Draft:Dynamic Gravitation

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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 ^[1] 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 ^[2]

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

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 ${\displaystyle r_{N}}$ and a velocity of ${\displaystyle v_{N}}$.

Orbital evolution equations are derived for dynamic gravitation. ^[3] 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) ${\displaystyle r={\frac {r_{1}}{1+{\epsilon }cos{\theta }}}}$,

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

The period of a Keplerian orbit is

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

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

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

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 ${\displaystyle r_{b}}$ as the instantaneous radius of the baseline orbit, and ${\displaystyle r}$ as the instantaneous radius of the eccentric orbit, their reciprocals can be expressed as

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

The baseline precession mode ${\displaystyle \lambda }$ is defined as

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

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

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

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

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

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

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

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

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

Consequently, the orbital equation can be expressed as

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

where ${\displaystyle \delta }$ is

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

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 ${\displaystyle \xi =const1}$, ${\displaystyle \xi _{L}=const2}$, ${\displaystyle \xi _{m}=const3}$, ${\displaystyle \xi _{M}=const4}$.

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

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

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

The evolution rates suffice the radial migration law:

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

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

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

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

The centripetal force of the dynamic gravitational law is ^[3]

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

where ${\displaystyle r_{0}}$ 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 (${\displaystyle r=r_{0}}$), and it asymptotically approaches the inverse linear law at large distances where ${\displaystyle r\gg r_{0}}$.

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

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

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

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

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

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

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

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

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

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

The period of the eccentric spiral orbit is

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

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

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