One of the most studied central force problems is Newton's universal law of gravitation. This law states that two spherical
particles of mass M_{1 }and M_{2}
attracted each other with a force F_{12} that is
inversely proportional to their separation squared and is directed
along a line connecting their centers

The direction of the force on M_{1} is indicated by
a unit vector "r-hat" pointing from M_{1 }to M_{2}.
The proportionality constant G depends on the system of units and takes
the value G = 6.67x10^{-11} m^{3}/kg^{.}s
in the metric system.

Because two gravitationally interacting particles orbit in a plane about their common center of mass, we choose a coordinate system with the center of mass at the origin. Intermediate mechanics texts show this two-body problem can be reduced to an equivalent one-body problem so this inverse square law model is very general.

Applying Newton's Second Law **F** = M_{1}**a**
to the orbiting mass and using units such that M_{2}G=1
the acceleration of the orbiting mass becomes

Rewriting this vector formula using Cartesian components and
recognizing that sin(θ)
= y/r, cos(θ) =
x/r, and r = (x^{2}+y^{2})^{1/2}
in the plane of the orbit gives the following coupled differential
equations for input into the Ejs model.