2D random walk of photons emitted from solar core (pdf document) download at fliiby.com

"Two-Dimensional Random Walk of a Photon Emitted from the Solar Core Billy Teets Physics 341 Statistical Mechanics, Spring 2006 I. Introduction & Background In the field of astronomy, one subject of interest is how energy that is created in stars is transported outwards where it is eventually radiated into space. In modeling the interiors of stars, astronomers have determined that there are three main areas of interest: the nuclear-fusing core, the radiative zone, and the convective zone (Figure 1). The core of a star is where the action takes place. As a result of gravitational collapse, core temperatures rise high enough (except in the case of brown dwarfs) that stars are able to start fusing hydrogen, their main constituent, into helium. At this point in most stars, gamma ray photons are released where they begin their journey to the outer layers. As the photon travels, it can be subjected to free-free absorption where an electron can gain kinetic energy by absorbing the photon, or bound-free absorption where an electron bound to an atom can absorb the photon, jump to a higher energy level, and then spontaneously re-emit the photon(s). The latter process is more likely to occur in the very outer layers of the star since interior temperatures are high enough to ionize the vast majority of the atoms. Thus, the photons interact with the stellar material and make their way through the star via a random walk. When the photons undergo a random walk as the main way of transporting their energy, they are in radiative zone of a star. As temperatures decrease with radius, convection eventually takes over as the main energy transportation mechanism. The photons can still undergo random walks in this zone, but most of their journey is due to convection. Figure 1. Interior of the Sun As with virtually all characteristics of a star, the stellar mass determines the structure of these zones (Figure 2). Large stars have virtually no convection zones, so photons will have to undergo a random walk until they reach the very outer layers of the star. However, the cores are thought to have convection since fusion is occurring at such a furious rate. Intermediate mass stars, like our sun, have radiative zones that are approximately half the solar radius (not including the core) with convection taking over the last 30 percent of the radius. Very low mass stars can have no radiative zones at all, leaving virtually all energy transport to convective processes. These interior models support observational evidence for how long stars of different masses live. High mass stars live the shortest lives since their cores cannot be replenished with hydrogen fuel by convection. Lower mass stars live the longest since virtually all the hydrogen in the star can be transported to the core where it will be “burned” into helium. One should also note that the long lives of lower mass stars are also a result of hydrogen fusion occurring at a slower rate due to lower mass and thus lower temperatures. Figure 2. Interiors of Different Mass Stars II. Calculation of the Random Walk The basis of this two dimensional random walk is that a photon created in the core will travel a small distance before it will likely interact with an atom/electron. Many textbooks state that a mean free path of approximately one centimeter is adequate for many solar models. When the photon interacts, it will spontaneously be re-emitted in a random direction. There limitations to these statements which will be discussed later. If we assume the photon travels a distance l between interactions and that it undergoes N interactions, then the total distance d that it travels from the starting point can be given by: Figure 3. Random Walk d l d • d = l1 • l1 + l1 • l2 + ... + l2 • l1 +l 2 •l2 + ... + LN −1 • LN + LN • LN d • d = d 2 = Nl 2 d =l N To determine the magnitude of d, we can dot it with itself. In doing so, we have to dot all of the l vectors with themselves and each other, resulting in N2 dot products. In the end, the only dot products that survive are the ones that involve the component being dotted with itself. The others will cancel out since the assumption has been made that there is no preferential direction of spontaneous photon emission. Therefore, the total distance traveled from the core is proportional to the square root of the number of interactions. A simulation has been coded in MatLab and the average results are given for 100 trials (Figure 4)*. Figure 4. Random Walk Simulations for Various Interaction Numbers Random Walk from the Solar Core 90 15 120 60 120 Random Walk from the Solar Core 90 60 60 10 150 5 30 150 40 30 20 180 0 180 0 210 330 210 330 240 270 300 240 270 300 100 Interactions, Average Distance = 8.67 cm 1000 Interactions, Average Distance = 28.57 cm Random Walk from the Solar Core 90 80 120 60 60 120 Random Walk from the Solar Core 90 500 60 400 300 150 40 30 150 200 30 20 100 180 0 180 0 210 330 210 330 240 270 300 240 270 300 10000 Interactions, Average Distance = 90.69 cm 100000 Interactions, Average Distance = 321.98 cm *The 100000 interaction simulation results are only from one trial due to limitations in computing power III. Limitations in the Simulation A. Density Profile In this simulation, we have assumed an average mean free path and thus an average density in the star. However, as one knows, the density is not constant throughout a star. One can use a program such as StatStar to model the density profile as a function of radius so that realistic densities can be incorporated into the simulation (Figure 5). Figure 5. StatStar Program Modeling Density vs. Radius B. Computing Power The main complication in a simulation such as this one is the amount of computations required to model even a modest amount of the total random walk in a star. For instance, the solar radius is 7x1010 cm. If we assume a mean free path of one centimeter, then the total number of interactions is approximately the square of the solar radius, resulting in approximately 1022 interactions! Knowing the photon’s speed and that its total distance traveled is approximately 1022 centimeters, the photon will spend approximately 3x1011 seconds, or 10,000 years, in the sun. A more modest mean free path of one millimeter increases the time by an order of magnitude. Increasing the size of the star and further increasing the total range of the radiative zone increases this time even more. In a word, modeling the entire path is an enormous challenge without the use of statistics. C. Photon Statistics One consideration not taken into account in this model is the relative time that a photon will remain absorbed before it is spontaneously re-emitted (such as Einstein A Coefficients). Here we have assumed that the excitation has an average lifetime of zero seconds which is not the case in the real world. Also, we have assumed that a photon’s energy remains unchanged on its journey. As stated before, hydrogen fusion in the core results in the creation and emission of gamma ray photons, specifically six per helium atom creation (Figure 6). Figure 6. Proton-Proton Chain: Fusion of Hydrogen into Helium in the Stellar Core When a single gamma ray photon is absorbed, it can be re-emitted as a single gamma ray photon or as multiple, lower-energy photons, each of which can have very different “absorption” lifetimes. The latter situation we know to be true of course since we see the entire electromagnetic spectrum instead of just gamma rays being emitted. IV. Applications & Conclusions One may wonder how a random walk in a star is useful. A prominent use of a random walk simulation in a star is in the field of supernova research. Supernovae come in several different classes, all but one of them being the result of a high mass star’s core collapsing. The other, known as Type 1a supernovae, is the result of a degenerate stellar core undergoing fusion as a result of accretion from its companion. All supernovae but the latter produce neutrinos during the core collapse in a process known as reverse-beta decay (Figure 7). These neutrinos typically do not interact with matter. However, when modeling these supernovae, astronomers find that if they do not require neutrinos to interact with the infalling matter, the supernovae do not explode. Therefore, the neutrinos must interact and will do so via a random walk. Consequently, understanding how neutrinos interact through random walks should improve the understanding of supernova physics. Figure 7. Reverse-Beta Decay How does understanding supernovae help our understanding of physics? As is well known, nuclear physics is a major field that influences our everyday lives, such as providing us with sources of electricity. A major area of study is determining how many of the heavier elements are produced and how we can produce them for study and possibly for beneficial use (Figure 8). Figure 8. Production of Heavier Elements in Supernovae via the Rapid Neutron-Capture Process V. MatLab Code for Random Walk Simulation % % % % Billy Teets Statistical Mechanics Computational Project This program will simulate a random walk of a photon from the solar core for a specified number of interactions. clear; clc; % Clear the memory and command window, respectively. mean_path = 100; solarradius = 7*10^10; % (in cm) % (in cm) % initialization of variables rho_new = 1; % normalized length of vector rho(1) = 0; % starting length of path (origin) theta(1) = 0; % starting angle totaliterations = 100; % number of trials to run interactions = 10000; % number of interactions per trial run c= 3*10^10; % speed of light in cm/s for y=1:totaliterations for x = 2:1:interactions thetanew(x) = 2*pi*rand(1); % calculation of random emission angle i(x) = rho(x-1)*cos(theta(x-1))+rho_new*cos(thetanew(x)); j(x) = rho(x-1)*sin(theta(x-1))+rho_new*sin(thetanew(x)); rho(x)=sqrt(i(x)^2+j(x)^2); % calculation of new vector length % with respect to the origin theta(x) = atan2(j(x),i(x)); % calculation of new vector angle % with respect to the origin end N(y)=rho(interactions); % ending distance of photon from origin after % specified number of interactions end average=sum(N)/totaliterations % average ending distance from core polar(theta,rho,'-b'); % polar plot of random walk title('Random Walk from the Solar Core'); total_time = mean_path*interactions/c % time it takes photon to undergo the % total number of interactions (in % seconds VI. Figure Citations 1. http://outreach.atnf.csiro.au/education/senior/astrophysics/stellarevolution_mainsequence.html 2. ©The Cosmic Perspective, 3rd ed. Bennett, Donahue, Schneider, Voit 3. http://spiff.rit.edu/classes/phys440/lectures/walk/walk.html 5. StatStar program courtesy of Eindhoven University of Technology, available at http://plasimo.phys.tue.nl/statstar/ 6. ©The Cosmic Perspective, 3rd ed. Bennett, Donahue, Schneider, Voit 7. ©The Cosmic Perspective, 3rd ed. Bennett, Donahue, Schneider, Voit 8. http://www.physics.vanderbilt.edu/volker/p330b/add_lecture_materials/wkb/wkb.html ..."

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