Orbital Precession without GR
© David Barwacz 11/24/2007
Abstract
The anomalous precession of the planet Mercury’s orbit puzzled scientist for decades. Einstein developed a curved space model of gravitation (General Relativity) and showed that the precession of the planets could be explained very accurately by this model. This fact combined with the prediction and subsequent observation of the gravitational bending of light rays, made GR one of, if not the most, and highly accepted theories in science.
In this paper I will show that the precession of the planets can be explained with nothing more then Special Relativity, Newtonian derived formulas and simple mathematical relationships. The result is extremely accurate.
Keywords: Planetary
Precession, General Relativity, Orbital Precession of mercury, Flat Space
History
The orbital precession of the planet Mercury has been known for decades. Prior to the advent of Special Relativity [1] attempts to explain the precession were generally discarded for various reasons. Attempts to explain the anomalous precession after the advent of SR and prior to the development of General Relativity appear absent in the literature. Recently a few papers have appeared in the literature attempting to explain the anomalous precession using some modifications/combinations of SR and Newtonian gravity. Biswas [2] has shown that a Lorentz covariant modification of the classic Newtonian gravitational potential can yield the correct precession. Others [4] have made more radical assumptions such as modifying the gravitation field equations to parallel the field equations of Electromagnetism [3,8] or assuming a relativistic Lagrangian that looks surprisingly like a Schwarzschild metric [4].
The fact that the relativistic effects are very small in
planetary motion would lead one to believe that some simple approximation involving
Newtonian mechanics combined with SR should at least give a good approximation
to orbital precession. It turns out that this can in fact be done with simple
mathematics and a couple of well known and universally accepted physical formulas.
No assumptions regarding the nature of the gravitational field or the geometry
of space are required.
Background physics
The energy of a relativistic particle in motion is given by:
1.10
These terms are all we will use as additional terms only become relevant at extreme velocities. The kinetic energy of the particle will then be:
1.20
From Newtonian mechanics the total velocity of an object moving under the influence of a central force is given by [6]:
1.30
Where 
and 
is the total constant
angular momentum and e is the eccentricity of the orbit.
Equations 1.20 and 1.30 are all the physics that is needed. The rest is mathematics.
The Math
To develop the necessary mathematical relationships needed we first explore the relationship between the Lagrangian operators on general power functions.
The Lagrangian operators are:
and 
In one dimensional mechanics, given the initial conditions one can determine both the position as a function of the time x(t) and the velocity as a function of time v(t). Similarly time can be written (by taking the inverse function) as a function of position t(x), and hence velocity can be written as a function of position v(x).
Take a general function of the form
, where n is a positive integer 
2 and a is a constant.
The function can be written as:
= 
where 
Applying 
to 
we find:
1.40
Applying 
to 
we find:
1.50
In 1.50, 
was written as 
From 1.40 and 1.50 it is clear that:
1.60
To extract the actually Lagrangian form we use the function
:
1.70
1.70 is valid regardless of any constants added to T so we can write:
Where 
is the total energy
which is constant 
is just –V(x), the
potential energy which is assumed to be a function
only of x.
Since operating on V(x) is
zero and operating on 
is zero we can write
1.70 as:
1.80
1.80 is in the form of the classical Lagrangian.
We can use 1.70 to generate the gravitational potential energy function of an orbiting body by noting that 1.30 expresses the total velocity and hence the total kinetic energy as a function of r. The problem has only one independent variable, r and hence we can use the above one dimensional reasoning.
Using:
and 
and substituting 1.30
for 
then differentiating
we get:
, which is the classic
Newtonian formula.
We just showed that, knowing the velocity as a function of r 1.70 allows one to generate the potential energy function.
We can now proceed to solve the problem of orbital precession.
Using 1.20 and 1.60 we get the equalities:
= 
and 
= 
Combining terms we get the equality:
= 
1.90
We don’t know v(r) exactly but for most problems such as the
orbit of the Planet Mercury we know that it very closely approximates the
classical v(r). We will then use 1.30 on the right side. Squaring 1.30 to get 
and differentiating we get.
= 
2.00
.
It is very small compared with the first and has no effect on the orbital precession but has some interesting properties that we shall discuss later.
What we want to solve 2.00 for is 
. The velocity v however appears in both terms on the left.
We would like to get it in a single term.
Define the following kinetic energy functions:
and 
2.00 becomes:
2.10
Next we define a potential function as follows:
To get this in the form of 1.60 we write:
where n=2 and 
Next we note that:
=
where n=2
=
Combining term gives:
2.20
2.20 can be solved exactly. We could either ignore the second term or combine it with the first effectively skewing the Gravitational constant. Interesting, it is a function of eccentricity and depending on the orbit, it can be either positive negative or zero. For an elliptic orbit, such as that of the Pioneer 10 space probe (e=1.737) it increases the gravitational constant. It may be at least partially responsible for the anomalous Doppler readings.
We can proceed to solve 2.20, ignoring the second term both because it is very small and because it has no effect on the precession.
In polar coordinates 2.20 results in two equations, one from
the r coordinate and one from the 
coordinate
The details of solving this equation for 
can be found in any text book and many online sites [ 6 ].
In summary the solution of the equation is
independent of the nature of the potential field and results in conservation of
angular momentum designated L. The following conversions are then used on the
radial equation:
And 
And 
Applying these conversions yields:
2.50
Where 
, the angular momentum per unit mass and 
. c is of course the speed of light. 
is a measurable
property of an orbit and is designated
the semi-latus rectum. It is related to the eccentricity.
Simplifying 2.50 yields:
2.60
This can be solved exactly by letting 
Doing the math gives 
, from which we can directly determine the
orbital precession.
.
For the planet mercury 
Giving 
The precession per revolution in radians is 
Converting to arc seconds the precession becomes: 0.103435965
Mercury orbits the sun 414.9378 times in one earth century, so the precession per century is: 42.9195 arc seconds per century in excellent agreement with observation and GR.
Conclusions
The precession observed in the orbit of Mercury can be
explained with nothing more then well accepted physical formulas and relatively
simple mathematics. No assumptions about the nature of the gravitational field
are required and space is treated as “flat”. A curious term appears in the
calculations that deserve some exploration. The 
term is orbital path
related. It skews the gravitational constant and this skewing can be positive
or negative or zero depending on the eccentricity of the orbit. For the Pioneer
space probe e=1.7372 and is positive resulting
in an increase in the gravitational force. Prior to being subjected to the
“sling shot effect” its eccentricity was most likely much smaller maybe even
negative. This would have resulted in an effective increase in G after it
reached its escape velocity.
I am not familiar with the orbital data of the craft but welcome any input from anyone who has more information.
References
1. Reflections on relativity “Anomalous Precession” http://www.mathpages.com/rr/s6-02/6-02.htm
2. T. Biswas, “Special Relativistic Newtonian Gravity”, Foundations of Physics Vol. 24, No. 4, 1994
3. Behera, Harihar; Naik, P.C., “A flat space-time relativistic explanation for the perihelion advance of Mercury” eprint arXiv:astro-ph/0306611. ARXIV, 2003astro.ph..6611B, http://arxiv.org/abs/astro-ph/0306611 2003
4. J. M. C. Montanus, “Flat Space Gravitation”, Foundations of Physics Vol. 35, No. 9, 2005
5. Bhimsen K. Shivamoggi,
“Perturbation Methods for Differential Equations”, (Springer Publishing,
6. “Motion under the Influence of
a Central Force” http://ads.harvard.edu/books/1989fcm..book/Chapter5.pdf
, 2004
7. W. Benenson, J. Harris, H. Stocker, H. Lutz
(Eds), “Gravitation and the Theory of Relativity”, in Handbook of Physics
(Springer Publishing,
8.
Robert L. Kurucz, “The Precession of
Mercury and the Deflection of Starlight from Special Relativity alone”,
(arXiv:astro-ph/0608434 v1 21 Aug 2006)