Stellar Astrophysics. dt dr. GM r. The current model for treating convection in stellar interiors is called mixing length theory:

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1 Stella Astophyscs Ovevew of last lectue: We connected the mean molecula weght to the mass factons X, Y and Z: = X + Y + μ 1 4 n 1 (1 + 1) = X μ 1 1 A n Z (1 + ) + Y z A Z We ntoduced the pessue scale heght and expessed t as: H = g Chs Engelbecht Boston College We deved the adabatc tempeatue gadent: ad 1 μm 1 = k H GM Spng 008 and asked: when wll the gas nsde a sta tansfe enegy though convecton athe than adaton? Stella Inteos The cuent model fo teatng convecton n stella nteos s called mxng length theoy: We fst deve the followng esult: Adabatc convecton wll occu when Stella Inteos The cuent model fo teatng convecton n stella nteos s called mxng length theoy: We fst deve the followng esult: Adabatc convecton wll occu when d ln d lnt < 1 d ln d lnt < 1 (whch takes the followng value n an deal monatomc gas): d ln 5 < d lnt Remembe ou bubble: we assume that t exsts almost n themal equlbum wth the suoundngs, that t expands adabatcally, and that bubble pessue emans equal to suoundng pessue:

2 Stella Inteos Ou devaton poceeds n 5 steps: Step 1: Consde changes on a small enough scale that we can descbe them wth a tuncated Taylo expanson: f f d ( + d ( + To sustan convecton: b) s) < Stella Inteos (Ths s often called the fundamental condton fo convecton. It also eques: d d < Step : We saw last week that adabatc expanson eques the elaton: d d = Step 3: Combnng these two equatons wth the deal gas law fo the suoundngs (and assumng the mean molecula weght emans constant), we fnd: d d < T Stella Inteos Step 4: The condton of pessue equalsaton allows smplfcaton to: d < T o, ewtng: ( ) s Stella Inteos We have now shown clealy why convecton wll occu when the adabatc tempeatue gadent s exceeded nsde a sta: bulk > ad d T 1 1 > When a small volume element (a bubble ) nsde the stella medum s petubed slghtly upwads n poston, t wll contnue sng a long as the tempeatue gadent meets ths condton. Ths easly educes to the adabatc 1 μmh GM ad 1 tempeatue gadent we deved eale: = k

3 Stella Inteos Stella Inteos When convecton occus nsde a sta, d T 1 1 > ad 1 μm 1 = k H GM Step 5: Smple algebac manpulaton of ths elatonshp leads to the desed condton: d ln < d lnt 1 whch becomes, fo = 5/3: d ln 5 < d lnt When convecton does not occu, = 3 4ac T L μm 1 = k H GM Stella Inteos Inspecton of these two gadents suggests when we should expect convecton to occu: ad Thee scenaos fo convecton: 3 = 4ac T The opacty of the stella medum becomes vey lage; 3 4 L A patal onsaton zone occus nsde the stella medum (lage C low (/) ad = -g/c ); the tempeatue dependence of the nuclea enegy geneaton ate ( ) becomes vey lage ( lage flux gadent). Mxng length theoy We defne the mxng length at any patcula pont nsde the stella medum, as the heght though whch a typcal bubble of mateal n the medum wll se befoe dsspatng and themalsng wth ts suoundngs. We defne a paamete as follows: CONCET 18 = H We wll now use the concept of mxng length to clafy the amount of convectve flux that wll ase n a convectve egon of the stella medum: (the pessue scale heght)

4 Mxng length theoy Mxng length theoy A quck emnde: the net foce (pe unt volume) on a body of densty 1 mmesed n a medum of densty s: Step 1: Detemne the excess tempeatue of a sng bubble, compaed wth the suoundng medum: Stat wth the fundamental condton fo convecton: < f net = g() Assumng pessue equalsaton, ths mples T > T Mxng length theoy T > T To mantan the convecton condton, we need > Let s call the excess tempeatue of the bubble Mxng length theoy Now watch caefully: we use the symbol hee to epesent the dffeence between some physcal popety, as assocated wth the bubble, compaed to ts value n the suoundng medum. (We wll encounte ths late n the couse as the Lagangan vaaton of the popety): T = Assumng adabatc expanson of the bubble: T = T = T = ad ( s)

5 Mxng length theoy Step : Detemne the heat flow deposted by a sng bubble, when t themalses wth the suoundng medum: The excess heat flow pe unt volume, fom one bubble s: q = ( C T ) Remembeng the defnton of the mxng length: q = C l Mxng length theoy Step 3: Detemne the convectve enegy flux contbuted by a sng bubble, when t themalses wth the suoundng medum: The added enegy cossng a unt aea, pe second, comng fom one bubble s: q = qv c t whee v c s the velocty of the convectve bubble, o, wtten as the convectve flux: F c = C lvc Mxng length theoy Sdestep: Detemne the convectve velocty of a bubble: Combnng the expesson fo buoyant foce and the vaaton of the deal gas law: kt f net = g g = μ f net = g = g T T m H the net foce on a convectve bubble can be wtten as Mxng length theoy Assumng a lnea dependence of ths foce wth T, and a neglgble ntal value of T, we wte the aveage foce as: f net 1 T = g T Calculate the wok done (pe unt volume) on the bubble as t moves along a mxng length, and assume that a facton of ths wok s conveted to knetc enegy of the bubble,.e. fnal fnet v = c 1 /

6 Mxng length theoy Usng the defnton of the mxng length n tems of the pessue scale heght, the expesson H =/(g) fo the pessue scale heght, and the deal gas law, a few steps of smple algeba (examnable) delve an expesson fo the convectve flux n the followng fom: F c = C lvc = C k μm H 3 / T g 1/ 3 / F c = C lvc Mxng length theoy Usng the defnton of the mxng length n tems of the pessue scale heght, the expesson H =/(g) fo the pessue scale heght, and the deal gas law, a few steps of smple algeba (examnable) delve an expesson fo the convectve flux n the followng fom: = C k μm 3 / H T g 1/ In the case whee all of the enegy tansfe occus though convecton, the convectve flux can be wtten n tems of the local lumnosty: L Fc = 4 3 / Mxng length theoy and the Lagangan vaaton n the tempeatue gadent can be wtten as : / 3 1/ 3 / L 1 μmh g = 4 C k T If we compae ths to the adabatc tempeatue gadent, usng known numbes fo the sun, we fnd that the ato s a mee (justfyng ou eale statement) Nuclea enegy geneaton Smple measuements of stella dstances and lumnous fluxes mply lumnostes of watt. The sun has a lumnosty of 4 x 10 6 W. Whch pocesses could possbly account fo these hghly poweful enegy outputs? gavtatonal contacton themal coolng chemcal eactons nuclea eactons?

7 Nuclea enegy geneaton Nuclea enegy geneaton Gavtatonal contacton: The val theoem tells us that V = q p = K Home execse (examnable): Stat wth the followng expesson fo the gavtatonal potental enegy of a volume element n the fom of a sphecal shell at adus, wth mass dm: GM dm du s = wth the coollay: U = Home execse: Stat wth the followng expesson fo the gavtatonal potental enegy of a volume element n the fom of a sphecal shell at adus, wth mass dm: GM dm du s = K Integate acoss the ente sta, assumng a constant densty equal to the mean densty, and deve the followng esult fo the total gavtatonal potental enegy: 3 GM U = 5 R Nuclea enegy geneaton Nuclea enegy geneaton Use ths esult to calculate fo how long the pesent sola lumnosty could have been sustaned by gavtatonal contacton. Themal coolng? U 3 GM = 5 R

8 Nuclea enegy geneaton Nuclea enegy geneaton Chemcal eactons: see homewok assgnment! Nuclea eactons? Stella specta mply H, He n abundance m He = u m H = u (4m H m He ) x (4m H ) Fo how long could hyogen fuson sustan the pesent sola lumnosty? Nuclea enegy geneaton Is hyogen fuson physcally feasble nsde the sun? Compae themal enegy wth the Coulomb bae: Ou cude estmates of the tempeatue at the cente of the sun wee odes of magntude below K. How do we esolve the stuaton? Consde the de Bogle elaton and deve the followng expesson fo the poton wavelength: h (40) = e μ m (we need to wok n the COM fame to analyse the two-body nteacton between the potons; hence we use the educed mass hee) Nuclea enegy geneaton Usng one de Bogle wavelength as the necessay dstance of appoach, the equed eacton tempeatue becomes: e μm T = 1 h k Nuclea eacton ates ae notoously complcated to calculate; the followng gaph llustates the competng nfluences of the patcle eneges on patcle populaton and tunnelng pobablty espectvely: 4 0

9 Nuclea enegy geneaton Nuclea eacton ates ae notoously complcated to calculate; the followng gaph llustates the competng nfluences of the patcle eneges on patcle populaton and tunnelng pobablty espectvely: Nuclea enegy geneaton Note the shap dependence of eacton ate on collson enegy: Nuclea enegy geneaton Note the shap dependence of eacton ate on collson enegy. The peak occus at the enegy / 3 0 E = bkt whee μ b 1/ m Z1Z 0 e ( ) h Nuclea enegy geneaton Thee ae many possble nuclea eactons avalable n the coes of stas. The geneal fom of the ate equatons s x 0 X X wth = x T fo -body eactons. The tempeatue dependence vaes wldly fo dffeent eactons.

10 Nuclea enegy geneaton The nuclea enegy geneaton ate pe unt mass, fo any specfc eacton, takes the fom 0 x = x x = 0 o X X T wth x = 1 Nuclea enegy geneaton Nuclea fuson eactons geneally occu as a chan of two-body eactons. Each two-body eacton must obey thee consevaton laws: electc chage bayon numbe lepton numbe The fst fuson eacton we look at s the poton-poton chan: CONCET 19 The I Chan Two moe poton-poton chans: slow step content.answes.com

11 Tple-alpha (essentally a thee-body eacton) outeach.atnf.cso.au content.answes.com A compason of eacton ate equatons: Homewok Assgnment 3 pp CNO 0 X fc) pp ( T , CNO XX CNO T6 CO oblems 10.1, 10.3, 10.4, 10.7, 10.8, 10.10, and ,3 Y f ) 3 ( T 41 8 Due date: Wednesday 7 Febuay, 1:00