APPENDIX D

APPENDIX D: SPIRAL GALAXY ROTATION

The main purpose of this appendix is to compare NCNG with GRLCDM as per section VIII(D). Eq. (5.5) forms the basis for obtaining the NG rotational velocity profile from a given distribution of neutral matter. The mass distribution of a suitable flat disc galaxy in Cartesian coordinates is approximated, \[\rho \left( {\bf{x}} \right) = \sigma \left( R \right)\delta \left( z \right){\rm{,}}\tag{D.1}\] where \({R^2} = {x^2} + {y^2}\) and \(\delta \) is the Dirac delta function. Next, take the Fourier transform of both \(\mathcal{A}_N({\bf{x}},q)\) and \(\rho({\bf{x}})\) to obtain the relation between transformed functions using Eq. (5.5), \[\begin{array}{l} {{\bar{\mathcal{A}}}_N}\left( {{\bf{p}},q} \right) = -4\pi G{Z_q}\,\overline{\rho}\left( {\bf{p}} \right){\left\{ {{{\bf{p}}^2} + L_q^{-2}} \right\}^{-1}},\\ \overline{\rho}\left( {\bf{p}} \right) = \int d^3x\,\rho({\bf{x}})\exp(i{\bf{x}} \cdot {\bf{p}}). \end{array}\tag{D.2}\] The expression for \(\overline \rho \) can be simplified through a tedious series of steps manipulating the integration in the x-y plane, \[\overline \rho \left( {\bf{p}} \right) = 2\pi \int_0^\infty {RdR{\kern 1pt} \,\sigma \left( R \right)} {J_0}\left( {aR} \right) \equiv \overline \rho \left( a \right){\rm{,}}\tag{D.3}\] where \({a^2} = p_x^2 + p_y^2\) and J is the Bessel function of indicated order. Take the inverse Fourier transform of Eq. (D.2) to obtain \({A_N}\), \[\begin{array}{c}\Phi \left( {R,q} \right) \equiv {A_N}\left( {x,y,z = 0,q} \right)\\ = – G{Z_q}\int_0^\infty {ada} \frac{{\overline \rho \left( a \right){J_0}\left( {aR} \right)}}{{{{\left( {{a^2} + L_q^{ – 2}} \right)}^{1/2}}}}{\rm{.}}\end{array}\tag{D.4}\] Next, obtain the rotational velocity by using Eq. (5.5) along with the assumptions that pressure gradients and viscosity can be ignored, \[v_{rot}^2\left( R \right) = R\sum\limits_{q = 1}^{{N_q}} {\frac{{d\Phi \left( {R,q} \right)}}{{dR}}} .\tag{D.5}\] The last step introduces dimensionless variables to simplify Eq. (D.5): \[\begin{array}{c}R = \eta {R_0}{\rm{,}}\\a = \chi R_0^{ – 1}{\rm{,}}\\{M_G} = \xi {M_0}{\rm{,}}\\\sigma \left( R \right) = {M_0}R_0^{ – 2}\Lambda \left( \eta \right){\rm{,}}\\{\lambda _q} = {R_0}/{L_q}{\rm{,}}\\{v_0} = {\left( {G{M_0}R_0^{ – 1}} \right)^{1/2}}{\rm{,}}\end{array}\tag{D.6}\] where \[\begin{array}{c}{R_0} = {\rm{ 1}}{{\rm{0}}^3}{\rm~{ light – years,}}\\{M_0} = {\rm{ 1}}{{\rm{0}}^9}{\rm~{ solar masses,}}\\{v_0} = {\rm{ 118}}{\rm{.5 ~km/s}}{\rm{.}}\end{array}\tag{D.7}\] The equations simplify as follows: \[\int_0^\infty {\eta d\eta \Lambda \left( \eta \right) = \xi /2\pi {\rm{,}}} \tag{D.8}\] \[\Delta \left( \chi \right) = 2\pi \int_0^\infty {\eta d\eta \Lambda \left( \eta \right)} {J_0}\left( {\chi \eta } \right){\rm{,}}\tag{D.9}\] \[v_{rot}^2\left( \eta \right) = v_0^2\eta F(\eta ){\rm{,}}\tag{D.10}\] \[F\left( \eta \right) = \int_0^\infty {{\chi ^2}d\chi } \Delta \left( \chi \right)\Psi (\chi ){J_1}\left( {\chi \eta } \right),\tag{D.11}\] \[\Psi (\chi ) = \sum\limits_{q = 1}^{{N_q}} {{Z_q}} {({\chi ^2} + {\lambda _q}^2)^{ – 1/2}}.\tag{D.12}\] Eq. (D.12) is constructed from universal constants and it is the same for all spiral galaxies. The delta function is governed by the distribution of mass for each spiral galaxy.
One suitable model for the delta function is akin to a Laplace transformation, \[2\pi \Lambda \left( \eta \right) = \sum\limits_{k = 1}^N {\alpha _k^2} {\xi _k}\,{\rm{exp}}\left( { – {\alpha _k}\eta } \right){\rm{,}}\tag{D.13}\] in which case, \[\Delta (\chi ) = \sum\limits_{k = 1}^N {{\xi _k}\alpha _k^3} {\left( {\alpha _k^2 + {\chi ^2}} \right)^{ – 3/2}}.\tag{D.14}\] The total mass of the galaxy is \(\xi = \sum\limits_{n = 1}^N {} {\xi _n}\). Note that \({\xi _n}\) must be positive.
There is another relation that connects the density of galactic mass directly with the rotation profile; \[ F(\eta) = \int_0^\infty \mathrm{d}\eta’\, \eta’ \, 2\pi \Lambda(\eta’) \, K(\eta, \eta’), \tag{D.15} \] \[ K(\eta, \eta’) = \int_0^\infty \chi^2 \, \mathrm{d}\chi \, \Psi(\chi) \, J_1(\chi \eta) \, J_0(\chi \eta’). \tag{D.16} \] The K function is the same for all galaxies and needs only to be calculated once.
The universal function, \(\Psi \), is featured in Fig. D.1 below.

Figure D.1 Graph of Eq. (D.12) using the standard model universal gravitational parameters of Appendix Z.

The primary plague of astronomy (unreliable ladder of distances) infects this study. Astronomers are wise enough to tabulate rotation observations in arc minutes or seconds (an angle relative to the center of the galaxy). In that case, the problems of the distance ladder are side-stepped. At this point, it is useful to convert the equations above to angles instead of distances. That is accomplished by using the relation (\(\eta \to {\Gamma _D}\,\omega \)) in Eq. (D.10) and Eq. (D.11) or Eq. (D.15), where \({\Gamma _D} = {D_{MPC}}/63.24\) and \({D_{MPC}}\) is the distance to the center of the spiral galaxy from earth in units of Mpc, and \(\omega \) has units of arc seconds.
The rotation velocities for all galaxies subject to equations (D1) to (D16) have a common feature, a finite radius beyond which \(d{v_{rot}}\left( \omega \right)/d\omega > 0\) out to vast radii. This feature is the hallmark of NCNG. The two competing theories (GRLCDM or MOND) always have asymptotic rotation velocities where \(d{v_{rot}}\left( \omega \right)/d\omega \le 0\). Thus, rotation velocities at large radii are of great interest.
Rotation velocities at larger radii are best observed using radio telescope arrays (relative doppler shift of 21 cm radiation from H1). Only two (VLA and Westerbork) have been used for this purpose. VLA has 1.93 times the maximum collection-area versus Westerbork—an advantage for VLA. The signal to noise error for an array governs the maximum radius for observable velocities. The subject of signal versus noise for radio telescopes is very arcane. For rotation velocities, the 21 cm doppler shift (signal) at the earthly array collectors is extremely weak and the large collection of noises is extremely strong by comparison. It is necessary to reduce the noise.
A single collector (dish) illustrates the technique. The collector focuses electromagnetic waves on an antenna which converts this information into a time dependent voltage. The voltage has both a weak signal and strong noise. The noise voltage has an average of zero. The voltage is subject to a series of steps.
Step one is a band-pass filter that passes only frequencies close to 1.489 GHZ (bandwidth \(\Lambda \nu \sim \)6 MHZ). Next, the voltage is squared. This leaves the resulting voltage with three parts: the square of the signal (average = constant with time), the square of the product of the signal and noise (average = 0), and the square of the noise (a random value between 0 and a large positive number). The final step is the integrator which is the average of the sum of the square voltages with time. The result is a voltage that equals the signal plus an error, where the error is proportional to \({(\Delta \nu \,t)^{ – 1/2}}\). One can find an integrating time (t) that produces a sufficiently small error compared to any signal. The error bars for the observations of [7] are obtained from the known integration errors.
The only suitable large radii observations are found in references [5], [7], and [15]. Reference [7] studied NGC 3198 and 2841 (observed in spring of 1982 using a primitive version of Westerbork). Reference [15] concerns only NGC 3198 but provides several data points at larger radii than [7] (using an upgraded Westerbork). Reference [5] provides higher resolution rotational velocity profiles for many galaxies including those of interest using VLA (but not larger radii than the Westerbork observations).
NGC 3198 and 2841 are important because [7] shows increasing velocities at largest radii for both (favoring NCNG). Furthermore, [7] velocity fits at largest radii for GRLCDM are mediocre. Nevertheless, the data of [5], [7], and [15] is not yet sufficient to decide which theory is correct. It may be necessary to observe rotation velocities for larger values of \(\eta \) for NGC 3198 and 2841. VLA should be able to do this by using a longer integration time. These two galaxies also share the benefit of very accurate distances due to the detection of cephid variables using the Hubble space-telescope.
A supercomputer is necessary to study the observations of [5], [7], and [15] in detail. However, there is a simple spiral-galaxy model which has \(N = 1\) in Eq. D.14—a single value \({\alpha _1}\). In that case, a pc can calculate the rotation velocity profiles. The one-a model has a simple relation for the rotation velocity profile,
\[v(\omega ,{D_{MPC}};{\alpha _1},\xi ) = {(\xi /{\xi _0})^{1/2}}v(\omega ,{D_{MPC}};{\alpha _1},{\xi _0}),\tag{D.17}\] where \(\eta = \omega {D_{MPC}}/63.24\). Eq. (D.13) gives the density of H1 for large radii for this model and allows an estimate of required integration time for necessary accuracy.
A one-a model for a hypothetical galaxy (that mimics NGC 3198) would require \({D_{MPC}} = 14.4,\) \({\alpha _1} = 0.07,\) and \({\xi _0} = 55.\) This hypothetical galaxy is used below to illustrate and compare the larger radii behavior of the rotation velocity profiles for NCNG and GRLCDM.
The GRLCDM version of the hypothetical galaxy results from the following set of equations: \[\begin{array}{l}\;\;\Delta (x) = {\xi _{GR}}\,\alpha _1^3{\left( {\alpha _1^2 + {x^2}} \right)^{ – 3/2}},\\{v_{GR}}(\eta ) = {[{v_M}{(\eta )^2} + {v_H}{(\eta )^2}]^{1/2}},\\{v_M}{(\eta )^2} = {v_0}^2\eta \int_0^\infty {dx\,x\Delta (x){J_1}} (x\eta ),\end{array}\tag{D.18}\] where \({v_0}\) is found in Eq. D.7 and \({v_H}\) below.
The spherically-symmetric dark matter halo is defined by the mass density, \(\rho (r) = {\rho _0}{[1 + {(r/{R_c})^2} + \gamma {(r/{R_c})^4}]^{ – 1}}\), where \({R_c}\) is measured in kly and \({v_{H0}} = {(4\pi G{\rho _0}{R_c}^2)^{1/2}}\) is a velocity measured in km/s. The dimensionless constant \(\gamma \) causes the dark matter halo to have a finite mass. Note that [7] uses \(\gamma = 0\) (infinite halo mass). The halo equations are as follows: \[\begin{array}{l}\kappa = {R_0}/{R_c}\\{v_H}{(\eta )^2} = {v_{H0}}^2{(\kappa \eta )^{ – 1}}H(\kappa ,\gamma ,\eta ),\\H(\kappa ,\gamma ,\eta ) = \int_0^{\kappa \eta } {dy\,{y^2}} {[1 + {y^2} + \gamma {y^4}]^{ – 1}},\\{\xi _H} = {\kappa ^{ – 1}}\,{({v_{H0}}/{v_0})^2}H(\kappa ,\gamma ,\infty ),\\\mu = {\xi _H}/{\xi _{GR}},\end{array}\tag{D.19}\] where \({\xi _H}\) is the finite mass of the dark matter halo in units of one billion solar masses and \(\kappa \)is dimensionless (\({R_0} = 1\) kly). It is thought that the universal ratio of dark matter mass to ordinary matter mass is 5.6 in the LCDM cosmology. One should not expect that \(\mu \) for this model would radically differ from 5.6.
The NCNG version of the hypothetical galaxy rotation velocity profile has three parameters: \({D_{MPC}}\), \({\xi _0}\), and \({\alpha _1}\). The GRLCDM velocity profile has six parameters: \({D_{MPC}}\), \({\alpha _1}\), \(\kappa \), \({v_{H0}}\), \({\xi _{GR}}\), and \({\xi _H}\). The two profiles cannot be the same, so the only path forward is to use the extra parameters of GRLCDM to fit the two profiles to coincide for small radii and see where they diverge. This task requires defining an error function, \[1 + E(\eta ) = {v_{GR}}(\eta )/{v_{NCNG}}(\eta ),\tag{D.20}\] where \(E({\eta _0}) = 0\) for the smallest observed radius (\({\eta _0}\)). Manipulations of Eq. (D.18) and Eq. (D.19) show that \[E = E(\mu ,\kappa ,\gamma ,\eta )\]. Thus, the task requires finding the set of parameters (\(\mu ,\kappa ,\gamma \)) that give the smallest E values for small radii. A tedious calculation of E gives the following result: \[\begin{array}{l}1 + E = {v_{NCNG}}({\eta _0}){v_{NCNG}}{(\eta )^{ – 1}}M{(\mu ,\kappa ,\gamma ,\eta )^{1/2}}M{(\mu ,\kappa ,\gamma ,{\eta _0})^{ – 1/2}},\\M(\mu ,\kappa ,\gamma ,\eta ) = G(\eta )I(\gamma ) + \mu {\eta ^{ – 1}}H(\kappa ,\gamma ,\eta ),\\I(\gamma ) = H(\kappa ,\gamma ,\infty ),\\G(\eta ) = \eta \int_0^\infty {dx\,x\,\alpha _1^3{{\left( {\alpha _1^2 + {x^2}} \right)}^{ – 3/2}}{J_1}} (x\eta ),\end{array}\tag{D.21}\] where \({v_{NCNG}}(\eta )\) is calculated using Eq. (D.1) through Eq. (D.17)—recall that \({D_{MPC}} = 14.4,\) \({\alpha _1} = 0.07,\) \({\xi _0} = 55\) for this hypothetical galaxy.
Eq. (D.21) gives \(E(\mu ,\kappa ,\gamma ,{\eta _J})\) for each value of eta, where \({\eta _J} = 0.22770(j + 1)\Delta \omega \) and \(\Delta \omega = 30\) arc seconds. The three parameters (\(\mu ,\kappa ,\gamma \)) need to minimize the errors for small values of eta. The error is automatically zero for 30 arc seconds. Three more constraints are needed, \[ 0 = E(\mu, \kappa, \gamma, 60^\circ) = E(\mu, \kappa, \gamma, 90^\circ) = E(\mu, \kappa, \gamma, 120^\circ). \tag{D.22} \] Eq. (D.22) is beyond my pc, so I have used \(\kappa = 0.05\) and for the figures below. FIG. D.2 below illustrates the rotation velocity profiles for NCNG and GRLCDM for this hypothetical galaxy. If NCNG is a correct theory, then one would have observed (using the 1982 Westerbork) the (solid) velocity profile. If GRLCDM is a correct theory, then one would have observed (using the 1982 Westerbork) the (dash) velocity profile for an (absurdly heavy) dark matter halo (\(\mu = 200\)), or the (dash-dot) velocity profile for a more reasonable halo (\(\mu = 5.6\)).
The two theories start diverging at 210 arc seconds—small compared to the 660 arc second limit of the 1982 Westerbork signal to noise ratio for this hypothetical galaxy. The NCNG profile is very similar to the actual 1982 NGC 3198 observations found in the source of [7]. Therefore, one can only realize that GRLCDM is a complete failure for NGC 3198.

Figure D.2 Rotation velocities: (blue) NCNG, (green) GRLCDM mu = 200, (red) GRLCDM mu = 5.6.

Figure D.3 The model of rotation velocities in FIG. D.2 extended to larger distances (not observable in 1982).

The equations above show that all NCNG rotation-velocity profiles will trend upward as \(\eta \) increases. However, GRLCDM profiles with finite-mass dark matter haloes will all trend downward. The surprise for me is that the two curves start diverging at a relatively small \(\eta \). Furthermore, the GRLCDM profile dives while the NCNG profile creeps upward. One should expect that other galaxies will repeat this result. The original rational for dark matter was to flatten out the rotation-velocity profiles. I believe that goal has not been met, and GRLCDM should be discarded as a viable theory.
The equations above are an ideal model for a spiral galaxy—cylindrical symmetry and a thin disk of matter. Real spiral-galaxies are messy. They have bars, arms, lumps, bulges, warps, and so on. Suitable galaxies for study (using the equations above) are hard to come by.
The most important requirements (for a suitable galaxy) are an accurate distance (\({D_{MPC}}\) determined using cephid variables) and an accurate rotation profile extending well beyond the likely theory-divergence point. One should discard near face-on or edge-on galaxies. As mentioned above, only two of the ten galaxies studied in [7] are suitable. NGC 3198 (Sb) has a faint bar and many faint arms wrapped around the galactic center—a reasonable model for cylindrical symmetry. Furthermore, there is very little warping except for the outermost observed \(\eta \). I believe that the model for FIG, D.2 is sound.
Eventually, it will be necessary to extend rotation profiles beyond the limits of 1982 since the rate of upward trend is an important constraint on the universal constants of Eq. D.12.