I am taking a multivariable calculus lecture online provided by MIT OpenCourseWare.
In lecture, professor used vectors to prove the Kepler's Second law. The second law says that:
A planet moves in a plane, and the radius vector (from the sun to the planet) sweeps out equal areas in equal times.
And as an additional information, first law is:
The planet’s orbit in that plane is an ellipse, with the sun at one focus.
Assume that the sun is located on the origin. The vector from the sun to the planet is r, and the Δr is the location change vector. And r+Δr is the new location over time t.
The professor stated that, in order for the second law to be proved, |r x dr/dt| must be a constant. Because, an parallelogram area created by two vectors is:
|a x b| (where a and b vectors).
The proffessor stated that this cross product has same direction and length on every point of the elipse trajectory. Direction, ok, that since we are in a plane and all the vector that we product of in the plane, it has same direction every point. But I don't understand about the length. How does the length be the same at every point?
Best Answer
Well if you assume that the object is acted on by a central force (gravity) then $\mathbf r(t)$ is parallel to $\mathbf a(t)$ for all times $t$. And we know that $$\dfrac{d}{dt}(\mathbf r \times \mathbf v) = \require{cancel}\color{red}{\cancelto{0}{\color{black}{\mathbf v \times \mathbf v}}} + \color{red}{\cancelto{0}{\color{black}{\mathbf r \times \mathbf a}}} = 0$$ because the cross product of parallel vectors is zero. Therefore $\mathbf r \times \mathbf v$ must be a constant vector.
Constant vectors have both constant direction and magnitude. Because the object is moving in an elliptical orbit, the only way that this vector is constant is if the object speeds up as it gets closer to the perihelion and slows down as it moves toward the aphelion. Thus is it able to sweep out equal areas in equal times.