Show that the validity of Kepler's first two laws for any

Verify that the positions of two particles can be written in terms of the CM and relative positions as r1 = R m2r/M and r2 = R m ir/M. Hence confirm that the total KE of the two particles can be expressed as T = 4 MR2 + 4,u,i2, where denotes the reduced mass = m 1m2/ M.

Although the main topic of this chapter is the motion of two particles subject to no external forces, many of the ideas [for example, the splitting of the Lagrangian into two independent pieces = Zan as in Equation (8.13)] extend easily to more general situations. To illustrate this, consider the following: Two masses m1 and m2 move in a uniform gravitational field g and interact via a potential energy U(r). (a) Show that the Lagrangian can be decomposed as in (8.13). (b) Write down Lagrange's equations for the three CM coordinates X, Y, Z and describe the motion of the CM. Write down the three Lagrange equations for the relative coordinates and show clearly that the motion of r is the same as that of a single particle of mass equal to the reduced mass it, with position r and potential energy U(r).

Two particles of masses m 1 and m2 are joined by a massless spring of natural length L and force constant k. Initially, m2 is resting on a table and I am holding m1 vertically above m2 at a height L. At time t = 0, I project m 1 vertically upward with initial velocity vo. Find the positions of the two masses at any subsequent time t (before either mass returns to the table) and describe the motion. [Hints: See Problem 8.2. Assume that vo is small enough that the two masses never collide.]

Using the Lagrangian (8.13) write down the three Lagrange equations for the relative coordinates x, y, z and show clearly that the motion of the relative position r is the same as that of a single particle with position r, potential energy U(r), and mass equal to the reduced mass ,u,.

The momentum p conjugate to the relative position r is defined with components px = and so on. Prove that p = pti. Prove also that in the CM frame, p is the same as Pi the momentum of particle 1 (and also p2).

Show that in the CM frame, the angular momentum Li of particle 1 is related to the total angular momentum L by P1 = (m2/ M)L and likewise 2 = (m 1/M)L. Since L is conserved, this shows that the same is true of L 1 and 2 separately in the CM frame.

a) Using elementary Newtonian mechanics find the period of a mass m1 in a circular orbit of radius r around a fixed mass m2. (b) Using the separation into CM and relative motions, find the corresponding period for the case that m2 is not fixed and the masses circle each other a constant distance r apart. Discuss the limit of this result if m2 oo. (c) What would be the orbital period if the earth were replaced by a star of mass equal to the solar mass, in a circular orbit, with the distance between the sun and star equal to the present earthsun distance? (The mass of the sun is more than 300,000 times that of the earth.)

Two masses m1 and m2 move in a plane and interact by a potential energy U(r) = ikr2. Write down their Lagrangian in terms of the CM and relative positions R and r, and find the equations of motion for the coordinates X, Y and x, y. Describe the motion and find the frequency of the relative motion.

Consider two particles of equal masses, m1 = m2, attached to each other by a light straight spring (force constant k, natural length L) and free to slide over a frictionless horizontal table. (a) Write down the Lagrangian in terms of the coordinates r1 and r2, and rewrite it in terms of the CM and relative positions, R and r, using polar coordinates (r, 0) for r. (b) Write down and solve the Lagrange equations for the CM coordinates X, Y. (c) Write down the Lagrange equations for r and 0. Solve these for the two special cases that r remains constant and that remains constant. Describe the corresponding motions. In particular, show that the frequency of oscillations in the second case is co = '2k/ml

Two particles of equal masses m1 = m2 move on a frictionless horizontal surface in the vicinity of a fixed force center, with potential energies U1 = 1-kr? and U2 = z k r 22 . In addition, they interact with each other via a potential energy U12 = i akr2, where r is the distance between them and a and k are positive constants. (a) Find the Lagrangian in terms of the CM position R and the relative position r = ri r2. (b) Write down and solve the Lagrange equations for the CM and relative coordinates X, Y and x, y. Describe the motion.

Consider two particles interacting by a Hooke's law potential energy, U = Ikr2, where r is their relative position r = r1 r2, and subject to no external forces. Show that r(t) describes an ellipse. Hence show that both particles move on similar ellipses around their common CM. [This is surprisingly awkward. Perhaps the simplest procedure is to choose the xy plane as the plane of the orbit and then solve the equation of motion (8.15) for x and y. Your solution will have the form x = A cos cot B sin cot, with a similar expression for y. If you solve these for sin cot and cos cot and remember that sine cos2 = 1, you can put the orbital equation in the form axe 2bxy cy2 = k where k is a positive constant. Now invoke the standard result that if a and c are positive and ac > b2, this equation defines an ellipse.]

a) By examining the effective potential energy (8.32) find the radius at which a planet (or comet) with angular momentum can orbit the sun in a circular orbit with fixed radius. [Look at dUeff I dr 1 (b) Show that this circular orbit is stable, in the sense that a small radial nudge will cause only small radial oscillations. [Look at d2 Ueff I dr2 1 Show that the period of these oscillations is equal to the planet's orbital period.

Two particles whose reduced mass is interact via a potential energy U = Zkr2, where r is the distance between them. (a) Make a sketch showing U (r), the centrifugal potential energy Ucf(r), and the effective potential energy Ueff(r). (Treat the angular momentum as a known, fixed constant.) (b) Find the "equilibrium" separation ro, the distance at which the two particles can circle each other with constant r. [Hint: This requires that dUeif I dr be zero.] (c) By making a Taylor expansion of Ueff(r) about the equilibrium point ro and neglecting all terms in (r ro)3 and higher, find the frequency of small oscillations about the circular orbit if the particles are disturbed a little from the separation ro

Consider a particle of reduced mass orbiting in a central force with U = krn where kn > 0. (a) Explain what the condition kn > 0 tells us about the force. Sketch the effective potential energy Ueff for the cases that n = 2, 1, and 3. (b) Find the radius at which the particle (with given angular momentum .e) can orbit at a fixed radius. For what values of n is this circular orbit stable? Do your sketches confirm this conclusion? (c) For the stable case, show that the period of small oscillations about the circular orbit is rose =Toth/,N/n 2. Argue that if ,N/n + 2 is a rational number, these orbits are closed. Sketch them for the cases that n = 2, 1, and 7.

In deriving Kepler's third law (8.55) we made an approximation based on the fact that the sun's mass Ms is much greater than that of the planet m. Show that the law should actually read 1.2 = [472/G 4s (m m)] a3, and hence that the "constant" of proportionality is actually a little different for different planets. Given that the mass of the heaviest planet (Jupiter) is about 2 x 1027 kg, while Ms is about 2 x 1030 kg (and some planets have masses several orders of magnitude less than Jupiter), by what percent would you expect the "constant" in Kepler's third law to vary among the planets?

We have proved in (8.49) that any Kepler orbit can be written in the form 'JO) = c/(1 + E cos 0), where c > 0 and E > 0. For the case that 0 < E < 1, rewrite this equation in rectangular coordinates (x, y) and prove that the equation can be cast in the form (8.51), which is the equation of an ellipse. Verify the values of the constants given in (8.52).

If you did Problem 4.41 you met the virial theorem for a circular orbit of a particle in a central force with U = krn. Here is a more general form of the theorem that applies to any periodic orbit of a particle. (a) Find the time derivative of the quantity G = r p and, by integrating from time 0 to t, show that G (t) G(0) =2(T) + (F r) t where F is the net force on the particle and ( f) denotes the average over time of any quantity f . (b) Explain why, if the particle's orbit is periodic and if we make t sufficiently large, we can make the left-hand side of this equation as small as we please. That is, the left side approaches zero as t oo. (c) Use this result to prove that if F comes from the potential energy U = krn, then (T) = n(U)12, if now (f) denotes the time average over a very long time.

An earth satellite is observed at perigee to be 250 km above the earth's surface and traveling at about 8500 m/s. Find the eccentricity of its orbit and its height above the earth at apogee. [Hint: The earth's radius is Re ',-z= 6.4 x 106 m. You will also need to know GMe, but you can find this if you remember that GMe/RZ = g.]

The height of a satellite at perigee is 300 km above the earth's surface and it is 3000 km at apogee. Find the orbit's eccentricity. If we take the orbit to define the xy plane and the major axis in the x direction with the earth at the origin, what is the satellite's height when it crosses the y axis? [See the hint for Problem 8.18.]

Consider a comet which passes through its aphelion at a distance I-max from the sun. Imagine that, keeping riiiax fixed, we somehow make the angular momentum smaller and smaller, though not actually zero; that is, we let 0. Use equations (8.48) and (8.50) to show that in this limit the eccentricity E of the elliptical orbit approaches 1 and that the distance of closest approach rmir, approaches zero. Describe the orbit with rmax fixed but f very small. What is the semimajor axis a?

(a) If you haven't already done so, do Problem 8.20. (b) Use Kepler's third law (8.55) to find the period of this orbit in terms of rmax (and G and Ms). (c) Now consider the extreme case that the comet is released from rest at a distance rmax from the sun. (In this case is actually zero.) Use the technique described in connection with (4.58) to find how long the comet takes to reach the sun. (Take the sun's radius to be zero.) (d) Assuming the comet can somehow pass freely through the sun, describe its overall motion and find its period. (e) Compare your answers in parts (b) and (d).

A particle of mass m moves with angular momentum about a fixed force center with F(r) = k 1 r3 where k can be positive or negative. (a) Sketch the effective potential energy Ueff for various values of k and describe the various possible kinds of orbit. (b) Write down and solve the transformed radial equation (8.41), and use your solutions to confirm your predictions in part (a).

A particle of mass m moves with angular momentum in the field of a fixed force center with F(r) = --k +A 1'2 r3 where k and A are positive. (a) Write down the transformed radial equation (8.41) and prove that the orbit has the form r (0) = 1 + E COS(80) where c, /3, and E are positive constants. (b) Find c and $ in terms of the given parameters, and describe the orbit for the case that 0 < E < 1. (c) For what values of fi is the orbit closed? What happens to your results as A 0?

Consider the particle of Problem 8.23, but suppose that the constant ? is a negative. Write down the transformed radial equation (8.41) and describe the orbits of low angular momentum (specifically,

[Computer] Consider a particle with mass m and angular momentum in the field of a central force F = k/r5/2. To simplify your equations, choose units for which m = = k = 1. (a) Find the value ro of r at which Ueff is minimum and make a plot of Ueff(r) for 0 < r < 5ro. (Choose your scale so that your plot shows the interesting part of the curve.) (b) Assuming now that the particle has energy E = 0.1, find an accurate value of r,,n, the particle's distance of closest approach to the force center. (This will require the use of a computer program to solve the relevant equation numerically.) (c) Assuming that the particle is at r = rmir, when 0 = 0, use a computer program (such as "NDSolve" in Mathematica) to solve the transformed radial equation (8.41) and find the orbit in the form r = r(0) for 0 < < 77r. Plot the orbit. Does it appear to be closed?

Show that the validity of Kepler's first two laws for any body orbiting the sun implies that the force (assumed conservative) of the sun on any body is central and proportional to 1/r2.

At time to a comet is observed at radius ro traveling with speed vo at an acute angle a to the line from the comet to the sun. Put the sun at the origin 0, with the comet on the x axis (at to) and its orbit in the xy plane, and then show how you could calculate the parameters of the orbital equation in the form r = c/[1 + E cos(0 8)]. Do so for the case that ro = 1.0 x 1011 m, vo = 45 km/s, and a = 50 degrees. [The sun's mass is about 2.0 x 1030 kg, and G = 6.7 x 10-11 Nm2/s2.]

For a given earth satellite with given angular momentum f, show that the distance of closest approach rmin on a parabolic orbit is half the radius of the circular orbit.

What would become of the earth's orbit (which you may consider to be a circle) if half of the sun's mass were suddenly to disappear? Would the earth remain bound to the sun? [Hints: Consider what happens to the earth's KE and PE at the moment of the great disappearance. The virial theorem for the circular orbit (Problem 4.41) helps with this one.] Treat the sun (or what remains of it) as fixed.

The general Kepler orbit is given in polar coordinates by (8.49). Rewrite this in Cartesian coordinates for the cases that E = 1 and E > 1. Show that if E = 1, you get the parabola (8.60), and if E > 1, the hyperbola (8.61). For the latter, identify the constants a, ,8, and 6 in terms of c and E.

Consider the motion of two particles subject to a repulsive inverse-square force (for example, two positive charges). Show that this system has no states with E < 0 (as measured in the CM frame), and that in all states with E > 0, the relative motion follows a hyperbola. Sketch a typical orbit. [Hint: You can follow closely the analysis of Sections 8.6 and 8.7 except that you must reverse the force; probably the simplest way to do this is to change the sign of y in (8.44) and all subsequent equations (so that F(r) = Ir2) and then keep y itself positive. Assume 0.]

Prove that for circular orbits around a given gravitational force center (such as the sun) the speed of the orbiting body is inversely proportional to the square root of the orbital radius.

Figure 8.13 shows a space vehicle boosting from a circular orbit 1 at P to a transfer orbit 2 and then from the transfer orbit at P' to the final circular orbit 3. Example 8.6 derived in detail the thrust factor required for the boost at P. Show similarly that the thrust factor required at P' is ),,,' = ./(Ri + R3)/2Ri. [Your argument should parallel closely that of Example 8.6, but you must account for the fact that P' is the apogee (not perigee) of the transfer orbit. For example, the plus signs in (8.67) should be minus signs here.]

Suppose that we decide to send a spacecraft to Neptune, using the simple transfer described in Example 8.6 (page 318). The craft starts in a circular orbit close to the earth (radius 1 AU or astronomical unit) and is to end up in a circular orbit near Neptune (radius about 30 AU). Use Kepler's third law to show that the transfer will take about 31 years. (In practice we can do a lot better than this by arranging that the craft gets a gravitational boost as it passes Jupiter.)

A spacecraft in a circular orbit wishes to transfer to another circular orbit of quarter the radius by means of a tangential thrust to move into an elliptical orbit and a second tangential thrust at the opposite end of the ellipse to move into the desired circular orbit. (The picture looks like Figure 8.13 but run backwards.) Find the thrust factors required and show that the speed in the final orbit is two times greater than the initial speed.