.. _physics-satelliteDynamicalFriction:

Dynamical Friction
==================

Class providing models of the satellite vector acceleration due to dynamical friction---the gravitational drag force that decelerates a satellite as it moves through the background distribution of dark matter particles and substructures. The Chandrasekhar dynamical friction formula scales as :math:`\mathbf{a}_\mathrm{df} \propto -\rho \ln\Lambda\, \mathbf{v}_\mathrm{sat}/v_\mathrm{sat}^3`, where :math:`\ln\Lambda` is the Coulomb logarithm. Dynamical friction drives orbital decay and eventual coalescence of satellite galaxies with their hosts, and is complementary to SIDM-driven deceleration.

**Default implementation:** ``satelliteDynamicalFrictionChandrasekhar1943``

Methods
-------

``acceleration`` → ``double precision, dimension(3)``
   Returns the satellite acceleration due to dynamical friction for ``node`` (in units of km/s/Gyr).

   * ``type(treeNode), intent(inout) :: node``

.. _physics-satelliteDynamicalFrictionChandrasekhar1943:

``satelliteDynamicalFrictionChandrasekhar1943``
-----------------------------------------------

A satellite dynamical friction class which uses the :cite:t:`chandrasekhar_dynamical_1943` formula to compute the acceleration of a satellite at radius :math:`r` from the center of the host due to dynamical friction:

.. math::

   \mathbf{a}_{DF} = -\frac{4\pi \mathrm{G}^2M_\mathrm{sat}\rho_\mathrm{host}(r)}{v_\mathrm{sat}^3}\ln
   \Lambda\left[\mathrm{erf}(x)-\frac{2x}{\sqrt{\pi}}\exp(-x^2)\right]\mathbf{v}_\mathrm{sat},

where :math:`M_\mathrm{sat}` and :math:`\mathbf{v}_\mathrm{sat}` are the satellite's mass and velocity, respectively, :math:`v_\mathrm{sat}=|\mathbf{v}_\mathrm{sat}|`, :math:`\rho_\mathrm{host}(r)` is the host's density profile, :math:`\ln\Lambda=`\ ``[logarithmCoulomb]`` is the Coulomb logarithm, and :math:`x\equiv v_\mathrm{sat}/\sqrt{2}\sigma(r)`, where :math:`\sigma(r)` is the velocity dispersion of the host halo at radius :math:`r`, approximated to be equal to the host virial velocity, :math:`v_\mathrm{vir}`.

**(Default implementation)**

**Methods**

* ``coulombLogarithm`` — Compute the Coulomb logarithm, :math:`\log \Lambda`, appearing in the :cite:t:`chandrasekhar_dynamical_1943` dynamical friction equation.

**Parameters**

* ``[logarithmCoulomb]`` (default ``2.0d0``) — The Coulomb logarithm, :math:`\ln \Lambda`, appearing in the :cite:t:`chandrasekhar_dynamical_1943` formulation of the acceleration due to dynamical friction.

.. _physics-satelliteDynamicalFrictionKaur2018:

``satelliteDynamicalFrictionKaur2018``
--------------------------------------

A satellite dynamical friction class which applies the core-stalling model of :cite:t:`kaur_stalling_2018` to another dynamical friction class. Specifically, the acceleration due to dynamical friction is given by:

.. math::

   \mathbf{a}_\mathrm{df} = \mathbf{a}^\prime_\mathrm{df} [S_\mathrm{trail}(r/r_*)+S_\mathrm{lead}(r/r_*)],

where :math:`\mathbf{a}^\prime_\mathrm{df}` is the acceleration provided by the child dynamical friction class, and :math:`S_\mathrm{trail}(x)` and :math:`S_\mathrm{lead}(x)` are the torque suppression factors defined by :cite:t:`kaur_stalling_2018`. The functional forms for :math:`S(x)` are taken from Figure 8 of :cite:t:`kaur_stalling_2018`---extrapolation in :math:`\log(S)` vs. :math:`x` is used to extend :math:`S(x)` to lower :math:`x` than shown in that figure, while for values of :math:`x` higher than that shown in that figure :math:`S(x)` is held constant at the maximum value shown.

The stalling radius, :math:`r_*`, is computed using equation (55) of :cite:t:`kaur_stalling_2018`. For dark matter halo profiles with a central cusp (for which that equation has no solution), the acceleration provided by the child dynamical friction class is returned unmodified.

.. _physics-satelliteDynamicalFrictionMassRatioThreshold:

``satelliteDynamicalFrictionMassRatioThreshold``
------------------------------------------------

A satellite dynamical friction class that wraps another dynamical friction implementation and suppresses the deceleration to zero whenever the satellite-to-host halo mass ratio falls below a threshold value. This prevents unphysically large dynamical friction forces on very low-mass satellites. The mass ratio threshold is set by ``[massRatioThreshold]``.

**Parameters**

* ``[massRatioThreshold]`` — The satellite-to-host mass ratio below which dynamical friction acceleration is truncated to zero.

.. _physics-satelliteDynamicalFrictionPetts2015:

``satelliteDynamicalFrictionPetts2015``
---------------------------------------

A satellite dynamical friction class which computes the Coulomb logarithm following the model of :cite:t:`petts_semi-analytic_2015`. The minimum impact parameter is taken to be :math:`b_\mathrm{min} = \max(r_{1/2,\mathrm{sat}}, G M_\mathrm{sat} / v_\mathrm{orbital}^2)`, where :math:`r_{1/2,\mathrm{sat}}` is the half-mass radius of the dark matter component of the satellite, and the maximum is :math:`b_\mathrm{max} = r_\mathrm{orbital}/|\gamma|` for :math:`|\gamma| > 1` (where :math:`\gamma` is the logarithmic density slope of the host at the satellite position), or :math:`r_\mathrm{orbital}` otherwise. The ``[logarithmCoulombApproximate]`` parameter controls whether the Coulomb logarithm is evaluated as :math:`\ln\Lambda` (for :math:`\Lambda \geq 1`, else zero) or the approximate form :math:`\frac{1}{2}\ln(1+\Lambda^2)`.

**Parameters**

* ``[logarithmCoulombApproximate]`` (default ``.false.``) — If true, the Coulomb logarithm term is evaluated as :math:`\frac{1}{2}\log(1+\Lambda^2)`. Otherwise, it is evaluated as :math:`\log\Lambda` for :math:`\Lambda \ge 1` and set to zero otherwise.

.. _physics-satelliteDynamicalFrictionZero:

``satelliteDynamicalFrictionZero``
----------------------------------

A satellite dynamical friction class in which the acceleration is always zero.