The answer is no. Galaxies are not point like mass. Actually in the central parts, the rotation is solid like while the rotation curve is essentially flat in the external parts.

you can use Kepler's third law to calculate orbital period of a star in galaxy. Only, you can calculate orbital rotation around galaxy center.

@Yvan Dutil and Mehmet Tanriver

If I introduce the deviation factor n in Kepler's third law as shown in eq. (2.10) of the article mentioned, I get correct orbital periods even for the stars far away from galactic center. What do you say about that?

Due to the presence of a huge amount of dark matter (80% of total???) in galaxies the use of Kepler's third law to calculate the orbital period of stars in a galaxy is impractical; it will lead to absolutely wrong values. In fact, this is the confirmation that dark matter really exists.

If the mass of the galaxy were only due to the mass of single stars, the outer stars would be going much more slowly than observed speed; only in this case the speed would fall off with distance as predicted by Kepler's laws, but only approximately because this is a multi-body problem.

Don't forget the presence of central super-massive black holes and smaller ones disseminated around in most galaxies.

@Cesare D. La Padula. If I introduce a deviation factor n in Kepler's third law as shown in Eq.(2.10) of the article listed under question, which is related to proper time of the orbiting star by Eq.(1.18) and satisfies Einstein's field equation as shown in Eq.(2,4), then it is possible for Eq.(2.10) to yield precisely same value as Eq.(2.9). This modified Kepler's law Eq.(2.10) can calculate orbital period of all stars in a galaxy regardless of distance. The proper time of a star is dependent on uneven distribution of galactic matter and the presence of cold dark matter. Do you notice any problem with this analysis?

You can't use the Kepler's third law to computed the period of stars in the Galaxy because usually the stars do not conserve their angular momentum, due axisymmetrical features (spiral arms and bars) present in the Galaxy. So even using the Eq (2.10) it would be just an approximation of a real stellar orbit.    

@Thiago. I think you are right. Eq.(2.10) would be an approximation as you say.

For a system to qualify as Keplerian, it must be gravitationally non-viscous. Our solar system would qualify if the planets are assumed to not gravitationally interact with each other. However, if there is an N-body system, such as a galaxy, in which particles, ie. stars, within the system interact with each other, then the system would be gravitationally viscous and therefore non-Keplerian. George O. Abell correctly calculated the mass of M31 using the tangential velocity of an outer star in the galaxy and Gaus's Law. His result was in line with the overall measured luminosity of the galaxy. 

well, how do the orbital period (galaktic year ~2,25–2,50×10^8 yıl) of  the sun according to the center of the galaxy compute? with the 3rd law of kepler, is not it?

@Mehmet and Bruce. Orbital period of the Sun from Kepler's 3rd law modified (Eq.2.10) gives 2.234x10^8 yrs.

What values are you using for v, R and P? Are you assuming the 3rd law to be correct and then backing out your values or are they independently measured?

@Bruce. As shown in my article "Supplement to periodic relativity" (link given under question), I have used v = 8 Kpc, v = 220 Km/s, and n = 0.56566 which is the deviation factor that modifies Kepler's third law and satisfies Einstein's field equation. Using this values P is calculated as given in previous answer. Kepler's 3rd law by itself cannot give this answer. My question is whether you see any problem with this modified version.

Vikram, I have read over the link. If I may, I would like to mention a couple of things. If velocity is measured as distance travelled over time and these measurements are using clocks and rulers in flat space, then the tangential velocity would be 2 \pi R /T where T is the measured period (again as measured by a clock in flat space and R is measured by a ruler in flat space). On the other hand, measuring tangential velocity using the shifting of spectral lines is highly questionable due to general relativistic effects of rotation, which you mention and utilize. The general relativistic effects due to local gravitational fields in a galaxy, in the flat segment of the rotation curve, would be negligible. That being said, I do not feel the metric you provide in 1.19 yields a solution to predicting rotation curves. Rather than attempting to describe the rotation curve of the Galaxy, perhaps compare to that of M31 or, better yet, NGC 3198, whose rotation curve has been studied to exhaustion.

@Bruce. The second solution of Einstein's field equation that I have utilized in the listed article yiled a constant k which I think provides a measure of uneven distribution of galactic matter at a given radial distance. When n is determined with this constant, it gives a very negligible deviation from the coordinate time. So the relativistic effect in the flat portion of the curve is negligible as you say. However this deviation factor also appear in the expression for the acceleration and there it behaves similar to the MOND function and in much more precise manner. MOND is an effective theory but theory given here is a physical theory because it explains why it is so. So the constant k plays the key role which cannot be obtained from the general relativity line element but only from the metric (1.19) which directly brings it up into the expression for the acceleration. This is not possible in general relativity because of the weak field approximation. I am looking into M31 and NGC 3198. But if this solution fits one galaxy then the perfect fit with all other galaxies is automatically assured.

Hello Vikram. The luminosity profile indicates a constant linear distribution of matter, (Shapley). However, the matter distribution is not linear unless one applies Fermi coordinates and a transformation into L-1 space. The relativistic effects in the flat part of the rotation curve are significant; however, the Lorenz factor is from rotational effects rather than gravitational. The metric for rotating systems has already been solved and matches the rotation curve of M31, NGC 3198 and about 870 other galaxies with a rather acceptable degree of precision.

Hello Bruce. Cold dark matter is not luminicent and makes up 29% of the matter. Visible matter is only 4%. Most of the CDM is in galaxy halos. Constant k accounts for CDM as well. Constant k is due to relativistic effect because it originates from Einstein's field equations as a constant of integration. Lorentz factor is from rotational effect ok. But the modified Lorentz factor in my theory which includes the deviation factor n is both rotational as well as gravitational which can be seen from Eq.(2.6). Here we can see that the deviation factor n is a ratio of Newtonian gravitational acceleration to the measured acceleration (v^2/r). This relation cannot be obtained in general relativity because of the weak field approximation. I plugged in the rotation curve data for M31 (Corbelli) and NGC3198 (Begeman) and they fit very well. The metric for rotating system cannot account for the proper time of the star. This is the only theory that can do it.

Thanks for your post. Although I disagree with many if your statements I wish you the best. It appears you have no further need of any advice I could give.

Forget Einstein,

Stars are driven by angular momentum of black hole. Mutual repulsion of spiral arms should be taken in account.

@Edgar. What do you mean by mutual repulsion of spiral arms?

Cold dark matter is not luminicent and makes up 29% of the matter. Visible matter is only 4%. Most of the CDM is in galaxy halos.

The 2013 Planck results indicate that dark matter represents 26.8% of the total matter and energy in the universe – while ordinary (baryonic) matter represents 4.9%. Accordingly, dark matter is considered to represent nearly 85% of all material mass while ordinary matter represents about 15%. See http://www.esa.int/Our_Activities/Space_Science/Planck/Planck_reveals_an_almost_perfect_Universe.

These are universal proportions – various galaxies and galaxy types are considered to have varying configurations of dark matter halos. In addition, galaxy clusters and filaments of the large scale ‘cosmic web’ structure is thought to have very larger dark matter components – I suspect greater than the total for all galactic halos…

Also, please see http://arxiv.org/abs/1101.3224.

Dear James:

Thank you for the accurate information. I wrote 29% from the top of my head and immediately remembered that it is about 26% but did not bother to correct it,

Following article is now published in "Progress in Physics"

Zaveri V.H., Periodic relativity: deflection of light, acceleration, rotation curves. Progress in Physics, 2015, v.11(1), 43-49.

https://arxiv.org/abs/0805.2233

��pR��

Dear colleagues,

I have recently derived a model based on gravitomagnetism in which the flat rotation curve agree with the model. This is available at

Article Flat rotation curve without dark matter: the generalized New...

I would appreciate comments about this model.

Arbab

Arbab,

In your article you say:

“Therefore, the dark matter exists because of theoretical prejudice of using the ordinary

Newtonian law of gravitation to account for all astronomical phenomena.”

But the fact is that the (Cold) Dark Matter (CDM) was primarily proposed to explain the structure formation in the universe. Later general relativity utilized the same dark matter to explain the flat rotation curves. The latest cosmological model is also named Lambda-CDM cosmological model. So if your theory eliminates the dark matter then you will have to explain how large scale structure is formed in the universe in the absence of the dark matter.

.�����

If you consider the galaxy as apoint or if you calculate it according to the center of the galaxy.

My following article shows that Kepler’s third law can be modified with a deviation factor n given by eqns.(43 and 50) to include the effect of the dark matter present in the galaxy. With this modified law (eqn.49) orbital period of a star can be calculated with respect to the center of the galaxy.

https://arxiv.org/abs/0805.2233

*�p��