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).

previous

Dabs: dust absorption model

next

Delt: delta line model