Resources

1. Introduction to Electrodynamics, 4th ed. by David J. Griffiths
2. Physics of the Galaxy and Interstellar Matter by H. Scheffler and H. Elsässer
3. Interstellar Grains by N.C. Wickramasinghe
4. Physical Processes in the Interstellar Medium by Lyman Spitzer, Jr.
5. The scattering of light, and other electromagnetic radiation by Milton Kerker
6. Mathematica 9 Help Documentation Center (the most valuable resource!)

Project Description

I will be exploring Mie theory and modeling this type of scattering for particles present in the interstellar medium (ISM). There are a number of parameters that can be easily derived from applying Mie theory to the ISM, including an asymmetry parameter (g), extinction coefficients, albedo, and phase functions. I will focus first on modeling the asymmetry parameter (g), but time permitting, will continue onto other properties as I move into Week 3 – as stated in my timeline (below).

While the space between stars and galaxies appears quite vast and barren given only the access of our eyes fixed quaintly at ground level, the ISM is teeming with a variety of matter and electromagnetic waves. These regions are rich with gas (atomic and molecular), dust, and are permeated by electromagnetic waves, or radiation, from starlight (and occasionally other sources). Observations of various astrophysical phenomena show that along a given line of sight, their is “extinction” of this radiation. Scattering and absorption account for these observations and occur due to the presence of various dust grains within the ISM. This is where Mie scattering fits into modeling the asymmetry parameter (g), a measure of the fraction of light scattered in the forward direction. Our efforts towards modeling this relationship as well as the values of other dust grain properties such as size, composition, etc, begins with a preliminary comprehension of Mie’s work.

In 1908, Mie was working well before the substantially assistive mechanisms of modern computational modeling. Although his successors refined the theory of scattering (for spherical particles) over subsequent years, Mie’s initial work is fitting for the relatively fundamental level of analysis in this course. As Kerker explains (5), the basic scattering functions can be derived from a process whereby the proper Ricati-Bessel function is chosen and scattering coefficients are derived:

and

where m indicates an index of refraction, and  (and their respective primes) are the Ricati-Bessel functions chosen, and  and  are constants given by wave parameters (k, etc). These scattering coefficients are then combined with a mathematical tool called a “Legendre polynomial” to give amplitude functions for the scattering. Other parameters can then be derived once these functions are modeled.

Tools

1. Mathematica
2. LaTeX

Timeline

Week 1 (4/7 – 4/13)

I will refresh and further extend my knowledge of Mathematica and continue my research into the computational and theoretical aspects of Mie scattering in the context of the interstellar medium (ISM). As my project is computationally based, I will focus heavily on bridging the divide between my theoretical knowledge of scattering in the ISM and my work in Mathematica.

Week 2 (4/14 – 4/20)

For Week 2 I plan on completing a working Mathematica model for the asymmetry parameter, g, and complimenting this work as needed with research into Bessel and Ricati-Bessel functions. Mie theory is founded in the rigors of these functions and gives solutions to Maxwell’s equations that illustrates the scattering of waves by spherical particles. Specifically useful with the Bessel and Ricati-Bessel functions are the eponymous differential equations. If this work is completed I will move on to Week 3 work.

Week 3 (4/21 – 4/27)

I will continue modeling. My initial research into the theory will be added to as needed. As time permits, I will continue on to explore the utility of Mie theory and scattering within the ISM as it relates to the derivation of other interstellar grain properties.

Week 4 (4/28 -5/4)

I will finalize any lingering modeling. My results will be nearly all in and I will begin to wrap up my blog – focusing on visualizations (plots and animations) and ensuring that they function properly within the blog space. This week will be mainly devoted to polishing off my blog posts (data and results especially), i.e. making all aspects of the blog look aesthetically pleasing.

Week 5 (5/5 – 5/11)

This week concludes my project as I finish commenting on my peer’s blog posts.

Collaborators

While I plan on working alone, Professor Magnes will be assisting and advising as this project progresses. Both of my predecessors focused on the theory and modeling of Rayleigh scattering – a related phenomena but still outside of the purview of my project. I intend to move in a direction much different than my predecessors who explored scattering of the Rayleigh domain. I do, however, anticipate that as my project develops, I will be reflecting on my results and the possibility exists to compare and contrast them in the context of the work done by these individuals. I cannot predict what these comparisons may entail, but as I conclude my project, these ties may reflect elements shared between background theory, computational processes, and my results.