Ashish Tewari - Foundations of Space Dynamics

The gradient of with respect to is the following:

(2.42)

On comparing Eqs. ( 2.40) and ( 2.42), we have

(2.43)

The acceleration of the mass due to the gravitational field created by the mass is therefore given by

(2.44)

and is independent of the test mass, .

The concept of gravitational potential between a pair of isolated masses, , can be extended to a system of point masses, where the net gravitational acceleration caused by point masses, , on the particle of mass is the vector sum of all the individual gravitational accelerations given by

(2.45)

or

(2.46)

where

(2.47)

is the net gravitational potential experienced by the particle due to the gravity of all the other particles.

The potential energy , , of the ‐particle system can be defined by

(2.48)

to be the net work done by the gravitational forces to assemble all the particles, beginning from an infinite separation, , where . Thus a finite separation of the particles results in a negative potential energy (a potential well), escaping from which requires a positive energy expenditure.

The gradient of with respect to gives the negative of the gravitational force, , on the particle as follows:

(2.49)

Hence the right‐hand side of Eq. (2.37)is expressed as follows:

(2.50)

A substitution of Eqs. ( 2.38) and ( 2.50) into Eq. (2.37)yields the important result that the total energy of the system is conserved:

(2.51)

or This is true for any system solely governed by gravity.

To demonstrate another constant of the ‐particle system, consider the vector product of Eq. (2.33)with , followed by summing over all particles:

(2.52)

Because , all the terms on the right‐hand side of Eq. (2.52)vanish, resulting in the following:

(2.53)