4.5: Energy Transport by Conduction
- Page ID
- 141615
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)a. Mean Free Path
Consider a simple monatomic gas where the kinetic energy per particle is 3kT/2 so that the speed is \[v=\left(\frac{3 k T}{m}\right)^{1 / 2}\label{4.4.1}\]
We will let the collisional cross section be just the geometric cross section, so that \[\sigma=\pi\left(r_1+r_2\right)^2\label{4.4.2}\]
where \(\mathrm{r_1}\) and \(\mathrm{r_2}\) are the radii of the two species of colliding particles. As we did with nuclear reaction rates, we get the collision frequency from the effective volume swept per unit time \(\sigma\mathrm{v}\) multiplied by the number density p/m. The time between collisions is just the reciprocal of the collision frequency, so that the distance traveled between collisions is \[l=v\left(\frac{\sigma v \rho}{m}\right)^{-1}=\frac{m}{\rho \sigma}\label{4.4.3}\]
and is known as the mean free path for collisions.
b. Heat Flow
The thermodynamic theory of heat says that the heat flux through a given area is proportional to the temperature gradient so that \[F_{\mathrm{cond}}=-\mathrm{K} \nabla T\label{4.4.4}\]
where Eddington1 gives the conductivity K as \[\mathrm{K}=\frac{C_V \rho l v}{3}=\frac{C_V(k T m / 3)^{1 / 2}}{\sigma}\label{4.4.5}\]
If we compare the maximum luminosity obtainable with the conductive flux to the total solar luminosity, we have \[\frac{L_{\text {cond }}}{L_{\odot}}=\frac{4 \pi R_{\odot}^2 F_{\text {cond }}}{L_{\odot}} \approx 10^{-12}\label{4.4.6}\]
With the gradient estimated as \(\mathrm{T_c/R_\odot}\), using the central temperature to make the conductivity as large as possible and taking the geometric cross section to be about 10-20 cm2, we still fail to carry the solar luminosity by at least 5 orders of magnitude. Thus, conduction can play no significant role in the energy transport in the sun. Indeed, that is true for all normal stars. However, in white dwarf stars, where the electrons are degenerate, the mean free path of the electrons is comparable to the dimensions of the star itself. Then conduction becomes so important that the internal temperature distribution is essentially isothermal.
If we combine equations \ref{4.2.14} and \ref{4.2.1} we can write the radiative flux \[F_{\mathrm{rad}}=\left(\frac{4 a c}{3 \bar{\kappa} \rho}\right) \nabla T\left(T^3\right)\label{4.4.7}\]
which has the same form as equation \ref{4.4.4}. Thus we may define a conductive opacity from the conductivity so that \[\kappa_{\mathrm{cond}}=\frac{4 a c T^3}{3 \mathrm{~K} \rho}\label{4.4.8}\]
Then, if necessary, the conductive and radiative fluxes can be combined by augmenting the mean radiative opacity, so that \[\frac{1}{\bar{\kappa}}=\frac{1}{\bar{\kappa}_{\mathrm{rad}}}+\frac{1}{\kappa_{\mathrm{cond}}}\label{4.4.9}\]