Predictions for a radiating 'first hydrostatic core' sink particle

Given the following parameters:

l = h = r (box radius)	.0069 (pc) = 2.14e+16 (cm)
\rho	2.03*10-16 (g cm-3)
n	1.2e+8 (cm-3)
T	10 (K)
\lambda	.3
\kappa_R	.23 (cm2 g-1)
\kappa_P	.4 (cm2 g-1)

We computed the various time-scales as:

t_{diff}	24 days
t_{coupling}	?
t_{sc}	33 kyr
t_{ff}	4 ky
t_{sim}	24 days

Here is the initial condition:

Prediction: By 24 days, expect the diffusion wave to hit the boundary. Depending on the energy injection rate compared to the diffusion rate, the shape of the curve might be different. For instance, if energy injection rate (Er) >> diffusion rate (Dr), might expect the gaussian to be increasing in height as well as width. If Er = Dr, might expect the profile to be flat, as it grows in width. If Er << Dr, expect a gaussian that grows in width over time, but not height.

Check: Make time curves of Erad(x), and check that the wave hits the boundary by t=24 days.

Prediction: The total thermal energy (and thus the total energy) should increase like t². This is because,

\frac{\partial e}{\partial t} = -\kappa_P \rho (4 \pi B - cE), where

cE = 4 \pi B + Lt (L is luminosity of FHSC)

\frac{\partial e}{\partial t} = \kappa_P \rho Lt

e \propto \kappa_P \rho L t^2 (density doesn't change given t_diff << t_sc)

This holds as long as 4 pi B << the energy injected each time step. That way we can effectively ignore changes to thermal energy induced by increasing B(T).

Check: Make curve of e_total(t), E_total(t).

Prediction: The total radiative energy should increase as:

\frac{dE_{rad}}{dt} = L

E_{rad}= Lt

Check: Make curve of Erad_total(t).

Prediction: The thermal energy, e, of a zone immediately adjacent to the kernel should increase linearly. This is because, the difference between the energy injected into the grid, E*, and 4piB, is some height X (where X = L*dt-B). So, right next to kernel, expect the coupling time to go like:

\int de=\kappa_P \rho X \int dt

Or,

\triangle e=\kappa_P \rho (L*dt-B) \triangle t

*This assumes the energy injection rate is << diffusion rate, so that we know the difference in height between Erad and B*.

Check: Make a time query of a cell near the kernel.

Prediction: Initially,

\frac{E_{rad}}{E_{gas}}=\frac{aT^4}{nKT}=4e-4

So the radiation will not induce appreciable changes in the temperature.

However, given the luminosity is,

L=3.9~e+29 ~erg~s^{-1},

I expect a swift increase in the radiative energy of the core such that, Erad >> Egas. More precisely, given the diffusion time from sink to boundary is 24 days, we can get a 'diffusion speed' and use it to calculate the time for the diffusion wave to hit any part of the core. At that point, the Erad >> Egas, and thus, we can expect significant heating of the gas (so long as the coupling time is short).

Consider this is the case for small pre-stellar cores, and perhaps why radiative feedback is so important. With high mass star formation it will inevitably take longer to heat the gas.

Here are my initial energy density budgets:

Erad	aT4=7.565e-11 erg/cm3
Egas	nKT=1.656e-7 erg/cm3

Here is the total initial energy budgets:

Erad	aT4=(7.565e-11 erg/cm3)*(7.7e+49)=6e+39 erg
Egas	nKT=1.656e-7 erg/cm3=1e+43 erg

And the total energy radiated by the star in a diffusion time:

Estar

L*tdiff= 8e+35 erg