I imagine this was a simple task for many of you during undergrad studies, but I am a lowly engineer attempting to derive Kepler's Law of Orbits. I've tried to do it as cleanly as possible. I've omitted a lot of the derivations for the stuff involving spherical coordinates, angular momentum not changing with respect to time, dot product/cross product identities, etc.
Note that I'm using little m as reduced mass. In my head, I am just thinking of them as the sun (M) and earth (m). Given that the earth has little bearing on the orbit of the sun.
I'm using r hat and theta hat for the respective directions in polar coordinates.
The one problem I am having with the derivation is regarding the constant of integration that is yielded when taking the anti-derivative of v cross L with respect to time. It seems rather important that it should equal the eccentricity times cosine of the angle. But I have no idea where this reasoning comes in to play. If anyone has any advice that would be greatly appreciated.
I've based the derivation on this HyperPhysics page: http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/keplerd.html
They have this sentence on the HyperPhysics page: "where we can argue that the vector constant of integration D must be in the plane of the orbit since the other two quantities lie in that plane." I have no idea how they got a Dcos(theta) out of this, since the overall direction of the integrated quantity is in the r-hat direction.
$$\vec{L}\cdot\vec{L}=L^2$$
$$(m\vec{r} \times \vec{v})\cdot \vec{L}=L^2 $$
$$\vec{r}\cdot(\vec{v} \times \vec{L}) = L^2 /m $$
Find v cross L
$$\frac{d}{dt}(\vec{v} \times \vec{L})=\vec{a} \times \vec{L}$$
Since L is constant with respect to time
$$\vec{a} \times \vec{L} = m(\vec{a} \times \vec{r} \times \vec{v}) = m((\vec{a} \cdot \vec{v})\vec{r} - (\vec{a} \cdot \vec{r})\vec{v})$$ $$\vec{a} = a\hat{r}, \vec{r}=r\hat{r}, \vec{v}=\dot{r}\hat{r}+r\dot{\theta}\hat{\theta}$$ $$\vec{a} \times \vec{L} = -mar^2\dot{\theta}\hat{\theta}$$ Where, $\vec{a} = \frac{-GM}{r^2}\hat{r}$ $$\vec{a} \times \vec{L} = +GMm\dot{\theta}\hat{\theta}$$ $$\frac{d}{dt}(\vec{v} \times \vec{L}) = GMm\frac{d}{dt}\hat{r}$$ Where, $\frac{d}{dt}\hat{r}=\dot{\theta}\hat{\theta}$ $$d(\vec{v} \times \vec{L}) = GMmd\hat{r}$$
$$\int{d(\vec{v} \times \vec{L})} = \int{GMmd\hat{r}}$$
$$\vec{v} \times \vec{L} = GMm(\int{d1})\hat{r}$$
$$\vec{v} \times \vec{L} = GMm(1+B)\hat{r}$$
$$\vec{r} \cdot (\vec{v} \times \vec{L}) = GMmr(1+B) = L^2/m$$
$$r = \frac{L^2 / m^2}{GM(1+B)} $$
Let $B = \epsilon cos\theta$ (This is the part I don't understand)
Noting that $$r(\theta) = a\frac{(1-\epsilon^2)}{1+\epsilon cos\theta}$$