Stars are commonly treated as if they were in thermodynamic equilibrium but the whole discussion of energy transport in stars indicates that this cannot generally be a correct assumption. Real stars radiate net energy into space and have temperature gradients and thus are not blackbodies. However, if the characteristic distance over which the temperature changes is large compared with the mean free paths of particles and photons, the idea of local thermodynamic equilibrium (LTE) may still be retained: particles and photons cannot quickly escape from their local "box" and within that local box an approximately constant temperature exists. It is common (particularly in discussions of stellar atmospheres) to define a temperature scale height $H_{\mathrm{T}}$ that is analogous to the pressure scale height introduced in Eq. (7.44),
$$
H_{\mathrm{T}}=\frac{T}{|d T / d r|}
$$
and to use this scale as the characteristic one for measuring how rapidly the temperature is changing. Show that in the deep interior of stars the LTE assumption is valid for both matter and photons, but that in the photosphere LTE is typically expected for matter but is problematic for photons. (Note: this result notwithstanding, the assumption of LTE is often used for stellar atmospheres. Notice also that this result is an inevitable consequence of the definition of the photosphere, since the photosphere is precisely where photon mean free paths must become long.) You may use the Standard Solar Model (see Table 10.1) for representative conditions in the solar interior, and you may use that for a typical atmospheric model of the Sun (for example, see Table 14.9 of Ref. [75]) the temperature changes from $5980 \mathrm{~K}$ to $6180 \mathrm{~K}$ over a distance of $20 \mathrm{~km}$, in a
surface region having density $2.5 \times 10^{-7} \mathrm{gcm}^{-3}$ and opacity $\sim 0.25$ $\mathrm{cm}^{-2} \mathrm{~g}^{-1}$ at the dominant photon wavelengths.