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For two pairwise interacting point particles, the gravitational potential energy is the work that an outside agent must do in order to quasi-statically bring the masses together (which is therefore, exactly opposite the work done by the gravitational field on the masses): = = where is the displacement vector of the mass, is gravitational force acting on it and denotes scalar product.
The gravitational potential (V) at a location is the gravitational potential energy (U) at that location per unit mass: =, where m is the mass of the object. Potential energy is equal (in magnitude, but negative) to the work done by the gravitational field moving a body to its given position in space from infinity.
There are various types of potential energy, each associated with a particular type of force. For example, the work of an elastic force is called elastic potential energy; work of the gravitational force is called gravitational potential energy; work of the Coulomb force is called electric potential energy; work of the strong nuclear force or weak nuclear force acting on the baryon charge is ...
A common misconception occurs between centre of mass and centre of gravity.They are defined in similar ways but are not exactly the same quantity. Centre of mass is the mathematical description of placing all the mass in the region considered to one position, centre of gravity is a real physical quantity, the point of a body where the gravitational force acts.
Conversely, as two massive objects move towards each other, the motion accelerates under gravity causing an increase in the (positive) kinetic energy of the system and, in order to conserve the total sum of energy, the increase of the same amount in the gravitational potential energy of the object is treated as negative. [1]
The potential energy, U, depends on the position of an object subjected to gravity or some other conservative force. The gravitational potential energy of an object is equal to the weight W of the object multiplied by the height h of the object's center of gravity relative to an arbitrary datum: =
The potential has units of energy per mass, e.g., J/kg in the MKS system. By convention, it is always negative where it is defined, and as x tends to infinity, it approaches zero. The gravitational field, and thus the acceleration of a small body in the space around the massive object, is the negative gradient of the gravitational potential ...
For this the gravitational force, i.e. the gradient of the potential, must be computed. Efficient recursive algorithms have been designed to compute the gravitational force for any N z {\displaystyle N_{z}} and N t {\displaystyle N_{t}} (the max degree of zonal and tesseral terms) and such algorithms are used in standard orbit propagation software.