Light trapping by the dark matter

Abstract

Considering the dark matter as self-gravitating Bose-Einstein condensate in a gravito-chemical equilibrium state, we have studied the light trajectory through the halo. Found that depending on the initial direction cosine of the light ray, there are three types of trajectories: (a) the refracted light i.e. light which can refracts away from the halo to the out side, (b) the trapped light i.e. light that is trapped into the halo i.e. an observer, situated outside of the halo, can not observe it and (c) the reflected light: in this case the halo behaves as a perfect reflector to the outside light. The light trapping can introduce asymmetry in the observed spectrum and modify the apparent number of stars in a dark matter dominated galaxy.

Appendices

Appendices

Gibb’s fundamental relation in thermodynamics

$$ dE=\sum \xi _{i}dX_{i} $$

tells us that a system can change its energy in several ways. The number of terms is equal to the number of degrees of freedom of the system. In the couples of “energy-conjugated” variables at the right hand side of the equation, the first quantity, $\xi _{i}$, is, in general, an intensive quantity and the second one, $X_{i}$, an extensive quantity. Within the framework of thermodynamics one often considers systems whose Gibbs fundamental relation reads

$$ dE=Tds-pdv+\sum \mu _{i} dn_{i} $$

The equation means, that the system under consideration, a gas for instance, can exchange energy in the following ways: in the form of heat $Tds$, by realizing work $-pdv$ or as chemical energy $\mu _{i} dn_{i}$. If the system is composed of only one substance and does not change its composition or phase, the above equation simplifies to:

$$ dE=Tds-pdv+ \mu dn $$

For other systems the Gibbs fundamental relation can contain many other terms of very distinct nature. The system which has the following Gibbs fundamental relation

$$ dE=Tds-pdv+ \mu dn+\phi dm $$

can exchange energy in the following forms: as heat, as chemical energy and as gravitational energy. Now, to each of the extensive quantities $X_{i}$, with the exception of the volume, corresponds a current with the current density $\textbf{J}_{X_{i}}$. If the current is flowing through a dissipative environment (e.g. through a viscous medium), i.e., if the current causes the production of entropy, the corresponding intensive quantity has different values at both ends of the medium or channel. The energy dissipation rate per volume is:

$$\begin{aligned} T\sigma =-\sum \textbf{J}_{X_{i}}.\nabla \xi _{i} \end{aligned}$$

(18)

Here, $\sigma$ is the rate of entropy production in the system. The term $\nabla \xi _{i}$ can be interpreted as the driving force for the current density $\textbf{J}_{X_{i}}$.

Now, consider an isothermal self gravitating system consists of $n$ number of particles of mass $m$. Here, two potentials, the chemical ($\mu$) and the gravitational $(\phi )$, are acting on the particles in opposite direction. $\phi$ is acting in the direction of increasing pressure and $\mu$ is driving the particles in the opposite direction. The current density for mass flow is $\textbf{J}_{m}$ and for the ‘number of particle’ flow is $\textbf{J}_{n}$. These two quantity is related as $\textbf{J}_{m}=m \textbf{J}_{n}$. Hence, the equation (18) reads as:

$$ T\sigma =- \textbf{J}_{n}.\left (m\nabla \phi +\nabla \mu \right ) $$

For equilibrium of the system, the entropy production rate must be zero. Therefore, we have the following condition

$$ \nabla \left (m\phi + \mu \right )=0 $$

or

$$\begin{aligned} \nabla \psi =0 \end{aligned}$$

(19)

Here, $\psi =\mu +m\phi$ is the gravito-chemical potential. We can use the equation (19) directly or we can modify it as $\nabla ^{2} \psi =0$ for our convenience.