diff --git a/doc/source/users/user_manual/models_spatial.rst b/doc/source/users/user_manual/models_spatial.rst index 8db16c7..5a65928 100644 --- a/doc/source/users/user_manual/models_spatial.rst +++ b/doc/source/users/user_manual/models_spatial.rst @@ -156,6 +156,42 @@ RadialGaussian * ``DEC`` is the Declination of the Gaussian centre (degrees) * :math:`\sigma` = ``Sigma`` (degrees) + RadialGeneralGaussian +~~~~~~~~~~~~~~ + + The ``RadialGeneralGaussian`` model describes a generalised Gaussian + intensity distribution, with the radial profile index as a model parameter. + + .. code-block:: xml + + + + + + + + + + ... + + + + and implements + + .. math:: + M_{\rm spatial}(\theta) = \frac{1}{2 \pi r^2 \eta \Gamma(2\eta)} \exp + \left[- \left(\frac{\theta^2}{r^2}\right)^\frac{1}{\eta} \right], + + where + + * ``RA`` is the Right Ascension of the Gaussian centre (degrees) + * ``DEC`` is the Declination of the Gaussian centre (degrees) + * :math:`r` = ``Radius`` (degrees) + * :math:`\eta` = ``R_Index`` + +The model normalisation is correct in the small angle + approximation and for :math:`\eta` order unity or smaller. + RadialShell ~~~~~~~~~~~ @@ -275,6 +311,54 @@ EllipticalGaussian from North * :math:`\phi` is the azimuth angle with respect to North. + EllipticalGeneralGaussian +~~~~~~~~~~~~~~~~~~ + + The ``EllipticalGeneralGaussian`` model describes a Gaussian intensity distribution + + .. code-block:: xml + + + + + + + + + + + + ... + + + + and implements + + .. math:: + M_{\rm spatial}(\theta, \phi) = \frac{1}{2 \pi r^2 \eta + \Gamma(2\eta)} \exp \left[ -\left(\frac{\theta^2}{2 r_\mathrm{eff}^2}\right)^\frac{1}{\eta} \right], + + with + + .. math:: + r_\mathrm{eff} = \frac{ab} {\sqrt{\left( a \sin (\phi - \phi_0) \right)^2 + + \sqrt{\left( b \cos (\phi - \phi_0) \right)^2}}} + + where + + * ``RA`` is the Right Ascension (degrees) + * ``DEC`` is the Declination (degrees) + * ``PA`` is the position angle, counted counterclockwise from North (degrees) + * :math:`a` = ``MinorRadius`` (degrees) + * :math:`b` = ``MajorRadius`` (degrees) + * :math:`\phi_0` is the position angle of the ellipse, counted counterclockwise + from North + * :math:`\phi` is the azimuth angle with respect to North + * :math:`\eta` = ``R_Index`` + + The model normalisation is correct in the small angle + approximation and for :math:`\eta` order unity or smaller. + Diffuse source ^^^^^^^^^^^^^^