# Ch. 5 - Basic Definitions Specific intensity/mean intensity Flux The K integral and radiation pressure Absorption coefficient & optical depth

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<ul><li> Slide 1 </li> <li> Ch. 5 - Basic Definitions Specific intensity/mean intensity Flux The K integral and radiation pressure Absorption coefficient & optical depth Emission coefficient & the source function Scattering and absorption Einstein coefficients </li> <li> Slide 2 </li> <li> Area at an angle of view from the normal to the surface through an increment of solid angle Normal AA to observer We want to characterize the radiation from Specific Intensity Assume no azimuthal dependence </li> <li> Slide 3 </li> <li> Specific Intensity Average Energy (E d ) is the amount of energy carried into a cone in a time interval dt Specific Intensity in cgs (ergs s -1 cm -2 sr -1 -1 ) Intensity is a measure of brightness the amount of energy coming from a point on the surface towards a particular direction at a given time, at a frequency For a black body radiator, the Planck function gives the specific intensity (and its isotropic) Normally, specific intensity varies with direction </li> <li> Slide 4 </li> <li> I vs I The shapes of I and I are different because d and d are different sizes at the same energy of light: d = -(c/ 2 ) d For example, in the Sun, I peaks at ~4500 while I peaks at ~8000 </li> <li> Slide 5 </li> <li> Mean Intensity Average of specific intensity over all directions If the radiation field is isotropic (same intensity in all directions), then =I Black body radiation is isotropic and =B </li> <li> Slide 6 </li> <li> Flux The flux F is the net energy flow across an area A over time t, in the spectral range integrating over all directions energy per second at a given wavelength flowing through a unit surface area (ergs cm -2 s -1 Hz -1 ) for isotropic radiation, there is no net transport of energy, so F =0 </li> <li> Slide 7 </li> <li> On the physical boundary of a radiating sphere if we define F =F out + F in then, at the surface, F in is zero we also assumed no azimuthal dependence, so which gives the theoretical spectrum of a star </li> <li> Slide 8 </li> <li> One more assumption: If I is independent of , then This is known as the Eddington Approximation (well see it again) </li> <li> Slide 9 </li> <li> Specific Intensity vs. Flux Use specific intensity when the surface is resolved (e.g. a point on the surface of the Sun). The specific intensity is independent of distance (so long as we can resolve the object). For example, the surface brightness of a planetary nebula or a galaxy is independent of distance. Use radiative flux when the source isnt resolved, and we're seeing light from the whole surface (integrating the specific intensity over all directions). The radiative flux declines with distance (1/r 2 ). </li> <li> Slide 10 </li> <li> The K Integral The K integral is useful because the radiation exerts pressure on the gas. The radiation pressure can be described as </li> <li> Slide 11 </li> <li> Radiation Pressure Again, if I is independent of direction, then Using the definition of the black body temperature, the radiation pressure becomes </li> <li> Slide 12 </li> <li> Luminosity Luminosity is the total energy radiated from a star, at all wavelengths, integrated over a full sphere. </li> <li> Slide 13 </li> <li> Class Problem From the luminosity and radius of the Sun, compute the bolometric flux, the specific intensity, and the mean intensity at the Suns surface. L = 3.91 x 10 33 ergs sec -1 R = 6.96 x 10 10 cm </li> <li> Slide 14 </li> <li> Solution F= T 4 L = 4 R 2 T 4 or L = 4 R 2 F, F = L/4 R 2 Eddington Approximation Assume I is independent of direction within the outgoing hemisphere. Then F = I J = I (radiation flows out, but not in) </li> <li> Slide 15 </li> <li> The Numbers F = L/4 R 2 = 6.3 x 10 10 ergs s -1 cm -2 I = F/ = 2 x 10 10 ergs s -1 cm -2 steradian -1 J = I= 1 x 10 10 ergs s -1 cm -2 steradian -1 (note these are BOLOMETRIC integrated over wavelength!) </li> <li> Slide 16 </li> <li> The K Integral and Radiation Pressure Thought Problem: Compare the contribution of radiation pressure to total pressure in the Sun and in other stars. For which kinds of stars is radiation pressure important in a stellar atmosphere? </li> <li> Slide 17 </li> <li> Absorption Coefficient and Optical Depth Gas absorbs photons passing through it Photons are converted to thermal energy or Re-radiated isotropically Radiation lost is proportional to absorption coefficient (per gram) density intensity pathlength Optical depth is the integral of the absorption coefficient times the density along the path (if no emission) </li> <li> Slide 18 </li> <li> Class Problem Consider radiation with intensity I (0) passing through a layer with optical depth = 2. What is the intensity of the radiation that emerges? </li> <li> Slide 19 </li> <li> Class Problem A star has magnitude +12 measured above the Earths atmosphere and magnitude +13 measured from the surface of the Earth. What is the optical depth of the Earths atmosphere at the wavelength corresponding to the measured magnitudes? </li> <li> Slide 20 </li> <li> There are two sources of radiation within a volume of gas real emission, as in the creation of new photons from collisionally excited gas, and scattering of photons into the direction being considered. We can define an emission coefficient for which the change in the intensity of the radiation is just the product of the emission coefficient times the density times the distance considered. Emission Coefficient Note that dI does NOT depend on I! </li> <li> Slide 21 </li> <li> The Source Function The source function is just the ratio of the absorption coefficient to the emission coefficient: Sounds simple, but just wait. </li> <li> Slide 22 </li> <li> Pure Isotropic Scattering The gas itself is not radiating photons only arise from absorption and isotropic re-radiation Contribution of photons proportional to solid angle and energy absorbed: J is the mean intensity: dI/d = -I + J v The source function depends only on the radiation field </li> <li> Slide 23 </li> <li> For pure isotropic scattering Remember the definition of J So J = j / Hey! Then J = S for pure isotropic scattering </li> <li> Slide 24 </li> <li> Pure Absorption No scattering all incoming photons are destroyed and all emitted photons are newly created with a distribution set by the physical state of the gas. Source function given by Planck radiation law Generally, use B rather than S if the source function is the Planck function </li> <li> Slide 25 </li> <li> Einstein Coefficients For spectral lines or bound-bound transitions, assumed isotropic Spontaneous emission is proportional to N u x Einstein probability coefficient, A ul j = N u A ul h (N u is the number of excited atoms per unit volume) Induced emission proportional to intensity = N l B lu h N u B ul h </li> <li> Slide 26 </li> <li> Induced (Stimulated) Emission Induced emission in the same direction as the inducing photon Induced emission proportional to intensity I = N l B lu I h N u B ul I h True absorptionInduced emission </li> <li> Slide 27 </li> <li> Radiative Energy in a Gas As light passes through a gas, it is both emitted and absorbed. The total change of intensity with distance is just dividing both sides by -k dx gives </li> <li> Slide 28 </li> <li> The Source Function The source function S is defined as the ratio of the emission coefficient to the absorption coefficient The source function is useful in computing the changes to radiation passing through a gas </li> <li> Slide 29 </li> <li> The Transfer Equation We can then write the basic equation of transfer for radiation passing through gas, the change in intensity I is equal to: dI = intensity emitted intensity absorbed dI = j dx I dx dI /d = -I + j / = -I + S This is the basic equation which must be solved to compute the spectrum emerging from or passing through a gas. </li> <li> Slide 30 </li> <li> Special Cases If the intensity of light DOES NOT VARY, then I =S (the intensity is equal to the source function) When we assume LTE, we are assuming that S =B </li> <li> Slide 31 </li> <li> Thermodynamic Equilibrium Every process of absorption is balanced by a process of emission; no energy is added or subtracted from the radiation Then the total flux is constant with depth If the total flux is constant, then the mean intensity must be equal to the source function: =S </li> <li> Slide 32 </li> <li> Simplifying Assumptions Plane parallel atmospheres (the depth of a stars atmosphere is thin compared to its radius, and the MFP of a photon is short compared to the depth of the atmosphere Opacity is independent of wavelength (a gray atmosphere) </li> <li> Slide 33 </li> <li> Eddington Approximation Assume that the intensity of the radiation (I ) has one value in all directions toward the outward facing hemisphere and another value in all directions toward the inward facing hemisphere. These assumptions combined lead to a simple physical description of a gray atmosphere </li> </ul>

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