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a) The luminosity density of an accretion disk, which extends from riso to max, is given by the blackbody Planck's formula max r -dr, ehv/kBT(r)-1 Ιναν "Iso where is the radiation frequency, T(r) is the temperature in the disc at radius r, h is the Planck's constant and kB is the Boltzmann's constant. Show that, in the limit 1, L, is proportional to v³e-hv/kBT("iso). kBT b) An AGN emits 1039W at one fourth of the Eddington luminosity limit. Determine the radius of the last stable orbit around the central black hole in AU. c) Consider a spherically symmetric cloud of ideal gas. The gas thermal energy supports it against gravitational collapse and the gas therefore obeys the hydrostatic equilibrium equation dp(r) dr GM(r)p(r) r2 where G is the gravitational constant, p(r) and p(r) are respectively the pressure and the density of the gas at radius r from the centre of the cloud and M(r) is the mass of the cloud interior to r. Show that M(r) can be expressed by the equation M(r) = - kBT(r)2 Gμm H dln p(r) dr dln T(r) + dr where T(r) is the gas temperature at r, kB is the Boltzmann constant, μ is the mean molecular weight of the gas and my is the mass of the hydrogen atom. Explain the importance of the above equation from an observational point of view.

a) The luminosity density of an accretion disk, which extends from riso to max, is given by the blackbody Planck's formula max r -dr, ehv/kBT(r)-1 Ιναν "Iso where is the radiation frequency, T(r) is the temperature in the disc at radius r, h is the Planck's constant and kB is the Boltzmann's constant. Show that, in the limit 1, L, is proportional to v³e-hv/kBT("iso). kBT b) An AGN emits 1039W at one fourth of the Eddington luminosity limit. Determine the radius of the last stable orbit around the central black hole in AU. c) Consider a spherically symmetric cloud of ideal gas. The gas thermal energy supports it against gravitational collapse and the gas therefore obeys the hydrostatic equilibrium equation dp(r) dr GM(r)p(r) r2 where G is the gravitational constant, p(r) and p(r) are respectively the pressure and the density of the gas at radius r from the centre of the cloud and M(r) is the mass of the cloud interior to r. Show that M(r) can be expressed by the equation M(r) = - kBT(r)2 Gμm H dln p(r) dr dln T(r) + dr where T(r) is the gas temperature at r, kB is the Boltzmann constant, μ is the mean molecular weight of the gas and my is the mass of the hydrogen atom. Explain the importance of the above equation from an observational point of view.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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a) The luminosity density of an accretion disk, which extends from riso to max, is given by the blackbody Planck's formula max r -dr, ehv/kBT(r)-1 Ιναν "Iso where is the radiation frequency, T(r) is the temperature in the disc at radius r, h is the Planck's constant and kB is the Boltzmann's constant. Show that, in the limit 1, L, is proportional to v³e-hv/kBT("iso). kBT b) An AGN emits 1039W at one fourth of the Eddington luminosity limit. Determine the radius of the last stable orbit around the central black hole in AU. c) Consider a spherically symmetric cloud of ideal gas. The gas thermal energy supports it against gravitational collapse and the gas therefore obeys the hydrostatic equilibrium equation dp(r) dr GM(r)p(r) r2 where G is the gravitational constant, p(r) and p(r) are respectively the pressure and the density of the gas at radius r from the centre of the cloud and M(r) is the mass of the cloud interior to r. Show that M(r) can be expressed by the equation M(r) = - kBT(r)2 Gμm H dln p(r) dr dln T(r) + dr where T(r) is the gas temperature at r, kB is the Boltzmann constant, μ is the mean molecular weight of the gas and my is the mass of the hydrogen atom. Explain the importance of the above equation from an observational point of view.
Transcribed Image Text:a) The luminosity density of an accretion disk, which extends from riso to max, is given by the blackbody Planck's formula max r -dr, ehv/kBT(r)-1 Ιναν "Iso where is the radiation frequency, T(r) is the temperature in the disc at radius r, h is the Planck's constant and kB is the Boltzmann's constant. Show that, in the limit 1, L, is proportional to v³e-hv/kBT("iso). kBT b) An AGN emits 1039W at one fourth of the Eddington luminosity limit. Determine the radius of the last stable orbit around the central black hole in AU. c) Consider a spherically symmetric cloud of ideal gas. The gas thermal energy supports it against gravitational collapse and the gas therefore obeys the hydrostatic equilibrium equation dp(r) dr GM(r)p(r) r2 where G is the gravitational constant, p(r) and p(r) are respectively the pressure and the density of the gas at radius r from the centre of the cloud and M(r) is the mass of the cloud interior to r. Show that M(r) can be expressed by the equation M(r) = - kBT(r)2 Gμm H dln p(r) dr dln T(r) + dr where T(r) is the gas temperature at r, kB is the Boltzmann constant, μ is the mean molecular weight of the gas and my is the mass of the hydrogen atom. Explain the importance of the above equation from an observational point of view.
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