Question: Optically thick Bremsstrahlung.

Oppositely directed jets of plasma that are bright radio sources originate in the nuclei of many radio galaxies. They emerge along the

axis of an ‘accretion disc’, as shown in Fig. 1. The radio jet emits synchrotron radiation and the accretion disc emits Bremsstrahlung,

but at radio wavelengths, the jet is normally much brighter than the disc, the latter being essentially invisible. In one such radio

galaxy, 3C84, the jet directed away from the observer (the ‘counterjet’) is obscured as a result of free-free absorption by the accretion

disc at frequencies below 22 GHz, i.e. the accretion disc has opacity greater than 1 for frequencies below 22 GHz. Assume that the

temperature of accretion disc T = 104 K, that its thickness is 1.4 pc and that it is viewed at θ = 45° to the axis.

Find the number density of electrons in the accretion disc.

Hints: Use the Gaunt factor g appropriate to radio wavelengths (given as below) and find an expression for the emissivity. Then consider

the criterion for opacity >1.

g(ν,T) = √3/2π [ln⁡((128ϵ_0^2 k^3 T^3)/(m_e e^4 ν^2 Z^2 ))-0.76]

Where ν is frequency, ϵ is permittivity of free space 8.854 x 10-12 Fm-1, k is Boltzmann’s constant 1.381 x 10-23 J K-1, m_e is mass of

electron 9.109 x 10-31 kg, e is charge on the electron 1.602 x 10-19 C and Z can be assumed as 1 for fully ionised gas.

Remember throughout this question that the calculations involve the Bremsstrahlung of the accretion disc and not the synchrotron radiation

from the jet. See note below diagram.
Figure 1: The jet and counterjet emerge along the axis of the accretion disc. Radiation from the counterjet is absorbed by the disc at

frequencies below 22GHz.

Note: In most radio galaxies, the counterjet is hidden as a result of relativistic aberration, which greatly diminishes the intensity of a

source moving away from the observer. 3C84 is an unusual case: its jets are thought to move too slowly for relativistic aberration to be

significant – it seems the counterjet is hidden at low frequencies only by free-free absorption.