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TWO CLAIRAULTIAN GRAVITATIONAL FORMULAE

Wells 2011/2016 refers to two well-known patterns of centripetal acceleration which (when applied in a Euclidean-Newtonian orbital system) produce "perihelion precession" aka "apsidal rotation" of the rate observed for the non-Newtonian precession of the orbits of Solar System planets in particular Mercury and Mars.

The formulae, (following the names used by Wells: "A: Alice" and "B: Bob") are expressed by Wells in terms of gravitational potential (in rotating frames of reference) but they can also be expressed in terms of gravitational acceleration by differentiating wrt distance $r$, giving (in an inertial frame of reference):-

$$ \frac{F}{m} = a_A = \frac{GM}{r^2} \left( 1 + \frac{K_A}{r^2} \right) = \frac{GM}{r^2} \left( 1 + \frac{3 GM P}{c^2 r^2} \right) ...[1A],$$

and

$$ \frac{F}{m} = a_B = \frac{GM}{r^2} \left( 1 + \frac{K_B}{r^2} \right) = \frac{GM}{r^2} \left( 1 + \frac{6 GM }{c^2 r} \right) ...[1B].$$

and, for comparison, the standard Newtonian formulae is:-

$$ \frac{F}{m} = a_K = \frac{GM}{r^2} ...[1N],$$

where $F$ is the centripetal attractive force and $a$ is the centripetal acceleration of the target test-particle (contra convention I show positive towards the source), G is Newton's Universal Gravitational Constant, $M$ is the (large) mass of the attracting body, $m$ is the (infinitessimal) mass of the orbiting test-particle, $c$ is the speed of light and $r$ is the distance between the centres of the two bodies.

Note: Here, and subsequently throughout, the formulae given apply for an inertial frame of reference (rather than a rotating frame of reference).

Historical note: Formulae of this pattern $ a = (GM/r^2) (1 + K/r^n) $ are sometimes labelled "Clairaultian" after the formulae proposed in 1745 by the French astronomer Clairault in attempts to explain anomalous features of the Moon's orbit. Similar formulae had been considered by Newton in his Principiae in the C17th in seeking an explanantion of the Lunar orbit.

The rate of apsidal rotation $\epsilon$ (in radians per complete periapse-to-periapse cycle) in cases A and B is given by the Einstein 1915 formula see this earlier question.

$$\epsilon = \frac{24 \pi^3 A^2}{T^2 c^2(1-e^2)} = \frac{2\pi *3 *4 \pi^2 A^3}{T^2 c^2A(1-e^2)} ...[2]$$

which, using the Keplerian equality $ GM = {4\pi A^3}{T^2} $ can be re-written in units of (extra) revolutions per periapse-to-periapse cycle as:-

$$\epsilon = \frac{3 * 4 \pi^2 A^3}{T^2 c^2A(1-e^2)}= \frac{3 GM}{c^2 P}...[3]$$

where $A$ is the orbit semi-major axis, $e$ is orbit eccentricity and $P$ is the orbit semi-latus rectum.


CIRCULAR ORBITS

Now it is interesting to consider what happens in the case of a circular orbit, i.e. when $r = A = P = constant$. In the Newtonian case we have no additional acceleration, the centripetal acceleration is constant and using the well-known formula $a = V^2/r$ we obtain for the constant, transverse velocity of the target:

$$ \frac{Vt_N^2}{P} = \frac{GM}{P^2} => Vt_N = \sqrt\frac{GM}{P}...[4N]$$

In case A, using the binomial approximation: $\sqrt{1+x} \approx {1+x/2}$ for $x<<1$ we obtain:-

$$ \frac{Vt_A^2}{P} = \frac{GM}{P^2}\left( 1 + \frac{3 GM P}{c^2 P^2} \right) => Vt_A = \sqrt{\frac{GM}{P}\left( 1 + \frac{3 GM}{c^2 P} \right)} \approx Vt_N\left( 1 + \frac{3 GM}{2 c^2 P} \right) ...[4A]$$

and for case B:-

$$ \frac{Vt_B^2}{P} = \frac{GM}{P^2}\left( 1 + \frac{6 GM}{c^2 P} \right) => Vt_B = \sqrt{\frac{GM}{P}\left( 1 + \frac{6 GM}{c^2 P} \right)} \approx Vt_N\left( 1 + \frac{3 GM}{ c^2 P} \right) ...[4B].$$

Now, in a circular orbit, eccentricity=zero and so we cannot talk of periapse precession or rotation of the line of apsides. However we can think of the A and B orbits as slowly-rotating versions of a Newtonian orbit N of identical radius ($P$). In the same time period ($T_N$), equivalent to the sidereal period of the Newtonian orbit, the various orbits show total circumferential travel distances of:-

$$ d_N = Vt_N * T_N ...[5N] $$ $$ d_A = Vt_A * T_N ...[5A] $$ $$ d_B = Vt_B * T_N ...[5B].$$

The angles swept by particles N, A, B in time period $T_N$ are:-

$$ \theta_N = 2\pi ...[6N] $$ $$ \theta_A = 2\pi * Vt_A/Vt_N ...[6A] $$ $$ \theta_B = 2\pi * Vt_B/Vt_N ...[6B] $$

Defining $\epsilon$ as the dimensioness fraction: $\frac{(\text{revolutions in time period } T_N) - (1\text{ revolution} )}{\text{1 revolution}}$, the $\epsilon$ values for particles A, B are therefore:-

$$ \epsilon_A = \frac{Vt_A - Vt_N}{Vt_N} = \frac{Vt_A}{Vt_N} - 1 ...[7A]$$

$$ \epsilon_B = \frac{Vt_B - Vt_N}{Vt_N} = \frac{Vt_B}{Vt_N} - 1 ...[7B]$$


THE DISCREPANCY

Inserting expressions for $\frac{Vt_A}{Vt_N }-1$ , $\frac{Vt_B}{Vt_N }-1$ derived from equations [4A], [4B] into equations [7A],[7B] respectively, produces:-

$$ \epsilon_A = \frac{3 GM}{2 c^2 P} ...[8A]$$

$$ \epsilon_B = \frac{3 GM}{ c^2 P} ...[8B]$$

Clearly equation [8B] agrees with equation [3] whereas equation [8A] gives a value of $\epsilon$ which is only 50% of that given in equation [3].

Note that orbital rotation rate formulae derived from General Relativity by Wells and other authors cited by him (Part 7, p.28) agree with equations [3] and [8B]. Some of those derivations were based on an Alice model and others were based on a Bob model. Thus it is equation [8A] (derived here for a circular, Alician orbit) which is the anomalous result.


THOUGHTS

Several possible explanations present themselves:-

(1) Something is wrong in my assumptions/reasoning/maths and so eqtn [8A] is incorrect.

(2) In case A, equation [3] does not apply but equation [8A] does - in which case (i) models of the Solar System using equation [1A] will only predict 50% of the observed non-Newtonian planetary precessions; (ii) models such as that used by Shaid-Saless & Yeomans - see my answer here which use the $\frac{3 GM}{c^2 r^2}$ extra-acceleration term along with other rotation-inducing terms will be affected pro-rata.

(3) In case A, equation [3] applies to non-circular orbits while equation [8A] applies to circular orbits. If true it seems strange to me that there is such a marked discontinuity between the circular and non-circular orbit rotation rates. Surely when the orbit is extremely-close to circular the rotation rate should be close to the circular-orbit rotation rate? Perhaps this would be resolved by conducting a more-detailed (higher-order) perturbation analysis which would show a transitional pattern, at low orbital eccentricity, between equations [3] and [8A]?

I would be grateful for any explanations or suggestions on how to resolve this apparent discrepancy. (I am not competent to carry out perturbation analysis myself).

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