Version 8 (modified by 9 years ago) ( diff ) | ,
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To implement angular momentum injection in our outflows, following Federrath 2014, we first define the following quantities:
these represent the total *accreted* mass (m_acc), center of mass (COM_acc), momentum (p_acc) and angular momentum (L_acc) from all the cells within the accretion volume surrounding the sink particle.
We then use these quantities to update the particle's mass, COM, and so on. To maintain conservation between grid zones and particle, this is done following:
where primes (') denote post-accretion quantities (i.e. m'-m represents the change in particle mass after an accretion event).
Now, angular momentum does not have the same form as above, namely:
which is somewhat unexpected. This is because the position of the sink can change ever so slightly over the course of the accretion step, and thus, the angular momentum can change slightly as well (to maintain conservation of the com).. Thus, we instead write the angular momentum, post-accretion, as:
That is, the cross product of the particle's post-accretion com and velocity. Given this may be a slight over (or under) shoot of the updated angular momentum, due to the fact that the particle may have lost (or gained) some angular momentum over the accretion time step, we create a new vector to absorb this difference, the 'spin' vector (S). We want the total angular momentum (L+S) to maintain conservation over the accretion time step. That is, we want:
and thus, we can define the spin vector using:
Thus, we can say the conserved accreted angular momentum per time step is:
Next, we would like to launch a fraction of the accreted angular momentum back along the spin axis of the particle, into the jets. To do this, we want to select only those components that are parallel to the jet, so we have the following prescription for the angular momentum for the outflow:
where f_a is the fraction of accreted angular momentum that should be launched in the jet.
Notes: calling it 'spin' might be slightly confusing.