Choosing the accretion luminosity for radiative feedback

This plot shows 'distance from the star' on the x-axis, and 'ke®/(GM/R)' on the y-axis, for different starting kinetic energies, ke(r). Ke(r) for the different curves is in terms of the 'freefall energy' at r, where freefall energy = GM/r. Thus, this plot *does not* show how the energy changes as the material falls in, it rather asks the question — if you start at r (x-axis), with energy ke(r) (different curves), what fraction of GM/R will your accretion energy be (y-axis). Note, the horizontal line shows GM/R, normalized to 1.

Let's start by considering the pink line, where a particle starts from rest a distance r away. That is,

From this, we see that even at small distances compared to a typical zone size (~1/100,000 pc), the 'error' is ~.001%. That means, unless your zone size is smaller than this size scale, wouldn't expect any noticeable deviation.

The other lines show if the particle didn't start from rest, but rather some fraction of the freefall speed it would have acquired falling to r from infinity. This shows that for any gas parcels falling onto the sink within the accretion volume, going any speed from 0 to freefall produces near agreement between the equations (again, until you get to extremely small scales).

What about for gas in the accretion volume going faster than freefall? First of all, this material should not be bound so it wouldn't be accreted anyway. But assuming it would be, here is a plot now showing curves corresponding to ke(r)>GM/r.

Here, GM/R is the x-axis. Again, this estimate holds to good accuracy over extremely small distances from the sink particle (in terms of typical simulation resolution).

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