User:Pao.christ/sandbox3

to be inserted in MOND before observational evidence

GRAVITOMAGNETISM vs. MOND and DARK MATTER.
An immediate  manipulation of  Milgrom's equation with the simple interpolating function yields

$$ \frac {GM (1+a_{0} /a) }{r^2}  = a  $$

which can immediately interpreted as an ordinary Newton equation with standard a = v^2/r but with an increased mass   $$M= M (1+a_{0} /a)$$ where the extra term  may be interpreted as dark matter. This renders the  alternatives between the modification of the equation of motion or the law of gravitation academic. In addition this does justice to the speculations about seasonal variations of dark matter. Indeed it would be hard to believe in a seasonal variation of (corrections to) Newton's law. A more physical approach is provided by gravitomagnetism. The basic equation  is

$$ \frac{v^2}{r} = \frac {GM}{r^2} + 2 h v $$

where the first term  of the r.h.s. represents   the Newtonian contribution which yields the Keplerian orbiting velocities and the last one  is due to  the gravitomagnetic field h, [h]= [1/T], of the core rotating with angular velocity $$\omega_{G}$$ (G= galaxy). This term has the form of a Coriolis force, which  appears  in non inertial frames, increasing  the tangential velocity in the rotating frame and results in a total velocity in the external inertial frame (we on earth in a first approximation) given by this velocity plus the velocity of rotation (as measured from the Doppler shift  of H21 lines in the orbital plane).

Now the gravitomagnetic force has been shown to be proportional to the angular velocity as

$$ h = \epsilon_{G} \; \omega _{G} $$

where the coefficient $$\epsilon_{G}$$ $$\epsilon_{G}\simeq \frac{GM}{c^2r}$$ shows because of the factor $$1/c^2$$ its relativistic origin and is a measure of the (relativistic) strength  of gravitation. The first relevant feature of gravitomagnetic forces is their different (with respect to the Newtonian term) r behaviour. They are indeed (as gravitational waves ) proportional to 1/r instead of $$1/r^2$$. Results are given by the positive root (in $$ \bar v $$) of the preceding equation. A very simple evaluation of the gravitomagnetic term alone would yield

$$ \bar v \simeq \frac{GM}{c^2} \omega_{G} $$

where the term of the r.h.s. in fraction, with the given numerical values of M33, yields $$10^{13}$$ to be compared to the term on the l.h.s. of $$10^{5}$$. Thus an $$\omega_{G}$$ of the order of $$10^{-8}$$ which means that a  rotation period of the core of some years would do the job. So the core would indeed rotate, as conjectured with a greater velocity than the outskirts. Of course this hypothised core angular velocity has nothing to do with the angular velocity of orbiting objects near the origin which only can be read off from the data. A plot of v for the values $$v= 10^{5} m/s, r= 10^{20} m$$ and $$M_{G} \simeq 10^{11} M_{S} \simeq 10^{40} kg $$ is reported in the Fig. 1).



Of course this treatment is admittedly and necessarily approximate in the sense that the core does not rotate all at the same angular velocity. It is also immediate to get for the centripetal gravitomagnetic acceleration $$ 2 h v \simeq 2 h^2 r \simeq 10^{-10} m /s^2 $$ i.e. obviously the same value demanded by MOND. It is also immediate to see   the absence of this gravitomagnetic effect on the earth motion due to the Sun rotation. Therefore large distances make the possibility of highlighting the 1/r behaviour of small gravitomagnetism comparable to the ordinary dominating $$1/r^2$$ Newtonian term. Thus the gravitomagnetic force, adding to the usual Newtonian one  in non inertial frames, provides the extra attraction which increases the velocities and  in a Newtonian context had to be attributed either to a bigger mass than that responsible for the luminosity or a modification of Newton's law. Although the presence of gravitomagnetism in unquestionable the angular velocity of the core, even if reasonable, appears as a free parameter.