Scharzschild equation provides a tool to easily find and analyze the solution of the radiative transfer cases for some simple source functions. First of all, the lambda operator is clearly linear, so\begin{equation}
\Lambda[a\,f(t) + b\,g(t)] = a\,\Lambda[f(t)] + b\,\Lambda[g(t)].
\end{equation}
Sunday, 2 June 2013
Saturday, 1 June 2013
Formal solution and Schwarzschild equation
The radiation transport equation (RTE) for time-independent plane-parallel atmosphere has well known form (e.g. Rutten, 2003):
\begin{equation}
\mu\,\frac{\mathrm{d}I_{\nu}}{\mathrm{d} \tau_{\nu}} = I_{\nu} - S_{\nu},
\end{equation}
where $I_{\nu}$ is specific intensity, $S_{\nu}$ source function, $\tau_{\nu}$ is the radial optical depth and $\mu$ is the cosine of the angle between the line of sight and vertical axis. Mathematically, this is a linear first order differential equation with constant coefficients. Physically, it describes propagation of radiation through an interacting medium. Properties of the medium are hidden in the source function and the optical depth. However, to make the first step toward the solution of RTE, we will assume that the optical depth is independent variable and that the source function is a know function of it.
\begin{equation}
\mu\,\frac{\mathrm{d}I_{\nu}}{\mathrm{d} \tau_{\nu}} = I_{\nu} - S_{\nu},
\end{equation}
where $I_{\nu}$ is specific intensity, $S_{\nu}$ source function, $\tau_{\nu}$ is the radial optical depth and $\mu$ is the cosine of the angle between the line of sight and vertical axis. Mathematically, this is a linear first order differential equation with constant coefficients. Physically, it describes propagation of radiation through an interacting medium. Properties of the medium are hidden in the source function and the optical depth. However, to make the first step toward the solution of RTE, we will assume that the optical depth is independent variable and that the source function is a know function of it.
Subscribe to:
Posts (Atom)