## Monte Carlo Simulations of Radiative Transfer: Basics of Radiative Transfer Theory (Part IIa)

SOURCES FOR CONTENT:

1. Chandrasekhar, S., 1960. “Radiative Transfer”. Dover. 1.
2. Choudhuri, A.R., 2010. “Astrophysics for Physicists”. Cambridge University Press. 2.
3. Boyce, W.E., and DiPrima, R.C., 2005. “Elementary Differential Equations”. John Wiley & Sons. 2.1.

Recall from last time , the radiative transfer equation

$\displaystyle \frac{1}{\epsilon \rho}\frac{dI_{\nu}}{ds}= M_{\nu}-N_{\nu}I_{\nu}, (1)$

where $M_{\nu}$ and $N_{\nu}$ are the emission and absorption coefficients, respectively. We can further define the absorption coefficient to be equivalent to $\epsilon \rho$. Hence,

$\displaystyle N_{\nu}=\frac{d\tau_{\nu}}{ds}, (2)$

which upon rearrangement and substitution in Eq. (1) gives

$\displaystyle \frac{dI_{\nu}(\tau_{\nu})}{d\tau_{\nu}}+I_{\nu}(\tau_{\nu})= U_{\nu}(\tau_{\nu}). (3)$

We may solve this equation by using the method of integrating factors, by which we multiply Eq.(3) by some unknown function (the integrating factor) $\mu(\tau_{\nu})$ yielding

$\displaystyle \mu(\tau_{\nu})\frac{dI_{\nu}(\tau_{\nu})}{d\tau_{\nu}}+\mu(\tau_{\nu})I_{\nu}(\tau_{\nu})=\mu(\tau_{\nu})U_{\nu}(\tau_{\nu}). (4)$

Upon examining Eq.(4), we see that the left hand side is the product rule. It follows that

$\displaystyle \frac{d}{d\tau_{\nu}}\bigg\{\mu(\tau_{\nu})I_{\nu}(\tau_{\nu})\bigg\}=\mu({\tau_{\nu}})U_{\nu}(\tau_{\nu}). (5)$

This only works if  $d(\mu(\tau_{\nu}))/d\tau_{\nu}=\mu(\tau_{\nu})$. To show that this is valid, consider the equation for $\mu(\tau_{\nu})$ only:

$\displaystyle \frac{d\mu(\tau_{\nu})}{d\tau_{\nu}}=\mu(\tau_{\nu}). (6.1)$

This is a separable ordinary differential equation so we can rearrange and integrate to get

$\displaystyle \int \frac{d\mu(\tau_{\nu})}{\mu(\tau_{\nu})}=\int d\tau_{\nu}\implies \ln(\mu(\tau_{\nu}))= \tau_{\nu}+C, (6.2)$

where $C$ is some constant of integration. Let us assume that the constant of integration is $0$, and let us also take the exponential of (6.2). This gives us

$\displaystyle \mu(\tau_{\nu})=\exp{(\tau_{\nu})}. (6.3)$

This is our integrating factor. Just as a check, let us take the derivative of our integrating factor with respect to $d\tau_{\nu}$,

$\displaystyle \frac{d}{d\tau_{\nu}}\exp{(\tau_{\nu})}=\exp{(\tau_{\nu})},$

Thus this requirement is satisfied. If we now return to Eq.(4) and substitute in our integrating factor we get

$\displaystyle \frac{d}{d\tau_{\nu}}\bigg\{\exp{(\tau_{\nu})}I_{\nu}(\tau_{\nu})\bigg\}=\exp{(\tau_{\nu})}U_{\nu}(\tau_{\nu}). (7)$

We can treat this as a separable differential equation so we can integrate immediately. However, we are integrating from an optical depth $0$ to some optical depth $\tau_{\nu}$, hence we have that

$\displaystyle \int_{0}^{\tau_{\nu}}d\bigg\{\exp{(\tau_{\nu})}I_{\nu}(\tau_{\nu})\bigg\}=\int_{0}^{\tau_{\nu}}\bigg\{\exp{(\bar{\tau}_{\nu})}U_{\nu}(\bar{\tau}_{\nu})\bigg\}d\bar{\tau}_{\nu}, (8)$

We find that

$\displaystyle \exp{(\tau_{\nu})}I_{\nu}(\tau_{\nu})-I_{\nu}(0)=\int_{0}^{\tau_{\nu}}\bigg\{\exp{(\bar{\tau}_{\nu})}U_{\nu}(\bar{\tau}_{\nu})\bigg\}d\bar{\tau}_{\nu} (9),$

where if we add $I_{\nu}(0)$ and divide by $\exp{(\tau_{\nu})}$ we arrive at the general solution of the radiative transfer equation

$\displaystyle I_{\nu}(\tau_{\nu}) = I_{\nu}(0)\exp{(-\tau_{\nu})}+\int_{0}^{\tau_{\nu}}\exp{(\bar{\tau}_{\nu}-\tau_{\nu})}U_{\nu}(\bar{\tau}_{\nu})d\bar{\tau}_{\nu}. (10)$

This is the mathematically formal solution to the radiative transfer equation. While mathematically sound, much of the more interesting physical phenomena require more complicated equations and therefore more sophisticated methods of solving them (an example would be the use of quadrature formulae or $n$-th approximation for isotropic scattering).

Recall also that in general we can write the phase function $p(\theta,\phi; \theta^{\prime},\phi^{\prime})$ via the following

$\displaystyle p(\theta,\phi;\theta^{\prime},\phi^{\prime})=\sum_{l=0}^{\infty}\gamma_{l}P_{l}(\cos{\Theta}). (11)$

Let us consider the case for which $l=0$ in the sum given by (11). This then would mean that the phase function is constant

$p(\theta,\phi;\theta^{\prime},\phi^{\prime})=\gamma_{0}=const. (12)$

Such a phase function is consistent with isotropic scattering. The term isotropic means, in this context, that radiation scattered is the same in all directions. Such a case yields a source function of the form

$\displaystyle U_{\nu}(\tau_{\nu})=\frac{1}{4\pi}\int_{0}^{\pi}\int_{0}^{2\pi}\gamma_{0}I_{\nu}(\tau_{\nu})\sin{\theta^{\prime}}d\theta^{\prime}d\phi^{\prime}, (13)$

where upon use in the radiative transfer equation we get the integro-differential equation

$\displaystyle \frac{dI_{\nu}(\tau_{\nu})}{d\tau_{\nu}}+I_{\nu}(\tau_{\nu})= \frac{1}{4\pi}\int_{0}^{\pi}\int_{0}^{2\pi}\gamma_{0}I_{\nu}(\tau_{\nu})\sin{\theta^{\prime}}d\theta^{\prime}d\phi^{\prime}. (14)$

Solution of this equation is beyond the scope of the project. In the next post I will discuss Rayleigh scattering and the corresponding phase function.

## Monte Carlo Simulations of Radiative Transfer: Basics of Radiative Transfer Theory (Part I)

SOURCE FOR CONTENT: Chandrasekhar, S., 1960. Radiative Transfer. 1.

In this post, I will be discussing the basics of radiative transfer theory necessary to understand the methods used in this project. I will start with some definitions, then I will look at the radiative transfer equation and consider two simple cases of scattering.

The first definition we require is the specific intensity, which is the amount of energy associated with a specific frequency $dE_{\nu}$ passing through an area $dA$ constrained to a solid angle $d\Omega$ in a time $dt$. We may write this mathematically as

$dE_{\nu}=I_{\nu}\cos{\theta}d\nu d\Sigma d\Omega dt. (1)$

We must also consider the net flux given by

$\displaystyle d\nu d\Sigma dt \int I_{\nu}\cos{\theta}d\Omega, (2)$

where if we integrate over all solid angles $\Omega$ we get

$\pi F_{\nu}=\displaystyle \int I_{\nu}\cos{\theta}d\Omega. (3)$

Let $d\Lambda$ be an element of the surface $\Lambda$ in a volume $V$ through which radiation passes. Further let $\Theta$ and $\theta$ denote the angles which form normals with respect to elements $d\Lambda$ and $d\Sigma$. These surfaces are joined by these normals and hence we have the surface across which energy flows  includes the elements $d\Lambda$ and $d\Sigma$, given by the following:

$I_{\nu}\cos{\Theta}d\Sigma d\Omega^{\prime}d\nu = I_{\nu}d\nu \frac{\cos{\Theta}\cos{\theta}d\Sigma d\Lambda}{r^{2}} (4),$

where $d\Omega^{\prime}=d\Lambda \cos{\Theta}/r^{2}$ is the solid angle subtended by the surface element $d\Lambda$ at a point $P$ and volume element $dV=ld\Sigma \cos{\theta}$ is the volume that is intercepted in volume $V$. If we take this further, and integrate over all $V$ and $\Omega$ we arrive at

$\displaystyle \frac{d\nu}{c}\int dV \int I_{\nu} d\Omega=\frac{V}{c}d\nu \int I_{\nu}d\Omega, (5)$

where if the radiation travels some distance $L$ in the volume, then we must multiply Eq.(5) by $l/c$, where $c$ is the speed of light.

We now define the integrated energy density as being

$U_{\nu}=\displaystyle \frac{1}{c}\int I_{\nu}d\Omega, (6.1)$

while the average intensity is

$J_{\nu}=\displaystyle \frac{1}{4\pi}\int I_{\nu}d\nu, (6.2)$

and the relation between these two equations is

$U_{\nu}=\frac{4\pi}{c}J_{\nu}. (6.3)$

I will now introduce the radiative transfer equation. This equation is a balance between the amount of radiation absorbed and the radiation that is emitted. The equation is,

$\frac{dI_{\nu}}{ds}=-\epsilon \rho I_{\nu}+h_{\nu}\rho, (7)$

where if we divide by $\epsilon \rho$ we get

$-\frac{1}{\epsilon_{\nu}\rho}\frac{dI_{\nu}}{ds}=I_{\nu}+U_{\nu}(\theta, \phi), (8)$

where $U(\theta,\phi)$ represents the source function given by

$U_{\nu}(\theta,\phi)=\displaystyle \frac{1}{4\pi}\int_{0}^{\pi}\int_{0}^{2\pi}p(\theta,\phi;\theta^{\prime},\phi^{\prime})I_{\nu}\sin{\theta^{\prime}}d\theta^{\prime}d\phi^{\prime}. (9)$

The source function is typically the ratio between the absorption and emission coefficients. One of the terms in the source function is the phase function which varies according to the specific scattering geometry. In its most general form, we can represent the phase function as an expansion of Legendre polynomials:

$p(\theta, \phi; \theta^{\prime},\phi^{\prime})=\displaystyle \sum_{j=0}^{\infty}\gamma_{j}P_{j}(\mu), (10)$

where we have let $\mu = \cos{\theta}$ (in keeping with our notation in previous posts).

In Part II, we will discuss a few simple cases of scattering and their corresponding phase functions, as well as obtaining the formal solution of the radiative transfer equation. (DISCLAIMER: While this solution will be consistent in a mathematical sense, it is not exactly an insightful solution since much of the more interesting and complex cases involve the solution of either integro-differential equations or pure integral equations (a possible new topic).)

## Monte Carlo Simulations of Radiative Transfer: Project Overview

The goal of this project was to simulate radiative transfer in the circumbinary disk GG Tau by using statistical methods (i.e. Monte Carlo methods). To provide some context for the project, I will describe the astrophysical system that I considered, and I will describe the objectives of the project

Of the many stars in the night sky, a significant number of them, when you are able to resolve them, are in fact binary stars. These are star systems in which two stars orbit a common center of mass known as a barycenter. For this reason, we regard GG Tau to be a close binary system; a system in which two stars are separated by a significant distance comparable to the diameters of each star.

The system of GG Tau has four known components with a hypothesized fifth component. There is a disk of material encircling the entire system which we call a circumbinary disk. Additionally, there is also an inner disk of material that surrounds the primary and secondary components referred to as a circumstellar disk.

The goal of this project was to use Monte Carlo methods to simulate radiative transfer processes present in GG Tau. This is accomplished through the use of importance sampling two essential quantities over cumulative distribution function of the form

$\displaystyle \xi=\int_{0}^{L} \mathcal{P}(x)dx\equiv \psi(x_{0}), (1)$

using FORTRAN code. I will discuss the mathematical background of probability and Monte Carlo methods (stochastic processes) in more detail in a later post. The general process of radiative transfer is relatively simple. Starting with an emitter,  radiation travels from its emission point, travels a certain distance, and at this point the radiation could either be scattered into an angle different from the incidence angle, or it can be absorbed by the material. In the context of this project, the emitter (or emitters) are the primary and secondary components of GG Tau. The radiation emitted is electromagnetic radiation, or light. This light will travel a distance $L$ after which the light is either scattered or absorbed. The code uses Eq.(1) (cumulative distribution function) to sample optical depths and scattering angles to calculate the distance traveled $L$. I will reserve the exact algorithm for a future post as well. In the next post, I will discuss the basics of radiative transfer theory as presented in the text “Radiative Transfer” by S. Chandrasekhar, 1960.

The following are the sources  used in this project

Wood, K., Whitney, B., Bjorkman, J., and Wolff M., 2013. Introduction to Monte Carlo Radiative Transfer.

Whitney, B. A., 2011. Monte Carlo Radiative Transfer. Bull. Astr. Soc. India.

Kitamura, Y., Kawabe, R., Omodaka, T., Ishiguro, M., and Miyama, S., 1994. Rotating Protoplanetary Gas Disk in GG Tau. ASP Conference Series. Vol. 59.

Piètu, V., Gueth, F., Hily-Blant, P., Schuster, K.F., and Pety, J., 2011. High Resolution Imaging of the GG Tauri system at 267 GHz. Astronomy & Astrophysics. 528. A81.

Skrutskie, M.F., Snell, R.L., Strom, K.M., Strom, S.E., and Edwards, S., 1993. Detection of Circumstellar Gas Associated with GG Tauri. The Astrophysical Journal. 409:422-428.

Choudhuri, A.R., 2010. Astrophysics for Physicists. 2.

Carroll B.W., Ostlie, D.A., 2007. An Introduction to Modern Astrophysics. 9,10.

Chandrasekhar, S., 1960. Radiative Transfer. 1.

Schroeder, D., 2000. Introduction to Thermal Physics. 8.

Ross, S., 2010. Introduction to Probability Model. 2,4,11.

I obtained the FORTRAN code from K. Wood via the following link. (For the sake of completeness the URL is: www-star.st-and.ac.uk/~kw25/research/montecarlo/montecarlo.html)

## Monte Carlo Simulations of Radiative Transfer: Overview

FEATURED IMAGE: This is one of the plots that I will be explaining in Post IV. This plot represents the radiation flux emanating from the primary and secondary components of the system considered.

I realize I haven’t posted anything new recently. This is because I have been working on finishing the research project that I worked on during my final semester as an undergraduate.

However, I now intend to share the project on this blog. I will be starting a series of posts in which I will attempt to explain my project and the results I have obtained and open up a dialogue as to any improvements and/or questions that you may have.

Here is the basic overview of the series and projected content:

Post I: Project Overview:

In this first post, I will introduce the project. I will describe the goals of the project and, in particular, I will describe the nature of the system considered.   I will also give a list of references used and mention any and all acknowledgements.

Posts II & III: Radiative Transfer Theory

These posts will most likely be the bulk of the concepts used in the project. I will be defining the quantities used in the research, developing the radiative transfer equation, formulating the problems of radiative transfer, and using the more elementary methods to solving the radiative transfer equation.

Post IV: Monte Carlo Simulations

This post will largely be concerned with the mathematics and physics of Monte Carlo simulations both in the context of probability theory and Ising models. I will describe both and explain how it applies to Monte Carlo simulations of Radiative Transfer.

Post V: Results

I will discuss the results that the simulations produced and explain the data displayed in the posts. I will then discuss the conclusions that I have made and I will also explain the shortcomings of the methods used to produce the results I obtained. I will then open it up to questions or suggestions.