Astro 162 – Planetary Astrophysics – Problem Set 2Due Thursday in class.Readings: Landstreet, sections 4.4–4.6Problem 1. Protostellar Disk Sizes and the Centrifugal BarrierThe gravitational collapse of a molecular cloud to form a star leads inevitably to theformation of a disk th at carries most of the angular momentum of the initial cloud. Inthis problem we will estimate roughly the sizes of these protostellar disks.Consider a Jeans-unstable clump of molecular gas of mass M = 1M, radius R = 0.1 pc,and spin rate 1 km s−1pc−1. (Clumps like this have been observed via millimeter-waveobservations.) As this cloud collapses inward to form a star, the outer radiu s of thecloud will decrease. The outer radius will not decrease indefinitely, however: It will stallat the centrifugal barrier. Estimate the radius of the centrifugal barrier in AUs, andcompare your answer to the sizes of protostellar disks displayed in class.Problem 2. Making Ice Cubes in the Protoplanetary Disk(a) Assume that all of the cosmically abu ndant oxygen in the minimum-mass solar nebulatakes the form of water (H2O) vapor.What is the mass density [g/cm3] in water vapor at a stellocentric distance of 5 AU?Use the parameters for the solar nebula described in class.(b) Estimate the m ean thermal speed of water molecules in the solar nebula at this samestellocentric distance. Give your answer in [km/s].(c) Consider the growth of an ice particle at 5 AU. Assume that the particle accreteswater molecules from the gas phase. Es timate the rate at which its radius increases,assuming that water molecules stick to the grain with nearly perfect efficiency due toelectrostatic forces and chemical bonds. Exp ress your answer in u nits of [cm/yr]. Is thiscalculation promising or discouraging for growing large bodies in the solar nebula, giventhat th e lifetime of the solar nebula is estimated to be 107yr?Problem 3. Make like a tree and ...If all the leaves of a tree fall to the ground, how thick is the layer of leaves on the ground?Give your answer in leaf thicknesses, to order-of-magnitude.You are lying below the tree. When you look up, the tree p resents a certain opticaldepth to visible light photons coming from the sun. What is the order-of-magnitude1of this optical depth? Why does this answer make sense from the perspective of thevegetation? (Be the tree.)The above two paragraphs are flip sides of the same question. You can try the secondparagraph before the first, or vice versa.Problem 4. Baby BoomLandfills are filled with our garbage. In the United States, what f raction of landfi lls isfilled with disposable d