Posts Tagged ‘potential energy diagram’

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

July 18, 2013

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.

A potential energy diagram is perfectly possible for a hyperbolic black-hole gravitational field

January 11, 2012

The “normal” Newtonian potential energy diagram derived from the inverse square relation versus the hyperbolic black hole gravitational potential energy diagram derived from the inverse (1/kr) relation

First of all, note that a potential energy diagram is perfectly possible for a hyperbolic black-hole gravitational field. The only trouble is with convention. Normally, one takes potential energy U as U = 0 at r = infinity. But, U keeps increasing forever with increasing r in the case of the HBH field. It does not level off to an asymptotic value. So, we would need to adopt a different convention, with U = 0 at r = 1. Then, we would have to remember that all U computed for the HBH case will need to be multiplied by -1 in order to be consistent with conventional usage.

We could represent how the ultra-massive universe excited inflaton HBH gravitational field collapses or transitions to the conventional inverse square field, thereby donating its potential energy to what is increasingly an inverse square gravitational universe, accelerating its expansion in the latter 2/3 of its evolution. We might use weighting factors. We could use a linear weighting factor or maybe an inverse square exponential form or even an hyperbolic expression.

Using x = r in the above diagram, let us try weight for the inverse square derived contribution, S = % and weight for the hyperbolic contribution, H = 1.00 – S, 0.00 < S < 1.00, so that U = S( -1/r +1) + (1.00 – S)ln(r), then total U will transition smoothly from the HBH hyperbolic potential energy to the inverse square potential. Here, 1 is a translation amount to let the inverse square derived curve superpose upon the hyperbolicaly derived. The potential energy lost by the HBH phase of the universe is made up by a gain in kinetic energy of expansion in the inverse square phase.

Remember, it is legitimate to think of Hubble expansion of spacetime carrying the objects embedded within it as a kinematic growth process. One need not always regard it as a “stretching” of spacetime, though for some other purposes, this may help.

Obviously, the resulting composite curve will have a significant positive slope on the right, connoting Dark Energy. But, the curves explicitly describe Dark Matter. So there is a strong link between DM and DE.

There have been misstatements and misinterpretations of Birkhoff’s Theorem. For instance, it has been shown by Kristin Schleich and Donald M. Witt (“A simple proof of Birkhoff’s theorem for cosmological constant”, arXiv: 09084110v2, 27oc09) that Birkhoff does not demand staticity in spherically symmetric solutions to Einstein’s vacuum field equations. Static solutions have heretofore been thought of as being required. There may be other misstatements and misinterpretations that are not yet recognized.

For instance, Birkhoff’s Theorem must actually leave black hole singular gravitational fields as an exception to the commonly quoted rigid rule that only asymptotically flat (commonly assumed meaning: inverse square) gravitational fields are allowed. Otherwise there is no way to measure or unequivocally determine that the center of a black hole is a singularity since electric charge and gravity are the only items the influence of which can escape the interior of a black hole.  Then the theories of Schwarzschild and Kretschmann that say such singularities are physically real are largely meaningless as unfalsifiable hypotheses.

There simply must be a measurable consequence of a true singularity at the center of a black hole or else its existence cannot be postulated. That the mathematics seems so very precise is not good enough. There must be a way to experimentally verify or falsify the equations.

If the gravitational singularity at the center of a super-massive galactic black hole results in a hyperbolic gravitational field, there is a way. By measuring the velocity distribution of stars in the surrounding galactic disk, it can be determined if they move with a constant velocity, v = (GM)^½ at large r, as they must if they move in a hyperbolic gravitational field. As a matter of fact, the velocities of stars in spiral galaxies do indeed move with constant velocity at large r. This can be seen as proof of a singularity at the center of a spiral galaxy’s black hole.

We often state that “The laws of physics must break down at the incredibly tight curvature of spacetime near the singularity of a black hole”. What does this mean? One thing it could mean is that Birkhoff’s Theorem breaks down too. The metrics to which Birkhoff applies probably are not strictly valid near the singularities that they themselves predict so, the “asymptotically flat” dictum may not be strictly true either. Otherwise, our cautionary statement is meaningless.

Besides, the intense benefit brought by the postulate of a hyperbolic galactic super-massive black hole gravitational field is too great to be ignored. It explains the anomalous stellar velocity distributions in galaxies, anomalous velocity distributions in galactic clusters, galactic lensing phenomena, temperature distributions within galaxies, Bullet Cluster type apparent offsets in the barycenters of colliding galaxy clusters, etc. It does everything that Dark Matter is supposed to do! So, Dark Matter is an unnecessary complication that violates Occam’s Rule.

Cosmologists will not like this idea. The LCDM model would have to be drastically revised. The consensus would have to change. Since journal editors and referees endorse only papers that conform to the consensus (they would have been comfortable with the Pope’s decision to censor Galileo), no-one will publish a paper that challenges the commonly accepted interpretation of Birkhoff’s Theorem. And, if Birkhoff does break down in the vicinity of a gravitational singularity, how can it ever be proven? One would have to develop a whole new physics of the ultra strong curvature near black hole singularities. Unless it also had consequences outside the black hole, such a theory could not be falsified so, it could not be admitted as a part of science.

Catch 22 says “Anyone who wants to get out of combat duty isn’t really crazy.” Hence, pilots who request a mental fitness evaluation are sane, and therefore must fly in combat. We would be better off flying in combat duty than trying to fly the hyperbolic super-massive galactic gravitational field into some journal’s pages.

The amazing thing is that the HBHG field does have consequences outside the BH and the event horizon.