Question

(3.) The first realization that supermassive black holes existed came from observational inference of the luminosity of QSOs and Active Galactic Nuclei. The Eddington Luminosity is the maximum rate at which an object of can radiate photon energy before the momentum transfer from the photons moving through the material in the object tear it apart (a.) Derive the relationship between the mass of an object in hydrostatic equilibrium, the mass of a proton, the Thomson cross section for photon-electron scattering, the gravita- tional constant G, and the Eddington Luminosity. (Hint: We did this in class.) (b.) A QS0 at high redshift is inferred to have absolute luminosity6x 10ergs. Use the Eddington Luminosity result from (a.) to infer an estimate for the mass of the black hole powering this monster

Answer 1

The ingredients we need to derive the Eddington luminosity a The mass of the central object, M Small cloud of "stuff" Distance R om the radiating objec Cloud has mass m The total luminosity, L A suitable opacity for radiation pressure against any surrounding material. We will see that this must have the dimensions of area per unit mass Luminous object with mass M and luminosiry L Balancing the two forces gives We wish to find the luminosity at which the gravitational force inwards balances with the radiation force outwards. The gravitational force is given simply via GMm LKmm R2 4TR2c GMm R2 And solving for this luminosity we get To calculate the radiation force first we need to get the radiation pressure at R L 1 4TGMc ra 2 Then to calculate the radiation force on the cloud, we need its opacity, K. Radiation pressure is force per unit area, opacity is the cross-sectional area per unit mass for radiation scattering radPradkm. Some important things to note here are There will still be electrostatic forces between the e s and the p+s; if we exert a radiation force on the cloud that is felt mostly by the electrons, they will drag the protons along with them. Thus the . The Eddington luminosity depends only on the mass of the approximation is to set K-PT/mp, and we get the following approximation for the Eddington luminosity: radiating object. . We have assumed spherical symmetry Now, we have yet not specified a particular value for k. In high-energy accretion scenarios we make a useful approximati on the basis that the accreting material is mostly ionized hydro and the opacity is provided by Thomson scattering. The cross-section will then come almost exclusively from radiation stars the opacity follows Kramer's Law, R ox p/T3.5. pressure on the electrons, but the mass lies almost exclusively. the protons Note that this approximation for K is not valid in all situations, especially in stars of all but the highest masses. E.g. in low mass