GG 304: Physics of the Earth and Planets

Homework 1: Math/Physics Review and Lowrie Chapter 1 Turn in for grading on Friday 1/16

Part A: Complete for discussion on Wed. 1/14 1. Derivatives and planet motion: The planets have elliptical orbits around the Sun described by,

1 e r  r (1) o 1 e cos

where r is the distance between the planet and the Sun, ro is the distance between the planet and the Sun at perihelion, e is the eccentricity of the orbit, and  is the angle from perihelion. (a)By taking the derivative of (1) with respect to , determine how quickly the planets distance r is changing with the angle . (b) At what values of  is r not changing (i.e., dr/d = 0)? (c) Briefly describe how to determine where r is changing most rapidly with respect to . (d) Extra credit: Do (c).

2. Vectors and plate motions at a triple junction. The Pacific, Cocos, and Nazca Plate meet at a ridge-ridge-ridge triple junction in the Pacific (see pp. 21 & 30 of Lowrie). Relative to the Pacific plate, the Cocos plate moves at a rate of VCo-Pa = 135 km/Myr in the direction 11.3 north of east, and the Nazca plate moves VNo-Pa 126 km/Myr in the direction 6.7  south of east. Again these are velocities described with the Pacific plate being stationary. (a) Find the velocity vector, VCo-Na, that describes the motion of the Cocos plate relative to the Nazca plate. (b) What is the dot product of VCo-Pa gVNa-Pa? Is it a scalar or vector? (c) What is the cross product VCo-Pa VNa-Pa? Is it a scalar or vector?

3. Rotational kinematics: A particle moving in a circular path, travels an arc length l = r, where r is the distance of the particle from the spin axis and  is the change in angle about the spin axis. (a) Approximately how far (l) do you in Honolulu (latitude=21.3N) travel in one hour as the Earth rotates (see Table 1.1 for radius of Earth)? (b) How fast does the Earth travel through space as it rotates around the sun (see Table 1.2)? GG 304: Physics of the Earth and Planets

Part B: Complete for discussion on Fri. 1/16 4. Integrals: Calculate the volume V of a sphere in terms of its radius a. The diagrams below will help you do this; they illustrate (left) a spherical coordinate system, (middle) an infinitesimal volume dV of a sphere and (right) a calculation of dV in spherical coordinates (i.e., simply the product of the length of each side).

The volume of the whole sphere is simply an integral of dV over the whole volume encompassed by the sphere. That is,

? ? ? V= 蝌 R2 sinq d R d q d f . (2) 0 0 0

Evaluate the above integral (filling in the correct values for the “?”s) and verify that V =4/3a3. 5. Integrals and angular momentum: Estimate the angular momentum, h = I (Eq. 1.5 of Lowrie), of the Sun as it spins around its own axis. First, you need to derive moment of inertia I of the Sun. Here are some tips on how to proceed (you can look it up in your physics book but please present a complete derivation). Approximate the Sun as a perfect sphere of radius a and of uniform density . You can imagine breaking this sphere into a bunch of small volumes dV with mass dm = dV. Each volume is a distance r from the Sun’s spin axis (see diagram above) and contributes a small amount dI = (dM)r2=(dV)r2 (after Eq. 1.4) to the total moment of inertia I. The total moment of inertia I is just the sum, or integral of dI over the volume of the sphere, but remember r changes throughout the sphere. Now you must figure out how to setup this integral; problem (4) should help you do this. After evaluating this integral you should get I = 2/5Ma2 where M = (4/3a3is the mass of the Sun. Now compute the Sun’s moment of inertia by substituting the Sun’s mass of 2.0 x 1030 kg and radius a = 6.9 x 108m. Now compute h. 6. Compare your estimate for the angular momentum of Sun to (a) that given in Table 1.3 (make sure your units are the same as those in Table 1.3). Is your estimate greater or less than the true value? What about the simple assumptions made in problem (5) above might be in error and thus account for this discrepancy?