Energy Transport. Chapter 3 Energy Balance and Temperature. Temperature. Topics to be covered. Blackbody - Introduction

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1 Energy Transpor Chaper Energy Balance and Temperaure Energy can be ransmied by:. Conducion. Radiaion. Convecion Asro 960 One mechanism usually dominaes In solids, conducion dominaes In space and enuous gases, radiaion dominaes Convecion is imporan in amospheres (and liquid ineriors) 4 Topics o be covered Temperaure Energy Balance and Temperaure (.) - All Conducion (..), Radiaion (.. and...) Convecion (..), Hydrosaic Equilibrium (...), Firs Law of Thermodynamics (...) and dadiabaic Lapse rae () (...) All o be discussed in lecure noes wih Ch. 4 (where i makes sense!) The emperaure of an objec is proporional o he average ranslaional kineic energy of is molecules. Noe ha one objec can have many emperaures 5 Radiaion and Planeary Science All solar sysem bodies are illuminaed by he sun Balance beween solar radiaion received (plus any inernal energy) and ha emied defines emperaure ulimaely equilibrium is reached which defines T Temperaure of bodies criical o behaviour of amospheres, surfaces and ineriors Blackbody - Inroducion Blackbody a hypoheical (idealized) body ha Absorbs all inciden radiaion (hence he erm black ) Emis he maximum possible radian energy in all wavelengh bands in all direcions No radiaion is refleced All bodies wih emperaures above absolue zero emi radiaion Max Planck hp://home.wanadoo.nl/paulschils/07.0.hml 6

2 The amoun of radiaion emied by a blackbody is uniquely deermined by is emperaure (Planck s law): The black body specific inensiy or brighness is defined (following discovery by Max Planck in 900) as eiher ( ) = hc Bλ T ( ) 5 hc / λ λ e kt or = hν Bν T hν / c e kt where c=.99x0 0 cm/s, h=6.57x0-7 erg s, k=.8x0-6 erg/s. Using cgs unis (λ in Angsroms) we have.9x0 Bλ ( T ) = 8.44 x0 / e 7 λt 5 λ Max Planck hp://home.wanadoo.nl/paulschils/07.0.hml 7 In he limi of small f: Classical Limi (small f, large λ) 4 λ ν kbt Bv ( T ) c Rayleigh-Jeans This equaion doesn involve Planck s consan was originally derived from purely classical consideraions. Classical physics predics he so-called ulraviole caasrophe an infinie amoun of energy being radiaed a high frequencies or shor wavelenghs (derived from he equipariion heorem). 0 Blackbody radiaion is isoropic; he radiance is independen of direcion Unis are J m - Hz - s - ser - (erg cm - Hz - s - ser - ) Recall 0 7 ergs = J hνν Bν ( T ) = hν / kt c e A he oher exreme for high f (or for shor wavelenghs), Planck s law simplifies o Wiens Law: hν k Bv ( T ) e c hc B λ 5 hc λ exp λk T hν BT Max Planck hp:// 8 hp://home.wanadoo.nl/paulschils/07.0.hml Characerisic shape for blackbody radiaion ploed using Planck s law The Wien displacemen law Using Planck s law and differeniaing o find he peak (ie. solve B/ λ=0), one can find he wavelengh of peak emission for a blackbody a emperaure T: λ m 897 ( μm K) = T known as he Wien displacemen law. This law makes possible he esimae of he emperaure of a radiaion source from knowledge of is emission specrum. Sharp shor wavelengh cuoff, seep rise o he maximum, genle dropoff oward longer wavelenghs ofen can use limiing expressions a high f (Wien Law) or low f (Rayleigh-Jeans Law) 9

3 The Wien displacemen law Consequence: solar radiaion (due o he emperaure of he sun) is concenraed in he visible and near-ir pars of he specrum planeary pa eayradiaion ada and ha of heir amospheres eesis largely confined o he IR Albedos When he sun illuminaes an objec, some of he radiaion is absorbed, and some scaered. The albedo (raio of refleced and scaered inensiy o inciden inensiy) varies wih wavelengh. A ν is he monochromaic albedo. The luminosiy observed depends on he geomery, specifically he phase angle. Earh (normalized) Objec Sun 6 The Wien displacemen law Albedos Noe he lack of overlap ha allows separaion of he radiaive ransfer problems of he earh and of he sun 4 The geomeric albedo is he raio of he flux refleced head-on (back o he sun) o he inciden flux The bond albedo is he raio of he oal flux refleced o he inciden. I incorporaes an inegral over phase angle F( ϕ = 0) A = 0 F inciden Ab = A0 q ph 7 The Sefan-Bolzmann law If we inegrae Planck s law jus above he surface of an objec and over all frequencies, we find: Marley e al. (999) F = 0 ν dν π 0 F = σ T F ( T ) B ( T ) dν where F is he flux (power/uni area) which is known as he Sefan-Bolzmann law 4 ν F = Flux, (power/uni area), T = Temp. in Kelvin, σ = 5.67 x 0-8 W/m K 4 (conduciviy) For non-ideal black body, F = σt 4 ε where ε = emissiviy <. Josef Sefan hp://home.wanadoo.nl/paulschils/07.0.hml 5 8

4 Phase Funcion: I( ϕ) φ = I(0) Sudarsky e al. (005) 9 Equilibrium emperaure Eros from NEAR Lsun 4 Fin = ( Ab ) πr F 4 R T 4πr ou = π εσ We can calculae he equilibrium emperaure by seing he wo equal o each oher. / 4 Fsun ( Ab ) Teq = r 4εσ The emperaure depends on he disance o he sun, bu no on he size of he objec. Muinonen e al. (00) 0 Equilibrium emperaure The sunli hemisphere of a plane absorbs radiaion: Lsun Fin = ( Ab ) πr 4πr Cross-secional area of plane Area over which solar radiaion is spread a disance r from sun If he plane roaes rapidly, is emperaure is uniform. In ha case, i emis radiaion: F ou = 4 4πR εσt We can calculae he equilibrium emperaure by seing he wo equal o each oher. Planeary Temperaures Teq Teff Tsurf Mercury 446 K 446 K K Venus Earh Moon Mars 5 Jupier 4 Saurn 8 95 Uranus Nepune

5 Albedos in he solar sysem Rocky surfaces: Icy bodies Gaseous planes: ~0. The Moon: 0.07 Venus: 0.75 Thermal radiaion is refleced in all direcions (slow roaor) so as seen a he Earh he hermal radiaion received is: Thus he raio of visible o hermal radiaion is: We can measure he visual albedo by comparing he refleced and emied radiaion. 5 Therefore if we can simulaneously measure he hermal and visible flux we can direcly measure he visible (and hence hermal) albedos. 8 Refleced visible ligh A v =0.0 A v =0.05 Hea Conducion Conducion is he ranspor of energy by collisions beween paricles. Conducion is imporan in he upper amosphere, where he mean free pah is long and collisions are imporan. Sunligh heas many surfaces during he day. The energy is ranspored downwards from he surface. The rae of flow of hea is known a he hea flux, Q. Q depends on he emperaure gradien, or and he hermal conduciviy K T. K T is a measure of he maerial s abiliy o conduc hea. IR emission 6 Unis of K T : erg s - cm - K - or J s - m - K - 9 Solar radiaion flux falling on an aseroid surface per square meer: Toal refleced visible luminosiy of he aseroid is given by: Energy no refleced is absorbed and hen re-emied a IR wavelenghs: Conducion as diffusion The energy ha goes ino a volume elemen per uni ime is: How much does his hea up he maerial? Combining his wih Assume aseroid is a opposiion wih he Earh and reflecs visible radiaion uniformly over is sunli hemisphere (π seradians). Visible radiaion deeced a he Earh is hen: d 7 We ge: or where This is known as he diffusion equaion Compare o he wave equaion: which has oscillaing soluions. The diffusion equaion has exponenially spreading soluions. 0 5

6 Thermal diffusion coefficiens C P (J/kgK) ρ (kg/m ) K T (W/mK) K d (m /s) Waer x 0-7 Iron x 0-5 Sone x 0-5 Typical Near-Earh Aseroid roaion period ~ 0 4 sec Longes known aseroid roaion period ~ 0 7 sec For Mars/Moon Z ~ 5 cm Z ~ 0 cm Z ~ 0 m 6