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 J2 parameter in the param.dat file to any value different than 0.

Parameters for the J2 forces:

  • J2

  • J2 radius (mean radius of mass distribution, \(R_{mean}\))

The potential energy of an additional gravitational multipole expansion is given in [ZdericMadigan20], Equation 1, as

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

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

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

The \(J_2\) value corresponds to [ZdericMadigan20]

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

The implementation of the force is done similarly as the Rotational Deformation force.