Version 12 (modified by 9 years ago) ( diff ) | ,
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Diffusion time estimate
This is an estimate for how long it should take radiation to diffuse through gas. The longer this timescale, the more confined the radiation is — and thus, the more it can act as a source of heating.
Starting from:
We get:
L is the size of our system, c is speed of light, kappa_R is the rosseland specific mean opacity, rho is density of the system, and lamba is a dimensionless parameter that seems to do with the length scale of gradients in radiative energy.
Offner et al '09 gives,
for T=10 K gas. Assuming our system is a protostellar core, we have L~.1 pc, rho~1 solar mass/L3. In cgs these parameters work out to be:
c | 3e+10 |
6.5e-20 | |
L | ½*3.08e+17 |
.23 |
(For L, the radiation is leaving from the center of the volume, so is going approximately 1 half the length). I am not completely sure on
, but from Offner's paper it should have units of,
where R has units of:
and so I gather an estimate for lambda might be:
which using our values gives:
Using all of these values in the formula above for the diffusion time gives,
or ~118 years.
I am not sure on this because lambda is not well constrained, and you can get very different estimates based on what you choose lambda to be (i.e tdiff = 4 hours when lambda =1, tdiff=87 days when lambda = .002, etc).
Compare this time to the 'free streaming limit':
or 57 days.
Coupling time estimate
This timescale provides an estimate for how long it should take the to 'heat up' (or cool down) from radiation (right?). This is controlled by the planck opacity,
. For low , the matter and radiation are relatively decoupled. Adjusting the degree of coupling is akin to adjusting the specific heat capacity of the material (right?).Starting from,
we have:
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