wiki:u/erica/radtimescales

Version 33 (modified by Erica Kaminski, 9 years ago) ( diff )

Diffusion time estimate for a 1 solar mass core, of r=.05 pc

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 lambda is the dimensionless 'flux limiter' that we will look at below.

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).

Let's now take a closer look at the flux limiter.

A closer look at the 'Flux Limiter'

is the 'flux limiter'. It comes into the equations in the diffusion term:

This is because instead of solving the following conservation equation,

(which would require an additional equation for F), we make the 'flux-limited' approximation that,

This then turns the conservation equation into a diffusion equation:

The value of controls whether the radiation is diffusing in the free-streaming limit (), i.e. at the speed of light, or is diffusing as it would in the optically thick limit (). The FLD approximation does well at these two limits, but not in between. Here is the functional form of ,

where,

Note that (see bottom of page for more detail), the mean free path is given by:

(i.e. given the opacity and the density of the material, one can easily compute the mean free path), and,

where h is the scale height. Thus, we can interpret R as the ratio of the mean free path (l) to the 'radiation scale height (h)' :

(note there is no frequency dependence now in the mean free path, as integrated over frequency space to give an 'average' opacity).

Graphically, we have:

So we see that for small , , and for large , . Let's examine these two limits closer.

Optically thick limit

From the graph above, we see that as , . Thus, we have:

That is, the mean free path is << scale height for the radiation, i.e. we're in the optically thick regime.

The rad diffusion equation in this limit becomes,

which is consistent with equation 6.59 in Drake's book describing optically thick, non-equilibrium radiation transfer.

Free streaming limit

In the other limit, , we have:

That is, the mean free path >> scale height, and we are in the free-streaming limit.

In this case, the rad diffusion equation becomes:

That is, the radiation diffuses instantly through the grid. Recall, how this radiative energy couples to the gas is given by the coupling term in the radiation equation, not shown here.

Estimating the flux limiter for a 1 solar mass core, with r=.05 pc

Estimating starts with approximating R.

Note that,

so if we make the approximation,

(by dimensional arguments and assuming the gradient is negative), we have:

which integrates to:

or,

From this equation, it is clear that h is the scale-height. Thus, we have,

By setting h to be the distance between the sink and box radius (h=r), we imagine it as the scale height for the radiation.

Now, using our parameters from above, the mean free path is:

Given the radius of the box is r=.05 pc, we have,

This moves us into the 'free-streaming' regime on the curve, and thus we can approximate . Thus, we have:

Back to the diffusion time estimate

Using all of these values in the formula above for the diffusion time gives,

Compare this time to the light crossing time:

Given these timescales, the diffusion wave probably won't be strongly peaked, but will be broad — traveling close to the speed of light. If my 'assumptions' were correct about a pre-stellar core (their average size and mass), then given their 'average density' and the rosseland mean opacity, we see that they are in the optically thin limit for diffusion. Given cores are not optically thin, this means that either the density assumption is way off (they should be centrally concentrated), or setting the scale height = box radius is off (perhaps for an optically thick core the scale height should be greater than the core size, so that the energy is roughly constant throughout the core).

Tackling this slightly differently could be starting from the optically thick limit, . Then, we should be able to predict the timescale over which we would see a well-defined 'diffusion wave' propagate through the grid.

Diffusion estimate for optically thick, 1 solar mass core

If we let,

(so that ), we have

From before,

and

This last equation is equivalent to setting the box radius = mean free path = scale height. Solving these equations equations gives:

So a (uniform density) core that contains a solar mass, has to be .0069 ~pc (or roughly 1,000 AU) to be nearly optically thick. That is an interesting result.

Recalculating the sims parameters, and diffusion time, gives:

.3
l = h = r (box radius) .0069 (pc) = 2.14e+16 (cm)
2.03*10-16 (g cm-3)
.23 (cm2 g-1)

Note this can be rewritten as,

The first factor is the light-crossing time:

The second factor is the ratio of the box radius to the mean free path. We set this to = 1 in our derivation above:

And the last factor is inverse lambda. Again, we constrained this in our derivation:

Taken together, in the optically thick limit, when r=l:

So for a simulation with these parameters, expect a diffusion wave to cross the grid in 24 days:

Coupling time estimate

This timescale provides an estimate for how long it should take the gas to 'heat up' (or cool down) from radiation. 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.

Starting from,

it is helpful to rewrite by expanding E out:

This shows that the total radiative energy (due to blackbody plus any sources/sinks) can change when there is a mismatch between the total radiative energy and the energy being radiated from a blackbody. Because I am writing the source as , I am ignoring any sinks of radiation (i.e. diffusion), and instead am only considering the source as coming from the protostar.

This then becomes,

(dropping the negative sign because I am not interested in which way the energy is flowing).

If I assume the radiation output from the protostar (its accretion energy) is constant over time, I can kill the difference of E* on the LHS. This leaves,

Plugging in for B(T) puts this in terms of temperature:

So the coupling time depends on the temperature difference you want to achieve, as well as the planck opacity, density, and radiation output from the protostar. Note, I may have made dropped a factor of 4pi/c in the case it isn't absorbed by 'a'.

Sound crossing time

The time it takes a sound wave to travel from the center of the prestellar core, to the outer edge is:

where the sound speed for 10 K molecular gas is ~.2 km/s and r=.0069 pc. This gives:

How long it takes the gas density to changes in response to changes in thermal energy corresponds to the crossing time, for some appropriate length scale.

Freefall time

Material in the core will fall onto the star if it is Jeans unstable (or it can fragment). An estimate for the time it will take to 'collapse' onto the star is given by the freefall time:

where rho is the mean density of the gas, calculated above. This gives,

That this value is very nearly the sound crossing time shows that a pre-stellar core is marginally jeans unstable.

Summary of timescales

A box that has r=.0069, filled with a solar mass worth of material at constant density and temperature (=10 K), and with the above quoted opacities, should have the following timescales of evolution:

24 days
?
33 kyr
4 ky

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