Orbits of One Body in a Central Potential
Joachim Köppen Kiel Jan 2022
Some brief explanations
- This simulation shows the orbit of a body in an attractive radial potential,
such as the gravitational field of a massive body.
- The Force exponent specifies the potential: -2 for gravity or an
attractive electrostatic field, +1 for the harmonic oscillator
- The Plummer sphere radial scale models the gravitational potential
from a self-gravitating cloud of stars and/or gas.
- One gives the initial conditions, and clicks start. Then the
trajectory for the time interval is shown, with the blue box denoting the
start position and direction
- As long as more is kept pressed, the models runs to show the subsequent
trajectory ...
- The time step determines the computational accuracy.
Choosing it too large results in inaccurate and bizarre, perhaps interesting but
wrong results. Choosing it too small gives reliable results, but for
reasons of computational economy only a short part of the orbit can be
shown ...
- The computational method can be chosen among a variety of methods.
- The results are plotted as the trajectory in the X-Y plane. The
Surface of Section (or Poincaré map) plots the points in the X-VX plane
whenever the orbiting body crosses this plane in the positive Y direction.
The Potential is plotted as a function
of distance from the centre, assuming a zero value at large distances; for positive
force exponents the potential is zero at the centre.
- In a gravitational field (exponent = -2) the orbit shows a precession,
i.e. the major axis of the orbit turns around the centre. This is
a purely computational effect, since the precession rate depends on the
chosen time step.
- But for exponents different from -2 the orbit shows a genuine precession,
as the precession rate does not change when a sufficiently small
time step is chosen.
- For exponents equal to -3 or smaller, the orbits get unstable. Even the
circular orbit is difficult to keep on its track, with very small time steps.
- There are only two cases where the orbit has no genuine precession:
exponent = -2 and +1.