ASTR 680Problem Set 5Due Thursday, April 191. (4 points) Derivation of linear overdensity dependence on expansion factor.Consider a spherical Newtonian explosion in vacuum. When the sphere has radius r, a particleon the edge of the explosion has speed v(r), and the mass of the material interior to the particleis M = Mcrit(1 + δ), where Mcritis the mass that would make v(r) exactly equal to the escapespeed at r and δ ¿ 1. By either following the motion of the particle (tougher) or arguing basedon conservation of energy (easier), demonstrate that as the explosion proceeds the overdensity δevolves as δ ∝ r, as long as δ ¿ 1.2. (4 points) Cooling flows.Suppose we model the hot gas in a cluster as a singular isothermal sphere (meaning that thenumber density scales with radius as n(r) ∝ r−2) out to a radius of rmax= 1 Mpc. Thetemperature is a constant T = 108K. The total mass in baryons (assumed to be fully ionizedhydrogen) is 1014M¯. To within a factor of 2, what is the radius inside of which the coolingtime is less than 1010years? To compute this, assume that the cooling occurs exclusively bybremsstrahlung, and define “cooling time” as the thermal energy per volume divided by theradiation rate per volume. Inside this radius, one might expect a cooling flow to do dramaticthings (such as produce lots of star formation) within a Hubble time.4. (8 points) Sunyaev-Zeldovich effect.Dr. Sane has had yet another realization. Clearly, the Sunyaev-Zeldovich mechanism does notwork. He indicates that although some photons might gain energy, others obviously lose energy;consider respectively a head-on collision of a photon with an electron (leading to a higher energyphoton), versus a tail-on collision which would decrease the photon energy. Stuart Vogel, directorof our Laboratory for Millimeter-wave Astronomy, has asked you to look into this.To demonstrate yet again that Dr. Sane’s grey matter is malfunctioning, write a simplecode to do the following. Suppose that the electrons have a temperature T , which we willassume is small enough that all kinematics may be treated nonrelativistically. Simplify furtherby requiring that all the electrons have the same speed v =pkT/me, rather than the actualMaxwell-Boltzmann distribution.Assume that in the electron rest frame, you have Thomson scattering, meaning that in thatframe the photon energy is not changed by scattering (and also meaning that the fractional changein energy is independent of the initial photon energy). Assume also that the photons are equallylikely to come from any direction relative to the electron velocity vector, and that in the electronrest frame the scattering is isotropic, meaning that in that frame there is an equal probabilityof scattering into any direction, independent of the initial direction (this is not quite true forThomson scattering).In your code, determine the energy change of the photon given the initial and final directionsrelative to the electron velocity. Then, use this to compute the average energy gain or loss as afunction of T ; note that for this answer you need to determine the rate of interactions with theelectron as a function of the initial photon direction.In addition to writing down the specific equations you use, please give me a hardcopy of aplot that shows the average fractional energy gain as a function of T , where T is linear and rangesfrom 107K to 108K. I will also need from you a couple of sentences indicating whether you feelyour answer makes sense. For example, how do you expect the average gain of energy to scalewith temperature; inversely, directly, linearly, quadratically, as the square root, or what? You donot need to send me your code, but if you do so before the due date I will try to look at it.Suggestion: In this problem and in many others it is valuable to build your intuition bydoing a simpler problem first. In this case, such a problem is one in which you assign a 50%probability to a photon having a head-on collision with an electron and bouncing straight back,and a 50% probability of a tail-on collision (again bouncing straight back). You can easily computeeverything analytically in that case. Then, consider the rates of such interactions and
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