Download Homework 1 Questions - Planetary Science | EART 160 and more Assignments Geology in PDF only on Docsity! EART160 Homework #1 Due Friday 16th Jan 2009 Show all your working for full credit 1. In class, we saw that the rate of growth of a planet can be written as 2 2 2 8 1s dM G R vR dt v (1) where M is the mass, t is time, and s are the densities of the solid planet and the planetesimal swarm, G is the gravitational constant, R is the planet radius and v is the relative velocity. a) Explain in words what style of accretion is represented by each of the two terms in the brackets, and what controls which style of accretion dominates at a particular time (3) b) Using the relationship between mass and density for a solid body and the chain rule, rewrite equation (1) in terms of dR/dt. (1) Your answer to part b) is a first-order differential equation, but it’s non-linear, so it’s hard to solve completely. Instead, we’ll consider the two end-member cases. c) Assume that you can neglect the second term in the brackets compared to the first term. Solve this simplified differential equation to obtain R as a function of t, subject to the boundary condition that R=R0 at t=0. (2) d) Now instead assume that you can neglect the first term in the brackets compared to the second. Solve this differential equation to obtain R as a function of t, subject to the same boundary condition. (3) e) Draw the growth curve for the body implied by your answer to d). (1) f) Also derive an expression for the time at which the radius of the body goes to infinity, and explain why this expression makes physical sense (i.e. what happens as you vary v, s and G?) (4) g) If the relative velocity is 1 km/s and the planetesimal density is 10-7 kg/m3, how long does it take for a planet to grow from a 100 km initial radius to a very large size? How does this timescale compare with the time it takes to blow away the nebular gas? (2) (16 total) 2. Here we’re going to consider a simplified solar system which consists of three components: hydrogen (H2), ice (H2O) and silicate (MgSiO3). For every 106 Si or Mg atoms, there are 2.1x107 O atoms available. a) For every 106 silicate molecules produced, how many O atoms are left over? (1)
