2015恒星结构与演化20209s

1、恒星结构与演化(2020.2-2020.6)第2部分恒星物理罗新炼仙林校区天文楼521房间： xlluo 89685982状态方程辐射传能第2部分恒星物理对流传能核过程请就你掌握的辐射相关知识做思维导图。+3ò1dBn dv¥1 = o kva 1- exp (- hn kT )+ kvs dTkò¥ dBn dvo dTdP = - Gm(r) rdrr2 恒星大气的简单模型 恒星质光关系 Eddington极限光度 更加广阔的应用恒星内部能量转移过程：辐射与热传导辅助方程 k4热传导不辐射转移1k能量平衡 需要具体光子与物质的相互作用。 计算过程需要

2、利用量子力学和量子场论方面的知识，相当复杂。 参考文献：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.需补充更新文献!恒星内部物质的不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简略m 434L = 96 k M b 院馆可借，物理写得比较好 有关不数据的网页?8请查更新的恒星内部物质的不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 (委内瑞拉).

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?作业：请找到类似这个，计算恒星(太阳)内部物质不. 恒星内部物质的不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随意取值做的.随意取值做的.1415high energyphoton16自由电子散射自由-自由吸收-自由吸收-吸收恒星内部不主要的来源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、在此略过了. 请在课外补充阅读这方面知识.不与质量密度，温度和化学组成有关，通常用幂率形式(Krammer formulae n =1, s = 7/2)来表示k = k0r Tn-s量纲为m2kg-1，或cm2g-1。总的不21(4) 自由自由吸收(3)自由吸收(2)电子的散射(吸收)(1) 自由电子的散射罗列了一些有关不的计算结果(1) 自由电子的散射如果电子是非简并且为非相对论性的，则不来自于Thomson 散射主要可用经典电动力学方法导出.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为æ ds ö3s2r2 qq )=+ cos) =T (1+ cos2

16、1;÷ ec (1216pè dW øunpolq 为散射角表示入射方向与出射方向的夹角.总散射截面即对全方位ö28p æe2sº= 0.6652´10-24cm2 1 barnç÷Tm c23èøe30略Thomson 散射不为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 (与入射光子的频率无关) Thomoson 散射仅在高温时比较重要，因为物质几乎完全电离。 但如果电子是相对论性的，则要考虑Compton散射。SR31 While for scattering, photon direction changed =>momentum changed => energy changedThomson 散射的特点局部稳恒磁场Synchrotron （同步） Radiationthe energy den

18、sity in themagnetic field辐射功率辐射能谱Kj is the modified Bessel function of the second kindBremsstrahlung （轫致辐射） Radiation略？谱特征辐射机制内容33以前课程中夹带了部分狭义相对论和广义相对论知识。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 (克莱因-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为入射光子 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 !作业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(选): 完成对Kompaneets Equation的调研。Homework(选): 一非简并系统,满足LTE条件,温度为T, 求自由电子对不同频率入射光子的散射截面,不,以及Rosseland 平均不随温度T的关系。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) 自由电子的散射不与质量密度，温度和化学组成有关，通常用幂率形式(Krammer formulae n =1, s = 7/2)来表示k = k0r Tn-s量纲为m2kg-1，或cm2g-1。sT= 0.2 ×(1+ X ) cm2k=g-1em mue 对更一般的情况（如电子为简并的或电子为相对论性的），自

29、由电子散射不透明度可表示为-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)电子的散射（吸收）(1) 自由电子的散射罗列了一些有关不的计算结果(2)电子的散射 (经典角度)Rayleigh scattering of photons on electrons bound inatoms or molecules.在经典物理学中，电子被认为在原子中而形成一个电偶极子。若忽略原子的大小(particles much smaller than the wavelength of the light)，在外加电磁波的波动电场作用下，电子振动方程为d 2 zdziwt= -g- Kz + eE e dtozmedt2引入w02

31、 = K / me，g = g / me，分别对应电子振荡的圆频率（angular frequency）和阻尼系数(damping constant).42周期性的外加电场力耗散力弹性回复力方程可以改写为43d 2 z + g dz + w2= eiwtdt2dto zm Eozee非齐次微分方程的解为（Dsolve）eiwteEz = OZ w2- w + iwg2meo 在经典理论中，作Larmor 公式运动的带电粒子必然要辐射能量，辐射功率为满足æ dW öæ deö2 e2w42 q22 e2d 2 z(非相对论)-ç÷ =

32、ç dt ÷=3 c3=(3 c3dt 2)=3222azc2èdtøèøNRW为经典振子的能量，令为振子极大值的振动能2æ dz ö1212W =mw2 z2mç dt ÷oèø044在运动周期取平均(za2= z02/2)代入Larmor 公式dW = - 2w2e2W3m c3dte解为W = W e-gt= W e-t /too其中阻尼系数（the damping constant）g为2w2e20.2223？wor w-1g=sec0l23mc3t = 1 g =

33、4.50l2 sec辐射系统的平均为完全由入射电磁辐射的波长、频率决定。2we2w4pe22g/w=3mc3=3 mc2 c=/ l<< 1rec通常电磁波345经典振子的自由振动将因其辐射而逐渐衰减，但其振荡能量指数衰减，衰减时标 t = ( g )远远长于振子自由振荡的周期。-1 带电粒子的散射截面（吸收来简单处理）可以写为时间内被吸收的辐射能量s = Ps =入射的辐射流量S 吸收的辐射功率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电子对圆频率为w的电磁辐射的吸收系数，可给出一个= 4p e2w2gw4s=1(w )m c (w2 - w ) + w2g 2(w2 - w ) + w2g 22222eoo存在三种极限情况1. w>>w0高频(短波)情况,回复力可以不考虑,回 到自由电子的Thomson散射.2.w<<w0低频 (长波)情况, 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作业：请查找Mie散射与Rayleigh散射的不同。sT电子对园频率为w的电磁辐射的吸收系数，可给出一个= 4p e2w2gw4s=1(w )m c (w2 - w ) + w2g 2(w2 - w ) + w2g 22222eoo存在三种极限情况1. w>>w0高频(短波)情况,回复力可以不考虑,回 到自由电子的Thomson散射.2.w<<w0低频 (长波)情况, s

36、(w) µ w4 µ l-4 的Reyleigh散射(main reason why the sky is blue).3.w»w0情 况，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为共振吸收散射截面，为原子物理，核物理中共振截面的普遍公式，不过等式右边要乘振子强度(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一个原子对辐射的总吸收为p e2¥òosv dv = m c = 2.65´10cmHz-22有点小错吗?e它同辐射阻尼系数无关。如果每个原子有 f 个电子，其本征频率均为n0 (w0)，则原子吸收系数应为f p e2 /me c。在量子力学的框架下，可推导出类似的结果，不过 f 1称为振子强度。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谱线自然宽度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)电子的散射（吸收）(量子力学角度)Quantum mechanic

42、ally:correction factoroscillator strengthp e2s=m cflulueindex “lu” 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-跃迁的散射截面g n / 4pe2s=fijR(n -n+ (g/ 4p )2)2m ceongn为在频率n0处内禀的线宽. fij是描述量子力学修正 的振子强度.可以近似理解为原子内经典振子的“数目”.56略对于氢原子能级nn间跃迁的振子强度为：26111

44、-3-3fn'n =3p g n2 - n¢2 (nn')gbb3n其中gn = 2n2为能级n的简并，含自旋。11(E - E )R3c3-3 = nn¢-3 =yn2n¢2R hcv3y即 fµ v-3gbb称为Gaunt因子（常用符号gi = gbb ）。n'nn'n在可见区， gbb 1，当能级差很大时， gbb < 1。57略通常情况下,在恒星内部传 能的过程中-吸收的贡献不重要（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的贡献 105 K-吸收对不 振子强度计算过程非常复杂。 吸收光子能量后，电子仅仅是跃迁到高能级（

46、没有电离），后续的退激发过程产生光子，使得吸收效果减弱。 因为吸收与能级结构有关，仅仅在某些特征频率上吸收较强，因此对总的不贡献不大。（碰撞致宽） 在T < 106 K，对总的不贡献显著（2倍）。在 T > 107 K时，在总的不中的贡献比较小（ 10%）58略在实际的恒星光谱中，一系列因素都会引起光谱线加宽如Doppler加宽，Stark加宽，碰撞加宽，Doppler加宽源于原子的运动Skipö1/ 2éAm c2 n -nùp e2æ Am c21s=× exp ê-u(o )2 úfçu÷

47、;è 2p kTn2p kTnDijm cøëûeoo碰撞加宽源于原子与周围的粒子碰撞后使得处于激发态上原子的降低g / 4pe2s=f c(n -n+ (g / 4p )2Cij)2m ceoc= 4p kr y+ n '4n4g5.3´108T1/ 2 Z *2 åcx mhjjjn, n 为初态和末态的主量子数, Z*为等效核电荷数, y为每个原子平均的自由电子数目,mj 为第j种成分的原子量, xj为每摩尔总j成分的丰度 .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-跃迁产生谱线,对不的贡献极其复杂，一般没有简单的表达式来表示。is the effective

49、 number of electrons which can make the transition from level n n.(2)电子的散射不与质量密度，温度和化学组成有关，通常用幂率形式(Krammer formulae n =1, s = 7/2)来表示k = k r Tn-s0量纲为m2 kg-1，或cm2g-1。恒星内部几乎完全电离，只在外区或内部较冷区域才重要。(4) 自由自由吸收-跃迁产生谱线,对不的贡献极其复杂，一般没有简单的表达式来表示。61(3)自由吸收(2)电子的散射(1) 自由电子的散射罗列了一些有关不的计算结果(3)自由(b-f)吸收(量子角度)光致电离(pho

50、toionisation)过程恒星内部满足高温的条件下，高能光子 (hnXr,n, 表示 r次电离原子处于能级n的电离电势)会使得重元素的内电子层的电子电离。以类氢原子为例其中为Rydberg 常数。低能态为分立能级，高能态为连续能态.62略光致电离（ photoionization ）截面Z 核电荷数，n电子为对应的能级，me电子质量，e电子电荷；gbf为Gaunt因子，ln 为极限波长63可以由量子力学来计算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 对于多于一个电子的离子来说，还须考虑核电荷被其它 显然电子的-自由不作用。参见彭依赖于处于不同原子中的-自电子的数目。当温度很高时，原子完全电离，则由不降为零。65 -自由不通常用Kramers公式来表示X, Z 分别氢和其它重元素的质量丰度, 当氢、氦或其它重元素部分电离时，即当T ³ 2 ´ 104 K时， “Kramers”不 透明度比较重要。 光致电离的反应率ærö1 æ T ö-3.5k= 4.34´1025 ×(1+ X )× Zcm2 g-1bfç g