(a) Argue qualitatively from the sketch that the angular momentum vector of the moon's orbit should precess about the normal to the ecliptic due to the forces shown. (Neglect the effect of the earth's equatorial bulge.) Will the precession rate be the same at all times of the year? (b) Consider the radial forces (along the sun-earth line) on a test mass m near the earth (e.g., the moon) in the «o«-inertial (accelerating about the sun) frame of the earth. Specifically, show that the total (net) force F* due to the sum of (i) the outward (away from sun) centrifugal force arising from the earth's orbital motion and (ii) the inward attractive gravitational force due to the sun (mass M) is


where RE is the earth's distance from the sun (1 AU) and R = RE + e is the radial distance of the mass m from the sun. How does the force vary at different positions in the moon's orbit? (Hint: see sketch) What does this force have to do with the ocean tides? Why do we neglect the gravitational force due to the (assumed spherical) earth? (c) Demonstrate that, in general, the angular velocity of precession, i.e., the angular velocity of the node, is q = t/(L sin 0) where t and L are the magnitudes of the torque applied to, and angular momentum of, the moon with origin at the earth, respectively, and where 0 is the angle between the angular momentum L and the ecliptic normal. (Make an appropriate vector diagram in spherical coordinates.) (d) For the special case where the moon and the sun are both 90° from the nodes, what is the torque on the moon about the earth center, and what is the implied angular velocity of precession? Useful values are: 0 = 5.1°; m = 7.35 x 1022 kg, vmoon orbit = 2.7 x 10-6 rad s-1. [Ans. —80° yr-1; this is a maximum value. Averaging over the lunar orbit and over all sun directions yields approximately the actual value of 19.4°/yr]

4.5 Measures of time

Problem 4.51. Explain the following statement: "It is fortunate indeed that solar and sidereal clocks are not synchronized. If they were, we could never observe certain regions of the sky". Could this situation occur in another star-planet system? If it could, would space-borne telescopes orbiting the planet make more of the sky accessible for observations? Explain.

Problem 4.52. (a) Use (14) to find the precise (to 0.01 s) GMST on 1995 Dec. 7 at 0 h UT. It happens that Dec. 7 is the 341st day of 1995. (b) Repeat for your birthday at 0 h UT in 2005. To calculate the parameter T, watch out for leap years, or use the Julian-date table in the Astronomical Almanac. [Ans. (a) 5 h 01 m 11.0 s]

Problem 4.53. (a) What is the approximate sidereal time at the instant shown in Fig. 3.1 at Palomar Mountain and in Moscow? Palomar is at longitude 118° W and Moscow at 38° E. (b) What is the local sidereal time at my home on my birthday at 1800 h zone time (6 PM EST) on 1995 Dec. 7 to the nearest second (of sidereal time, not arcseconds)? I live in Belmont MA at longitude 71° 08' 17" W. In the preceding problem, or in the 1995 Astronomical Almanac, you will find that GMST at 0 h UT on Dec. 7 (i.e., at Greenwich at the beginning of the day) is 5h01m11.0s. You should correct this for (i) clock time noting that Boston time is 5 h earlier than Greenwich UT and that the sidereal day is only 86 164.1 s long, and (ii ) for longitude. (c) What is the sidereal time at your home at 6 PM on your birthday this year (as you read this)? [Ans. (b): —23 h]

Problem 4.54. How many days have elapsed between your birth and your 20th birthday? How many SI seconds (SI = TT)? Try using the Julian date table in the

Astronomical Almanac.

Problem 4.55. (Some knowledge of relativity would be helpful for this one, but it is not necessary.) Demonstrate that a terrestrial (on earth) clock should run more slowly than Barycentric Coordinate Time (TCB) by 49 s per century. Take into account separately the two effects mentioned in the text: (a) the earth clock is in the gravitational potential well of sun, and (b) the earth is moving fast (1.0 x 10-4c) relative to the solar system barycenter. Hints: (a) a photon leaving an atom in a gravitational potential well at radius r from a non-rotating mass M with frequency vr will arrive at infinite distance with the lesser frequency v such that

An observer detecting it at infinity concludes therefore that the originating atom (clock) is running slower than his local clock. In the case of weak gravity, the fraction inside the parentheses is much less than unity. (b) Apply the relativistic time dilation ratio y = [1 - (v/c)2]-1/2. Note that we mix two theories here, general and special relativity; in fact GR embraces both effects. [Ans. (a) + (b) « 50 s]

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