.. _physics-blackHoleBinaryRecoil:

Black Hole Binaries Recoil
==========================

Class providing models of the gravitational wave recoil velocity (in km s\ :math:`^{-1}`) imparted to the remnant black hole when two black holes of given masses and spins merge. Asymmetric gravitational wave emission during coalescence gives the merged black hole a "kick" that can eject it from the nucleus or even from its host galaxy. The recoil velocity depends sensitively on the mass ratio and the spin magnitudes and orientations of the merging black holes.

**Default implementation:** ``blackHoleBinaryRecoilZero``

Methods
-------

``velocity`` → ``double precision``
   Computes the recoil velocity of the given pair of merging black holes.

   * ``class(nodeComponentBlackHole), intent(inout) :: blackHole1, blackHole2``

.. _physics-blackHoleBinaryRecoilCampanelli2007:

``blackHoleBinaryRecoilCampanelli2007``
---------------------------------------

A black hole binary recoil class that computes the recoil velocity during a black hole binary merger due to the emission of gravitational waves, following the formulae derived in :cite:t:`campanelli_large_2007`

.. math::

   V_\mathrm{recoil}=V_\mathrm{m}\mathbf{\hat{e_1}}+V_\perp(\cos\xi\mathbf{\hat{e_1}}+\sin\xi\mathbf{\hat{e_2}})+V_\parallel\mathbf{\hat{e_2}}

with:

.. math::

   V_\mathrm{m}=A\frac{q^2(1-q)}{(1+q)^5}(1+B\frac{q}{(1+q)^2})

.. math::

   V_\perp=H\frac{q^2}{(1+q)^5}(\alpha^\parallel_2-q\alpha^\parallel_1)

.. math::

   V_\parallel=K\cos(\theta-\theta_0)\frac{q^2}{(1+q)^5}(\alpha^\perp_2-q\alpha^\perp_1)

where :math:`\theta` is defined as the angle between the inplane :term:`component` of :math:`\Delta` and the infall direction at merger. :math:`q` is the mass ratio of the black holes as :math:`q=M_{\bullet,1}/M_{\bullet,2}` and :math:`\alpha_i=\mathbf{S_i}/M_{\bullet,i}` depends of the spin and mass of the black hole :math:`\xi` measures the angle between the unequal mass and the spin contribution to the recoil velocity in the orbital plane. :math:`\mathbf{\hat{e_1}} , \mathbf{\hat{e_2}}` are orthogonal unit vectors in the orbital plane. Our method assumes the spin of the second black hole is randomly generated, while that of the first is aligned with the angular momentum of the system. The constants used are retrieved from the articles by: :cite:t:`koppitz_recoil_2007` for :math:`H=(7.3\pm 0.3)10^3` km/s, :cite:t:`gonzalez_maximum_2007` for :math:`A=1.2 \times 10^4` km/s :math:`B=-0.93`, :cite:t:`gonzalez_supermassive_2007` for :math:`K\cos(\delta\theta)=(6,-5.3)10^4` km/s and :math:`K=(6.0\pm 0.1)10^4` km/s.

.. _physics-blackHoleBinaryRecoilZero:

``blackHoleBinaryRecoilZero``
-----------------------------

A null implementation of the black hole binary recoil velocity class that always returns zero recoil velocity. Useful for isolating other merger dynamics or for models where gravitational wave recoil is negligible.

**(Default implementation)**