Comments and hints:

3.14: Use the orbit integral approach to solve this problem. Determine r explicitly as a function of using the initial condition that r is at the inner turning point and = 0 at t = 0. The ``precession of the ellipse'' means that the orbit is approximately an ellipse, but doesn't quite close: the angle at which r returns to its minimum value changes with each radial oscillation by some amount , which is to be determined.

The actual situation for the precession of the orbit of Mercury is complicated. The observed precession is quite large, 5601 arc seconds (5601") per century. Most of the precession is associated with the revolution of the earth-based frame of observation (5025") and orbital perturbations caused by the other planets (532"). The remaining 43" quoted in the problem is the "anomalous" precession left after these purely Newtonian effects are removed. This precession was explained by Einstein on the basis of general relativity, and is one of the classic tests of the general theory. It is large for Mercury because of Mercury's closeness to the Sun, and can be measured fairly well because of the large eccentricity of Mercury's orbit.

3.23a Be careful deriving the equations of motion! The crossed E and B fields of the magnetic monople and electric charge lead to an E×B flow of momentum around the symmetry axis of the system, hence to an angular momentum directed along the axis. The system will therefore act like a gyroscope, a fact apparently first noted by J.J. Thomson.

The angular momentum has the magnitude ge/c in Gaussian units. Were we to quantize this angular momentum using the Bohr quantization condition, we would find that , n = 0, 1, 2, ..., i.e., the possible charges e are integer multiples of a basic unit if there is even a single monopole in the universe. Alternatively, given e, the condition determines the very large monopole charge. A quantum mechanical argument first given by Dirac shows that n should properly be replaced by ½n, the "Dirac quantization condition".

3.26: This problem can be solved using either the orbit integral approach or by solving the equation of motion in the variable u = 1/r directly. See Goldstein 3-50. Hint: In either approach you need to determine the radial turning point r₀. It simplifies the solutions to take = 0 at that point.

Hint: To focus your discussion of the astrophysical application of this result, determine how the potential V scales with the mass and size of the system, hence, how v² scales with those quantities. You can think of the mass distribution as being continuous for large systems.

Another approach to detecting dark matter in spiral galaxies uses the "rotation curves" which relate the observed orbital velocities of disk stars moving around the galactic center to their radial distances from the center. The observed rotation curves are said to be "flat": the velocities do not fall off as rapidly with distance as would be expected from the observed (that is, luminous) mass in the galaxy. To see how this works, compare the velocity distribution for a star moving around a point "galaxy" to that obtained when the star is embedded in a uniform mass distribution.

Hint: Think about which variable to take as independent. The correct choice immediately gives you a constant of the motion, and allows you to obtain an ordinary differential equation in which you can separate the variables. The integral follows. How can you determine the constant of integration?

Send comments or questions to: ldurand@hep.wisc.edu

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© 1997, 1998, Loyal Durand