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Is Kepler’s Second Law Wrong?

-- | January 9, 2014

Question: I am suspicious about Kepler’s area law. Such law should not exist. I am finding “the cycling velocity of the celestial body is constant and not the swept out area”. How to prove that area 1/2*r*Vp=constant? In addition I discover r*Vp^2=Constant where (r=distance to the sun; Vp=revolving velocity of the body around the sun). The data of the planets (r;Vp) confirm this constant,then if r*Vp^2=Constant,may not allow area r*Vp to be constant.
Here are some sample for r*Vp^2=1,32725E+11 km^3/sec^2
Earth r=149597890 km Vp=29,78607371 km/Sec
Mars r=227939150 km Vp=24,13051171 km/sec
You may use the known data to confirm r*Vp^2=CT.
Then how to prove Kepler’s are law r*Vp=Ct.That should be wrong — Necat

Answer: I think that your fundamental assumption, that the velocity of the celestial body is constant, is the part that is the source of confusion regarding Kepler’s second law. Let me point you to a very nice and detailed derivation of Kepler’s laws that explains both the math and the physics behind the proof of Kepler’s three laws. To summarize the derivation, you can show with a simple geometrical argument that the rate of sweeping out of area in the orbit of a celestial body is proportional to the angular momentum of the celestial body. Since Newton’s Laws tell us that the rate of change of angular momentum torque of the forces acting on the body, which is zero for celestial bodies orbiting a star like our Sun. Since the rate of change of angular momentum is zero, that angular momentum must be constant, which then says that the rate of change of swept-out area for the orbit of the celestial body must be constant. This then leads to Kepler’s Second Law, that celestial objects in orbit sweep out equal areas in equal time.

Jeff Mangum

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