Gravitational Potential

The Sun Planet Potential JavaScript model plots the gravitational potential U(r) from the surface of Sun to two astronomical units (AUs). The planet is located at 1 AU and potential plot passes through the center of the planet assuming a constant mass density for the planet. Because Earth has a mass approximately 3 million times smaller than the mass of Sun, the program allows the user to zoom in to observe the potential near the planet. The user can also vary the planet's mass and observe the effect of planet mass on the potential.

Theory

The gravitational potential at point r is the sum the potential from Earth and Sun. Assuming a spherically symmetric mass distribution with constant mass density we calculate the potential from Sun, Earth, and inside Earth using:

Potential Equation

In this simulation we use natural units rather than MKS so that the distance from Sun to Earth, the mass of Sun, and Newton's gravitational constant G are unity.

Constants used to compute potential
Variable	MKS value	Natural Units
Astronomical Unit AU	1.496 x 10¹¹ m	1
G	6.67 x 10^-11 m³/kg/s²	1
Sun Radius	6.86 x 10⁸ m	4.65 x 10^-3
Earth Radius	6.37 x 10⁶ m	4.25 x 10^-5
Sun Mass	1.99 x 10³⁰ kg	1
Earth Mass	5.97 x 10²⁴ kg	3.00 x 10^-6
g = GM_Earth/R²_Earth	9.8 m/s²	1.66 x 10³