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Satellite Orbit Decay

A satellite of mass m is placed in a circular orbit of radius r₀ around the earth. To begin with, the satellite can move in a circular orbit if the necessary centripetal force is provided by earth’s gravity:

where v is the initial tangential velocity given to satellite by the launching device. The kinetic energy of the satellite is and the total (initial) energy is,

The total energy is equal to kinetic energy in magnitude but negative as the satellite is bounded to earth’s gravitational field.

Suppose now the satellite is resisted by a drag force F_fr = Av², as it moves in the orbit; A is assumed to be a constant. Because of the drag force, the total mechanical energy of the satellite decreases; the work done to overcome resistance is lost as heat. Since total mechanical energy E is negative, decrease of E means it becomes more negative. That is, the radius r of the orbit decreases

At some instant of time t, let the orbit of the satellite changes in time dt, from r to r + dr, where dr is negative. The corresponding change in kinetic energy of the satellite is given by,

Since dr is negative, the kinetic energy increases by,

Let us now calculate this kinetic energy change by using work-energy theorem. That is, we have

dK = F . dl

where F is the total force acting on the satellite and dl is the displacement vector.

The force F consists of two parts: the gravitational force which acts radially inwards, and the drag force F_fr = Av² which acts opposite to direction of velocity v = dl/dr.

Due to drag force, satellite changes the path from a circle to an inward spiral and continuously shifts itself from circle of radius r to (r – dr) in time dt. The line element PQ represents the segment dl (of the actual path) traversed by satellite during the time dt. The line PR is radial, PR = dr.