EINSTEIN'S USE OF INVERSE DISTANCES
IN THE CALCULATION OF
MERCURY'S MOTION OF THE PERIHELION
by
Introduction
On November 18th 1915 Albert Einstein presented a paper entitled "Explanation of the Motion of Mercury's Perihelion by the Theory of General Relativity" as a session report of the Prussian Academy of Sciences in Berlin containing two quantitative evidences for of the correctness of General Theory of Relativity:
Figure 1 PRINCIPLE OF PERIHELIC ROTATION Source: LANDAU and LIFSCHITZ (1967) 
The graph shows planetary orbits close to KEPLERellipses with minimum (perihelion) and maximum (aphelion) radii between a planet and the sun as the centre of gravitation. In a complete turn from perihelion to perihelion of a pure elliptic orbit this angle would exactly be 2p = 360°. Instead, Mercury shows a tiny increase of this angle by Df = 43 " in 100 Earth years (grossly exaggerated in the graph) resulting in a perihelic turn due to the extended motion potential of EINSTEIN's second approximation. 
The delivery of this, as it seems, hurriedly accomplished paper marcs the end of lengthy and futile calculations as documented in a recently discovered EINSTEINBESSOmanuscript (KLEIN et al., 1995) and the onset of a triumphant scientific breakthrough (PAIS, 1982).
• In his euphoria about the results EINSTEIN wanted to inform the scientific public as fast as possible.
• There was only one week between the presentation (18 Nov.) and the publication, making it very improbable that EINSTEIN had any chance of proof reading of his paper.
• Accordingly, there are eight printing errors in the formulas as stated by KOX et al. (1996)
EARMAN and JANSSEN (1993) have very well analysed and commented the first part of EINSTEIN's perihelion paper, dealing with the mathemetical foundation of his first and second approximation.
The present paper focuses on the last part of EINSTEIN's epochmaking paper, referred to as the 'perihelion paper'. It intends to show how EINSTEIN succeeded in calculating the tiny perihelic turn of Mercury by using the inverses of the maximum and minimum distances between Mercury and the sun as roots of the equation of planetary motion. Some steps which EINSTEIN, supposedly for the sake of succinctness, has left out, are inserted here. Numerical results of intermediate calculations are included to enhance transparency.
First Approximation: The Orbit of Mercury as a NEWTONian KEPLEREllipse
The Table 1 shows the characteristics of the Mercury orbit. The data confirm that Mercury is the closesttosun and, therefore, fastest planet of the solar system.
Table1
CHARACTERISTICS OF THE ORBIT OF MERKURY
SemiMajor Axis 
a 
5.79*107km 
Perihelion 
r1 
4.59*107km 
SemiMinor Axis 
b 
5.66*107km 
Aphelion 
r2 
6.97*107km 
Semilatus Rectus 
p 
5.54*107km 
1/Perihelion 
a1 
2.18*108km1 
Numerical Excentricity 
e 
0.2056 
1/Aphelion 
a2 
1.43*108km1 
Its highest speed in the perihelion is 59 km/sec, its lowest in the aphelion is 39 km/sec. Accordingly, one turn around the sun  a Mercury year  has the short duration of 88 Earth days (T = 7.6*10^{6}sec). With a numerical excentricity of e = 0.2056 the orbit of Mercury has a pronounced elliptic shape.
The Figure 2 shows the parameters of a KEPLERellipse in the polar coordinates r and f. In agreement with KEPLER's first law the gravitational and coordinate centre  the sun  is located, by convention, at the right focus of the ellipse.
The distance r from the sun is determined by the formula r = p / (1 + e cos f). Inversely, as a fact which greatly simplifies further calculations, there is a linear relationship between the inverse of the distance 1/r and the angle f.
Figure 2
CHARACTERISTICS OF A KEPLERELLIPSE
r1 perihelion r2 aphelion p elliptic parameter (semilatus rectus) a semiaxis major b semiaxis minor 
EINSTEIN's first approximation includes the orbit of Mercury as a KEPLERellipse according to NEWTON's law of gravitation
(1) 
where F stands for the force of gravitational attraction and G for the NEWTONian gravitational constant. MS and mM symbolize the masses of the sun and Mercury respectively. From the energetic equivalence of acceleration and gravitational force depending on different distances (r0 and r1) we can derive the law of energy conservation (2) and, accordingly, estimate the specific energy A of Mercury




(2) 
which amounts to A = 1 150 km²/sec², as derived from A = GM/2a. The negative sign of A confirms that the orbit of Mercury has an elliptic shape with 0 < e < 1. This can easily be checked by the numerical excentricity e = [1 + 2A.p/GM]½ attaining e = 0.2056 for Mercury's orbit.
The term GM/r is defined as the gravitational potential F. GM as a constant of the solar system (GM = 1.33*10^{11} km^{3}sec^{2}) can be expressed as the gravitational constant G times the mass M of the sun. EINSTEIN does not mention GM in his paper. Instead he introduces a new constant named Alpha (a) which has the value of 2GM/c², where c2 denotes the square of vacuum light velocity (c = 3 . 10^{5} km sec^{1}). By his conversion of Alpha into 2a .[2ap/(Tc)]² , applying KEPLER's third law, EINSTEIN confirms this equivalence. The dimension of a = 2.95 km indicates that Alpha is equal to the Schwarzschildradius of the solar system (SCHWARZSCHILD 1916b).
Derived from (2), we obtain the square of the planet's timedependent velocity v² comprising the radial (dr/dt)² and the angle (azimutal) component r²(df/dt)².




(3) 
The latter angle component stems from the area constant B as an expression of KEPLER's second law. This, in turn, expresses the law of conservation of angular momentum as a product of the vectors r x v.
(4) 

(5) 
Equation (5) serves to single out the radial component (dr/dt)² and to shift the quotient B²/r² to the righthand side of equation (3).
(6) 
To make this expression independent from time we replace (1/dt)² by B²/r4(1/df)². The transfer of B² to the righthand side of the equation and the expression x = 1/r lead to the classical equation of planetary motion (7).
(7) 
Now there comes the main reasoning. If the variable r of the distance between Mercury and the sun is already expressed in terms of x = 1/r , why shouldn't the fixed points, i.e. the minimum and maximum distances, the perihelion and the aphelion, be expressed in terms of their inverses a1 = 1/r1 and a2 = 1/r2 ? Otherwise we would have a mixture of references as it was the case in the EINSTEINBESSO manuscript.
The following equations show how the inverses of the minimum (a1 ) and maximum (a2 ) distances from the sun are related to the parameters e and p = a(1e2) of a KEPLERellipse.
Two equations  the one from calculating the lengths of r1 and r2
the other from KEPLER's third law
provide a bridge between the energy constant A, the area constant B and GM of (7) on the one side and the parameters a and p of the ellipse on the other.
leading to 


Thus we can reformulate equation (7) in terms of r1 and r2 just to compare, and then in terms of a1 and a2




(8) 
demonstrating in (8) that the two roots a1 and a2 enabled EINSTEIN to formulate a factorized equation according to the theorem of VIETA. With this he had a description of a planetary orbit in its simplest form. Expressing all by inverse distances virtually makes the physical constants A, B and GM obsolete  including NEWTON's gravitational constant G.
The integral of this function with limits at the aphelion and the perihelion quantifies the angle between the two apexes.
(9) 
By assigning 1 to a, (a1+a2 ) to b and a1a2 to c, this integral becomes a standard format
as it is found in integration tables. Accordingly, integration furnishes
.
The result is trivial: In a pure elliptic orbit the perihelion and the aphelion are at the apexes of the major axis, i.e. located on a straight line. However, the method employed by EINSTEIN and presented here shows a reliable way to determine a deviation from this rule.
Second Approximation: Extending the Potential of Gravitation
Due to EINSTEIN's second approximation the gravitational potential F gets GM .B2/(c2r3) as an additional component. This gives a chance to quatify a slightly further turn from perihelion to perihelion than the one of 360°. In (10) we obtain this second component by multiplying the potential GM/r with (B/r)² , which is equal to the angle component of the velocity equation (4).
(10) 
The equation shows how the 1/r3 term can be traced back to multiplying 1/r with 1/r2.
Due to the conversion of GM = a . c2/2 the square of light velocity c2 is cancelled out and therefore disappears in EINSTEIN's formulation. The square of light velocity in the denominator of the remaining components is cancelled by dividing by B2 which, by virtue of B2 Þ (GM/c2).p in relativistic terms, has also c2 in its denominator.
With the aim to express the angle f as a function of inverse distances between Mercury and the sun and vice versa, we can proceed exactly as in the NEWTONian model




(11) 
arriving at
(12) 
with (11) showing the equation of planetary motion (E11) of EINSTEIN's perihelion paper. The only difference with the NEWTON model, as EINSTEIN put it, is that there is an additional cubic component ax3.
By setting equation (12) equal to zero, there are three roots (x1,x2,x3) to determine. The first two roots (x1,x2) remain a1 and a2. The third root x3 can be found by
.
Accordingly, x3 = 1/a  (a1+a2) is equal to 0.339 km1, roughly the inverse of the Schwarzschildradius. The factorized equation is then
(13) 
The integral of (14) determines the angle f between perihelion and aphelion:
(14) 
Since the diminuend in {1 a[(a1 + a2) + x]}½ is much smaller than 1 we can swap this term to the enumerator. There it gives {1 + a/2 . [(a1 + a2) + x]} as shows (15) in its expanded form.
(15) 
The integral can now be split into two components
.
Integration furnishes
,
with the result of
,
in full agreement with the result presented by EINSTEIN. In contrast to the assumption made by KLEIN et al. (1995) and by EARMAN and JANSSEN (1993) that EINSTEIN, as in the EINSTEINBESSO manuscript, had employed the rather difficult method of contour integration in his perihelion paper, it turns out that the integral can be calculated in a very conventional way provided the inverses of the perihelion (a1) and the aphelion (a2) are used as roots of the equation of planetary motion.
The remainder of the calculation of the perihelic shift is quite straightforward.
1. Duplicating the value of f for a full turn minus 2p to obtain Df, the additional shift
(16) 
2. Transforming (a1+a2) into 2/p = 2/[a(1e²)],
(17) 
3. Transforming a into 2a .[2ap/(Tc)]²
(18) 
4. Transforming rad into arc seconds and multiplying Df with TE/TM*100 = 415 i.e. the number of Mercuryyears in 100 years of the Earth,
(19) 
leading to a perihelic turn of Mercury of 43 arc seconds per one hundered Earth years.
Discussion
An attempt was made to reconstruct EINSTEIN's ingenious calculation of the perihelic turn in the last part of his paper. It reveals  perhaps to the dismay of many  his excellent command of ellipse geometry conducing him to the right path. Although written in a rather cryptic style like an 'internal memo' for a limited circle of enlightened scientists, the publication of the perihelion paper prompted a wealth of publications and a vivid discussion on the subject.
But what prevented other authors in the field of celestial mechanics (e.g. FLAMM 1916; SCHWARZSCHILD 1916a,b; FREUNDLICH 1916; ADLER et al. 1965; WEINBERG 1972; GUTHMANN 1994), with the exception of MØLLER (1969), to use EINSTEIN's extremely elegant and simple formulations in terms of inverse radii and, accordingly, to calculate the integral the simple way it is shown here?
Could it be that the conflict between the knowledge of the correct solution on the one side and the application of strict mathematical rules on the other, prevented EINSTEIN's colleagues from using his formulations? Specifically, there must have been the question of how EINSTEIN, starting from his equation (E11)
, has managed to arrive at
by means of integration.
The snag is that the bracket in front of the integral in EINSTEIN's paper, stemming from
[1a(a1+a2)]½, apparently has undergone a transcription error (a instead of a/2) misleading anyone who does not dare to mend his writing. As late as 1996, KOX et al. (1996) 'officially' corrected this error. Accordingly, the integral should have been
.
The integral shows as well that, in EINSTEIN's writing, the third root of the motion equation consists of the parenthesis (1ax) times the bracket [1a(a1 +a2)], where, after integration, he had to neglect the superflous and very small term of (a2p/8) .(a1+ a2)2.
Some authors (e.g. ADLER et al., 1965) have declared EINSTEIN's equation (E11) as 'not particularily enlightening' or 'cumbersome' or even 'labyrinthine', looking for ways out, e.g. BINET's formula, to attain the magic 43 arc seconds all the same. This reminds on PHAEDRUS' fable about the hungry wolf and the grape hanging too high, where  out of frustration  the wolf declares the grape as immature and too sour for consumption.
Another striking aspect in EINSTEIN's paper is the complete absence of NEWTON's gravitational constant G. If nothing else, the EINSTEINBESSO manuscript of 191314 shows how often both scientists were busy in redefining this constant, thus proving EINSTEIN's critical attitude towards G. Years later, in 1916, the struggle seemed to be settled with EINSTEIN's gravitational constant k = 8pG/c2 (EINSTEIN, 1916). In this context, equation (8)
,
where only inverse distances appear, must have been highly intriguing to EINSTEIN who constantly was looking for the very simple solutions Nature could 'think' of. This might, as well, be an explanation for the fact that from three equivalent expressions of Df to quantify the perihelic turn
,
EINSTEIN chose the one that did not contain G. The irony is that, by introducing ax3 in his second approximation, EINSTEIN was forced to reintroduce the gravitational constant all the same.
Posterity has shown that it still employs the NEWTONian gravitational constant G. Could it be that we haven't yet integrated in our minds the real implications of General Relativity?
Summary
The present paper comments on the second part of the epochmaking publication of Albert EINSTEIN Explanation of Mercury's motion of the perihelion by the theory of General Relativity (1915) in which EINSTEIN publishes the corrected degree of light inclination and the secular perihelic motion of Mercury.
A novelty is the constant Alpha (a) which is equal to the Schwarzschild radius of the solar system. As derived from EINSTEIN's calculations, Alpha is equal to 2GM /c²; where G is the gravitational constant, M the mass of the sun and c the vacuum light velocity. For calculation of the additional angle Df to quantify the perihelic turn, EINSTEIN uses x as the inverse of the variable distances r between Mercury and the sun, a1 for the inverse of the perihelion and a2 as the inverse of the aphelion as roots of the equation of planetary motion.
EINSTEIN shows that his first approximation contains NEWTON's model of a KEPLERellipse. In his second approximation the gravitational potential F gets an additional cubic component. Expressed as ax3, this component allows for an additional shift of the perihelion. This method facilitates integration and converts physical constants such as the energy A, the area constant B and the gravitational constant G into terms of inverse distances.
The fact that other authors in the field of celestial mechanics did not use EINSTEIN's formulation in terms of a,a1,a2 is traced back to a disturbing transcription error. The absence of NEWTON's G in the perihelion paper is attributed to EINSTEIN's critical attitude towards the gravitational constant.
Acknowledgement
For the author, being a nonprofessional with the desire to understand a piece of work EINSTEIN had done, it was essential to get help and enouragement along the road of learning. He therefore is very grateful for the personal communications and the assistance he got from Peter C. Aichelburg, Heinz Dehnen, Michel Janssen, Rainer Kraupner, Rainer Lübbe, Heiner Mühlig, Gerhard Musiol, Reinhard Pflug, Christoph P. Pöppe, Steffen Polster and Jürgen Renn who strongly  and rightly  suggested him to digest the historical evidences in the EINSTEINBESSO manuscript and in the EARMANJANSSEN publication.
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Dr. Christian Strutz, Steigstr. 26 D88131 LINDAU
Lindau, März 1999
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