Accretion discs and the Accretion mechanism

/ nihan10

How one of the most efficient energy production mechanism in the Universe works.

 

An accretion disk is a structure formed by the orbital motion of diffuse objects around a central body. The central body can be anything from young stars (called protostars), to neutron stars and also black holes. However, the most efficient energy is produced in the accretion disks surrounding black holes, particularly the super-massive black holes (SMBH) which are resident at the center of almost all the galaxies in the universe. When these SMBHs are accreting matter onto themselves, they release such tremendous amounts of energy that they often outshine their host galaxies. In such cases, these galaxies are called active galaxies, and the SMBHs, together with the accretion disk are called as the active galactic nuclei (AGN). As an example of the energy output in the case of AGNs, a subclass of them, called Quasars, release, typically, energies of the order 10 ^ {46} \ \textrm{ergs/s}  . An example of such an AGN is given in the following image taken from the Hubble Space Telescope:

An active galaxy with a visible jet from the AGN

An active galaxy with a visible jet from the AGN

 

So how are such high energies generated in the accretion disks surrounding AGNs ? Lets take a look:

The Principle of Accretion:

The total principle can be stated most compactly as follows:

“The (gravitational) potential energy is converted to rotational kinetic energy as the gas falls closer to the black hole (due to collisions between particles), which is then converted into heat via friction between the particles. This heat is then radiated away as electromagnetic radiation.”

Having said that, lets go into a bit more detail.
For every massive body in the universe, there is associated with it its very own gravitational field. The same is the case for the SMBH at the center of the galaxy. In this gravitational field, the accreting material (which is gas) is arranged into orbits around the black hole in an analogous manner to that of the solar system. The gas is arranged into a disk shape due to the intrinsic angular momentum of the central black hole, combined with collisions between the particles. This disk is perpendicular to the angular momentum vector of the black hole. This is illustrated with the following artist’s impression:

Thus, when the gas falls from outer orbits into the inner orbits, its gravitational potential is converted into kinetic energy. If this inflow of the gas is not stopped, then the gas will just fall into the black hole without radiating anything away.
However, the thing that stops this direct inflow of gas is the fact that the gas surrounding the black hole possess some amount of angular momentum. Thus, in accordance with the conservation of angular momentum, there is the formation of a “barrier” due to the angular momentum of the gas which prevents it from falling directly into the black hole.
Next, the force of friction can be assumed to be much smaller than the gravitational force produced by the black hole. Thus, the disk will rotate locally with Keplerian velocity, i.e.

\Omega (R) = \left( GM/R^3 \right) ^{1/2}

where \Omega \longrightarrow \textrm{angular momentum}

Since a Kepler disk rotates differentially (different angular velocity for different radii R), the gas will be heated by internal friction. In addition, the friction will also cause a slight decrease in the rotational velocity of the particles, causing them to move inwards. The energy source for the heating of the gas is provided by this inward motion, i.e. conversion of potential energy to (rotational) kinetic energy, and then to heat through internal friction.

According to the Virial theorem , half of the potential energy will be released as the kinetic energy, which in this case is the rotational energy. The other half of the energy is available for conversion into heat.

To get a quantitative feel of the phenomena, lets do a bit of a “back of the envelope” type of calculation to see what results we may derive.

Temperature Profile of a geometrically thin, optically thick accretion disk:

When a mass m falls from a radius of r + \Delta r to radius r, the change in its energy is:

\Delta E = \displaystyle \frac{GMm}{r} - \frac{GMm}{r + \Delta r}

\approx \displaystyle \frac{GMm}{r} \frac{\Delta r}{r}

where M \longrightarrow \textrm{mass of the black hole}

According to Virial theorem, half of this energy is available to be converted into heat, i.e. \Delta E/2 . Thus, if we assume that this energy is emitted locally, then the luminosity can be given by:

\Delta L = \displaystyle \frac {GM\dot{m}} {2 r^2} . \Delta r

\textrm{where} \ \dot{m} = \displaystyle \frac{dm}{dt} \longrightarrow \textrm{time rate of change of in falling mass, called accretion rate}

[remember that luminosity is given by units J/s and that is how we get the accretion rate into the above equation]
For the stationary case, \dot{m}  is independent of radius, because otherwise, the matter would accumulate at some radius. Thus, equal amount of matter per unit time flows through any cylindrical radius.
If the disk is optically thick, the local emission corresponds to that of a black body. Thus, we can apply Stefan-Boltzmann law to relate the luminosity to the temperature of the accretion disk as follows:

F = \sigma T^4

with \displaystyle F \longrightarrow \textrm{flux in units of } \frac{J}{m^2 s}

Thus, for a thin ring between r and r + \Delta r  , the luminosity can be given by

\Delta L = 2 \times 2 \pi r \Delta r \sigma T^4

where the extra factor of 2 comes due to the fact that there are two sides to the ring so formed.
Now, equating the two expressions for the luminosity, we get the temperature profile as a function of the distance from the black hole, i.e. as a function of the radius:

\displaystyle T(r) = \left( \frac {GM \dot{m}}{8 \pi \sigma r^3}\right)^{1/4}

A more detailed analysis also takes into account the effects due to friction, as well as the amount of energy used in heating up the gas itself. On taking into account all these extra considerations, the formula obtained is, apart from a numerical factor, the same as the one obtained above. Thus, the complete formula is:

\displaystyle T(r) = \left( \frac{3GM \dot{m}}{8 \pi \sigma \ r^3} \right)^{1/4}

which is valid for r \gg r_s
where r_s \longrightarrow \textrm{Schwarzchild radius}

Scaling the above equation with r_s and using the fact that \displaystyle r_s = \frac{2GM}{c^2}  (for this expression, see this), we get

\displaystyle T(r) = \left( \frac{3GM \dot{m}}{8 \pi \sigma {r_s}^3} \right)^{1/4} \left( \frac{r}{r_s} \right)^{-3/4}

\textrm{and then finally we get} \ T(r) = \displaystyle \left( \frac{3c^6}{64 \pi \sigma G^2} \right)^{1/4} \dot{m}^{1/4} M^{-1/2} \left( \frac{r}{r_s} \right)^{-3/4}

Lets see what we can understand from this equation of the temperature profile of the accretion disk:

  • First of all, the temperature profile of the disk does not depend on the detailed mechanism of the dissipation, because viscosity terms do not appear explicitly in this equation. This allows us to obtain quantitative predictions from the model of the geometrically thin, otically thick accretion disk. (Note that the exact source of the viscosity is still unknown. Molecular viscosity seems too small to fit the bill. The leading candidates are turbulent flows in the disk, magnetic fields produced by the SMBH, and hydrodynamic instabilities)
  • The temperature of the disk increases inwards as  \propto r^{-3/4}  as expected. Thus, the emission spectra of the AGN is a combination of a number of different blackbodies consisting of rings with different radii at different temperature. Thus, the spectrum does not have a Planck shape, but a much broader energy distribution.
  • For a fixed ratio \displaystyle \frac{r}{r_s}  , the temperature increases with an increase in the accretion rate \dot{m} . This was again expected because the accretion rate was proportional to the locally emitted energy.
  • For a fixed ratio \displaystyle \frac{r}{r_s}  , the temperature of the accretion disk decreases with an increase in the mass of the black hole M. This is a bit unexpected, but has a feasible explanation to it. As the mass of the black hole increases, the tidal forces at any fixed radius are correspondingly lower, which in turn results in reduced amount of friction and in flow of gas, which in turn reduces the temperature. Most importantly, it implies that the maximum temperature in the accretion disks of AGNs is lower than the corresponding temperatures for neutron stars and stellar mass black holes. This is backed up by observations, in which neutron stars and stellar mass black holes are found to emit in the hard X-ray regions (hence called X-ray binaries), while the AGN thermal radiation peaks out at the UV range.

Thus, we have seen how accretion disks are used to convert the gravitational potential energy of the central body into kinetic, and then ultimately into electromagnetic energy. This analysis of accretion disks and their mechanism involves a few approximations, but is well suited towards describing many instances of this phenomena.

Sources:

Wikipedia
Scholarpedia
Extragalactic Astronomy and Cosmology – Peter Shneider

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