Light is bended by the gravity - this is very old phenomenon, once confirmed the general theory of relativity. The Einstein formula for the light bending is:
ϒ=4*G*M/(r*c2)
where ϒ is the angle of the deviation of the light near the star, M is the mass of star, G is gravitational constant, c is speed of light and r is the shortest distance between the light and star.
From this formula it follows that the light, being traveled near the star, influenced the star, too, transferring part of its pulse to star. While for passing light this is truly negligible, what about the light trapped inside the star itself? It is well known fact, that the gamma quantum generated during the fusion in the Suns core, will spend millions of years till it is emitted by the Sun. During all this time the quantum of the light will be subject of the gravitational pull of other stars (Galaxy in general to explain the additional force added to the usual gravity which may help to explain dark matter partially). This force will mean that some kind of gravitational mass equivalent is added to the star. According to the weak equivalence principle the amount of inertial mass of the star does not change at such process.
The evaluation of the importance of such additional force from the photons back onto the barionic matter may be done as follows:
For the formal consideration (just to have the formula) the photon is treated as having mass m. Then deviation of the light near the star would be:
Vp=a*t, a=F/m, F=G*m*Ms/(r2) → a=G*Ms/(r2)
here
Vp is the perpendicular component of the velocity of the photon of formal mass m, t is time of flight near the star, a is the formal acceleration of the photon of formal mass m, Ms
t=2r/c and
ϒ=Vp/c=G*Ms*2r/(r2*c2)=2G*Ms/(r*c2)
The same formal approach may be used to evaluate the influence of the galactic pull onto the all the photons inside the star (multiplying later the result by 2 to account to general theory of relativity).
The photons generated inside the star during the fusion are not leaving it immediately but essentially trapped inside for millions of years. During all this time all the numerous trapped photons are generating the pull toward the center of galaxy, which may be estimated as follows:
Using the same formal approach the formal "force" onto the photon is:
The force on the photon toward the center of Galaxy:
Here F is the force onto the photon toward the center of Galaxy, a is the acceleration created by such force, Vp is the perpendicular component of the velocity the photon obtained during the time Δt of its stay inside the star.
What would be the change of pulse of the photon during such stay? It may be evaluated assuming the velocity of the photon equals to c/(n)1/2
where n is the effective refraction coefficient for the light in the star medium (since the interior of star is enormously dense and hot plasma, this value is not 1)
The change of pulse of photon is:
Δp=Vp*n*P/c
here Δp is the change of pulse of photon, P is the total pulse of photon, Vp is the obtained perpendicular component of the velocity, c/n
is the velocity of light inside the star. The obtained velocity Vp is considered very small compare to the initial velocity - despite million of years, the photon inside the star is moving as a photon only for small time periods - it is absorbed and re-emit almost instantly. Using the formula for Vp it is possible to obtain:
Δp=G*M*Δt*n*P/(r2*c)
where Δp is the full change of pulse during time period Δt, G is the gravitational constant, M is the mass of Galaxy, n is the effective refraction coefficient, P is the full pulse of the photon, r is the distance from the star to the Galaxy center, c is speed of light.
But Δp/Δt is the effective force (toward the center of the Galaxy) expressed through pulse of photon instead of time Δt. For one photon:
F=Δp/Δt=G*M*P*n/(r2*c)
The pulse of photon is P=n*E/c for the refracted light (Minkowski formula [1]) and:
F=Δp/Δt=G*M*e*(n)2/(r2*c2)
Where e is the energy of one photon. For the total force exhibited by all the photons this force would be:F=Δp/Δt=G*M*E*(n)2/(r2*c2)
Where E is the total energy of the photons inside the star. Now it is possible to calculate the ratio of this force to the gravitational force exhibited by the star (as derived from weak equivalence principle E=mc*c). The gravitational force
F/Fg=E*(n)2/(Ms*c2)
Adding multiple 2 from Einstein's formula:
F/Fg=2*E*(n)2/(Ms*c2) This value may be somehow estimated using the data for Sun. Mass of the Sun is
Still there is no additional gravitational pull for the Sun, which lives for billion of years. The dark matter however usually associated with the presence of young stars in the sleeves of Galaxy. For the star with the big initial mass the life time may be just 10 millions of years. It means that such star will emit the equivalent of 10-7 of its mass per year as radiation and if the photons are still trapped inside for 10 millions of years that means that the ratio of forces
References.
1. https://en.wikipedia.org/wiki/Abraham%E2%80%93Minkowski_controversy
The evaluation of the importance of such additional force from the photons back onto the barionic matter may be done as follows:
For the formal consideration (just to have the formula) the photon is treated as having mass m. Then deviation of the light near the star would be:
Vp=a*t, a=F/m, F=G*m*Ms/(r2) → a=G*Ms/(r2)
here
Vp is the perpendicular component of the velocity of the photon of formal mass m, t is time of flight near the star, a is the formal acceleration of the photon of formal mass m, Ms
is the mass
of the star, G is gravitational constant, r is the effective distance between
the star and the photon, t is the effective time of flight of the photon near
the star. Knowing the values of the acceleration and time it would be possible to evaluate the perpendicular component of velocity:
Vp=G*Ms*t/(r2)t=2r/c and
ϒ=Vp/c=G*Ms*2r/(r2*c2)=2G*Ms/(r*c2)
The same formal approach may be used to evaluate the influence of the galactic pull onto the all the photons inside the star (multiplying later the result by 2 to account to general theory of relativity).
The photons generated inside the star during the fusion are not leaving it immediately but essentially trapped inside for millions of years. During all this time all the numerous trapped photons are generating the pull toward the center of galaxy, which may be estimated as follows:
Using the same formal approach the formal "force" onto the photon is:
F=G*m*M/(r2)
Here G is gravitational constant, M is the effective mass of the Galaxy, r is the distance between the photon and the Galaxy center, m is the formal "mass" of the photon (the idea of such approach is that since it allows to obtain Einstein formula with accuracy of factor 2, it will allow to evaluate this pull with the same accuracy - later final formula to be multiplied by 2).The force on the photon toward the center of Galaxy:
F=ma, a=G*M/(r2), Vp=at=G*M*Δt/(r2)
Here F is the force onto the photon toward the center of Galaxy, a is the acceleration created by such force, Vp is the perpendicular component of the velocity the photon obtained during the time Δt of its stay inside the star.
What would be the change of pulse of the photon during such stay? It may be evaluated assuming the velocity of the photon equals to c/(n)1/2
where n is the effective refraction coefficient for the light in the star medium (since the interior of star is enormously dense and hot plasma, this value is not 1)
The change of pulse of photon is:
Δp=Vp*n*P/c
here Δp is the change of pulse of photon, P is the total pulse of photon, Vp is the obtained perpendicular component of the velocity, c/n
is the velocity of light inside the star. The obtained velocity Vp is considered very small compare to the initial velocity - despite million of years, the photon inside the star is moving as a photon only for small time periods - it is absorbed and re-emit almost instantly. Using the formula for Vp it is possible to obtain:
Δp=G*M*Δt*n*P/(r2*c)
where Δp is the full change of pulse during time period Δt, G is the gravitational constant, M is the mass of Galaxy, n is the effective refraction coefficient, P is the full pulse of the photon, r is the distance from the star to the Galaxy center, c is speed of light.
But Δp/Δt is the effective force (toward the center of the Galaxy) expressed through pulse of photon instead of time Δt. For one photon:
F=Δp/Δt=G*M*P*n/(r2*c)
The pulse of photon is P=n*E/c for the refracted light (Minkowski formula [1]) and:
F=Δp/Δt=G*M*e*(n)2/(r2*c2)
Where e is the energy of one photon. For the total force exhibited by all the photons this force would be:F=Δp/Δt=G*M*E*(n)2/(r2*c2)
Where E is the total energy of the photons inside the star. Now it is possible to calculate the ratio of this force to the gravitational force exhibited by the star (as derived from weak equivalence principle E=mc*c). The gravitational force
Fg=G*Ms*M/(r2) where G is gravitational constant, Ms is the mass of star, M is the mass of Galaxy, r is the distance between the star and Galaxy. The ratio of those forces
is:F/Fg=E*(n)2/(Ms*c2)
Adding multiple 2 from Einstein's formula:
F/Fg=2*E*(n)2/(Ms*c2) This value may be somehow estimated using the data for Sun. Mass of the Sun is
2*1030 kg, energy release is 3.9*1026
Watt and assuming photons are trapped inside for 10 millions years ( 3.15*1014
s) the ratio would be:
F/Fg=(n)2*1.4*10-6
which is very small if n=1. However, the star matter is relative not investigated and the effective refraction coefficient may be very high (in metals, for example it is supposed to be infinity). Essentially the light inside the Sun may be traveling very slow. If in the highly conductive full ionized plasma value of n is 100, the added force may jump to 1.4 % and become noticableStill there is no additional gravitational pull for the Sun, which lives for billion of years. The dark matter however usually associated with the presence of young stars in the sleeves of Galaxy. For the star with the big initial mass the life time may be just 10 millions of years. It means that such star will emit the equivalent of 10-7 of its mass per year as radiation and if the photons are still trapped inside for 10 millions of years that means that the ratio of forces
now is 2*n2
That means that for the star of larger mass the pull toward the center of Galaxy, associated with the light matter may be many times larger than the gravitational pull from the ordinary matter. But this trapped light still makes contribution to inertial mass E=m*c*c (according to weak equivalence principle), which would mean that it will be rotating faster compare to pure barionic mass body like planet. This may explain the dark matter, at least partially.
In a broad sense this idea is in today track of unification of matter and wave behavior. In addition to being full with trapped for long time quanta (pure wave), inside the star the process of tunneling of baryonic particles takes place (during the fusion). However, during the tunneling the baryonic particle is pure wave (the energy is negative, what is inconsistent with particle). Thus inside the star more energy is in wave form and waves are attracted gravitationally differently, so the overall star may have larger than possible orbital speed without presence of dark matter. More experimentation with plasma in fusion reactors may be necessary to understand the behavior of stars.
References.
1. https://en.wikipedia.org/wiki/Abraham%E2%80%93Minkowski_controversy
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