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Sectional Curvatures Determine the Conditions of Dynamical Equilibrium in Kepler's Problem

Updated: Aug 5, 2021

We have advanced the idea that gravitational phenomena can be explained in terms of their sectional curvatures. All sectional curvatures vary between r"/r, radial curvature, and (C-r'^2)/r^2, where C is a constant and the primes denote differentiation with respect to space or time coordinates.


As a prime example, we consider the Kepler problem. The angular velocity is given by Bernoulli's equation:


r(d\theta/dt)={C+(M/R^3)r^2\cos( 2\theta)+ 2(\Omega r)^2\sin^2(\theta)}^1/2,


where C is the total energy, R the planetary radius, r<R, the radial distance, and \theta is the angle that the radius vector r makes with the vertical axis (latitude). The \cos(2\theta) term is what represents the deformation, or tidal force, that is described by the second harmonic, and the sin^2\theta term is the centrifugal terms with \Omega, the spin frequency.


Here, we have replaced the second harmonic, representing the tidal deformations,

P_2(cos\theta)=(1/2)(cos^2\theta-1),

by the double angle formula

= (3/4)(cos2\theta+1/3),

and drop the constant term because it is independent of the angle, and hence of the spinning planet.


At constant r, the time derivative of the Bernoulli equation expressed in terms of the angular velocity yields


(d^2\theta/dt^2)=(-M/R^3+\Omega^2)\sin(2\theta).

This term is proportional to the radial curvature, q"/q, where the prime denotes the time derivative and q is any generic distance. The radial curvatures vanishes when Kepler's third law holds. This is valid for any angle, \theta. Kepler's third law ensures that the motion is geodesic.


Under this condition the tangential curvature reduces to

{C-(r d\theta/dt)^2}/r^2= -M/R^3<0.

The tangential curvature is equal to the negative density. The tangential curvature is defined as

{C-(dq/dt)^2}/q^2

for any generic distance, q, and C is a constant.

Therefore, Bernoulli's equation determines, therefore, the tangential sectional curvature.


Thus, the Kepler problem has vanishing rotation acceleration, when Kepler's III law is satisfied. In this case the tangential curvature is constant. Note that the internal potential is proportional to (M/R^3) r^2, while the external potential is proportional to M/r^3 R^2.


Now, if we consider radial motion, at any given angle, and disregard the centrifugal motion, the velocity is

dr/dt= {C-M/R^3 r^2P_2(cos\theta)}^1/2,

where P_2 is the second-order Legendre polynomial that describes the tidal deformation. The potential is a sole function of the angle \theta, indicating an axis of symmetry between the line joining the two bodies.


The relation between the radial and tangential curvatures, at \theta=constant,

d^2r/dt^2/r+[C-(dr/dt)^2]/r^2=0,

shows that they balance one another under the validity of Kepler's III law. Each component


d^2r/dt^2/r=[C-(dr/dt)^2]/r^2=M/R^3 cos(2\theta)+2w^2\sin^2\theta,\]

depends only on the angle \theta. Terms in the potential that do not depend on the angle are deemed compressional forces. Such forces vanish if the fluid is incompressible.


There are two values of the angle that determine the shape of the distorted planet: \theta=0, \pi/2. At these angles, the second harmonic,

P_2=(3\cos^2\theta-1)/2,

reduce to 1, for \theta=0, and -1/2 at \pi/2. Thus the distorted planet takes the shape of an oblate spheroid with long semi-axis lying along the line connecting the two bodies, and short semi-axes which are equal. For equilibrium, the sum of the axes must vanish. This is analogous to the vanishing of the Ricci tensor as a condition of dynamic equilibrium.


Unlike Einstein's theory, the vanishing of the sum of the sectional curvatures has absolutely nothing to do with the "emptiness" of the universe. Rather, it is a condition of dynamic equilibrium. In the rotational case treated above, one sectional curvature vanishes, which determines Kepler's III law, and the other sectional curvature was constant. This is also a case of dynamic equilibrium. So Kepler's III law establishes dynamical equilibrium by requiring the centrifugal acceleration to vanish and the rotational sectional curvature to be constant and equal to the negative density.



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