(5 pts: quantitative) Gravitational Focusing Imagine you are a planetary embryo orbiting the early Sun at a semi-major axis of a = 1 AU. You orbit in a swarm of much smaller planetesimals with random relative velocities vos = 0.5 km/s on top of their circular velocities ~ 30 km/s. (Remember that Voo is the velocity of planetesimals relative to distant planetary embryos.) Assume they orbit in nearly circular orbits, that only gravity affects their motions, and the Sun has its current mass. (a) What is the vertical thickness 2h of the swarm of planetesimals perpendicular to the orbit plane (km)? Remember h/r = Voo/Vc. (b) If the nebula has a surface mass density E = 100 kg/m, seen from above the nebula, what is the average mass density p, spread evenly through the height of the nebula? Give your answer in kg/m3. (c) Consider three spherical embryos of radius 200 (a Vesta-like body), 2000 (a Moon-like body) and 4000 km (a Mars-like body). Assume they have the same density as Earth. Find the gravitational focusing factors for each, F. = (1 + v2 /v2 ). What is the collisional radius (b) versus the physical radius (R) of each (km)? (d) Now calculate the growth rate, dM/dt, in kg/s, and the growth timescale, (growth = M/(dM/dt), in years for each. Which one has a faster growth timescale, and why