Template:ASTR508/Assignment11

Assume the following:

• The Rosseland mean opacity is related to the density and temperature of the gas through a power-law relationship,

${\kappa }_R={\kappa }_0{\rho }^{\alpha }{T}^{\beta };$

• The pressure of the gas is given by the ideal gas law;
• The gas is in hydrostatic equilibrium so $p=g\mathrm{\Sigma }$ where $g$ is the surface gravity; and
• The gas is in radiative equilibrium with the radiation field so the flux is constant with respect to $z$ or $\mathrm{\Sigma }$.

Calculate the temperature of the gas as a function of $\mathrm{\Sigma }$ for a general opacity law.

Now use the specific opacity law

${\kappa }_R=3.68\times 10^{22}{g}_{ff}(1-Z)(1+X)\frac{\rho }{1{\mathrm{g}\mathrm{c}\mathrm{m}}^3}{\left(\frac{T}{1\mathrm{K}}\right)}^{-7/2}{\mathrm{c}\mathrm{m}}^2{\mathrm{g}}^{-1}$

and take the value of the gravitational acceleration and effective temperature appropriate for the Sun to get an estimate for the run of temperature with column density for the Sun.

Now let's try to estimate the continuum spectrum under the following assumptions

• The opacity is a power-law function of wavelength: ${\kappa }_{\nu }\propto {\nu }^{-\delta }$.
• The emission from the atmosphere is thermal.
• The radiation is coupled to the atmosphere at optical depths ${\tau }_{\nu }$ greater than unity and completely decoupled at shallower depths where $d{\tau }_{\nu }={\kappa }_{\nu }d\mathrm{\Sigma }/(\cos \theta )$.

What does the spectrum ($I_{\nu }$) look like for $\delta =0$ as a function of $\theta$? What about $\delta =2$ and $\delta =3$?