To completely describe the radiative transfer problem, we must take emission into account as well as absorption. The source function \(S\) is defined so that \(S \mathrm{d} \tau\) is the increase in intensity due to emission in passing through a region of optical depth \(\mathrm{d} \tau .\) This means that the radiative transfer equation should be written $$\mathrm{d} I | \mathrm{d} \tau=-I+S$$ (a) If \(S\) is a constant, solve for \(I\) vs. \(\tau\), assuming an intensity \(I_{0}\) enters the material. (b) Discuss your result in the limits \(\tau \ll 1\) and \(\tau \gg 1\).

In radiative transfer, two key processes govern how light or radiation propagates through a medium: emission and absorption.
Emission refers to the process where the medium itself generates radiation, increasing the overall intensity of the radiation field. In the exercise, this is represented by the source function, denoted as S. The term \(\text{S d} \tau\) signifies the increase in radiation intensity due to emission over a small optical depth \(\text{d} \tau\).
Absorption, on the other hand, refers to the reduction in radiation intensity as it passes through the medium. The medium absorbs part of the radiation, causing a decline in its intensity.
Understanding the balance between these two processes is crucial when solving the radiative transfer equation.

Optical depth, symbolized as \(\tau\), is a measure of how opaque a medium is to radiation passing through it.
It's a dimensionless quantity that represents the amount of absorption and scattering that radiation undergoes as it travels through a material.
The higher the optical depth, the less transparent the medium is, and thus, the more radiation is absorbed or scattered.
In the radiative transfer equation, \(\text{d} I / \text{d} \tau = -I + S\), \(\tau\) indicates the cumulative effect of absorption that radiation has encountered up to a specific point in the medium.
Optical depth influences solutions for the intensity I, illustrating how radiation is modified as it travels through the medium.

The radiative transfer equation is a first-order linear differential equation.
Such equations take the general form: \[ \frac{\text{d} y}{\text{d} x} + p(x)y = q(x) \]
where \(y\) is the function we are solving for, and \(p(x)\) and \(q(x)\) are given functions.
For the radiative transfer equation, we write \[ \frac{\text{d} I}{\text{d} \tau} + I = S \]
Here, I is the intensity we want to find, and S is constant, simplifying our functions. Solving such an equation involves techniques like finding an integrating factor.

To solve the first-order linear differential equation, we use an integrating factor. This method transforms the equation into one where both sides can be integrated directly.
The integrating factor \( \text{is found as} e^{\tau}\).
Multiplying through by this factor converts the equation:
\[ e^{\tau} \frac{\text{d} I}{\text{d} \tau} + e^{\tau} I = S e^{\tau} \]
Recognizing that the left side is a derivative of a product allows us to write
\[ \frac{\text{d}}{\text{d} \tau} \big(I e^{\tau}\big) = S e^{\tau}\big. \]
Integrating both sides with respect to \( \tau\) simplifies to:
\[ I e^{\tau} = S e^{\tau} + C \]
Solving for I, we determine how intensity changes with optical depth, considering emission and absorption.