Ge 133 - Problem Set # 3, due Oct. 27thA) The goal of this problem is to understand Spectral Energy Distributions (SEDs), the spectra emitted bya star plus a disk. Using some simple assumptions, you’ll generate your own model SED. For this problem,assume the star has the properties adopted by Chiang & Goldreich (an effective temp e rature of 4000 K, amass of 0.5 solar masses, and a radius of 2.5 solar radii). For the disk, assume an opaque disk extendingfrom the stellar surface out to 100 AU. We will also assume that the dust in the disk radiates as a perfectblackbody. For this problem, use the following form of the blackbody equation:Bλ(T ) =2πhc2(λ)51ehc/(λkT )− 1[erg/s/cm2/cm]where everything is in cgs units.(1) Equation (4) of Chiang & Goldreich tells you what the temperature versus radius is for a flat, geometri-cally thin disk. Calculate this flat disk temperature at a distance of 1 AU and compare it to what you knowabout the Earth. Why do these numbers differ?(2) Real disks are flared, and for the interior, optically thick part of the disk the temperaur e can beapproximated as T(interior)∼150/R3/7K, where R=distance from the central star (in AU). Assuming thatwe are viewing the star and disk from directly above (i.e. along the pole) at a distance of 10 parsecs (a parsecis 3 × 1018cm), use your favorite plotting program to plot the total emission seen from the star a lone, thedisk alone, and the star plus disk at wavelengths from 0.1 to 100 microns. Use a log-log scale , use cgs unitsfor the y-axis, and use microns for the x-axis.(3) The Spitzer Space telescope observing bands are centered at 3.6, 4.5, 5.8, 8.0, 24.0, 70.0, and 160.0microns. For a circularly symmetric dis k, write down the equation for the disk flux versus radius (distancefrom the central star). Call this function F(R). The “characteristic” radius for the disk emission, or firstmoment of the emission, would be < R >= (RR ∗ F (R)dR)/RF (R)dR (so things are normalized). Usingyour favorite mathematics package, calculate < R > at 4.5, 24, and 70 microns. For disks at 140 pc, how bigwould a telescope have to be to provide a diffraction limit of < R > at 4.5, 24, and 70 microns? In realityof course we’d have to add the stellar flux as a point source. What do these calculations tell you?(4) Assume that there is no disk from the stellar surface until an orbital distance of 4 AU, but that the dis kresumes after that. (This might happen if a giant planet has formed and clear e d out part of the disk). Plotthe emission as a function of wavelength for this disk geometry as well as for that calculated in part (b).What do you notice?B) Gravitational Focusing. One of the most important concepts in pairwise accretion is the “g ravitationalfo c using” caused by the deflection of a small body by a massive b ody. This deflection works to increa se thecross-s ection for collision of the two bodies. Your goal is to reproduce a calculation first done by Safronov.Try this, at first, with no books, no notes, no discussion with anyone. Really. Consider a test particleapproaching a body of mass M and radius R . The impact parameter is b and the velocity at infinite distanceis v∞(i.e. in the absence of gravity, the two bodies would approach within a distance of b of each otherand with relative velo city v∞, see the figure at the end of the problem set). In the a bs ence of gravity, thecross-s ection for collision would be πR2. We would like to k now by how much this cross-sec tio n increasesdue to the effects of gravity.1. What is the escape velocity from the large body? (you’ll need this for later).2. Draw a diagram of a grazing c ollision between the two bodies. (A grazing collision means that if thepoint mass had just slightly more energy it would not collide with the massive body) Pay particularattention to where the two bodies touch.3. For the case you just drew, use the principle of conservation of angular momentum to write an e quationrelating b, v∞, R, and vimp, the velocity of the object at impact. (Hint: Write down the total angular1momentum in the ‘before’ picture, shown above, and the total angular momentum at the moment ofimpact, and set these equal)4. Write a n energy conservation equation relating v∞, vimp, R, and M.5. What is the maximum value for b for which an impact will occur? Write your answer in the form ofb2= R2(1 + f) where f is ca lled the focusing or Safronov parameter, which should be expressed interms o f the escape velo c ity.6. (you can start collabor ating again at this point) How large does an object need to be before gravitationalfo c using significantly increases the impact (and thus accretion) cross-section. The answer to thisquestion is not obvious without some additional information that you don’t neccessarily possess, so doyour best and make reasonable estimates where neccessary. Justify your answer carefully.7. An important concept that we will discuss next week is that of dynamical friction, in which largebodies are slowed by the presence of a swarm of smaller bodies. This process leads to an equipartitionof energy between small and large bodies such that mv2= MV2where the lowercase values re fer tosmall bodies and the uppercase to large bodies. Explain how this dynamical friction could greatlysp e e d up the growth of large bodies.M v∞ b R m
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