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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,

    $\displaystyle 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

    $\displaystyle 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

    $\displaystyle 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

    $\displaystyle 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

    $\displaystyle \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

    $\displaystyle 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

    $\displaystyle 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

    $\displaystyle \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)

    $\displaystyle 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

    $\displaystyle \dot{\varpi}= \frac{3\,\epsilon}{5}\left(\frac{R}{a}\right)^2 n,
$

    where $ n$ is the mean orbital angular velocity of the satellite.


next up previous
Next: Rotating reference frames Up: Orbits in central force Previous: Perihelion precession of Mercury
Richard Fitzpatrick 2016-03-31