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What is the Law of Force at the Center of a Kepler Ellipse?

It is very well-known that Newton subsumed his law of inverse square from the Kepler's work on elliptical motion. The source of gravitational attraction was at one of the foci of the ellipse. But what happens when we move the source to the center of the ellipse?


For one thing the true anomaly becomes the eccentric anomaly. Now there has been a great deal of work done on dual laws: one law can can be converted into another by complex squaring. That is, an ellipse when it is squared becomes another ellipse with its distance displaced by an amount ea, where e is the eccentricity and a is the semi-major axis. The duality then becomes one between the inverse square law and the harmonic oscillator. This juxtaposed the two arch-rivals Newton and Hooke. But what type of motion can we expect when the source is transferred to the center?


Certainly not that of a lissajous figure. When an ellipse is squared, it is not the same ellipse just displaced by an amount ae. So what does Hooke's law have to do with the motion of planetary bodies about the sun when the sun is displaced from the focus to the center of the ellipse?


In fact, none... The search for a solution to Kepler's equation is a well-trodden path that had occupied many a mathematician from the time of Kepler. If the sun were to be displaced from the focus to the center, we would certainly not see the earth behaving as a mass tied to a spring. We would expect that an inverse law would still apply, but which one?


Our working hypothesis is that the eccentricity does not change the nature of the law of force, just whether it is attractive or repulsive. An ellipse has a positive semi-major axis, whereas an hyperbola has a negative one, but the force laws are identical. And so too for the intermediary case of a parabola. So all three are governed by an inverse square law when the source is at the focus.


We know that Kepler's II law of equal areas in equal times applies to the product of the square of the radial coordinate and the time derivative of the true anomaly. Strangely enough, it also applies to the time derivative of the eccentric anomaly so that there will be conservation of angular momentum no matter where the source is located. The eccentric and true anomalies are related by an equation of aberration where the eccentricity plays the role of a relative acceleration.


If we set the eccentricity equal to one, things becomes simpler. We know that a circle and straight line are related by inversion. So too is a parabola and a cardioid. Where there's one there is also its inverse. Now the definition of the eccentric anomaly is give by an equation that describes a limacon. Why we don't see the cusp or the loop is due to the fact that the eccentricity of planets are small in comparison to unity. The exception, of course is Mercury, where the eccentricity is e=0.21, but still too small to make the limacon noticeable...or maybe not. That it, it shows up in the unaccounted for advance of its perihelion.


A limacon of eccentricity unity is a cardioid whose inverse is the parabola. Taking the time derivative of the equation of a cardioid,

r=a(1-cos E),

where r is the radial coordinate, a is a constant and E is the eccentric anomaly. we get a relation between the rate of change of the radial coordinate and the corresponding change in E

dr/dt/sinE=a dE/dt.

For sin E we introduce the radial equation

sin E=(2r/a-r^2/a^2)^{1/2}.

But we know that dr/dt must have the form

dr/dt=(V-h^2/r^2)^{1/2}

where V is the scalar potential. Factoring out the necessary term gives

sin E=(r/a)^2(2a^3/r^3-a^2/r^2)^{1/2}.

This the ratio of the last two equations will reduce to a^2/r^2, and when equated with the right-hand side gives

r^2 dE/dt=constant,

which is Kepler's aerial law.

It also specifies the the potential as

V=2a^3/r^3,

whose derivative gives a force law proportional to the inverse-fourth of the radius---and not the inverse square, as in Newton's law!


Moreover, if we set a=h^2/GM, we can write the force as

F=6(a/r)^2 GM/r^2,

and Kepler's aerial theorem becomes

(GM a)^{1/2}=r^2 dE/dt,

which shows its constancy in time.


Finally, Einstein's modification of the equation for the trajectory involved precisely a force component with an inverse fourth dependency on the radial distance. Thus, it is seen that the advance of the perihelion is nothing other than the force law for a cardioid, or limacon, since the eccentricity does not change the magnitude of the law of force.


We can even determine its numerical value by observing that the h^2/GM is the equivalent of the Bohr radius, corresponding to the inverse square law. It is the semi-latus rectum of the ellipse, p. Multiplying it by the fine structure constant, GM/hc gives the Compton wavelength h/c, since h is the specific angular momentum. Again multiplying by the fine structure constant gives r_s=GM/c^2, which is precisely the coefficient in the inverse fourth order force law. 3 times the ratio of the lengths 3r_s/p is precisely the advance of the perihelion.


As a generalization we observe that these are examples of sinusoidal spirals which are planar projections of the asymptotic lines of a Plucker conoid. Sinusoidal spirals of index n is a trajectory of a mass M attracted to a central force which norm


GM/r^(2n+3)

n=-1/2 corresponds to a parabola (ellipse, hyperbola) with an inverse square law. Its inverse n=1/2 corresponds to a cardioid (limacon) with an inverse fourth law as its central force.


The sinusoidal spiral of index n is the field line of the complex field z^n which when combined with its inverse, z^(-n) is the Joukowski transform

w=z^n+z^(-n)

whose corresponding force law for n=1/2 is Einstein's solution to the advance of the perihelion of Mercury.

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