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.

initial x
initial y
initial vx
initial vy

time step
time interval
elapsed time

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