Neddy, I know what you want to express now. Below is the explanation I gave.so

F(0) / F(c) = (R + c*T) / R

now what?

In step1, in the integral calculation, the force used is based on the gravitational force when the instantaneous velocity v of the object m at any position is 0, and the average gravitational force reflects the equivalent gravitational force at different positions of the object. . The velocity of gravity relative to m is c-0.

The real situation is that the object has an instantaneous velocity v at any position. We introduced boundary conditions and finally obtained the relationship between gravity and distance and velocity according to the linear relationship between the equivalent gravity and the velocity v of m. .

Then you may ask, the premise of our deduced average gravity is that the instantaneous velocity of any position of the object is 0. But it is not. Will the change of this premise affect the correctness of the derivation?

In fact, we can continue to maintain the position change velocity v of the object, and let the instantaneous velocity of the object at any position continue to be 0. We only need to adjust the instantaneous velocity of the gravity at any position from the original c to the current c-v. Let's look at the Newtonian gravitational model. This model reflects the relationship between gravitation and distance. The velocity of gravitation relative to the object is not directly reflected, but is implicit in the gravitational constant G. So after the gravitational velocity changes now, if you continue to establish only the relationship between gravitation and distance, then G needs to be changed. We need to derive G after the gravitational velocity becomes c-v.

Because the gravitational velocity changes from c to c - v, it does not affect the linear relationship between the average gravitational force and the object velocity derived previously. So we get K = G*M*m/R^2/ (c*T) through the boundary conditions v=0, v=c, and then substitute it into F(v) = G*M*m/R^2 - K*v *T, get F(v) = G*M*m/R^2 * ((c - v)/c). If v is a constant, then we can write it like this F(v) = (G* (c - v)/c )*M*m/R^2, where G*(c - v)/c is the new gravitational constant.

Now can you understand why here F(0) / F(c) = (R + c*T) / R is right ?