Rotational Deformation

The rotational deformation force can be enabled with the Use Rotational Deformation parameter in the param.dat file.

Parameters for the tidal forces:

  • Star fluid Love Number

  • Star spin_x

  • Star spin_y

  • Star spin_z

  • Star Ic

and the initial conditions:

  • k2f

  • Sx

  • Sy

  • Sz

  • Ic

When a viscous body is rotating, then its shape is transformed into a symmetric oblate ellipsoid. The potential energy of a system containing two oblate bodies 0 and 1 is give in [Moy71], Equation 158, [Moy03] or [CorreiaLaskarFaragoBoue11] as

\[U = -\frac{Gm_0 m_1}{r} \left[ 1 - \sum_{i=0,1} \sum_{n=1}^{\infty} J_n \left( \frac{R_i}{r} \right)^n Pn(\hat{r} \cdot \hat{\Omega_i}) \right],\]

with the mass \(m\), the physical radius \(R\), the distance between the bodies \(r\), the rotational angular velocity \(\Omega\) and the Legendre polynomial of degree \(n\), \(Pn(x)\).

Further, we truncate the order \(n\) to two and introduce the \(J_2\) parameter [CorreiaLaskarFaragoBoue11], [BolmontRaymondLeconte+15]

\[J_{2,i} = k_{2f,i} \frac{\Omega^2_i R^3_i}{3G m_i}\]

and

\[J_{2,\star} = k_{2f,\star} \frac{\Omega^2_\star R^3_\star}{3G m_\star},\]

with \(k_{2f,i}\) the second potential Love number (fluid Love number).

We follow the description in [BolmontRaymondLeconte+15] and define the following quantities (In [BolmontRaymondLeconte+15], \(C_\star\) and \(C_i\) are reversed):

\[C_{\star} = \frac{1}{2} G m_i m_\star J_{2,\star} R^2_\star\]

and

\[C_{i} = \frac{1}{2} G m_i m_\star J_{2,i} R^2_i.\]

Then the force due to the rotational deformation is given as [BolmontRaymondLeconte+15]:

\[\begin{split}\mathbf{F_R} = \left\{ - \frac{3}{r^5_i} \left( C_\star + C_i\right) \right. \\ \nonumber \left. + \frac{15}{r^7_i} \left[ C_\star \frac{(\mathbf{r}_i \cdot \mathbf{\Omega}_\star)^2}{\mathbf{\Omega}_\star^2} + C_i \frac{(\mathbf{r}_i \cdot \mathbf{\Omega}_i)^2}{\mathbf{\Omega}_i^2} \right] \right\} \mathbf{r}_i \\ \nonumber - \frac{6}{r^5_i} \left( C_\star \frac{\mathbf{r}_i \cdot \mathbf{\Omega}_\star}{\mathbf{\Omega}_\star^2} \mathbf{\Omega}_\star + C_i \frac{\mathbf{r}_i \cdot \mathbf{\Omega}_i}{\mathbf{\Omega}_i^2} \mathbf{\Omega}_i\right).\end{split}\]

Rotational deformation spin evolution

Additionally to the acceleration on the particles, the rotational deformation force generates a torque, which changes the spin of the particles and of the central star.

This rotational deformation torque is given as

\[\mathbf{N}_R = \mathbf{r} \times \mathbf{F}_{R\theta},\]

with the transverse component of the tidal force \(\mathbf{F_{r\theta}}\).

It can be written following [BolmontRaymondLeconte+15] as

\[\mathbf{N}_{R\star} = - \frac{6}{r^5_i} C_\star \frac{\mathbf{r}_i \cdot \mathbf{\Omega}_\star}{\mathbf{\Omega}_\star^2} \left( \mathbf{r}_i \times \mathbf{\Omega}_\star \right)\]

and

\[\mathbf{N}_{Ri} = - \frac{6}{r^5_i} C_i \frac{\mathbf{r}_i \cdot \mathbf{\Omega}_i}{\mathbf{\Omega}_i^2} \left( \mathbf{r}_i \times \mathbf{\Omega}_i \right).\]

By using heliocentric coordinates, the spin evolution is takes the form [BolmontRaymondLeconte+15]

\[\frac{d}{dt} (I_\star \mathbf{\Omega}_\star) = - \sum_{j=1}^N \frac{m_\star}{m_\star + m_j} \mathbf{N}_{R\star}\]

and

\[\frac{d}{dt} (I_i \mathbf{\Omega}_i) = - \frac{m_\star}{m_\star + m_i} \mathbf{N}_{Ri},\]

with the moment of inertia \(I\).