Exercises

1. Demonstrate that if a particle moves in a central force field with zero angular momentum, so that ${\bf h}\equiv {\bf r}\times\dot{\bf r}={\bf0}$, then the particle's trajectory lies on a fixed straight line that passes through the origin. [Hint: Show that $d/dt({\bf r}/r)={\bf0}$.]

2. Demonstrate that $h=r^{\,2}\,\skew{5}\dot{\theta}$ is the magnitude of the angular momentum (per unit mass) vector ${\bf h} = {\bf r}\times \dot{\bf r}$. Here, $r$ and $\theta$ are plane polar coordinates.

3. Consider a planet in a Keplerian elliptical orbit about the Sun. Let ${\bf r}$ be the planet's position vector, relative to the Sun, and let ${\bf h} = {\bf r}\times \dot{\bf r}$ be its angular momentum per unit mass. Demonstrate that the so-called eccentricity vector,
   $${\bf e} = \frac{\dot{\bf r}\times {\bf h}}{G\,M} - \frac{\bf r}{r},$$
   can be written
   $${\bf e} = e\,\cos\theta\,\,{\bf e}_r-e\,\sin\theta\,\,{\bf e}_\theta,$$
   where $M$ is the solar mass, $e$ the orbital eccentricity, and $r$, $\theta$ are plane polar coordinates in the orbital plane (with the perihelion corresponding to $\theta=0$). Hence, show that ${\bf e}$ is a constant vector, of length $e$, that is directed from the Sun toward the perihelion point.

4. Given the Sun's mean apparent radius seen from the Earth ($16'$), the Earth's mean apparent radius seen from the Moon ($57'$), and the mean number of lunar revolutions in a year ($13.4$), show that the ratio of the Sun's mean density to that of the Earth is $0.252$. (From Lamb 1923.)

5. Prove that the orbital period of a satellite close to the surface of a spherical planet depends on the mean density of the planet, but not on its size. Show that if the mean density is that of water then the period is 3 h. 18 m. (From Lamb 1923.)

6. Jupiter's satellite Ganymede has an orbital period of 7 d. 3 h. 43 m. and a mean orbital radius that is $15.3$ times the mean radius of the planet. The Moon has an orbital period of 27 d. 7 h. 43 m. and a mean orbital radius that is $60.3$ times the Earth's mean radius. Show that the ratio of Jupiter's mean density to that of the Earth is $0.238$. (From Lamb 1923.)

7. Halley's comet has an orbital eccentricity of $0.967$ and a perihelion distance of 55,000,000 miles. Find the orbital period and the comet's speed at perihelion and aphelion.

8. Show that the velocity at any point on a Keplerian elliptical orbit can be resolved into two constant components: a velocity $n\,a/\sqrt{1-e^{\,2}}$ at right angles to the radius vector, and a velocity $n\,a\,e/\sqrt{1-e^{\,2}}$ at right angles to the major axis. Here, $n$ is the mean orbital angular velocity, $a$ the major radius, and $e$ the eccentricity. (From Lamb 1923.)

9. The latus rectum of a conic section is a chord that passes through a focus; it is perpendicular to the major axis (or the symmetry axis, in the case of a parabola or a hyperbola). Show that, for a body in a Keplerian orbit around the Sun, the maximum value of the radial speed occurs at the points where the latus rectum (associated with the non-empty focus, in the case of an ellipse) intersects the orbit, and that this maximum value is $e\,h/[r_p\,(1+e)]$. Here, $h$ is the angular momentum per unit mass, $e$ the orbital eccentricity, and $r_p$ the perihelion distance.

10. A comet is observed a distance $R$ astronomical units from the Sun, traveling at a speed that is $V$ times the Earth's mean orbital speed. Show that the orbit of the comet is hyperbolic, parabolic, or elliptical, depending on whether the quantity $V^{\,2}\,R$ is greater than, equal to, or less than 2, respectively. (Modified from Fowles and Cassiday 2005.)

11. Consider a planet in an elliptical orbit of major radius $a$ and eccentricity $e$ about the Sun. Suppose that the eccentricity of the orbit is small (i.e., $0<e\ll 1$), as is indeed the case for all of the planets. Demonstrate that, to first order in $e$, the orbit can be approximated as a circle whose center is shifted a distance $e\,a$ from the Sun, and that the planet's angular motion appears uniform when viewed from a point (called the equant) that is shifted a distance $2\,e\,a$ from the Sun, in the same direction as the center of the circle. [This theorem is the basis of Ptolemy's model of planetary motion (Evans 1998).]

12. How long (in days) does it take the Sun-Earth radius vector to rotate through $90^\circ$, starting at the perihelion point? How long does it take starting at the aphelion point? The period and eccentricity of the Earth's orbit are $T=365.24$ days, and $e=0.01673$, respectively.

13. If $\theta$ is the Sun's ecliptic longitude, measured from the perigee (the point of closest approach to the Earth), show that the Sun's apparent diameter is given by
    $$D\simeq D_1\,\cos^2(\theta/2) + D_2\,\sin^2(\theta/2),$$
    where $D_1$ and $D_2$ are the greatest and least values of $D$. (From Lamb 1923.)

14. Show that the time-averaged apparent diameter of the Sun, as seen from a planet describing a low-eccentricity elliptical orbit, is approximately equal to the apparent diameter when the planet's distance from the Sun equals the major radius of the orbit. (From Lamb 1923.)

15. Consider an asteroid orbiting the Sun. Demonstrate that, at fixed orbital energy, the orbit that maximizes the orbital angular momentum is circular.

16. Demonstrate that for a Keplerian orbit

    $$\cos E = \frac{\cos\theta+e}{1+e\,\cos\theta},$$

    $$\sin E = \frac{(1-e^{\,2})^{1/2}\,\sin\theta}{1+e\,\cos\theta},$$

    $$\cos \theta = \frac{\cos E-e}{1-e\,\cos E},$$

    $$\sin \theta = \frac{(1-e^{\,2})^{1/2}\,\sin E}{1- e\,\cos E},$$

    where $E$, $\theta$, and $e$ are the elliptic anomaly, the true anomaly, and the eccentricity, respectively.

17. Derive Equations (4.102)–(4.104).

18. Derive Equations (4.105)–(4.107).

19. A parabolic Keplerian orbit is specified by Equation (4.102), which can be written
    $$P + \frac{P^{\,3}}{3} = {\cal M},$$
    where $P$ is the parabolic anomaly, and ${\cal M} = (G\,M/2\,r_p^{\,3})^{1/2}\,(t-\tau)$ is termed the parabolic mean anomaly. Here, $M$ is the solar mass, $r_p$ the perihelion distance, and $\tau$ the time of perihelion passage. Demonstrate that the preceding equation has the analytic solution
    $$P = \frac{1}{2}\,Q^{\,1/3} - 2\,Q^{-1/3},$$
    where
    $$Q =12\,{\cal M}+4\sqrt{4+9\,{\cal M}^{\,2}}.$$

20. Consider a comet in an elliptical orbit about the Sun. Let $x$ and $y$ be Cartesian coordinates in the orbital plane, such that $x=y=0$ corresponds to the Sun, and the $x$-axis is parallel to the orbital major axis. Demonstrate that

    $$x = a\,(\cos E-e),$$

    $$y =a\,(1-e^{\,2})^{1/2}\,\sin E,$$

    where $a$ is the orbital major radius, $e$ the eccentricity, and $E$ the eccentric anomaly.

21. Consider a comet in a parabolic orbit about the Sun. Let $x$ and $y$ be Cartesian coordinates in the orbital plane, such that $x=y=0$ corresponds to the Sun, and the $x$-axis is parallel to the orbital symmetry axis. Demonstrate that

    $$x = r_p\,(1-P^{\,2}),$$

    $$y =2\,r_p\,P,$$

    where $r_p$ is the perihelion distance, and $P$ the parabolic anomaly.

22. Consider a comet in an hyperbolic orbit about the Sun. Let $x$ and $y$ be Cartesian coordinates in the orbital plane, such that $x=y=0$ corresponds to the Sun, and the $x$-axis is parallel to the orbital symmetry axis. Demonstrate that

    $$x = a\,(e-\cosh H),$$

    $$y =a\,(e^{\,2}-1)^{1/2}\,\sinh H,$$

    where $a$ is the orbital major radius, $e$ the eccentricity, and $H$ the hyperbolic anomaly.

23. Consider a comet in an elliptical orbit about the Sun. If $r_1$ and $r_2$ are the radial distances from the Sun of two neighboring points, $C_1$ and $C_2$, on the orbit, and if $s$ is the length of the straight line joining these two points, prove that the time, $t$, required for the comet to move from $C_1$ to $C_2$ is
    $${\mit\Delta} t =\frac{T}{2\pi}\left[ (\eta -\sin\eta) -(\xi - \sin\xi)\right],$$
    where

    $$\sin(\eta/2) =\frac{1}{2}\left(\frac{r_1+r_2+s}{a}\right)^{1/2},$$

    $$\sin(\xi/2) = \frac{1}{2}\left(\frac{r_1+r_2-s}{a}\right)^{1/2}.$$

    Here, $T$ and $a$ are the period and the major radius of the orbit, respectively.

24. Consider a comet in a parabolic orbit about the Sun. If $r_1$ and $r_2$ are the radial distances from the Sun of two neighboring points, $C_1$ and $C_2$, on the orbit, and if $s$ is the length of the straight line joining these two points, prove that the time required for the comet to move from $C_1$ to $C_2$ is
    $${\mit\Delta} t = \frac{T}{12\pi}\left[\left(\frac{r_1+r_2+s}{a}\right)^{3/2}-\left(\frac{r_1+r_2-s}{a}\right)^{3/2}\right],$$
    where $T$ and $a$ are the period and the major radius of the Earth's orbit, respectively.

25. Consider a comet in a hyperbolic orbit about the Sun. If $r_1$ and $r_2$ are the radial distances from the Sun of two neighboring points, $C_1$ and $C_2$, on the orbit, and if $s$ is the length of the straight line joining these two points, prove that the time, $t$, required for the comet to move from $C_1$ to $C_2$ is
    $${\mit\Delta} t =\frac{T}{2\pi}\left[ (\sinh\eta-\eta) -(\sinh\xi-\xi)\right],$$
    where

    $$\sinh(\eta/2) =\frac{1}{2}\left(\frac{r_1+r_2+s}{a}\right)^{1/2},$$

    $$\sinh(\xi/2) = \frac{1}{2}\left(\frac{r_1+r_2-s}{a}\right)^{1/2}.$$

    Here, $a$ is major radius of the orbit, and $T$ is the period of an elliptical orbit with the same major radius. (From Smart 1951.)

26. A comet is in a parabolic orbit that lies in the plane of the Earth's orbit. Regarding the Earth's orbit as a circle of radius $a$, show that the points at which the comet intersects the Earth's orbit are given by
    $$\cos\theta = -1 + \frac{2\,r_p}{a},$$
    where $r_p$ is the perihelion distance of the comet, defined at $\theta=0$. Demonstrate that the time interval that the comet remains inside the Earth's orbit is the faction
    $$\frac{2^{1/2}}{3\pi}\left(\frac{2\,r_p}{a}+1\right)\left(1-\frac{r_p}{a}\right)^{1/2}$$
    of a year, and that the maximum value of this time interval is $2/3\pi$ year, or about 11 weeks.

27. The orbit of a comet around the Sun is a hyperbola of eccentricity $e$, lying in the ecliptic plane, whose least distance from the Sun is $1/n$ times the radius of the Earth's orbit (which is approximated as a circle). Prove that the time that the comet remains within the Earth's orbit is $(T/\pi)\,(e\,\sinh \phi-\phi)$, where $e\,\cosh\phi = 1-n\,(1-e)$, and $T$ is the periodic time of a planet describing an elliptic orbit whose major radius is equal to that of the hyperbolic orbit. (From Smart 1951.)

28. Consider a comet in a hyperbolic orbit focused on the Sun. The impact parameter, $b$ is defined as the the distance of closest approach in the absence of any gravitational attraction between the comet and the Sun. Demonstrate that $b=h/(2\,{\cal E})^{1/2}$, where $h$ is the comet's angular momentum per unit mass, and ${\cal E}$ its energy per unit mass. Show that the relationship between the impact parameter, $b$, and the true distance of closest approach, $r_p$, is
    $$r_p = \frac{2\,b}{\alpha+\sqrt{\alpha^{\,2}+4}},$$
    where $\alpha=G\,M/({\cal E}\,b)$, and $M$ is the solar mass. Hence, deduce that if the comet is to avoid hitting the Sun then
    $$\alpha< \frac{b}{R} -\frac{R}{b}$$
    (assuming that $b>R$), where $R$ is the solar radius.

29. Spectroscopic analysis has revealed that Spica is a double star whose components revolve around one another with a period of 4.1 days, the greatest relative orbital velocity being 36 miles per second. Show that the mean distance between the components of the star is $2.03\times 10^6$ miles, and that the total mass of the system is $0.083$ that of the Sun. The mean distance of the Earth from the Sun is $92.75$ million miles. (From Lamb 1923.)

30. Consider the binary star system discussed in Section 4.16. Show that

    $${\bf h} = {\bf r}\times \dot{\bf r},$$

    $${\cal E} = \frac{1}{2}\,v^{\,2}-\frac{G\,M}{r},$$

    where $M=m_1+m_2$ and ${\bf v}=\dot{\bf r}$, are integrals of the the reduced equation of motion, (4.110) (in other words, ${\bf h}$ and ${\cal E}$ are constants of the motion). Demonstrate that, in the center of mass frame, the net angular momentum and energy of the system are

    $${\bf L} = \mu\,{\bf h},$$

    $$E = \mu\,{\cal E},$$

    respectively, where $\mu=m_1\,m_2/M$ is the reduced mass.