This paper is a development of earlier papers [1] and [2] . There the concept was introduced that if gravitons exist, and given that they are bosonic relativistic particles, like photons but of spin 2, then, like photons traveling in a gravitational field, the gravitons traveling in the field would experience a gravitational redshift as they go from a lower negative potential near the source mass to a higher potential going toward an orbiting mass. This energy loss $\delta {\xi }_g$ due to the gravitational redshift is expressed by,

$\delta {\xi }_g\left(r\right)=-\frac{GM\left(r\right)m}{r^2}\delta r,$(1)

where $G$ is Newton’s gravitational constant, $M\left(r\right)$ is the baryonic source mass at $r$, $m$ is the relativistic mass of the gravitons and $\delta r$ is a small change in the position of the gravitons. Integrating Equation (1) from the origin of the system of masses to the position $r$ of the gravitons and multiplying by a coupling coefficient $K_g\left(r\right)$ of the gravitons to the mass at position $r$ yields the total energy loss $\Delta {\Xi }_g\left(r\right)$ given by,

$\begin{array}{c}\Delta {\Xi }_g\left(r\right)=K_g\left(r\right){\displaystyle {\int }_0^r\delta {\xi }_g\left(s\right)}\\ =-k_{\text{}f}\frac{M\left(r\right)}{M_b}{\displaystyle {\int }_0^r\frac{GM\left(s\right)m}{s^2}\delta s},\end{array}$(2)

where the coupling coefficient

$K_g\left(r\right)=k_{\text{}f}\frac{M\left(r\right)}{M_b},$(3)

where $k_{\text{}f}$ is a coupling constant, $M\left(r\right)$ is the baryonic mass within radial position $r$ and $M_b$ is the total baryonic mass of the galaxy.

In addition to gravitons experiencing reduction of their energy due to gravity, it is hypothesised that even in galaxies, gravitons are also susceptible to the expansion of the universe and undergo an energy loss $\delta {\xi }_e$ due to cosmological redshift of their energy, which is assumed to take the form,

$\delta {\xi }_e=-m{c}^2{k}_e\frac{\delta r}{R_0},$(4)

where $m$ is the relativistic mass of the gravitons, $c$ is the speed of gravitons (speed of light) in vacuum, $k_e$ is a coupling constant, $\delta r$ is the change in distance from the galaxy center and $R_0=c/H_0$ is the radius of the universe where $H_0$ is the Hubble constant at the epoch of interaction. Integrating Equation (4) from the origin of the system of masses to the position $r$ of the gravitons yields the total energy loss $\Delta {\Xi }_e\left(r\right)$ given by,

$\Delta {\Xi }_e\left(r\right)={\displaystyle {\int }_0^{\Delta {\Xi }_e\left(r\right)}\delta {\xi }_e}=-{\displaystyle {\int }_0^{r}m{c}^2{k}_e\frac{\delta s}{c/H_0}}=-m{k}_{e}c{H}_{0}r.$(5)

Adding $\Delta {\Xi }_g\left(r\right)$ from Equation (2) and $\Delta {\Xi }_e\left(r\right)$ from Equation (5), while treating them as potential energies, they can be combined with the gravitational potential energy in the form,

$-\frac{mGℳ\left(r\right)}{r}=-\frac{mG}{r}\left(M\left(r\right)+\frac{k_{\text{}f}M\left(r\right)r}{M_b}{\displaystyle {\int }_0^{\infty }\frac{M\left(s\right)}{s^2}\text{d}s}+\frac{k_{e}c{H}_0{r}^2}{G}\right),$(6)

where the total galaxy mass $ℳ\left(r\right)$ is defined by,

$ℳ\left(r\right)=M\left(r\right)+\frac{k_{\text{}f}M\left(r\right)r}{M_b}{\displaystyle {\int }_0^r\frac{M\left(s\right)}{s^2}\text{d}s}+\frac{k_{e}c{H}_0{r}^2}{G}.$(7)

Then, since by the virial theorem the kinetic energy is minus half the potential energy, with Equation (6), the standard form for the equation of circular motion in a galaxy becomes,

$\frac{1}{2}m{v}^2\left(r\right)-\frac{mGℳ\left(r\right)}{r}=-\frac{mGℳ\left(r\right)}{2r},$(8)

which expresses the graviton redshift theory (GRST) version of the energy equation of motion of an object of mass $m$ in a circular orbit in the galaxy,

$\begin{array}{l}\frac{1}{2}m{v}^2\left(r\right)-\frac{GM\left(r\right)m}{r}-k_{\text{}f}\frac{M\left(r\right)}{M_b}{\displaystyle {\int }_0^r\frac{GM\left(s\right)m}{s^2}\delta s}-m{k}_{e}c{H}_{0}r\\ =-\frac{GM\left(r\right)m}{2r}-k_{\text{}f}\frac{M\left(r\right)}{2M_b}{\displaystyle {\int }_0^r\frac{GM\left(s\right)m}{s^2}\delta s}-m\frac{k_{e}c{H}_0}{2}r.\end{array}$(9)

Moving all terms except $v^2\left(r\right)$ from the left hand side (l.h.s.) to the right hand side (r.h.s.) of Equation (9), while changing $\delta s$ to $\text{d}s$, and simplifying, yields the expression for the rotational velocity at the radial distance $r$ from the galaxy center, given by,

$v^2\left(r\right)=\frac{GM\left(r\right)}{r}+\frac{k_{\text{}f}M\left(r\right)}{M_b}{\displaystyle {\int }_0^{\infty }\frac{GM\left(s\right)}{s^2}\text{d}s}+k_{e}c{H}_{0}r.$(10)

Equation (10) is the new equation which will be used to fit the rotation curves of spiral galaxies and to derive a distribution form for the baryonic Tully-Fisher relation (BTFR).

This paper relates to the study of the dynamics of spiral galaxy rotation curves found in [3] and will access the data provided by the SPARC data base [4] to which that study refers.

Assuming that the galaxy rotation velocity is constant in a neighborhood of radial position $r_0$ which is near the furthest radial position of the galaxy, take the derivative with respect to $r$ of the velocity in Equation (10), assuming the mass is constant so $\text{d}M\left(r\right)/\text{d}r=0$ at $r_0$, and setting the result to zero at $r_0$ gives,

$0=-\frac{GM\left(r_0\right)}{r_0^2}+\frac{k_{\text{}f}G{\left(M\left(r_0\right)\right)}^2}{M_b{r}_0^2}+k_{e}c{H}_0,$(11)

which can be solved for $k_e$, expressed by

$k_e=\frac{GM\left(r_0\right)}{c{H}_0{r}_0^2}\left(1-\frac{k_{\text{}f}M\left(r_0\right)}{M_b}\right).$(12)

To obtain a formula for $k_{\text{}f}$, solve for it using Equation (10) evaluated at $r=r_0$, expressed by,

$k_{\text{}f}=\frac{v^2\left(r_0\right)-\frac{GM\left(r_0\right)}{r_0}-k_{e}c{H}_0{r}_0}{\frac{M\left(r_0\right)}{M_b}{\displaystyle {\int }_0^{r_0}\frac{GM\left(s\right)}{s^2}\text{d}s}},$(13)

which after substitution for $k_e$ using Equation (12) reduces to,

$k_{\text{}f}=\frac{v^2\left(r_0\right)-\frac{2GM\left(r_0\right)}{r_0}}{\frac{M\left(r_0\right)}{M_b}\left({\displaystyle {\int }_0^{r_0}\frac{GM\left(s\right)}{s^2}\text{d}s}-\frac{GM\left(r_0\right)}{r_0}\right)}.$(14)

The next section will define the updated quadratic equation for the baryonic mass distribution which includes the cosmological term.

With the additional cosmological term, an updated quadratic equation for the baryonic mass distribution $M\left(r\right)$ can be obtained, as first described in [2] . The second term on the r. h. s. of Equation (10) can be split into two parts as follows,

$\frac{k_{\text{}f}M\left(r\right)}{M_b}{\displaystyle {\int }_0^r\frac{GM\left(s\right)}{s^2}\text{d}s}=\frac{k_{\text{}f}M\left(r\right)}{M_b}{\displaystyle {\int }_0^{r_i}\frac{GM\left(s\right)}{s^2}\text{d}s}+\frac{k_{\text{}f}M\left(r\right)}{M_b}{\displaystyle {\int }_{r_i}^r\frac{GM\left(s\right)}{s^2}\text{d}s},$(15)

where $r_i<r$ and in the second term on the r.h.s., in the integrand, it is assumed that $M\left(s\right)=M\left(r\right)$ for $r_i\le s\le r$. In other words, for a small enough interval $\text{Δ}r=r-r_i$, the galaxy mass at that interval is the mass at radial position $r$. Thus in the above Equation (15), in the second term on the r.h.s., the mass $M\left(s\right)$ in the integrand can be moved out of the integral, giving the square of mass $M\left(r\right)$ in the form,

$\frac{k_{\text{}f}M\left(r\right)}{M_b}{\displaystyle {\int }_{r_i}^r\frac{GM\left(s\right)}{s^2}\text{d}s}=\frac{k_{\text{}f}{\left(M\left(r\right)\right)}^2}{M_b}{\displaystyle {\int }_{r_i}^r\frac{G}{s^2}\text{d}s}.$(16)

Combining the results of Equations (15) and (16) into Equation (10), after simplification, yields the equation quadratic in baryonic mass $M\left(r\right)$,

$\left(\frac{k_{f}G}{M_b}{\displaystyle {\int }_{r_i}^r\frac{\text{d}s}{s^2}}\right){\left(M\left(r\right)\right)}^2+\left(\frac{G}{r}+\frac{k_{f}G}{M_b}{\displaystyle {\int }_0^{r_i}\frac{M\left(s\right)}{s^2}\text{d}s}\right)M\left(r\right)+k_{e}c{H}_{0}r-{\left(v\left(r\right)\right)}^2=0.$(17)

By defining the three parameters $a\left(r\right)$, $b\left(r\right)$ and $c\left(r\right)$ in the form,

$a\left(r\right)=\frac{k_{f}G}{M_b}{\displaystyle {\int }_{r_i}^r\frac{\text{d}s}{s^2}},$(18)

$b\left(r\right)=\frac{G}{r}+\frac{k_{f}G}{M_b}{\displaystyle {\int }_0^{r_i}\frac{M\left(s\right)}{s^2}\text{d}s},$(19)

$c\left(r\right)=k_{e}c{H}_{0}r-{\left(v\left(r\right)\right)}^2,$(20)

we obtain solutions for $M\left(r\right)$ in the familiar form,

$M\left(r\right)=\frac{1}{2a\left(r\right)}\left(-b\left(r\right)+\sqrt{{\left(b\left(r\right)\right)}^2-4a\left(r\right)c\left(r\right)}\right),$(21)

where the positive square root was chosen to keep the mass positive and it is apparent that the quantity under the square root is always non-negative since $a\left(r\right)$ is positive and $-c\left(r\right)$ is positive since ${\left(v\left(r\right)\right)}^2>2k_{e}c{H}_0$ probably in most phy-sical cases¹.

Knowing the galaxy total baryonic mass $M_b$ and the observed rotation curve with radial positions and velocities $\left(r_i,v_i\right)$, Equation (10) along with Equations (12) and (14) for $k_e$ and $k_{\text{}f}$, respectively, and with the distribution of the baryonic mass $M\left(r\right)$ given by Equations (18) to (21), an iterative fit of the equation of motion to galaxy data can be obtained.

From Equation (10), move the cosmological term to the l.h.s. of the equation, factor out the baryonic mass $M\left(r\right)$ from the first and second terms on the r.h.s., divide the l.h.s. by the resulting factor to $M\left(r\right)$, multiply the l.h.s. by the unity factor $v^4/v^4$, and factor out $v^2\left(r\right)r$, with some simplification, to obtain the baryonic Tully-Fisher relation (BTFR),

$\begin{array}{c}M\left(r\right)=\left(\frac{v^2\left(r\right)r-k_{e}c{H}_0{r}^2}{G{v}^4\left(r\right)\left(1+\frac{k_{\text{}f}r}{M_b}{\displaystyle {\int }_0^r\frac{M\left(s\right)}{s^2}\text{d}s}\right)}\right)v^4\left(r\right)\\ =\left[\frac{r\left(1-\frac{k_{e}c{H}_{0}r}{v^2\left(r\right)}\right)}{G{v}^2\left(r\right)\left(1+\frac{k_{\text{}f}r}{M_b}{\displaystyle {\int }_0^r\frac{M\left(s\right)}{s^2}\text{d}s}\right)}\right]v^4\left(r\right)\\ =A\left(r\right)v^4\left(r\right),\end{array}$(22)

where the BTFR normalization $A\left(r\right)$ [5] [6] is defined by,

$A\left(r\right)=\frac{r\left(1-\frac{k_{e}c{H}_{0}r}{v^2\left(r\right)}\right)}{G{v}^2\left(r\right)\left(1+\frac{k_{\text{}f}r}{M_b}{\displaystyle {\int }_0^r\frac{M\left(s\right)}{s^2}\text{d}s}\right)}.$(23)

Although the standard BTFR is applied only at the flat rotation part of the galaxy at location $R_f$ with velocity $V_f$, the GRST version of the BTFR Equation (23), after fitting $v\left(r\right)=v_{obs}\left(r\right)$ for any $r$ of the galaxy, is extended to the entire observed rotational velocity and baryonic mass range of the galaxy.

It has been observed in numerous reports that when the BTFR is expressed in the form,

$V_f^4=\frac{M_b}{A}=a_{0}G{M}_b,$(24)

where $A$ is the normalization, $a_0$ is an acceleration, and $M_b$ is the total baryonic mass of the galaxy, then the values of $a_0$ for many spiral galaxies falls into the approximate range 1 × 10⁻¹⁰ m·s⁻² to 1.5 × 10⁻¹⁰ m·s⁻², which is about $0.15c{H}_0$ to $0.3c{H}_0$. This forms the basis of the modified Newtonian dynamics theory (MOND) [7] . Inverting Equation (23) to solve for the velocity $v^4\left(r\right)$ in terms of $M\left(r\right)/A\left(r\right)$, the cosmological acceleration $c{H}_0$ can be factored out in the form,

$v^4\left(r\right)=\left(\frac{1}{A\left(r\right)}\right)M\left(r\right)=a_0\left(r\right)GM\left(r\right),$(25)

where the acceleration $a_0\left(r\right)$ is given by,

$a_0\left(r\right)=\left(\frac{v^2\left(r\right)}{r}\right)\frac{1+\frac{k_{\text{}f}r}{M_b}{\displaystyle {\int }_0^r\frac{M\left(s\right)}{s^2}\text{d}s}}{1-\frac{k_{e}c{H}_{0}r}{v^2\left(r\right)}}.$(26)

Although the cosmic acceleration $c{H}_0$ appears in the acceleration $a_0\left(r\right)$ of Equation (26), because $k_e$ is inversely proportional to $c{H}_0$, the cosmic acceleration drops out from $a_0\left(r\right)$ and from any term in any equation that contains the factor $k_{e}c{H}_0$.

The GRST can generate a distributed version of the fundamental modified Newtonian dynamics (MOND) acceleration constant $a_0$ which is the foundation of that theory [7] [8] . MOND theory is defined using an interpolating function $\mu \left(x\right)$ which transforms the real rotation acceleration $v^2/r$ in a galaxy into the Newtonian acceleration $GM/r^2$ based on the galaxy baryonic mass $M$, expressed by,

$\mu \left(\frac{g\left(r\right)}{a_0}\right)g\left(r\right)=g_N\left(r\right),$(27)

where

$\begin{array}{l}g\left(r\right)=\frac{v^2\left(r\right)}{r}\hat{r},\\ g_N\left(r\right)=\frac{GM\left(r\right)}{r^2}\hat{r},\\ g\left(r\right)=\left|g\left(r\right)\right|=\frac{v^2\left(r\right)}{r},\\ a_0=1.22\times {10}^{-10}\text{m}\cdot {\text{s}}^{-2},\end{array}$(28)

where $g\left(r\right)$ is the acceleration of an orbit in a galaxy, $g_N$ is the Newtonian acceleration and $M\left(r\right)$ is the baryonic mass. A sticky point is that $a_0$ is not a constant but varies around $\left(1.27\pm 0.30\right)\times {10}^{-10}\text{m}\cdot {\text{s}}^{-2}$ for the standard $\mu \left(x\right)$ in the analysis of different galaxies [8] . Using the GRST, now with a cosmological term, the acceleration $a_0$ becomes a variable applicable to any distance $r$ in a galaxy and settling to around the MOND value for any particular galaxy. This can be shown by using Equation (10), setting the galaxy acceleration $g\left(r\right)=v^2\left(r\right)/r$. Dividing Equation (10) by $r$ yields,

$g\left(r\right)=\frac{v^2\left(r\right)}{r}=\frac{GM\left(r\right)}{r^2}+\frac{2k_{\text{}f}M\left(r\right)}{r{M}_b}{\displaystyle {\int }_0^r\frac{GM\left(s\right)}{s^2}\text{d}s}+2k_{e}c{H}_0.$(29)

The Newtonian acceleration is the standard and using Equation (29) becomes,

$g_N\left(r\right)=\frac{GM\left(r\right)}{r^2}=\frac{\frac{v^2\left(r\right)}{r}\left(1-\frac{k_{e}c{H}_{0}r}{v^2\left(r\right)}\right)}{1+\frac{k_{\text{}f}r}{M_b}{\displaystyle {\int }_0^r\frac{M\left(s\right)}{s^2}\text{d}s}}.$(30)

The MOND theory can be expressed in terms of the BTFR, dividing Equation (25) by $r^2$,

$\frac{v^4\left(r\right)}{r^2}=a_0\left(r\right)\frac{GM\left(r\right)}{r^2},$(31)

which, using Equations (29) and (30) can be put into the form,

${\left(g\left(r\right)\right)}^2={\left(\frac{v^2\left(r\right)}{r}\right)}^2=a_0\left(r\right)\frac{GM\left(r\right)}{r^2}=a_0\left(r\right)g_N\left(r\right).$(32)

From Equation (32) we solve for $a_0\left(r\right)$ in the form,

$a_0\left(r\right)=\frac{{\left(g\left(r\right)\right)}^2}{g_N\left(r\right)},$(33)

which, substituting for $g_N\left(r\right)$ from Equation (30), after simplification, becomes,

$\begin{array}{c}{a}_0\left(r\right)=\frac{{\left(g\left(r\right)\right)}^2}{g_N\left(r\right)}={\left(g\left(r\right)\right)}^2\frac{1+\frac{k_{\text{}f}r}{M_b}{\displaystyle {\int }_0^r\frac{M\left(s\right)}{s^2}\text{d}s}}{\frac{v^2\left(r\right)}{r}\left(1-\frac{k_{e}c{H}_{0}r}{v^2\left(r\right)}\right)}\\ =\left(\frac{v^2\left(r\right)}{r}\right)\frac{1+\frac{k_{\text{}f}r}{M_b}{\displaystyle {\int }_0^r\frac{M\left(s\right)}{s^2}\text{d}s}}{1-\frac{k_{e}c{H}_{0}r}{v^2\left(r\right)}},\end{array}$(34)

where substitution for $g=v^2\left(r\right)/r$ was made using Equation (28).

The standard form of the interpolating function $\mu \left(x\right)$ equation in MOND is defined,

$\mu \left(x\right)=\frac{x}{\sqrt{1+x^2}},$(35)

where

$x=\frac{g}{a_0}=\frac{v^2\left(r\right)/r}{a_0},$(36)

where the magnitude of the galaxy radial acceleration $g\left(r\right)$ at position $r$ is related to the Newtonian radial acceleration magnitude $g_N\left(r\right)$ by the definition,

$\mu \left(\frac{g\left(r\right)}{a_0}\right)g\left(r\right)=g_N\left(r\right).$(37)

For the GRST the acceleration $a_0$ takes on a continuous property, Equation (34), where $a_0=a_0\left(r\right)$ at radial position $r$. For this analysis, the acceleration distribution $a_0\left(r\right)$ will be used instead of constant $a_0$ in the MOND interpolating function Equation (37).

For the GRST, the interpolating function ${\mu }_g\left(x\right)$ is defined using Equations (27) and (34),

${\mu }_g\left(\frac{g\left(r\right)}{a_0\left(r\right)}\right)g\left(r\right)=\left(\frac{g\left(r\right)}{a_0\left(r\right)}\right)g\left(r\right)=g_N\left(r\right),$(38)

where,

$\begin{array}{ll}{\mu }_g\left(\frac{g\left(r\right)}{a_0\left(r\right)}\right)\hfill & =\frac{1-\frac{k_{e}c{H}_{0}r}{v^2\left(r\right)}}{1+\frac{k_{\text{}f}r}{M_b}{\displaystyle {\int }_0^r\frac{M\left(s\right)}{s^2}\text{d}s}}.\hfill \end{array}$(39)

With the BTFR baryonic mass input of $M_b=2.464\times {10}^{10}{\text{M}}_{\odot }$ for NGC 3198, which is matched by the GRST, the total graviton redshift (gravitational and cosmological) relativistic mass at $r_0=40.1\text{kpc}$ determined by the model is $M_g=18.513\times {10}^{10}{\text{M}}_{\odot }$ (see Table 1 ). The BTFR acceleration at normalization $A=50$ is $a_0=1.507\times {10}^{-10}\text{m}\cdot {\text{s}}^{-2}$. Table 2 shows the parameters determined by the iterative fitting using total rotation energy Equation (10), coupling constants Equations (12) and (13), and the quadratic equations to determine the baryonic mass, Equations (18) to (21). The normalization parameter for the fit was $48.68{\text{M}}_{\odot }\cdot {\text{s}}^4\cdot {\text{km}}^{-4}$ which compares well to the default BTFR normalization. The GRST acceleration at the $r_0$ is $a_0=1.548\times {10}^{-10}\text{m}\cdot {\text{s}}^{-2}$.

Figure 1 shows the rotation curves for the disk, gas and graviton gravitational and cosmological redshifts and the sum of these fitting to the galaxy rotation velocity. Figure 2 shows the baryon mass distribution for $M\left(r\right)$ which was derived by iteration of the suite of equations including the rotation Equation (10) and the mass quadratic Equation (21). Figure 3 shows the masses for the baryons and graviton gravitational and cosmological redshifts, the sum of these masses equal to the galaxy mass, and also equal to the galaxy mass determined using the observed rotation velocities and positions.

Comparing with the dark matter paper [9] which determined for NGC 3198, for a maximum disk mass model, a disk mass $M_{disk}=3\times {10}^{10}{\text{M}}_{\odot }$ and a spherical halo model dark matter mass $M_{dm}=12\times {10}^{10}{\text{M}}_{\odot }$ for a total galaxy mass of $M_{total}=15\times {10}^{10}{\text{M}}_{\odot }$ at radius $R_f=30\text{kpc}$. On the other hand, with the GRST model the total galaxy fit to radius 44.08 kpc predicts a baryon mass $M_{b30}=2.321\times {10}^{10}{\text{M}}_{\odot }$ at 30 kpc radius and a graviton redshift (gravitational and cosmological) relativistic mass of $M_{g30}=12.587\times {10}^{10}{\text{M}}_{\odot }$ for a total galaxy mass at 30 kpc of $14.908\times {10}^{10}{\text{M}}_{\odot }$. This is a fair match to the dark matter study considering the slight difference in baryonic mass between the two models.

Comparing with the MOND paper [8] where the acceleration constant $a_0=0.9\times {10}^{-10}\text{m}\cdot {\text{s}}^{-2}$ for a NGC 3198 distance of $D=11.2\text{Mpc}$, the GRST acceleration $a_0\left(r=30\text{kpc}\right)=1.475\times {10}^{-10}\text{m}\cdot {\text{s}}^{-2}$. Figure 4 displays a plot for the GRST version of the MOND standard interpolating function $\mu \left(x\right)$ of Equation (35) where $x=g\left(r\right)/a_0\left(r\right)$ along with the GRST interpolating function ${\mu }_g\left(r\right)=g\left(r\right)/a_0\left(r\right)$ given by Equation (39). The mean absolute difference (MAE)² between the GRST version of the standard MOND $\mu \left(x\right)$ and the GRST ${\mu }_g\left(x\right)$ for the fit to NGC 3198 is 2.974 × 10⁻⁴.

In [8] , in Fig. 8 of that paper at radial distance 30 kpc, the Newtonian velocity for the stellar disk is about 50 km·s⁻¹ with a mass to light of 0.76 whilst the gas velocity is about 49 km·s⁻¹ implying a total baryonic mass of baryonic mass $M_{mond}=3\times {10}^{10}{\text{M}}_{\odot }$. With an acceleration $a_0=0.9\times {10}^{-10}\text{m}\cdot {\text{s}}^{-2}$ and using the simple interpolating function, ${\mu }_s\left(x\right)=x/\left(1+x\right)$, ${\mu }_s\left(g\left(r=30\text{kpc}\right)/a_0\right)\approx 0.202$, implies a rotation velocity $v\left(r=30\text{kpc}\right)=145.6\text{km}\cdot {\text{s}}^{-1}$, which is within the range of the observed rotation data. Using the GRST version of the standard interpolating function ${\mu }_g\left(x\right)$ of Equation (39), with the same mass $M_{mond}$, with an $a_0=1.12\times {10}^{-10}\text{m}\cdot {\text{s}}^{-2}$, the rotation velocity at 30 kp is 146.1 km·s⁻¹ which agrees with the observed rotation velocity at that position.

Figure 5 shows the distribution of the acceleration $a_0$ for NGC 3198. The value of the acceleration at near the end of the galaxy at $r_0=40.1\text{kpc}$ is $a_0\left(r_0\right)=1.548\times {10}^{-10}\text{m}\cdot {\text{s}}^{-2}$.

The addition of the graviton interaction with the expanding universe in the form of a cosmological redshift to its energy as it travels in the galaxy was discovered in the process of assessing the value for the graviton gravitational redshift coupling constant $k_{\text{}f}$. If the graviton cosmological redshift constant $k_e$ is dropped in Equation (11), since $M\left(r_0\right)=M_b$, this would imply that $k_{\text{}f}=1$, a constant, which is problematical because fits do not work well for NGC 3198 using this value of the coupling constant. The value for $k_{\text{}f}$ is peculiar to each galaxy. Thus, the idea of the graviton cosmological redshift energy loss for galaxies was born. And, the remarkable aspect is that the role of these two types of graviton redshifts in galaxies is analogous to the roles of graviton energy redshifts in the expansion of the universe. The GRST, with gravitational and cosmological graviton interactions, seems to solve the mystery of dark matter and dark energy in spiral galaxies and the expanding universe.

The GRST equation of motion does well at explaining the extra mass found in a spiral galaxy. In algorithmic form, the equation of motion giving $v^2\left(r\right)$, the formulas to determine the coupling coefficients $k_{\text{}f}$ and $k_e$ and the quadratic equation to determine the baryonic mass distribution $M\left(r\right)$ provide a powerful tool to handle the dynamics of spiral galaxies. It is notable that both the BTFR and the MOND theory are derivable in continuous form with this theory. The mass in the relation $M\left(r\right)=A\left(r\right)v^4\left(r\right)$ represents the baryonic mass throughout the galaxy, not just at the galaxy edge. Likewise for the MOND relation, the acceleration $a_0\left(r\right)=v^4/GM\left(r\right)$ is continuous for the galaxy and there is no need to determine a transition radius from high to low acceleration. Finally, the new cosmic normalization parameter $\mathcal{A}$ was introduced in the discussion section as a possible model for the BTFR acceleration $a_0$.

¹Addendum: In [2] there was an extra factor of the gravitation constant G which must be removed in one of the terms of both Equation (13) and Equation (15) of that paper.

²The mean absolute error (MAE) is defined as the average of the sum of the absolute values of the corresponding pairwise differences of two sets of variables $\left\{x_i\right\}$ and $\left\{y_i\right\}$.