Addendum 13: Active Galaxies: Chapter 22

First define some astronomical and physics constants and dimensional units that may be needed below

1. Luminosity of an AGN. AGN's radiate at many frequencies, from the radio to gamma region of the spectrum. A convenient measure of the received power (called flux density) is the Jansky, define as 1 Jy = 10^-26 W m^-2 Hz^-1. To determine the total received flux, we integrate over all frequencies. As a simplifying assumption, we can assume many AGN spectra are flat, i.e, the flux density is constant over many decades of frequency.

Example: Calculate the luminosity [in solar luminosities] of an AGN at a redshift z = 0.07 with 1 Jy flux density and a flat spectrum from radio (10⁶ Hz) to x-rays (10¹⁵ Hz).

or

2. Maximum luminosity of a radiating black hole with infalling matter. The maximum luminosity of a black hole radiating due to infalling matter of mass m is simply the gravitational energy of the mass dropped from infinity to the event horizon. We previously (Addendum 4, section 1a) showed that the gravitational potential energy of two masses separated bya distance r is

But the event horizon (Schwarshild radius) is (addendum 8):

So, substituting:

In other words, the maximum available energy is 50% of the rest mass. (This is in contrast to thermonuclear fusion, in wihhc only .7% of the rest mass is converted to energy!). The maximum luminosity (assuming all the infalling kinetic energy is converted to radiation) is the time derivation of the energy:

Note that this result does not depend on the black hole's mass.

Example: A typical AGN has a luminosity of 10⁴⁰ W. What minimum mass accretion rate, in solar masses per year, are required to explain this luminosity?

Call

Note: This is a minimum mass accretion rate, since it assumes all the available energy assocaited with infall is converted to radiate energy

3. Gravitational Lensing and lensing galaxy masses. From addendum 8, section 3, we derived the bending angle of light rays whose path approaches within distance b from a mass M:

where

Solving for M, we obtain:

This means that images of very distant objects can be distorted by masses (galaxies) near the object's line of sight, forming partial or entire rings, as shown in the image below. This effect ,called gravitational lensing, can be used to determine the mas of the lens (intervening galaxy).

Example: In the image above, the partial circular arcs have an effective angular diameter of 10 arcsec. Assume that the light passes the edge of the galaxy's disk, about 50 kpc. What is the mass of the galaxy?

bending angle

Much of this is dark matter!