This section recapitulates some of the basic concepts and equations of radiation theory. Further details
can be found in any standard introductory textbook on astrophysics. One such classic text is
Unsöld’s *The New Cosmos*[74]. However, there are numerous suitable textbooks. Assume
some radiation passing through a surface and consider an element of the surface of area
$dA$
(Figure 1). Some of the radiation will leave the surface element within a beam of solid angle
$d\omega $ at an
angle $\theta $
to the surface. The amount of energy entering the solid angle within a frequency range
$\left[\nu ,\nu +d\nu \right]$ in a
time $dt$
will be:

$$d{E}_{\nu}={I}_{\nu}cos\theta \phantom{\rule{2.43306pt}{0ex}}dA\phantom{\rule{0.3em}{0ex}}d\nu \phantom{\rule{0.3em}{0ex}}d\omega \phantom{\rule{0.3em}{0ex}}dt$$ | (1) |

where ${I}_{\nu}$ is the **specific intensity**
of radiation at the frequency $\nu $
in the direction of the solid angle, with dimensions of
$W\phantom{\rule{1em}{0ex}}{m}^{-2}\phantom{\rule{1em}{0ex}}H{z}^{-1}\phantom{\rule{1em}{0ex}}s{r}^{-1}$.

The intensity including *all* possible frequencies, the **total intensity**
$I$, can be
obtained by integrating over all frequencies:

$$I={\int}_{0}^{\infty}{I}_{\nu}\phantom{\rule{0.3em}{0ex}}d\nu $$ | (2) |

From an observational point of view we are generally more interested in the energy flux or **flux**
(${L}_{\nu},L$) and the **flux density**
(${F}_{\nu},F$)^{1}.
Flux density gives the power of the radiation per unit area and hence has dimensions of
$W\phantom{\rule{1em}{0ex}}{m}^{-2}\phantom{\rule{1em}{0ex}}H{z}^{-1}$ or
$W\phantom{\rule{1em}{0ex}}{m}^{-2}$.
Observed flux densities are usually extremely small and therefore (especially in radio
astronomy) flux densities are often expressed in units of the **Jansky** (Jy), where 1 Jy
$=1{0}^{-26}\phantom{\rule{1em}{0ex}}W\phantom{\rule{1em}{0ex}}{m}^{-2}\phantom{\rule{1em}{0ex}}H{z}^{-1}$.

If we consider a star as the source of radiation, then the flux emitted by the star into a solid angle
$\omega $ is
$L=\omega {r}^{2}\phantom{\rule{1em}{0ex}}F$, where
$F$ is the flux density
observed at a distance $r$
from the star (it is also usual to refer to the total flux from a star as the **luminosity**,
$L$).
If the star radiates isotropically then radiation at a distance
$r$ will be distributed evenly on
a spherical surface of area $4\pi {r}^{2}$
and hence we get the relationship:

$$L=4\pi {r}^{2}\phantom{\rule{1em}{0ex}}F$$ | (3) |

The situation is slightly more complicated for an extended luminous object such as a nebula or galaxy.
The **surface brightness** is defined as the flux density per unit solid angle. The geometry of the
situation results in the interesting fact that the observed surface brightness is *independent* of the
distance of the observer from the extended source. This slightly counter-intuitive phenomenon can be
understood by realising that although the flux density arriving from a *unit* area is inversely
proportional to the distance to the observer, the area on the surface of the source enclosed by a unit
solid angle at the observer is *directly* proportional to the square of the distance. Thus the two effects
cancel each other out.

^{1}You should be aware - and beware - that different authors define the terms flux density, flux and intensity differently,
and they are sometimes used interchangeably!

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