The stars of a galaxy

Counting the stars of a galaxy is very hard. Look in fact at the following figure.  
   
On the left you can see a group of stars. On the right the group of stars has been moved away to a distance twice as large. The apparent distances between the stars have then become half large, but the size of each star did not change. The apparent dimension of each star depends in fact on the telescope and the quality of the night, not on its distance from us.  

If we keep on moving the stars away, at a certain point their apparent images begin to overlap, until the object appears as a confused stain of light. This effect is larger when the number of stars of the system is larger, as you can see now.  

Suppose, for example, that you take a box and you fill it with stars. The following figures show you this experiment and the scheme explains how to interpret them. The left figure shows the side of the box, the right figure shows its front face.  

     

Let us keep the density of stars fixed, and let us just change the dimensions of the box. Let us start with a small box and let us fill it with stars. Looking at it front side you have the effect of projection of the stars onto the sky. 
     
Here you can see that, doubling the length of the box, you also increase the density of stars in the sky. Looking at a galaxy these two effects are both present: there are many stars "behind" a given area of the sky, and their images overlap because of their distance and the scattering of their light due to the Earth atmosphere. In the case of the space telescopes the effect of the atmosphere is removed, but the enlargement caused by the optics remains, even if it is much smaller. In space, the possibility to resolve the stars of a galaxy can be extended to farther objects, but in any case it is limited.  

The galaxies are so far, and they contain so many stars, that their images gather in very small areas of the sky, to the extent that all together they appear as diffuse nebulae. With ground based instruments, it is possible to resolve just the stars that are found in low density regions of nearby galaxies, for example their outer parts; or dwarf galaxies can be studied, since they contain relatively few stars.  

So how to estimate the number of stars contained in a galaxy? One could for example compute the total luminosity of the galaxy and divide it by the typical luminosity of one star. The total luminosity of a galaxy is found with a procedure identical to that followed for the stars. First of all, one finds how much light we receive from it (the apparent luminosity) and then rescales it according to the inverse square law. In the case of the spiral galaxies, you saw that their distance can be found by means of the Cepheids. With this procedure, it is found that the Andromeda galaxy has a total luminosity which is some dozen billion times that of the Sun (just considering the visible radiation). If it were composed by stars all equal to the Sun, then it would contain some 10 billion stars. The true number depends on how the stars are distributed in luminosity. For example, a galaxy 100 times brighter than the Sun (of course, it does not exist! ) can be made of 20 stars as luminous as the Sun, plus 160 stars as luminous as 1/2 the Sun. But it could also contain 150 stars whose luminosity is 1/3 solar, plus 200 stars whose luminosity is 1/4 solar. In the first case it would contain 180 stars, in the second one 350, that is a number almost two times the former.  

Actually most of the stars of any system are less luminous than the Sun, so the estimate that we made is a lower limit. A realistic estimate could be some hundred billions of stars. This figure is representative for a spiral galaxy, but as you know, different types of galaxies exist. Moreover, within the same class brighter and fainter objects are comprised. Realistic limits go from some million stars for the dwarf galaxies up to the giant elliptical galaxies with a number of stars 100 times higher than that of M31.  

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