Changes between Version 12 and Version 13 of u/erica/radtimescales
- Timestamp:
- 03/30/16 14:23:19 (9 years ago)
Legend:
- Unmodified
- Added
- Removed
- Modified
-
u/erica/radtimescales
v12 v13 62 62 [[latex($\frac{dE}{dt} = {\kappa_P\rho}(4\pi B - cE)$)]] 63 63 64 we have: 64 it is helpful to rewrite by expanding E out: 65 66 [[latex($\frac{d (\frac{4 \pi}{c} B+ E_*)}{dt} = {\kappa_P\rho}(4\pi B - c(\frac{4\pi}{c}B+E_*)$)]] 65 67 66 68 69 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 [[latex($E_*$)]], I am ignoring any sinks of radiation (i.e. diffusion), and instead am only considering the source as coming from the protostar. 67 70 71 This then becomes, 68 72 73 [[latex($ \frac{\triangle (\frac{4 \pi}{c} B+ E_*)}{\triangle t} = {\kappa_P\rho}cE_*$)]] 69 74 75 (dropping the negative sign because I am not interested in which way the energy is flowing). 70 76 77 If I assume the radiation output from the protostar (its accretion energy) is constant over time, I can kill the difference of E* on the LHS. This leaves, 78 79 [[latex($ t = \frac{\triangle B(T)}{\kappa_P \rho c E_*}$)]] 80 81 Plugging in for B(T) puts this in terms of temperature: 82 83 [[latex($ t = \frac{a(T_2^4-T_1^4)}{\kappa_P \rho c E_*}$)]] 84 85 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.