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 17th 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. EA, the energy while in
motion, is greater than zero for both. The difference, EK (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 EK=EKS+EKX for the system. Mass can be derived as E/c2 for
each term, while velocity can be derived from kinetic energy and MA. The systemic
kinetic energy is typically asymmetrically distributed, with EKX=EAS*EK/EA. 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: FCS=2EKS/rS,
FCX=2EKX/rX,. 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 EAS~E0S.
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 EKX is proportionate to 1/r2. At low velocities this is equivalent to
VX being proportionate to 1/r, but not at high velocities.