7 Stellar Structure III. introduc)on to Astrophysics, C. Bertulani, Texas A&M-Commerce 1

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1 7 Stellar Structure III ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 1

2 Fundamental physcal constants a radaton densty constant J m -3 K -4 c velocty of lght m s -1 G gravtatonal constant N m 2 kg -2 h Planck s constant J s K Boltzmann s constant J K -1 m e mass of electron kg m H mass of hydrogen atom kg N A Avogadro s number mol -1 σ Stefan Boltzmann constant W m -2 K -4 (σ = ac/4) R gas constant (k/mh) J K -1 kg -1 e charge of electron C L lumnosty of Sun W M mass of Sun kg T eff effectve temperature of sun 5780 K R radus of Sun m Parsec (unt of dstance) m ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 2

3 7.1 - The equaton of radatve transport We assume for the moment that the condton for convecton s not satsfed, and we wll derve an expresson relatng the change n temperature wth radus n a star assumng all energy s transported by radaton. Hence we gnore the effects of convecton and conducton. The equaton of radatve transport wth gas condtons s a functon of only one coordnate, n ths case r. Photons nteract wth free electrons, cause atomc transtons and onze atoms. The net radaton flux s gven by: ( T ) 4π 1 di 4π 1 d σ = 3 κρ dr 3 κρ dr 4 F rad = (6.59) Where I s the ntensty of radaton and κ s the opacty. ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 3 3

4 Rosseland mean opacty Assume where for a star composed of gas- radaton mxture. Usng the Stefan-Boltzmann law, radaton pressure due to a photon gas s P rad = at4 3 Then where. Thus, (7.3) (7.4) Now we average over all radaton frequences ν: wth κ β = P gas P L = 4π r 2 ac 3ρκ P = P gas + P beng the Rosseland mean opacty L dt dr = 4, one can show that the ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 4 rad (7.1) (7.2) a = 4σ 3c 1 = κ π act π r σt = 4π r 2 c ρκ F = 1 di ( ν, T) κ dt 1 β = dν dp dr rad L 4πr 2 P at rad = P 3 P ν (7.8) = 4π 3 4 (7.6) 1 di κρ dr (7.5) (7.7)

5 Radaton transport Where We thus can wrte or I( ν, T ) s the Planck functon for the ntensty of blackbody radaton: 3 2h ν I( ν, T) = 2 hν / kt c e L( r) 4 ac 3 F rad = = T 2 4πr 3 κρ dt dr = 3 4ac κρ L( r) 3 2 T 4πr 1 dt dr and ths s the equaton for radaton transport. (7.9) (7.11) (7.10) ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 5

6 7.2 - From atoms to atmospheres Chemcal composton How can we descrbe the relatve abundances of elements (nucle) of dfferent speces and ther evoluton n a gven sample (say, a star, or the Unverse)? Number densty n We could use the number densty = number of nucle of speces per cm 3 Dsadvantage: tracks not only nuclear processes that create or destroy nucle, but also densty changes, for example due to compresson or expanson of the materal. Mass fracton X s fracton of total mass of sample that s made up by nucleus of speces (7.12) n = X m ρ ρ : mass densty (g/cm 3 ) m mass of nucleus of speces (CGS only!!!) wth m A m u and m u m = 12C / 12 = 1/ N A as atomc mass unt (AMU) (7.13) (7.14) ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 6

7 Mass fracton and abundance n = X A ρ N A call ths abundance Y (7.15) note: f the mass s expressed n the CGS unt (grams) then 1 [g] = N A. m u (we neglect the bndng energes and the mass of the electrons n the atoms) so n = Y ρ N X note: Abundance has no unts wth A Y = only vald n CGS A (7.16) (7.17) The abundance Y s proportonal to number densty but changes only f the nuclear speces gets destroyed or produced. Changes n densty are factored out. of course X = 1 but, snce Y = X /A < X Y < 1 (7.18) (7.19) ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 7

8 Mass fracton and abundance Mean molecular weght µ A n AY A = = = = average mass number = n Y (7.21) µ = 1 Y Electron Abundance Y e As matter s electrcally neutral, for each nucleus wth charge number Z there are Z electrons: Y Z Y and as wth nucle, electron densty e A e e = (7.21) can also wrte: Y e = (7.20) Z Y AY ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 8 n 1 Y prop. to number of protons prop. to number of nucleons (7.23) = ρ N So Y e s rato of protons to nucleons n sample (countng all protons ncludng the ones contaned n nucle - not just free protons as descrbed by the proton abundance ) or For 100% hydrogen: Y e = 1 For equal number of protons and neutrons (N = Z nucle): Y e = 0.5 For pure neutron gas: Y e = 0 Y (7.22)

9 7.3 - Cross sectons Bombard target nucle wth projectles: Defnton of cross secton: (7.24) # of reactons = σ x # of ncomng projectles per second and target nucleus per second and cm 2 or n symbols: rate = σ j Unts for cross secton: wth j as partcle number current densty. j = n v wth partcle number densty n) (7.25) 1 barn = cm 2 ( = 100 fm 2 or about half the sze (cross sectonal area) of a uranum nucleus) ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 9

10 Complcaton (1) Determne σ The cross secton s a measure of how lkely a photon gets absorbed when an atom s bombarded wth a flux of photons. It depends on: Oscllator strength: a quantum mechancal property of the atomc transton Needs to be measured n the laboratory - not done wth suffcent accuracy for a number of elements. Lne wdth The wder the lne n wavelength, the more lkely a photon s absorbed (as n a classcal oscllator). E ΔE excted state has an energy wdth ΔE. Ths leads to a range of photon energes that can be absorbed and to a lne wdth Atom photon energy range Hesenbergs uncertanty prncple relates that to the lfetme τ of the excted state ΔE τ =! need lfetme of fnal state (7.26) ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 10

11 Determne σ The lfetme of an atomc level n the stellar envronment depends on: The natural lfetme (natural wdth) lfetme that level would have f atom s left undsturbed Frequency of Interactons of atom wth other atoms or electrons Collsons wth other atoms or electrons lead to deexctaton, and therefore to a shortenng of the lfetme and a broadenng of the lne Varyng electrc felds from neghborng ons vary level energes through Stark Effect depends on pressure need local gravty, or mass/radus of star Doppler broadenng through varatons n atom velocty thermal moton mcro turbulence depends on temperature Need detaled and accurate model of stellar atmosphere! ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 11

12 Complcaton (2) Atomc transtons depend on the state of onzaton! The number densty n determned through absorpton lnes s therefore the number densty of ons n the onzaton state that corresponds to the respectve transton. To determne the total abundance of an atomc speces one needs the fracton of atoms n the specfc state of onzaton. Notaton: I = neutral atom, II = one electron removed, III = two electrons removed.. Example: a CaII lne orgnates from sngly onzed Calcum Frequently used notaton: n s the densty of atoms n the -th state of onzaton, that s wth electrons removed. g s the degeneracy of states for the -ons E s the energy requred to remove electrons from a neutral atom, creatng an -level on. n e s the electron densty ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 12

13 7.4 - The Saha Equaton In Local Thermodynamcal Equlbrum (LTE) the dstrbuton of atoms over varous states of exctaton and onzaton s descrbed by Boltzmann s relaton for the relatve populatons of the ground atomc level n 1 and the n th level n n (per unt volume), n 1 n n = g 1 " exp E E % n 1 $ ' g n # kt & wth correspondng energes E 1 and E n and statstcal weghts g 1 and g n : Saha s equaton s a relaton, based on Boltzmann s dstrbuton, for the rato of number denstes of two dfferent onzaton states of an atomc speces. For free electrons and large enough temperatures, we wll assume a Boltzmann dstrbuton. The Saha equaton follows as Saha s equaton s a relaton, based on Boltzmann s dstrbuton, for the rato of number denstes of two dfferent onzaton states of an atomc speces. For free electrons and large enough temperatures, we wll assume a Boltzmann dstrbuton. The Saha equaton follows form n +1 n e n = g +1 g e g (7.27) " exp χ % " % $ 'exp$ p2 'd 3 p # kt & # 2mkT & (7.28) ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 13

14 The Saha Equaton whch, when ntegrated over the free electron momentum p, yelds n +1 n e n = g +1 g e g! 2πm e kt $ # & " h 2 % where n s the number densty of atoms n the th onzaton state, n +1 s the number densty n the next onzaton state, n e s the number densty of free electrons, χ s the onzaton energy dfference between the th and +1 states, and the factors g, g +1, and g e are the correspondng statstcal weghts (degeneracy) of the dscrete energy states. For free (unbound) electrons, g e = 2, owng to two spn states. Generally, one can use g n = 2n 2 (n s the prncpal quantum number) for bound energy states. For the more complex atomc systems, a quantum treatment s usually necessary to calculate the statstcal weghts. 3/2! exp χ $ # " kt % & (7.26) Let us see what we obtan for the onzaton fracton of hydrogen n the solar photosphere, where E = 13.6 ev, T onz = E/k ~ 160,000 K, T gas ~ 6,000 K, and n H ~ cm 3. Thus, the actual gas temperature s much smaller than the temperature equvalent of the onzaton energy. Navely one would expect a very low onzaton fracton of hydrogen n the solar photosphere. Hydrogen s the domnatng element, hence the electron densty should be related to the hydrogen densty by ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 14

15 The Saha Equaton n e = n +1 n + n +1 n H (7.27) The rato of level partton sums s approxmately 0.5, because the frst excted level of atomc hydrogen s hgh compared to the thermal energy. Ths means that, approxmately, n s the number of atoms n the ground state and n +1 s the number of atoms ether onzed or n the frst excted state. Insertng the numbers n Saha s equaton then gves the onzaton fracton ξ = n +1 n + n +1 n +1 n e n n H = ξ 2 1 ξ ~ 10 7 ξ ~ (7.28) where the ntermedate step was obtaned from Eq. (7.26) wth χ = 13.6 ev. The result above seems small ndeed, but repeatng the calculaton for a slghtly hotter star wth photospherc temperature T = 12,000 K and the same gas densty yelds an onzaton fracton ξ(t = 12,000 K) ~ 0.3, much hgher than what the onzaton energy would suggest. ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 15

16 7.5 Contrbutons to opacty Electron scatterng (Thompson): where r e s the classcal electron radus. Free-free transtons: Bound-free transtons κ ν Z ρt ν 2 1/ 2 3 (7.30) κ = 8π 3 r 2 e m H (7.29) Bound-bound transtons: contrbute sgnfcantly for T < 10 6 K (e.g., the Sun) 3 κ ν ν (7.31) + other atomc processes Opactes are very complcated to calculate. The fgure shows numercal calculatons of opactes (n cm 2 /g) as a functon of the temperature (n Kelvns) for several denstes (n g/cm 3 ). ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 16

17 7.6 - Solvng the equatons of stellar structure Hence we now have four dfferental equatons, whch govern the structure of stars (note n the absence of convecton). dm(r) dr = 4πr 2 ρ(r) dp(r) dr = GM(r)ρ(r) r 2 dl(r) dr = 4πr 2 ρ(r)ε(r) dt ( r) dr 3ρ( r) κ ( r) = L( r) πr σt ( r) Informaton needs to be complemented wth: P = P (ρ, T, chemcal composton) The equaton of state κ = κ (ρ, T, chemcal composton) ε = ε (ρ, T, chemcal composton) ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 17 17

18 Boundary condtons Two of the boundary condtons are farly obvous, at the center of the star M = 0, L = 0 at r = 0 At the surface of the star ts not so clear, but we use approxmatons to allow soluton. There s no sharp edge to the star, but for the the Sun ρ(surface) ~ 10-4 kg m -3. Ths s much smaller than mean densty ρ(mean) ~ 1.4 x 10 3 kg m -3 (whch we derved). We know the surface temperature (T eff = 5780K) s much smaller than ts mnmum mean temperature ( K). Thus we make two approxmatons for the surface boundary condtons: ρ = T = 0 at r = r s.e. that the star does have a sharp boundary wth the surroundng vacuum. ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 18 18

19 Use of mass as the ndependent varable The above formulae would (n prncple) allow theoretcal models of stars wth a gven radus. However from a theoretcal pont of vew t s the mass of the star whch s chosen, the stellar structure equatons solved, then the radus (and other parameters) are determned. We observe stellar rad to change by orders of magntude durng stellar evoluton, whereas mass appears to reman constant. Hence t s much more useful to rewrte the equatons n terms of M rather than r. If we dvde the other three equatons by the equaton of mass conservaton, and nvert the latter, we get dr dm = 1 4πr 2 ρ dp dm = GM 4πr 4 (7.32) dl dm = ε dt dm = 3κ R L 64π 2 r 4 act 3 (7.34) (7.35) (7.33) Wth boundary condtons: r = 0, L = 0 at M = 0 ρ = 0, T = 0 at M = M s We specfy M s and the chemcal composton and now have a well defned set of relatons to solve. It s possble to do ths analytcally f smplfyng assumptons are made, but n general these need to be solved numercally 19 on a computer. ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 19

20 Role of nuclear abundances If there are no bulk motons n the nteror of the star, then any changes of chemcal composton are localsed n the element of materal n whch the nuclear reactons occurred. So star would have a chemcal composton whch s a functon of mass M. In the case of no bulk motons the set of equatons we derved must be supplemented by equatons descrbng the rate of change of abundances of the dfferent chemcal elements. Let C X,Y,Z be the chemcal composton of stellar materal n terms of mass fractons of hydrogen (X), helum, (Y) and metals (Z) [e.g. for solar system X = 0.7, Y = 0.28, Z = 0.02] (C X,Y,Z ) M = f (ρ,t,c X,Y,Z ) t Such a model evolves as (7.36) (C X,Y,Z ) M,t0 +δt = (C X,Y,Z ) M,t0 + (C X,Y,Z ) M t (7.37) In solvng the equatons of stellar structure the eqns approprate to a convectve regon must be swtched on whenever the temperature gradent reaches the adabatc value, and swtched off when all energy can be transported 20 by radaton. ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 20

21 Ex: - Solar model T(eV) n e (cm -3 ) r/r 54 1x boundary poston depends on transport measured wth helosesmology 182 9x x Solar model : J.N. Bahcall et al, Rev. Mod. Phys. 54, 767 (1982) x radaton convecton Fgure: Jm Baley, Sanda Transport depends on opacty, composton, n e, T ntroduc)on to Astrophyscs, C. Bertulan, Texas A&M-Commerce 21