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The Sun as a Star Internal Structure and Evolution Simon Jeffery, Armagh Observatory Monday, 17 September 12 Outline • Properties of Stars • Equations of Stellar Structure • Plasma Physics • Solar Models • Equations of Stellar Evolution • Future of the Sun Monday, 17 September 12 1. Properties of Stars Monday, 17 September 12 Monday, 17 September 12 ω Cen - Kitt Peak Monday, 17 September 12 ω Cen - HST Monday, 17 September 12 The Hertzsprung-Russell diagram Spectral type <> colour <> effective temperature Magnitude <> luminosity White Dwarfs <> Dwarfs <> Subgiants <> Giants <> Supergiants Monday, 17 September 12 Spectral Type O 50 kK B 20 kK Hγ Hβ Hα A 10 kK F 7500K G 6000K K 4000K M 3500K Monday, 17 September 12 Image Credit: N.A.Sharp, NOAO/NSO/Kitt Peak FTS/AURA/NSF Monday, 17 September 12 Stellar Properties The Sun M stellar mass (M / M) M = 1 M = 1.99 1030 kg R stellar radius (R / R) L stellar luminosity (L / L) R = 1 R = 6.96 108 m L = 1 L = 3.86 1026 W Teff effective temperature (K) = ∜L/4πσR2 g surface gravity = GM/R2 X,Y,Z mass fractions of H, He and other elements t age Monday, 17 September 12 Teff = 5780 K g = 274 m s-2 X = 0.71 Y = 0.265 Z = 0.025 t ~ 4.6 109 y Big Questions What is the Sun made of? What happens inside the Sun? Why does the Sun shine? What holds the Sun up? How did the Sun form? How long will the Sun last? How will the Sun end? Can we “see” inside the Sun? Where did the Sun’s constituents come from? What will happen when the Sun runs out of fuel? What governs the size and brightness of the Sun? ... Ask the same questions for any star .... but the answers are NOT the same for every star! Monday, 17 September 12 2. Equations of Stellar Structure Objective Assumptions Conservation Laws Mass Continuity Energy Conservation Total Energy of a Star Hydrostatic Equilibrium Virial Theorem Energy Transport Monday, 17 September 12 a stellar interior Monday, 17 September 12 definitions Surface: r=R Structure: r radius m(r) mass within r l(r) flux through r T(r) temperature at r P(r) pressure at r [ ρ(r) r M,L,0,Teff density at r ] Centre: r=0 0,0,Pc,Tc Monday, 17 September 12 X,Y,Z m,l,P,T assumptions / approximations Isolated body: Thermodynamic equilibrium Conservation Laws: Mass Energy Angular Momentum Forces: self-gravity internal pressure Approximations: spherical symmetry (1D) ** no rotation ** no magnetic fields ** no chemical diffusion ** ** Many models now take some or all of these into account, but in setting up the basic equaitons, and for comparing with other stars, it is more practical to make these approximations. Monday, 17 September 12 conservation of mass (or the continuity equation) Consider a spherical shell of radius r thickness δr (δr <<r) density ρ Its mass (volume x density): δm = 4 π r2 ρ δr As δr→0: dm/dr = 4 π r2 ρ Also: m= 4 π ∫ r2 ρ dr Monday, 17 September 12 δr r ρ conservation of energy Consider a spherical volume element dv=4 π r2 dr Conservation of energy demands: energy out = energy in + energy produced or lost If ε is the energy produced/lost per unit mass, then l+δl = l + ε δm ⇒ dl/dm = ε Since dm = 4πr2 ρ dr dl/dr = 4πr2 ρ ε We will consider the nature of energy sources, ε, later. Monday, 17 September 12 l+δl δr l r εδm hydrostatic equilibrium Consider forces at a radius r acting on an element of thickness δr, area δA and mass δm = ρ δr δA A sphere of mass m acts as a gravitational mass situated at the centre, giving rise to an inward force: δm g = δm Gm/r2 P+δP g ρ zP δr If a pressure gradient (dP/dr) exists, there will be a nett inward force δr δA dP/dr ≡ δm / ρ dP/dr The sum of inward forces is then δm ( g +1/ ρ dP/dr ) = - δm d2r/dt2 r In order to oppose gravity, pressure must increase towards the centre. For hydrostatic equilibrium, forces must balance: dP/dr = - Gm ρ / r2 Monday, 17 September 12 energy transport Stars shine. What does this tell you about them ? A temperature difference between the centre and surface of a star implies a temperature gradient, and hence a “flux” of energy. How can energy be transported ? What methods are important in stars ? How does the choice of energy transport affect the temperature gradient – or vice versa ? Monday, 17 September 12 radiative equilibrium If transported by radiation (photons), then the flux density F obeys Fick’s law of diffusion: F = -D ∇Urad Urad =aT4 is the radiation energy density. For radiative transfer, identify the diffusion coefficient D = c/3κρ. In spherical symmetry ∇ = d/dr. Hence F = - c/3κρ d(aT4)/dr Multiply by 4πr2 to obtain a total flux, whence l= 4πr2 (-c / 3κρ) d(aT4)/dr ⇒ dT/dr = - 3 κ ρ/4acT3 l/4πr2 Monday, 17 September 12 convective equilibrium An element of gas has pressure P and density ρ. Displace upward by distance δr. Allow to expand adiabatically until pressure within is equal to pressure outside. Release the element. If it continues to move upwards, the layer is convectively unstable. If it remains stationary or sinks, the layer is stable against convection. It may be shown that the condition for convective equilibrium is related to the temperature gradient assuming some equation of state (eg P = ρkT/µmH) so that: d ln T ⌘ r < rad ⌘ d ln P Monday, 17 September 12 1 3. Plasma Physics Monday, 17 September 12 Stellar Structure Equations We have derived four time-independent first-order differential equations in four variables, with one independent variable r. They can also be rewritten in terms of any of the other variables; m is useful. dm dr dl dr dP dr dT dr Monday, 17 September 12 2 = 4⇡r ⇢ 2 = 4⇡r ⇢" = = Gm⇢ r2 Gm⇢ T r 2 r P dr dm dl dm dP dm dT dm 1 = 4⇡r2 ⇢ =" = = Gm 4⇡r4 Gm T r 4 4⇡r P Boundary Conditions and Constitutive Relations Four first order differential equations require four boundary conditions: Centre: m(r=0) = 0, l(r=0) = 0 Surface: P(r=R) = 0, T(r=R) = Teff To close the system, we need expressions for: ρ ε κ ∇ density nuclear energy generation opacity energy transport Plasma Physics In practice, these are all functions of P, T, and Xi. Monday, 17 September 12 Equation of State ρ = n/N0µ P=1/3E PR=aT4/3 8 log T/K P=nkT 6 P=KNRn5/3 4 P=KURn4/3 P=2/3E -4 0 4 log ρ/kg m-3 8 Stellar Structure: TCD 2009: 7. Monday, 17 September 12 12 photon-electron interactions bound-bound bound-free free-free hν = ½m(v22-v12) ½mv2 χion n=3 hν = χ3-χ2 n=2 Excitation energy hν = χ2 hν = χion+½mv2 n=1 Stellar Structure: TCD PYA301: 8. Monday, 17 September 12 Opacity In regions where specific atoms are ionized, opacity - due to bound-free transitions has a local maximum. H+He H+He He+ He+ Fe,… C,N,O κ= κ(ρ,T,Xi) e-s Stellar Structure: TCD PYA301: 8. Monday, 17 September 12 Proton Rest mass: mp = 1.672 x 10-27 kg = 1.00728 u E = mpc2 = 1.5027 x 10-10 J + + The electrostatic force like charges repel!! Stellar Structure: TCD PY3A03: 9. Monday, 17 September 12 Atomic Mass Unit 1u = m(12C)/12 Nuclear Reaction Rates The cross-section <σv> for a fusion reaction is represented by the product of the particle energy distribution and the tunnelling probability ∝exp(-E/kT) ∝exp(-(EG/E)1/2) E0 ∝exp(-E/kT - (EG/E)1/2) Δ Stellar Structure: TCD PY3A03: 9. Monday, 17 September 12 Proton rest mass: mp = 1.00728 u E=mpc2 = 1.5027 x 10-10 J + +- νe d + Deuterium nucleus rest mass: md = 2.01355 u Eexcess = δmc2 = (2mp - md)c2 = 1.502 x 10-13 J Stellar Structure: TCD PY3A03: 9. Monday, 17 September 12 Proton rest mass: mp = 1.00728 u E=mpc2 = 1.5027 x 10-10 J + + νe d + 3He + Helium-3 nucleus rest mass: m3He = 3.01603 u Eexcess = δmc2 = (3mp - m3He )c2 = 8.697 x 10-13 J Stellar Structure: TCD PY3A03: 9. Monday, 17 September 12 Proton Helium-4 nucleus rest mass: mp = 1.00728 u rest mass: m4He = 4.00260 u E=mmpc2 = 1.5027 x 10-10 J Eexcess = δmc2 = (4mp - m4He )c2 + + 1.4 x 1010 y d 6s + + 3He 106 y 3He + Stellar Structure: TCD PY3A03: 9. Monday, 17 September 12 J 4He νe d + -12 + + + = 3.956 x 10 =24.7MeV νe + Nuclear energy production Stellar Structure: TCD PY3A03: 9. Monday, 17 September 12 Convection MHD model for opacity-driven convection zone in the Sun showing superganulation -- click above !!! http://www.pa.msu.edu/~steinr/research.html#convection Monday, 17 September 12 4. Solar Models Monday, 17 September 12 Solutions - 3 families Approximate solutions Polytropes Cowling point source model ** Clayton model 1D Numerical Solutions Henyey (difference) schemes (Lagrangian) 3D Numerical Solutions Stellar Structure: TCD PY3A03: 10. Monday, 17 September 12 The Clayton model Guess a form for the Pressure profile P(r) Use this as starting point for an approximate solution. Outline: Tackle equations sequentially, starting with mass continuity and hydrostatic equilibrium to obtain the pressure structure. Use an equation of state to obtain the temperature structure. Introduce an opacity κ(ρ,T) to obtain the power flow from the transport equation . Compare this with the power flow obtained from the nuclear power generation ε(ρ,T) and energy conservation. Stellar Structure: TCD PY3A03: 10. Monday, 17 September 12 Clayton model for 1 solar mass star Stellar Structure: TCD PY3A03: 10. Monday, 17 September 12 Standard solar model Spherically symmetric quasi-static model Constrained by mass, luminosity, radius, age and composition... Necessary to tune physics (eg mixing length for convection, diffusion of 3He, ...) in order to match constraints Model can be tested... Sun is only star for which internal properties can be measured directly... Stellar Structure: TCD PY3A03: Monday, 17 September 12 The Solar Interior Monday, 17 September 12 pulsation across the HR diagram • ~ hours • main-sequence stars: β Cep, PG1716, δ Sct • horizontal-branch stars: RR Lyr, PG1716 • ~ minutes • main-sequence: ro-Ap, Sun-like • hot subdwarfs: EC14026, PG1159 • white dwarfs: DAV, DBV, CVs • < 0.1s • neutron stars ? Monday, 17 September 12 the sun tens of millions of modes periods ~ 2 - 20 min colour shows energy at each frequency for every mode Monday, 17 September 12 add rotation, magnetic fields, 3d convection, ... • differential rotation, mapped by helioseismology • tachocline - shear layer at radiative / convective interface Gough & McIntyre 1998, Garaud & Garaud 2008 Monday, 17 September 12 Measuring neutrinos Helioseismology says solar models are right. Stellar Structure: TCD PY3A03: Monday, 17 September 12 Solar Neutrinos Solution (particle physics): all neutrinos generated in the Sun are electron neutrinos if neutrinos have small mass, they may change type on path from Sun to Earth from electron to µ or τ neutrinos vacuum oscillations: ✘ matter oscillations: ✔ Above: The Sun in neutrinos from Super K2 Stellar Structure: TCD PY3A03: Right: Sudbury Neutrino Observatory Monday, 17 September 12 Other Main-Sequence Stars How do overall properties vary with mass? How does internal structure vary with mass? What happens as hydrogen is converted to helium? What do observations tell us? Stellar Structure: TCD PY3A03: Monday, 17 September 12 F - K dwarfs: 1.8 - 0.7 M About 13% of stars in Galaxy Convective envelopes and radiative cores H-burning mostly by PP-chains Convective envelopes associated with starspots, magnetic fields, and other activity About one third of F - K stars are predicted to have at least one planet Subdwarf F - K stars (sdF, ...) are metal-poor or Population II main-sequence stars. They lie below the main-sequence because the lower opacity makes the temperature gradient steeper, and hence reduces their radii. Well-known F - K stars: The Sun Alpha Centauri Stellar Structure: TCD PY3A03: Monday, 17 September 12 M dwarfs: 0.7 - 0.1 M M dwarfs make up some 80% of stars in the Galaxy They are completely convective As a consequence, M dwarfs have strong magnetic fields, which drives a lot of activity, including giant spots and giant flares and bright X-ray coronae. Hydrogen-burning is by the PP chains; Brown Dwarfs (L, T, ...) are found below H-burning limit, so are not real stars; limited deuterium burning operates. Stellar Structure: TCD PY3A03: Monday, 17 September 12 B - A stars: 20 - 1.8 M A and B stars make up less than 1% of the stars in the Galaxy Exercise: which emits more light in total: B stars or G stars ? A and B stars have convective cores and radiative envelopes. Hydrogen-burning is by the CNO-cycle. They do not have active atmospheres (no spots, flares, etc.) Some have strong surface magnetic fields Some important sub-types: Ap stars: chemically peculiar atmospheres produced by radiatively driven diffusion Be stars: B stars show hydrogen-emission lines, from a shell or disk HgMn stars: B stars with strong lines of mercury and manganese (see Ap stars) Well-known B and A stars: Sirius A Pleiades Stellar Structure: TCD PY3A03: Monday, 17 September 12 O stars: >20 M The brightest main-sequence stars. O stars have convective cores and radiative envelopes. Hydrogen-burning is by the CNO-cycle. O stars are always observed close to sites of current or recent star formation. Because of their high luminosities, O stars have very strong stellar winds, observed as asymmetric emission/absorption lines. Material is driven off the star at velocities of several hundred km s-1, with mass-loss rates of 10-7 10-8 M yr-1. With lifetimes of ~107 yr, an O star can lose most of its mass during its main-sequence lifetime. Well-known O stars: P Cygni Trapezium cluster in Orion Stellar Structure: TCD PY3A03: Monday, 17 September 12 5. Equations of Stellar Evolution Monday, 17 September 12 time So far, none of our equations have used time: t. An equilibrium solution of the time-independent equations is simply a model of stellar structure, not evolution. Why does a star change during its evolution ? Change in size (expansion / contraction) Change in entropy (heating / cooling) Change in chemical composition nucleosynthesis mixing Monday, 17 September 12 size If a region of a star changes in volume rapidly, then the material must obey laws of motion Normally we consider changes in stellar dimensions to be very slow, but in the case of explosions, this equation becomes very important. Monday, 17 September 12 entropy If a region of a star changes in volume it will release or absorb energy according to the 1st law of TD dE = -T dS = - (dU + PdV) Hence the energy equation needs to consider dL/dm = ε – T dS/dt Where ε represents other sources/sinks of energy within a unit volume Monday, 17 September 12 nucleosynthesis Nuclear reactions produce energy and transmute elements from one species to another A reaction i(j,k)l destroys species i and j, and creates species k,l at a rate rij. The rate of change of abundance of species i will then be given (in mass fraction) by dxi/dt = - (1+δij) rij / NA Where the bracket allows for more than one particle of species I being destroyed in a reaction. Thus for the pp chain: dX/dt = -4 rpp / NA dY/dt = - dX/dt Monday, 17 September 12 mixing Convection provides an efficient energy transport process as well as a chemical transport mechanism. In stellar evolution calculations it is normally assumed that a fully convective region is fully mixed. During a timestep, nucleosynthesis may transform the mixture in part of a convection zone, or else a convective region may move to incorporate material of different composition. The new material must then be mixed throughout the convection zone. In 1D, the composition in a convective region <Xi> = ∫ [Xi(m) + δt dXi(m)/dt] dm / ∫ dm BUT.... Monday, 17 September 12 Monday, 17 September 12 6. Future of the Sun and other Stars Monday, 17 September 12 Hertzsprung Russell diagram Monday, 17 September 12 Evolution of a 1 M star Stellar Structure: TCD PY3A03: Monday, 17 September 12 main-sequence stars H H > He Monday, 17 September 12 giant-branch stars H H > He He Monday, 17 September 12 3D convection in a rotating red giant Velocity structures in a rotating model star with convection in equilibrium in its unstable outer half, PPM simulation by Porter, Anderson, Habermann, and Woodward on two 64-processor SGI Origins at NCSA, 1997. Monday, 17 September 12 white dwarfs H He Monday, 17 September 12 horizontal-branch stars H H > He He He > C/O Monday, 17 September 12 asymptotic giant-branch stars H H > He He He > C/O C/O Monday, 17 September 12 planetary-nebula formation Stellar Structure: TCD PY3A03: 13. Monday, 17 September 12 Evolution of a 5 M star Stellar Structure: TCD PY3A03: Monday, 17 September 12 white dwarfs H He C/O Monday, 17 September 12 The Sun’s Final Show Stellar Structure: TCD PY3A03: 13. Monday, 17 September 12 Evolution of 1 M star Stellar Structure: TCD PY3A03: 13. Monday, 17 September 12