The classical velocity orbit formula is

where is the orbital velocity and is the central mass. It is NOT . It is . So the issue is what is a reasonable model of mass distribution of a galaxy in terms of radius . The most reasonable hypothesis is that it is assuming it’s two dimensional. Then the expected velocity rotation curve is

So all the graphs that people present showing anomalous rotational velocities are severely misleading because the observed galaxy rotations are actually extremely close to the above relation. A discrepancy between the observed velocities and the above relation can be easily explained by GRAVITATIONAL REDSHIFT. Gravitational redshift formula is approximately (see Gravitational redshift)

So the approximate velocity due to gravitational redshift is just this multiplied by the speed of light. I will illustrate by a simple procedure in R how to get a from actual data so that the relation has statistical significance with on actual data which can be found here:Galaxy rotation dataGalaxy rotation dataGalaxy rotation data. So take for example the data for DDO 64

r v verr

1 1.5 6.3 4.9

2 4.5 7.6 3.0

3 7.5 9.9 2.6

4 10.5 13.8 1.4

5 13.5 9.1 2.5

6 16.5 16.6 3.2

7 19.5 14.2 2.5

8 25.5 13.3 1.2

9 28.5 22.4 2.4

10 31.5 25.2 5.0

11 34.5 32.0 4.0

12 40.5 30.8 1.7

13 43.5 41.6 2.9

14 46.5 44.6 2.3

15 49.5 36.7 4.0

16 52.5 51.2 3.9

17 55.5 44.9 1.8

18 58.5 53.6 13.0

19 61.5 48.1 9.8

20 64.5 59.8 9.0

The columns are radius, velocity, velocity error. We conside a power law relationship so in R summarize linear regression of log-velocity versus log-radius:

> summary(lm(log(v)~log(r))) Call: lm(formula = log(v) ~ log(r)) Residuals: Min 1Q Median 3Q Max -0.59982 -0.09126 0.01336 0.19051 0.54301 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.02706 0.21837 4.703 0.000177 *** log(r) 0.66710 0.06551 10.183 6.75e-09 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.2843 on 18 degrees of freedom Multiple R-squared: 0.8521, Adjusted R-squared: 0.8439 F-statistic: 103.7 on 1 and 18 DF, p-value: 6.752e-09

So we see that the relationship that fits (with very significant p-value) is a power law . This is a bit higher than the simple classical model suggests, i.e. . So let’s assume that this deviation is due to gravitational redshift, which by the same model will be based on the mass-by-radius formula for a two-disc, where the constant is different in varoous occurrences above since we are interested in the power of in this exercise. Well, a little experimentation gives us:

> summary(lm(log(v-0.0059*r^2)~log(r))) Call: lm(formula = log(v - 0.0059 * r^2) ~ log(r)) Residuals: Min 1Q Median 3Q Max -0.64957 -0.08524 -0.00853 0.19738 0.35968 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.27580 0.20315 6.280 6.38e-06 *** log(r) 0.50058 0.06095 8.214 1.68e-07 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.2645 on 18 degrees of freedom Multiple R-squared: 0.7894, Adjusted R-squared: 0.7777 F-statistic: 67.46 on 1 and 18 DF, p-value: 1.681e-07

So if we consider the action of gravitational redshift to lower the velocities we recover approximately quite easily. It is not hard to check that this is true about the other examples as well. The significance of this exercise is that in fact there is no missing mass at all if we make the assumption that the mass is approxmately distributed as which can be obtained from the luminous matter without the need for any mysterious dark matter. We do not need to modify gravity laws (which is what many people had done to explain the anomalous rotation curves). We don’t need to introduce dark matter either. The problem is that the ‘expected Keplerian velocities’ are a red herring since they assume that objects distributed inside the galaxy can treat the center with the same mass. This assumption is silly. A point at a given radius from the center should experience the center of mass as a function of radius obviously. So this tells us that standard Newtonian gravity with Einstein’s gravitational redshift as correction WITHOUT dark matter explains the galaxy rotations without any problems. There is no evidence of dark matter from galaxy rotation velocities.

Another example to ensure that we are not making an unjustified extrapolation. Data for UGC 1281:

#UGC,1281

5.0,2.5,2.2

7.0,11.2,2.8

9.0,7.9,3.1

13.0,15.6,1.2

15.0,12.9,3.3

17.0,8.8,2.3

23.0,9.6,1.8

25.0,12.1,4.8

27.0,8.9,1.3

29.0,7.0,3.5

31.0,14.1,5.0

33.0,18.3,1.4

35.0,16.2,4.0

37.0,14.1,4.5

39.0,12.1,2.9

41.0,20.4,2.3

43.0,19.3,1.8

45.0,24.3,2.1

47.0,25.4,5.8

49.0,17.8,6.3

51.5,33.5,3.7

54.5,30.1,2.7

60.5,26.5,1.2

63.5,33.6,3.1

66.5,45.8,1.9

69.5,37.8,5.5

Without any gravitational redshift adjustment, we get a law :

> summary(lm(log(v-0.00*r^2)~log(r))) Call: lm(formula = log(v - 0 * r^2) ~ log(r)) Residuals: Min 1Q Median 3Q Max -0.79011 -0.22565 0.00927 0.20012 0.73392 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.2390 0.3605 0.663 0.514 log(r) 0.7416 0.1040 7.129 2.28e-07 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.3665 on 24 degrees of freedom Multiple R-squared: 0.6793, Adjusted R-squared: 0.6659 F-statistic: 50.83 on 1 and 24 DF, p-value: 2.279e-07

With a loss of goodness-of-fit but without losing significance of the power law fit, we can obtain :

> summary(lm(log(v-0.0035*r^2)~log(r))) Call: lm(formula = log(v - 0.0035 * r^2) ~ log(r)) Residuals: Min 1Q Median 3Q Max -1.01665 -0.23224 0.02452 0.20987 0.70148 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.7161 0.4291 1.669 0.108115 log(r) 0.5051 0.1238 4.080 0.000431 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.4363 on 24 degrees of freedom Multiple R-squared: 0.4095, Adjusted R-squared: 0.3849 F-statistic: 16.65 on 1 and 24 DF, p-value: 0.0004306

We are treating gravitational redshift with mass distribution for simplicity. Our central point here is that galaxy rotation curves in the classical model with mass distribution should follow rather than so the anomaly in these rotation curves is small perturbation in the power law from expected . Gravitational redshift can account for the discrepancy. There should be no need to modify gravitational laws and no need to hypothesize dark matter to explain the rotation curves.