## Topological Phase Transition III: Solar Surface Eruptions and Sunspots

Tian Ma & Shouhong Wang, Topological Phase Transition III: Solar Surface Eruptions and Sunspots, 2017, hal-01672381

The physical and mathematical reasons for solar surface eruptions and sunspots are not satisfactorily understood. This paper is aimed to provide a new theory for the formation of the solar surface eruptions, sunspots and solar prominences, based on the recently developed statistical theory of heat by the authors, and on the theory and notion of topological phase transitions.

1. The most important ingredient of the study is the recently developed theory of heat [T. Ma & S. Wang, Statistical Theory of Heat, 2017, hal-01578634]. We derived the energy level temperature formula, showing that the temperature is essentially the average energy level of system particles. We also obtained the photon number entropy formula, demonstrating that the entropy is the number of photons in the gap between system particles, and the physical carrier of heat is the photons.

Another important component of the theory is the vibratory mechanism of photon absorption and radiation:

a particle can only absorb and radiate photons while experiencing vibratory motion. The higher the frequency of the vibration of the particle, the larger the absorbing and radiating energy. The vibration or irregular motion of particles in a system is caused by collisions between particles and by absorbing and radiating photons.

2. This above mechanism shows immediately that for particles in high speed vibration and irregular motion, the rate of photon emission and absorption increases, leading to the number density of photons to increase, and further causing the particle energy levels to elevate. Hence, the photon absorption and emission induce the concentration of temperature, which we call the anti-diffusive effect of heat:

Due to the higher rate of photon absorption and emission of the particles with higher energy levels, the photon flux will move toward to the higher temperature regions from the lower temperature regions.

By the Stefan-Boltzmann law, the reversed heat flux measuring the anti-diffusive effect is expressed as

$\displaystyle \bigg(\frac{\text{d}T}{\text{d}t}\bigg)_{ADE}=\beta_0T^4,$

where ${\beta_0}$ is the heat effect coefficient.

3. Then by the Fourier law, we derive the following law for heat transfer for the solar atmosphere:

$\displaystyle \frac{\partial T}{\partial t}+(u\cdot \nabla)T=\kappa \Delta T+\beta_0T^4+\beta_1(\pmb{E}^2+\pmb{H}^2). \ \ \ \ \ (1)$

Here on the right-hand side, the first term represents the usual diffusion of heat, the last term is the heat source due to the solar electromagnetic fields. Importantly the second term represents the anti-diffusive effect of heat, and it is this anti-diffusive effect that leads to the formation of sunspots, the solar flares and the prominences.

4. The full model governing Sun’s surface plasma fluid combines the fluid dynamical equations, the above new heat equation (1), and the Maxwell equations. They are given as follows:

$\displaystyle \rho\bigg[\frac{\partial u}{\partial t}+(u\cdot\nabla)u\bigg]= \mu\Delta u-\nabla p +\rho_e(\pmb{E}+ u\times \pmb{H})-g\pmb{k}\rho(1-\alpha T), \ \ \ \ \ (2)$

$\displaystyle \frac{\partial T}{\partial t}+(u\cdot \nabla)T=\kappa \Delta T+\beta_0T^4+\beta_1(\pmb{E}^2+\pmb{H}^2), \ \ \ \ \ (3)$

$\displaystyle \frac{\partial \pmb{H}}{\partial t}=-\frac{1}{\mu_0}\text{curl }\pmb{E}, \ \ \ \ \ (4)$

$\displaystyle \frac{\partial \pmb{E}}{\partial t}=\frac{1}{\varepsilon_0}\text{curl }\pmb{H}-\frac{1}{\varepsilon_0}\rho_eu, \ \ \ \ \ (5)$

$\displaystyle \mbox{div }\pmb{H}=0, \ \ \ \ \ (6)$

$\displaystyle \mbox{div } \pmb{E}= \rho_e, \ \ \ \ \ (7)$

$\displaystyle \frac{\partial \rho}{\partial t}=-\text{div}(\rho u). \ \ \ \ \ (8)$

5. As the anti-diffusive term in (3) counteracts with the diffusion term, we are able to prove a temperature blow-up theorem, which shows that there exist ${x_0\in\Omega}$ and ${t_0>0}$, such that the temperature ${T}$ blows up at ${(x_0,t_0)}$ with blow-up time estimated as

$\displaystyle t_0=\frac{|\Omega|^3 }{3 a^3\beta_0}, \qquad a=\int_\Omega T_0(x) dx, \ \ \ \ \ (9)$

where ${T_0}$ is the initial value of temperature, ${ \Omega=S^2\times(r_0,r_1)}$, ${r_0}$ is the solar radius, and ${r_1=r_0+h}$ with the thickness of solar atmosphere ${h}$.

6. The sunspots can now be clearly explained by the anti-diffusive effect of heat and the temperature blow-up that we just mentioned. We summarize this explanation as follows:

• Due to thermal fluctuations, the temperature in the solar atmosphere is nonhomogeneous, leading to elevated temperature in some local areas. The anti-diffusive effect of heat then shows that the higher temperature regions absorb more photons from their surrounding areas, leading to their temperature decreasing, and consequently generating sunspots;
• The anti-diffusive effect of heat makes the temperature around the sunspot areas increasing rapidly, generating the temperature blow-up, and leading to solar eruptions.

In fact, equation (2) dictates the behavior of mass ejections. When the temperature ${T}$ blows up at ${(x_0,t_0)}$, the maximal forces acting on the particles near ${x_0}$ are just ${\nabla p}$. Hence, in the neighborhood of ${(x_0,t_0)}$, (2) can be approximatively expressed as

$\displaystyle \frac{\text{d} u}{\text{d} t}=-\frac{1}{\rho}\nabla p. \ \ \ \ \ (10)$

By the gaseous equation of state: ${p=R\rho T/m}$, where ${R}$ is the gas constant and ${m}$ is the particle mass, the equation (10) is written as

$\displaystyle \frac{\text{d} u}{\text{d} t}=-\frac{R}{m}\nabla T. \ \ \ \ \ (11)$

The temperature blow-up shows that

$\displaystyle \lim\limits_{t\rightarrow t_0}|\nabla T(x_0,t)|=\infty.$

Therefore we deduce from (11) that

$\displaystyle \lim\limits_{t\rightarrow t_0}\bigg|\frac{\text{d} u(x_0,t)}{\text{d} t}\bigg|=\infty, \ \ \ \ \ (12)$

which represents the high speed gas explosion and particle ejections. The ejection direction is

$\displaystyle \vec{r}=-\lim\limits_{t\rightarrow t_0}\frac{\nabla T(x_0,t)}{|\nabla T(x_0,t)|}. \ \ \ \ \ (13)$

• It is clear that the temperature blow-up generates solar flares.
• By the Maxwell equations (4) and (5) can be written as

$\displaystyle \frac{\partial^2 \pmb{E}}{\partial t^2}+\frac{1}{\varepsilon_0\mu_0}\text{curl}^2 \pmb{E}=-\frac{\rho_e}{\varepsilon_0}\frac{\partial u}{\partial t}, \ \ \ \ \ (14)$

$\displaystyle \frac{\partial^2 \pmb{H}}{\partial t^2}+\frac{1}{\varepsilon_0\mu_0}\text{curl}^2 \pmb{H}= \frac{\rho_e}{\varepsilon_0}u. \ \ \ \ \ (15)$

In view of (12) and (13), the equations (14) and (15) generate very strong electromagnetic radiation in the ${\vec{r}}$ direction.

• The eruption (11) leads also to a huge current jet ${J=\rho_eu}$ in the ${\vec{r}}$ direction.Also, (5) leads to the approximately the Ampère law:

$\displaystyle \text{curl } \pmb{H}=\rho_e u, \ \ \ \ \ (16)$

which gives rise to violent magnetic loops, perpendicular to the direction ${\vec{r}}$, leading to the solar prominences.

• Astronomical observations show that sunspots and solar flares occur periodically in an 11-year cycle. The blow-up time (9) links the initial temperature ${T_0}$, the solar eruption period, and the anti-diffusive effect coefficient ${\beta_0}$. Such a link is applicable to all stars. For the Sun, we can easily estimate

$\displaystyle \beta_0=2.85\times 10^{-22}/(K^3\cdot s).$

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

Tian Ma, Shouhong Wang