The mass of the galaxies 

You can have a first estimate of the mass of a galaxy by knowing approximately how many stars it contains. For example you have seen that M31 contains about 100 billion stars. If they were all like the Sun, the mass of M31 would be 100 billion times that of the Sun. But even for the mass, you have the problem already met when trying to count the stars. That is, you have to know the fraction of stars that falls in each mass interval. Of 100 stars of a galaxy, 50 stars could be more massive than the Sun and 50 less massive. But also 10 could be more massive and 90 less massive. For the two cases, the total mass is obviously different.  

If, in order to compute the total luminosity, we adopted some distribution in luminosity of the stars, we can use this information to estimate also the distribution in mass. In any case the result is rather uncertain. Moreover, you have seen that a galaxy does not contain only stars, but also gas and dust. It is not therefore enough to estimate how much mass is present in stars.

Both for the case of the planets and that of the stars, you have seen that the best method for estimating their masses is to exploit their gravitational effect. When two objects orbit each other their mass can be computed on the basis of Kepler's laws. To this aim, the distance of the system and the shape of the orbits must be known.  

The same principle can be used in order to estimate the mass of a spiral galaxy. In the case when one of the two bodies is much more massive than the other, Kepler's third law states that  
a3 = k * M * P2  
where a is the mean distance between the two bodies, M is the mass of the larger body, P the orbital period and k a constant. If the orbits are circular, the period is computed as  
P = 2 * pi * a / v  
where v is the speed on the orbit. We can then calculate the mass of a spiral galaxy when we know how fast a star moves, at a given distance form the center of the galaxy. Actually we will find the amount of mass contained inside the orbit of the star. In order to have the best results we should take the farthest observable stars from the center.  

In order to help the calculation, we can make a comparison with the orbit of the Earth around the Sun. Combining the two previous expressions we then find  
M/MSun = (a/aEarth) * (v/vEarth)2  
We then need the speed of a star and its distance from the center of any galaxy. But we can then use the Sun inside the Milky Way! The Sun moves at about 250 km/s at a distance of about 8 kpc from the center of our Galaxy. 8 kpc means 8,000 parsec, and you know that 1 parsec=206265 Astronomical Units. The Astronomical Unit is just the mean Earth-Sun distance. The Earth moves at about 30 km/s on its orbit. With these data we can thus write:  
M/MSun = ( 8000 * 206265 / 1 ) * ( 250 / 30 )2  
One immediately sees that this is a very large number. In fact one finds 114.6 billion. Between the Sun and the center of the Galaxy a mass of about 100 billion solar masses is contained. If you want to convert to kg, take into account that the mass of the Sun is 2*1030kg (that is 2,000 billion of billion of billion kg) and multiply the two numbers!  

According to Kepler's third law the orbital speeds must decrease when moving away from the gravitational center. One therefore expects that the stars farther from the center of a spiral galaxy will rotate slower and slower when going outwards. But it happens that the measured speeds remain constant out to where any galaxy's stars are visible.  

Therefore there is matter even beyond the limit where the star light ends. Hence this is dark matter. There are several hypotheses on the nature of this invisible matter (for example that it is composed by sub-stellar bodies) but at the moment this subject is still much debated.  

In order to have a better estimate of the mass of a galaxy, a model is needed, which takes into account the different components and that is able to predict the observed motions of stars and gas. With these calculations it is found that the spiral galaxies can reach some thousand billion solar masses.  

The smaller dwarf galaxies "just" contain some million solar masses, while the giant elliptical galaxies can reach 100 times the mass that we estimated.  

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