Mass of the Galaxy's Blackhole

Astronomers have been studying the motion of the stars in the region of Sgr A*, a very bright source of radio waves located in the center of our galaxy. Optical invesitigation in this region(wave lengths around 5x10-7m) is not possible because light is strongly absorbed in the dusty environment of the galactic center. However, at infrared wavelengths ( about 2x10-6m) it is possible to see through the dust and investigate the motion of the stars. Recall that the infrared part of the spectrum was discover more than 200 years ago by William Herschel.
A more detailed description is found here.

Using the Doppler shift it is possible to determine the velocities along the line of sight for stars moving near Sgr A*. In addition, the surveying technique of parallax is refined enough now that one can also determine the positions of the stars. Two articles(ajp620_744_2005_black_hole_SgrA.pdf,  blackhole_arXiv_0810.4674v1.Jan31_2009.pdf),  describing more than twenty years of studying stellar motions near Sgr A* have come to the conclusion that these stellar motions are possible only if a massive object exists near the galactic center of mass 4x106 solar masses! This massive object emits no electromagnetic radiation. It is not the direct source of the radio emissions. Astronmers have concluded that these stellar orbits are conclusive proof of the existence of a massive black hole(MBH) near the galactic center. The stars in the vicinity of the MBH are labelled in the diagram below. These orbits correspond to the celestial coordinates of declination and right ascension ( Crowe's 2nd book ). The star S2 has a short orbital period of about 15 years! This means that during the observation time of the astronomers they have seen a complete orbit for S2. Compare this to Saturn's period of about 29 years.

The Galactic Center is about 8.3kpc ( 26,000 light years) away from the solar system.

Orbits of stars near the Massive Black Hole in the Galaxy Center: source = arXiv_0810.4674v1, Jan. 31, 2009


Table of Orbital Parameters for the stars near the massive black hole (MBH)




Orbit of S2 fitted to a Keplerian elliptical orbit, an example of Kepler's 3rd Law to obtain the mass of the black hole
semi-major axis = 919 AU = 919x150x109 m = 1.378x1014 m

period = 14.53yr=14.53*365.25d/yr*24hr/day*3600sec/hr = 4.58x108 s

Kepler's 3rd Law: T2/a3 = 4*(pi)2/G/M  and from this we can calculate the mass
of the black hole:
G = 6.67x10-11 Nm2/kg2
M = 4*(pi)2*a3/G/T2  = 7.3x1036 kg

solar mass = 1.99x1030 kg
Using only S2 to get the mass they obtained in 2005
(black hole mass)/(solar mass) = 3.7x106  (from all their 2005 data M = 3.7x106 solar mass)
From their 2009 data they obtain M = 4.3x106 solar masses.

distance of closest approach = a(1 - eccentricity) = 919AU(1-0.867) = 126 AU for S2                                                                                                     
for the star S16 the distance of closest approach = 1680AU(1-0.974) = 43 AU
compare this to the solar system size, for Pluto a = 39 AU.
The speed of S16 at the distance of closest approach is 11,700 km/s! Compare to the earth's orbital speed of 30 km/s.

The event horizon for a black hole is at the Schwarzchild radius. No object closer than this distance can escape the black hole. For the M = 6.6x1036 kg the Schwarzchild radius is:

Black hole Schwarzchild radius = Rs = 2GM/c2 = 9.75x109m = 0.065 AU
This radius is smaller than mercury's orbital radius, 0.387AU.
Compare this radius to the distance to the moon, 3.84x108m.
Or compare to the radius of the sun, 6.95x108m.