Normal distribution applied to Milky Way galaxy M-Sigma relation and bulge star data

Image

This superposition is supposed to show how the M-sigma relation could be applied to a given galaxy (the Milky Way). The vertical blue lines represent the positions of + and – sigma, the standard deviation of the normal distribution. The horizontal green line is positioned at the points where the blue lines intersect the distribution curve. Values are read at the green line from the vertical velocity dispersion axis.

It is seen that the M-sigma velocity dispersion for the Milky Way is about 100-103 km/s which we can use to estimate the M-sigma mass of the MW central supermassive black hole.

Msigma

This graph was made by the author of  “The M-Sigma Relation” in Wikipedia. I am trying to track his identity. No luck yet.

Take a look:

Two ten-billion-solar-mass black holes at the centers of giant elliptical galaxies

McConnell, Nicholas J.; Ma, Chung-Pei; Gebhardt, Karl; Wright, Shelley A.; Murphy, Jeremy D.; Lauer, Tod R.; Graham, James R.; Richstone, Douglas O.

Nature, Volume 480, Issue 7376, pp. 215-218 (2011)

 “Observational work conducted over the past few decades indicates that all massive galaxies have supermassive black holes at their centres. Although the luminosities and brightness fluctuations of quasars in the early Universe suggest that some were powered by black holes with masses greater than 10 billion solar masses, the remnants of these objects have not been found in the nearby Universe. The giant elliptical galaxy Messier 87 hosts the hitherto most massive known black hole, which has a mass of 6.3 billion solar masses. Here we report that NGC 3842, the brightest galaxy in a cluster at a distance from Earth of 98 megaparsecs, has a central black hole with a mass of 9.7 billion solar masses, and that a black hole of comparable or greater mass is present in NGC 4889, the brightest galaxy in the Coma cluster (at a distance of 103 megaparsecs). These two black holes are significantly more massive than predicted by linearly extrapolating the widely used correlations between black-hole mass and the stellar velocity dispersion or bulge luminosity of the host galaxy. Although these correlations remain useful for predicting black-hole masses in less massive elliptical galaxies, our measurements suggest that different evolutionary processes influence the growth of the largest galaxies and their black holes.”

    


  Of course, M-sigma works best when it is confined to galaxies of a given class. Maybe giant ellipticals constitute another such class.

  The M-sigma relation (those widely used correlations) may be written[i],[ii] :

  (1)        

         a)   M  =  Mbh  =  3.1 (σ/200 km s-1)4 x 108  Mʘ  =  M.     

  A current study, based on published black hole masses in nearby galaxies, gives[iii]

         b)   M  =  Mbh  =  1.9 (σ/200 km s-1)5.1 x 108  Mʘ  =  M.

  (2)                  Solar mass [iv]  =   Mʘ   =    1.98855 x 1030 kg                                   

  (3)                    M   =  Mbh  =   M●   =   r* v2/κG  from eqn. (2) in the paper, by the Postulate

                 Mbh  =  r* σ2/κG  =  3.1×108 (σ/200,000 m s-1)4 Mʘ       

                                      vσ  =  “σ” by the Postulate

  or

  (4)                  Mbh  =  r* σ 2/κG  =  3.1×108 (σ/200,000 m s-1)5 Mʘ   =   M  

                 Milky Way mass, Mmw  =  7×1011 M[v]

                                                or     = 1–1.5×1012 M[vi]

                                             with “Dark Matter “ contributing  

   We cannot have it both ways. Either bulge stars obey standard Kepler (SK) or adapted Kepler (AK). Which?  Is it a mixture of SK and AK as in eq. (6) of

   the paper? The Author of the paper dislikes the mixture, as it appears in eq. (6). But, such questions are good. They make for lots more research.

   So, astrophysicists, cosmologists and their grad students should love the Postulate.

[ii]     Ferrarese, F. and Merritt, D. (2000), A Fundamental Relation between Supermassive Black Holes and Their Host Galaxies, The Astrophysical Journal, 539, L9-L12

[iii]    McConnell, N. J. et al. (2011), Two ten-billion-solar-mass black holes at the centres of giant elliptical galaxies, Nature, 480, 215-218

[v]        Milky Way Mass  7×1011 M   Reid, M. J. et al. (2009). “Trigonometric parallaxes of massive star-forming regions. VI. Galactic structure, fundamental parameters, and noncircular motions”. The Astrophysical Journal 700: 137–148, solar mass M  =  1.9891×1030 kg,   Mmw  =  1.4 x 1042 kg   computed by conventional methods

[vi]    Milky Way Mass including “Dark Matter” 1–1.5×1012 M McMillan, P. J. (July 2011). “Mass models of the Milky Way”. Monthly Notices of the Royal Astronomical Society 414 (3): 2446–2457.  solar mass M  =  1.9891×1030 kg,  Mmw  =  2-3 x 1042 kg  by conventional methods

Wikipedia, rotation velocity  =  v,  AVD

 

Estimates for the mass of the Milky Way vary, depending upon the method and data used. At the low end of the estimate range, the mass of the Milky Way is 5.8×1011 solar masses (M), somewhat smaller than the Andromeda Galaxy. Measurements using the Very Long Baseline Array in 2009 found velocities as large as 254 km/s for stars at the outer edge of the Milky Way, higher than the previously accepted value of 220 km/s. As the orbital velocity depends on the total mass inside the orbital radius, this suggests that the Milky Way is more massive, roughly equaling the mass of Andromeda Galaxy at 7×1011 M within 50 kiloparsecs (160,000 ly) of its center. A 2010 measurement of the radial velocity of halo stars finds the mass enclosed within 80 kiloparsecs is 7×1011 MBut, we cannot apply standard Kepler or a correlation diagram based on unadapted Kepler, to stars that obviously do not follow Kepler’s laws, as is exemplified by the flat MW velocity dispersion diagram. But we go ahead anyway as if we haven’t a clue and do not understand. Most of the mass of the Galaxy appears to be matter of unknown form which interacts with other matter through gravitational but not electromagnetic forces; this is dubbed dark matter. A dark matter halo is spread out relatively uniformly to a distance beyond one hundred kiloparsecs from the Galactic Center. Mathematical models of the Milky Way suggests that the total mass of the entire Galaxy lies in the range 1-1.5×1012 M.

 

Galactic rotation,    velocity = v,  AVD

The stars and gas in the Galaxy rotate about its center differentially, meaning that the rotation period varies with location. As is typical for spiral galaxies, the distribution of mass in the Milky Way Galaxy is such that the orbital speed of most stars in the Galaxy does not depend strongly on their distance from the center. Away from the central bulge or outer rim, the typical stellar orbital speed is between 210 and 240 km/s. Hence the orbital period of the typical star is directly proportional only to the length of the path traveled. This is unlike the situation within the Solar System, where two-body gravitational dynamics dominate and different orbits have significantly different velocities associated with them. The rotation curve (shown in the figure) describes this rotation.

If the Galaxy contained only the mass observed in stars, gas, and other baryonic (ordinary) matter, the rotation speed would decrease with distance from the center. However, the observed curve is relatively flat, indicating that there is additional mass that cannot be detected directly with electromagnetic radiation. This inconsistency is attributed to dark matter. Alternatively, a minority of astronomers propose that a modification of the law of gravity may explain the observed rotation curve. The constant rotation speed of most of the Galaxy means that objects further from the Galactic center take longer to orbit the center than objects closer in. But, in fact, they orbit faster than they would if they followed Kepler’s 3rd  law. This is actually the problem. If they orbited according to Kepler’s 3rd, they would orbit so slowly as they neared the galactic rim that the spiral arms would wrap backward multiple times around the galactic center like the mainspring of an old windup clock. So, we can actually see the anomalous velocity dispersion at work when we observe a spiral galaxy.

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