Problems

4.2 Gravity

Problem 4.21. (a) If the entire mass of the sun were compressed into a sphere the size of the earth, what would be its density relative to that of the earth? (This is typical of a white dwarf star.) (b) If your scale indicates 80 kg when you weigh yourself on earth, what would it read if you weighed yourself with it on the surface of the white dwarf? (c) Could you pick up a penny? See Appendix for the masses and sizes of the earth and the sun. A penny weighs about 2 g. [Ans. (b) ~107 kg]

Introduction to Problems 4.22-25. The moon's interaction with the earth's oceans causes the earth to slow down and the moon to gain energy. Energy is dissipated by the movement of the tides across the earth, but angular momentum of the moon-earth system must be conserved. The moon thus moves into a higher orbit and gains angular momentum while the earth loses angular momentum. We explore this evolution in these problems. We neglect the effect of the sun's gravity and other factors affecting the earth spin and lunar orbit. The long-term lengthening of the earth's spin period due primarily to the tidal effect is (dP/At)0 ^ 1.7 ms per century. The current lengthening of the moon's orbital period is about (dp/dt)o ^ 35.3 ms per century corresponding to a changing earth-moon distance of 38.2 mm yr-1. The current sidereal periods (relative to fixed stars) of the earth spin and moon orbit are respectively P0 = 86 164 s and p0 = 27.322 d (1 d = 86400 s). The earth mass is M = 6.0 x 1024 kg and the ratio of masses (earth to moon) is M/m = 81.3. The earth radius is R = 6.4 x 106 m and the mean moon-earth center-to-center distance is currently r0 = 3.8 x 108 m. The moment of inertia of the earth is —MR2/3. In the following, let all angular momenta be about one axis and ignore moon spin. Take the earth to be a solid body (except for the tides). Use Newtonian expressions for circular orbits where M ^ m. Neglect any effects of the sun's gravitational field. Let Q represent the angular velocity (rad/s or s-1) of the earth spin and rn that of the moon's orbital motion.

Problem 4.22. Earth-moon: Spindown of the earth. (a) If the moon was once part of the earth (but was later somehow ejected while conserving system angular momentum), what would have been the rotation period of the earth before the ejection? (b) If the slowdown rate of the earth spin dP/dt is taken (improbably) to be constant at its current value since it had that period, how long would it take the earth to spin down to its current period? How does this compare to the age of the earth, 4.5 x 109 yr? Ignore the nature of the ejection process. (c) What does our simplistic assumption that dP /dt = constant imply about the rate of change of the angular velocity of the earth dQ/dt (rad s-2), i.e., its angular acceleration, as a function of Q? How does the torque on the earth spin depend on Q in this approximation? Your answer could be compared to physical models of the torque exerted on the earth by lunar tides. [Ans. —4 h; —1010 yr; a —Q2]

Problem 4.23. Earth-moon continued. Compare the relative slowing rates to theory. (a) Find an expression for the ratio of the angular acceleration of the earth spin to that of the moon orbit, [(dQ/dt)/(dw/dt)], in terms of M, m, G, R, w, under the sole assumption that angular momentum is conserved. Proceed as follows. Write an expression for the total system angular momentum L; include only earth spin and moon orbital motion. Use Kepler's third law to eliminate the orbital radius r of the moon, and finally set dL/dt = 0. Find the numerical value of the ratio for the current angular velocity of the moon. (b) Find the actual ratio observed today, from the data on the periods given in the "Introduction" above, and compare to the theoretical value just derived. Hint: What is the relation between dP /dt and dQ/dt? [Ans. —40, —40]

Problem 4.24. Earth-moon continued. Evolution of angular velocities under angular momentum conservation. (a) Find an expression for Q in terms of M, R, G, m, and L the total angular momentum (actually its component along the earth spin axis). Under the assumption of strict angular momentum conservation for all time at the current value L = L0, create a plot of Q vs. m for the earth-moon system from m ~ 10-7 s-1 to at least m = 10-3 s-1 (i.e., p = 2^/m = 1.7 h). Program your calculator, calculate 3 points per decade of m (or more if the function changes rapidly), and tabulate your results before plotting. Find analytically and/or empirically, and show on your plot, the position (m, Q) where (i) the rotations are synchronous, Q = m (two places), (ii) the angular accelerations are equal (see previous problem), (iii) the angular momenta of the earth and moon are equal, (iv) the spins are those of today, and (v) the earth spin is zero, Q = 0. At each point label or tabulate the equivalent rotational periods in hours or days. (b) For the two cases where Q = m, find the distances to the moon from the earth center, in units of earth radius; i.e., find r /R for each case. Compare to today's values. [Ans. synchrony at P ~ 5 h and ~50 d at r/R ~ 2 and 90; Ref. Counselman, ApJ 180, 307, 1973]

Problem 4.25. Earth-moon continued. Energy evolution. (a) Find an expression for the total mechanical energy E of the system, including earth spin, moon kinetic, and moon potential energies, the latter going to zero at infinity, as a function of M, m, R, L, and m where L is the total (conserved) angular momentum. Eliminate Q with the expression for Q(L) from the previous problem, 4.24a. (b) Plot log E vs. log m for the actual values of M, m, R, and L0 for values of log m ranging from -7 to -2. Program your calculator to do this. (c) Demonstrate empirically with your calculator that the energy reaches an extremum (max. or min.) at the two points where Q = m (see previous problem). (d) Demonstrate analytically that at an energy extremum the frequencies Q and m are equal. Hint: justify and use dE(Q, m)/dm = 0 and dL(Q, m)/dm) = 0. (e) If the moon were initially at the inner-orbit extremum (with high m = Q), what are the possible scenarios for its evolution as it experiences tidal friction? (f) Qualitatively, what would be the expected effect on the earth-moon system of ocean tides raised by the sun's gravity after outer synchrony is reached? [Ans. (b) Sample point: when m = 356 x 10-6 s-1 at the inner, rapid synchrony point, E « +4.2 x 1030J]

4.3 Apparent motions of stars

Problem 4.31. Draw a horizon coordinate system for an observer in the northern hemisphere at latitude +60°. Show the tracks of five stars: a star that is always north of the observer, a star that never sets, a star that rises and sets in the north and another which rises and sets in the south, and a star never visible to the observer. Indicate with small arrows the range of track locations that satisfies each of these five conditions.

Problem 4.32. (Library problem) (a) Refer to the star charts of Norton's 2000 Star Atlas to find the boundaries of the constellation Aquarius. Determine from the rate of motion of the first point of Aries (the vernal equinox) along the ecliptic when the vernal equinox will move into the constellation Aquarius. (b) In what constellation is the sun on your birthday in ~2000? The Sun's daily coordinates are in the Astronomical Almanac, or you can deduce it from the "Index Maps" in Norton. (c) Is it possible on your birthday to see the stars of the astrological sign of the zodiac for your birthday? If not, could you go to the other side of the earth, to China, to see the stars of your sign on your birthday? If not, how might you manage to succeed in this endeavor? Does precession help? (d) Could you see or observe these stars easily at other times of the year? [Ans. Precession does help, even today!]

Problem 4.33. Make sketches of the apparent annual motion of a given star at distance 32.6 LY (=10 pc) due to parallax and also due to aberration as the earth circles the sun (assume a perfectly circular orbit) for the following cases: (a) a star lying on the ecliptic (at ecliptic colatitude 0 = 90°) at ecliptic longitude $ = 90°. (Here, $ is measured from the vernal equinox in the same direction as right ascension a; i.e., our star is at a « 6 h), (b) a star at the north ecliptic pole 0 = 0°, and (c) a star 30° from the ecliptic equator at 0 = 60° and at $ = 90°. In each case, sketch the stellar track as viewed from the earth in the ecliptic coordinate system indicating magnitudes and directions of angular displacements. Also indicate the star locations on the tracks for the dates Mar. 21, June 21, Sept. 21, and Dec. 21. (d) What are the practical consequences of parallax and aberration for an observer? What measurements would one make in order to construct the track? [Ans. Tracks are (a) lines, (b) circles, (c) ellipses]

Problem 4.34. Find, from the earth's precessional motion (Fig. 4b) and the current J2000.0 coordinates, the approximate J2100.0 coordinates for four hypothetical stars: (a) on the ecliptic at a = 0 h, (b) on the ecliptic at a = 6 h, (c) at the north celestial pole, (d) at the north ecliptic pole. In each case, first write down the J2000.0 celestial coordinates, a,8. Obtain your answers by visualizing and sketching the motion of the celestial coordinate system in the locale of the star. Work solely in degrees (not h m s). Make use of flat space and small angle approximations. In each case, give the (approximate) great circle distance (in degrees) that corresponds to the shift of coordinates of the star. [Ans. the shifts are ~1.5°; ~1.5°; ~0.6°; ~0°]

4.4 Lunar and planet motions - eclipses

Problem 4.41. (a) Confirm that the four cyclic periods given in the text lead to similar eclipses at intervals of 18 y 11.3 d (or 10.3 d). Explain the role of each period in bringing this about. (b) Show that the length of the saros is about 1350 yr. Hint: consider that the sun is susceptible to an eclipse for ~35 d while it is near a node. (c) How do the shadow tracks of totality on the earth change location for sequential eclipses in a given saros? Find approximate magnitudes in km and directions of the shifts in the E-W and N-S directions. Hint: consider spin of the earth and the change of sun-moon alignments as the saros evolves. [Ans. (c) ~ 15 000 km E-W; ~300 km N-S for shadows at low geographic (or ecliptic) latitudes.]

Problem 4.42. The moon is currently receding from the earth at the rate of about 38 mm/yr. About how long will it take for total eclipses to no longer occur because the moon is never large enough (in angle) to cover the sun? Because of the elliptical orbits of the earth and moon, the current largest angular diameter of the moon (as viewed from earth) is about 33' 30" and the minimum sun diameter is about 31' 32". The current minimum perigee distance of the moon is 3.564 x 108 m; assume that it moves away from the earth at the aforementioned 38 mm/year. [Ans. ~109 yr]

Problem 4.43 (challenging problem). Here we derive the average precession rate of the moon's orbit.

^ Ecliptic

Node

To sun

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