4.1.9. Dbb: disk blackbody model

We take the model for a standard Shakura-Sunyaev accretion disk. The radiative losses from such a disk are given by

Q = \frac{3GM\dot{M}(1-\sqrt{r_i/r}) }{ 8\pi r^3},

where Q is the loss term in W m^{-2} at radius r, M the mass of the central object, \dot{M} the accretion rate through the disk and r_i the inner radius. If this energy loss is radiated as a black body, we have

Q = \sigma T^4

with \sigma the constant of Stefan-Boltzmann and T(r) the local temperature of the black body. The total spectrum of such a disk is then obtained by integration over all radii. We do this integration numerically. Note that for large r, T\sim r^{-3/4}.

Warning

A popular disk model (diskbb in XSPEC) assumes this temperature dependence over the full disk. However, we correct it using the factor (1-\sqrt{r_i/r}) in Q which corresponds to the torque-free condition at the inner boundary of the disk.

The photon spectrum of the disk is now given by

N(E) = \frac{8\pi^2E^2r_i^2\cos i }{ h^3c^2} f_d(E/kT_i,r_o/r_i),

where i is the inclination of the disk (0 degrees for face-on disk, 90 degrees for an edge-on disk), E the photon energy, r_o the outer edge of the disk, and T_i is defined by

(1)T_i^4 = 3GM\dot{M}/8\pi r_i^3\sigma

and further the function f_d(y,r) is defined by

f_d(y,r) = \int_{1}^{r} \frac{x{\mathrm d}x }{ e^{y/\tau} - 1}

where \tau(x) is defined by \tau^4(x) = (1-1/\sqrt{x})/x^3.

In addition to calculating the spectrum, the model also allows to calculate the average radius of emission R_e at a specified energy E_r. This is sometimes useful for time variability studies (softer photons emerging from the outer parts of the disk).

Given the fit parameters T_i and r_i, using (1) it is straightforward to calculate the product M\dot{M}. Further note that if r_i=6GM/c^2, it is possible to find both M and \dot{M}, provided the inclination angle i is known.

The parameters of the model are:

norm : Normalisation A (=r_i^2\cos i), in units of 10^{16} \mathrm{m}^2. Default value: 1.
t : The nominal temperature T_i in keV. Default value: 1 keV.
ro : The ratio of outer to inner disk radius, r_o/r_i
ener : Energy E_r at which the average radius of emission will be calculated
rav : Average radius R_e of all emission at energy E_r specified by the parameter above. Note that this is not a free parameter, it is calculated each time the model is evaluated.

Recommended citation: Shakura & Sunyaev (1973).