This tutorial is meant to complement the first lecture of the course *High Energy Astrophysics*. Here you can practice your knowledge of the topic after completing Lecture 1.

Sometimes you will find some collapsable/expandable paragraphs marked in blue (questions) or orange (quizzes). These are quick tests to check your knowledge as you follow the tutorial. Click on the paragraph to read the answer.

# Radiative Transfer Theory Basics

## Radiative Transfer Approximation

Electromagnetic radiation is described by the Maxwell equations. Energy is transported by a transverse wave in the electric and magnetic fields. The Maxwell equations need to be solved in order to find how these waves interact and propagate. When photons propagate through macroscopic media, the solutions of the Maxwell equations become too complex to solve for all practical purposes. There are two simplifications that can be done without losing generality:

- The first is that of
*geometric optic*s, with photons approximated as light rays. Here the scale of a system greatly exceeds the wavelength of radiation : . The wave nature of photons is considered only to the extent that it introduces light bending on surfaces and changing of refraction index. - If the index of refraction is everywhere (or at least it can be safely approximated as
everywhere), then we talk about*radiative transfer*, which represents a further simplification with respect to geometric optics.

Specific Intensity Definition

**Exercise **1

Let’s consider a pinhole camera, which is nothing but a small hole in a box, able to generate an image without using a lens. Let’s assume that the hole is circular with a diameter *L* so that the rays have to cross a distance

What is the meaning of *“specific*” in the specific intensity and flux?

*Specific *means that the quantity is calculated per unity frequency. That’s why you see

In this exercise, we want to calculate the flux

where

**Solution 1**

Before reading the full solution, consider the following hints.

Hint 1

Start with the general relation between specific flux and specific intensity and consider that, given that

Hint 2

Remember the definition of solid angle:

Solution

The relation between specific flux and intensity is

The solid angle is

The pinhole area is then

### Exercise 2

A thin disk of material is emitting radiation with specific intensity

### Solution 2

As usual, let’s start with some hints.

Hint 1

Start by using the usual expression relating the specific flux and specific intensity:

Hint 2

Consider the angle

Hint 3

Remember the expression for the solid angle

Solution

Let’s start with

## The Constancy of Intensity in Free Space

If the rays propagate in free space, then the (specific) intensity does not change with distance. This might sound counter-intuitive at first. Indeed we know very well that the inverse square law tells us that, in free space, the flux of an object decreases as the square of the distance. So why do we say that the surface intensity remains constant? The keyword here is *surface*, meaning that we are observing an extended object and not a point source. Remember that the apparent size of an extended object, i.e., its solid angle, decreases with the square of the distance so that you squeeze more and more surface elements into a given solid angle. So the “number” of surface elements emitting light in the field of view increases as the square of the distance, whereas the flux we receive decreases by the square of the distance. The two effects cancel each other out and we have a constant surface intensity.

In the image on the left, the spheres represent an extended object emitting light at distances

Surface Intensity and the Moon

It might be worth insisting on this point a bit further with an even simpler example. Imagine walking in the countryside and looking at a row of nearly identical trees in perspective, like in the figure below.

Suppose you are close to the first tree, about 5 meters away, whereas the last tree you see is at about 500 meters. Since the flux scales as

This happens because all trees are roughly illuminated in the same way, so they all have the same surface brightness. Their flux decreases as

Look at the picture in the panel below. The trees here are illuminated by a flashlight. Their intensity clearly decreases with distance. Why?

Solution

The reason is that the illumination does not come from a source at infinity – that would provide a uniform illumination – but from a flashlight. Therefore the amount of light reaching each tree scales as

### Exercise 3

The specific intensity (or surface brightness) of an object remains constant in free space. Give now an expression that describes how the specific intensity changes when there is some absorbing material along the line of sight (to be clear: you need to write the radiative transfer equation when only absorption is present and solve the equation by giving an expression for

(NOTE: you can also choose to write and solve the transport equation in terms of the optical depth

Do you remember the definition of the absorption coefficient?

Think in microscopic terms…

The absorption coefficient is a consequence of the interaction with microscopic particles with number density

Now imagine having a box with absorbers. In the image below the cross-section of each absorber is represented by a small red circle. Select the correct picture below by answering the question.

### Solution 3

Let’s start by writing the equation of transport for the absorption-only case. This means we will set the emission coefficient

What are the dimensions of the absorption coefficient?

The differential expression for the equation of transport is therefore

## Source Function

In the previous exercise, we mentioned that you can use the optical depth in place of the absorption coefficient

Can you write the transfer equation as a function of the optical depth?

Hint 1

Remember to add the emission coefficient

Solution

The ratio between emission coefficient and absorption coefficient is called *source function*

The source function is not only a useful quantity to rewrite the equation of radiative transport in a simpler way, but it has also a profound physical meaning. The source function, having the dimensions of a specific intensity, can be thought of as the local input of radiation. For example when the optical depth

The Meaning of the Source Function

Let’s illustrate the meaning of the source function with an astrophysical example. An astrophysical source has intrinsic specific brightness

Hint 1

Use the equation of transport with the approximation of a constant source function

Solution

When

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