 1970 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Technical Report 32-7382 The Motion of (48) Doris and the Mass of Jupiter J. William Zielenbach JET PROPULSION LABORATORY CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA April 15, 1970 Prepared Under Contract No. NAS 7-1 00 National Aeronautics and Space Administration Preface The work described in this report was performed by the Mission Analysis Division of the Jet Propulsion Laboratory. JPL TECHNICAL REPORT 32-1382 iii The major portion of this work was completed while the author was a student in Washington, D.C. However, much of the final computation and critical analysis was performed in the Mission Analysis Division at JPL. The author wishes to acknowledge the assistance of Raynor L. Duncombe of the United States Naval Observatory (USNO) and Douglas A. OHandley of JPL, who directed the research. Paul M. Janiczek of the USNO and Jay H. Lieske and J. Derral Mulholland of JPL contributed much advice and constructive criticism. Paul Herget of the Cincinnati Observatory made available the observation files of the Minor Planet Center. Francis J. Heyden, S.J., provided the facilities of the Georgetown College Observatory. Computing support came from the staff members and facilities of Georgetown University, the USNO, and JPL. Finally, the author appreciates the effort of those who responded to requests for positions remeasured from old plates: Th. Weimer, Algiers; G. v. Herk, Leiden; J. Dommanget, Uccle; A. Rohrmann, Heidelberg; R. Carrasco, Madrid; J. A. Bruwer, Johannesburg; H. Hirose, Tokyo; Y. Vaisala, Turku; C. Cristescu, Bucharest; L. Galkin, Crimean Astrophysical Observatory, H. J. M. Abraham, Mt. Stromlo; J. Fhbrer, Barcelona; H. L. Giclas, Lowell; Betty Mintz, USNO; and the Goethe Link Observatory. iv 32-I382 8 ntroduction ................ 1 II . Numerical Integration ............. 2 A . Equations To Be Integrated ........... 2 B . Method of Integration ............ 4 111 . Numerical Integration (Details of Application) ....... 6 1V . Least-Squares Differential orrection ......... 9 A . Conditional Equations ....... .... 9 B . Formation and Solution of Normal Equations ...... 9 V . Differentias Correcfion Coefficients ..........11 bservations ............14 A . Observed Angles ..............14 B . Observation Time .............16 C. Observatory Location ............17 VI1 . Discussion of bservations ............17 A . Collection and Selection ............17 B . Distribution ...............19 C. Weighting .............. 19 D . Residual Analysis . ............19 E . Catalog Corrections .............24 VIII . Discussion of Final Results .... ..... 24 Appendix A . Modifications to Newcomb's Theory of the Sun .....30 Appendix B . Eckert-Brouwer Differential Correction Coefficients ...31 Appendix C . Observations and Residuals .......34 References ........ 48 1. Physical constants ..... ....... 9 2 . Reciprocal solar masses ............ 9 3 . Observatories ............... 18 ~~1~~6REPORT 32-7382 Contents (contd 1 Tables (contd) 4 . Analysis of residuals by observatory and type ......23 5 . Comparison of published and rereduced observations ....24 6 . Definitive elements for (48) Doris based on the system of masses in Table 1 ...........27 7. Correlations in the solution for Keplerian elements .....29 8 . Normal matrices for solutions with ar/% and variational equations, and with &/am and Eckert-Brouwer set 3 .............29 A.1 . Replacements in Newcomb's theory of the sun ......30 B.1 . Eckert-Brouwer set 1 .............32 B.2 . Eckert-Brouwer set 2 .............32 8.3 . Eckert-Brouwer set 3 .............33 Figures 1 . Comparison of a predictor-only run with a predictor-corrector run. both of eighth order and 4-day stepsize ...... 7 2 . Effect of stepsize-predictor-corrector run at 8 days. minus one at 4 days ........... 8 3 . Right-ascension residuals for reference orbit .......20 4 . Distribution of observations ...........21 5 . Equatorial x and y coordinates of earth and (48) Doris at observation times ..........22 6 . Numerically integrated partial derivatives ax/ax, .....25 7 . Finite difference Ax/Ax,- numerically integrated ax/ax. .................26 8 . Declination residuals for reference orbit ........28 vi JPL TECHNICAL REPORT 32-7382 Abstract A definitive orbit is obtained for (48) Doris based upon the provisional reciprocal mass of Jupiter. Numerically integrated variational equations for the coordinates of Doris with respect to its initial rectangular coordinates and velocities and the mass of Jupiter are used to differentially correct the orbit of Doris and the mass of Jupiter. The reciprocal mass obtained, using 617 observations over a 110-yr time span, is 1047.340 20.016. JPL TECHNICAL REPORT 32- 1382 vii he Motion of (4 oris and the Mass of Jupiter 1. Introduction planets. Moreover, the general location of the belt of asteroids between Mars and Jupiter makes the asteroids Jupiter is the most massive planet in the solar system highly susceptible to Jovian perturbations. The para- and has an important gravitational influence on the motion graphs that follow indicate the reasons why certain of all other bodies in the solar system. Since the gravi- asteroids are more suitable than others for determining tational effect is dependent upon the mass of Jupiter, the mass of Jupiter by classical astronomical methods. this mass must be determined for accurate representation of planetary and interplanetary motions. From the theory of general perturbations (Ref. 2, p. 467), it is known that the disturbing function 9 for any object The mass of the planet itself is not easily determined perturbed by a body of mass m’ is given by from the motion of its satellites because the size and shape of the disk make it difficult to measure their posi- tions with respect to the center of the planet. The 9 = k2 interaction of the inner satellites is quite complicated, and relatively few observations have been made of the cos (in + kw + mM + yn’ + k’w’ + m’M’) outer, less perturbed satellites. Consequently, investi- gators have tended to analyze the gravitational effect of where a, e, i, 0, 0, and M are the usual Keplerian ele- the whole Jupiter system at interplanetary distances. ments for the perturbed body and the primed quantities are the elements of the perturbing planet. The analytical Newcomb (Ref. 1) pointed out that perhaps the best expressions for the time variations of the elements of the determination of the mass by classical astronomical perturbed body are obtained by integration. Since methods would be that afforded by the motion of minor the mean anomaly M can be written as planets (asteroids) because their positions can be more precisely observed than can those of comets or major M=nt+n 1 where n is the mean motion and t is the time elapsed results are critically analyzed, and the various compu- since M equaled U, the trigonometric term can be written tational aspects of the problem are discussed with the benefit of hindsight and with an eye to future research. cos [(in + j'n')t + e(i,j',k,k',m,m',s2,sz',","',o,d)l which, when integrated, will have a term involving II. Numerical integration (in + Yn') in the denominator. If there is a near com- Numerical solution of differential equations has be- mensurability of the mean motions n and n' for indices come common with the advent of electronic computers. i and j', the resulting coefficient of the theory is large and This is especially true for cases of perturbed motion for the period of the trigonometric term is long. which no complete analytical formulation is available. A typical example is the calculation of planetary orbits by In 1873, G. W. Hill (Ref. 3) drew attention to the fact the method of special perturbations. This section begins that the Hecuba group of minor planets has nearly 2:l with a description of the basic differential equation of commensurabilities of mean motions with Jupiter. This motion to be integrated, along with its partial derivatives gives rise to periodic perturbations of large amplitudes, with respect to various parameters. It concludes with a whose periods are short enough to be observed within brief description of the techniques used in the integration a reasonable length of time. He pointed out that, since and a presentation of general starting conditions. proximity to Jupiter greatly affects the magnitude of the perturbations, asteroids with large semimajor axes, as The equation of motion and its derivatives have been well as highly elliptic orbits whose aphelia are near expressed in a center-of-mass (c.m.) frame because the Jupiter, would be most desirable (as long as the mutual amount of computation required to evaluate E' for one inclination of the two orbits is small). These criteria are object perturbed by n planets and the sun is roughly fulfilled to a greater or lesser degree by the 13 minor (n + 1)/(2n + 1) of that required in heliocentric co- planets he recommended for future observation and ordinates. Transforming any quantity from barycentric analysis. to heliocentric merely involves subtracting the appro- priate barycentric value for the sun. Depending upon the Minor planet (48) Doris is one of this group. Its long number of equations being integrated and the bodies to period term is about 72 yr, and there has