Exercises

1. Derive Equations (5.2) and (5.7).

2. Prove that in the case of a central force varying inversely as the cube of the distance,
   $$r^{\,2} = A\,t^{\,2}+B\,t+C,$$
   where $A$, $B$, $C$ are constants. (From Lamb 1923.)

3. The orbit of a particle moving in a central field is a circle that passes through the origin; that is, $r=r_0\,\cos\theta$, where $r_0>0$. Show that the force law is inverse fifth power. (Modified from Fowles and Cassiday 2005.)

4. The orbit of a particle moving in a central field is the cardoid $r=a\,(1+\cos\theta)$, where $a>0$. Show that the force law is inverse fourth power.

5. A particle moving in a central field describes a spiral orbit $r=r_0\,\exp(k\,\theta)$, where $r_0$, $k>0$. Show that the force law is inverse cube, and that $\theta$ varies logarithmically with $t$. Demonstrate that there are two other possible types of orbit in this force field, and give their equations. (Modified from Fowles and Cassiday 2005.)

6. A particle moves in the spiral orbit $r=a\,\theta$, where $a>0$. Suppose that $\theta$ increases linearly with $t$. Is the force acting on the particle central in nature? If not, determine how $\theta$ would have to vary with $t$ in order to make the force central. Assuming that the force is central, demonstrate that the particle's potential energy per unit mass is
   $$V(r)=-\frac{h^{\,2}}{2}\left(\frac{a^{\,2}}{r^{\,4}}+ \frac{1}{r^{\,2}}\right),$$
   where $h$ is its (constant) angular momentum per unit mass. (Modified from Fowles and Cassiday 2005.)

7. A particle moves under the influence of a central force per unit mass of the form
   $$f(r)=-\frac{k}{r^{\,2}} + \frac{c}{r^{\,3}},$$
   where $k$ and $c$ are positive constants. Show that the associated orbit can be written
   $$r = \frac{a\, (1-e^{\,2} )}{1+e\,\cos(\alpha\,\theta)} ,$$
   which is a closed ellipse for $e<1$ and $\alpha= 1$. Discuss the character of the orbit for $\alpha\neq 1$ and $e<1$. Demonstrate that
   $$\alpha = \left(1-\frac{\gamma}{1-e^{\,2}}\right)^{-1/2},$$
   where $\gamma = c/(k\,a)$.

8. A particle moves in a circular orbit of radius $r_0$ in an attractive central force field of the form $f(r) = -c\,\exp(-r/a)/r^{\,2}$, where $c>0$ and $a>0$. Demonstrate that the orbit is only stable provided that $r_0<a$.

9. A particle moves in a circular orbit in an attractive central force field of the form $f(r) = -a\,r^{\,-3}$, where $a>0$. Show that the orbit is unstable to small perturbations.

10. A particle moves in a nearly circular orbit of radius $a$ under the action of the radial force per unit mass
    $$f(r)= -\frac{\mu}{r^{\,2}}\,{\rm e}^{-k\,r},$$
    where $\mu>0$ and $0<k\,a\ll 1$. Demonstrate that the so-called apse line, joining successive apse points, rotates in the same direction as the orbital motion through an angle $\pi\,k\,a$ each revolution. (From Lamb 1923.)

11. A particle moves in a nearly circular orbit of radius $a$ under the action of the central potential per unit mass
    $$V(r)= -\frac{\mu}{r}\,{\rm e}^{-k\,r},$$
    where $\mu>0$ and $0<k\,a\ll 1$. Show that the apse line rotates in the same direction as the orbital motion through an angle $\pi\,k^{\,2}\,a^{\,2}$ each revolution. (From Lamb 1923.)

12. Suppose that the solar system were embedded in a tenuous uniform dust cloud. Demonstrate that the apsidal angle of a planet in a nearly circular orbit around the Sun would be
    $$\pi \left(1- \frac{3}{2}\,\frac{M_0}{M}\right),$$
    where $M$ is the mass of the Sun, and $M_0$ is the mass of dust enclosed by a sphere whose radius matches the major radius of the orbit. It is assumed that $M_0\ll M$.

13. Consider a satellite orbiting around an idealized planet that takes the form of a uniform spheroidal mass distribution of mean radius $R$ and ellipticity $\epsilon$ (where $0<\epsilon\ll 1$). Suppose that the orbit is nearly circular, with a major radius $a$, and lies in the equatorial plane of the planet. The potential energy per unit mass of the satellite is thus (see Chapter 3)
    $$V(r) = -\frac{G\,M}{r}\left(1+ \frac{\epsilon}{5}\,\frac{R^{\,2}}{r^{\,2}}\right),$$
    where $r$ is a radial coordinate in the equatorial plane. Demonstrate that the apse line rotates in the same direction as the orbital motion at the rate
    $$\dot{\varpi}= \frac{3\,\epsilon}{5}\left(\frac{R}{a}\right)^2 n,$$
    where $n$ is the mean orbital angular velocity of the satellite.