Analysis of slope of mass-luminosity curves for different subsets of binaries – dark matter, MOND or something else governs the accelerated rotation of galaxies?

Abstract.

Analysis of mass-luminosity curves for different subsets of binaries (both visual and eclipsing spectroscopic binaries) revealed the deviation in slopes  for relatively close binaries (averaged around 3.6*10^-4 light years) compare to relatively far spaced binaries (averaged around 5.6*10^-3 light years). The slope for close binaries is larger, what means that for the same luminosity of the main sequence stars the determined from Kepler law gravitational mass is smaller (or gravity between stars is stronger). This observation is opposite to the MOND idea (the far the stars the higher shift from 1/r² law to 1/r law for gravity) – that would be opposite effect. The idea of dark matter seems to be confirmed once more (as if some dark mass is hanging around the star, thus making the mass seemingly larger), but a new concept of some kind of gravity enhancement by the mass itself may also be relevant – the closer the binary the higher local concentration of mass and higher value of G in the gravity law.

 

 

 

 

 

 

 

 

 

 

 

 

Introduction.

One of the unsolved problem of modern science is the observed deviations of the galaxy rotation curves from the predicted ones. The phenomenon is observed only on large scales and that is why it is so difficult to understand. At the same time such phenomenon is expected to reveal itself on all scales and all objects, including the simplest ones, where the gravity may be probed – binary star. Indeed, the simplest atom – hydrogen atom allowed to create quantum mechanics (including quantum electrodynamics due to Lamb shift) and from history of science perspective it is expected that the investigation of the simplest objects may lead to the most efficient theories. Hydrogen atom was especially simple binary system  because both masses were quantized with high accuracy. Binary stars, of course, may have all the possible variations of masses of both stars, but still it is  a simplest model object for applications of law of mechanics. Any deviation from simple Newton laws (Einstein modifications for close stars would be necessary) which is visible on galactic scale (dark matter problem) must reveal itself despite possibly in miniscule amounts on this simple objects.

               The long and unsuccessful search for dark matter started to reveal different ideas. One of them is MOND, and at modified Newton gravity the binaries with high deviation between stars would start feel this deviation from Newton law and attract each other stronger [1].  

Main part.

In order to test the idea of the change of gravity law for the binaries as  a function of separation between them I decided to go the same way as for the testing of the additional gravity created by photons [2,3]. That is, the mass-luminosity curve will have a different slope for the different subsets for binaries (subset of binaries with close luminosities versus subset  binaries with different luminosities would reveal any additional force connected to the photons trapped inside the stars, for example). The comparison of subset of binaries with relatively far separation between star versus subset of binaries with small separation would reveal any deviation from Newton law as a function of distance.

               I manually chose several visual binaries which are close to the Sun (the close the star, the better accuracy of all measurements) and plotted separately relatively close binaries versus relatively far binaries. (two eclipsing spectroscopic binaries were added to close binaries to have points with masses between 3 and 4 Suns)

 

Fig 1. Mass-luminosity relation for binaries with relatively far semi-major axis (average~ 5.6*10^-3 ly) and relatively small semi-major axis (average ~ 3.6*10^-4 ly).

Table 1 Distant binaries.

Name of binary	Mass in Suns	Ln(Luminosity), Luminosity is in Suns
Andromeda Groombridge 34	0.38	-3.816
	0.15	-7.07
Eta Cassiopea	0.972	0.208
	0.57	-2.81
24 Comae Berenices	4.4	5.155
	3.3	3.173
61 Cygnus	0.7	-1.877
	0.63	-2.465
Mu Cignus	1.31	1.79
	0.99	0.34
Gamma Delphinus	1.57	1.93
	1.72	3.034
Epsilon Lirae 1	2.03	3.18
	1.61	2.13
Epsilon Lirae 2	2.11	3.367
	2.15	3.466
36 Ophiuchus	1.7	-0.6
	0.71	-2.41

 

 

Table 2 Close binaries.

Name of binary	Mass in Suns	Ln(Luminosity), Luminosity is in Suns
Xi Bootes	0.9	-0.5
	0.66	-2.8
Sirius	2.063	3.23
	1.018	-2.88
Alfa Centarous	1.1	0.418
	0.907	-0.69
Alfa Comae Berenices	1.237	0.542
	1.087	0.56
Beta Delphinus	1.75	3.18
	1.47	2.08
Delta Equaleus	1.192	0.81
	1.187	0.728
Zeta Herculesis	1.45	1.879
	0.98	-0.48
99 Herculesis	0.94	0.673
	0.46	-1.966
Sigma Herculesis	2.6	5.44
	1.5	2.0
Beta Leonis minor	2.11	3.58
	1.35	1.76
Psi Centari*	3.114	4.95
	1.909	2.89
Chi 2 Hidrae*	3.605	5.84
	2.632	4.19
70 Ophiuchus	0.9	-0.53
	0.7	-2.04

·        * - eclipsing spectroscopic binaries (obviously close binaries)

Slopes of the curves are different! It means that for close binaries the effective gravitational constant would be larger. Indeed, the visual binaries gives the masses as:

M₁+M₂=4*π²R³/(G*T²)                                                                                       (1)

M₁, M₂ – masses of the stars, R- semi-major axis, G – gravitational constant, T is the period of the binary.

And similar formula for the eclipsing spectroscopic binaries:

M₁+M₂=T²*(V₁+V₂)³/(2*π*G)                                                             (2)

Here V₁, V₂ – maximum velocities of the stars

 Assuming that the absolute luminosity determines the inertial mass of the star (indeed, any deviation from gravitation law is small and should not influence the evolution of the star), it is possible to see, that higher slope corresponds to smaller deduced gravitational mass for close binary compare to far binary (if the gravitational constant is the same). Assuming the equivalence principle holds, it means that the gravitational constant for close binaries is different from the gravitational constant for far binaries (larger for close binaries). This observation is exactly opposite to what is expected for MOND – in this case the far binaries would be attracted stronger. It looks like some additional mass is present in addition to the star masses which forces them to go closer (almost like the dark matter is present).

However, why would the dark matter be present only for close binaries and not for all of them (in this case on average the slopes should be the same)? More plausible idea is that gravity constant depends upon the mass of the star itself – the gravity enhancing field is created by ordinary matter, which is stronger for higher concentration of the matter in the space.

What is the problem with dark matter being considered as some kind of exotic particles being able to gravitate but not react in any other way with usual barionic and non-barionic (light, for example) matter? In principle such matter is possible, but all the previous experimental evidence tells that the less particle interact with barionic matter the less it contribute to gravity. Indeed, any ions and molecules are easy to catch and they contribute to gravitation tremendously so far. Electrons are less interacting with matter and also less heavy. Neutrinos are kind of particles that are almost not interacting with barionic matter but they are also do not have significant contribution to the gravity. It plausible to assume that other types of particle exist which would interact with matter even less, but they also would contribute to the gravity even less. The idea of any type of particle which would be not interacting with ordinary matter but contribute to the gravity even more than barionic matter is out of this sequence and seems not obvious.

In addition the recent discovery of ultra-diffuse  galaxies with diluted stars concentration and completely devoid of dark matter [4] poses even more questions: how the dark matter may be separated from the ordinary matter [5] if they interact gravitationally? Why would not dark matter be attracted back for billions of years and completed the usual setup: dark matter halo around the visible galaxy?

At the same time the dark matter is absent in ultra-diffuse galaxies only – may be the concentration of ordinary matter plays some role? The ordinary matter changes the gravity constant through some kind of gravity enhancing field?

From the slope of the curves it is possible to roughly evaluate how gravitational constant G changes with distance.

We have two equations:

Y=3.7978*ln(x)-0.1622 – far binaries (distance ~56.29*10^-4 light years, l.y.)

Y=4.653ln(x)-0.0421 – for close binaries (distance ~3.63*10^-4 l.y.)

For mass m=2 from the first equation y=2.4702. This value is assumed to be correlated with inertial mass which determined by star evolution and it is assumed that small change in gravity law can not influence the luminosity (the luminosity dependence  upon the heavy metal composition is neglected). Substituting into second equation we got m=1.716 (instead of two). The equivalence principle should not be violated for close  binaries compare to far binaries, so it means that the mass of the star is not enough for such luminosity.

               It may be simpler explanation, of course for such deviation – both stars were formed from the same cloud, which was much denser for close binaries (that is why they are closer) compare to very diluted cloud for far binaries. In addition to the stars, huge amount of planets and asteroids are hanging around each star (because the initial cloud was dense), effectively creating invisible but quite real barionic matter (“dark matter” in the very original sense). Assuming the observations of the brightness variation exclude such explanation (constant dimming of the star due to interstellar objects), the other explanation is that the gravity constant is different. From equations (1) and (2) it follows that G would be larger for close binaries (and G=K/m law holds). For close binaries G is 2/1.716=1.166 times larger.

               Influence of the mass to the gravity may be written in a formula similar to Coulomb law:

F=(1/[4πεε_o])*q₁*q₂/r²                                                                               (3)

Where q₁, q₂ are electrostatic charges, r is the distance between charges, ε is the permittivity of space (due to dipole nature of the medium the force is weakened), ε_o is the permittivity of free space.

               For gravity it would be:

               F=(ε_g/[4πε_go])*m₁*m₂/r²                                                             (4)

Where m₁,m₂ are masses, r is the distance between masses, ε_g is the gravitoelectric permittivity of space (due to the absence of antigravitation it always enhances the force) and ε_go is the gravitoelectric permittivity of free space (the notations would be suitable for gravitoelectromagnetism [6,7]).

In this equation ε_g moved up to numerator compare to formula (3) because the gravity is enhanced, not weakened as in the case of electricity.

With loose similarity to Debye length [8] the dependence of such field may be written in a way like this:

ε_g=1+δ*{ΣM_i*exp(-r_i/ξ)}/{ΣM_i}                                    (5)

Here M_i are masses around the point (actually all masses in Universe, but due to exponential decay only closest masses are necessary), r_i are distances to the point of interest, ξ is the decay length, δ is some empirical constant (how strongly gravitational constant is enhanced). Formula (5) would drop to 1 in infinity (no influence of mass) and to some enhanced value near the star.

Simplifying even further to evaluate the value of the effect in the Solar system:

G=G_o*exp(-r/ξ)                                                               (6)

And 1.166=[exp(-3.6*10^-4/ξ)]/[exp(-5.6*10^-3/ξ)]

The decay length would be 0.034 l.y. (3.2*10¹⁴ m) and for the Pluto orbit (5.9*10¹² meters) change of gravitational constant of 2% is expected (G=0.98G_o).

               This is quite large a change and should be easily noticeable if the Cavendish experiment is performed on Pluto orbit or on the Pluto surface (because the planets are small compare to Sun, the only real player in Solar system is Sun). For example, the Cavendish experiment performed on Moon surface would lead to only around 4*10^-8 relative change – not enough with modern accuracy of Cavendish experiment. The previously published idea of Cavendish experiment near the surface of the Sun would be helpful in the case the accuracy will be good enough.

               It is interesting to note, that the idea of quantum vacuum being influenced by different fields with corresponding change of gravity constant or electric field constants is not new and was already discussed [6,9]. In [6] the weakness of gravity is hypothesized to be due to the existence of Higgs boson “gravitational antiparticle” (second quantization is predicted), so that virtual pairs particle-gravitational antiparticle would weaken the field in exactly the same way as virtual electron-positron pairs are weakening the electric field in quantum vacuum explanation of speed of light value. If there is no gravitational antiparticle in nature, the presence of the mass is expected to polarize the quantum vacuum in such a way, that popping out of quantum vacuum particles are all bosons with the same positive sign of mass (all attracting each other). In this case if the boson condensation of all of them is avoided (collapsing the mass into the black hole as described in [6,9]), the virtual particles would be increasing the strength of the gravitational filed, not weakening it as in the case of electromagnetism. This would be exactly what is observed in this article. The enhancement length seems to be enormous – but this is in the range what is expected for dark matter (actually the real length may be higher, because more accurate experiments are necessary.

Conclusion.

The discovered deviation in the  mass-luminosity curves is a hint, that the gravity constant is not valid for the free space and becomes stronger in the presence of classical barionic matter. Such behavior is exactly opposite for what is expected by MOND and formally in line with dark matter hypothesis (the non-barionic unseparable and mass induced field is in broad sense would be “dark matter”). However, such observation is more consistent with old definition of field, not matter. To confirm or reject the observation made here the more accurate data on numerous binaries would be necessary (because the “googled” data can not be considered accurate in modern science). The article may be of some interested for visual binaries specialists who are trying to decrease the scattering in the mass-luminosity curve (the idea is that the scattering is not really the experimental error, which would be much smaller in the time of space-based telescopes, but rather some underlying physical mechanism, which may give different slopes for different subsets of binaries). To my best knowledge, nobody so far analyzed mass-luminosity curves from this perspective.

 

References.

1. McCulloch, M.E., Lucio, J.H. Testing Newton/GR, MoND and quantised inertia on wide binaries. Astrophys Space Sci 364, 121 (2019). https://doi.org/10.1007/s10509-019-3615-z

https://link.springer.com/article/10.1007/s10509-019-3615-z

2. https://vixra.org/pdf/2005.0250v1.pdf

3. https://vixra.org/pdf/2007.0195v1.pdf

4. https://www.discovermagazine.com/the-sciences/hubble-reveals-new-evidence-for-controversial-galaxies-without-dark-matter

5. https://astronomy.com/news/2019/03/ghostly-galaxy-without-dark-matter-confirmed

6. https://en.wikipedia.org/wiki/Gravitoelectromagnetism

7. https://tipikin.blogspot.com/2019/12/quantum-vacuum-application-to-gravity.html

8. https://en.wikipedia.org/wiki/Debye_length

9. https://tipikin.blogspot.com/2020/03/unification-of-gravitational-and.html

 

 

 

Weak equivalence principle check for non-barionic matter using eclipsing spectrometric binaries. No evidence for dark matter.

 

Abstract.

Weak equivalence principle (the bodies are gravitating equally per inertial mass irrespective of the chemical composition) was confirmed for barionic matter with very high accuracy. However, a priory it is not clear, how to check weak equivalence principle for the mixture of barionic and non-barionic matter (light is inside the ordinary matter). For example, how fast would the sphere full of photons fall in the Earth gravity field? The experiment is not possible on Earth. However, such verification is possible for stars using the observational data on binary stars. In this article the analysis of the mass-luminosity was made for similar stars forming binary versus different stars forming binary and the slopes were found the same with accuracy of 6%. That would be the accuracy of confirmation of the equivalence principle for non-barionic matter (actually a mixture of barionic and non-barionic matter with around 0.14% of non-barionic matter ratio). While some violations of weak equivalence principle are still possible (the idea of strong gravitation of slow light) the scale of such violations is clearly well below the level expected for explanation of dark matter.

Introduction.

In order to check the weak equivalence principle for non-barionic matter, it would be necessary to find the object where such form of energy would be present in great amount. The only such object which is relatively easy to find is a star.  Indeed, the star should burn some matter and transform it into the light. The light can not leave star instantly and trapped inside for many thousands of years (possibly millions of years), slowly diffusing toward chromosphere. During such a process the light is absorbed and re-emitted again, and during the short life time the photons are gravitating independently of the surroundings and thus may be considered as the non-barionic matter trapped inside the barionic matter. If the light would gravitate differently, the obtained additional pulse would contribute back to barionic matter at re-absorption, thus making the overall gravitation of the mixture different from pure barionic matter. The total mass loss due to the thermonuclear synthesis in the star is around 1.4% of initial mass and the shortest lifetime for largest known stars is around 10 million years. Therefore, on average around 0.14% of total mass is emanating from the large star per million of years and assuming the light is trapped inside for around 1 million years too, the total energy kept inside the star as photons of all kinds (non-baryonic matter) would be around 0.14% of its barionic mass. 

              The idea is to use the data on binary stars and to compare the mass-luminosity curve for the stars with close masses and the mass-luminosity curve for the stars with as much difference in mass as possible.

There are many binary stars which are visible as double stars with resolved period and axis and ratio of inertial masses (through measurements of the velocities of stars). Many parameters of such stars are published in Internet.

The usual formula applied to the stars from the third Kepler Law:

T²=4π²*a³/[G(m₁+m₂)]         (1)

Here T is the period of rotation of one star around the second one, a is semi-axis, m₁ and m₂ are masses of the stars (assuming gravitational mass is equal to inertial mass) and G is gravitational constant.

However, the light theoretically may have much higher gravitational pull compare to the inertial mass from E=mc² relation (it is assumed that the inertial mass of light being emitted and reabsorbed inside star is still according to E=mc², as it was proved by Einstein himself). The presence of slow light may modify the gravitational pull, making it much stronger for the star which has more trapped light (and other non-baryonic matter). While the exact amount of trapped light is difficult to calculate (not much is known about the light content of the interior of fully ionized plasma), it is obvious that this amount is correlated with luminosity of the star - the higher the luminosity, the higher the amount of trapped light and the higher the additional gravitational pull on the star (the higher the deviation between the gravitational and inertial mass).

In the derivation of the formula (1) the gravitational masses are always comes as a product [1]:

F=G*M₁*M₂/r²

Here M₁ and M₂ are gravitational masses. Assuming the added pull is proportional to luminosity which is proportional to mass (whether gravitational or inertial), it is possible to assume:

F=G*K₁*K₂*m₁*m₂/r²

Here K₁ and K₂ are multiplicity coefficients, the value of K may be especially high to ultra-bright star (because due to very short life time the ultra- bright star should emit more light per second and as a consequence has more light “on hold”, ready to be emitted but so far trapped inside). If weak equivalence principle hold, K=1. It is important that both coefficients for binaries are always a product.

The modified third Kepler Law:

T²=4π²*a³/[G*K₁*K₂*(m₁+m₂)]

Here m₁ and m₂ are inertial masses. When K₁=K₂=1, the third Kepler Law for baryonic matter is obtained.

To determine the masses from the observation of binaries we need: T, a, and ratio of masses m₁/m₂=n. Since the ratio of masses is determined through the Doppler shift of spectra of stars, it is a ratio of inertial masses. We have two equations for masses m₁, m₂

G*K₁*K₂*(m₁+m₂)=4π²*a³/T²

m₁/m₂=n

Then:

m₂=4π²*a³/[G*T²*K₁*K₂*(n+1)]

m₁=4π²*a³*n/[G*T²*K₁*K₂*(n+1)]

Suppose we decided to determine the inertial masses from the visual binaries with two distinct masses m₁>>m₂ taken in different combinations.   How it would influence the mass-luminosity correlation?

It is possible to show that for very strong effect (K is large) the slope of mass-luminosity curve will depend upon the choice of stars in pair (Kepler third law is not valid any more).

 Lets  consider three cases:

1.Binary m₁ and m₁

2.Binary m₂ and m₂

3.Binary m₁ and m₂

In the first case the value of m1 is (because n=1)

m₁=m₁(old)/[K₁*K₁], here m₁(old)=4π²*a³/[G*T²*2]

Here m₁(old) is real inertial mass. K₁ is large and the value of m₁ is shifted strongly toward smaller mass compare to real inertial mass.

In the second case the value of m₂ (n is equal to 1)

m₂=m₂(old)/[K₂*K₂]

If K₂ is smaller (closer to 1)  the mass of smaller star will be actually equal to inertial mass

In the third case the value of m₁ is

m₁=m₁(old)/[K₁*K₂], m₁(old)=m₁(old)=4π²*a³*n/[G*T²*(n+1)]

Since both coefficients K₁ and K₂ are here, one is small and one is big, the shift down compare to the real inertial mass is smaller compare to the case of the big equal masses, but still present.

m₂=m₂(old)/[K₁*K₂]

The smaller mass is becoming too small for this type of star, well below the real inertial mass for smaller star.

This idea may be immediately checked. If the mass-luminosity curve is plotted using first only stars with close masses, it will be compressed  toward y-axis because of K₁*K₁ and K₂*K₂ coefficients along the x-axis (the slope will be larger). If the same curve is plotted using the stars with different masses  the slope will be smaller. In addition since the same stars now would be in pairs with different masses the scattering will be much larger (the same star like Sun in pair with another Sun-like star would give almost the inertial mass, but in pair with blue giant  a much smaller mass, thus creating additional to the experimental error scattering). In [2] this idea was checked for visual binaries from publication, which is 70 years old. The results showed that indeed the slope for the mass-luminosity curve was higher for close masses.

The results were checked with the help of visual binaries using the modern data from Wikipedia. The slope for the close masses was higher again. However, the most prominent effect is expected for the ultra bright stars with masses 30-100 of Sun mass. For them the percentage of trapped light should be tens of thousands times more compare to Sun and smaller stars (because the total amount of light trapped inside is inversely correlated with life time of star and ultra bright stars are very short lived).

In this case the only way to verify the idea it to use data on spectroscopic binaries. According to [1] the sum of masses is determined by the formula:

m₁+m₂=[P/(2*π*G)]*[(V₁+V₂)³/Sin³(i)]

and ratio of masses is determined through the ratio of velocities: m₁/m₂=V₂/V₁

Here P is the period of binary, G is gravitational constant, V₁ and V₂ are semi-amplitudes of velocities (they marked K₁ and K₂ in Wikipedia articles on binaries), Sin(i) is the sin of the angle between the axis of the rotation and Earth-binary direction. For very important subset of spectroscopic binaries called eclipsing spectroscopic binaries both stars are eclipsing each other thus guarantee that the angle i is close to 90 degrees and that allowed determination of masses of such stars using the known astronometric data. I used binaries: 1 Persei, Theta 1 Orioni 3, Prismis 24-1, NGC 3603-A1, CD Crucis for the brightest stars with close masses and WR22, LY Aurigae, AO Cassiopei for the largest stars with different masses. For the smaller masses the stars from the visual binaries were used (except for stars smaller than Sun). The results are below:

With accuracy of 6% the slopes are the same. Intercept on both curves put on zero. Therefore, the expected from the preliminary results higher slope for the close masses is not confirmed for the ultra bright stars (where the effect should be the largest). While the weak equivalence principle still may be violated due to stronger gravitation of slow light (the error is rather large), the effect on rotation of Galaxy is negligible and by no means may be responsible for the explanation of large scale phenomena like dark matter.

 

                 

References.

1.      https://www.astro.caltech.edu/~george/ay20/Ay20-Lec4x.pdf

2.      https://vixra.org/pdf/2005.0250v1.pdf

3.       

Supernova mechanism of transfer of thermonuclear energy into the energy of accelerated rotation in galaxy. Violation of weak equivalence principle for the supernova remnants.

In addition to the mechanisms responsible for transformation of the thermonuclear energy into the energy of the rotation of the stars in galaxy [1-4] another important mechanism would be the supernova explosion (and, to the lesser degree, nova explosions and other types of star explosion). The overall idea is that during such event a lot of energy is transformed into the light, neutrinos and ultra-relativistic matter like cosmic rays. All of them will be moving with speed of light and have twice the cold barionic matter gravitational pull [1,5]. 

Indeed, the total energy release of supernova would be 2*10exp(44) Joules. The remnants of the star will be moving away with the velocity of 20000 km/h (5.6*10exp(3) m/s). Assuming the shed mass is equal to 10 masses of Sun, which would be 2*10exp(31) kg, it means that the energy stuck in relatively cold barionic matter is only 6.3*10exp(38) Joules, and the rest of the released energy would be in light, neutrinos, cosmic rays and other ultra-relativistic matter. The associated inertial mass would be (from 

E=mc² formula) equal to 2*10exp(27) kg. This effective mass will be gravitating twice as strong as cold barionic matter. Age of Milky way is 13.51 billions of years and supernova comes every 30 years, thus making the total number of supernovas equal to 4.5*10exp(8). The total inertial mass of created ultra-relativistic particles (light and neutrinos and cosmic rays combined) would be 9*10exp(35) kg. The total mass of stars in Milky way is 50 billions  Suns [6] (dark matter excluded) and total inertial mass thus estimated is 10exp(41) kg.

The ratio of the double gravitating matter created by supernovas to the total mass of the stars is not high - only 10exp(-6). However, evaluation of this amount may change a lot if other explosions are taken into the consideration. For example nova explosions take place 2500 more frequently compare to the supernova. If the similar release of ultra-relativistic particles takes place this means up to 0.25% of doubly gravitating matter was created during the Milky Way lifetime. 

From different ideas of the presence of the violating weak equivalence principle matter (light inside the stars, ultra-relativistic matter inside the stars, formation of ultra-relativistic matter during violent explosion of the stars) it seems that the accelerated rotation of galaxies (dark matter problem) may be solved rather by accurate accounting for all those small contributions, rather than by postulating of the predominant presence of invisible interacting only gravitationally particles.

References.

1. http://tipikin.blogspot.com/2019/10/stars-are-full-of-trapped-light-may.html

2. http://tipikin.blogspot.com/2019/12/light-matter-attraction-as-driving.html

3. http://tipikin.blogspot.com/2019/09/accelerated-rotation-of-star-because-of.html

4. http://tipikin.blogspot.com/2020/06/gravitation-of-ultra-relativistic.html

5.  D.Fargion "Deflection of Massive Neutrinos by Gravitational Fields" // Lettere al Nouvo Cimento, Vol.31, No 2, 1981

https://www.researchgate.net/publication/

227245454_Deflection_of_massive_neutrinos_by_gravitational_fields

6. https://en.wikipedia.org/wiki/Milky_Way