The Astrophysics Spectator
The Astrophysics Spectator
The characteristics of Keplerian orbits are easily derived from Newtonian physics, which is why it is commonly encountered in undergraduate textbooks on mechanics.
The problem of planetary motion motivated Newton to develop his inverse square rule for gravitational attraction between two bodies and to invent calculus. With these he was able to derive Kepler's observationally-derived laws of planetary motion.
On this page we present the equations of Keplerian motion. The aim of this page is to uncover the basic principals of Keplerian motion. We do not derive the equations on this page, but we provide enough detail to enable the knowledgeable reader to derive them for himself.
The basic equation of motion for an object feeling the gravitational field of another object is
| m1 |
|
= - |
| , |
where the subscripts indicate the characteristics of the first and second objects. The terms are the gravitational constant G, mass m, position vector x, and separation r. The equation of motion for the second object is identical to this equation with the indexes interchanged. The two equations of motion fully describe the motion of both bodies.
The center of mass of the two objects is defined as
where M = m1 + m2 is the total mass of the system. Under the equations of motion, which have left sides that sum to zero, the center of mass moves at a constant velocity. The two objects orbit the center of mass, with their motion confined to a plane that passes through the center of mass. From the equations of motion one can derive two equations that describe the motion in this plane; one equation describes the evolution of the orientation of the bodies, where is the angle relative to a coordinate axis in the orbital plane, and the remaining equation describes the evolution of the separation r of the bodies.
| ( r2 |
| ) | = | 0, | ||||||
| - r ( |
| )2 | = | - GM r2. |
The first equation is a consequence of the conservation of angular momentum, while the second is a consequence of the conservation of energy.
The first equation can be written in terms of a constant of motion L as
| r2 |
| = L. |
The constant L is related to the angular momentum of the system. This equation is the origin of the Kepler law of equal areas being swept in equal times during an orbit; the area swept in a given time is given by the right-hand side of this equation integrated over the time-interval, but from the left-hand side we see that this equals the constant L times the time interval. Because the force term in the equations of motion do not appear in this equation, we see that this equation holds for any isotropic gravitational field; an example of this equation holding for a gravitational field not following the inverse square rule is for the orbit of a star in a globular star cluster.
From the equation for dr/dt and for d/dt, one can write and solve an equation for r in terms of . 1 This equation of motion for a bound orbit is
| r = |
| , |
where a is the semimajor axis and e is the eccentricity of the orbit. By definition, the semimajor axis is the average of the minimum and maximum values of r. The value of e is limited by 0 ≤ e < 1. The minimum separation between the orbiting objects is a (1-e) at = 0, and the maximum separation is a(1+e) at = . The constant L is related to the semimajor axis and the eccentricity by
The equation for r with gives the Kepler law that planets move in closed elliptical orbits around the Sun. This is true regardless of the mass of the planet, and is true of binary stars when effects of general relativity can be ignored. But this equation depends on the presence of the inverse-square rule for the gravitational field; if the field is different, as is the case in a galaxy or globular star cluster, the stars will have a different orbit, and this orbit is not generally closed.
While one cannot find a simple solution for the angle and separation of the bodies with time, one can find a simple relationship between the semimajor axis of a closed orbit and the period for its orbit. For a circular orbit, the relationship is easily derived from the relationship of r with and the conservation equation for L, but even with e > 0, a solution exists for the definite integral over angle from 0 to 2 , and the resulting equation is independent of e:
This is the relationship between semimajor axis and orbital period found by Kepler. It contains a term that is the sum of the masses of the two orbiting objects, so it is valid for planets only in the limit that the mass of the planets are much smaller than the mass of the Sun. Current measurements of orbital period and semimajor axis are fine enough to show the effects of the planet's mass and the gravitational attraction of the other planetary bodies.
1 The trick in solving the differential equation for r is to make a variable change: let r = 1/u, and solve the equation for u. The resulting differential equation is that for a harmonic oscillator.
