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i.e., dA/dt=constant

Also, while deriving we get that,

dA/dt=L/2m where L is the angular momentum of the planet, and m is its mass.

Now, in order for dA/dt to be constant, L should be constant.

I think this is possible only when the following two conditions are fulfilled:

i) the eccentricity e of the orbit is zero, and

ii) the sun is exactly at the centre of the orbit.

Now let us consider a case where these two conditions are not fulfilled-

The earth revolves around the sun in an elliptical orbit (e is not equal to zero), and the sun is not exactly at the centre of the orbit. Now, at most of the points on the orbit(almost all) the gravitational force of the sun on the earth has a tangential component. This means that there is a net non-zero torque on the earth. And this essentially means that the angular momentum of the earth is not constant, i.e. L is not constant. So,now if we have a look at the eq. above, we find that dA/dt is not constant.

Can somebody please tell what the matter is?