Astronomy 111, Fall October 2011

Transcription

1 Astonomy 111, Fall Octobe 011 Today in Astonomy 111: moe details on enegy tanspot and the tempeatues of the planets Moe about albedo and emissivity Moe about the tempeatue of sunlit, adiation-cooled sufaces Heat conduction and intenal heat geneation Intenal tempeatues of ocky planets How small could diffeentiated bodies be? Tempeatue image of Mas, made by Mas Odyssey s Themal Emission Imaging System (ASU/JPL/NASA) 4 Octobe 011 Astonomy 111, Fall Geometic and Bond albedo The geometic albedo is the atio of the flux eflected head-on (towad the Sun) to that incident The Bond albedo is the atio of the total flux eflected and scatteed in all diections, to that incident Bumpy sufaces tend to eflect light back the way it came The Moon and Mecuy, fo example, ae moe than 10 times bighte full than half So thei geometic and Bond albedoes ae simila Obseve Bightness of Mecuy as a function of phase angle, fom SOHO (Mallama et al 00) 4 Octobe 011 Astonomy 111, Fall 011 Geometic and Bond albedo Albedo geneally vaies with wavelength At a paticula wavelength the atio of eflected and scatteed light to that incident is A, the monochomatic albedo The Bond albedo, which h we ll call A b, is what one usually wants to use in solaheating calculations Natually, it s the one that s hadest to measue Planet o moon Geometic albedo Bond albedo Mecuy Venus Eath Moon Mas Jupite Io 06 Euopa 068 Ganymede 044 Callisto 019 Satun Titan 0 Enceladus 10 Uanus Neptune Pluto Octobe 011 Astonomy 111, Fall (c) Univesity of Rocheste 1

2 Astonomy 111, Fall Octobe 011 Sunlight-heated sufaces and thei tempeatues So fa, we have assumed sola-heated bodies to have unifom suface tempeatue, but this of couse isn t quite ight Because of the high incidence angle of sunlight in the pola egions of most planets, it s colde thee Shadow of plane with aea S has aea Scos with sunlight incident at Scos S angle But it still emits blackbody adiation fom its full aea Thus vaiation of tempeatue with latitude And: most slowly-otating bodies get much wame on the sunlit side than the dak side Thus vaiation of tempeatue with angle between vetical and Sun 4 Octobe 011 Astonomy 111, Fall Sunlight-heated sufaces and thei tempeatues Example: a slow otato Suppose a planet with no atmosphee, in cicula obit 1 AU fom a sta just like the Sun, and with unifom albedo, otates with peiod equal to its obital peiod, so that it always shows the same face to the Sun Neglecting the conduction of heat, what is the distibution of tempeatue on its sunlit side? Conside a ibbon of the suface at an angle fom the sub-sola point, as seen fom the cente of the planet, and with infinitesimal angula width d R sin Sunlight R 4 Octobe 011 Astonomy 111, Fall Sunlight-heated sufaces and thei tempeatues Evey point on the ibbon, then, eceives the same flux of sunlight The aea of the ibbon, on the suface, is ds R sinrd R sin d, and the pojected aea, pependicula to the diection of sunlight that is, the aea of the ibbon s shadow is ds ds cos R cos sin d Thus the powe absobed by the ibbon is L dpin 1Ab fds 1Ab R cos sin d 4 4 Octobe 011 Astonomy 111, Fall (c) Univesity of Rocheste

3 Astonomy 111, Fall Octobe 011 Sunlight-heated sufaces and thei tempeatues Meanwhile, the ibbon is emitting blackbody adiation, in the amount 4 dpout fibbonds Ts R sin d So if the ibbon is in equilibium, its tempeatue is given by dp in dpout L 4 1Ab R cos sind T sin s R d Ab L 14 Ts cos T0 cos 4 4 Octobe 011 Astonomy 111, Fall Sunlight-heated sufaces and thei tempeatues Using paametes of the Moon (Bond albedo 011, emissivity 1), 14 1 A b L T0 38 K 4 Not fa fom the tempeatue at the Moon s subsola point! ( ) Data fom Jessica Sunshine and the EPOXI team T (K) 4 Octobe 011 Astonomy 111, Fall Enegy tanspot in planets Enegy is tanspoted pimaily by conduction, adiation, o convection Usually one mechanism dominates Tanspot in solids is usually dominated by conduction Radiation usually dominates in space and tenuous gases Convection and adiation ae usually most impotant in the inteios of stas, and in planetay atmosphees and coes, but conduction is often significant Conduction is most often applicable in teestialplanetay inteios, which we will now discuss 4 Octobe 011 Astonomy 111, Fall (c) Univesity of Rocheste 3

4 Astonomy 111, Fall Octobe 011 Heat conduction Heat conduction is the tanspot of enegy by collisions between paticles (in a gas o an electical conducto), o by exchange of lattice vibations (in an insulating solid) The ate at which heat flows is called the heat flux, f T, -1 - which like adiation flux has the units eg sec cm Definition: iti if two plana, unifom-tempeatue t sufaces ae sepaated by infinitesimal distance dz and ae infinitesimally diffeent in tempeatue by dt, and the medium sepaating the planes is unifom, then the heat flux though the sufaces is dt ft zt z dz whee T is called the themal conductivity 4 Octobe 011 Astonomy 111, Fall Heat conduction In thee dimensions we would need to make the heat flux a vecto, and speak of the gadient instead of a deivative: ft T T But the adial component of the gadient in spheical coodinates is just d/, so fo spheical symmety the flux is adial and has magnitude dt ft T Themal conductivity of common mateials at T = 300 K: Mateial T (10 5 eg sec -1 cm -1 K -1 ) Quatz 3 Basaltic ock 45 Stainless steel 16 4 Octobe 011 Astonomy 111, Fall Intenal heat geneation We have aleady discussed two souces of intenal heat in planetay bodies, accetion heating and adioactive heating, in lectue on 0 and Septembe Radioactive heating is elevant fo planetay inteios today Fom table shown on Septembe: the adioactive heating ate pe unit mass, = ad,of cabonaceous chonite meteoitic mateial is eg sec gm today; eg sec gm 46 Gy ago 4 Octobe 011 Astonomy 111, Fall (c) Univesity of Rocheste 4

5 Astonomy 111, Fall Octobe 011 Intenal heat geneation Conside the powe dq geneated inside a spheically symmetic object; specifically, the heat geneated within a spheical shell with adius and thickness : if the mass density is, dq dm4 o dq 4 But if the tempeatue is constant, as much heat must flow out of this shell as is geneated thee: dq d d dt 4 ft 4 T 4 1 d dt Poisson s equation (fo spheical symmety) T 4 Octobe 011 Astonomy 111, Fall The AST 111 calculus palette, page To solve diffeential equations, need as many bounday conditions values of the solution as the ode of diffeential equation, to evaluate integation constants d Fist ode: f x g x dx f x g x dx G x C C f x0 G x0 d Second ode: fo example, f x gx dx f x g x dx dx G x C dx x CxD f x0 x0 Cx0 D two equations in the f x1 x1 Cx1 D unknowns C and D 4 Octobe 011 Astonomy 111, Fall Tempeatue of the inteio of a ocky planet Poisson s equation is a diffeential equation we can solve fo T, given a planet with mass M and a pesciption fo the density, heating ate, and themal conductivity T It is a second-ode diffeential equation, so we need two bounday conditions, conveniently povided by the suface tempeatue, t set by sola heating and the total tl adioactive heating powe Pad M (see lectue, Septembe), hee assumed unifom and modified fo non-blackbodies: 14 1 A L M Ts 16 4 R and the fact that the cental tempeatue must be finite 4 Octobe 011 Astonomy 111, Fall (c) Univesity of Rocheste 5

6 Astonomy 111, Fall Octobe 011 Tempeatue of the inteio of a ocky planet Let us assume fo simplicity a ocky sola-system body (mass M, adius R) with unifom density, and take the themal conductivity T to be independent of tempeatue Integate the Poisson equation twice, and apply the bounday conditions: d dt T d dt T 3 dt C 3 T 4 Octobe 011 Astonomy 111, Fall Tempeatue of the inteio of a ocky planet dt C 3 T C T D, 6T whee C and D ae integation constants At = 0: T appoaches infinity unless C = 0 At = R: T Ts R D DTs R 6T 6T Thus T T s 6 R T 4 Octobe 011 Astonomy 111, Fall Tempeatue of the inteio of a ocky planet 3 Taking density to be 3M 4 R ; the heating ate to be that in cabonaceous chonites today, eg sec gm ; and themal conductivity T eg sec cm K, as appopiate fo silicate ocks; we get: Body Obital adius, AU Mass M, gm Radius R, km Albedo A b T(R), K T(0), K Eath Moon Vesta T(0) too high fo Eath and Moon; about ight fo Vesta 4 Octobe 011 Astonomy 111, Fall (c) Univesity of Rocheste 6

7 Astonomy 111, Fall Octobe 011 (K) T() ( Tempeatue of the inteio of a ocky planet Vesta Moon Eath /R Remembe, each body is consideed unifom in density hee When tempeatues exceed 000 K, they ae oveestimates: T inceases linealy with inceasing T fo liquid metals and convection is impotant fo heat tanspot in liquids, too 4 Octobe 011 Astonomy 111, Fall (K) T() ( Tempeatue of the inteio of a ocky planet Vesta Moon Eath /R Nevetheless this demonstates a few impotant points: If Eath ween t aleady diffeentiated, it would become so vey quickly If the Moon has any liquid metal in its coe, it s just baely liquid Vesta is solid though and though, and pobably has been fo quite some time 4 Octobe 011 Astonomy 111, Fall The smallest diffeentiated bodies What about ealie times? At the time of CAI fomation 4568 Gy ago, the adioactive heating powe and poto-sola luminosity wee eg sec gm, 33 1 L 5 5 L eg sec Conside a small unifom sphee with non-poous, cabonaceous-chonite composition, in an obit like that 3 of 1 Cees: 7 gm cm, Ab 005, 77 AU and suppose it is just baely massive enough that mafic mineals melt in its cente: T K, so that on the aveage T eg sec cm K 4 Octobe 011 Astonomy 111, Fall (c) Univesity of Rocheste 7

8 Astonomy 111, Fall Octobe 011 (K) T() ( The smallest diffeentiated bodies /R Solving (iteatively) fo the R which gives T(0) = 100 K, we get 5 R cm 16 M 7010 gm Ts 16 K Thus it is possible that nonpoous bodies as small as a few km in size melted in thei centes and became diffeentiated, if they fomed ealy enough in the Sola system s histoy 4 Octobe 011 Astonomy 111, Fall 011 (c) Univesity of Rocheste 8