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sun-part2 - n ounceme'fis Fri March 11‘“ we will talk...

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Unformatted text preview: n ounceme'fis Fri. March 11‘“, we will talk about solar neutrinos Fri. March 18‘“, we reinforce our ideas about stellar evolution ReadBennett et al. Ch: 14 ' NASA, NOAO, ESAand The Hubble Heritage Team (STSCI/ Midterm #1 - High: 95 - Median: 71 Number of students score AST 203 (Spring 2011) The Sun continued... (Kutner, Ch. 6, Bennett el al. Ch. 14) AST 203 (Spring 2011) Radiation Transport The Sun's energy is produced in the core. Photons absorbed and re-emitted by atoms as the carry energy to the surface. Photons escape at the surface—this is the light we see. Radiation transport theory describes this outward dance of absorption and re-emission and how light and matter interact. Understanding absorption allows us to interpret the shape of absorption lines in the spectra, yields detailed information about the conditions in the Sun. AST 203 (Spring 2011) Number Density vs. Mass Density We are used to mass density: mass M” volume V In Astronomy, we often encounter number density: number of particles N n : — : _ volume V we can relate the total mass, M, and the number of particles, N, as M : Nm, where m is the mass of a particle. AST 203 (Spring 2011) Take atoms in solar atmosphere as tiny spheres with radius 7". Any photon that hits one of these spheres is absorbed. Each sphere has a cross-section of 0 = 7TT2 Consider photons passing through a cylinder of cross-section A and length l . number density of atoms = n. AST 203 (Spring 2011) Total # of atoms in the cylinder: N : nV : n (Al) Take the number density of atoms to be small no two atoms lie along the same line of sight of the incoming photons—Le. no atoms are blocked by other atoms. This means that the total absorbing cross section is atot : N0 : nAla Saying that no atoms blocks another means 0101 < A AST 203 (Spring 2011) Radiation Transport Fraction of incoming radiation that is absorbed is f : Utot A = nal —optical depth, 7 : ml Saying am < A means that 7' << 1—optically thin material. For optically thin materials/atmospheres, if T = 0,01 then 1% of the incident radiation is absorbed. AST 203 (Spring 2011) Radiation Transport In general cross section is a function of wavelength. Atoms can absorb photons whose energy is equal to the difference in the atom energy levels At wavelengths away from these atomic transitions, we expect the radiation to pass through with little absorption. We sometimes write the cross-section as 0A or 0V Useful quantity: column density, nl AST 203 (Spring 2011) Radiation Transport It is sometimes convenient to define the absorption coefficient, 7) KAI—2710'), 1 This is simply the number of absorptions per unit length. 1/m average distance a photon travels before being absorbed 1 1 LA : — : — mean free path K) naA We can then rewrite the optical depth as —L— H l T : : A LA A “The optical depth may be thought of as the number of mean free paths from the original position to the surface, as measured along the ray's path”—Carrol| and Ostlie AST 203 (Spring 2011) As | becomes long, we also break the approximation that now line of sight passes through multiple atoms something different is needed here. AsT 203 (Spring 2011) Radiation Transport Not optically thin —> 7 no longer fraction of radiation absorbed. Divide atmosphere into thin slabs d7 << 1 —> amount of radiation [0 , [1, [2, ,,I" absorbed in each slab is simply IdT Change in the intensity through a slab is d] : —I d7 4i l7 d7 The minus sign indicates that radiation is removed as it passes through the layer. AsT 203 (Spring 2011) Radiation Transport Now that we know what happens when we pass through one slab, we can integrate through all layers. I / 7' d] _/:—/ d7“, ID I 0 1n(I) —1n(I0) : —7' This integrates to or I = [De—T AST 203 (Spring 2011) Radiation Transport This is an exponential decay: By T m 1, most of the radiation is absorbed—we only see the outermost layers. AST 203 (Spring 2011) Radiation Transport Recall that we can approximate the exponential as emzl—ka: f0r|x|<<1 So if the optical depth is small, we can write I : log—T N [0(1 — 7') —) fraction of intensity removed (absorbed) is the optical depth— optically thin limit. AST 203 (Spring 2011) The Solar Limb Center of Sun looks bright while the edges look dark—limb darkening. How do we explain this? 2003110128 05:24 UT (SOHO/NASA) AST 203 (Spring 2011) Limb Darkening We see to equal optical depths, T m 1 when we look toward the edge, we are going through the atmosphere at an oblique angle—we do not see as deep. Since we see deeper layers when looking toward the center, we are seeing hotter material, and therefore it appears brighter. AST 203 (Spring 2011) Aside: Random Walk Bennett et al. Ch. 14; see also Shu Ch. 5 and Carroll and Ostlie Ch 9) We've been talking about photons in the solar atmosphere, but radiation transport is important in the interior of the Sun as well. Photons moving outward from center of the Sun are continually absorbed and re-emitted. Photon path to the surface of the Sun is a series of short jumps Jump length is the mean free path random walk. AST 203 (Spring 2011) Aside: Random Walk Mean free path in Sun < 0.5 cm Light travel time across solar radius is t : RQ/c : 2.3 8 Actual time for photon to random walk out of the Sun is ~100,000 years. (random_walk.avi) AST 203 (Spring 2011) Reading For the next two lectures, it is strongly suggested that you take a look at the books on reserve in the library—in particular, our recommended text, Bennett et al. Cosmic Perspective, Ch. 14, 15, and S4, and Shu's The Physical Universe, Chapter 6. We are not covering Chapters 7 and 8 in our text (Special and General Relativity). We will introduce concepts from there as we need them. AST 203 (Spring 2011) ...
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