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Review: Conserving angular momentum over accretion

Enforcing mass and linear momentum conservation means that angular momentum will generally not be conserved over the accretion step. We can see this by first noting that (by mass conservation) the sum of the mass to be accreted from the kernel (LHS) is equal to the mass of the sink particle after the accretion takes place (RHS; post-accretion quantities are denoted by a prime):

\Sigma m_i = M'_{sink}$

Similarly, momentum conservation says that:

\Sigma m_i v_{i} = P'_{sink} = M'_{sink} v'_{sink}$

Note, this last equation gives the velocity of the center of mass of the accreted material, which must equal the velocity of the sink particle, given momentum conservation.

From here, we might be tempted to write:

\Sigma m_i r_i \times v_i = L'_{sink}

However, we already know that the mass of the sink and velocity of the sink are M'_{sink} and V'_{sink}, respectively. Thus we have to write:

\Sigma m_i r_i \times v_i = M'_{sink} R' \times v'_{sink}$

and ask for what R' is this equation valid. Rearranging this equation (and calling the LHS L_{acc}), we can write:

\frac{L_{acc}}{M'_{sink}}= R' \times v'_{sink}$

In general, L_{acc} and v'_{sink} will not be perpendicular, thus there will be no solution for R'.

What if we instead let R' equal the center of mass of the accreted material? Then, the equation for angular momentum conservation becomes:

\Sigma m_i r_i \times v_i = \Sigma m_i r_i \times v_{sink}

Equality then requires that v_i = v_{sink}, which also will not generally be true.

Thus, the sink angular momentum cannot strictly be set equal to the accreted angular momentum, if the angular momentum is to be conserved across the accretion step. Instead, we need an additional vector which can absorb the difference in the angular momentum. This is the reason behind devising a 'spin' angular momentum vector for the sink particle. With the spin vector, the total angular momentum across the accretion step can be conserved.

Using a spin angular momentum vector to conserve angular momentum

As discussed above, the sink angular momentum following accretion (L'_{sink}) is set by the accreted mass and linear momentum from the kernel. It is given by:

L'_{sink} = M'_{sink} R' \times v'_{sink}

As shown above, this angular momentum vector will generally differ from the total angular momentum accreted from the kernel (L_{acc}):

L_{acc} = \Sigma m_i r_i \times v_i \neq L'_{sink}

Thus, we can't simply set the particle's spin axis equal to L_{acc}. Instead, we want to look for some function s such that the total angular momentum across accretion is conserved. We call this s vector the particle's spin. The spin can be found by enforcing conservation of the total angular momentum across the accretion step:

L'_{sink} + S'_{sink} = L_{sink} + S_{sink} + L_{acc}

This equation shows that the updated total angular momentum of the sink = the old total angular momentum + the accreted angular momentum. By rearranging this equation, we can solve for s', which absorbs any excess angular momentum over L_{acc}:

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Review: Conserving angular momentum over accretion

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