.. _physics-hiiRegionDensityDistribution:

HII Region Density Distributions
================================

Class providing models for the distribution of hydrogen number density :math:`n_\mathrm{H}` within H**ii** regions---the probability distribution of gas density within the photoionized nebula. The cumulative density distribution is returned between specified minimum and maximum :math:`n_\mathrm{H}` values, and is used to model the density-dependent emission line ratios and ionization structure of the nebula. Different implementations (e.g.\ log-normal) capture the clumpy, inhomogeneous nature of real H**ii** regions, which affects nebular line luminosities and the ionizing photon budget.

**Default implementation:** ``hiiRegionDensityDistributionLogNormal``

Methods
-------

``cumulativeDensityDistribution`` → ``double precision``
   Return the cumulative distribution of HII region hydrogen density between a minimum and maximum :math:`n_\mathrm{H}`.

   * ``double precision, intent(in ) :: densityHydrogenMinimum, densityHydrogenMaximum``

.. _physics-hiiRegionDensityDistributionDeltaFunction:

``hiiRegionDensityDistributionDeltaFunction``
---------------------------------------------

A class for the distribution of hydrogen density in a HII region in which the distribution is a delta function. Specifically:

.. math::

   p(n_\mathrm{H}) = \delta(n_\mathrm{H} - n_\mathrm{H,0}),

where :math:`n_\mathrm{H,0}=`\ ``[densityHydrogen]``.

**Parameters**

* ``[densityHydrogen]`` (default ``100.0d0``) — The density of hydrogen, :math:`n_\mathrm{H}`, in HII regions (in units of cm\ :math:`^{-3}`).
* ``[spin]`` (default ``0.03687d0``) — The fixed value of spin in a :math:`\delta`-function spin distribution.

.. _physics-hiiRegionDensityDistributionLogNormal:

``hiiRegionDensityDistributionLogNormal``
-----------------------------------------

A class for the  distribution of hydrogen density in a HII region in which the distribution is a lognormal, specifically:

.. math::

   p(n_\mathrm{H}) = \left\{ \begin{array}{ll} \frac{1}{\sqrt{2\pi} n_\mathrm{H} \sigma} \exp\left(-\frac{1}{2}\left[\frac{\log(n_\mathrm{H})-\log(n_\mathrm{H,0})}{\sigma}\right]^2\right) & \hbox{ if } n_\mathrm{H,min} < n_\mathrm{H} < n_\mathrm{H,max}, \\ 0 & \hbox{ otherwise.} \end{array} \right.

**(Default implementation)**

**Parameters**

* ``[densityHydrogenReference]`` (default ``250.0d0``) — The parameter :math:`n_\mathrm{H,0}` in the log normal distribution of HII region densities.
* ``[sigma]`` (default ``1.8d0``) — The parameter :math:`\sigma` in the log normal distribution of HII region densities.
* ``[densityHydrogenMinimum]`` (default ``1.0d1``) — Minimum value of hydrogen density in HII regions.
* ``[densityHydrogenMaximum]`` (default ``1.0d5``) — Maximum value of hydrogen density in HII regions.