Tweaking Kepler’s Laws Aran David Stubbs Abstract

Tweaking Kepler’s Laws

Aran David Stubbs

Abstract

This is a re-examining of Kepler’s Laws in a relativistic universe, from an energy basis.

Main Article

In the early 17^th century, Kepler deduced several laws of motion based on our solar system, 3 of which are still in general use. Newton extended these to form his theory of universal motion. Einstein adjusted Newton’s work, adding a relativistic correction under extreme conditions without changing the basis from which the work was derived. Each of these treated mass and velocity as separate primary properties, although Einstein showed the interrelationship.

A better approach is to look at the laws of motion on the basis of energy, with mass and velocity as derived aspects. While similar lines of reasoning can be applied to tachyons, for now we will just deal with luxons (particles moving at the speed of light) and tardyons (those moving slower than light).

Kepler started with the assumption that the sun is motionless, and all other objects in the solar system move around it. A better perspective is that the center of mass for the sun and any other single object remains fixed, with the sun and that object moving around the center of mass. Aside from Jupiter, the center of mass of such a pair is within the sun, and even in Jupiter’s case it is just outside.

From the conservation of energy, the total energy of a closed system is constant. In our simple model, we will consider the sun and 1 other object as having rest energy, kinetic energy, and potential energy. While in reality the sun is constantly losing mass due to nuclear reactions and the solar wind, we will ignore these effects. Potential energy varies with distance, so one way to measure it has zero potential at zero distance. That has the unwanted consequence of infinite potential energy at infinite distances, so a common work-around is to treat potential energy as zero at infinite distance, and negative as objects approach. Since nothing can get closer to the sun than the suns “surface”, minimum distance is around 700,000 km.

E₀, the rest energy, is zero for luxons and >0 for tardyons. E_A, the energy while in motion, is greater than zero for both. The difference, E_K (the Kinetic energy) is similarly greater than zero for both. Using a second subscript S for the sun, and X for another object, kinetic energy is E_K=E_KS+E_KX for the system. Mass can be derived as E/c² for each term, while velocity can be derived from kinetic energy and M_A. The systemic kinetic energy is typically asymmetrically distributed, with E_KX=E_AS*E_K/E_A. This is equivalent to showing the centripetal force on each is twice the kinetic energy divided by the distance from the objects center of mass to the systems center of mass: F_CS=2E_KS/r_S, F_CX=2E_KX/r_X,. These are vectors outward from the systems center of mass through the objects center of mass, and of equal magnitude.

The systems Kinetic Energy is inversely related to the distance between the centers of mass of the 2 objects. Since gravitational attraction varies with total energy (E_A), this is slightly more complicated than Kepler thought. Even if Jupiter were orbiting just beyond the Roche limit, the Sun’s velocity would be small, so we can approximate E_AS~E_0S. Under those conditions the sun would have about 0.1% of the total kinetic energy. Under less extreme conditions, where the orbiting body has small rest energy, the sun has trivial kinetic energy. Then E_KX is proportionate to 1/r². At low velocities this is equivalent to V_X being proportionate to 1/r, but not at high velocities.