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Modelling an accretion disk
Ioana Ciuc, Department of Physics, University Of Durham, South Road, Durham, DH1 3LE


Black holes are fundamentally the simplest

manifestation of space and time and also, as
S. Chandrasekhar deemed, the perfect one. In
the general frame, black holes are described by
the Kerr metric which, in turn, is given by their
mass and angular momentum. Their strong gravity
environment can be mapped to three regions of
interest: the event horizon, beyond which nothing
can escape, the ergosphere, where even light
and spacetime bend, and the innermost stable
circular orbit orbit, which is the last radius at
which a particle can still orbit the black hole.

Figure 6: Binder et al. 2006

Chandra 0.358 keV image
of NGC 300 X-1. The source
is bounded by the green
boundary, whereas in red we
have the contribution from
the background region. It is
found that the properties of
NGC 300 X-1 are consistent
with that of a high-mass
X-ray binary a very rare
one since the companion
Figure 2: Dotani et al. 1997
The spectra of Cyg X-1 corresponding to two different ASCA observations are
shown together with the model that fits the observations best. The model for
the disk is relativistic with a power law contribution corrected for smearing of
the Fe-K line.

Figure 3: The profile of the relativistic temperature for a Kerr black

hole, spinning at different rates given by the spin parameter. The nonrotating case is also modelled, both in Newtonian and General Relativity,
to notice any trends. It can be clearly seen that as the spin increases,
the temperature (evaluated at its maximum value) also increases.

the innermost stable circular orbit (ISCO) radius.

We have so far modelled the variation of
the temperature and radius with the spin
parameter under a relativistic regime, and used
the results to map the accretion spectrum
as can be seen from Figure 3, 4 and 5.


Figure 1: Depiction of a black hole.




There are two cases of accretion onto black

holes, involving either the massive black holes
at the centers of galaxies (quasars) or the
black holes in binary systems (BHB) where
the companion is the source of accretion. In
both cases, as the orbiting matter loses angular
momentum through a dissipative mechanism, it
descends on lower orbits at the expense of its
gravitational energy. The heat released in the
process radiates at X-ray frequencies for BHBs.
The loss of angular momentum is most likely to

Figure 4: The variation of spectrum geometry with spin was modelled.

As can be seen, there exists a clear trend that with increasing spin,
the spectrum deviates towards higher values of maximum local energy.
A profile of the efficiency of a black hole as a function of the spin
parameter is needed to make further inference.

be due the turbulences induced by tangled

magnetic fields and global oscillations in the
disk that can be modelled as the viscous stress

component of the stress energy tensor T obeying:

( T ) =

Various analytic models and numerical methods

are used to fit the observed spectra of accretion
disks; the most well-known is the ShakuraSunyaev (&Novikov-Thorne, 1973) thin disk
model which is geometrically thin (H/r << 1),
optically thick (electron and ion temperatures are
equal) and in which the gravitational energy
is radiated efficiently at each radial location.

Figure 5: The radius at the innermost stable circular orbit is modelled as a

function of the spin parameter. Notice that the negative values of the spin
simply mean that the disc rotation is retrograde (as it is the case for
quasars). It was assumed that G=c=1 so that the GM/c2 = M.



Since light cannot escape a black hole, the

possibility of direct observation is forbidden,
however, there are indirect means to infer its
properties through the emission of its accretion
disk. The basic Newtonian model of such a thin
disk assumes the following temperature T profile:

With M the mass, t the time, rin the inner

radius and r the radial distance. When general
relativity is used in the model, several effects
take place, including changes in the rate of
conversion of gravitational energy to heat and in

The contribution from the line emission will be

integrated in the relativistic model for which
a color-corrected blackbody will be assumed.
Spectra of various X-ray sources will then be
approximated using our fitting model. A remarkable
source is the binary NGC 300 X-1, in which
the companion is a Wolf-Rayet star (WR 41),
an evolved massive star which loses mass
through wind-driven mechanisms. Furthermore,
an estimate of the black hole spin will be given
for the same sources (we chose specifically
the ones for which we know the distance, the
inclination and the mass) by following a relatively
simple method: we find the radius at ISCO by
first determining the solid angle of the black
hole which is given by the flux and temperature,
then use our model to infer the black hole spin.


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