Draft:Dynamic Gravitation

Three centuries subsequent to Isaac Newton's contributions (1642-1726), classical physics has achieved a level of sophistication that enables it to provide solutions to numerous unresolved challenges, including the interpretation of the rotation curve, the prediction of orbital precession, the modeling of gravitational lensing, the mechanism of bullet clusters, the formation of spiral, ring, bar, X and shell structures, as well as the principles governing leading arms, tidal tails, cosmic voids, and superstructures.

Newton's law of universal gravitation is governed by the inverse square law. It is derived in an inertial reference frame, wherein time-invariant masses exist, and no continuous external transfer of angular momentum occurs. The centripetal force can be expressed as

(1) $F_{N}={\frac {GMm}{r^{2}}}$,

where G is the gravitational constant, M and m are the central and revolving masses, respectively, and r is the galactic distance. For the central-force problem, the orbit is circular, called a Newtonian orbit, with a radius of $r_{N}$ and a velocity of $v_{N}$.

Orbital evolution equations are derived for dynamic gravitation. When a radial perturbation is applied to the circular path, it gives rise to an elliptical path with constant angular momentum. The elliptical orbit is referred to as a Keplerian orbit, where the radial and rotational degrees of freedom (DOF) are independent. In the radial DOF, a restoring force arises analogous to that of a mass-spring system. The rotation and radial oscillation are synchronized, resulting in a stationary orbit (no orbital precession). The orbit of a Keplerian orbit can be expressed as

(2) $r={\frac {r_{1}}{1+{\epsilon }cos{\theta }}}$,

where ${\epsilon }$ is the eccentricity, and $\theta$ is the revolving angle that is always monotonic and ascending, even during reverse evolution.

The period of a Keplerian orbit is

(3) $T_{K}={\frac {T_{N}}{(1-{\epsilon }^{2})^{\frac {3}{2}}}}$,

where $T_{N}$ is the period of the Newtonian orbit, or

(4) $T_{N}={\frac {2{\pi }r_{N}}{v_{N}}}$.

The static gravitation becomes dynamic during secular evolution when the central mass, rotating mass, or angular momentum change over time. Orbital evolution takes place accordingly. The Newtonian orbit transforms into a concentric spiral orbit (baseline orbit), while the Keplerian orbit evolves into an eccentric spiral orbit. Denoting $r_{b}$ as the instantaneous radius of the baseline orbit, and $r$ as the instantaneous radius of the eccentric orbit, their reciprocals can be expressed as

(5) $u={\frac {1}{r}}$, and $u_{b}={\frac {1}{r_{b}}}$.

The baseline precession mode $\lambda$ is defined as

(6) $\lambda ={\frac {m}{L}}{\sqrt {\frac {GM}{u_{b}}}}$.

The evolution rate of the baseline galactic distance ($r_{b}$) is defined as

(7) $\xi ={\frac {1}{2r_{b}}}{\frac {\mathrm {d} r_{b}}{\mathrm {d} \theta }}.$

The evolution rate of the central mass (M) is specified as

(8) $\xi _{M}={\frac {1}{4M}}{\frac {\mathrm {d} M}{\mathrm {d} \theta }}.$

The evolution rate of the self-mass (m) is prescribed as

(9) $\xi _{m}={\frac {1}{2m}}{\frac {\mathrm {d} m}{\mathrm {d} \theta }}$.

The evolution rate of angular momentum (L) is given as

(10) $\xi _{L}={\frac {1}{2L}}{\frac {\mathrm {d} L}{\mathrm {d} \theta }}$.

Consequently, the orbital equation can be expressed as

(11) $[{\frac {\mathrm {d^{2}} u}{\mathrm {d} \theta ^{2}}}+2\xi {\frac {\mathrm {d} u}{\mathrm {d} \theta }}+{\lambda }^{2}(u-u_{b})]{\frac {\mathrm {d} u}{\mathrm {d} \theta }}={\lambda }^{2}\delta$.

where $\delta$ is

(12) $\delta ={\frac {(\epsilon ^{2}-1)u_{b}^{2}+u^{2}}{2u_{b}}}{\frac {\mathrm {d} u_{b}}{\mathrm {d} \theta }}+\epsilon u_{b}^{2}{\frac {\mathrm {d} \epsilon }{\mathrm {d} \theta }}$.

On a velocity curve where a local velocity slope can be approximated as a constant on a log-log scale, these evolution rates can be considered as constants, or $\xi =const1$, $\xi _{L}=const2$, $\xi _{m}=const3$, $\xi _{M}=const4$.

When the radial perturbation is infinitesimal, $\epsilon \rightarrow 0$, $u\rightarrow u_{b}$, and $\delta \rightarrow 0$. In this scenario, Eq. (11) becomes a concentric spiral orbit and can be solved as

(13) $r_{b}=r_{1}e^{2{\xi }{\theta }}$,

where $r_{b}$ is the baseline galactic distance, $r_{1}$ is the initial galactic distance, and $\xi$ is the evolution rate of the baseline galactic distance.

The evolution rates suffice the radial migration law:

(14) $\xi =\xi _{L}-\xi _{m}-\xi _{M}$.

The corresponding rotational velocity ($v_{b}$) is responsive to $\xi _{M}$ as

(15) $v_{b}=v_{1}e^{2{\xi _{M}}{\theta }}$,

where $v_{1}$ is the initial velocity.

The centripetal force of the dynamic gravitational law is

(16) $F_{D}={\frac {GMm}{r_{0}r}}$,

where $r_{0}$ is the Newtonian radius, which corresponds to a circular orbit at the onset of the evolution. The dynamic gravitation is reduced to the inverse square law when orbital evolution is absent ($r=r_{0}$), and it asymptotically approaches the inverse linear law at large distances where $r\gg r_{0}$.

The eccentric spiral orbit in Eq. (11) can be solved as

(17) $r={\frac {r_{1}e^{2{\xi }{\theta }}}{1+{\epsilon e^{{\frac {3}{2}}\xi \theta }}cos{\phi }}}$,

where $\epsilon$ is orbital eccentricity, and $\phi$ is radial phase angle that is related to the revolving angle $\theta$ as

(18) ${\phi }={\frac {1}{\xi }}(1-e^{-\xi \theta })$.

The baseline precession mode ${\lambda }$ in Eq. (6) can be alternatively expressed as

(19) ${\lambda }=e^{-{\xi }{\theta }}$.

For an arbitrary eccentricity ${\epsilon }_{d}$, a compliance ratio is introduced as

(20) $\sigma ={\sqrt {\frac {2\epsilon _{d}}{ln{\frac {1+\epsilon _{d}}{1-\epsilon _{d}}}}}}$.

The eccentric spiral orbit with an arbitrary eccentricity in Eq. (11) can be expressed as

(21) $r={\frac {\sigma r_{1}e^{2{\xi }{\theta }}}{1+{\epsilon _{d}}cos{\phi }}}$.

The period of the eccentric spiral orbit is

(22) $T_{D}={\frac {T_{b}}{(1-{\epsilon _{d}}^{2})^{\kappa }}}$,

where $T_{b}$ is the circular period of the baseline orbit, and $\kappa$ is a fitting constant approximately 0.9247. When $\epsilon _{d}=0$, the eccentric orbit is reduced to a concentric orbit, or the baseline orbit. $T_{b}$ is calculated as

(23) $T_{b}={\frac {2\pi r_{b}}{v_{b}}}$.

Orbital Precession

Spiral orbits exhibit orbital precession. For orbits with minimal eccentricity, the precession can be expressed as

(24) $\Delta \beta =2\pi ({\frac {1}{\lambda }}-1)$.

For orbits with an arbitrary eccentricity,

(25) $\Delta \beta =2\pi [{\frac {1}{\lambda }}(1-{\epsilon _{d}^{2}})^{\kappa -{\frac {3}{2}}}-1]$,

The baseline precession mode ${\lambda }$ varies with orbital evolution. It is greater than unity if the revolving object drifts inward, otherwise outward. When the baseline precession is assessed at the pericenter or apocenter, the baseline precession mode ${\lambda }$ is referred to as the harmonic precession mode ${\overline {\lambda }}$, which is

(26) ${\overline {\lambda }}={\frac {2\pi }{2\pi +\Delta \beta }}$.

The baseline precession mode is

(27) ${\lambda }={\overline {\lambda }}(1-{\epsilon _{d}^{2}})^{\kappa -{\frac {3}{2}}}$.

For example, Earth’s perihelion precession is 11.45 arcsec per year, or $\Delta \beta =11.45$ arcsec. The harmonic precession mode is ${\overline {\lambda }}=1-8.835\times 10^{-6}$, and the baseline precession mode is $\lambda =1+1.385\times 10^{-4}$. The fact that the harmonic precession mode ${\overline {\lambda }}<1$ implies Earth is undergoing a reverse evolution and is spiraling outward. Additionally, $\lambda >1$ suggests that Earth experienced an inward orbital evolution and remains within the Newtonian orbit where $\lambda =1$.

Gravitational Lensing

The light deflection angle predicted by the inverse square law is

(28) $\theta _{N}={\frac {2GM}{c^{2}R}}$,

where GM is the gravitational constant, c is the speed of light in vacuum, and R is the radius of the Sun.

The center of the Sun is specified as the origin (x=0, y=0). The gravitational lens can be modelled as an ellipse that circumscribes the Sun and contacts the Sun at the minor radius of the ellipse (x=0, y=R). Suppose the eccentricity is $\epsilon _{d}$ and a light ray parallel to the major axis (x) grazes the Sun at the minor radius (x=0, y=R). The ray of light is only affected by the gravity in the vicinity of the Sun and then follows a straight line. Assume the lateral deflection of the ray is dy while the effective distance is dx. Beyond the effective lens point (dx, R-dy), the ray of light becomes a straight line at a deflection angle of $\theta _{d}$ from the major axis (x).

As derived , the eccentricity of the lens is

(29) $\epsilon _{d}=1-{\sqrt {{\frac {R}{dx}}tan\theta }}$.

The relationship between the deflection angle $\theta _{d}$ and that predicted from the inverse square law $\theta _{N}$ is

(30) ${\frac {\theta _{d}}{\theta _{N}}}={\sqrt {\frac {({\frac {R}{dx}})^{2}-(1-\epsilon _{d})^{2}}{({\frac {\sigma R}{dx}})^{2}-(1-\epsilon _{d})^{2}}}}.$

If the half length of the effective lens (dx) is assumed to be equal to the radius of the Sun, or $dx=R$, it can be simplified as

(31) ${\frac {\theta _{d}}{\theta _{N}}}={\sqrt {\frac {1-(1-\epsilon _{d})^{2}}{\sigma ^{2}-(1-\epsilon _{d})^{2}}}}$.

Likewise, Eq. (29) becomes

(32) $\epsilon _{d}=1-{\sqrt {tan\theta }}$.

Combining Eq. (31) and Eq. (32), one obtains

(33) ${\frac {\theta _{d}}{\theta _{N}}}={\sqrt {\frac {1-tan\theta }{\sigma ^{2}-tan\theta }}}.$.

Considering the deflection angle is typically very small, $\theta \ll \sigma \ll 1$, the ratio is simplified to

(33) ${\frac {\theta _{d}}{\theta _{N}}}={\frac {1}{\sigma }}$.

The deflection angle predicted from the inverse square law is 0.875 arcsec , which results in the eccentricity of the lens $\epsilon _{d}=0.9979$ in Eq. (32), and the compliance ratio $\sigma =0.5387$ in Eq. (20). As a result, the deflection angle predicted by the dynamic gravitational law is

(33) $\theta _{d}={\frac {\theta _{N}}{\sigma }}=1.62$ arcsec.

The observed deflection angle is 1.66 arcsec $\pm$10%. It is noted that the half length of the effective lens is assumed to be the radius of the Sun, or $dx=R$.

Other Applications

The interpretation of the rotation curve, the mechanism of bullet clusters, and the formation of galactic structures are illustrated in terms of dynamic gravitation.