PX282 - C12 - temperature structure

P_{r a d} = \frac{4 π}{3 c} ⟨ I ⟩

this concept of $P_{r a d}$ can be linked to the flux, $F$ , to relate $T$ with $z$ and $τ$
considering a small block of material with flux coming out of a star
the block has a thickness. $d z$
$P_{r a d} (z)$ and $P_{r a d} (z + d z)$ on the surfaces
incoming flux: $F$ , outgoing flux: $F - d F$
over a distance, $z$ :
- $F$ decreases
- $P_{r a d}$ decreases
the change in $P_{r a d}$ can be related to the change in $F$
for the photon: $E = p c$ and $F = \frac{d E}{d t}$

\frac{d p}{d t} = \frac{d E}{c} = F o r c e

the force, ie: the difference in $P_{r a d}$ , is: $F o r c e = \frac{d P_{r a d}}{d z} d z$ :

\frac{d P_{r a d}}{d z} d z = \frac{1}{c} d F

\frac{d F}{F} = - ρ κ d z

where, $\frac{d I}{d z} = - ρ κ I$ , and $F = \int I \cos θ d Ω$

\frac{d P_{r a d}}{d z} d z = - \frac{1}{c} F ρ κ d z

\frac{d P_{r a d}}{d τ} = \frac{F}{c}

P_{r a d} = \frac{1}{c} F τ + A

- where, $A$ is the constant of integration

\frac{4 π}{3 c} ⟨ I ⟩ = \frac{1}{c} F τ + A

solving for $T$ structure near the surface
using boundary conditions:
- at the top of the atmosphere: $τ = 0$ , $⟨ I ⟩ = \frac{I_{o u t}}{2}$ (no $I$ heading inwards)

\frac{2 π I_{o u t}}{3 c} = A = \frac{2 F}{3 c}

∴ \frac{4 π}{3 c} ⟨ I ⟩ = \frac{F τ}{c} + \frac{2 F}{3 c}

\frac{4 π}{3} ⟨ I ⟩ = F (τ + \frac{2}{3})

near the surface (photosphere):
$F = σ T_{e f f}^{4}$
$⟨ I ⟩ = S = B = \frac{σ T^{4}}{π}$

\frac{4 π}{3} B = σ T_{e f f}^{4} (τ + \frac{2}{3})

∴ T^{4} = \frac{3}{4} T_{e f f}^{4} (τ + \frac{2}{3})

key points:
- $T = T_{e f f}$ when $τ = \frac{2}{3}$
- the 'surface' that is seen is at $τ \approx \frac{2}{3}$

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