Vocabulary

Colours. These images are taken through broad-band filters, typically^1.1 V, Band R, and so each individual exposure might as well be in black and white, the relative blackness being a measure of how much, or how little, of the light in that wavelength band fell on that part of the photograph/CCD. Images taken trough different filters are then combined to produce a colour image, scaling the different bands trying to mimic the colour your eye would see.

Sometimes though, one of the filters used is a narrow-band filter, for example H$\alpha$ or O[III]. These filters block all light which is not in a narrow range around the hydrogen H$\alpha$ transition, or a transition in doubly ionised oxygen. Combining such a narrow-band image with images made with others filters produces a false colour image, i.e. the colour of the object is not how your eye would see it, but the colour coding has been chosen to bring out a particular feature - for example H$\alpha$ emission. The really pretty colour pictures of Planetary Nebulae are false colour images.

Luminosity, flux and surface brightness

A star emits a certain amount of energy per unit time. For example the Sun has a luminosity $L=1L\odot=3.4\times 10^{26}$W. This quantity is not observable however: what we can measure is how much of this energy we receive, per unit time, per unit surface area (assuming you put the surface perpendicular to the radiation!), at a given distance from the source. The observable quantity, energy received per unit time per unit of surface area, is called flux. Clearly the flux will depend both on the luminosity of the star, and on its distance. How?

Consider a sphere of radius $r$, with the star at the centre. Since the star will distribute all of its energy equally over the surface of the sphere, the flux $F=L/(4\pi r^2)$. (As an exercise, find the unit of flux).

It is the flux of the star that determines how bright it appears to be. Indeed, star A may be brighter than star B, either because it is more luminous, $L_A\gg L_B$, or because it is nearer to us, $r_A\ll r_B$.

Although $F$ is in principle observable, in practise astronomers express brightness in terms of magnitudes $m$, where the apparent magnitude $m=-2.5\log_{10} (F)+C$, where $C$ is some constant that depends on the magnitude system used. Since the $F$ depends on distance, so does $m$, it is not an intrinsic quantity of the star - hence `apparent magnitude'. The absolute magnitude, $M$, is the apparent magnitude when $r=10$pc.

Since a galaxy contains many stars, we can also talk about its luminosity, and the flux we receive from it. However, because galaxies are spatially extended^1.2, we can also try to measure what fraction of the flux comes from the centre of the galaxy and what fraction comes from the outer parts, say. This gives rise to the term surface brightness.

Consider a small patch of galaxy, namely that part contained in a (small) solid angle $d\Omega$. Observationally we can measure the flux, $dF$, of light, coming from the part of the galaxy contained within $d\Omega$. The quantity $dF/d\Omega$ is called surface brightness (SB), and so it has dimensions energy/time/area/steradian.^1.3

The surface brightness of a galaxy does not depend on its distance. To see this, assume that the patch of galaxy contained within $d\Omega$ contains $N$ stars, with mean luminosity $L$, when it is at a distance $r$. The surface brightness SB= $N\,L/(4\pi r^2)/d\Omega$. Now increase the distance to the galaxy by a factor of 2. The flux from each individual star will decrease by a factor $2^2$, since $F\propto 1/r^2$. However the number $N$ of stars within $d\Omega$ will increase by a factor $2^2$, since the physical size of the patch enclosed by $d\Omega$ will double when $r$ doubles. Hence SB is distance independent^1.4, therefore whereas with brightness there was an intrinsic quantity (luminosity), and an observable one (flux), there is only one surface brightness.

A galaxy's SB depends on its distribution of stars. Assume a face-on galaxy at distance $r$ has a surface density $\Sigma$ of stars (in stars/pc$^2$, say), all with the same luminosity $L$. The surface area of galaxy $dS$, contained within a solid angle $d\Omega$, is $dS=d\Omega\,r^2$. Therefore the number, $dN$, of stars within $d\Omega$ is $dN=\Sigma\,dS=\Sigma\,d\Omega\,r^2$. The flux received from a single star is $L/(4\pi r^2$, and therefore the flux received from all $N$ stars within $d\Omega$ is $dF=(L/(4\pi r^2))\times (\Sigma\,d\Omega\,r^2)\,=\Sigma\,L\,d\Omega/(4\pi)$. The surface brightness is the flux per unit solid angle, is $df/d\Omega=\Sigma\,L/(4\pi)$ is distance independent, as we saw before.