2. Radiation Field Basics I. Specific Intensity

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1 . Raiation Fiel Basics Rutten:. Basic efinitions of intensity, flux Enegy ensity, aiation pessue E Specific ntensity t Pencil beam of aiation at position, iection n, caying enegy E, pasg though aea, between the times t an t + t, in the fequency ban between an +. s, n s is nomal to Units of : /m /s/h/s egs/cm /s/h/s

2 s s E p p No souces o ks of aiation. Pencil beam of aiation pasg though at p an at p. Raiant enegy pasg though both aeas is the same: ' ' t E ''t ' Soli angle subtene by at p, subtene by at p : ' ' / ; Ω ' / Theefoe: ' Specific intensity inepenent of istance when no souces o ks. Sola Limb Dakening Assume plane paallel atmosphee Measue at iffeent positions on sola isk > get

3 Mean ntensity is a vey impotant quantity. t etemines level populations an ioniation state thoughout atmosphee. Like, it is a function of position. Fo a plane paallel atmosphee no epenence with :,, We ll late use moments of the aiation fiel. is the th moment Moment opeato M opeating on f : M n [ f ] f n R What is at istance fom a sta with unifom specific intensity as its suface? fo < < < < fo > < w is the ilution facto R / w At lage, w R /

4 Flux Ω Monochomatic Flux, : The net flow of aiant enegy pe secon though an aea in time t in fequency ange. The flux enables us to calculate the total enegy, E, pasg though a suface in a given time, i.e., integate ove all iections. The enegy tanspot can be positive o negative. Ω Ω ˆ,, A n t n t E E This is use fo specifying the enegetics of aiation though stella inteios, atmosphees, SM, etc. n pinciple, flux is a vecto. n stella atmosphees, the outwa aial iection is always implie positive, so that / / / / + + With both the outwa flux, +, an the inwa flux,, positive. sotopic aiation has + an. Axisymmety: +

5 The flux emitte by a sta pe unit aea of its suface is + whee is the intensity, aveage ove the appaent stella isk, eceive by an obseve. This equality is why that flux is often witten as F, so that F, with F calle the Astophysical Flux. This explains the often confug factos of that ae floating about in efinitions of flux: Monochomatic Flux o just the Flux; F Astophysical Flux They ae elate by F. n tems of moments of the aiation fiel, the fist moment is efine as the Eington Flux, H. Fo plane paallel geomety: H F Flux enegy/secon pe aea Luminosity enegy/secon Stella Luminosity L A R Assume B an integate: L L R B R σ T

6 Unesolve Souces Relate enegy obseve to at stella suface: Enegy eceive pe etecto aea, fom anulus: f ω ω soli angle of anulus Anulus aea R : S R S f ω S /D ntegate ove ω: R / D R / D R, α angula iamete α R, Unesolve > measue flux nvese squae law. Know α, get absolute flux at sta R R,, Enegy Density The enegy flow in a beam of aiation is V s c t n E t The flow has velocity c photons an tavels a istance s in time t s/c though volume V s. Thus, each beam caies E /c V. f multiple beams pass though a small volume V, integation ove V an ove all beam iections gives the aiant enegy E containe in V as banwith as: E V c V Ω

7 Fo sufficiently small V, the intensity is homogeneous, so the two integations V, Ω ae inepenent. The enegy ensity is u c Raiation Pessue Each photon has momentum p h/c. Component of momentum nomal to a soli wall pe time pe aea is p E c t Re-wite in tems of an integating ove soli angle gives: n p h/c h/c s c t p c sotopic aiation has p u /3. Raiation pessue is analogous to gas pessue, being the pessue of the photon gas.

Physics 122, Fall December 2012

Physics 122, Fall December 2012 Physics 1, Fall 01 6 Decembe 01 Toay in Physics 1: Examples in eview By class vote: Poblem -40: offcente chage cylines Poblem 8-39: B along axis of spinning, chage isk Poblem 30-74: selfinuctance of a