ASTR 680Problem Set 3Due Thursday, March 12, 20091. 8 points Assume you have a black hole of initial mass M0and some specified initial spin,and that it accretes matter at the innermost stable circular orbit. Write a computer programto calculate the dimensionless spin parameter j = a/M = J/M2of the hole as a function of itscurrent mass M. For this I’ll need an e-mail copy of your code, which must be able to compile andrun on the astro machines (I’m not going to install anything!). I’ll also need hardcopy of graphsshowing j versus M for the two cases below.(a) If it starts with j = 0 and always accretes matter in the prograde direction, then to within0.1% how much mass does it accrete to get to j=0.5, 0.9, and the extremal value 1.0?(b) Suppose you start with a near-extremal Kerr hole (j = 0.999) and accrete matter in retrogradeorbits at the innermost stable circular orbit. To within 0.1%, how much mass does it accrete toget to j=0.99, 0.9, 0.5, and 0?For this problem you need the following formulae. The specific angular momentum (i.e., perunit rest mass) of a particle in a circular geodesic at radius r around a black hole of mass M andspin parameter a = jM isuφ= ±√Mrr2∓ 2a√Mr + a2rr2− 3Mr ± 2a√Mr1/2. (1)Here the upper sign is for prograde orbits and the lower sign is for retrograde orbits. The specificenergy of a particle in that same circular orbit is−ut=r2− 2Mr ± a√Mrr(r2− 3Mr ± 2a√Mr)1/2(2)where again the upper sign is for prograde orbits and the lower sign is for retrograde orbits. Theradius of the innermost stable circular orbit isrISCO= Mn3 + Z2∓ [(3 − Z1)(3 + Z1+ 2Z2)]1/2o, (3)where once again the upper sign is for prograde and the lower is for retrograde. Here we useZ1= 1 +1 − j21/3h(1 + j)1/3+ (1 − j)1/3i(4)andZ2=3j2+ Z211/2. (5)2. 8 points Dr. Sane has a new model for active galactic nuclei. He asserts that galaxies atz ∼ 2 − 5 (the peak of AGN activity) had proto-molecular clouds at their center, with typicaltemperatures of 100 K and typical densities of n = 102cm−3. In his model white dwarfs, withinitial masses Minit= 0.6 M⊙and velocities relative to the medium of 20 km s−1, accrete from theclouds until their masses exceed the Chandrasekhar mass MCh= 1.4 M⊙and they collapse in freefall into neutron stars with radii 10 km from an initial radius of 1000 km. The gravitational energyreleased powe rs the AGN; in addition, angular momentum conservation means that the energy isoften released in jets along the rotation axis. State at least three reasons why this model cannotexplain AGN. Recall that typical observed energy outputs are ∼ 1044erg s−1, that the activephase might last for < 109yr, that variability timescales are ∼minutes to ∼years, and that manyof these sources have persistent and large-scale jets. You must make quantitative calculationssupporting at least two of the problems you bring up. The other can be qualitative, i.e., somethingthat Dr. Sane’s model probably can’t account for but is more difficult to
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