Rotation curves of spiral galaxies [1] have been suspected not to confirm to gravitational forces due to galaxies’ visible mass as per the Newton’s Law of gravitation, which is known to work well in our day-to-day experience on earth as well for planetary orbits in our solar system. In order to explain the observed rotation curves, it has been proposed and long believed that there is significant amount of invisible “dark matter” surrounding almost all spiral galaxies. There was no other existing theory which could explain the rotation behavior in a satisfactory manner, although modification of the laws of Newtonian dynamics has been proposed [2]. Recently, a new unified electro-gravity (UEG) theory is established, which has been successfully applied to model an elementary particle and the Casimir Effect [3,4], where a new gravitational force proportional to electromagnetic energy density, is introduced. In this paper, the simple UEG theory of [3,4], applicable for the spherically symmetric structure of an elementary particle, would be extended for the non-spherical structure of a spiral galaxy. The energy density due to star lights in the galaxy would contribute to a new gravitational force, which could support the observed stellar rotation around the galaxy. A constant rotation speed beyond certain radial distance would require a $1/r$-dependent gravitational acceleration, in the given region. When the UEG theory of [3,4] is properly modified for the non-spherical structure of a spiral galaxy, the required $1/r$-dependent acceleration may result, although the stellar light radiation from the galaxy exhibit an approximate $1/r^2$ dependence, in the given region. This is possible, because the energy density of the actual light radiation may need to be redistributed, based on the physical asymmetry of the spiral galaxy. The UEG field may be defined in proportion to the redistributed, effective energy density.

The required UEG constant $\gamma$ of proportionality, between the UEG field and the associated effective energy density, may be deduced from the new UEG model using measured data from galaxy survey as well as data for selected individual galaxies. The results may be compared with the UEG constant deduced from [3,4], for validation or verification of the new UEG model. The functional trends established from the new UEG model may be compared, for validation of the model, with those from the empirical Tully-Fisher Relation (TFR) [5] and the Modified Newtonian Dynamics (MOND) model [2,6]. The trends predicted from the UEG model would explicitly depend upon the spiral galaxy’s aspect ratio (ratio of the scale lengths in radius and thickness), because the new model is formulated based on the spherical asymmetry of the galaxy. This is distinct from the the MOND model, where there may not be such definitive interrelation between the galaxy’s aspect ratio and the rotation speed. The functional dependence of the galaxy’s aspect ratio on the surface brightness and rotation velocity, as required for the UEG galaxy model to reproduce the rotation curves, may be compared with available relevant measurements, for another independent validation of the basic UEG galaxy model.

As mentioned earlier, the galaxy’s UEG force field is to be defined in proportion to an effective distribution of energy density, not the actual energy density of stellar radiation as was the case in [3,4] for a spherical structure. The effective energy density may be obtained by suitable redistribution of the galaxy’s light radiation, in proportion to the distribution of the Newtonian gravitation potential of the galaxy, as discussed in the following section. The divergence of the resulting UEG force field surrounding the galaxy would be equivalent to having a fictitious “dark-matter” distribution [7], on the basis of the conventional Newtonian gravitation, which may be needed in order to explain the observed rotation behavior of the spiral galaxies, as well as formation and evolution of the galaxies. Beyond a sufficiently large radial distance from the galactic center, the galaxy would “look” like a point body with a spherically symmetric distribution of the Newtonian potential, and with a $1/r^2$ dependence of its light intensity. In this far region the radial UEG field would also be spherically symmetric, and therefore the field would be directly proportional to the $1/r^2$-dependent light’s energy density, without any need for redistribution of the energy density as per the proposed model. This spherically symmetric, $1/r^2$-dependent radial UEG field in the far region is associated with zero field divergence, and therefore with no dark matter. In contrast, the intermediate region where the conventional Newtonian potential exhibits strong spherical asymmetry, the UEG field (with the expected $1/r$ dependence) would be associated with a strong divergence, which would emulate heavy presence of the dark-matter in the region. The UEG field in the innermost region, where the spherical symmetry of the Newtonian potential is generally restored, would be associated with only minimal presence of the dark-matter, much smaller in magnitude than the intermediate region. The intermediate region of heavy dark-matter presence would at least include the smallest spherical region which encloses most of the galaxy’s mass and light sources, and may extend much farther.

Section 2 presents the theoretical concepts and an analytical formulation of the theory. The results for flat rotation velocity deduced from the model are validated with measured data for a galaxy survey as well as for an individual galaxy, in Section 3, Section 4. The Tully-Fisher Relation (TFR) and the Modified Newtonian Dynamics (MOND) model are studied in Section 5, in relation to the present UEG galaxy model, for further validation of the model. Possible extension of the theory to other astrophysical problems are discussed in Section 6, followed by general conclusion from the study in 7.

We first estimate the $\gamma$ based on (8), using an average data point from the I-band measurement of the galaxy survey [9]. As suggested above, the data point is located approximately at the statistical center of the survey samples.

Similarly, we estimate the $\gamma$ from the K-band measurement of [9]. Note that an effective radius, $R_e$, is provided in [9] for the K-band measurements. The effective radius, defined as the radius of a sphere that encloses half of the total luminosity, would be $1.678$ times the scale radius R used in our modeling, assuming an exponential light profile.

Measurements in the K-band overestimates the luminosity and the energy density, leading to underestimation of the $\gamma$. On the other hand, measurements in the I-band underestimates the energy density, leading to overestimation of the $\gamma$. Accordingly, the above results estimate a useful range for the value of the $\gamma$, which is consistent with the value of the $\gamma =0.6\times {10}^3$ $\left({\mathrm{ms}}^{-2}\right)/\left({\mathrm{Jm}}^{-3}\right)$ deduced from the UEG model [3,4] in the Appendix A for an elementary particle.

The best estimate for $\gamma$ is assumed to be the average of the two estimates in the $I-$ and $K-$ bands.

The above estimate closely agrees with the $\gamma$ from the particle model [3,4] in the Appendix A. Considering that we used a first-order approximation in the UEG modeling of (1,2), such agreement is remarkable. This means that the ideal conditions we assumed in the first-order UEG modeling of (1,2) are remarkably valid for the central data point of [9] used in our estimation.

Measured data for the surface brightness distribution $\mu \left(r\right)$ of a specific galaxy is first properly fitted with an exponential, and then an effective surface brightness distribution ${\mu }_e\left(r\right)$, as defined in (4). The data using mixed units, such as magnitude, arcsec, light-years, may be converted to suitable standard units. The ${\mu }_e$ distribution can then be related to the rotation velocity v using (7).

The UEG constant $\gamma$ is deduced using the amplitude b, or its equivalent parameter $s_0$, of the effective surface brightness distribution ${\mu }_e\left(r\right)$, the flat rotation velocity v and the distance d of the galaxy. Suitable correction factors may be needed to relate the K- and U-band measured magnitudes to a common reference of solar bolometric magnitude of $4.83$. This assumes the solar magnitudes in the K- and U-bands are $3.28$ and $5.56$, respectively.

Using the U-band (assumed ≃ U’-band) surface-brightness data [10] for the galaxy NGC-2403, presented in Figure 2, we estimate the amplitude parameter $s_0=32.9$. This parameter, together with the galaxy’s distance $d=11.4$MLyr [11] and flat rotation velocity $v=1.35\times {10}^5$m/s [12], would provide an estimate for the ${\gamma }_u=0.81\times {10}^3$$\left({\mathrm{ms}}^{-2}\right)/\left({\mathrm{Jm}}^{-3}\right)$, using the above relation (14). Similarly, using the K-band data [13] for the same galaxy NGC-2403, presented in Figure 3, we estimate the amplitude parameter $s_0=430$. This would provide an estimate for the ${\gamma }_k=0.51\times {10}^3$$\left({\mathrm{ms}}^{-2}\right)/\left({\mathrm{Jm}}^{-3}\right)$, using (14). An average of these two estimates for the $\gamma$ would lead to the best estimate for the $\gamma =0.66\times {10}^3$$\left({\mathrm{ms}}^{-2}\right)/\left({\mathrm{Jm}}^{-3}\right)$ from the available data for the galaxy NGC-2403. This is close to the $\gamma =0.63\times {10}^3$$\left({\mathrm{ms}}^{-2}\right)/\left({\mathrm{Jm}}^{-3}\right)$ deduced from the galaxy survey in (12) or the $\gamma =0.60\times {10}^3$$\left({\mathrm{ms}}^{-2}\right)/\left({\mathrm{Jm}}^{-3}\right)$ from particle model [3,4] in the Appendix A. Such remarkable agreement implies that any deviation from the basic model of (1-4) due to differences in the surface brightness ${\mu }_0$ (see Section 5)) of the individual galaxy NGC-2403 from the “average” galaxy used in the estimation (12), is minimal. The ${\mu }_{0,k}$ are estimated to be roughly equal to $16.75$$\left({\mathrm{mag}/\mathrm{arcsec}}^2\right)$ in both cases ([9], Figure 3), which is consistent with the above expectation.

Combining (5,7) and assuming an approximately constant ${\mu }_0$, a Tully-Fisher Relation (TFR) [5] may be deduced, where the total luminosity L would be proportional to the fourth power of the flat rotation velocity v. As mentioned before, the above condition of an approximately constant ${\mu }_0$ is satisfied by a large group of high surface brightness (HSB) galaxies that were believed to confirm to the Freeman’s Law [8].

However, the Freeman’s Law is no longer believed to be strictly valid, and galaxies are measured to exhibit a broad range of amplitudes ${\mu }_0$ covering variations among the HSB galaxies as well as extending to low surface brightness (LSB) galaxies with lower values of ${\mu }_0$. For a general treatment to closely model the variation in the amplitude ${\mu }_0$, we may introduce a new parameter $\alpha$ for fitting the $1/r$ profile of ${\mu }_e$ with the exponential profile of $\mu$ in (4). The unit reference value of $\alpha$ is expected to apply for an “average” HSB galaxy, as assumed in the basic model of (4) and in the estimations of (12,15). The ${\mu }_e$ may be adjusted to a smaller or larger value, relative to the $\mu (r=R)$, with a proportional adjustment of the parameter $\alpha$, which would represent a smaller or large value of the UEG force, respectively, as per (7).

The variable factor $\alpha$ is accommodated in the gravitational potential model of (1,2), Figure 1, by recognizing the galaxy thickness $z_0$ to be an active variable, like the scale radius R or the surface brightness ${\mu }_0$, for parametrization of galaxy characteristics. In the potential model of (1), an approximately uniform (spherically) potential would be established for all radial distances less than a variable threshold radius $R_t$, dependent on a variable thickness $z_0$, not less than the ideal fixed threshold radius $r=R$ assumed in (2). Accordingly, the effective energy density $W_{\tau e}$ would match with the actual energy density $W_{\tau }$ for all the radial distances less than the variable threshold radius, not the ideal reference threshold $r=R$ assumed in (2). Consequently, the $W_{\tau e}(r=R)$ would no longer be equal to $W_{\tau }(r=R)$ as ideally assumed in (2), but now be equal to $\alpha W_{\tau }(r=R)$, with the variable factor $\alpha$ proportional to the normalized galaxy thickness $R/z_0$.

The model of (1,2) may be revised as follows, as explained above.

Using the above revisions and (3), the relation (4) between the surface density $\mu$ and effective surface density ${\mu }_e$, and the resulting expression for the luminosity L (5) using (7), may also be revised.

The TFR (16), which was established based on the simple assumption of an approximately constant ${\mu }_0$, would still be valid for a range of different surface brightness ${\mu }_0$, if ${\mu }_0{\alpha }^2$ in (18) is approximately a constant. This condition, of having a larger value of the $\alpha$ for a lower ${\mu }_0$, means there would be relatively more contribution from the UEG force as the surface brightness ${\mu }_0$ reduces. This trend better represents observed characteristics among the HSB galaxies, extending to LSB galaxies as well. The higher UEG contribution for a lower surface brightness ${\mu }_0$ would be equivalent to having relatively more “dark matter” contribution for a LSB galaxy [14], as per the current dark-matter paradigm.

The above TFR of having the luminosity proportional to the fourth power of the velocity v, is also consistent with prediction from an alternate model using a modified Newtonian dynamics (MOND) [2,6].

As derived in (17,18), the parameter $\alpha$, which proportionately represents the equivalent distribution $W_{\tau e}$ or ${\mu }_e$, is proportional to the normalized galaxy scale $R/z_0$. Accordingly, the condition (19) of a constant factor ${\mu }_0{\alpha }^2$, required for the validity of the TFR or MOND, would be satisfied if the normalized scale parameter $(z_0/R)$ is proportional to the square-root of the surface brightness ${\mu }_0$. This general trend, of having the normalized galaxy thickness $z_0/R$ to be smaller for a lower surface brightness ${\mu }_0$, may seem to be a sensible characteristic. The specific required relationship between the galaxy thickness and the surface brightness may be compared and verified with the measured data in [15].

Using the above required relationship (19) between the ${\mu }_0$ and the normalized scale $z_0/R$ in (18,7) would translate to another galaxy scaling relationship between the absolute thickness $z_0$ (not normalized to R) and the flat rotation velocity v.

Accordingly, the galaxy thickness $z_0$ is required to be proportional to the square of the flat rotation velocity $v=v_{max}$. This required relationship is clearly verified from the measured data of [15], as presented in the Figure 4. It is significant to note that the above two required relations (a) between the galaxy normalized thickness $z_0/R$ and the surface brightness ${\mu }_0$, and (b) between the thickness $z_0$ and the flat rotation velocity v, are independently predicted from the UEG model of (17,18), based on the observed TFR (19,16), but could not have been anticipated either from the TFR of [5] or the MOND [2,6]. Verification of the above predictions from [15] is a significant development, which strongly validates the new UEG model of (1,17), as applied to the non-spherical structure of a galaxy.

The UEG theory for spiral galaxies is expected to be extended as well to model excess gravitational mass observed in galaxy clusters. The UEG acceleration produced due to the Cosmic Microwave Background (CMB) radiation surrounding a galaxy cluster could support the measured velocity dispersion and gravitational lensing in the galaxy cluster, emulating the effect of the hypothetical dark matter. For example, the effective gravitational mass $M=1.2\times {10}^{15}{M}_0$ of the Virgo cluster [17,18], measured at the outer boundary of the cluster of diameter $d=4.6Mpc$ (± 8 degrees, observed at the cluster’s average distance of $R=16.5$Mpc) would be associated with an effective gravitational acceleration $a=4GM/d^2$ $=3.18\times {10}^{-11}m/s^2$, where $G=6.67\times {10}^{-11}{m}^3{s}^{-2}k{g}^{-1}$ is the universal gravitational constant and $M_0=2.0\times {10}^{30}kg$ is the reference solar mass. Assuming that the hypothetical dark matter content of the cluster is typically about $90\%$ of the total gravitational mass [19], the required non-Newtonian acceleration is about $a^{\prime }=0.9a=2.86\times {10}^{-11}m/s^2$. This can be emulated by the UEG acceleration $a_{UEG}=\gamma W_{\tau \left(CMB\right)}=2.54\times {10}^{-11}m/s^2$ due to the energy density $W_{\tau \left(CMB\right)}=(4\sigma /c)T^4$ of the CMB radiation surrounding the cluster, where $\sigma =5.670\times {10}^{-8}W/\left(m^2{K}^4\right)$ is the Stefan-Boltzman constant, $T=T_0{(1+z)}^4$ the CMB temperature around the cluster with a red-shift $z=0.00385=H_{0}R/c$, $T_0=2.{725}^{0}K$ the current CMB temperature, $H_0\simeq 70km{s}^{-1}/Mpc$ the current Hubble constant, $\gamma =600(m/s^2)/(J/m^3)$ the UEG constant [3,4] from Appendix A, and $c=3\times {10}^8$ the speed of light. The required non-Newtonian acceleration $a^{\prime }$ is quite close to the UEG acceleration $a_{UEG}$ within $12\%$ of difference, suggesting validity of the UEG theory. The small difference may, however, be accommodated by having the measured distance R (hence cluster diameter d) larger about $6\%$ ($R\simeq 17.6$MPc). This is very likely, considering that the measurement of the distance R for the Virgo cluster, based on measurement of distance of individual galaxies in the cluster, has a significant spread [18].

Similarly, the effective gravitational mass $M=5.1\times {10}^{14}{M}_0$ of the Bullet cluster [20,21] would be associated with an effective gravitational acceleration $a=4GM/d^2$ $=7.92\times {10}^{-11}m/s^2$, measured at a selected spherical boundary of diameter $d\simeq 1.9Mpc$ (expected to be somewhat larger than an estimated $d\simeq 1.5Mpc$ of the cluster’s 2-D gravitational-lensing boundary). The required non-Newtonian acceleration $a^{\prime }=0.9a=7.13\times {10}^{-11}m/s^2$ can be emulated by the UEG acceleration $a_{UEG}=\gamma W_{\tau \left(CMB\right)}=7.14\times {10}^{-11}m/s^2$ due to the energy density $W_{\tau \left(CMB\right)}=(4\sigma /c)T_0^4{(1+z)}^4$ of the CMB radiation surrounding the cluster, with measured redshift $z=0.3$. Like for the Virgo cluster, the required non-Newtonian acceleration $a^{\prime }$ for the Bullet cluster is reasonably emulated by the UEG acceleration $a_{UEG}$, reconfirming the success of the UEG theory. Similar to the modeling for the spiral galaxies, the distribution of the UEG force, and equivalently of the gravitational mass for the non-spherical structure of the bullet cluster, is expected to be most concentrated around the region of strong Newtonian gravitation, which would clearly be confined around the radial axis connecting the Bullet sub-clusters. The analysis maybe followed further, based on a useful simplified model that the region of strong Newtonian potential due to the Bullet sub clusters constitute a cylindrical space enclosing the sub clusters, with the cylinder’s cross-sectional diameter close to the diameter of each sub cluster (assumed symmetrical), and the length of the cylinder extending somewhat beyond the axial ends of the clusters. The uniform UEG acceleration due to the uniform CMB radiation would be associated with equivalent gravitational mass, enclosed within in a spherical volume of a given radius, to increase with the square of the radius. This mass, to be concentrated in the cylindrical volume described above, as per the proposed UEG theory, would result in linearly increasing density of the gravitational mass in the cylinder, with increasing radial distance away from the center. This would result in the peak density towards the axial ends of the cylindrical region, which is radially offset away from the Newtonian mass centers of the sub clusters. This predicted offset of the gravitational mass concentration would be consistent with the gravitational lensing measurements of the Bullet cluster [22]. Any finer improvement of the analysis may involve proper modeling of the Newtonian potentials due to the separate gaseous and visible (galaxies) mass concentrations, as well as including additional contributions from any equivalent UEG gravitational mass outside of the idealized cylindrical region discussed above.

The above success of the UEG model for the Virgo and the Bullet clusters, is expected to extend as well to other galaxy clusters. Using median values of cluster mass $M=5\times {10}^{14}{M}_0$ (${10}^{14}$ to ${10}^{15}{M}_0$) and diameter $d=6Mpc$ (2 to $10Mpc$) [19], the calculated acceleration $a=0.78\times {10}^{-11}m/s^2$ at the outer boundary of a median cluster is lower than the UEG acceleration $a_{UEG}=2.5\times {10}^{-11}m/s^2$ (assuming redshift $z\sim 0$, see calculations above). The difference between the accelerations a and $a_{UEG}$ would be reduced, with somewhat larger M and/or lower d. The calculations suggest there is adequate UEG acceleration to emulate the clusters’ gravitational effects, without any hypothetical dark matter.

The role of the hypothetical dark matter (possibly dark energy) in a cosmological model could also be similarly emulated by the UEG acceleration due to the CMB radiation (as well as current and future star lights). Specific details of the cosmological modeling based on the new UEG theory is beyond the scope of the present work, and is separately explored [23].

The estimate of the UEG constant $\gamma$ from measured data from a galaxy survey [9], based on the new UEG model, agrees well with an accurate value derived from the UEG model in the Appendix A for an elementary particle [3,4]. This is based on a statistically average data point from the survey samples. Direct analysis of measured brightness profile and rotation curve of a specific selected galaxy is also illustrated to provide a similar estimate for the $\gamma$, that is consistent with the estimate from the galaxy survey. Further, the UEG galaxy model confirms to the TFR [5,16] for varying range of galaxy amplitudes, and is consistent with results from a modified Newtonian dynamics (MOND) [2,6] model. The required condition for the agreement between the UEG model, TFR and MOND is supported by measured relations of the galaxy thickness with the surface brightness and the rotation velocity [15], which may be considered as an independent validation of the UEG model. The above studies strongly support validity of the new UEG model, established for the non-spherical structure of a disk galaxy. The UEG theory is intended to serve as a theoretical substitute for the current “dark-matter” hypothesis. Further, the UEG model may similarly be applicable to model excess gravitation observed in galaxy clusters, as well as to model cosmological expansion of the universe.

The UEG theory, which is shown to be consistently applicable to model elementary particles [3,4] and spiral galaxies, and is extended as well to galaxy clusters and possibly for cosmological modeling, could provide a new unified theoretical paradigm for a broad range of physical concepts, covering both small and large size scales of nature, and spherically symmetric as well as asymmetric structures.