Updated: Aug 5, 2021
Every school boy, or girl, knows that the planets follow Keplerian ellipses. If the sun were placed at the center of the ellipse, the force acting on a planet would be Hookean. Only when the sun is displaced to a focus, does Newton's inverse square law emerge.
The angle involved in the expression for the ellipse is known as the true anomaly. Rather, the equation that is used to measure the position of a planet, known as Kepler's equation, uses the eccentric anomaly. The true anomaly is measured at the focus in the direction of increasing from the perihelion, while the eccentric anomaly is measured from the center of an ellipse.
The equation of the ellipse is given by
r= L/(1+e cos v), (*)
where L is the semi-latus rectum, e the eccentricity, and v, the true anomaly. The is the second figure in the above. Closely related to Kepler's equation, which was originally of Hindu origin taken over by 9th Century Islamic astronomers, is the equation expressing the radial distance as
r=a(1-e cos E), (**)
where a is the semi-major axis related to L by L=a(1-e^2), and E is the eccentric anomaly. This is the first figure above.
Whereas (*) is a conic section, (e<1, an ellipse), (**) is a cardioid. Since e is approximately .2
for Mercury, there would be no loop on the trajectory. The figure shows how the motion of the planets look from earth. Mercury looks like a cardioid, Venus a pentagram, etc.
Although I have not been able to find any mention that (**) is a cardioid, there is a definite relation between the anomalies v and e. The tangents of their half-angles are related to a longitude Doppler law if the eccentricity is treated as a relative velocity. And a relative velocity it is. It arises as an integration constant in the expression of the velocity.
The acceleration is in the same direction as the radial vector
dV/dt= -k(cos v, sin v, 0)/r^2, (***)
where V is the velocity, and k is the gravitational parameter, GM. If we use the conservation of angular momentum, dv/dt=h/r^2, and dividing (***) by this quantity, the r^2's cancel, and
dV/dv=-R(cos v, sin v, 0)
where R=k/H, which has dimensions of a velocity! Now integrating gives
v=R(-sin v, cos v, 0)+ c
where c is an integration vector. It shows that the velocity vector moves in a circle about a center c. Incorporating it into the solution, we define e=|c|/R, and the expression for the velocity, assuming the center lies on the positive y-axis is
v=R(sin v, e+cos v, 0).
Since the angular momentum is h=x X v, where x=r(cos v, sin v, 0), we get on solving for r
This shows conclusively that e=|c|/R is a relative velocity, and it is relates the two anomalies by
tan(v/2)=(1+e/1-e)^1/2 tan E/2,
that is by a longitudinal Doppler effect! This is related to the old problem of the parallax where two observers distanced at two points on earth will measure the same object in the sky with different coordinates. So we have no reason to prefer an ellipse to a cardioid, and, in fact the mean anomaly
M=E-e sin E,
uses the eccentric anomaly, not the true anomaly to measure celestial objects. Moreover, as the line connecting Venus and the earth rotates, it first gives the image of a cardioid and then goes on to give a much more complicated image, finally winding up as a five star image.
I have search far and wide to find any reference to (**) as a cardioid. The image I have reproduced can be found on reddit. And while we are here, what happens when we suppose that the rotatum, and not the angular momentum is conserved. The two are related by
where w=dv/dt is the angular speed. Instead of Kepler's II law, that the radius vector sweeps out equal areas in equal times, we have that the velocity vector sweeps out equal areas in velocity space in equal times. The magnitude of the rotatum is
and if we divide Newton's law (***) by it, (writing r=V/w),
dV/dv= -(kw^2/R)(cos v, sin v, 0).
Since R has dimensions of power, the coefficient has dimensions of a velocity.
Now R=V X dV/dt, take its modulus and solving for the velocity
V=(kw^2/R)(1+e cos v)^1/2.
The form of the velocity is shown in the first figure above. So the two are related by a longitudinal Doppler effect. In this sense, it is relative to our point (position) of view.
The conservation of rotatum comes from the vanishing of the normal component of the jerk.
If we use the conservation of angular momentum, the vanishing of the tangential component of the acceleration resulting in the conservation of the angular momentum, h, we get for the radial component
F_S cos v=V^2/p,
where F_S is the force directed to the source at one of the foci, and p is the radius of curvature. Multiplying both sides by the square of r cos v, and using the definition of the radius of curvature,
1/ (p cos^3) v=u+u",
where u=1/r, and the double prime the derivative with respect to the true anomaly gives the equation of the orbit as
If the left-hand side is constant, implying an inverse square law, the solution is an ellipse,
u=(1+e cos v)/L.
So when we go from the conservation of angular momentum to that of rotatum, we move from configuration to velocity space where the velocity orbit is no longer circular. The superposition of an ellipse and cardioid for e=0.5 is shown below.
The two anomalies become proportional to one another when e=1, i.e. a parabola.
We can even still go further as ask about the central force for the cardioid. The force will be given by
p cos^3 v=F r^2/h^2 (****)
where p is the radius of curvature. For the true anomaly, the left-hand side is a constant, and we get Newton's inverse-square law.
Now for the eccentric angle, the law of force is
p cos^3 E=F r^2/h^2 (*****)
and the left-hand side is 1/r^2. This is Einstein's modification of the equation of the orbit. So if the total force is a linear combination of the two, i.e.,
we come out with Einstein's modified equation, where u=1/r, and the prime is the derivative with respect to the true anomaly.
The true anomaly (****) would give only the first term in the equation of the orbit. Whereas the eccentric anomaly (****) would give the second term, or the 1/r^4 law. However, the two angles are related to the relativistic velocity addition law
cos v=(cos E-e)/(1-e cos E),
so this must be taken into account when combining the two central force laws.
Apart from a numerical factor, the coefficient of the second term in (*******) can be deduced on dimensional grounds insofar as whatever is need to multiply h^2 must have dimensions of distance squared. It must be the inverse of a velocity squared, and the only invariant velocity is that of the speed of light.