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When calculating injection from outflows, we generally have some prescription for the mass, momentum, and possibly energy injection as a function of position within the outflow kernel. However, we would like the numerical injection to satisfy certain requirements.
- The total mass injected should equal
- The center of mass of the injected mass should equal
- The total momentum injected should be (bipolar)
- The scalar momentum injected should equal
- The angular momentum injected should equal
First, let's consider a simpler example. Let's assume we want to inject a certain amount of mass
around a point (in 2D) with a kernel given by
where our normalized kernel
for .
Here
is the distance from a sink particle.For a grid with 10 points in x and y over the interval x=[-1,1] y=[-1,1] - and for a sink particle at [.05,.05] the result is
Note the center of mass of the injected material is not .05,.05 and the injected mass is not 1.0. This is because of discretization error.
We would like to find a solution for
that is close to this, but subject to the constraints involving the center of mass and the total mass injected.If we consider each zone inside the kernel, we can write the constraints as the following matrix equation
This can be expressed as
and we can solve this underconstrained problem using the pseudo-inverse of AWhile this solves our constraints - it looks nothing like our desired profile. What we would like to do is to solve
, in a way that does not minimize , but that minimizes where is our vector of initial given by our kernel.We can do this using Matlab's lsqlin function - or we can use the prescription given in https://see.stanford.edu/materials/lsoeldsee263/08-min-norm.pdf (page 8-13 through 8-15)
This gives
Now for the more complicated case, we have terms in involving momentum, and we could simultaneously solve for
and , though it is probably more important to have be close to the desired values. Errors in and could lead to unphysical velocities. We cannot solve for and at the same time, since there are non-linear terms that involve the product of and .However, if we separate out the solution for
using the constraints on total mass and mass moment, we can treat those as constants when solving for . This then gives us the matrix equationwhere
Taking into account particle motion and orientation
If we are explicit about our constraints, we have
Change in mass of particle
Change in mass moment of particle
Change in momentum of particle (due to loss of mass)
Change in internal angular momentum of particle
Amount of outward momentum in jet
which can be reduced to
An alternative approach is to try to match the amount of outward momentum in the jet for each direction independently.
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