Ge/Ay133 – Problem Set #2 Due October 13th, 20111 Angular Momenta(a) Verify eq. (1.1) (page 3) in Armitage, and use it to estimate the total angular momentum of the spinningsun, and how much angular momentum the sun would have if it were spinning on the verge of breakup (i.e.the outer layers are essentially in orbit). Tr e at the sun as a uniform spher e.(b) Verify that the total angular momentum in the planets dwarfs that of the sun (also page 3 of Armitage)by summing the orbital angular momenta of all of the planets (ignore s pins; they’re insignificant). The dataneeded for parts (a) and (b) may b e found in Table 1.1 and pages 2,3 of Armitage.(c) Calculate the angular momentum of a spherical uniformly-rotating gas cloud that has a density of 100hydrogen molecules per cm3, a radius of 10 parsecs (1 parsec is 3 × 1016m) and a r otation period of 100million years (same formula as for the Sun).(d) What do you make of any of this (i.e. can we use any of this information to help us understand theprocess of planet formation and what do you think it tells us)? Note that many of the problems in this classwill have a similar “what do you make of any of this” type ques tion. The reason is that the calculationsabove are designed to make you think about the bigger picture of what is happening. So here is where youshould do that thinking in case you haven’t yet.2 The Minimum Mass Solar NebulaAll of the first ideas about planetary system formation were derived from properties of our own solar system.The hypothetical starting point for our solar system is called the Minimum Mass Solar Nebula (MMSN).Below, you’ll derive its to tal mass, as well as its mass surface density.The chemical compositions of solar system planets is a strong function of their dista nce from the sun. Thisis largely because if a chemical is va porized, it is not available for c onstructing the solid body of the planet.Thus solar system materials are often ca tegorized as either ‘gas’, ‘ice’ or ‘rock’, based on their volatility. Itis reaso nable to assume that planets retain all of their original ‘rock’, but have lost some fraction of theiroriginal ‘ice’ and ‘gas’. The table below shows the real or estimated rock masses of the solar system planets,in Earth mas ses.Body Rock Mass Ice mass Gas mass Total original mass Total current massAll Terre strial 2Jupiter 10Saturn 10Uranus 3Neptune 3(a) The table b e low shows the cosmic abundances, by numb e r, of the most common elements. We’ll assumethese represent the original abundances in the solar s ystem, and use these numbers to ca lc ulate the originalmass of the solar system. Assume all of the O, Mg, Si and Fe go into silicate rocks (use Mg2SiO4, Fe2SiO4,and SiO2for simplicity) and that the remaining O , as well as all available C and N, g o into H2O, CH4andNH3(ices). Calculate cosmic ice/rock, He/rock and H/rock mass fractions and use these to fill in the next3 columns of the chart, in Earth masses. (These are approximations, so don’t sweat the details.)1Atom Abundance by #H 1C4 × 10−4N 1 × 10−4O 7 × 10−4Mg4 × 10−5Si 4 × 10−5Fe3 × 10−5He 8 × 10−2(b) Look up the cur rent masses o f the planets and fill in the last column. Are the differences between thelast 2 columns what you ex pected? Explain.(c) Now sum up the original and current masses of all of the planets and express the two totals in solarmasses. How much mass has been lost? How do you think it was re moved?(d) Now let’s estimate the MMSN surface density, Σ. Take the total original mass of each planet and spreadit out into an annulus that extends halfway to each of its neighbors. P lot the result, using g/cm2and AU,and log-log spa c e.(e) Do a linear fit to the MMSN in log-log space and find the exponent, p, ass uming Σ = Σ0(RAU)p(f) One way to think about a phys ic al basis for the Titius-Bode relationship is to assume that you haveapproximately the same total mass in each ‘planet forming z one.’ More quantitatively, this means that themass within an octave of radius (a fancy way of saying a factor of two) is constant. Write down the integralexpression for this statement of the Titius-B ode relationship.(g) If the Titius-Bode relationship is true, what exponent would you derive for the mass surface density?(h) Assuming the total mass of the MMSN is the same as that derived above (in question 3), calculate theTitius-Bode mass surface density and add it to your plot. Is it similar to the MMSN you derived?(i) In the Chiang & Goldreich paper we’ll be covering shortly, the assumed mass surface density is Σ =Σ0R−3/2AU, where Σ0= 103g cm−2is the mass s urface density at 1 AU. Add this to your plot.(j) How different is the Chiang & Goldreich surface density from the one you derived? Can you think of anyreasons why the surface density of the MMSN mig ht have varied with time?3 Exoplanetary PiddlingGo to exoplanets.org, and look around. From the various distributions you see, dig into one system, takea look at it, and tell us what’s interesting about it. Be brief, and use plots as is
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