## Topological Phase Transitions II: Spiral Structure of Galaxies

Tian Ma & Shouhong Wang, Topological Phase Transitions II: Spiral Structure of Galaxies, 2017, hal-01671178

The aim of this paper is to derive a new mechanism for the formation of the galactic spiral patterns.

1. There are three types of galactic structures: the spiral, the elliptical, and the irregular. The existing theory of the formation of the spiral galactic structure is the density wave theory by Chia-Ciao Lin and Frank Shu in 1964. They proposed that the spiral arms, being non material, are caused by the non-homogeneous velocity of stars and nebulae, similar to a traffic jam in a highway. However, the reasons behind the nonhomogeneous velocity of stars and nebulae are still not clear, and the density wave theory is not completely satisfactory.

In essence the physical origin of the density wave theory is not clearly understood. This is resolved in this paper using the recent development by the authors on gravitational field particle and gravitational radiation.

2. Due to the presence of dark matter and dark energy, the Einstein general theory of relativity can be uniquely modified using PID to take into account the effect of dark energy and dark matter phenomena, and to preserve the Einstein’s two fundamental principles: the principle of equivalence and the principle of general relativity. The new field equations are given by

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=-\frac{8\pi G}{c^4}T_{\mu\nu}-\nabla_\mu\Phi_\nu, \ \ \ \ \ (1)$

where

•  ${\{g_{\mu\nu}\}}$ is the Riemannian metrics of 4-dimensional space-time, representing the gravitational potential, depicting the curved space-time;
•  ${\{\Phi_\nu\}}$ is the dual gravitational potential, representing the gravitational field particle and carrying the field energy, which is similar to the electromagnetic interaction field particle: the photon.

The gravitational field particle is described by the dual field ${\{\Phi_\mu\}}$, which is regarded as the graviton.

3. Motivated by the new statistical theory of heat by the authors, we have the following conclusions for the gravitational field particle:

the absorption and radiation of gravitons could generate a gravitational temperature field, representing the average energy level of massive matter, reminiscent of the photons yielding the temperature in thermodynamical systems.

Also, the gravitational temperature ${\mathcal{T}}$, which we call G-temperature, satisfies a diffusion equation given by

$\displaystyle \frac{\partial \mathcal{T}}{\partial t}+\frac{1}{\rho}(P\cdot\nabla)\mathcal{T}=\kappa \Delta \mathcal{T} +Q, \ \ \ \ \ (2)$

where ${Q}$ represents the gravitational source.

The new gravitational temperature field provides the needed key source for the formation of different galactic patterns, which was entirely missing in existing theories such as the pioneering work of Lin and Shu.

4. With the G-temparature equation at our disposal, we can derive the momentum form of the astrophysical fluid dynamical model coupling the diffusion equation of G-temperature field. This new astrophysical galactic dynamics model is a dissipative system. Hence the mechanism of the formation of different galactic structures is of characteristic for dissipative systems, in contrast with the density wave theory.

5. The spatial domain of the model is the ring domain in polar coordinates

$\displaystyle \Omega=\{(r,\theta)~|~0\leq \theta\leq 2\pi,~r_0

and we take the transformation

$\displaystyle x_1=r\theta,\quad x_2=r-r_0.$

The Rubin rotational curve suggests that the momentum density ${\overline{P}}$ in the steady state solution of the model is

$\displaystyle (\overline{P},\overline{\mathcal{T}},\overline{p})=\left((\rho_0v_0,0), \mathcal{T}_0-\beta x_2, -\rho_0g\int(1-\alpha\overline{\mathcal{T}})\text{d}x_2 \right) \ \ \ \ \ (3)$

where ${v_0}$ is the constant velocity in the Rubin rotational curve. Then the nondimensional deviation system from the basic solution is given by

$\displaystyle \frac{1}{\text{Pr}}\bigg[\frac{\partial P}{\partial t}+(P\cdot\nabla)P\bigg] =\Delta P -a\frac{\partial P}{\partial x_1}-\frac{1}{\text{Pr}}\nabla P+\sqrt{\text{Ra}}\mathcal{T}\vec{k}, \ \ \ \ \ (4)$

$\displaystyle \frac{\partial \mathcal{T}}{\partial t}+(P\cdot\nabla)\mathcal{T} = \Delta \mathcal{T}-a\text{Pr}\frac{\partial \mathcal {T}}{\partial x_1}+\sqrt{\text{Ra}}P_2, \ \ \ \ \ (5)$

$\displaystyle \text{div}P=0. \ \ \ \ \ (6)$

Here Pr is the Prandtl number, ${a}$ is the Rubin number, and ${\ell}$ is the ratio between the disk width and the halo radius. These are non-dimensional parameters given by

$\displaystyle \text{Ra}=\frac{g\alpha\rho_0\beta}{\kappa\mu}r^4_0, \qquad \beta=\frac{T_0-T_1}{r_1-r_0},\quad a=\frac{r_0 v_0}{\mu},\quad \ell=\frac{r_1-r_0}{r_0}. \ \ \ \ \ (7)$

For this galactic dynamics system, there are three crucial parameters for the formation of different galactic structures:

• Ra represents the G-temperature gradient,
• ${a}$ represents the average velocity of stars in the galaxy, and
• ${\ell}$ represents the relative ratio between the inner and outer radii of galactic disk.

5. Mathematically, using the dynamical transition theory and the geometric theory of incompressible flows, both developed by the authors, we can demonstrate that both topological phase transition and dynamic phase transition occur at the same critical control parameters. In particular we derive the following conclusions on the formation of galactic structures, dictated by the parameters Ra, ${a, \ell}$:

• if ${\text{Ra} or ${\ell}$ is small, then the galaxy is elliptic; and
• if ${\text{Ra}>R_c(a)}$ and ${\ell}$ is relatively large, then the galaxy is spiral.

This paper is part of the research program initiated recently by the authors on theory and applications of topological phase transitions, including

• quantum phase transitions,
• electromagnetic eruptions on solar surface,
• boundary-layer separation of fluid flows, and
• interior separation of fluid flows.

Tian Ma, Shouhong Wang