Kepler's Third Law
Kepler's third law defines the ideal relationship between elliptic orbital time
and mean orbital distance. It states that the square of the orbital time is porportional to the cube of the mean orbital distance. In numerical terms, Kepler's Third law can be expressed in the form of a simple, compact mathematical equation relating orbital time and mean distance:
${\left(\frac{{t}_{2}}{{t}_{1}}\right)}^{2}=\text{}{\left(\frac{{d}_{2}}{{d}_{1}}\right)}^{3}$
If it takes time (t_{1}) to orbit a body at mean distance (d_{1}), then it
would take time (t_{2}) to orbit the same body at mean distance (d_{2}).
Here, the mean orbital distance refers to the mean taken over the eccentric anomaly of the orbit, which in this case, equates to the semimajor axis, or 1/2 the longer or major axis of the orbital ellipse.
This program is based on the above equation, where the time and distance may be expressed in any convenient units.
The body being orbited generally refers to the sun, a planet or a star, compared to which, the mass of the lesser satellite is insignificant.
Knowing any three variables, we may directly compute the unknown fourth variable. The following equations, derived from the above, are used by the program to compute the individual unknown variables.
The equation assumes the ideal situation where the mass of the primary sphere is so extremely greater than the lesser sphere that the center of mass is at or extremely close to the center of the primary sphere. In other words, we are ignoring the mass of the lesser body as insignificant and treating the center of mass as being at the center of the primary body.
${t}_{1}\text{}=\text{}\frac{{t}_{2}}{\sqrt{{\left(\frac{{d}_{2}}{{d}_{1}}\right)}^{3}}}$
${t}_{2}\text{}=\text{}{t}_{1}\cdot \sqrt{{\left(\frac{{d}_{2}}{{d}_{1}}\right)}^{3}}$
${d}_{1}\text{}=\text{}\frac{{d}_{2}}{\sqrt[3]{{\left(\frac{{t}_{2}}{{t}_{1}}\right)}^{2}}}$
${d}_{2}\text{}=\text{}{d}_{1}\cdot \sqrt[3]{{\left(\frac{{t}_{2}}{{t}_{1}}\right)}^{2}}$
© Jay Tanner  2021
