PRECESSION OF PERIHELIA

LE VERRIER'S AND EINSTEIN'S PREDICTIONS COMPARED

James Constant

grav@coolissues.com

No theory, classical or relativistic, can predict the precession of perihelia unless it meets the necessary and sufficient requirements of elliptic orbits set by Newton's law of gravitation. To date, no exact Keplerian solution exists for the planetary n-body problem for predicting the precession of perihelia. The present theory presents a necessary but not sufficient Keplerian solution which it compares with Le Verrier's and Einstein's theories

Introduction

The discovery and attempts to find classical causes for the precession of the perihelion of the planet Mercury are well known.¹ In the 19th century, the French astronomer Le Verrier found that the perihelion of Mercury advanced by 43 arc- seconds/100y, an amount which could not be accounted for in terms of the gravitational forces exerted on Mercury by the other known planets. Le Verrier assumed another planet, which he named Vulcan, existed between Mercury and the Sun. However, observations failed to establish that Vulcan existed. Another explanation put forth by astronomers was that Newton's law of gravitation was not correct but, if additional terms were added, these could account for the excess rate of turning of the orbit of Mercury. For example, it is known that if a repulsive force C/r³ is added to Newton's law it would provide a correct value for precession of the perihelion of planet Mercury.² A different explanation is offered by Einstein's theory of gravitation which calls for an extra term A/r⁴ which, if added to Newton's law, gives the excess turning rate of 43 arcseconds/100y. Importantly, A was not adjusted to give this value. It emerged uniquely out of Einstein's theory. Thus, classical explanations, which could not be found since the 19th century were replaced by Einstein's theory in the 20th century.

In what follows, I revisit the classical ideas of perturbations as causes adding terms to Newton's law of gravitation to find the excess turning rate of a planet. These include causes for effects by other planets and the Sun's oblateness.

Precession of Perihelia

Newtonian theory requires that Newton's law of gravitation must be corrected for perturbations due to other planets.

F = F_o(1-s) . . . . . . . . . . . . s<<1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . (1)

in which s is a perturbation due to other planets affecting a planet's motion. The negative sign occurs by the fact that perturbing effects act as a repulsive force. Each planet follows a near elliptical orbit whose focus is nearly, with the exceptions of Mercury and Pluto, at the center of the Sun. Equation (1) says that a repulsive force +GM_om_os/r² must be added to Newton's force F_o.

The equation of a planet's orbit is

<<1 . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . .(2)

which is an ellipse if =0 but is a precessing ellipse if . In equation (2) r' is radial distance, L is the semi-latus rectum, a is the semi-major axis, is the eccentricity, and is angle.

Consider now the motion in n planetary cycles. In this case angle increases to which brings the planet back to the same point if =0 but advances or retards the perihelion () by an amount respectively if . Expanding the cosine term and neglecting higher order terms, the planet's orbit at the perihelion () is

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(3)

Necessary and Sufficient Requirements for Precession

Sommerfeld has calculated that, in a hydrogen atom, an electron's relativistic mass produces a precession of its elliptic orbit³, which I adapt for planetary mass

. . . . . . . . radians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .(4)

in which G is the gravitational constant, M is the Sun's mass, c is the speed of light, a is the semi major axis, v is the average orbital speed, and n is the number of cycles. I cite equation (4) because it suggests application for planetary precession of perihelia. However, while the Keplerian prediction of atomic precession is highly successful observationally it depends from a single perturbation, relativistic mass, which is of lesser importance in Solar planetary precessions where perturbations by other planets are more important. Moreover, it should be kept in mind that while the hydrogen atom is a 2-body problem which has an exact Keplerian solution, the planetary system is a 10-body problem which lacks such exact solution.

In view of equation (3), application of central force equation (1) to the differential equation of the orbit⁴ produces a result for the value of r at the perihelion (), for small values

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .. . . . . . . . . . . . (5)

By constructing a right triangle at the perihelion, I find for small values

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . . . . .(6)

and, combining equations (5) and (6) yields the solution, for small values

. . . . . . . . . . .radians . . . . . . . . . . . . . . . . . . . .. .. . . . . . . . . . . . .(7)

in which all terms under the square root are presumably known constants from observation. For example, terms ,r_p and L are presumed known orbital parameters and term s is determined by setting the square root equal to 574.1 arcsec/100y, the observed precession of Mercury, a presumed constant, and solving for s. The number of cycles n and the precession per cycle are unknown terms, to be determined for each planet. In effect, equation (7) is a single equation with two unknowns n and . Lacking a second equation, it is impossible to obtain exact values for n and .

Equation (7) is necessary but not sufficient to predict the precession of perihelia. Sufficiency not only requires a second equation but also requires making assumptions whether orbital terms and perturbations remain fixed over the number of cycles n. The first equality in equation (7) implies that if the precession per cycle is known then multiplying by the number of cycles n gives the precession per n cycles. However, the second equality requires that product is a constant since the square root is constant. The determination of this constant is made by computing the known values for each term in the square root.

There is only one set n,, among the many sets, solutions of equation (7), that determine numbers n and for a planet. Absent a second equation relating n and ,, such determination is not possible. It is only the product that equation (7) determines. The terms in the square root of equation (7) are but initial conditions which evolve in time. The initial conditions, orbital parameters and perturbations, and thus the precession per cycle , change on every cycle. For example, equations (5) and (6) show that the perihelion increases on each cycle. The precession per cycle might be considered as an average value over the time span of n cycles.

Le Verrier's Theory Compared

By the use of elliptical functions Le Verrier in 1859 calculated the precession of Mercury's perihelion, without proof, as follows

. . . . . . . . . . . . . . .radians . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . (8)

in which is the precession per cycle of the ith planet. He provided the following values for the contributions of the various planets to the advance of Mercury's perihelion in arcseconds/100y

These predictions, use n=100y/T arbitrarily, without proof, where T is the orbital period. The 100y time span is used because observations were available for Mercury going back to 1765⁶. It has now become an arbitrary convention to use the 100y time span when making predictions for planetary precessions. There is also a 4.809 arcsec/100y difference between Le Verrier's and modern calculations of Mercury's perihelion which raises a question about the accuracy of predictions of planetary precessions.

Thus, according to Le Verrier, the orbit of Mercury precesses by about 526.7 arcseconds/100y due to the disturbing effects of the other planets, primarily Venus, Earth, and Jupiter. The observed modern precession is about 574.1arcsec/100y or about 43 arcseconds/100y greater. This is believed to occur because the simple 2-body Keplerian orbit is disturbed by the presence of gravitating matter outside the orbit. Even though these effects are strictly Newtonian, the calculations are far from trivial. The questions raised are, first, is equation (8) an exact solution of the n-body planetary problem, second, are equations (7) and (8) identical, third, are perturbations from sources other than planets important and, fourth, is equation (8) provable and consistent. The answers to all four questions are no.

It is well known, since Newton's time, that solutions to the n-body problem are not integrable and over the long term become chaotic. We can only settle for short term approximations ad hoc or by computer, such as those NASA uses when planning space missions⁷, or long term solutions in special cases, such as the Lagrange 3-body configuration⁸. Accordingly, at best equation (8) is a short term approximation or at worst a wrong solution to the n-body planetary problem.

Unless the square root in equation (7) can be expanded as the finite sum of equation (8), and terms identified with terms in equation (8), there can be no correspondence between equations (7) and (8). Since the square root in equation (7) can be expanded as an exponential function, different from equation (8), there is no correspondence between terms of equations (7) and terms of equation (8). Accordingly, equation (8) is not identical with equation (7) which represents a necessary Keplerian solution to the planetary n-body problem.

Equation (8) does not include perturbations from sources other than planets, such as the Sun's oblateness and relativistic mass. For example, it is known that the Sun's oblateness requires that an r^-3 term is added to Newton's law of gravitation and that this gives Mercury an extra precession of 3.4 arcsec/100y, so that only 39.6 arcsec/100y would be the resulting error between the observed and predicted values of precession. This known fact alone produces an 8% disagreement with Einstein's prediction error of 43.03 arcsec/100y.⁹

By provable is meant can the sum in equation (8) be proven in terms of Newton's and Kepler's theories, as a valid approximation to the Solar n-body problem. The presumption that disturbances act as a sum of perturbations is made without proof. In contrast, Newton's and Kepler's theories prove, in equation (7), that disturbances act as a square root of a function s of perturbations. While equation (8) allows making exact Keplerian predictions for disturbances by each planet, equation (7) bars making exact predictions. Accordingly, Le Verrier's equation (8) is unprovable in terms of Newton's and Kepler's theories.

By consistency is meant does equation (8) make the same predictions when using the entire sum or using a single term to predict precession. Using the entire sum, Le Verrier's Keplerian prediction of 526.7 arcsec/100y for all planets falls short of the 574.1arcsec/100y observation by an amount of 43 arcsec/100y. It is not known whether the inconsistency between prediction and observation is due to an inadequacy of equation (8), as an approximation to the Solar n-body problem, or to the actual precession of Mercury. Using a single term, Einstein's relativistic prediction of 43 arcsec/100y for Mercury is arbitrarily added to Le Verrier's Keplerian prediction of 526.7 arcsec/100y for the remaining planets. It is inconsistent to apply Einstein's relativistic prediction for one planet, Mercury, but not apply Einstein's relativistic prediction to all other planets. There is no proof that Einstein's relativistic prediction is identical to Le Verrier's Keplerian predictions for all other planets. Accordingly, Le Verrier's Keplerian equation (8) is inconsistent between prediction and observation and with Einstein's relativistic theory.

Einstein's Theory Compared

There are important experimental as well as theoretical reasons for replacing Einstein's theory by classical explanations. Einstein's theory of gravitation, based on Riemann's geometry, predicts the existence of a repulsive force A/r⁴ which, with an unequivocal way to calculate A, claims to predict a correct value for precession of the perihelion of planet Mercury¹⁰. While there are doubts¹¹ and caveats¹² that it can predict correct values for Mercury and other planets, and while Einstein's theory must be used in its weak version approximating Newton's theory, its strongest support is claimed to be its prediction of the correct value for precession of the perihelion of planet Mercury. However, as expressed in equation (7), this claim disagrees with the requirements of elliptic orbits set by Newtonian theory.

Einstein uses Leverrier's method to obtain Mercury's precession. As stated previously, Le Verrier's method falls short of being an exact solution of the n-body planetary problem for precession of perihelia, violates Keplerian necessary solution equation (7), is exclusive of non-planetary perturbations and is unprovable in terms of Newton's and Kepler's theories, and is inconsistent with Einstein's relativistic theory. Einstein's theory computes a constant value for a planet's precession based on a simplified version of Riemann's geometry approximating Newton's theory, with relativistic result over n cycles

. . . . . . . . . . radians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(9)

in which G is the gravitational constant, M is the Sun's mass, c is the speed of light, L is the semi-latus rectum and n is the number of cycles of a planet's orbit¹³. Thus, without proof, equation (9) assumes that the computed angle is proportional to the number of cycles n and the precession per cycle , that the relativistic precession per cycle is constant over the span of n cycles, and that computed angle is a variable which can be determined without reference to known initial conditions, orbital parameters and perturbations. While Einstein applies equation (9) to Mercury, he does not apply it to all other planets disturbing Mercury's orbit. Thus, Einstein's relativistic prediction is inconsistent, first, because actual observations produce average values for precession per cycle and, second, is inconsistent with the Keplerian result obtained by using Le Verrier's entire sum to predict precession.

. . . . . . . . . . . . . radians . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . (10)

In contrast, the present theory shows that the computed angle is a constant for each planet and that initial conditions, orbital parameters and perturbations, alone determine its value, and that values for n and are unknown unless a second equation becomes available. Accordingly, one cannot compute the precession per cycle, as all theories arbitrarily do, and then simply multiply by the number of cycles to obtain the precession per n-cycles. Rather, in a first step, one compares the results of a theory with the angle computed by Newtonian theory in Keplerian equation (7), a necessary condition. In a second step, one produces a second equation which, when combined with equation (7), gives values for n and , to establish sufficiency of the theory. The present theory explains the first step but leaves the second step open.

Summary of the Theory

A planet's perihelion is a function of initial conditions, orbital parameters and perturbations mainly by other planets, which define a characteristic computed angle, constant for each planet. I conclude that no theory, classical or relativistic, can predict the precession of perihelia unless it meets the necessary and sufficient requirements of elliptical orbits set by Newton's law of gravitation.

¹ Fred Hoyle, Astronomy and Cosmology, W. H. Freeman and Company 1975 pages 476-477

² Herbert Goldstein, Classical Mechanics, Addison-Wesley Press, Inc. 1951 page 91 problem number 7

³ Henry Semat, Atomic Physics, Rinehart & Co., Inc. Sixth Printing 1950 page 200.

⁴ Goldstein, above page 76

⁵ http://www.schoolsobservatory.org.uk/study/sci/cosmo/internal/genrel.htm

⁶ Steven Weinberg, Gravitation and Cosmology, John Wiley & Sons, Inc. 1972 page 198

⁷ Jeremy Jones, How do space probes navigate large distances with such accuracy? Scientific American, August 2006 page 100

⁸ http://www.bookrags.com/N-body_problem

⁹ Weinberg above page 200

¹⁰ Hoyle, above page 477 Weinberg above page 198

¹¹ Louis Brillouin, Relativity Reexamined, Academic Press 1970 page 99. Examples, with large errors and opposite signs, in the solar system where Einstein's predictions conflict with experiments.

¹² Weinberg above pages 199-201

¹³ Weinberg above page 197 equation (8.6.11)

Copyright © 2006 by James Constant

By the same author: http://www.coolissues.com/gravitation/sameauthor.htm