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Angular Momentum and Sink Particles

Looked over Federrath Outflow Model and noticed significant issues with how we handle angular momentum in sink particles.

Both of us update the mass, momentum, and center of mass the same

M'=M+\sum{\Delta m_i}

M'V'=MV+\sum{\Delta m_i v_i}

M'R' = M'R + \sum{\Delta m_i \left ( r_i - R \right )}=MR + \sum{\Delta m_i r_i }

But our treatment of angular momentum has glossed over some subtleties

S'=S+\sum{\Delta m_i \left ( r_i -R \right ) \times v_i}

A better treatment is to conserve total angular momentum.

J=L+S

where

L=MR \times V

and S is the spin.

J'=J+\sum{\Delta j_i} = J +\sum{\Delta m_i r_i \times v_i}

S'=J'-L'

S'=S + MR \times V - M'R' \times V' + \sum{\Delta m_i r_i \times v_i}

S' = S+MR \times V - \left( MR + \sum{\Delta m_i r_i} \right ) \times \frac{MV + \sum{\Delta m_i v_i}}{M+\sum{m_i}}+\sum{\Delta m_i r_i \times v_i}

which if we approximate to first order in \frac{\sum{\Delta m_i}}{M}

S' \approx S+MR \times V - \left( MR + \sum{\Delta m_i r_i} \right ) \times \left ( V + \frac{\sum{\Delta m_i v_i}}{M} \right ) \left (1 -\frac{\sum{\Delta m_i}}{M} \right )+\sum{\Delta m_i r_i \times v_i}

S' \approx S+MR \times V - MR \times V - \sum{\Delta m_i r_i } \times V - R \times \sum{\Delta m_i v_i} + R \times \sum{\Delta m_i} V+\sum{\Delta m_i r_i \times v_i}

S' \approx S +\sum{\Delta m_i \left (r_i - R \right ) \times \left (v_i - V \right )}

astrobear

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Angular Momentum and Sink Particles

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