The Solution of Dark Matter Problem by the Galactic Electrostatic Field

1. Introduction

It is a fact that most spiral galaxies of the nearby Universe do have flat rotation curves but not those in the remote Universe [1]. Since more than half a century flat rotation curves of some hundreds of spiral galaxies have been well measured. On the contrary, it is known that the gravitational pull of galactic materials leads to decreasing rotation curves, not flat. Therefore, a problem developed during 90 years which was termed Dark Matter problem in spiral galaxies.

In other terms, the matter amount reckoned by astronomers generates an attractive force which is not sufficient to account for the observed flat rotation curves of stars and clouds orbiting the Galactic center. In a vast and persistent literature, imaginary material entities called Dark Matter are posited in such a way to exert an attractive force on material bodies. This paper presents a solution of dark matter problem : the attractive force is generated by the Galactic electrostatic field named here ${\overrightarrow{E}}_g$ (g on foot for Galactic) and, consequently, Dark Matter as material bodies of any stock does not exist at all.

The force is necessarily attractive because the Galactic electrostatic field is directed toward the Galactic center [2,3] and rotating stars and clouds globally retain positive electric charge deposited by the extinguished cosmic rays.

Stars and clouds in the Galaxy retain a notable amount of positive electric charge deposited by cosmic-ray nuclei pervading the Galaxy. Consider as an example the $Sun$, a typical G star. Charge deposition by cosmic rays occurs in two different modes in two separated stellar regions : the first process is the stopping of low-energy cosmic rays in the solar wind region and the second process is the stopping of all cosmic rays of low and high energies in the $Solar$ body by ionization and nuclear interactions. By solar body is meant all the high density material residing within photosphere. The electric charge per second deposited in the solar wind region and in the $Sun$ will be designated by $Q_{sw}$ (s for solar w for wind) and $Q_{ds}$ (ds for dense $Sun$), respectively.

Several fundamental prerequisites of empirical nature are inputs (1), (2), (3) to this work. (1) Cosmic rays above energies of a few $GeV$ have a prevailing positive charge. (2) The electric charge deposition by cosmic nuclei in the solar wind region mentioned above also apply to any stars as stellar winds are universal [4]. (3) Sun is a common G star with a typical stellar wind. It occupies a central position in the mass spectrum of stars, frequently termed the Salpeter initial function or any of its variants. $Sun$ lifetime and stellar wind are regarded as typical of G star category.

This paper is organized as follows : in $Section$ Section 2 it is defined the volume occupied by the solar wind designated by $V_{sc}$ ($sc$ on foot for solar cavity) where charge deposition of cosmic rays takes place. The adopted stationary condition of the electric charge flows within $V_{sc}$ is based on potassium data of terrestrial meteorites [5] : the negative charge entering the solar cavity has to equalize the positive charge to avoid an unlimited accumulation. In  Section 3 the equation governing the rotational motion of stars and clouds in the Galaxy is written down. Gravity and the electrostatic field ${\overrightarrow{E}}_g$ cooperate to bind stars and clouds in the Galaxy. In  Section 4 the radial dependence of the Galactic electric field relevant to this work is given. In  Section 5 data of the visible mass of the Galaxy are summarized : they determine the attractive gravitational force. In  Section 6 measurements of empirical rotation curves are compared with the computed rotation curves. In  Section 7 some features of the rotation curves data are debated. Compendium and conclusion are in  Section 8.

The focus of this work is on the rotation curves in the outer Galaxy, in the radial Galactocentric range, $8.5$≤r ≤ 18 $kpc$ where the Galactic electrostatic field is known better than that in the inner Galaxy. In this radial interval the basic discriminant is rising or decreasing rotation curves.

2. Charge Balance Around Single Stars and Multiple Stellar Systems

The region occupied by the solar wind has been measured in two different locations of the nearby circumstellar space by the Voyager Probes $V1$ and $V2$ at 94 $AU$ [6] and 84 AU [7], respectively. This region might be approximated by a sphere of nominal radius $R_{sc}$ (sc for solar cavity). Crudely and nominally, $R_{sc}$ ≡ (94 + 84)/2 = 89 AU = $1.331\times {10}^{13}$ m. The corresponding spherical volume, $V_{sc}$≡ (4/3) $\pi$$R_{sc}^3$ = $9.8769\times {10}^{39}$$m^3$ is a surrogate of the non spherical volume delimited by the termination shock positions observed by the Voyager mission.

According to Potassium abundance in terrestrial meteorites [5] the cosmic-ray intensity in the last 2 billion years has maintained approximately constant. This datum is regarded as solid input here to state, or posit, the quasi stationary state in the flows of the electric charge in the volume $V_{sc}$.

At the time interval $\delta$t from an arbitrary initial zero time, the positive electric charge deposited by cosmic rays in the volume $V_{sc}$ amounts to : ($I_{sw}$ + $I_{ds}$) $\delta$t. Recent calculations [8] reports $I_{ds}$ = $1.1\times {10}^3$ $Coulomb$/s (hereafter C/s) above 5 $GeV$ and $I_{sw}$ = $1.96\times {10}^{12}$C/s in the range 3 $MeV$- 20 $GeV$ of the cosmic-ray spectrum where measurements are available.

The neutralization of the positive charge excess during the arbitrary time span $\delta$t in the volume $V_{sc}$, namely, $I_{sw}$$\delta$t + $I_{ds}$$\delta$t promotes a quiescent electron current designated by $I_{ne}$ ($ne$ on foot for neutralization) and called neutralization current. The current $I_{ne}$ is qualitatively represented by the thick blue arrows in Figure 1 and it enters the solar cavity coming from the nearby interstellar space, beyond the radius $R_{sc}$ transporting the negative electric charge $Q_{ne}^{-}$ = $I_{ne}$$\delta$t in the arbitrary time span $\delta$t. Thus, the total electric charge contained in the solar cavity in the arbitrary time interval $\delta$t is given by :

$$(I_{ne}+I_{sw}+I_{ds})\delta t+q_{sc}\left(0\right)=R\left(\delta t\right)$$

(1)

where $q_{sc}$ (0) is the total electric charge in the volume $V_{sc}$ at initial time t = 0. Any arbitrary time may be chosen as initial time, or zero time, taking into account the $Sun$ age of about $4.5673$$Ga$. The charge within the volume $V_{sc}$ at time $\delta$t is denoted by R($\delta t$).

The stationary state during the time interval $\delta$t posits, $I_{ne}$ + $I_{sw}$ + $I_{ds}$ = 0 and, therefore, $q_{sc}$(0) = R($\delta t$) where R($\delta t$) designates the residual electric charge at the time $\delta t$. For instance, during the temporal span surveyed by Potassium data [5] of ≈ 2 $Ga$ (Giga $annum$), the residual charge R($2Ga$) remained constant and the time t = 0, i.e. the initial time of the calculation is 2 $Ga$ back in time from the present epoch.

In the steady condition, regardless of the complexity of the environment, the neutralization charge $Q_{ne}^{-}$ entering the solar cavity from the adjacent space has to equalize, or nearly equalize, the positive electric charge deposited by cosmic nuclei in the same volume.

Due to a deferred neutralization, a residual charge R($\delta t$) permanently remains on the star and stellar wind region as extensively advocated in $Sections$ 3, 4 and 5 of ref. [8]. This residual charge is positive as it derives from the extinction of positively charged cosmic nuclei. By this charge stars experience an attractive force toward the $Galactic$ center $\Omega$ because the Galactic electric field has a centripetal direction [2,3]. The neutralization process always lags behind the electric charge deposition of the cosmic rays which penetrate the solar cavity at light velocity. Thus, any negative charge transfer through the neutralization process, from the exterior to the interior of the solar cavity, is necessarily slow and it is deferred face to the rapid charge deposition.

3. Electrostatic and Gravitational Attractive Forces in the Galaxy

The Galactic electrostatic field ${\overrightarrow{E}}_g$ (see next $Section$ Section 4) exerts an attractive force on stars and clouds imbued with positive electric charge. This electric force adds to the gravity force. As the radial field intensity $E_g$(r) ≡ $E_r$ increases with the distance from the solar circle of $8.5$ $kpc$ up to $r_g$ = 15 $kpc$, the attractive force increases with the radial distance. Beyond the radius of 15 $kpc$ the electrostatic force diminishes as cosmic-ray sources fade away.

Consider a generic star of mass m and electric charge q in circular motion about the Galactic center in the disk plane. For sake of simplicity, admit a common barycenter for both charge and mass distributions. The electric and gravity forces in a circular motion are : q/m$E_r$ + GM(r)m/$r^2$ where G is the gravitational constant, r the distance of the star from the Galactic center, $\Omega$, and M(r) is the visible mass of the Galaxy within the radial distance r. Accordingly, the velocities squared of the stars $v^2$(r) about the center-of-mass of the Galaxy are given by :

$$v^2\left(r\right)={\displaystyle \frac{GM\left(r\right)}{r}}+{\displaystyle \frac{qr{E}_r}{m}}={\displaystyle \frac{GpM}{r}}+{\left(4\pi {ϵ}_{0}G\right)}^{1/2}r{\zeta }_s{E}_r$$

(2)

The mass M(r) can be expressed as a fraction p of the total mass of the Galaxy, M, within the distance r. Hence, M(r) = p(r)M with p≤ 1 while the ratio q/m is expressed by the dimensionless parameter ${\zeta }_s$1 (s on foot for a generic star or cloud). Notice that the quantities v(r), M(r), m and r in Equation (2) are measured with fair precision. For the $Sun$ the circular velocity $v_s$ = 230 $km$/s is adopted and, accordingly, $v_s^2$ = $5.29\times {10}^{10}$$m^2$/$s^2$.

For instance, with M = $3.0\times {10}^{41}$$kg$ and p(8.5 $kpc$) = $0.7$, the factor GM($r_s$)/$r_s$ = $6.6742\times {10}^{-11}$ ($0.7\times 3\times {10}^{41}$)/($8.5\times 3.085677\times {10}^{19}$) = $5.29\times {10}^{10}$$m^2$/$s^2$. The Equation (2) gives the radial dependence of the star velocity revolving around $\Omega$ with a circular motion. The velocity v depends on the functions $E_r$, M, p, the radius r and the dimensionless parameter ${\zeta }_s$ which expresses the electric charge residing on a given star or a gaseous complex.

The distance of the Galactic center from the $Sun$, $r_s$ (s on foot for $Sun$), is a critical parameter to build rotation curves from observations. Some measurements [10] provide $r_s$ = $8.34$±$0.16$ and a velocity $v_s$ of 240 $km$/s while others [11] using RR Lyrae stars in the direction of the $Galactic$ $Bulge$ measure $r_s$ = $8.1$ ± $0.6$ $kpc$. It seems that the $Sun$ velocity of 220 $km$/s around the Galactic center $\Omega$, recommended by the $Astronomical$ International $Union$ [12], does not exactly coincide with recent measurements of $v_s$ which would suggest a larger value of about 245 $km$/s [13]. Thus, the adopted values of $v_s$ = 230 $km$/s and $r_s$ = $8.5$ $kpc$ in the Equation (2) is compatible with current data.

Stars, $HI$ clouds and molecular clouds are used in the measurements of rotation curves (see $Sections$ Section 6, Section 7 and $Appendix$ Appendix A). It is known that multiple stellar systems and related stellar winds inhabit gaseous complexes such as High Mass Star Forming regions, $HII$ regions and others. Accordingly, the electric charge stored by a generic star, designated by ${\zeta }_s$, (see footnote 1) is also attributable to clouds and gaseous complexes without a fine distinction.

4. The Galactic Electrostatic Field Acting on Stars and Clouds

The Galactic field strength $E_g$ versus radius used in this work is shown in Figure 2. It is calculated elsewhere [14] in four different conditions regarding the $Galactic$ charge distribution originated and regulated by the motion of cosmic rays (see also ref. [2]). The interpolation of the radial electric field adopted here is the following :

$$E\left(r\right)=0.05V/m(3-\mathrm{a})$$

$$E\left(r\right)\equiv E_r=\alpha {\left[{\displaystyle \frac{r_{kpc}}{8.5}}\right]}^{\delta }V/m(3-\mathrm{b})$$

with $\alpha$ = $0.15$V/m and $\delta$ = $1.35$. Equation (3-a) applies to r≤ 1 $kpc$ while (3-b) in the range 1 ≤r≤ 15 $kpc$. In the interval 15 ≤r≤ 18 $kpc$ beyond the nominal disk rim of 15 $kpc$ [15,16], the radial field intensity $E_r$ has unsurprisingly a rapid decrease. The following parabolic functions imitate the radial decrease of the electric field :

$$E_r=7.3144-0.78785r+0.02145r^2(3-\mathrm{c})$$

$$E_r=5.866-0.6015r+0.0155r^2(3-\mathrm{d})$$

equation (3-c) applies in the range, 15 ≤r ≤ 17 $kpc$ while (3-d) in 17 ≤r ≤ 18 $kpc$. The negative sign of the field strength in Figure 3 labels its centripetal direction.

The $Galactic$ mass adopted in this work is $1.48\times {10}^{41}$$kg$ (see next  Section 5). Accordingly, Equation (2) can be practically rearranged in this way :

$$v(km/s)=100(km/s){[{\displaystyle \frac{21.62989M_{41}\times p}{r_{kpc}}}+0.265908{\zeta }_s{r}_{kpc}{E}_r]}^{1/2}$$

(4)

where $M_{41}$ is the $Galactic$ mass in unit of ${10}^{41}$ $kg$ and the numerical constant 0.265 908 is from : ${\left(4\pi {ϵ}_{0}G\right)}^{1/2}$×$3.085677\times {10}^{19}$m = ${10}^{10}$m/s (${0.265908)}^{1/2}$= 100 $km$/s× (${0.265908)}^{1/2}$ . For example, with the Galactic mass M of $1.48\times {10}^{41}$$kg$ = $7.4\times {10}^{10}$$M_s$ at the arbitrary radius $r_{kpc}$ = 9 $kpc$, ${\zeta }_s$ = 2.5, electric field strength from (3-b), $0.15$$\times {(9/8.5)}^{1.35}$ = 0.162 V/m and an arbitrary p($9kpc$) = 0.65, the resulting rotation velocity would be, 100 ( km/s)${[2.3119+0.9692]}^{1/2}$ = $181.14$$km$/s.

5. The Galactic Mass Based on Optical, Radio and Infrared Data

The function $p\left(r\right)$ is empirically known since many decades. In the simplest approach it is the sum of an exponential disk and a spherical halo.

Two exponential disks are usually distinguished; a thin disk containing a major fraction of matter (about 85 per cent) and a thick disk of tenuous matter (about 15 per cent of the total mass) (see, for example, ref. [17]). Here, to simplify, a single disk instead of two is adopted with its characteristic radius $R_d$. Traditionally, measurements during three decades indicate that $R_d$ ranges from 2 $kpc$ to 4.5 $kpc$. Likely, $R_d$ depends on the celestial object used in the measurements (referred to as tracer) besides other parameters.

Matter in the spherical halo consists of stars and includes a substructure named $Galactic$$Bulge$. Shapes and intensities from $Bulge$ light is used to classify spiral galaxies and many functions are used for its interpolation. Accordingly it is set : $p\left(r\right)$ = (${\rho }_d\left(r\right)$ + ${\rho }_h\left(r\right)$)/($m_d$ + $m_h$) where ${\rho }_d\left(r\right)$ and ${\rho }_h\left(r\right)$ designate, respectively, the disk and halo masses within the radius r and $m_d$ + $m_h$ the total $Galactic$$mass$.

The mass contained in the exponential disk within the radius r, $m_d$(r), can be adequately represented by :

$$m_d\left(r\right)=2\pi {\Sigma }_0{R}_d^2\left[1-(1+r/R_d)exp\left(-r/R_d\right)\right]$$

(5)

where $R_d$ is the characteristic radius of the exponential disk and ${\Sigma }_0$ the surface brightness density extrapolated at the $Galactic$ center $\Omega$, at r = 0. At the solar ringF the observed surface brightness ${\Sigma }_0$ ranges from 35 to 55 $solar$$masses$/$p{c}^2$.

Models of the halo mass indicate that it amounts to 9-12 per cent of the disk mass regardless of the details i.e. spherical or axisymmetric haloes with a bar or without it. Moreover, to resolve the Dark Matter problem the critical radial range is the outer Galaxy.

The Galactic halo extends up to and beyond 30 $kpc$ and has a characteristic radius b. Some studies admit an axisymmetric halo where r is the radial distance and z the Galactic elevation. A radial cutoff, $R_c$, is also contemplated in some models. An example of mass profile, $m_h$(r), is the following one :

$$m_h\left(r\right)=[C_h/{(1+u/b)}^{1.8}]\times exp(-u^2/R_c^2)$$

(6)

where $C_h$ is a constant, u is a combination of the elevation z and radial r distance, namely, u≡ ( $r^2$ + $z^2$/$q^2$${)}^{1/2}$, $R_c$ the radial halo cutoff and q the axial ratio. Ultimately, the radial mass profiles (5) and (6) are carved on the observed profiles of the Galactic optical light.

As an example, a measurement of the Galactic mass subdivided in exponential disk, halo and gas by a sample of 16 000 G dwarf stars [18] gives, $m_d$ = $1.18\times {10}^{41}$$kg$, $m_h$ = $0.12\times {10}^{41}$$kg$ and $m_{gas}$ = $0.236\times {10}^{41}$$kg$. This measurement adopts a solar ring of 8 $kpc$. With $r_s$ = 8.5 $kpc$ it becomes $m_d$ = $1.18\times {10}^{41}$$kg$ and $m_h$ = $0.12\times {10}^{41}$$kg$. According to the studies [10]-[13] on the solar circle radius $r_s$ quoted in Section 3, the value $r_s$ = $8.5$$kpc$ is adequate and, therefore, the $Galactic$ mass M from [18] becomes, M = $m_d$ + $m_h$ = $1.329\times {10}^{41}$$kg$ + $0.150\times {10}^{41}$$kg$ = $1.48\times {10}^{41}$$kg$ = $7.4\times {10}^{10}$ solar masses. From the same source [18]$R_d$ = $2.15$$kpc$. Recently, by a different method, a Galactic mass of $7.5\times {10}^{10}$ solar masses [19] has been also estimated in agreement with [18].

An example of function p(r) is displayed in Figure 4 for for an exponential disk with characteristic length $R_d$ = 4.5 $kpc$.

The literature reports a number of interpolations of the visible Galactic mass; they are only slightly different from those given above. These slightly different representations of the matter distributions, however, do not alter the basic traits of the rotation curves obtainable by Equation (4).

The results of the calculation via Equation (4) are shown in Figure 5 for M = $1.48\times {10}^{41}$$kg$, $R_d$ = $2.15$$kpc$ and also $R_d$ = $4.5$$kpc$ (blue curves). As it is visually realized, the effect of the electric field $E_r$ relative to that of gravity (black curves) for r≤$8.5$$kpc$ is quite modest becoming instead notable, with a prominent increasing trend, in the radial range $8.5$≤r≤ 15 $kpc$. Beyond 15 $kpc$ rotation velocity curves decrease as the radial electric field strength falls off.

The dependence of the rotation curves on the electric charge ${\zeta }_s$ is shown in Figure 6 for ${\zeta }_s$ = 1.5, 2.5 and 3.0. Here extreme values of M and $R_d$ ($3.0\times {10}^{41}$$kg$ and $R_d$ = $4.5$$kpc$, respectively) are used to offer the global view spanned by the calculation. The black curve in Figure 6 refers to the rotation curve forged only by gravity, purposely omitting the electric field $E_r$ to evidence the relative role of the two forces.

6. Data on Rotation Curves of Stars and Clouds in the Milky Way Galaxy

Numerous measurements of rotation curves of stars and clouds about the Galactic center are available. The measurements are based on the emission at 21 cm by HI gas [20,21], infrared emission of CO dispersed in molecular clouds [22], the motion of halo stars reconstructed in the optical band [23], ethanol, hydroxyl and water $MASERs$ in clouds excited by O and B stars [10], the motion of planetary nebulae [24,25] and Carbon stars of type N [26].

Data take advantage of rotational models of the Galactic disk as the 9 parameters used to determine v(r) are not all measured. These parameters are the three dimensional velocity, the distance and position of stars or clouds. Due to this shortcoming caution is mandatory in the scrutiny of the available measurements. In order to build the rotation curves two crucial parameters are the distance of the $Sun$$r_s$ from the Galactic center $\Omega$ and the circular velocity of the $Sun$ about $\Omega$ called here $v_s$($r_s$) and concisely, $v_s$ (s for $Sun$).

The Galaxy is not a rigid body but any parts rotate with a circular velocity $\Theta$(R) depending on the radial distance R from the Galactic center $\Omega$. An expression of the rotation velocity $\Theta$(R) used in the data analysis of the rotation velocity curves is, $\Theta$(R) = $v_s$ (R/$r_s$${)}^{\alpha }$ where $\alpha$ is the rotation index. A second expression is the polynomial function : $\Theta$(R) = $v_s$ + $a_0$(R -$r_s$) + $b_0$(R -$r_s$${)}^2$ which depends on the constants $a_0$ and $b_0$.

In order to fully appreciate the comparison of the rotation curves based on Equation (4) shown in Figure 5 and Figure 6 and experimental data [20]–26], the essential properties of gaseous matter and stars imbued with positive electric charge attracted by the electrostatic field ${\overrightarrow{E}}_g$ are here summarized.

(1) The electric field intensity increases with the radius r between 8.5 and 15 $kpc$ by more than a factor of 2 regardless of the approximation of the charge distributions which generate the field ${\overrightarrow{E}}_g$ in the Milky Way Galaxy [14]. The increment of the electric field entails the increment of the restoring force on stars and clouds toward the Galactic center. This characteristic feature, for an arbitrary given radius r between 8.5 and 15 $kpc$, lifts rotation velocities of any charged bodies (stars and clouds) : from Keplerian descending rotation curves to slightly increasing up to the radial distance $r_g$ = 15 $kpc$ as shown in Figure 5 and Figure 6.

(2) Inevitably, the radial electric field strength $E_r$ diminishes beyond the nominal Galactic rim $r_g$ = 15 $kpc$. As it is well known since long time, there are stars and gaseous materials beyond 15 $kpc$, in the Galaxy (see, for example, [27,28]) but their total mass fraction is negligible face to the mass enclosed within the nominal disk rim of 15 $kpc$. Let us remind here that the nominal disk border $r_g$ = 15 $kpc$ is not a working hypothesis but a solid empirical datum anchored to the drop of the optical luminosity of the Galaxy beyond 14-16 $kpc$ (see, for instance, [15,16]).

(3) The electric charge captured by a stellar category (for example Carbon stars) is expressed by the variable ${\zeta }_s$, the charge-to-mass ratio of the star, which is expected not to depend on the radial distance beyond a certain characteristic distance from the Galactic center $\Omega$. This characteristic distance is not exactly evaluated in this work but it is certainly minor than $r_s$ = $8.5$$kpc$. Thus, it is admitted an average constant value ${\zeta }_s$ for a given stellar category in the interval $8.5$ ≤r≤ 15 $kpc$.

The observational data [20]–26] on the rotation curves in the radial range $8.5$≤r≤ 18 are shown in Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 along with the rotation curves of the Equation (4) with ${\zeta }_s$ = $2.5$ (blue and green curves). The particular value ${\zeta }_s$ = $2.5$ in Equation (4) is motivated by the fair agreement with the classical observational data on rotation velocity curves shown in Figure 7 [20].

The order of magnitude of the electric charge stored around stars is based on the total electric charge transported by $Galactic$ cosmic rays, about 2.5 $\times {10}^{31}$C [2,3] and the number of stars in the Galaxy, about 100-200 billion from a $Salpeter$ initial mass function with low and high mass cuts of $0.4$-20 $M_s$ and an index of 2.35. Thus, $2.5\times {10}^{31}$ C/$(1-2)\times {10}^{11}$ = $(1.25-2.5)\times {10}^{20}$C. This estimate assumes that the dominant channel for cosmic-ray extinction occurs in stellar winds.

7. Galactic Rotation Curves and Charge Retention in Stars and Clouds

Some aspects in the comparison between measurement and calculation of the rotation curves in spiral galaxies are here highlighted. Rotation curves in spirals are designated by v(r).

According to the explanation of the Dark Matter $problem$ offered here via Equation (4), material bodies orbiting the Galactic center $\Omega$ have the same value ${\zeta }_s$ = $2.5$ or values quite close to $2.5$. Equivalently stated, they retain an electric charge proportional to their masses. It would seem implausible that two diverse categories of celestial bodies (age, chemical composition, dynamical status, environment), like planetary nebulae and blue giant horizontal branch stars might have exactly the same mass-to-charge ratio, for instance the adopted value ${\zeta }_s$ = $2.5$, thereby exhibiting the same rotation velocity curves. Plausibly, the average value of the parameter ${\zeta }_s$ not only changes, depending on the particular category of celestial body examined but, likely, it will fluctuate with a characteristic dispersion $\sigma$(${\zeta }_s$) around its average value. The fluctuations are caused by plenty of mechanisms which may favor or restrain the accumulation and release of electric charge (see $Seg.$$12.2$ of ref. [2] and [8]).

Generally a variation of the parameter ${\zeta }_s$ versus time is expected for the same object. A star like the $Sun$ at $r_s$ = $8.5$$kpc$ takes about 240 $\times {10}^6$ years to complete a turn about the Galactic center $\Omega$. A star at the radial distance of 15 $kpc$ takes approximately half billion years to complete a full circular turn on the Galactic midplane. Some young celestial bodies aged (10-20) $\times {10}^6$ years, for instance H$II$ regions illuminated by O and B stars , might have ${\zeta }_s$ values which vary on time intervals shorter than their intrinsic ages.

Following these premises it has to be noted $D_1$-$D_5$:

($D_1$) If the value ${\zeta }_s$ for a given category of objects and for a given disk radius r fluctuates due to intrinsic environmental causes, then, in turn, the corresponding rotation velocities will also fluctuate as dictated by Equation (4).

An example of rotation curve obtained with Carbon stars of type N [26] is shown in Figure 11. It is quickly realized a large dispersion in the measurements : at the radial distance of $12.5$ $kpc$ rotation velocities jerk from 20 to 280 $km$/s. It is a common practice that even with this large velocity fan, average rotation velocity curves v(r) are anyway forced and superimposed to the data.

($D_2$) If the average values ${\zeta }_s$ for two categories of celestial bodies differ in a sensible way, then rotation velocity curves will be different as required by Equation (4). This effect differs from that of $\sigma$(${\zeta }_s$) mentioned in the point $D_1$.

It has been remarked by others that the rotation curves v(r) based on Carbon star of type N in the Milky Way Galaxy lie below the corresponding rotation curves obtained via HI cloud emission. For example, by comparing the $HI$ data [20] displayed in Figure 7 with those in Figure 11 [26] emerges a systematic difference of 30-50 $km$/s in the radial interval 10 ≤r≤ 15 $kpc$. In order to explain the velocity gap, besides possible instrumental effects, one can legitimately speculate that Carbon stars of type N retain average ${\zeta }_s$ values inferior to those of the HI clouds.

($D_3$) Often in the literature average rotation curves are superimposed a$priori$ in the attempt at framing vast arrays of observational data of the unseizable Dark Matter by simple interpolation functions. Let us remind here that more than one thousand rotational curves of spiral galaxies are empirically known and tabulated. The practice of averaging rotation curves tends to destroy precious details of the data which would remain unexplained by the notion of Dark Matter recurrent in the literature, which is too simplistic and exerts only an attractive force.

For example, as asserted by others (page 17 ref. [30]) : " Clearly, real galaxies are more complex than the simple URC prescription suggests and other factors than luminosity must contribute to the detailed shape of a rotation curve as well " ($URC$ is for Universal Rotation Curve).

Frequently average rotation curves in spirals are imposed both at a particular radial distance and on an entire selected range of radial distances.

The sentiment here is that some recent works on rotation curves using trigonometric parallaxes herald decreasing rotation curves in a vast radial range but, as a matter of fact, increasing rotation curves in the range 9-15 $kpc$ are observed (see, for example, data points from 773 $Classical$$Cepheids$ in Figure 3 of ref. [31] in the limited range 9-14 $kpc$ which display an increasing tendency). A second example comes from the study of the proper motion of 3500 $Classical$$Cepheids$ (see Figure 3 in ref. [32]) where rotation curves in the radial range $9.5$-$15.5$$kpc$ show an increasing tendency). A third example is based from a sample of 2401 blue horizontal branch halo stars [33] where the first two data points in Figure 15, 16 and 17 mark an increasing tendency.

Rich data samples based on trigonometric parallaxes of 23000 red giant stars [34] are mutually inconsistent with the rotation curves based on 16000 primary red clump giants and 5700 halo K giants [35] as depicted in Figure 12. Notice further that both these experiments [34,35] assume the mechanical equilibrium of the Galaxy and use stellar dispersion velocities to determine rotation curves. The ironic side here is that if a straight line is anyway forced in the range 4-18 $kpc$ through the blue curve labeled $R_d$ = $2.15$ in Figure 5, a decreasing rotation curve looms out.

($D_4$) If, for instance, the value ${\zeta }_s$ = $0.1$ were inserted in Equation (4) or even a lower value, no significant effect of the electrostatic field on the rotation curves v(r) would result. In this case gravity would govern v(r) (black curves in Figure 5 and Figure 6).

($D_5$) Rising rotation curves in the Milky Way Galaxy are in accordance with the existence of the $Galactic$ electrostatic field as advocated in this work. Rising rotation curves in the radial range from $8.5$ to 15 $kpc$ have been assessed during four decades of measurements as explicitly vented by consummate, top astronomers [23] : " All the rotation velocities are in agreement in the sense that they show the tendency of increasing with the galacto-centric distance. ". This statement applies to the radial range $8.5$ to 15 $kpc$. Of the same token [22] : " All nine of the above methods for determining the rotation curve to large R show that the rotation curve does not fall out to the last measured point, at a distance (generally) near to 2 $R_0$. These results indicate that beyond ≈ 2$R_0$, the mean of the binned points lie above a flat rotation curve by ≈ 10-20 per cent of the circular velocity. ". Here $R_0$ signifies 8.5 $kpc$, the $Sun$ distance from the Galaxy center.

8. Compendium and Conclusion

The attractive electrostatic force2 on stars and clouds in the Milky Way Galaxy rests on two pillars : (1) the Galactic electrostatic field ${\overrightarrow{E}}_g$ and (2) the positive electric charge deposited by cosmic rays around stars in the stellar wind regions [8,37].

A pervasive, permanent electrostatic field was introduced in 2020 in the search for the correct acceleration mechanism of cosmic rays in the Milky Way Galaxy from energies of a few $GeV$ up to $2.4\times {10}^{21}$$eV$ [2,3,14]. As cosmic-ray sources mainly reside in the molecular and atomic clouds, they predominantly store negative electric charge made of orbital electrons abandoned by the accelerated cosmic-ray nuclei. This negative charge of quiescent electrons of -$2.5\times {10}^{31}$C, designated in ref. [2] by $Q_w$ ( w for $widow$$electrons$), inhabits the Galactic core in cold molecular and atomic clouds, while cosmic-ray nuclei with global positive charge, $Q_{cr}$ = - $Q_w$ spreads out around the core, due to their overwhelming high energies. These spatially disjoint positive and negative electric charge distributions, essentially and ultimately, generate the electrostatic field ${\overrightarrow{E}}_g$ [2,3,14] which is directed from periphery to the $Galactic$ center.

The electrostatic field ${\overrightarrow{E}}_g$ generates the additional attractive force, which constrains stars and clouds impregnated with positive electric charge, to rotate about the Galactic center with a velocity greater than that caused by gravity only.

The agreement between the increment of the rotation curve in Figure 5 beyond $8.5$$kpc$ and the increment displayed by the observational data [20] is evident. On the contrary, calculations based on $Celestial$$Mechanics$ yield decreasing rotation curves beyond 4-5 $kpc$, i. e. Keplerian curves like those in Figure 5 and Figure 6 (black curves), in apparent disagreement with the data [20]-[26].

The result of this work adheres to the severe, inescapable results from experimental physics in the quest for Dark Matter during half a century. $In$$primis$$et$$ante$$omnia$ all the experimental quests of the dark matter in laboratories, caverns and particle accelerators have produced null results over a period of more than 50 years with heterogeneous techniques and in diverse areas. See, for example, the important null results of LUX experiment [38] and $XENON100$ [39] along with those at accelerators gleaned by ATLAS and CMS detectors at the Large Hadron Collider [40,41] and many other null results of the same two experiments and others.

Historically the $missing$$mass$ or Dark Matter problem firstly emerged in 1932 [42] in the empirical study of stellar motions close to the Galactic midplane : gravity force measured by optical observations was insufficient to constrain stars in the Galactic midplane as they were actually observed. Some years later the computed dispersion of the velocity distribution of galaxies in the $Coma$ cluster of 80 $km$/s appeared to suffer the same fault, as the observed velocity dispersion was much larger, about 1000 $km$/s.

Thus, Dark Matter problem since 1932, walked on two imaginary wheels ; rotation curves of stars and clouds in spiral galaxies is one wheel. The violation of $Virial$$Theorem$ in galaxy clusters is the second wheel e. g. observed galaxy velocities in galaxy clusters require an attractive binding potential greater than the gravitational potential inferred from optical observations. Since then, a plethora of empirical studies on other spiral galaxies and galaxy clusters reiterate the deficit of gravitational force, accompanied by an explosive proliferation of futile, mystique papers on unreal material bodies.

A recently published (2022) outsize work [37] recognizes that any galaxy clusters have permanent and ubiquitous electrostatic fields designated by ${\overrightarrow{E}}_c$ (c on foot for cluster) generated by the motion of cosmic rays in cluster galaxies. From [37] it turns out that the elusive Dark Matter in galaxy clusters is an effect of the cluster electric fields ${\overrightarrow{E}}_c$ .

The quartet of empirical and logical evidence : (1) the account of flat rotation curves in spiral galaxies by the galactic electric fields ${\overrightarrow{E}}_g$ reported in this work; (2) the vast null results of any forms of Dark Matter from laboratories, caverns and accelerator experiments ; (3) the absence of astronomical evidence of low luminosity material bodies postulated ad hoc to account for the deficit of gravitational force in galaxies (see for example [43] along with many others); (4) and the account of $Dark$ $Matter$ in galaxy clusters by cluster electrostatic fields ${\overrightarrow{E}}_g$ , leads to the simplest conclusion that Dark Matter does not exist at all ; Dark Matter is only an electrostatic effect.