2015恒星結(jié)構(gòu)與演化20209s

1、恒星結(jié)構(gòu)與演化(2020.2-2020.6)第2部分恒星物理羅新煉仙林校區(qū)天文樓521房間： xlluo 89685982狀態(tài)方程輻射傳能第2部分恒星物理對流傳能核過程請就你掌握的輻射相關(guān)知識做思維導(dǎo)圖。+3ò1dBn dv¥1 = o kva 1- exp (- hn kT )+ kvs dTkò¥ dBn dvo dTdP = - Gm(r) rdrr2 恒星大氣的簡單模型 恒星質(zhì)光關(guān)系 Eddington極限光度 更加廣闊的應(yīng)用恒星內(nèi)部能量轉(zhuǎn)移過程：輻射與熱傳導(dǎo)輔助方程 k4熱傳導(dǎo)不輻射轉(zhuǎn)移1k能量平衡 需要具體光子與物質(zhì)的相互作用。 計算過程需要

2、利用量子力學(xué)和量子場論方面的知識，相當(dāng)復(fù)雜。 參考文獻：Carson., 1984, ApJ, 283, 466.Iglesias & Rogers, 1992, ApJ, 397, 711.Rogers & Iglesias, 1992, ApJS, 79, 507.Alexander & Ferguson, 1994, ApJ, 437, 879.Seaton., 1994, MNRAS, 266, 805.Iglesias & Rogers, 1996, ApJ, 464, 943.需補充更新文獻!恒星內(nèi)部物質(zhì)的不The luminosity of a s

3、tar of a given mass is essentially determined by the opacity of the matter and not by its nuclear reactions.Opacity is thus a key factor in stellar properties.6簡略m 434L = 96 k M b 院館可借，物理寫得比較好 有關(guān)不數(shù)據(jù)的網(wǎng)頁?8請查更新的恒星內(nèi)部物質(zhì)的不9The Opacity ProjectThe name Opacity Project (OP) 1 refers to an international colla

4、boration that was formed in 1984 to calculate the extensive atomic data required to estimate stellar envelope opacities and to compute Rosseland mean opacities and other related quantities. It involved research groups from France, Germany, the United Kingdom, the United States and Venezuela (委內(nèi)瑞拉).

5、The approach adopted by the OP to calculate opacities is based on a new formalism of the equation of state 2 and on the computation by ab initio methods of accurate atomic properties such as energy levels, f-values and photoionization crosssections 3. The OP final results are discussed by Seaton. 4.

6、 OP opacitieshave been recently revised to include inner-shell contributions 5. The new data and a suite of easy-to-use codes to compute Rosseland means and radiative accelerations 6 can be downloaded as a tar file below.10Reference1. The Opacity Project Team, 1995, The Opacity Project Vol. 1, Insti

7、tute of Physics Publications, Bristol, UK2. Hummer D.G., Mihalas D., 1988, ApJ 331, 794. Abstract3. Seaton M.J., 1987, J. Phys. B 20, 6363. Abstract4. Seaton M.J., Yu Yan, Mihalas D., Pradhan A.K., 1994, MNRAS 266, 805. Abstract5. Badnell N.R., Bautista M.A., Butler K., Delahaye F., Mendoza C., Palm

8、eri P., Zeippen C.J., Seaton M.J., 2004, MNRAS, in press (astro-ph/0410744).6. Seaton M.J., 2004, MNRAS, in press (astro-ph/0411010).the OPAL group at the Lawrence Livermore National Laboratory11?作業(yè)：請找到類似這個，計算恒星(太陽)內(nèi)部物質(zhì)不. 恒星內(nèi)部物質(zhì)的不13Log R osseland O pacity (cm 2/gm )Ross-Aller (standard mix)Nov 6, 20

9、07X= 0.60000 Y= 0.37800 Z= 0.02200LANL T-4隨意取值做的.隨意取值做的.1415high energyphoton16自由電子散射自由-自由吸收-自由吸收-吸收恒星內(nèi)部不主要的來源Skip 18The calculations of opacities is one ofthe most difcult problems in stellarastrophysics.Extensive opacities can befound in tabular forms.19注意: Line widths, Oscillator Strengths 等重要的概念

10、在此略過了. 請在課外補充閱讀這方面知識.不與質(zhì)量密度，溫度和化學(xué)組成有關(guān)，通常用冪率形式(Krammer formulae n =1, s = 7/2)來表示k = k0r Tn-s量綱為m2kg-1，或cm2g-1?？偟牟?1(4) 自由自由吸收(3)自由吸收(2)電子的散射(吸收)(1) 自由電子的散射羅列了一些有關(guān)不的計算結(jié)果(1) 自由電子的散射如果電子是非簡并且為非相對論性的，則不來自于Thomson 散射主要可用經(jīng)典電動力學(xué)方法導(dǎo)出.22Net effect: scattering of the incident EM radiation by charged particleR

11、adiate electromagnetic waves in all directionsoscillateOscillating charged particle will be moving in an accelerationAn electromagnetic wave incident upon a charged particle23Consider an EM wave E = E0 sin(w t ) acted on an electron. Assume the resultant motion of the electron is nonrelativistic (ne

12、glect the B- field). The Force on charge (equation of motion)F = eE + e v ´ B » eE = eE sin(wt) = m r0ed = e × rcUsing the dipole approximation,Oscillating electron will radiate energy at the same frequency as thatof the incident waveve4e4dP1&&E0 sin qdW = 4pc3(d ´ n)=(E

13、´ n)=2222pp2 3234m c4m ceeq is the direction between E andn28略 Thomson scattering- the scattering of a low energy photon (h n << mec2) by an electron.2s= 8p æe2ö= 8p2 »´-252T3 ç m c2 ÷3 rec6.710cmèeøc4pIncident radiation flux on electron isThe diffe

14、rential cross section isS =E 20æ ds öe4º 1 dPsin2 q =2 q=2ç dW ÷recsinS dWm2c4èø poleThis result is valid for radiation polarized along a specific direction e = E / E(polarized radiation)The differential cross section for unpolarized radiation is (skip here思考)æ

15、; dsö3sr 2= ec (1+ cos2q) =T (1+ cos2q)ç÷è dW øunpol16p2q is the scattering angle between the incident k and exit direction nThe total scattering cross section obtained by integration over all directions.peanut29略非偏振方向的微分散射截面為æ ds ö3s2r2 qq )=+ cos) =T (1+ cos2

16、1;÷ ec (1216pè dW øunpolq 為散射角表示入射方向與出射方向的夾角.總散射截面即對全方位ö28p æe2sº= 0.6652´10-24cm2 1 barnç÷Tm c23èøe30略Thomson 散射不為k = sT= 1+ X sT= 0.2 × (1 + X ) cm2 g-1em m2mueuclassical electron radius Thomson 散射 Thomson scattering does not change the f

17、requency of the radiation. Thomson cross-section is wavelength-independent (與入射光子的頻率無關(guān)) Thomoson 散射僅在高溫時比較重要，因為物質(zhì)幾乎完全電離。 但如果電子是相對論性的，則要考慮Compton散射。SR31 While for scattering, photon direction changed =>momentum changed => energy changedThomson 散射的特點局部穩(wěn)恒磁場Synchrotron （同步） Radiationthe energy den

18、sity in themagnetic field輻射功率輻射能譜Kj is the modified Bessel function of the second kindBremsstrahlung （軔致輻射） Radiation略？譜特征輻射機制內(nèi)容33以前課程中夾帶了部分狹義相對論和廣義相對論知識。pfypip2xConsider: a photon with frequency n collides with an restielectron and scatters with a new frequency nf in q direction. Find the emergent

19、frequency as a function of the photons scattering angleThe four- momentum can be written asp4p2 = me 1,0,0,0p4 = meg 41, v4 cosf,-v4 sinf,0p = hn 1,1,0,0iip= hn 1, cosq ,sinq ,0ffpi + p2 = pf + p4The conservation of four-momentumThe scalar invariantWhere= p hp= m2p jjkpjjkjh = DiagonalMatrix1,-1,-1,

20、-1(p + p- p )×h × (p + p - p )= m2So, we havei2fi2fWe can get the relationThink about: if electron is not rest The kinetic energy of the recoiled electron The relation between q, f and ni .n=n ifhn1+i (1- cosq )m c2eThe quantum electro dynamical (QED) correction for the unpolarized radiati

21、on isö2 æ nænnödsæö 1ç i + f -sin2 q ÷ç÷ f =r2ç dW ÷ç÷çn÷ecnnèøunpol2èi øèøfiwhere ni and nare the frequencies of the incident and thefscattered photons.The total cross section is called the

22、 Klein-Nishina (克萊因-仁科) cross section.2(e +1) ù14e íê1-ln(2e +1) +-1 ìéü= 3s1sýúûKNTe 22ee +1)28îë2(2þwhere e = ( wi / me c2 = h ni / me c2為入射光子 的能量)ì26e2ïsT (1- 2e+e<< 1)= ï5síïïîKN3 s T (ln 2e+ 0.5

23、)e>> 18 e35n=n ifhn1+i (1- cosq )m c2eæ ds ö= r 2+2 q3s2ç÷ec (1cos) =T (1+ cos q )è dW øunpol216phni2m c2e510-21510-10.52200200-2-1012The differential cross section1 barn = 10-24 cm2 at low energies, Thomson holds at higher energies, Thomson formula breaks down36T

24、he total cross section If electron is rest, the energy change for photon isEi(1- cosq)Ef - EiDEm c2E= -e» -iEim c2EE(1- cosq)1+iiem c2e If electron is relativistic (ge) , the energy change for photon is(1- b)E(1- b)Eg2E=» ei eie Ebfiéb ùE2EE1+(1- cosq) - e cosqi i+1- êi+ e &

25、#250; cosqg m c 2geg m c2g m c2gë eee ûeeee= veb1-1/ gwhere=2eecEnergy transfer from relativistic (high energy) electron to low energy photon is very efficient. inverse Compton scattering processApplication in astrophysics: Thermal Comptonization37If Ei << me c2 and ge >>1 at q

26、 = 0 directionCheck !作業(yè)If Ei << me c2 and average over qE =EifE1+i (1- cosq )m c2eEµ g E How to understand the g2 here? electron is relativistic with Lorentz factor ge The incident photon energy in Lab system is Ei2fei Transform to electron rest frame= Eig e (1- be cosq )Ee-r-f Scattering

27、 occurs in the electron rest frameE' Efe-r-f Transform back to the Lab systemE= E' g (1+ b cosq ')ffeeEµ g E2So, we getfei38Homework(選): 完成對Kompaneets Equation的調(diào)研。Homework(選): 一非簡并系統(tǒng),滿足LTE條件,溫度為T, 求自由電子對不同頻率入射光子的散射截面,不,以及Rosseland 平均不隨溫度T的關(guān)系。We used the Lorentz- transformation of mo

28、mentum-energy four- vectors hereThomson cross-section for electron scatteringElectron Scattering at High EnergiesThe total cross section(1) 自由電子的散射不與質(zhì)量密度，溫度和化學(xué)組成有關(guān)，通常用冪率形式(Krammer formulae n =1, s = 7/2)來表示k = k0r Tn-s量綱為m2kg-1，或cm2g-1。sT= 0.2 ×(1+ X ) cm2k=g-1em mue 對更一般的情況（如電子為簡并的或電子為相對論性的），自

29、由電子散射不透明度可表示為-1-1ö0.86 ùéùé-2æröTæöæTk = 0.2 × (1+ X ) ê1+ 2.7 ´1011ê1+ cm2g-1úúúûç÷ç K ÷çK ÷e-4.5´1038g cmøèøèøêëèúûê&#

30、235;41(4) 自由自由吸收(3)自由吸收(2)電子的散射（吸收）(1) 自由電子的散射羅列了一些有關(guān)不的計算結(jié)果(2)電子的散射 (經(jīng)典角度)Rayleigh scattering of photons on electrons bound inatoms or molecules.在經(jīng)典物理學(xué)中，電子被認為在原子中而形成一個電偶極子。若忽略原子的大小(particles much smaller than the wavelength of the light)，在外加電磁波的波動電場作用下，電子振動方程為d 2 zdziwt= -g- Kz + eE e dtozmedt2引入w02

31、 = K / me，g = g / me，分別對應(yīng)電子振蕩的圓頻率（angular frequency）和阻尼系數(shù)(damping constant).42周期性的外加電場力耗散力彈性回復(fù)力方程可以改寫為43d 2 z + g dz + w2= eiwtdt2dto zm Eozee非齊次微分方程的解為（Dsolve）eiwteEz = OZ w2- w + iwg2meo 在經(jīng)典理論中，作Larmor 公式運動的帶電粒子必然要輻射能量，輻射功率為滿足æ dW öæ deö2 e2w42 q22 e2d 2 z(非相對論)-ç÷ =

32、ç dt ÷=3 c3=(3 c3dt 2)=3222azc2èdtøèøNRW為經(jīng)典振子的能量，令為振子極大值的振動能2æ dz ö1212W =mw2 z2mç dt ÷oèø044在運動周期取平均(za2= z02/2)代入Larmor 公式dW = - 2w2e2W3m c3dte解為W = W e-gt= W e-t /too其中阻尼系數(shù)（the damping constant）g為2w2e20.2223？wor w-1g=sec0l23mc3t = 1 g =

33、4.50l2 sec輻射系統(tǒng)的平均為完全由入射電磁輻射的波長、頻率決定。2we2w4pe22g/w=3mc3=3 mc2 c=/ l<< 1rec通常電磁波345經(jīng)典振子的自由振動將因其輻射而逐漸衰減，但其振蕩能量指數(shù)衰減，衰減時標(biāo) t = ( g )遠遠長于振子自由振蕩的周期。-1 帶電粒子的散射截面（吸收來簡單處理）可以寫為時間內(nèi)被吸收的輻射能量s = Ps =入射的輻射流量S 吸收的輻射功率2 q2a2P =c33 The incident power acted on unit area (Energy flux) in the EM wave can be given b

34、y Poynting vector,c4pc4pS =EB =E246W = W e-gtog2we22 e2w4p/ w =r/ l << 13mc33 mc2 c3ecsT電子對圓頻率為w的電磁輻射的吸收系數(shù)，可給出一個= 4p e2w2gw4s=1(w )m c (w2 - w ) + w2g 2(w2 - w ) + w2g 22222eoo存在三種極限情況1. w>>w0高頻(短波)情況,回復(fù)力可以不考慮,回 到自由電子的Thomson散射.2.w<<w0低頻 (長波)情況, s(w) µ w4 µ l-4 的Reyleigh散

35、射(main reason why the sky is blue).478pe43 m 2c4ed 2 z + g dz + w2eiwtz =E edt2dtomozeg = 2w2e23mc348作業(yè)：請查找Mie散射與Rayleigh散射的不同。sT電子對園頻率為w的電磁輻射的吸收系數(shù)，可給出一個= 4p e2w2gw4s=1(w )m c (w2 - w ) + w2g 2(w2 - w ) + w2g 22222eoo存在三種極限情況1. w>>w0高頻(短波)情況,回復(fù)力可以不考慮,回 到自由電子的Thomson散射.2.w<<w0低頻 (長波)情況, s

36、(w) µ w4 µ l-4 的Reyleigh散射(main reason why the sky is blue).3.w»w0情 況，近似有w2- w02 »2 w (w- w0)2p2e2ö28pæw2g/ 2pe2s »= 0ç÷v4(w-w )2 +g2m c(w-w)2 + (g/ 2)2m c23èeøoeo為共振吸收散射截面，為原子物理，核物理中共振截面的普遍公式，不過等式右邊要乘振子強度(oscillator strength) fij498pe43 m 2c4ed

37、 2 z + g dz + w2eiwtz =E edt2dtomozeg = 2w2e23mc3g/w0(w /w0)4= 0.05g/w = 0.050g/w0 = 0.1g/w0 = 0.1Left: The cross section is shown (in units of the Thomson cross section T)for two values of the damping constant.Right: The same curves, plotted double-logarithmically, reveal the 4scaling for low frequ

38、encies, that is, the regime of Rayleigh scattering.the typical resonance behavior at50s=sw4(w)T (w2 -w2 )2 +w2g2oNear the resonance, the scattering cross section is reasonably approximated by the Lorentz profile.51s » dw2= 2p2e2g/ 2p 0vT 4(w-w )2 +g2m c (w-w)2 + (g/ 2)2oeoLorentz profileg/w0 =

39、0.1 resonance profiles=sw4(w)T (w2 -w2 )2 +w2g2o一個原子對輻射的總吸收為p e2¥òosv dv = m c = 2.65´10cmHz-22有點小錯嗎?e它同輻射阻尼系數(shù)無關(guān)。如果每個原子有 f 個電子，其本征頻率均為n0 (w0)，則原子吸收系數(shù)應(yīng)為f p e2 /me c。在量子力學(xué)的框架下，可推導(dǎo)出類似的結(jié)果，不過 f 1稱為振子強度。Profile functiongLorentzian shape ofstandard resonancep e1 =2434p()s w=l20m c g0e2e2

40、7; rwDw » g =ww22 ec 0003m c3ce52s = 2p2e2g/ 2pvm c(w-w)2 + (g/ 2)2eo譜線自然寬度g = 2w2e23m c3eLine profileAbsorption lines are due to bound-bound atomic transitions offixed energy.Why are the lines not infinitely narrow?Natural broadeningPressure or collisional broadening Doppler broadening1. Natur

41、al broadeningDE »Quantum mechanical effect - Heisenbergs uncertainty principle:DtAn electron in an excited state only exists there for a short amount of time. Therefore there is an associated uncertainty on the Energy.2wr r2mhwks=exp(i ks n × er ) sfkskc(2)電子的散射（吸收）(量子力學(xué)角度)Quantum mechanic

42、ally:correction factoroscillator strengthp e2s=m cflulueindex “l(fā)u” stands for transition lowerupper levelOscillator strengths flu can be obtained by: Laboratory measurements Solar spectrum Quantum mechanical computations (Opacity Project etc.) Allowed lines:flu»1,<<1 e.g. He I 1s2 1S®

43、;1s2sflu=210-14 Forbidden :flu3S55l/ÅLineflu1215.7Ly a0.411025.7Ly b0.07972.5Ly g0.036562.8H a0.644861.3H b0.124340.5H g0.042wr r2mhwks=exp(i ks n × er ) sfkskc-躍遷的散射截面g n / 4pe2s=fijR(n -n+ (g/ 4p )2)2m ceongn為在頻率n0處內(nèi)稟的線寬. fij是描述量子力學(xué)修正 的振子強度.可以近似理解為原子內(nèi)經(jīng)典振子的“數(shù)目”.56略對于氫原子能級nn間躍遷的振子強度為：26111

44、-3-3fn'n =3p g n2 - n¢2 (nn')gbb3n其中g(shù)n = 2n2為能級n的簡并，含自旋。11(E - E )R3c3-3 = nn¢-3 =yn2n¢2R hcv3y即 fµ v-3gbb稱為Gaunt因子（常用符號gi = gbb ）。n'nn'n在可見區(qū)， gbb 1，當(dāng)能級差很大時， gbb < 1。57略通常情況下,在恒星內(nèi)部傳 能的過程中-吸收的貢獻不重要（可以忽略）. The elements of the Fe group, which are in significant am

45、ounts, have many bound electrons which largely contribute to the opacity in Pop. I stars. Various improvements in the calculation of atomic physics for the partially ionized Fe ions have led to an opacity increase by a factor of 3 for T equal to a few的貢獻 105 K-吸收對不 振子強度計算過程非常復(fù)雜。 吸收光子能量后，電子僅僅是躍遷到高能級（

46、沒有電離），后續(xù)的退激發(fā)過程產(chǎn)生光子，使得吸收效果減弱。 因為吸收與能級結(jié)構(gòu)有關(guān)，僅僅在某些特征頻率上吸收較強，因此對總的不貢獻不大。（碰撞致寬） 在T < 106 K，對總的不貢獻顯著（2倍）。在 T > 107 K時，在總的不中的貢獻比較?。?10%）58略在實際的恒星光譜中，一系列因素都會引起光譜線加寬如Doppler加寬，Stark加寬，碰撞加寬，Doppler加寬源于原子的運動Skipö1/ 2éAm c2 n -nùp e2æ Am c21s=× exp ê-u(o )2 úfçu÷

47、;è 2p kTn2p kTnDijm cøëûeoo碰撞加寬源于原子與周圍的粒子碰撞后使得處于激發(fā)態(tài)上原子的降低g / 4pe2s=f c(n -n+ (g / 4p )2Cij)2m ceoc= 4p kr y+ n '4n4g5.3´108T1/ 2 Z *2 åcx mhjjjn, n 為初態(tài)和末態(tài)的主量子數(shù), Z*為等效核電荷數(shù), y為每個原子平均的自由電子數(shù)目,mj 為第j種成分的原子量, xj為每摩爾總j成分的豐度 .59略 >> 0Rayleigh scattering << 0Clos

48、e to the resonance 0 ,The quantum description of the atomic transitions leads toa modified expressionthe width of the levels60the oscillator forces and are determined in laboratory for most atomsthe number of the atoms considered on level n by volume u-躍遷產(chǎn)生譜線,對不的貢獻極其復(fù)雜，一般沒有簡單的表達式來表示。is the effective

49、 number of electrons which can make the transition from level n n.(2)電子的散射不與質(zhì)量密度，溫度和化學(xué)組成有關(guān)，通常用冪率形式(Krammer formulae n =1, s = 7/2)來表示k = k r Tn-s0量綱為m2 kg-1，或cm2g-1。恒星內(nèi)部幾乎完全電離，只在外區(qū)或內(nèi)部較冷區(qū)域才重要。(4) 自由自由吸收-躍遷產(chǎn)生譜線,對不的貢獻極其復(fù)雜，一般沒有簡單的表達式來表示。61(3)自由吸收(2)電子的散射(1) 自由電子的散射羅列了一些有關(guān)不的計算結(jié)果(3)自由(b-f)吸收(量子角度)光致電離(pho

50、toionisation)過程恒星內(nèi)部滿足高溫的條件下，高能光子 (hnXr,n, 表示 r次電離原子處于能級n的電離電勢)會使得重元素的內(nèi)電子層的電子電離。以類氫原子為例其中為Rydberg 常數(shù)。低能態(tài)為分立能級，高能態(tài)為連續(xù)能態(tài).62略光致電離（ photoionization ）截面Z 核電荷數(shù)，n電子為對應(yīng)的能級，me電子質(zhì)量，e電子電荷；gbf為Gaunt因子，ln 為極限波長63可以由量子力學(xué)來計算It can be seen that they are nearly constant and nearlyunear the threshold.This feature has

51、the advantage that in the calculation of the Rosseland mean an appropriate average g may be used, and the frequency dependence can beregarded as simply n-3.The free-electron energy expressed in Rydbergs per Z2.64略sn = 64p4 × m e10 × Z 4 × 1n( )eg( , n, l, Z )(b-f )33ch6n5v3bf 對于多于一個電子的離子來說，還須考慮核電荷被其它 顯然電子的-自由不作用。參見彭依賴于處于不同原子中的-自電子的數(shù)目。當(dāng)溫度很高時，原子完全電離，則由不降為零。65 -自由不通常用Kramers公式來表示X, Z 分別氫和其它重元素的質(zhì)量豐度, 當(dāng)氫、氦或其它重元素部分電離時，即當(dāng)T ³ 2 ´ 104 K時， “Kramers”不 透明度比較重要。 光致電離的反應(yīng)率ærö1 æ T ö-3.5k= 4.34´1025 ×(1+ X )× Zcm2 g-1bfç g