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MEMOIRS






OF THE



NATIONAL ACADEMY OF SCIENCES



"Volume

SEVENTH MEMOIR



WASHINGTON

GOVERNMENT PRINTING OFFICE

1910



NATIONAL, ACADEMY OF SCIENCES.



Volume X.



SEVENTH MEMOIR.



TABLES OF MINOR PLANETS DISCOVERED BY JAMES C. WATSON.




PART I.

T A. B I., E S OF

hy- k*v-

(93) MINERVA. 36/ (115) TIIYRA. W6 (139) JUEWA.

(101) HELENA. 30.7 (119) ALTHAEA. 3f$ (101) ATHOR.

(103) HERA. 33? :>>(128) NEMESIS. ZSS' (174) PHAEDRA. 30 1

(105) ARTEMIS. 236 (133) CYRENE. 2&~ (179) KLYTAEMNESTRA. JfS

BY

ARMIN O. LEUSCHNER,

\\

WITH THE ASSISTANCE OP

B. T. CRAWFORD, FRAN*K ROSS, BURT 1^. XEWKIRK,
ADELAIDE M. KOBE, ESTELLE GT..ANCY, AND OTHERS,



BEING IN PART A CONTINUATION OF PREVIOUS INVESTIGATIONS BY
E. BECKER, W. S. EICHELBERGEU, WILLIAM MCKNIGHT HITTER, and G. K. LAWTON.



193



670853



NATIONAL ACADEMY OF SCIENCES.
MEMOIRS.



VOLUME X.
SEVENTH MEMOIR.



Published under the Watson Trust.

TRUSTEES:

SIMON NEWCOMB, Chairman.
LEWIS BOSS.
WILLIAM L. ELK IN.



194



Page.

Preface by SIMON XEWCOMB 197

Introduction 199

Explanation of the tables 201

General plan of the tables 201

General tables 201

Tables of Jupiter's mean anomaly, g' (Table A) 201

Traverse tables (Table B) 201

Elements (Table C) 202

Special tables for the twelve planets 202

Arguments g and g' of the perturbations 203

Tables of eight planets: (105) Artemis, (115) Thyra, (128) Nemesis, (133) Cyrene, (139) Juewa, (161) Athor,

(174) Phaedra, (179) Klytaemnestra .- 204

Nonsecular portion of the perturbations 204

Secular portion of the perturbations 204

Arrangement of the tables 205

Directions for computing the perturbations ndz, 3 log r=log (1+v), and <J/3 205

Example

Tables of three planets: (101) Helena, (103) Hera, (119) Althaea 208

Nonsecular portion of the perturbations 208

Secular portion of the perturbations 208

Arrangement of the tables 209

Directions for computing the perturbations nSz, 8 log r=log (l+v), and <?/? 209

Example 209

Tables of (93) Minerva 210

Nonsecular portion of the perturbations 211

Secular portion of the perturbations 212

The long-period term in ndz , 212

Arrangement of the tables 213

Directions for computing the perturbations nSz, 3 log r=log (!+K), and o;3 213

Example 213

Directions for the computation of a geocentric position referred to the mean equinox and equator for the '

beginning of the year, from M, 3 log r, and 3p 215

Example 215

Table A. Jupiter's mean anomaly 217

Table B. Traverse tables 218

Table C. Elements 234

Tallies of (105) Artemis 236

Tables of (115) Thyra 246

Tables of (128) Nemesis '. 255

Tables of (133) Cyrene .' 2fi5

Tables of (139) Juewa : 278

Tables of (161) Alhor 291

Tables of (174) Phaedra 301

Tables of (179) Klytaemnestra 313

Tables of (101) Helena i.... 327

Tallies of (103) Hera 338

Tables of (119) Althaea 348

Tables of (93) Minerva 359

195



TABLES OF MINOR PLANETS DISCOVERED BY JAMES C. WATSON.

By ARMIN O. LEUSCHNER.
PREFACE.

By the will of JAMES C. WATSON, who died in 1880, a fund was bequeathed in trust to the
National Academy of Sciences for the purpose of promoting astronomical research. Objects
specifically designated were the awarding of a medal not oftener than once in two years for
important astronomical works and the construction of tables of the minor planets discovered
by the testator. The expenditures were to be made under direction of a board of three trustees.
The first board was named in the will, the members being J. E. HILGARD, JOHN H. C. COFFIN,
and SIMON NEWCOMB. At the present time, March, 1908, the members of the board are:
SIMOX XEWCOMB, chairman; WILLIAM L. ELKIN, and LEWIS Boss.

The construction of the tables was long delayed by the difficulty of finding computers
competent to carry the work through in a satisfactory way. To lessen this difficulty an
arrangement was made with Prof. E. BECKER, director of the Strassburg observatory, for sup-
plying a complete form for computing the perturbations according to HANSEN'S method, in
which the eccentric anomaly was taken as the independent variable. An example was also
supplied by Professor BECKER in the form of a computation of the perturbations of Eurynome by
Jupiter. Tables of Minerva were completed and published on this plan by Dr. W. S. EICHEL-

BERGER.

The conclusion subsequently reached was that the system of employing the eccentric
anomaly was not a desirable one, and that it was better to adhere to the use of the time as the
fundamental variable. Several experts were engaged in computing the perturbations of different
asteroids by Jupiter under the general direction of the writer, but in no case except that of
Minerva were the processes of tabulating the perturbations and correcting the elements brought
to a satisfactory conclusion.

The slow progress of the work made it evident that it must be prosecuted in a more sj's-
tematic way under the personal direction of a leader who would bring it to a conclusion. After
a careful survey of the field, Prof. ARMIN O. LEUSCHNER, of the University of California, was
selected as the leader, and all the papers were placed in his hands. He undertook to construct the
tables with the aid of the students and assistants in the Berkeley Astronomical Department of the
University of California. The system agreed upon was that the tables should be carried only to the
degree of precision necessary for finding ephemerides. The only perturbations then required
would be those of the first order by Jupiter, though approximate quantities of higher order
would in some cases be advisable. The construction of even these approximate tables proved
vastly more laborious than had been expected, owing to circumstances set forth by Professor
LEUSCHNER in the introduction. As the general outcome of the work up to the present time,
tables of twelve of the asteroids in question are herewith presented, with the hope that thej"
will be sufficiently accurate for as long a period as if entire theoretical precision had been aimed
at in their construction. The remaining tables are in an advanced state, and it is expected
that they will be completed and published at no distant day with the single exception of Aethra
which must await rediscovery.

SIMON NEWCOMB.

WASHINGTON, 1908, March.

197



INTRODUCTION.

The results recorded in these pages are the outcome of an effort to supplement the author's
lectures on celestial mechanics in the University of California by extensive numerical application
in the field of perturbations.

During the summer of 1901 the WATSON trustees of the National Academy of Sciences
agreed to engage students and graduates of the University of California in the work of computing
the perturbations of the minor planets discovered by WATSON on condition that the author
would assume the immediate direction of the work and sole responsibility to the trustees for its
success. The original scope of the undertaking as fixed by Prof. SIMON NEWCOMB, chairman
of the WATSOX trustees, was to embrace the numerical development of the perturbations,
including terms only of the first order with respect to the mass of Jupiter, by HANSEN'S method;
a correction of the elements by means of the differences between the computed and observed
positions for all available oppositions; and the construction of tables to facilitate the computa-
tion of positions to the nearest minute of arc, from the date of discovery to 1930.

As the undertaking is now nearing completion, it is deemed advisable to make the tables
available to astronomers in advance of the details of the investigations. The present series,
containing the tables of twelve planets, is to be followed by several others, which will contain
the tables of the remaining planets, and possibly also the detailed investigations.

Many difficulties of a theoretical as well as a practical nature were encountered during the
progress of the work, necessitating departures from the general program for individual planets,
particularly for planets of the Hecuba type.

It is unnecessary to enter here upon a discussion of the progress and organization of the
investigation. It is sufficient to say that all computations were done independently by two
computers, frequently by different methods, and that the system of checks used makes it highly
probable that the results are free from numerical error. Particular attention was paid to the
investigation of the differences between the theoretical and observed positions. The origin
of all unusually large residuals has been traced. When they occur they are, in general, accounted
for by higher order perturbations of Jupiter, or by perturbations of other major planets, or by
the fact that sufficiently accurate initial elements of the disturbed planet were not available
for the rigid computation of the coefficients of the perturbations. Corrections to the perturba-
tions, due to the differences between the initial and final elements of a planet may he included,
if it be deemed necessary, in subsequent series. Much useless labor has been caused by
erroneous identifications of the WATSON planets on the part of the observers. A comparison
of the tables with future observations will be necessary to decide whether all erroneous
observations have been eliminated.

Actual work was commenced in August, 1901, with Dr. RUSSELL TRACY CRAWFORD and
Dr. FRANK ELMORE Ross as computers. They continued in the work for one year, and in that
time computed, under the author's direction, the perturbations of ten planets in duplicate, by
HANSEN'S method. Tables of seven of these are included in the present series: (105) Artemis,
(128) Nemesis, (133) Cyrene, (139) Juewa, (161) Athor, (174) Phaedra, and (179)-.Klytaemnestra.
Since then the computations have been carried on almost entirely by university students, except
that Dr. BURT L. NEWKIRK was appointed in September, 1903, a special assistant and was
assigned with Miss ADELAIDE M. HOBE chiefly to take charge of some twelve piece computers
in revising perturbations, correcting elements, and constructing tables. Doctor NEWKIRK
and Miss HOBE have also developed the perturbations of (115) TJiyra and (93) Minerva. The
tables of Minerva, however, contained in this series, are based on a previous investigation by

Dr. W. S. ElCHELBERGER.

199



200 MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES.

Several interruptions, of which one lasted for nearly a year, were caused by the resignation
of assistants who were called to permanent positions elsewhere. Thus Doctor CRAWFORD, Doctor
Ross, Doctor NEWKIRK, and Miss HOBE were lost in turn to the work. Other interrupt ions arose
from the necessity of temporarily employing the WATSON computers as assistants in the regular
departmental work of instruction. So far three distinct sets of computers have been trained
in succession for the work. At present, the investigations are progressing with Miss ESTELLE
CLANCY, a graduate student in the university, as chief computer.

The original program, which included only those planets for which the investigation of the
general perturbations had not been undertaken by other astronomers, has since been extended
by the trustees, at their own initiative, to embrace the publication of tables of all of the
twenty-two WATSON planets excepting (132) Aethra, which is lost. Investigations, however.
by Mr. A; J. CHAMPREUX, which are under way, point to the possibility of deciding the fate of
this planet.

To enable the author to prepare all the available material for 'publication, the WATSON,
trustees placed in his hands such original computations and manuscripts as were in their pos-
session. These included, in the main, perturbations and tables of (103) Hera and (ll!H Althaea
by WILLIAM MCKNIGHT RIOTER; perturbations and tables of (93) Minerva by Dr. W T . S.
EICHELBERGER; perturbations of (101) Helena by RITTER, tables by EICHELBERGER, and
fragments of a comparison between theory and observation by G. K. LAWTOX: development
of the perturbations and partial comparison between theory and observation for (79) Eurynoim
by Prof. E. BECKER, continued by JJoasrtTR.

In regard to these planets, the trustees desired that certain discrepancies between theory
and observation should be investigated, that such revision of the results be made as might seem
necessary, and that the tables which in general gave the theoretical positions to a tenth of a
second of arc, should be abridged to correspond to the limits of accuracy fixed by the method
of investigation used in each case.

This task proved far more laborious than was anticipated. In the case of (93) Minerva
the perturbations were checked by a complete independent development. Yet, it has ulti-
mately been possible to preserve, in the main, the original results of the author's predecessors
in the work, and thus to secure for them the credit which is their due. No change whatever
has been made in EICHELBEUGER'S results on (93) Minerva.

The aim of the trustees was not that a theoretical study should be undertaken of the rela-
tive merits of the various methods of developing the perturbations of the minor planets, but
that tables of the WATSON planets be produced in the most expeditious manner. It was there-
fore not within the scope of this work to arrange for practical application to the minor planets
such analytical methods us have been made available by POIXCARE, E. W. BROWN, and others,
in a manner similar to that employed by BRENDEL in his "Theorie der Kleinen Plum-ten."
which is based on GYLDEN'S researches.

Furthermore, HAXSKX'S and BOHLIN'S methods have so far been found eminently suitable
for the objects aimed at. Nevertheless, the details of the investigation to be published later
will furnish abundant material for studies of a purely analytical nature.

The length of time required to develop perturbations by HANSEN'S method is not such an
important factor after all in these considerations, for after gaining the necessary experience
CRAWFORD and Ross were able to completely develop the perturbations of the first order in
normal cases in from forty to sixty hours.

For three planets of the Hecuba type the development of the perturbations remains to be
made. Preparatory to this, special tables for the group J have been computed on the basis
of the method employed by BOHLIN for the group J in his "Formeln und Tafeln zur gruppen-
weisse Berechnung der allgemcinen Storungen benachbarter Planeten" and "Sur le deVeloppe-
ment des Perturbations Plantaires."

The tables for the group J were barely completed when similar tables by H. vox ZEIPEL
appeared in the Memoires de 1' Academic Imperiale des Sciences de St. PStersbourg, VIII serie,



MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES. 201

Classe Physicomathematique, Volume XII, No. 11, under the title " Angeniiherte Jupitersto-
rungen fiir die Hecuba-Gruppe."

A thorough comparison of v. ZEIPEL'S and the author's tables for the group remains to
be undertaken before either are applied to the development of the perturbations of the three
planets belonging to the Hecuba type.

Acknowledgment is due to Prof. SIMON NEWCOMB, chairman of the WATSON trustees of
the National Academy of Sciences, for his constant efforts in promoting this investigation.

Acknowledgment is also due to Professor BOHLIN for facilitating the construction of the
special tables for the group by placing his revised computations for tables of group J in
the author's hands; to Director CAMPBELL, of the Lick Observatory of the University of Cali-
fornia, and to Superintendents ASA WALKER and W. J. BARNETTE, of the United States
Xaval Observatory, for furnishing especially needed observations, and finally to the author's
coworkers, particularly to Messrs. CRAWFORD, Ross, and NEWKIRK, and Misses HOBE and
GLANCY, for their untiring devotion to the numerical work connected with this undertaking.

EXPLANATION OF THE TABLES.
GENERAL PLAN OF THE TABLES.

The tables consist of general tables and special tables. The general tables are those tables
which are required for all planets in the computation of a geocentric position. The tabular
values were computed to one more figure than given in the tables, to insure correctness of the
last tabulated figure. Decimals of a degree are used throughout and computations may be
readily conducted on the basis of these tables with the aid of BREMIKER'S five-place logarithmic
tables.

The perturbations, excepting those of (93) Minerva, are developed with the argument
(igi'g'), and a uniform plan is adopted for the special tables of the various planets, except for
those planets for which the development of the perturbations and the construction of tables
was originally undertaken elsewhere. In these latter cases the original plan of tabulation has
been adhered to to avoid laborious and unnecessary transformations.

GENERAL TABLES.

In the general tables are included tables of Jupiter's mean anomaly from 1863, the year of
discovery of the first WATSON* planet, to 1930: Traverse Tables giving the products a sin A and a
cos A; and a table of Elements. All other general tables necessary for the computation of a
geocentric position may be found in BAUSCHINGER'S "Tafeln zur Theoretischen Astronomic" . ,

and in TIET^EN'S "Tafel zur Berechnung der Wahren Anomalie," which may be used in connec-
tion with the tables here given. The particular tables to be employed at various stages of the
computation of a geocentric position are referred to in the example given on page 215.

TABLES OF JUPITER'S MEAN ANOMALY, tj' (TABLE A).

The values of g' are tabulated from 1863 to 19^ They are taken from HILL'S Tables of*
Jupiter and Saturn, Tables VIII to^ff The values of g' contained in g-g', Table I of (93) ^
Minerva, however, do not correspond to the values of g' in Table A, since the value of g" used
by EICIIELBERGER in building the table g g' is the undisturbed mean anomaly of Jupiter.
It can be obtained by multiplying Arg. I of HILL'S Tables for the date +6 d .6 by the mean
motion 299". 1283756.

It should also be noted that the dates for which the values of g' and g are given in Tables A
and I refer to Berlin mean time, except in case of (93) Minerva, for which Table I refers to
Greenwich mean time.

TRAVERSE TABLES (TABLE B).

These tables are to facilitate the formation of the products of the form a sin A and a cos A,
which occur in the periodic parts of the perturbations, and are designated, at the foot of the

a Astronomical Papers prepared for the use of the American Ephemeris and Nautical Almanac. Vol. VII.



202 MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES.

tables of each planet, by o, sin ig and 6f cos ig or a t sin u and & 4 cos t. They are to be entered
with o=a< and A =ig or t- as arguments. The products are tabulated for every degree of A
from 1 to 90, and for every unit of a from 1 to 100. For a = 1 , the products are given to five
decimals; from a =2 to a = 9 to four decimals, and from a = 1 1 to a = 100 to one decimal. This
latter part of the Traverse Tables has been copied from Table II of The American Navigator
by NATHANIEL BOWDITCH. The a,- and b t coefficients are tabulated to one decimal of the
adopted unit of 0?001 for noz and u/cos i, and to one decimal of 0.00001 for <Jlog r. The
products may be taken directly from the tables by double interpolation for values a i and 6< up
to 100.0 units in each case, i. e., for coefficients not exceeding 0?1 or 0.001, respectively. If the
coefficients be larger, the products may be conveniently found in parts with the aid of that
portion of the table which is given to four and five decimals. Example : Required the product

a sin . 4 = - 2479.6 X sin 306?38; Unit of a=0?001

Since the angle is in the fourth quadrant, the algebraical sign of the product is +. The
numerical part may be taken from the Traverse Tables in the form:

(1000X2 + 100X4 + 79.6) cos 36?3X)



1000X2 cos 36?38 =1610.1 Table B, page L' Is

100X4 cos36. 38 =322.0 Tal.le I!, pane -M!)

79. 6 cos 36. 38 = 64.1 Table B, page 230



-2479.6 sin 306. 38 =1996.2 units-l?962



With BREMIKEK'S five-place tables the product is found to be 1996.3, which agrees with the
former result within the accuracy of the computation.

ELEMENTS (TABLE C).

In this table are given the final elements of the twelve planets and the quantities m and g
for computing the magnitude at opposition, the latter being taken from the Berliner Astro-
nomisches Jahrbuch.

SPECIAL TABLES FOR THE TWELVE PLANETS.

The special tables for the twelve planets are arranged in three groups.

The first group contains eight planets for which the argument of the developments is
(igi'g'), and for which the perturbations are tabulated in the form:

-i(a t sin ig + bi cos ig)+cT

where the coefficients a it b ( , and c are functions of coefficients of the original developments;
a^ and 6< are also functions of g', while c is a function of g.

The second group contains three planets for which the argument of the developments is
also (ig-i'g"), but in which all terms having (ig-i'g 1 ) or a multiple of (ig-i'g') as argument
are combined in the tables for particular values of i and i' under a single argument (ig-i'g').

If we denote the three components ndz, log (1 +i/) and 3,3 ,- by the symbol f, then the per-
turbations are tabulated in the form

-



where i is the numerical designation of the various arguments and c r is a constant.

The third group contains but one planet, (93) Minerva, for which the argument of the
developments is (ii'fi)ei'(g'/*g), and for which the perturbations are tabulated in the form

Jfflf sin ie + 2 ( bt cos it + (Jb a ) t

where the a, and b ( are functions of coefficients of the original developments and of the argu-
ment N^s-g'-fte sin s, and where (Jb a ) t is the nontrigonometrical secular part of the per-
turbations. Explicit directions for the use of the tables and an example are given with each
group.



MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES. 203

The special tables for each planet are preceded by the adopted elements, a list of the
auxiliary quantities needed in computing a geocentric position, and by the developments of nz,
M, and u/cos i.

The arrangement of the tables for the different planets has been rendered as uniform as
seemed expedient, without too extensive transformation, considering the diversity of the
original plans adopted by EICHELBERGER, HITTER, and the author. The angles throughout
are expressed in decimals of a degree. Table I of every planet gives the mean anomaly. Tables
II-IV give the nonsecular portion of the perturbations. In the tables these are designated as
periodic terms. Table V gives the secular portion of the perturbations. Table VI gives the
constants for the equator. For the first group, containing eight planets, the perturbation in
the third component is tabulated in the form u/cos i, for all others in the form dl=au/cos i.
The unit of the tabular values is printed at the head of each table.

For (93) Minerva, for which s was kept explicit in the plan adopted for tabulation by
EICHELBERGER, a Table VII is added, giving the reduction of the mean to the eccentric anomaly.
This table also contains part of the argument N of Tables II-IV of Minerva. Table I of Minerva
contains the other part of this argument.

Tables for the equation to the center and the logarithm of the radius-vector are not given,
as BAUSCHINGER'S "Tafeln zur Theoretischen Astronomic" and TIETJEN'S "Tafel zur Berech-
nung der Wahren Anomalie" answer all requirements.

The perturbations of the first group, containing eight planets, were developed to the nearest
second of arc, while the tables give the perturbations to one decimal of the adopted units 0?001
and 0.00001. These tabular values were computed only to the last figure given in the tables.
The last figure is, therefore, not exact, but was retained to insure greater exactness of the per-
turbations to the nearest 0?001 and 0.00001 . The tables, therefore, give the perturbations of the
first order well within the originally contemplated limit of one minute of arc. For the remaining
four planets the accuracy is still greater, the perturbations having been developed to the nearest
tenth of a second of arc and the values in the tables having been computed to one more decimal
than tabulated. The mean anomaly and the constants to the equator have been computed
to one and two more places than tabulated to facilitate later correction of the elements.

ARGUMENTS g AND g' OF THE PERTURBATIONS.

The perturbations are based on the initially adopted elements of a minor planet and Jupiter,



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