The Supermassive Black Hole at the Center of Our Galaxy: 
how we know it's there and what its mass is


the observations




the theory:  Kepler's 3rd law

    1) Use the data above to estimate/determine (for as many stars as pssible)

             a) the period (in years) of the star's orbit about the galactic center (GC);
                    the GC is marked with a figure of a star

             b) the size of the orbit's apparent semimajor axis in angular units (seconds of arc)
                    (the semimajor axis is one-half of the longest axis of the elliptical orbit)

             c) the size of the orbit's apparent semimajor axis in linear distance units (astronomical units)

                   In order to accomplish this, you will need to the small angle approximation (or trigonometry)
                   in addition to knowing the distance of the Milky Way's center from Earth (26,000 lilght years)
  
    2) Once you have collected ordered pairs of data (period & semimajor axis) for several stars' orbits,
                use Kepler's third law to determine the mass of the black hole that these stars are orbiting:

                                 (abh  +  astar)3 =   (Mbh + Mstar) P2  
            
                which approximates to (because the mass of the black hole is so much greater than the mass of a star)

                                 (astar)3   =  Mbh P2   


                A plot of the square of the period vs. the cube of the semimajor axis will therefore have a slope of  Mbh
                where the black hole is in solar units.


data collected from the observations


star
 major axis of the orbit,
angular size (")
 semimajor axis of the orbit,
linear size (au) *
 orbital period (yr)
SO-1
 0.65
 2600
 39
SO-2
 0.16
 640
 13
SO-4
 0.82
 3300
 70
SO-19
 0.41
 1600
 18
SO-20
 0.47
 1900
 29


* assuming a distance from galactic center to Earth = 8000 pc


conversion from angular size to linear size


from either right-triangle trigonometry or the small angle approximation

    w(radians)  =  2a(au)/d(au)

where  w(radians)  =  angular size of the major axis in radians

          2a (au)  =  length of the major axis in au

             d (au)  =  distance from galactic center to earth in au

But 1 radian  =  206,265 seconds of arc   and  1 pc  =  206,265 au.  Therefore,

          w (arcsec)  =   2a(au)/d(pc)

where  w(arcsec)  =  angular size of the major axis in seconds of arc

             d (pc)  =  distance from galactic center to earth in parsecs

Therefore,  a(au)  =  w(arcsec) * d(pc)/2



a graph of  astar3  (on the vertical axis)   vs.  P2  (on the horizontal axis) would then be expected to be linear with a slope of     (the mass of the black hole) and an intercept of zero

graphical fit to the data: finding the black hole mass






The slope of the graph is 7.2 x 106 solar masses, which is the mass interior to the star's orbits.  (Note that the smallest star orbit has a semimajor axis of only 640 au, roughly 15 times that of Pluto's orbit around the sun.)  Because the mass is concentrated in such a small volune, it is presumed to exist in the form of a black hole at the center of the galaxy.