Changes between Version 12 and Version 13 of u/erica/radtimescales


Ignore:
Timestamp:
03/30/16 14:23:19 (9 years ago)
Author:
Erica Kaminski
Comment:

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  • u/erica/radtimescales

    v12 v13  
    6262[[latex($\frac{dE}{dt} = {\kappa_P\rho}(4\pi B - cE)$)]]
    6363
    64 we have:
     64it is helpful to rewrite by expanding E out:
     65
     66[[latex($\frac{d (\frac{4 \pi}{c} B+ E_*)}{dt} = {\kappa_P\rho}(4\pi B - c(\frac{4\pi}{c}B+E_*)$)]]
    6567
    6668
     69This 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 [[latex($E_*$)]], I am ignoring any sinks of radiation (i.e. diffusion), and instead am only considering the source as coming from the protostar.
    6770
     71This then becomes,
    6872
     73[[latex($ \frac{\triangle (\frac{4 \pi}{c} B+ E_*)}{\triangle t} = {\kappa_P\rho}cE_*$)]]
    6974
     75(dropping the negative sign because I am not interested in which way the energy is flowing).
    7076
     77If I assume the radiation output from the protostar (its accretion energy) is constant over time, I can kill the difference of E* on the LHS. This leaves,
     78
     79[[latex($ t = \frac{\triangle B(T)}{\kappa_P \rho c E_*}$)]]
     80
     81Plugging in for B(T) puts this in terms of temperature:
     82
     83[[latex($ t = \frac{a(T_2^4-T_1^4)}{\kappa_P \rho c E_*}$)]]
     84
     85So 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.