Outflows

Particles can launch outflows following the description in ​Federrath et al 2014

The basic problem is to apply the following outflow function, but in a discrete way that is numerically symmetric and exact.

The change in rho has the following functional form d \rho=\rho_0 \mathcal{R}(r) \Theta(\theta)

as does the change in momentum d\mathbf{p}=d \rho \mathbf{\mathcal{V}}(\theta)

Since we want the total mass to equal some fraction of the accreted mass, as well as the top and bottom mass injection to be symmetric, we must have

\displaystyle{\sum_{top}{d \rho_i}}=\displaystyle{\sum_{bottom}{d \rho_ i}} = \frac{f \dot{M}}{2}

And since we also must have the total momentum be balanced, we need to have a magnitude constraint

\displaystyle{\sum_{top}{|dp^j_i|}}+\displaystyle{\sum_{bottom}{|dp^j_i|}} = f \dot{M}\mathcal{V}_0

as well as the symmetry constraint

\displaystyle{\sum_{top}{dp^j_i}}+\displaystyle{\sum_{bottom}{dp^j_i}} = \mathbf{0}

and directional constraint

\displaystyle{\sum_{top}{\mathbf{dp}_i}} \times \mathbf{S} = \mathbf{0}

It is easier to solve the system if the magnitude constraint takes the form of

\displaystyle{\left | \sum_{top}{\mathbf{dp_i}}\right | }+\displaystyle{\left | \sum_{bottom}{\mathbf{dp}_i} \right | } = \gamma f \dot{M}\mathcal{V}_0

where \gamma is the analytic solution for the fraction of the scalar momentum in the z direction.

If we introduce scaling parameters for the density and momentum for the top and bottom…

d \rho_\pm=\alpha_{\pm} d \rho

d p^j_\pm=\beta^j_\pm d p^j

and plug these into the above equations, we can solve for \alpha_\pm and \beta_\pm

\alpha_\pm = \frac{f \dot{M}}{2\displaystyle{\sum_\pm{d \rho}}}

The momentum equations give us 4 constraints for two variables.

For reference on another way astrobear has implemented sinks in the past, see SinkParticle