Exercises

1. Consider the secular evolution of two planets moving around a star in coplanar orbits of low eccentricity. Let $a$, $e$, and $\varpi$ be the orbital major radius, eccentricity, and longitude of the periastron (i.e., the point of closest approach to the star) of the first planet, respectively, and let $a'$, $e'$, and $\varpi'$ be the the corresponding parameters for the second planet. Suppose that $a'>a$. Let $h=e\,\sin\varpi$, $k=e\,\cos\varpi$, $h'=e'\,\sin\varpi'$, and $k'=e'\,\cos\varpi'$. Consider normal mode solutions of the two planets' secular evolution equations of the form $h(t)=\skew{3}\hat{e}\,\sin(g\,t+\beta)$, $k(t)=\skew{3}\hat{e}\,\cos(g\,t+\beta)$, $h'(t)=\skew{3}\hat{e}\,'\,\sin(g\,t+\beta)$, and $k'(t)=\skew{3}\hat{e}'\,\cos(g\,t+\beta)$, where $\skew{3}\hat{e}$, $\skew{3}\hat{e}\,'$, $g$, and $\beta$ are constants. Demonstrate that
   $$\left(\begin{array}{cc} \skew{3}\hat{g}-q\,\alpha,& q\,\alpha\,\b... ...t{e}'\end{array}\right)= \left(\begin{array}{c}0\\ [0.5ex]0\end{array}\right),$$
   where $\skew{3}\hat{g}= g/[(1/4)\,\epsilon\,n\,\alpha\,b^{(1)}_{3/2}(\alpha)]$, $\epsilon=m/M$, $n=(G\,M/a^{\,3})^{1/2}$, $\alpha=a/a'$, $q=m'/m$, and $\beta=b_{3/2}^{(2)}(\alpha)/b_{3/2}^{(1)}(\alpha)$. Here, $m$, $m'$, and $M$ are the masses of the first planet, second planet, and star, respectively. It is assumed that $M\gg m, m'$. Hence, deduce that the general time variation of the osculating orbital elements $h(t)$, $k(t)$, $h'(t)$, and $k'(t)$ is a linear combination of two normal modes of oscillation, which are characterized by
   $$\skew{3}\hat{g} = \frac{1}{2}\left(q\,\alpha+\alpha^{\,3/2}\pm\le... ...,\alpha-\alpha^{\,3/2})^2+4\,q\,\alpha^{5/2}\,\beta^{\,2}\right]^{1/2}\right),$$
   and
   $$\frac{\skew{3}\hat{e}\,'}{\skew{3}\hat{e}} = \frac{q\,\alpha-\ske... ...\alpha\,\beta} = \frac{\alpha^{\,3/2}\,\beta}{\alpha^{\,3/2}-\skew{3}\hat{g}}.$$
   Demonstrate that in the limit $\alpha\ll 1$, in which $b_{3/2}^{(1)}(\alpha)\simeq 3\,\alpha$ and $b_{3/2}^{(2)}(\alpha) \simeq (15/4)\,\alpha^{\,2}$, the first normal mode is such that $g\simeq (3/4)\,\epsilon'\,n\,\alpha^{\,3}$ and $\skew{3}\hat{e}\,'/\skew{3}\hat{e}\simeq -(5/4)\,\alpha^{\,3/2}/q$ (assuming that $q\gg \alpha^{\,1/2}$), whereas the second mode is such that $g\simeq (3/4)\,\epsilon\,n'\,\alpha^{\,2}$ and $\skew{3}\hat{e}\,'/\skew{3}\hat{e}\simeq (4/5)\,\alpha^{-1}$. Here, $\epsilon'=m'/M$ and $n'=(G\,M/a'^{\,3})^{1/2}$. (Modified from Murray and Dermott 1999.)

2. The gravitational potential of the Sun in the vicinity of the planet Mercury can be written
   $${\mit\Phi}(r) = -\frac{G\,M}{r}-\frac{G\,M\,h^{\,2}}{c^{\,2}\,r^{\,3}},$$
   where $M$ is the mass of the Sun, $r$ the radial distance of Mercury from the center of the Sun, $h$ the conserved angular momentum per unit mass of Mercury, and $c$ the velocity of light in vacuum. The second term on the right-hand side of the preceding expression comes from a small general relativistic correction to Newtonian gravity (Rindler 1977). Show that Mercury's equation of motion can be written in the standard form
   $$\ddot{\bf r} + \mu\,\frac{\bf r}{r^{\,3}} = \nabla{\cal R},$$
   where $\mu=G\,M$, and
   $${\cal R} = \frac{\mu\,h^{\,2}}{c^{\,2}\,r^{\,3}}$$
   is the disturbing function due to the general relativistic correction. Demonstrate that when the disturbing function is averaged over an orbital period it becomes
   $$\overline{\cal R}= \frac{\mu\,h^{\,2}}{c^{\,2}\,a^{\,3}\,(1-e^{\,2})^{\,3/2}},$$
   where $a$ and $e$ are the major radius and eccentricity, respectively, of Mercury's orbit. Hence, deduce from Lagrange's planetary equations that the general relativistic correction causes the argument of the perihelion of Mercury's orbit to precess at the rate
   $$\skew{3}\dot{\omega} =\frac{3\,\mu\,h^{\,2}}{c^{\,2}\,n\,a^{\,5}\,(1-e^{\,2})^{\,2}}= \frac{3\,\mu^{\,3/2}}{c^{\,2}\,a^{\,5/2}\,(1-e^{\,2})},$$
   where $n$ is Mercury's mean orbital angular velocity. Finally, show that the preceding expression evaluates to $0.43''\,{\rm yr}^{-1}$.

3. The gravitational potential in the immediate vicinity of the Earth can be written
   $${\mit\Phi}(r,\vartheta) =- \frac{\mu}{r}\left[1-J_2\left(\frac{R}... ...s\vartheta) - J_3 \left(\frac{R}{r}\right)^3 P_3(\cos\vartheta)+\cdots\right],$$
   where $\mu=G\,M$, $M$ is the terrestrial mass, $r$, $\vartheta$, $\phi$ are spherical coordinates that are centered on the Earth, and aligned with its axis of rotation, $R$ is the Earth's equatorial radius, and $J_2= 1.083\times 10^{-3}$, $J_3= -2.112\times 10^{-6}$ (Yoder 1995). In the preceding expression, the term involving $J_2$ is caused by the Earth's small oblateness, and the term involving $J_3$ is caused by the Earth's slightly asymmetric mass distribution between its northern and southern hemispheres. Consider an artificial satellite in orbit around the Earth. Let $a$, $e$, $I$, and $\omega$ be the orbital major radius, eccentricity, inclination (to the Earth's equatorial plane), and argument of the perigee, respectively. Furthermore, let $n=(\mu/a^{\,3})^{1/2}$ be the unperturbed mean orbital angular velocity.

   Demonstrate that, when averaged over an orbital period, the disturbing function due to the $J_3$ term takes the form

   $$\overline{\cal R}= -\frac{3\,J_3}{2}\,\frac{\mu\,R^{\,3}}{a^{\,4}... ...e}{(1-e^{\,2})^{\,5/2}}\left(\frac{5}{4}\,\sin^2 I-1\right)\sin I\,\sin\omega.$$
   Hence, deduce that the $J_3$ term causes the eccentricity and inclination of the satellite orbit to evolve in time as

   $$\frac{de}{d(n\,t)} =-\frac{3\,J_3}{8}\left(\frac{R}{a}\right)^3 \frac{(5\,\cos^2 I-1)}{(1-e^{\,2})^{\,2}}\,\sin I\,\cos\omega,$$

   $$\frac{dI}{d(n\,t)} = \frac{3\,J_3}{8}\left(\frac{R}{a}\right)^3 \frac{e\,(5\,\cos^2 I-1)}{(1-e^{\,2})^{\,3}}\,\cos I\,\cos\omega,$$

   respectively. Given that the (much larger) $J_2$ term causes the argument of the perigee to precess at the approximately constant (assuming that the variations in $e$ and $I$ are small) rate
   $$\frac{d\omega}{d(n\,t)} = \frac{3\,J_2}{4}\left(\frac{R}{a}\right)^2\frac{(5\,\cos^2 I-1)}{(1-e^{\,2})^{\,2}},$$
   deduce that the variations in the orbital eccentricity and inclination induced by the $J_3$ term can be written

   $$e \simeq e_0- \frac{J_3}{2\,J_2}\,\frac{R}{a}\,\sin I_0\,\sin\omega,$$

   $$I \simeq I_0+\frac{J_3}{2\,J_2}\,\frac{R}{a}\,\frac{e_0}{1-e_0^{\,2}}\,\cos I_0\,\sin\omega,$$

   respectively, where $e_0$ and $I_0$ are constants. (Modified from Murray and Dermott 1999.)

4. Demonstrate that, when averaged over an orbital period, the kinetic and potential energies of an object of mass $m$ executing a Keplerian orbit of major radius $a$ about an object of mass $M$ (in a frame of reference in which the latter object is stationary) are
   $$\langle K\rangle = \frac{\mu}{2\,a},$$
   and
   $$\langle U\rangle = -\frac{\mu}{a},$$
   respectively, where $\mu= G\,(M+m)$.

5. Using the notation of Section 10.6, show that when an artificial satellite interacts with the Earth's upper atmosphere, its orbit-averaged perigee and apogee distances evolve in time as

   $$\left\langle\frac{\skew{0}\dot{r_p}}{r_p}\right\rangle = -\frac{C_D\,A\,a\,\rho(a)}{m}\,n\oint {\rm e}^{\,\alpha\,\cos E}\,\frac{(1+e\,\cos E)^{1/2}}{(1-e\, \cos E)^{1/2}}\,(1-\cos E)\,\frac{dE}{2\pi},$$

   $$\left\langle\frac{\skew{0}\dot{r_a}}{r_a}\right\rangle = -\frac{C_D\,A\,a\,\rho(a)}{m}\,n\oint {\rm e}^{\,\alpha\,\cos E}\,\frac{(1+e\,\cos E)^{1/2}}{(1-e\, \cos E)^{1/2}}\,(1+\cos E)\,\frac{dE}{2\pi},$$

   respectively. Demonstrate that in the limit $\alpha\gg 1$, in which the difference between the apogee and perigee distances is much greater than the scale height of the atmosphere, the previous expressions reduce to

   $$\left\langle\frac{\skew{0}\dot{r_a}}{r_a}\right\rangle = -\frac{C_D\,A\,a\,\rho(r_p)}{m}\,n\left(\frac{r_a}{r_p}\right)^{1/2}\left(\frac{2}{\pi\,\alpha}\right)^{1/2},$$

   $$\left\langle\frac{\skew{0}\dot{r_p}}{r_p}\right\rangle =\frac{1}{4\,\alpha}\left\langle\frac{\skew{0}\dot{r_a}}{r_a}\right\rangle.$$

6. Consider an artificial satellite in a circular orbit of radius $a$ around the Earth. Assume that the altitude, $h=a-R$, of the orbit is much less that the terrestrial radius, $R$. Using the notation of Section 10.6, show that the time, $\tau$, for the orbit to decay from an initial altitude $h$ to zero altitude is
   $$\frac{\tau}{T} = \frac{m}{C_D\,2\pi\,R\,A\,\rho_0}\,\frac{H}{R}\left[\exp\left(\frac{h}{H}\right)-1\right],$$
   where $T =2\pi\,(R^{\,3}/\mu)^{1/2}$.

7. Consider a large dust grain orbiting the Sun. Making use of the notation of Section 10.7, demonstrate that the time evolution of the grain's orbital major radius, $a$, and eccentricity, $e$, under the action of solar radiation pressure is such that
   $$a\,e^{-4/5}\,(1-e^{\,2}) = C,$$
   where $C$ is a constant. Hence, deduce that the time required for a dust grain in an orbit of initial major radius $a$ and initial eccentricity $e$ to spiral into the Sun is
   $$\tau = \frac{8\pi\,m\,c^{\,2}\,a^{\,2}}{5\,L\,A}\,e^{-8/5}\,(1-e^{\,2})^{\,2}\int_0^e \frac{y^{\,3/5}}{(1-y^{\,2})^{3/2}}\,dy.$$