The theory of the gravitational redshift of gravitons continues to be relevant in the study of galactic phenomena. In this paper the process is investigated at a deeper level in spiral galaxy rotation velocities. It is notable that only the baryonic mass of the galaxy is a free parameter, with other parameters obtained from the observation, essentially just the radial velocity distribution $v\left(r\right)$, with the final radial distance $r_f$ and velocity $v\left(r_f\right)$ used to determine the graviton coupling coefficient $k_f$ [1] .

The SPARC database [2] [3] is used in the spiral galaxy analysis. It provides the baryonic mass, rotational velocity and surface brightness magnitudes for each galaxy. The velocity distributions derived from SPARC mass to light ratio analysis are also presented as a comparison to the Newtonian rotational velocity curves derived using the graviton redshift derived baryonic mass distribution. For the two galaxies presented here, NGC 2403 and NGC 2841, it is notable that the agreement with the standard method is quite good, because the graviton theory derived baryonic mass accounts for the bulge, disk and gas mass without any special adjustments.

In a new field for this theory, gravitational lensing is investigated using Einstein rings [4] . It is found that nothing new needs to be added to the galaxy version of the theory, except that, since galactic rotation velocities were not obtained, the coupling coefficient $k_f$ is determined using the lens mass.

For the layout of this paper, in the next section the rotation velocity relation for spiral galaxies is revisited and the comparisons are made to the predictions of the standard theory. In the section after that, the surface brightness magnitues for those galaxies are considered. In the next section and subsections the gravitational lens relations are described and the connection to the galaxy rotation energy with the graviton redshift energy boost is established. These results are discussed in the final section.

The gravitational potential at radial position r in the field of a galaxy of total baryonic mass $M_b$, including graviton energy loss [1] , can be expressed by,

$\Phi \left(r\right)=-\frac{GM\left(r\right)}{r}-k_f\frac{M\left(r\right)}{M_b}{\displaystyle {\int }_0^r}\frac{GM\left(s\right)}{s^2}\text{d}s,$ (1)

where $M\left(r\right)$ is the baryonic mass within r, and $k_f$ is a coupling coefficient. The negative sign on the second term on the right hand side of Equation (1) represents the energy loss due to the gravitational redshift of gravitons traveling outward against the field of the galaxy of mass $M\left(r\right)$ toward an orbiting object of mass m. The equation for the total energy of the mass m in the galaxy is given by,

$\frac{1}{2}m{v}^2\left(r\right)+m\Phi \left(r\right)=E_0\mathrm{,}$ (2)

where $v\left(r\right)$ is the velocity at r and E₀ is the total energy. For circular motion, $E_0=-GmM\left(r\right)/2r$. The equation for circular motion in the field, from Equation (2), is given by,

$v^2\left(r\right)=\frac{GM\left(r\right)}{r}+2k_f\frac{M\left(r\right)}{M_b}{\displaystyle {\int }_0^r}\frac{GM\left(s\right)}{s^2}\text{d}s\mathrm{.}$ (3)

Equation (3) can be put into a form quadratic in the mass $M\left(r\right)$, expressed by,

$v^2\left(r\right)=\frac{GM\left(r\right)}{r}+2k_f\frac{M\left(r\right)}{M_b}{\displaystyle {\int }_0^{r_1}}\frac{GM\left(s\right)}{s^2}\text{d}s+2k_f\frac{{\left(M\left(r\right)\right)}^2}{M_b}{\displaystyle {\int }_{r_1}^r}\frac{G}{s^2}\text{d}s\mathrm{,}$ (4)

where, for the last term on the right hand side, the mass $M\left(r\right)$ has been taken out of the integral because $r_1$ is defined such that, for $r_1<s\le r$, $M\left(s\right)\approx M\left(r\right)$. Given an estimate for the total galaxy baryonic mass $M_b$, a mass distribution can be determined [5] by iteratively solving Equation (4) for $M\left(r\right)$ until the mean error between the predicted and observed rotational velocities is within acceptable bounds. The galaxy total baryonic mass is obtained from the SPARC mass to light derived velocity estimates for the bulge and disk models and from the independently determined gas mass observations. The baryonic mass is estimated using the final position R and final bulge, disk and gas velocities $V_{bulge}\mathrm{,}{V}_{disk}$ and $V_{gas}$, respectively, by,

$M_b=\frac{R}{G}\left(V_{bulge}^2+V_{disk}^2+V_{gas}^2\right)\mathrm{.}$ (5)

Figure 1 shows for galaxy NGC 2403 the rotation velocity curves for the Newtonian, graviton redshift and the combined total velocity fitted to the observed rotation velocities and assumed baryon mass. Also shown on the figure are the rotation curve components for the gas mass and the disk stellar mass based on photometric observations, and the combined Newtonian velocity distribution.

Figure 2 shows for galaxy NGC 2841 the rotation velocity curves for the Newtonian, graviton redshift and the combined total velocity fitted to the observed rotation velocities and assumed baryon mass. Also shown on the figure are the rotation curve components for the gas mass and the stellar masses for the bulge and disk based on the photometric observations, and the combined Newtonian velocity distribution. Table 1 summarizes the parameters used in making these galaxy fits.

The mass to light ratio $\Upsilon \left(r\right)$ within radial position r in a galaxy is based on the observed surface brightness magnitude $\mu \left(r\right)$ with source reference ${\mu }_0$, which is expressed by,

$\Upsilon \left(r\right)={10}^{\left(\mu \left(r\right)-{\mu }_0\right)/2.5}\frac{M_{stars}\left(r\right)}{4\pi r^{2}Area\left(r\mathrm{,}d\right)B_0}\mathrm{,}$ (6)

where $B_0$ is the detector calibration zero point reference (count), where count ~ energy/time/area, $M_{stars}\left(r\right)$ is the mass of the stars within r, given by,

$M_{stars}\left(r\right)=M\left(r\right)-\frac{r{V}_{gas}^2\left(r\right)}{G}\mathrm{,}$ (7)

where $V_{gas}\left(r\right)$ is the molecular gas velocity at r, and the area $Area\left(r\mathrm{,}d\right)$ in square arcseconds spanned by a radius r of a galaxy at distance d from an observing detector is given by,

$Area\left(r\mathrm{,}d\right)={\left(\frac{r}{d}\frac{180}{\pi }3600\right)}^{\text{}2}\mathrm{.}$ (8)

In terms of Equations (6)-(8), the luminosity $L\left(r\right)$ at the radial position r from the center of a galaxy is defined by,

$L\left(r\right)=\frac{M_{stars}\left(r\right)}{\Upsilon \left(r\right)}\mathrm{.}$ (9)

The brightness $B\left(r\right)$ is the luminosity thru the surface area defined by,

$B\left(r\right)=\frac{L\left(r\right)}{4\pi r^2}\mathrm{.}$ (10)

In terms Equations (6)-(10) the surface brightness magnitude $\mu \left(r\right)$ is defined by,

$\mu \left(r\right)=-2.5\log \left(\frac{B\left(r\right)}{Area\left(r\mathrm{,}d\right)B_0}\right)+{\mu }_0\mathrm{,}$ (11)

where $B_0$ is the detector calibration zero point reference count, d is the distance from the observing detector to the source and ${\mu }_0$ is a source surface brightness reference.

Figure 3 shows for NGC 2403 the surface brightness magnitudes (SB) for the observed and predicted SB fit based on the predicted stellar mass. The reference offset was $Z_0=27.204$ to fit to the interpolated curve to the observed SB curve. (The interpolation aligns the observed SB radial positions with the observed galaxy velocity positions.) Figure 4 shows for NGC 2841 the surface brightness magnitudes for the observed and predicted surface brightness magnitudes fit based on the predicted stellar mass. The reference offset was $Z_0=26.941$ to fit to the interpolated curve. The detector calibration zero point value used is the same for both results at $B_0=7.5\times {10}^{23}$ count per observation.

The gravitational potential $\Phi \left(r\right)$ describing the interaction of a photon with a galaxy having total baryonic mass $M_b$ including graviton interacion is defined by,

$\Phi \left(r\right)=-\frac{G{M}_b}{r}+k_g{\displaystyle {\int }_0^r}\frac{GM\left(u\right)}{u^2}\text{d}u\mathrm{,}$ (12)

where G is the gravitational constant, $M_b$ is the total baryonic mass of the galaxy, $k_g$ is the graviton to photon coupling coefficient and $M\left(r\right)$ is the baryonic mass at radial position r. The positive value of the second term in Equation (12) indicates it is an increase in the graviton energy (blue shift) as they travel from the photons toward the galaxy mass. The deviation in the path of the photon as it travels from the distant star, passing a distance b at closest approach to the galaxy and continuing on to the detector at the earth is given by [6] ,

${\nabla }_{\perp }\Phi \left(r\right)=\frac{\partial \Phi \left(r\right)}{\partial b}=\frac{\partial \Phi \left(r\right)}{\partial r}\frac{\partial r}{\partial b}\mathrm{.}$ (13)

Assume,

$r=\sqrt{b^2+z^2}\mathrm{,}$ (14)

where b is the radial distance from the galaxy center which is beyond the bounds of the galaxy baryonic mass and z is on a line perpendicular to b which extends from the photon source to the observing detector. ¹With this definition of the nearest approach b to the galaxy, assuming $\partial k_g/\partial r=0$, the derivative of $\Phi \left(r\right)$ with respect to r is given by,

$\frac{\partial \Phi \left(r\right)}{\partial r}=\frac{G{M}_b}{r^2}+k_g\frac{GM\left(r\right)}{r^2}=\frac{G{M}_{lens}}{r^2}\mathrm{,}$ (15)

where the lens mass is defined by,

$M_{lens}=\left(1+k_g\right)M_b\mathrm{,}$ (16)

And $M\left(r\right)=M_b$ because r is greater than the galaxy radius $R_{gal}$. The derivative of r with respect to b, using (14), is given by

$\frac{\partial r}{\partial b}=\frac{b}{r}\mathrm{.}$ (17)

Substituting from Equations (15) and (17) into Equation (13) yields,

${\nabla }_{\perp }\Phi \left(r\right)=\frac{bG{M}_{lens}}{r^3}\mathrm{.}$ (18)

Refer to Figure 5 for an illustration of the ray paths associtated with a gravitational lens system, where it is shown that the total deflection angle of the photon traveling from the source to the observer is given by,

$\theta \left(b\right)=\alpha \left(b\right)+\beta \left(b\right)\mathrm{.}$ (19)

This total deflection angle $\theta \left(b\right)$ is obtained initially by integrating Equation (18) along the line z, expressed by,

$\begin{array}{c}\hat{\alpha }\left(b\right)=\frac{2}{c^2}{\displaystyle {\int }_{-\infty }^{\infty }}\text{d}z{\nabla }_{\text{}\text{}\perp }\Phi \left(r\right)\\ =\frac{2bG{M}_{lens}}{c^2}{\displaystyle {\int }_{-\infty }^{\infty }}\frac{\text{d}z}{{\left(b^2+z^2\right)}^{3/2}}\\ =\frac{2G{M}_{lens}}{c^{2}b}{\displaystyle {\int }_{-\infty }^{\infty }}\frac{\text{d}x}{{\left(1+x^2\right)}^{3/2}}\\ =\frac{4G{M}_{lens}}{c^{2}b}\mathrm{,}\end{array}$ (20)

where there is a change of variable $x=z/b$. In Equation (20), the first factor of 2 comes from general relativity theory and the second factor of 2 comes from the

integral of $\text{d}x{\left(1+x^2\right)}^{-3/2}$ with x going from $-\infty$ to $+\infty$.

Then, from Equation (20) the angle $\alpha \left(b\right)$ is given by,

$\alpha \left(b\right)=\frac{D_{LS}}{D_S}\hat{\alpha }\left(b\right)=\frac{D_{LS}}{D_S}\frac{4G{M}_{lens}}{c^{2}b}\mathrm{,}$ (21)

where $D_L$ is the angular diameter distance from the observing detector to the lens, $D_S$ is the angular diameter distance from the detector to the source, and $D_{LS}$ is the angular diameter distance from the lens to the source. The impact parameter of the light path to the lens is b, given by

$b=D_L\theta \left(b\right)\mathrm{.}$ (22)

For a source which is on the center line thru the detector and lens, the angle $\beta \left(b\right)=0$, which, by combining Equations (19), (21) and (22), implies that,

$\theta \left(b\right)=\alpha \left(b\right)=\frac{D_{LS}}{D_S}\hat{\alpha }\left(b\right)=\frac{D_{LS}}{D_S}\frac{4G{M}_{lens}}{c^2{D}_L\theta \left(b\right)}.$ (23)

In this case, where $\beta \left(b\right)=0$, from Equation (23) the angle ${\theta }_E=\theta \left(b\right)$ of the Einstein ring is given by,

${\theta }_E=\sqrt{\frac{D_{LS}}{D_L{D}_S}\frac{4G{M}_{lens}}{c^2}}\mathrm{,}$ (24)

which is the standard expression. For $\beta \left(b\right)\ne 0$, use Equations (19), (21), (22) and (24) to derive the quadratic equation in $\theta$,

${\theta }^2\left(b\right)-\beta \left(b\right)\theta \left(b\right)-{\theta }_E^2=\mathrm{0,}$ (25)

which has two solutions,

$\theta {\left(b\right)}_{\pm }=\frac{1}{2}\left(\beta \left(b\right)\pm \sqrt{{\beta }^2\left(b\right)+4{\theta }_E^2}\right)\mathrm{.}$ (26)

The fitting of spiral galaxy rotation curves using the theory of graviton gravitational redshift in the galaxy field may be perceived as not adding anything new to the explanation of high velocity rotations which flatten out at large radial distances, since these are adequately explained by other dark matter theories, particularly, in principle, in the framework of extended gravity [17] , although gravitons are more likely to exist than dark matter particles and the graviton redshift theory can explain binary star system decay without the emission of gravitational waves [18] . Also, showing how the stellar mass distribution derived from the graviton redshift method can be used to explain the galaxy surface brightness magnitudes does not offer anything extraordinary.

However, it is in gravitational lensing where this theory may have revealed its significance, because it has been shown how the graviton redshift theory can explain the lens mass when dark matter theory cannot, as in the cases of the ring systems in JWST-ER1 and COSMOS 0018 + 3845. The fundamental property about graviton gravitational redshift is that the total galaxy mass $M_{gal}\left(r\right)\propto r$ at large distance, since the gravitational redshift continues to act at larger distance r and add more relativistic mass to the system. Assume that most of the baryonic mass $M_b$ is within a radius $R_e\le R_{gal}$. Then, for $r>R_e$, expand Equation (48) into the form,

$\begin{array}{c}{M}_{gal}\left(r>R_e\right)\approx M_b+2k_{f}r\left({\displaystyle {\int }_0^{R_e}}\frac{M\left(u\right)}{u^2}\text{d}u+M_b{\displaystyle {\int }_{R_e}^r}\frac{\text{d}u}{u^2}\right)\\ =M_b+2k_{f}r\left(\frac{M_b}{R_a}+\frac{M_b}{R_e}-\frac{M_b}{r}\right)\\ =\left(1-2k_f\right)M_b+Br,\end{array}$ (50)

where $\frac{M_b}{R_a}-{\displaystyle {\int }_0^{R_e}}\frac{M\left(u\right)}{u^2}\text{d}u$ for some distance $R_a$ and $B=2k_f{M}_b\left(\frac{1}{R_a}+\frac{1}{R_e}\right)$. All of the lens systems in this study displayed this approximately $M_{gal}\left(r\right)\propto r$ relation, although the integrations were terminated at the galaxy radius $R_{gal}$.

The behavior of extending the graviton interaction distance beyond the galaxy radius would result in a stream of concentric rings at the lens site from a single source beyond the lens galaxy (or galaxy cluster.) This needs further study.

¹For our purposes it will be assumed that there is no mass in the thin shell region of space around the distance b from the galaxy center, even though graviton redshift generates a relativistic mass in what is traditionally empty space.