EXPL 48. In this exercise you will derive Kepler's First Law, that planets
travel in elliptical orbits. We begin with the notation. Place the coordinate
system so that the sun is at the origin and the planet's closest approach to
the sun (the perihelion) is on the positive \(x\) -axis and occurs at time
\(t=0\). Let \(\mathbf{r}(t)\) denote the position vector and let
\(r(t)=\|\mathbf{r}(t)\|\) denote the distance from the sun at time \(t\). Also,
let \(\theta(t)\) denote the angle that the vector \(\mathbf{r}(t)\) makes with
the positive \(x\) -axis at time \(t\). Thus, \((r(t), \theta(t))\) is the polar
coordinate representation of the planet's position. Let
\(\mathbf{u}_{1}=\mathbf{r} / r=(\cos \theta) \mathbf{i}+(\sin \theta)
\mathbf{j} \quad\) and \(\quad \mathbf{u}_{2}=(-\sin \theta) \mathbf{i}+(\cos
\theta) \mathbf{j}\)
Vectors \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) are orthogonal unit vectors
pointing in the directions of increasing \(r\) and increasing \(\theta\),
respectively. Figure 12 summarizes this notation. We will often omit the
argument \(t\), but keep in mind that \(\mathbf{r}, \theta, \mathbf{u}_{1}\), and
\(\mathbf{u}_{2}\) are all functions of \(t .\) A prime in. dicates
differentiation with respect to time \(t\).
(a) Show that \(\mathbf{u}_{1}^{\prime}=\theta^{\prime} \mathbf{u}_{2}\) and
\(\mathbf{u}_{2}^{\prime}=-\theta^{\prime} \mathbf{u}_{1}\).
(b) Show that the velocity and acceleration vectors satisfy
$$
\begin{array}{l}
\mathbf{v}=r^{\prime} \mathbf{u}_{1}+r \theta^{\prime} \mathbf{u}_{2} \\
\mathbf{a}=\left(r^{\prime \prime}-r\left(\theta^{\prime}\right)^{2}\right)
\mathbf{u}_{1}+\left(2 r^{\prime} \theta^{\prime}+r \theta^{\prime
\prime}\right) \mathbf{u}_{2}
\end{array}
$$
(c) Use the fact that the only force acting on the planet is the gravity of
the sun to express a as a multiple of \(\mathbf{u}_{1}\), then explain how we
can conclude that $$
\begin{aligned}
r^{\prime \prime}-r\left(\theta^{\prime}\right)^{2} &=\frac{-G M}{r^{2}} \\
2 r^{\prime} \theta^{\prime}+r \theta^{\prime \prime} &=0
\end{aligned}
$$
(d) Consider \(\mathbf{r} \times \mathbf{r}^{\prime}\), which we showed in
Example 8 was a constant vector, say D. Use the result from (b) to show that
\(\mathbf{D}=r^{2} \theta^{\prime} \mathbf{k} .\)
(e) Substitute \(t=0\) to get \(\mathbf{D}=r_{0} v_{0} \mathbf{k}\), where
\(r_{0}=r(0)\) and \(v_{0}=\|\mathbf{v}(0)\|\). Then argue that \(r^{2}
\theta^{\prime}=r_{0} v_{0}\) for all \(t\).
(f) Make the substitution \(q=r^{\prime}\) and use the result from (e) to obtain
the first-order (nonlinear) differential equation in \(q\) :
$$
q \frac{d q}{d r}=\frac{r_{0}^{2} v_{0}^{2}}{r^{3}}-\frac{G M}{r^{2}}
$$
(g) Integrate with respect to \(r\) on both sides of the above equation and use
an initial condition to obtain
$$
q^{2}=2 G
M\left(\frac{1}{r}-\frac{1}{r_{0}}\right)+v_{0}^{2}\left(1-\frac{r_{0}^{2}}{r^{2}}\right)
$$
(h) Substitute \(p=1 / r\) into the above equation to obtain
$$
\frac{r_{0}^{2} v_{0}^{2}}{\left(\theta^{\prime}\right)^{2}}\left(\frac{d p}{d
t}\right)^{2}=2 G
M\left(p-p_{0}\right)+v_{0}^{2}\left(1-\frac{p^{2}}{p_{0}^{2}}\right)
$$
