Sunday, January 3, 2016

Calculating Orbits: Kepler’s First Law

Kepler’s First Law

Kepler’s First Law tells us that planetary orbits are ellipses.  The following derivation, taken from Wiesel, is very clever: it solves a second order differential equation as if by magic.  We begin with our second order differential equation, which is an expression for the acceleration of a planet taken from Newton’s Law of Gravity:
We will now cross each side of this equation on the left with the vector $\vec{H}$:
Consider the left side of this equation and note the following
because $\vec{H}$ is constant with respect to $t$.
Now consider the vector portion of the right hand side
By the bac-cab formula
Clearly, $\vec{r}\cdot\vec{r}=r^2$.  It is also true that $\vec{r}\cdot\vec{r}=rDr$.  This is clear when we resolve $\vec{v}$ into a coordinate system in which $\vec{r}$ lies along one of the axes.  It follows that
It is at this point in the derivation that the god comes out of the machine.  Note
which can be written as
Therefore, the vector whose derivative we’ve taken is constant.  Choose the vector $\vec{e}$ so that
and as a consequence
There is yet another god in the machine, however.  Dot both sides of this equation with $\vec{r}$ so that
By the application of the triple scalar product, the left hand side may be transformed as follows

If we let $\nu$ be the angle between $\vec{r}$ and $\vec{e}$, the right hand side becomes
and consequently,

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