CLASSICALLY -- ELECTROMAGNETIC RADIATION Maxwell (1865) Electrmagnetic Radiatin http://apd.nasa.gv/apd/astrpix.html Classically, an electrmagnetic wave can be viewed as a self-sustaining wave f electric and magnetic field. 1 E B = + 4π j c t i B= 0 1 B E = - c t i E = 4πρ These equatins imply the existence f a prpagating self - sustaning wave. A change in B creates a changing E, which creates a changing E which creates a changing B which creates a changing B etc. Crudely, ne can say that a changing B prduces a changing E, but that implies an ut f phase scillatin which is nt the case. Electrmagnetic radiatin is characterized by a frequency ν and a wavelength λ. The prduct f wavelength and frequency is the speed f light. The time fr ne wavelength t pass at speed c is 1/ν, s c/ν = λ. ν λ= c c = 2.998. x 10 10 cm s -1 Wavelength is measured in units f length that smetimes vary depending upn what srt f radiatin yu are talking abut. m, cm, and mm fr radi emissin Angstrms fr x-rays and near ptical light: A = 10 micrn = µ = 10-6 m = 10-4 cm = 10,000 A fr infrared and micrwave 8 cm Frequency is measured in Hertz = s -1 kilhertz (khz) MegaHertz, etc as n yur radi (MHz) "ptical" light is apprximately 4000-7000 A ν = c λ = 2.99 10 10 cm (5000)(10 8 cm) sec = 6 1014 Hz (B and E scillatins are actually in phase as shwn)
Classically, electrmagnetic radiatin is prduced whenever electric charge is accelerated. Examples: Electrns flwing in a current up and dwn in a radi antenna micrwaves 1A Electrns clliding with nuclei and each ther in a ht gas - emissin depends n temperature Electrns spiraling in a magnetic field 7000 A 6000 A 5000 A 4000 A The light we can see is a very small part f the whle electrmagnetic spectrum. Transparency f the Earth s Atmsphere Blackbdy Radiatin 1 µ = 10-6 m = 10-4 cm = 10,000 A Mst electrmagnetic radiatin, except fr ptical light and radi waves, des nt make it t the surface f the Earth. In physics, a black bdy is an idealized bject that absrbs all electrmagnetic radiatin that falls nt it. N radiatin passes thrugh it and nne is reflected. Similarly, a black bdy is ne that radiates energy at every pssible wavelength and that emissin is sensitive nly t the temperature, i.e., nt the cmpsitin.!
Blackbdy Radiatin Blackbdies belw arund 800 K (530 C) prduce very little radiatin at visible wavelengths and appear black (hence the name). Blackbdies abve this temperature, hwever, begin t prduce radiatin at visible wavelengths starting at red, ging thrugh range, yellw, and white befre ending up at blue as the temperature increases. The term "blackbdy" was intrduced by Gustav Kirchhff in 1860. Tday the term has a technical meaning, an emitter r absrber whse spectrum depends nly n its temperature and nt its cmpsitin. " Experimentally, the distributin f intensity with wavelength is well determined As the temperature Is raised yu get mre energy (flux) at every wavelength, but the peak als mves http://highered.mheducatin.cm/lcweb/cgi/pluginpp.cgi?it=swf::800::600::/sites/dl/free/0072482621/220727/blackbdy_nav.swf::blackbdy Radiatin Interactive The sun as seen frm the Earth 1 nm = 10 A = 1000 µ! 1 µ = 10,000 A ultravilet is blcked The sun s radiatin is t fair apprximatin a black bdy with a temperature arund 5800 K Why d these curves lk like they d? The classical slutin t blackbdy radiatin assumed that electrns vibrating at any frequency had ~kt f energy t put int radiatin at that frequency. It ignred the fact that the radiatin had energy that depended n its frequency. There was, in fact, mre rm (phase space) fr radiatin with shrt wavelengths, hence its emissin was preferred. The fact that the prbability fr emitting shrt wavelength radiatin increased withut bund did nt vilate the cnservatin f energy, because there was n relatin between energy and wavelength. But this was ttally at dds with what was seen
Prblem: Divergent fr large values f! Classically the intensity f radiatin having frequency ν was given by the Rayleigh-Jeans frmula (e.g., Feynman, Leightn and Sands, Vl 1 p 41.5) I ν = 2ν 2 kt c 2 where I ν dν is the flux f radiatin emitted by a blackbdy f temperature T (erg cm 2 s 1 ) with a frequency in the range ν t ν +dν. k is Btzmann's cnstant and c the speed f light. If yu pened an ven yu wuld be verwhelmed by x-rays and gamma-rays puring ut (at all temperatures). Optical light t wuld be emitted at all temperatures. Thery gt the behavir at lng wavelengths crrect but was wrng fr shrt wavelengths lg flux At a cnstant T lg frequency http://www.cv.nra.edu/curse/astr534/blackbdyrad.html PLANCK - 1900 The slutin t the dilemma psed by the classical slutin was t require that electrmagnetic radiatin be quantized, that is it cmes in individual particle-like packets f energy called phtns. Each phtn has an energy prprtinal t the frequency f the radiatin. High frequency (shrt wavelength) radiatin thus had greater energy and was increasingly hard t prduce at a given temperature. x-rays have mre energy than ptical light and are harder t prduce. E γ = hν This particle like prperty f light als meant that light carried mmentum and culd exert a pressure. Energy = mmentum speed (fr relativistic particles there E γ = pc = hν p = h ν c = hλ is n 1 2 ut frnt as in 1/2 mv 2 ) where h is Planck s cnstant h = 6.626 x 10-27 erg sec
WITHOUT PROOF Fr a blackbdy with temperature T the emitted flux as a functin f frequency was erg cm 2 s Hz e x 1+ x if x <<1 s exp( hν hν ) 1 kt kt I ν 2ν 2 kt if hν << kt c 2 but fr hν kt I ν 2hν 3 c 2 exp( hν kt ) 0 sin http://en.wikipedia.rg/wiki/planck%27s_law Blackbdy (Thermal) Radiatin Aside: e is the base f Napierian lgarithms. It is als knwn as the "expnential functin" As T rises: Intensity e = 2.71828... It can be taken t a pwer like any ther number e 0 = 1 e = 0 etc. The "natural lgarithm" f a number, ln, is the pwer t which e is raised t get that number. mre radiatin at all wavelengths shift f peak emissin t shrter wavelength greater ttal emissin (area under the curve) classic quantum cut-ff
Intensity I = Pwer (erg/sec) radiated fr a range f frequencies and +d thrugh unit surface area, da Flux(ν)= I ν dν da Rewriting in terms f the wavelenth λ = c/ν I λ = 2hc2 1 λ 5 hc λkt e 1 We are interested in the emissin summed ver all wavelengths F(T) = I λ dλ = 0 2π 5 k 4 15h 3 c 2 T 4 r F(T) = σt 4 erg cm 2 s 1 where σ is the Stephan-Bltzmann cnstant σ = 5.6704 x 10 5 erg/(cm 2 s K 4 ) i..e., when multiplied by T 4 the units are thse f flux. The maximum ccurs where di = 0, which is dλ λ max = 0.28978 cm T = 2.8978 107 A T maximum slpe = 0 Fr ur purpses, yu nly need t knw 1) Each square cm f a blackbdy radiatr with temperature T emits T 4 erg s -1 2) Mst f the emissin ccurs at a wavelength given by = area under curve λ max = 0.2899 cm T = 2.899 107 A T σ is the Stefan Bltzmann radiatin cnstant 5.6704 10 5 erg s cm 2 K 4
Frm Nick Strbel s Astrnmy Ntes http://en.wikipedia.rg/wiki/randm_walk 7 2.8987 10 A λ = T DIFFUSION TIME FOR THE SUN The sun - a typical star Hw lng des it take? τ Diff 0.1 cm R 2 number cllisins c = R2 c time between each Intensity (6.9 10 10 cm) 2 s (0.1 cm)(3 10 10 cm) = 1.6 1012 s 50,000 years
THE LUMINOSITY OF THE SUN L = 4π R 2 σt 4 T= 5800 K = 4(3.14)(6.96 1010 cm) 2 (5.67 10 5 erg)(5800 K) 4 cm 2 s K 4 = 3.90 10 33 erg/s (Culd have gtten 5800 K frm Wien's Law) L= Area σt 4 L = 4π R 2 σt 4 The actual value is 3.83 x 10 33 erg/s Frm Nick Strbel s Astrnmy Ntes If radius is held cnstant,
red giants On the main sequence, apprximately R M 0.65 white dwarfs S M R = R M 0.65 This implies mre massive main sequence stars are less dense Anther Example f a Blackbdy The Universe Z = 1100 30 10 2 0
2.73 K 3000 K 1100 i.e., the temperature at recminatin divided by 1+z at recmbinatin * T = 2.7249 2.7251 K And anther example f blackbdy radiatin Planetary Temperatures A picture f the universe when it was nly 379,000 years ld (WMAP 2003) http://lasp.clrad.edu/~bagenal/1010/sessions/13.light.html
Sunlight L Received frm sun: 4πd Absrbed: f times this Reflected: (1 - f) times this Reradiated: 4π R σ T 2 4 P P f L In steady state: 4πd π R 2 π R 2 P = 4π R σ T 2 2 4 2 P P P R P T Earthshine f L = 16π d σ P 2 Assume planet is rapidly rtating 1/ 4 1/ 4 L n. b., Tp d and independent f R p Fr Earth: 1/4 ( 3.83 10 ) 33 f ( ) T P = 16π ( 1.49 10 ) 13 2 ( 5.67 10 ) 5 = 281 K f = 1 (8 C, 46 F) = 249 K f = 0.633 (-24 C, -12 F) But actually the Earth s average temperature is abut 288 K (15 C)
Fr ther planets that rbit the sun ne can take L t be cnstant and the calculatin is the same except that the temperature varies as 1/ d. 1 AU T P = 281 f 1/4 d 1/ 2 http://data.giss.nasa.gv/gistemp/ In last 100 years temperature has increased abut 0.9 K (r 0.9 C r 1.6 F). In the next century it is expected t increase several mre degrees K (http://en.wikipedia.rg/wiki/glbal_warming) if absrbed like the Earth Fr example, fr Mars at 1.52 AU T = 281 f P = 1/ 2 1/4 1 1/4 228 K f = 1 (-45 C -49 F) = 200 K f = 0.6 (-73 C -99 F) = 217 K 1.52 actually measured 218 = 228 K f f = 0.84 (-56 C -69 F) crrect f fr Mars
f T p = p L 16π d 2 σ f = T p Earth f Earth 1/4 1/4 1/4 AU = 281 f p d 1/2 ( ) 1/4 1 AU d 1/2 1/2 = 281 0.28 0.7233 fr Venus; nb nly 28% f the light it receives reaches the surface = 240 K (fr Earth we gt 247 K withut greenhuse; 288 K with it) S Venus, with its measured albed, shuld in fact be cler than the Earth, even thugh 2 1 φ Venus = 0.7233 φ Earth = 1.91φ Earth This is because nly 28% f the light gets thrugh s the flux at the base f Venus' atmsphere is VENUS 0.28 0.63 1.91 = 87% that f Earth fr any planet arund the sun But the bserved temperature n Venus is 730 K. The atmspheric pressure is abut 90 Earth atmspheres, mstly made f CO 2 This is htter than the planet Mercury and htter than the melting pint f lead. (850 F) The mist greenhuse effect ccurs when sunlight causes increased evapratin frm the ceans t the pint that the gradient f water vapr in the earth s atmsphere des nt decrease rapidly with altitude (it currently des). As a result water is present at high altitude where it can be brken brken dwn int hydrgen and xygen by ultravilet radiatin. The hydrgen escapes and the water is permanently lst frm the earth. Kasting (1988) shwed that this happens when the luminsity frm the sun exceeds a minimum f 1.1 times its present value. Cluds may increase this threshld value. A true runaway greenhuse effect happens when the luminsity f the sun is 1.4 times greater than nw. The ceans cmpletely evaprate. The extra water vapr in the atmsphere increases the greenhuse effect which raises the temperature still mre leading t faster evapratin... On the ther hand, belw a certain temperature the carbn dixide cndenses ut f the atmsphere and there is n greenhuse effect. This happens fr fluxes abut 55% that f the present sun at the Earth s rbit. This may be why Mars is s cld. Tgether these cnditins restrict the Habitable Zne f ur present sun t 0.95 t 1.37 AU. Mars is at 1.52 AU. Kasting et al. February, 1988 Scientific American Hw Climate Evlved n the Terrestrial Planets
Frm Nick Strbel s Astrnmy Ntes Stars that are t big dn t live lng enugh fr life t develp (3 by?). Stars that are t small have life znes that are t clse t the star and the planets becme tidally lcked (0.5 0.7 slar masses??). Fr main sequence stars nly (red giants and white dwarfs wuld have different tables) BACK TO THE STARS The fact that the stars are all blackbdy radiatrs allws astrnmers t prepare very useful tables that fr example give the blmetric crrectin and interesting physical quantities such as the radius and temperature
If knw L and T then als knw R. M cmes frm ther measurements - TBD