LTE -> Local Thermodynamic Equilibrium
I'm particularly confused about what it means to emit as a blackbody to within 1%. What does this have to do with local thermodynamic equilibrium?
1. Matter accreting into a black hole will form a thin, rotating accretion disc which is hottest at its midplane and coolest at its surface. Some distance away from a certain black hole, in the midplane of the accretion disc, the number density of all nuclei is n = 1.087 x 10^12 cm^-3 and the temperature is T = 18990 K. The surface of the disc at that radius, however, emits as a blackbody (to within 1%) with a lower temperature of Teff = 2000 K. Let's calculate some physical parameters in this accretion disc, always assuming that the disc is electrically neutral, that it is made only of hydrogen and helium, that the number density of helium nuclei is exactly one-tenth that of hydrogen nuclei, and that all helium is singly ionized. Info on partition functions and statistical weights: U = 1 for H+ or He++ and the statistical weight for H, He+ or He' in level n is gn = 2n^2. In units where energy is measured in eV (electron-Volts), the Boltzmann constant is 8.617343 x 10^-5 eV K^-1, Planck's constant is h = 4.13566733 x 10^-15 eV s, and the electron mass is me = (510998.91/c^2) eV.
A) [1 mark] Given the information above, without calculations, argue that LTE is likely to apply in the midplane of the accretion disc.
B) [0.5+1 marks] Given that the first excited state of hydrogen (n = 2) is 10.1988 eV above the ground state, do you expect a significant fraction of neutral hydrogen in the n = 2 state at this midplane location in this accretion disc, in light of the value of kT in units of eV at this location? Then do the calculation to find that fraction and compare to your prediction.
C) [0.5+2 marks] Given that the ionization energy of neutral hydrogen is E = 13.5984 eV, do you expect the fraction of hydrogen nuclei in the form of H+ to be higher or lower than the fraction of neutral hydrogen in the n = 2 state? Then do the calculation to determine the fraction nH+/nH (and nHo/nH as well) and compare to your prediction. You need only calculate the partition function to 1% accuracy.
D) [1+1 marks] How accurate is the assumption that no helium nuclei are doubly ionized or neutral at this point in the quasar accretion disc? Assume ionization energies of 24.581 eV for He and 54.403 eV for He+.
