.. SPDX-FileCopyrightText: 1992-2026 NWO-I/SRON Space Research Organisation Netherlands
..
.. SPDX-License-Identifier: CC-BY-4.0

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

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

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

.. math:: Q = \sigma T^4

with :math:`\sigma` the constant of Stefan-Boltzmann and :math:`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 :math:`r`,
:math:`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 :math:`(1-\sqrt{r_i/r})` in :math:`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

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

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

.. math::
   T_i^4 = 3GM\dot{M}/8\pi r_i^3\sigma
   :label: tidisk

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

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

where :math:`\tau(x)` is defined by
:math:`\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 :math:`R_e` at a specified
energy :math:`E_r`. This is sometimes useful for time variability
studies (softer photons emerging from the outer parts of the disk).

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

The parameters of the model are:

| ``norm`` : Normalisation :math:`A` (:math:`=r_i^2\cos i`), in units of
  :math:`10^{16}` :math:`\mathrm{m}^2`. Default value: 1.
| ``t`` : The nominal temperature :math:`T_i` in keV. Default value: 1
  keV.
| ``ro`` : The ratio of outer to inner disk radius, :math:`r_o/r_i`
| ``ener`` : Energy :math:`E_r` at which the average radius of emission
  will be calculated
| ``rav`` : Average radius :math:`R_e` of all emission at energy
  :math:`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) <https://ui.adsabs.harvard.edu/abs/1973A%26A....24..337S/abstract>`_.