Chapter 7

A particle \(P\) of mass \(m\) moves under the repulsive inverse cube field \(\boldsymbol{F}=\left(m \gamma / r^{3}\right) \widehat{\boldsymbol{r}}\). Initially \(P\) is at a great distance from \(O\) and is moving with speed \(V\) towards \(O\) along a straight line whose perpendicular distance from \(O\) is \(p\). Find the equation satisfied by the apsidal distances. What is the distance of closest approach of \(P\) to \(O\) ?

A particle \(P\) of mass \(m\) moves under the attractive inverse square field \(\boldsymbol{F}=-\left(m \gamma / r^{2}\right) \widehat{\boldsymbol{r}}\). Initially \(P\) is at a point \(C\), a distance \(c\) from \(O\), when it is projected with speed \((\gamma / c)^{1 / 2}\) in a direction making an acute angle \(\alpha\) with the line \(O C\). Find the apsidal distances in the resulting orbit. Given that the orbit is an ellipse with \(O\) at a focus, find the semi-major and semi-minor axes of this ellipse.

A particle of mass \(m\) moves under the attractive inverse square field \(\boldsymbol{F}=-\left(m \gamma / r^{2}\right) \widehat{\boldsymbol{r}}\) Show that the equation satisfied by the apsidal distances is $$ 2 E r^{2}+2 \gamma r-L^{2}=0 $$ where \(E\) and \(L\) are the specific total energy and angular momentum of the particle. When \(E<0\), the orbit is known to be an ellipse with \(O\) as a focus. By considering the sum and product of the roots of the above equation, establish the elliptic orbit formulae $$ L^{2}=\gamma b^{2} / a, \quad E=-\gamma / 2 a $$

A particle \(P\) of mass \(m\) moves under the simple harmonic field \(\boldsymbol{F}=-\left(m \Omega^{2} r\right) \widehat{\boldsymbol{r}}\), where \(\Omega\) is a positive constant. Obtain the radial motion equation and show that all orbits of \(P\) are bounded. Initially \(P\) is at a point \(C\), a distance \(c\) from \(O\), when it is projected with speed \(\Omega c\) in a direction making an acute angle \(\alpha\) with \(O C\). Find the equation satisfied by the apsidal distances. Given that the orbit of \(P\) is an ellipse with centre \(O\), find the semi-major and semiminor axes of this ellipse.

A particle \(P\) moves under the attractive inverse square field \(\boldsymbol{F}=-\left(m \gamma / r^{2}\right) \widehat{\boldsymbol{r}}\). Initially \(P\) is at the point \(C\), a distance \(c\) from \(O\), and is projected with speed \((3 \gamma / c)^{1 / 2}\) perpendicular to \(O C\). Find the polar equation of the path make a sketch of it. Deduce the angle between \(O C\) and the final direction of departure of \(P\).

A comet moves under the gravitational attraction of the Sun. Initially the comet is at a great distance from the Sun and is moving towards it with speed \(V\) along a straight line whose perpendicular distance from the Sun is \(p\). By using the path equation, find the angle through which the comet is deflected and the distance of closest approach.

A particle \(P\) of mass \(m\) moves under the attractive inverse cube field \(\boldsymbol{F}=-\left(m \gamma^{2} / r^{3}\right) \widehat{\boldsymbol{r}}\), where \(\gamma\) is a positive constant. Initially \(P\) is at a great distance from \(O\) and is projected towards \(O\) with speed \(V\) along a line whose perpendicular distance from \(O\) is \(p\). Obtain the path equation for \(P\). For the case in which $$ V=\frac{15 \gamma}{\sqrt{209} p} $$ find the polar equation of the path of \(P\) and make a sketch of it. Deduce the distance of closest approach to \(O\), and the final direction of departure.

A particle \(P\) of mass \(m\) moves under the central field \(\boldsymbol{F}=-\left(m \gamma^{2} / r^{5}\right) \widehat{\boldsymbol{r}}\), where \(\gamma\) is a positive constant. Initially \(P\) is at a great distance from \(O\) and is projected towards \(O\) with speed \(\sqrt{2} \gamma / p^{2}\) along a line whose perpendicular distance from \(O\) is \(p\). Show that the polar equation of the path of \(P\) is given by $$ r=\frac{p}{\sqrt{2}} \operatorname{coth}\left(\frac{\theta}{\sqrt{2}}\right) $$ Make a sketch of the path.

A particle of mass \(m\) moves under the central field $$ \boldsymbol{F}=-m \gamma^{2}\left(\frac{4}{r^{3}}+\frac{a^{2}}{r^{5}}\right) \widehat{\boldsymbol{r}} $$ where \(\gamma\) and \(a\) are positive constants. Initially the particle is at a distance \(a\) from the centre of force and is projected at right angles to the radius vector with speed \(3 \gamma / \sqrt{2} a\). Find the polar equation of the resulting path and make a sketch of it. Find the time taken for the particle to reach the centre of force.

A particle of mass \(m\) moves under the central field $$ \boldsymbol{F}=-m\left(\frac{\gamma e^{-\epsilon r / a}}{r^{2}}\right) \widehat{\boldsymbol{r}}, $$ where \(\gamma, a\) and \(\epsilon\) are positive constants. Find the apsidal angle for a nearly circular orbit of radius \(a\). When \(\epsilon\) is small, show that the perihelion of the orbit advances by approximately \(\pi \epsilon\) on each revolution.

A planet of mass \(m\) moves in the equatorial plane of a star that is a uniform oblate spheroid. The planet experiences a force field of the form $$ \boldsymbol{F}=-\frac{m \gamma}{r^{2}}\left(1+\frac{\epsilon a^{2}}{r^{2}}\right) \widehat{\boldsymbol{r}} $$ approximately, where \(\gamma, a\) and \(\epsilon\) are positive constants and \(\epsilon\) is small. If the planet moves in a nearly circular orbit of radius \(a\), find an approximation to the 'annual' advance of the perihelion. [It has been suggested that oblateness of the Sun might contribute significantly to the precession of the planets, thus undermining the success of general relativity. This point has yet to be resolved conclusively.]

Suppose the solar system is embedded in a dust cloud of uniform density \(\rho\). Find an approximation to the 'annual' advance of the perihelion of a planet moving in a nearly circular orbit of radius \(a\). (For convenience, let \(\rho=\epsilon M / a^{3}\), where \(M\) is the solar mass and \(\epsilon\) is small.)

A uniform flux of particles is incident upon a fixed hard sphere of radius \(a\). The particles that strike the sphere are reflected elastically. Find the differential scattering cross section.

A uniform flux of particles, each of mass \(m\) and speed \(V\), is incident upon a fixed scatterer that exerts the repulsive radial force \(\boldsymbol{F}=\left(m \gamma^{2} / r^{3}\right) \widehat{\boldsymbol{r}}\). Find the impact parameter \(p\) as a function of the scattering angle \(\theta\), and deduce the differential scattering cross section. Find the total back- scattering cross-section.

An Earth satellite has a speed of \(8.60 \mathrm{~km}\) per second at its perigee \(200 \mathrm{~km}\) above the Earth's surface. Find the apogee distance above the Earth, its speed at the apogee, and the period of its orbit.

A spacecraft is orbiting the Earth in a circular orbit of radius \(c\) when the motors are fired so as to multiply the speed of the spacecraft by a factor \(k(k>1)\), its direction of motion being unaffected. [You may neglect the time taken for this operation.] Find the range of \(k\) for which the spacecraft will escape from the Earth, and the eccentricity of the escape orbit.

A spacecraft travelling with speed \(V\) approaches a planet of mass \(M\) along a straight line whose perpendicular distance from the centre of the planet is \(p\). When the spacecraft is at a distance \(c\) from the planet, it fires its engines so as to multiply its current speed by a factor \(k(0

A body moving in an inverse square attractive field traverses an elliptical orbit with major axis \(2 a\). Show that the time average of the potential energy \(V=-\gamma / r\) is \(-\gamma / a\). [Transform the time integral to an integral with repect to the eccentric angle \(\psi\).] Deduce the time average of the kinetic energy in the same orbit.

A body moving in an inverse square attractive field traverses an elliptical orbit with eccentricity \(e\) and major axis \(2 a\). Show that the time average of the distance \(r\) of the body from the centre of force is \(a\left(1+\frac{1}{2} e^{2}\right)\). [Transform the time integral to an integral with respect to the eccentric angle \(\psi .]\)

A spacecraft is 'parked' in an elliptic orbit around the Earth. What is the most fuel efficient method of escaping from the Earth by using a single impulse?

A satellite of mass \(m\) moves under the attractive inverse square field \(-\left(m \gamma / r^{2}\right) \widehat{\boldsymbol{r}}\) and is also subject to the linear resistance force \(-m K \boldsymbol{v}\), where \(K\) is a positive constant. Show that the governing equations of motion can be reduced to the form $$ \ddot{r}+K \dot{r}+\frac{\gamma}{r^{2}}-\frac{L_{0}^{2} e^{-2 K t}}{r^{3}}=0, \quad r^{2} \dot{\theta}=L_{0} e^{-K t} $$ where \(L_{0}\) is a constant which will be assumed to be positive. Suppose now that the effect of resistance is slight and that the satellite is executing a 'circular' orbit of slowly changing radius. By neglecting the terms in \(\dot{r}\) and \(\ddot{r}\), find an approximate solution for the time variation of \(r\) and \(\theta\) in such an orbit. Deduce that small resistance causes the circular orbit to contract slowly, but that the satellite speeds up!

It is possible to 'see' the advance of the perihelion of Mercury predicted by general relativity by direct numerical solution. Take Einstein's path equation (see Problem 7.13) in the dimensionless form $$ \frac{d^{2} v}{d \theta^{2}}+v=\frac{1}{1-e^{2}}+\eta v^{2} $$ where \(v=a u\). Here \(a\) and \(e\) are the semi-major axis and eccentricity of the non-relativistic elliptic orbit and \(\eta=3 M G / a c^{2}\) is a small dimensionless parameter. For the orbit of Mercury, \(\eta=2.3 \times 10^{-7}\) approximately. Solve this equation numerically with the initial conditions \(r=a(1+e)\) and \(\dot{r}=0\) when \(\theta=0\); this makes \(\theta=0\) an aphelion of the orbit. To make the precession easy to see, use a fairly eccentric ellipse and take \(\eta\) to be about \(0.005\), which speeds up the precession by a factor of more than \(10^{4}\) !

Confirm the approximate solution for small resistance obtained in Problem \(7.26\) by numerical solution of the governing simultaneous ODEs. First write the governing equations in dimensionless form. Suppose that, in the absence of resistance, a circular orbit with \(r=a\) and \(\dot{\theta}=\Omega\) is possible; then \(\gamma=a^{3} \Omega\) and \(L_{0}=a^{2} \Omega\). On taking dimensionless variables \(\rho, \tau\) defined by \(\rho=r / a\) and \(\tau=\Omega t\), and taking \(L_{0}=a^{2} \Omega\), the governing equations become $$ \frac{d^{2} \rho}{d \tau^{2}}+\epsilon \frac{d \rho}{d \tau}+\frac{1}{\rho^{2}}-\frac{e^{-2 \epsilon \tau}}{\rho^{3}}=0, \quad \rho^{2} \frac{d \theta}{d \tau}=e^{-2 \epsilon \tau} $$ where \(\epsilon=K / \Omega\) is the dimensionless resistance parameter. Solve these equations with the initial conditions \(\rho=1, d \rho / d \tau=0\) and \(\theta=0\) when \(\tau=0 .\) Choose some small value for \(\epsilon\) and plot a polar graph of the path.