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R KVI EWS OF MODERN PH YSI CS VOLUME 19, NUM HER 4 OCTOBER, f 947'. .'.xe .Xe..a1:ivity . I:I:ec1:in '. . . .ane1:ary .V. :otions

G. M. CLEMENCEU. 5. aval Observatory, Washington, D. C.INTRODUCTION' 'T is well known that, according to the generaltheory of relativity, ' the elHptical orbit of aplanet referred to a Newtonian frame of referencerotates in its own plane in the same direction asthe planet moves, with a speed that is given bybee 12m 2u'

c'T'(1')In this formula 8~/ts is the amount of rotation(commonly called the motion of the perihelion)per revolution of the planet about the sun, a ishalf the major axis of the ellipse, c is the velocityof light, T is the time required for one revolutionof the planet, and e is the eccentricity of theellipse; if a, c, and rare measured in centimetersand seconds. The fraction of a revolution throughwhich the perihelion advances during one revolu-tion of the planet is represented by b~/y, a di-mensionless number. For comparison with ob-servations it is convenient to express b&u/y inseconds of arc per century. Table I giVes thetheoretical effects derived from the formula forthe five inner planets under the name of co',based on a value of the solar parallax of 8".790.The last column gives for each planet the motionof the perihelion multiplied by the eccentricityof the orbit; the size of this quantity is a measureof the angular displacement of the planet whenit is at perihelion, and hence this is the quantitythat fixes the accuracy with which the effect canbe determined by analysis of observations.It is at once evident that the effect can bedetected most easily in the motion of Mercury.Indeed, Einstein's announcement of the generaltheory of relativity in its definitive form~ wasimmediately hailed by some astronomers as ex-plaining a previously unaccountable discrepancybetween the observed and theoretical motions of' See, e.g. , A. S. Eddington, The iVathematicel Theory ofAe4tivity (Cambridge University Press, Teddington-, Eng-land, 1924), second edition, p. 89.2A. Einstein, "Die Grundlage der allgemeinen Rela-tivitatstheorie, "Ann. d. Physik 49, /69 (1916). 3

this planet. Others were, however, intuitivelyopposed to relativity, and they directed attentionto a small discrepancy yet remaining as evidencethat the theory of relativity could not be cor-rect; the relativists contended that the smallremaining. discrepancy was due to errors eitherin the observations or in the classical theory ofthe motion. In justice it should be said that thequestions involved are not simple ones, but arecomplicated by three ca,uses: (1) Observationsof Mercury are among the most dificult inpositional astronomy. They have to be made inthe daytime, near noon, under unfavorable con-ditions of the atmosphere; and they are subjectto large systematic and accidental errors arisingboth from this cause and from the shape of thevisible disk of the planet. (2) The planet's pathin Newtonian space is not an ellipse but anexceedingly complicated space-curve due to thedisturbing effects of all' of the other planets.The calculation of this curve is a dificult andlaborious task, and significantly different resultshave been obtained by different computers.(3) The observations cannot be made in theNewtonian frame of reference. They are referredto the moving equinox, that is, they are affectedby the precession of the equinoxes, and the deter-mination of the precessional motion is one of themost difhcult problems of positional astronomy,if not the most dif6eult. In the light of @11 thesehazards it is not surprising that a difference ofopinion could exist regarding the closeness ofagreement between the observed and theoreticalmotions.I am not aware that relativity is at presentregarded by physicists as a theory that may bebelieved or not, at will. Nevertheless, it may beof some interest to present the most recentevidence on the degree of'agreement between theobserved and theoretical motions of the planets,which is the object of this article. The evidenceis of two kinds: (I) a discussion by the author'' G. M. Clemence, "The motion of Mercury 1/651937,"Astro. Pap. Am. Ephemeris 11, 1 (1943).61


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G. M. CLEM ENCEof observations of Mercury from 1765 to 1937which was intended to exhaust the useful obser-vational evidence available on the motion ofMercury at that time. In this discussion therelativity effect in the motion of Mercury isconfirmed, and some slight evidence of the effectis found in the motion of the earth as well.(2) a discussion by H. R.Morgan' of observationsof the sun, in which he concludes that the effectis present in the motion of the earth.In order to render the subject more readilycomprehensible, the results are presented hereunder a different form from that heretoforepublished and the numerical values have beenaltered slightly in three ways. Doolittle's calcu-lation' of the Newtonian motions, .with certaincorrections, is used instead of Newcomb's, somenew. values of the planetary ma.sses are intro-duced, and Oort's most' recent value' of theprecession is adopted. The observational resultsremain unchanged. It may be remarked that theeffect of the alterations has been to make theagreement between observations and theoryslightly worse instead of better, but not sig-nificantly so.THE OBSERVED MOTIONS OF THE PERIHELIA OFMERCURY AND THE EARTHUnfortunately, the observational material isso extensive and the methods of analysis so com-

plex that it is not practicable here to presentany evidence that will enable the reader to forman independent judgment of the errors involved.All that can be done is to give a very brief de-scription of the methods employed, and thenumerical results. This is not to minimize theimportance of the error estimates, which are, ofcourse, the most critical feature of the entirework; the interested reader will, it is hoped, finda sufficient discussion of the errors in thereferences.The term "probable error, " whenever it isused in what follows, is not to be understood asmeaning the quartile error obtained by multiply-4 H. R. Morgan, "The earth's perihelion motion, "Astro. J. 50, 127 (1945}.~ Eric Doolittle, "The secular variations of the elementsof the 'orbits of. the four inner planets computed for theepoch 1850.0 G.M.T.,"Trans. Am. Phil. Soc.22, 37 (1925).6 J. H. Oort, "The constants of precession arid ofgalactic rotation, " Bull. Astro. Inst. Netherlands 9, 424(1943).

TABLE I.Theoretical values of the advance of the periheliaper century.P1anet

MercuryVenusEarthMarsJupiter

43".038.633.841.350.068".8470.0590.0640.1260.003

ing the standard deviation by 0.6745. It is wellknown that the quartile error measures only theaccidental discordances of a set of data, noallowance being made for systematic errors,which in an analysis of a very extended seriesof observations are likely to be much more im-portant than the accidental discordances. Theprobable errors given here are in every instancelarger than the quartile errors, and they corre-spond more nearly to what Dorsey' has calledthe "dubiety. " I have obtained them by addingto the quartile errors the quantity which, asnearly as I could judge, represents the size ofthe largest systematic error that could affect theresults. It is difficult to define in precise languagea quantity that depends on a multiplicity ofpersonal judgments, nevertheless the attemptmust be made. By probable error I mean thatquantity which, when added to and subtractedfrom a.result, gives a range within which theprobability for the inclusion of the true value isone-half; more precisely, the probable error isintended to measure the discordance of allfuture determinations in addition to those inthe past.The observations of Mercury are of twodifferent kinds: observations of its sphericalcoordinates on the celestial sphere when it is onthe meridian, and observations of the time atwhich its disk is tangent to the disk of the sunwhen Mercury crosses the face of the sun. Themeridian observations extend from 1765 to 1937and number about 10,000 in each coordinate.Observations of 17 transits have been used, ex-tending from 1799 to 1940.The observed coordinates are not discusseddirectly, but instead the small differences be-tween the observed coordinates and those calcu-lated from a theory of the motions are used.

N. E.Dorsey, "The velocity of light, "Trans. Am. Phil.Soc. 34, 1 (1944).


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RELATIVITY EFFECT IN PLANETARY MOTIONSEach of these differences gives rise to an equationof condition, the unknown quantities being cor-rections to the constants used in the calculatedcoordinates. These equations are collected intogroups extending over about ten years each andsolved by the method of least squares. Thenumber of unknown quantities is twelve, one ofthem being the correction to the assumed, ortabular, position of the perihelion. In principle,a number of corrections at successive epochs tothis assumed position of the perihelion are ob-tained, and the sum of the corrections gives thecorrection to the assumed motioN of the peri-helion. The procedure followed with the transitsof Mercury is much the same, except that thewhole series of transits furnishes only two equa-tions of condition because transits can occur onlyin two narrow regions of Mercury's orbit. Thesetwo additional conditions are imposed on thefinal results of the meridian observations, andanother adjustment is made by least squares.Since observations of Mercury do not give theabsolute position of the planet in space but onlythe direction of a line from the planet to theobserver, they depend equally on the position ofMercury and the position of the earth, and themotion of the earth's perihelion may be intro-duced also as an unknown to be determined.Determination of motion made in this way is forseveral reasons inferior in -accuracy to thatobtained from observations of the sun, but thedeter'mination has some value.The observations of the sun are more numerousthan are those of Mercury, but they extend overabout the same length of time. The analysis issimpler because fewer unknowns have to bedetermined, but the principles involved are thesame.For the total observed rate of motion ofMercury's perihelion at 1850, referred to themoving equinox, I have found, in seconds of areper Julian century of 36,525 mean solar days,5599.74&0.41. For the earth I have foundfrom observations of Mercury 6182.0&3.6, andMorgan has obtained from observations of thesun 6183.9&1.2.Weighting the last two determi-nations in accordance with their assigned prob-able errors gives, as the definitive result to beused here for the observed motion of the earth' sperihelion, 6183.7~1.1.

T&BI,E II. Contributions to the motion of the perihelia ofMercury and the earth.Cause Motion of perihelion

m &6 000 DDD &1000000408 000 & 1 000829 890 % 8003 088000 & 3 0001 047.89& 0.088499 & 422 800 & 80019 500 & 800

MercuryVenusEarth.MarsJupiterSaturnUranusNeptuneSolar oblatenessMoonGeneral precession (Julian century, 1850)

Mercury0.025w0. 00277.856+0.6890.038+0.082.586&0.DD158.584+0'.007.802%0,010.141&0.000.042+0.000.010+0.025025.645+0.50

Earth8"75&2".8845.49+0.897.69&0.1696.85%0.018.74+0.00.57&0.00, 18%0.0D.DD+0.07.68%0.05025.65&0.5

SumObserved motion 5557.18 &0.85 6179.1 &2.55599.74 +0.41 6188.7 &1.1DifferenceRelativity effect 42.56 +0.9448.08 &0.08 4.6 +2.78.8 +0.0

and for convenience the relativity effect isomitted from the upper part of the table. Thediscrepancy between the observed motion a,ndthe incomplete theory may then be compareddirectly with the relativity effect, which is givenon the last line of the table.The contributions of the planets are directlyproportional to their several masses, which arenot all known with the desired accuracy. Thequantities denoted by nz ' are the reciprocalsof the adopted masses, the sun's mass beingtaken as unity, and the attached probable errorsgive rise to the probable errors associated withthe theoretical contributions to the motions.In the case of Mercury each planetary contrjbu-tion (except that of Mercury itself) is the sumof three parts: the motion of the perihelion inthe plane of the orbit, the contribution arisingfrom the motion of the node, and the contributionfrom the motion of the ecliptic. These last twoeffects arise from the way in which the longitudeof the perihelion is measured; from the equinoxalong the ecliptic to the nodeand then alongthe orbit of Mercury to the perihelion. The

THE THEORETICAL MOTIONS OF THE PERIHELIAThe theoretical motions of the perihelia, re-ferred to the moving equinox, are obtained byadding together the parts contributed by thegravitational actions of the several planets (andin the case of the earth the portion arising fr'omthe non-sphericity of the earth-moon system),the rotational oblateness of the sun, the generalprecession in longitude, and the relativity effect.The separate contributions are shown in Table I I,!


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G. M. CLEMENCEfigures given depend on the calculations ofDoolittle, ' but his values of the masses havebeen altered.The probable errors of the masses are my ownestimates. It is evident from Table II that theuncertainties in the masses of Mercury andVenus contribute most to the uncertainty of the6nal results; indeed, until these two masses arebetter determined, the motions of the periheliaof Mercury and the earth can be observedmore accurately than they can be calculated.A thorough discussion of meridian observationsof Venus would probably give a value of the massof Mercury with a probable error about 6.ftypercent of that given here. Increased accuracyin the mass of Venus must await the completionof the theory of the motion of Mars now inprogress. The most uricertain of the probableerrors is that attached to Mercury; de Sitter' in1938 estimated it to be fifty percent larger thanthe value given here. The estimation of thisprobable. error is a very difFicult matter and itmay be that de Sitter's guess is better than mine;in this case the penultimate line of the tablewould read 4'!6+3'!7 instead of 4".6+2".7.The effect of the rotational oblateness of thesun is to produce a small additional contributionto the perihelion motions of the planets. Thegeneral theory of such effects has been discussedby Brouwer. ' It is known" that if the sun werea homogeneous gas sphere the resulting contribu-tion to the centennial motion of Mercury'sperihelion would be 1".2. For the actual sun thisvalue must be multiplied by 4Z/3, X being adimensionless constant depending on the interior

8W. de Sitter, edited and completed by Dirk Brouwer,"On the system of astronomical constants, " Bull. Astro.Inst. Netherlands 8, 213 (1938).' Dirk Brouwer, "The motion of a particle with negligiblemass under the ravitational attraction of a spheroid, "Astro. J. 51, 223 1945).~ F. Tisserand, Traits de Mecanique Celeste 4, 537(1896).

constitution. The value of E is very small for ahighly concentrated gas sphere, which the sun isbelieved to be; Russell" has given empiricalvalues deduced from the observed motions ofdouble stars, the more reliable of which range upto 0.02. The latest theoretical determination isthat of Motz, "who finds 0.006. I adopt the lattervalue with a probable error of twice its amount,which gives for the centennial perihelion motionof Mercury 0'!010&0".02; the probable error isvery uncertain. The effect on the earth is muchsmaller than for Mercury.The precession is that resulting from Oort'slatest discussion the attached probable erroris my estimate.The probable errors attached to the theoreticalrelativity-effects correspond to a probable errorin the solar parallax of ~0'!003.

CONCLUSIONThe theoretical relativity effect in the motionof Mercury's perihelion is 43".03&0".03; thevalue obtained by subtracting all other knowneffects from the total observed motion is 42".56&0'!94. For the earth's perihelion the corre-sponding 6gures are 3".8&0".0 and 4".6&2".7.The con6rmation by observation of the relativityeffect is regarded as satisfactory for both Mer-cury and the earth.As soon as the gravitational theory of Mars is

placed on a sound basis, the relativity effect inthe motion of this planet should be easily de-tected with higher precision than has beenfound for the earth.I am indebted to Professor Schilt and ProfessorSchwarzschild of Columbia University for valu-able aid in connection with this work."H, N. Russell, Note on ellipticity in eclipsing binaries,Astrophys. J. 90, 641 (1939).~ Lloyd Motz, "The apsidal motion in binary stars builton+a point-source convective-core model with varyingguillotine factor, "Astrophys. J. 94, 253 (1941).

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