Comparison of numerical integration of the motion of Saturn with Hill's tables.

Abstract

view Abstract Citations References Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Comparison of numerical integration of the motion of Saturn with Hill's tables. Clemence, G. M. Abstract The integration by the Selective Sequence Electronic Calculator of the motion of Saturn 1653-2060 has been compared with positions of Saturn taken from H ll's tables, at intervals of 1920 days. The rectangular equatorial coordinates referred to the mean equinox and equator 1950.0, given by the integration, were transformed to heliocentric longitude, latitude, and radius vector, referred to the mean equinox and ecliptic of date. Errors are indicated in all three of the coorodinates extracted from the tables. The salient features of the curve showing the error in the longitude are (I) a term with period the same as that of Saturn's revolution, with a coefficient that increases continuously with the time before and after the epoch of the tables, 1850, reaching 2" in 200 years, which is attributed to an error in Hill's secular perturbations, and (2) a term of long period, about 900 years, with coefficient about 4", attributable to an error in Hill's value of the great inequality. Solutions by the method of least squares give a correction to the mean longitude of Hill's tables -4'.'44 0(09 P 2(84 sin(5g - 2g') 2'(78 cos(5g - where P is the time in centuries after 1850, g is the mean anomaly of Saturn, and g' that of Jupiter. After removing these terms from the differences, another solution for the secular terms gives a correction to the true longitude of Hill's tables -0(03 sing 0(47 cos g - 0(57 P sing o'(o6 Pcosg - 0(25 P2 sing 0(11 P1 cosg. In both of the expressions above, the first two terms are without significance, so far as errors in Hill's tables are concerned; they simply represent the elements of the orbit that must be adjusted simultaneously with solutions for the other terms. When these additional terms are removed, the residuals still show strongly systematic trends, reaching I", which so far have not been amenable to analysis. The reality of the long-period and secular terms cannot be doubted, but it is possible that the solution for the secular terms has been affected by the presence of other terms of similar period. The curve for the error in the radius vector is strongly periodic, with semiamplitude about 0.00005 astronomical units. The errors cannot be connected with a single periodic term in Hill's theory; they must be of a more complicated nature. The curve for the error in the latitude also shows evidence of periodicity. The size of the errors is much smaller than for the other two coordinates, the average difference without regard to sign being o'( 10. It is probable that they are associated with errors in Hill's secular perturbations. The small size of the errors in the latitude must be associated with the relatively small size of the latitude perturbations themselves, which results from the disturbing planets being nearly in the plane of Saturn's orbit, and they constitute strong external evidence that no gross error has been committed in the numerical integration. The general conclusion is that Hill's tables of Saturn contain important errors, which increase with the time, reaching 6" during the next century. The cause is not completely known; to ascertain it would probably involve the repetition of much of Hill's work. This work was supported by the Office of Naval Research. U.S. Naval Observatory, Washington, D. C. Publication: The Astronomical Journal Pub Date: April 1951 DOI: 10.1086/106605 Bibcode: 1951AJ.....56R..35C full text sources ADS |