.. _J2:

J2: additional gravitation multipole expansion
==============================================

An additional gravitational multipole expansion term can be added to the simulation with the J2 force.

The J2 force can be enabled by setting the :literal:`J2` parameter in the :ref:`param.dat<ParamFile>` file to any value different than 0.


Parameters for the J2 forces:

 - :literal:`J2`
 - :literal:`J2 radius` (mean radius of mass distribution, :math:`R_{mean}`)



The potential energy of an additional gravitational multipole expansion is given in :cite:p:`ZdericMadigan2020`, Equation 1, as 

.. math::

    U_i = -\frac{Gm_{\star} m_i}{r} \left[ 1 - J_2 \left( \frac{R_{\text{mean}}}{r} \right)^2 P_2(\hat{r} \cdot \hat{\Omega}_{\text{system}}) \right],

with the mass of the central star :math:`m_{\star}`, the mass of a particle i :math:`m_i`, the mean radius of the mass distribution 
:math:`R_{\text{mean}}`, the distance between the body and the star :math:`r`, the rotational angular velocity of the N-body system
:math:`\Omega_{\text{system}}` and the Legendre polynomial of degree :math:`2`, :math:`P_2(x) = \frac{1}{2} (3x^2 - 1)`. 

The value of :math:`\Omega_{\text{system}}` is assumed to be (0,0,1), corresponding to be spin aligned to the N-body system.

The :math:`J_2` value corresponds to :cite:p:`ZdericMadigan2020`

.. math::

    J_{2} = \frac{1}{2.0 m_{\star} R_{\text{mean}}^2} \sum_{i=0}^n \left(m_i a_i^2 \right)


The implementation of the force is done similarly as the :ref:`RotationalDeformation` force.