2. Basic Assumptions for Stellar Atmospheres

Transcription

1 2. Basic Assumptions for Stellar Atmospheres 1. geometry, stationarity 2. conservation of momentum, mass 3. conservation of energy 4. Local Thermodynamic Equilibrium 1

2 1. Geometry Stars as gaseous spheres! spherical symmetry Exceptions: rapidly rotating stars Be stars v rot = km/s (Sun v rot = 2 km/s) John Monnier (CHARA/MIRC) 2

3 1. Geometry Stars as gaseous spheres! spherical symmetry Exceptions: rapidly rotating stars Be stars vrot = km/s (Sun vrot = 2 km/s) Gray 2005 For stellar photospheres typically: Δr/R << 1 3

4 1. Geometry Stars as gaseous spheres! spherical symmetry Exceptions: rapidly rotating stars Be stars vrot = km/s (Sun vrot = 2 km/s) Gray 2005 For stellar photospheres typically: Δr/R << 1 Sun: Ro = 700,000 km photosphere Δr = 500 km Δr/R = 7 x 10-4 chromosphere Δr = 3000 kmδr/r = 4 x 10-3 corona Δr/R ~ 3 4

5 As long as Δr/R << 1 : plane-parallel symmetry consider a light ray through the atmosphere: Δr/R << 1 Δr/R ~ 1 path of light lines of constant temperature and density Plane-parallel symmetry very small curvature (α β) Solar photo/cromosphere, dwarfs, (giants) Spherical symmetry significant curvature (α β) Solar corona, supergiants, expanding envelopes (OBA stars), novae, SN 5

6 Homogeneity & stationarity We assume the atmosphere to be homogeneous Counter-examples:? 6

7 Homogeneity & stationarity We assume the atmosphere to be homogeneous Counter-examples: sunspots, granulation, non-radial pulsations clumps and shocks in hot star winds magnetic Ap stars 7

8 Homogeneity & stationarity We assume the atmosphere to be homogeneous Counter-examples: sunspots, granulation, non-radial pulsations clumps and shocks in hot star winds magnetic Ap stars and stationary Most spectra are time-independent: / t = 0 (if we don t look carefully enough ) Exceptions:? 8

9 Homogeneity & stationarity We assume the atmosphere to be homogeneous Counter-examples: sunspots, granulation, non-radial pulsations clumps and shocks in hot star winds magnetic Ap stars and stationary Most spectra are time-independent: / t = 0 (if we don t look carefully enough ) Exceptions: explosive phenomena (SN), stellar pulsations, magnetic stars, mass transfer in close binaries 9

10 2. Conservation of momentum & mass consider a mass element dm in a spherically symmetric atmosphere The acceleration of the mass element results from the sum of all forces acting on dm, according to Newton s 2 nd law: d dm v(r,t) = df = F dt i i 10

11 2a. Hydrostatic equilibrium d dm v(r,t) = df = F dt i i Assuming a hydrostatic stratification: v(r,t)=0 0 = df i i 11

12 2a. Hydrostatic equilibrium dm d v(r,t) = df i = F dt i Assuming a hydrostatic stratification: v(r,t)=0 0 = i df i 12

13 2a. Hydrostatic equilibrium d dm v(r,t) = df = F dt i i Assuming a hydrostatic stratification: v(r,t)=0 0 = df i i Mr dm Gravitational forces: dfgrav = G = g( r) dm 2 r 13

14 2a. Hydrostatic equilibrium d dm v(r,t) = df = F dt i i Assuming a hydrostatic stratification: v(r,t)=0 0 = df i i Mr dm Gravitational forces: dfgrav = G = g( r) dm 2 r dp Gas Pressure forces: df p, gas = A P( r + dr) + A P( r) = A dr dr 14

15 2a. Hydrostatic equilibrium d dm v(r,t) = df = F dt i i Assuming a hydrostatic stratification: v(r,t)=0 0 = df i i Mr dm Gravitational forces: dfgrav = G = g( r) dm 2 r dp Gas Pressure forces: df p, gas = A P( r + dr) + A P( r) = A dr dr Radiation forces: df = g ( r) dm = absorbed photon momentum rad rad dt 15

16 In equilibrium: dp 0 = dfi = g( r) dm A dr + g i raddm dr and substituting dm = A ρ dr: dp g( r) ρ Adr A dr + grad ρ Adr = 0 dr dp dr = ρ( r)[ g( r) g rad ] Hydrostatic equilibrium in spherical symmetry 16

17 Approximation for g(r) The mass within the atmosphere M(r) M(R) << M(R)=M * g(r) = G M r / r 2 M(R) = M(r) = M * g(r) = G M * /r 2 17

18 Approximation for g(r) The mass within the atmosphere M(r) M(R) << M(R)=M * g(r) = G M r / r 2 M(R) = M(r) = M * g(r) = G M * /r 2 Example: take a geometrically thin photosphere 4π 4π Δr ΔM phot ρ R + Δr R = R + R 3 3 R 4π 3 Δr 2 ρ R [ ] = ρ 4π R Δr 3 R [( ) ] ρ [ (1 ) ] 18

19 Example: the Sun R = 7 x cm, Δr = 3 x 10 7 cm, ρ ~ m H N = 1.7 x x g/cm 3 : Δ M phot ρ 4π R 2 Δr ΔM phot = 3 x g ΔM phot /M =?? 19

20 Example: the Sun R = 7 x cm, Δr = 3 x 10 7 cm, ρ ~ m H N = 1.7 x x g/cm 3 : Δ M phot ρ 4π R 2 Δr ΔM phot = 3 x g ΔM phot /M =

21 Example: the Sun R = 7 x cm, Δr = 3 x 10 7 cm, ρ ~ m H N = 1.7 x x g/cm 3 : Δ M phot ρ 4π R 2 Δr ΔM phot = 3 x g ΔM phot /M = Moreover in plane-parallel symmetry: Δr/R << 1 g(r) = const g = G M * /R 2 Surface gravity log(g): critical parameter in stellar model atmospheres Main sequence log g = 4 (cgs: cm/s 2 ) supergiant 3.0 to 0.0 white dwarf 8 Sun 4.44 Earth

22 Red Giants Subgiants Main-sequence stars Huber et al. (2013) 22

23 Barometric formula Goal: derive basic solution for density profile in atmosphere based on hydrostatic equilibrium Assume: plane-parallel symmetry g rad << g dp dr = ρ( r) g( r) 23

24 Barometric formula Goal: derive basic solution for density profile in atmosphere based on hydrostatic equilibrium Assume: plane-parallel symmetry g rad << g dp dr = ρ( r) g( r) Ideal Gas: P ρkt ρrt = = = N gkt m µ µ H Sound Speed: k: Boltzmann constant = 1.38 x erg K -1 m H : mass of H atom = 1.66 x g µ: mean molecular weight per free particle = <m> / m H R = k N A = x 10 7 erg/mole/k 24

25 Barometric formula dp dr = ρ( r) g( r) k dρ dt [ T + ρ ] = gρ m µ dr dr H 25

26 Barometric formula dp dr = ρ( r) g( r) k dρ dt [ T + ρ ] = gρ m µ dr dr H Assume 1 dρ 1 dt >> ρ dr T dr 26

27 Barometric formula dp dr = ρ( r) g( r) k dρ dt [ T + ρ ] = gρ m µ dr dr H Assume 1 dρ 1 dt >> ρ dr T dr 27

28 Barometric formula dp dr = ρ( r) g( r) k dρ dt [ T + ρ ] = gρ m µ dr dr H Assume kt 1 dρ 1 dt [ + ] = g m µ ρ dr T dr H 1 dρ 1 dt >> ρ dr T dr 1 ρ dρ dr = gm H µ kt and: scale height 28

29 Barometric formula dp dr = ρ( r) g( r) k dρ dt [ T + ρ ] = gρ m µ dr dr H Assume kt 1 dρ 1 dt [ + ] = g m µ ρ dr T dr H 1 dρ 1 dt >> ρ dr T dr 1 ρ dρ dr = gm H µ kt and: scale height solution: ρ( r) = ρ( r ) e 0 r r H 0 29

30 Barometric formula dp dr = ρ( r) g( r) k dρ dt [ T + ρ ] = gρ m µ dr dr H Assume kt 1 dρ 1 dt [ + ] = g m µ ρ dr T dr H 1 dρ 1 dt >> ρ dr T dr 1 ρ dρ dr = gm H µ kt and: scale height solution: ρ( r) = ρ( r ) e 0 r r H 0 H(Sun) = 150 km 30

31 dp dr = ρ( r) g( r) ρ( r) = ρ( r ) e 0 r r H 0 Approximate solution: Barometric formula exponential decline of density scale length H ~ T/g explains why atmospheres are so thin 31

32 Conservation of Mass & Momentum Let us now consider the case in which there are deviations from hydrostatic equilibrium and v(r,t) 0. Newton s law: d dm v(r,t) = dfi dt i 32

33 Conservation of Mass & Momentum Let us now consider the case in which there are deviations from hydrostatic equilibrium and v(r,t) 0. Newton s law: d v( r + Δ r, t + Δt) v( r, t) v(r,t) = lim dt Δt 0 Δt d dm v(r,t) = dfi dt i Taylor expansion v v v( r + Δ r, t + Δ t) = v( r, t) + Δ r + Δt r t and Δ r = v Δt 33

34 Conservation of Mass & Momentum Let us now consider the case in which there are deviations from hydrostatic equilibrium and v(r,t) 0. Newton s law: d v( r + Δ r, t + Δt) v( r, t) v(r,t) = lim dt Δt 0 Δt d dm v(r,t) = dfi dt i Taylor expansion v v v( r + Δ r, t + Δ t) = v( r, t) + Δ r + Δt r t and Δ r = v Δt v v v Δ t + Δt d = lim v v v(r,t) r t = + v dt Δt 0 Δt t r In general ("material derivative"): 34

35 For the assumption of stationarity / t = 0 d dt v(r,t) v = v r 35

36 For the assumption of stationarity / t = 0 d dt v(r,t) v = v r d v dm v(r,t) = dm v = df dt r i i 36

37 For the assumption of stationarity / t = 0 d dt v(r,t) v = v r d v dm v(r,t) = dm v = df dt r i i dm = Aρ dr 37

38 For the assumption of stationarity / t = 0 d dt v(r,t) v = v r d v dm v(r,t) = dm v = df dt r i i dp dm = Aρ dr g( r) dm A dr + g rad dm dr 38

39 For the assumption of stationarity / t = 0 d dt v(r,t) v = v r d v dm v(r,t) = dm v = df dt r i i dp dm = Aρ dr g( r) dm A dr + g rad dm dr dp dr dv = ρ[ g( r) grad + v ] dr new term 39

40 Equation of continuity (mass conservation) d dt. 2 M = M = 4π r ρ v v = 4π. M r 2 ρ 40

41 Equation of continuity (mass conservation) d dt. 2 M = M = 4π r ρ v v = 4π. M r 2 ρ.. dv 2 M M dρ 2v v dρ = = dr 4π r ρ 4π r ρ dr r ρ dr ρ v dv v 2 d 2 ρ = 2ρ v dr r dr 41

42 Equation of continuity (mass conservation) d dt. 2 M = M = 4π r ρ v v = 4π. M r 2 ρ.. dv 2 M M dρ 2v v dρ = = dr 4π r ρ 4π r ρ dr r ρ dr dp dv From: = ρ[ g( r) grad + v ] dr dr and from the equation of state: dp dr kt dρ 2 = = v soun d mh µ dr dρ dr ρ v dv v 2 d 2 ρ = 2ρ v dr r dr Assuming again that the temperature gradient is small 42

43 2 2 2 dρ v ( vsound v ) = ρ[ g( r) grad 2 ] dr r Hydrodynamic equation of motion 2 mvesc ( r) mm 2 with escape velocity : = > G = mrg vesc ( r) = 2gr 2 r 43

44 2 2 2 dρ v ( vsound v ) = ρ[ g( r) grad 2 ] dr r Hydrodynamic equation of motion 2 mvesc ( r) mm 2 with escape velocity : = > G = mrg vesc ( r) = 2gr 2 r re-write g ( v ) ( )[1 4 ] dρ rad v sound v = ρg r 2 dr g( r) vesc ( r) When v << v sound : practically hydrostatic solution (v = 0) for density stratification. This is reached well below the sonic point (where v = v sound ). example: v sound = 6 km/s for solar photosphere, 20 km/s for O stars v esc = 100 to 1000 km/s for main sequence and supergiant stars! (v sound /v esc ) 2 << 1 44

45 Recap from last lecture: Δr/R << 1: Plane-parallel symmetry Solar photo/chromosphere, dwarfs, (giants) Δr/R ~ 1: Spherical symmetry Solar corona, supergiants, expanding envelopes (OBA stars), novae, SN for geometrically thin atmosphere: Δ M phot ρ 4π R 2 Δr Example: Solar Photosphere: Δr/Rsun ~ 7 x 10-4 ΔM/Msun ~

46 Recap from last lecture: Δr/R << 1: Plane-parallel symmetry Solar photo/chromosphere, dwarfs, (giants) Δr/R ~ 1: Spherical symmetry Solar corona, supergiants, expanding envelopes (OBA stars), novae, SN for geometrically thin atmosphere: Δ M phot ρ 4π R 2 Δr Example: Solar Photosphere: Δr/Rsun ~ 7 x 10-4 ΔM/Msun ~ Surface Gravity: g(r) = const g = G M * /R * 2 Star log(g) A dwarf:? K subgiant:? B supergiant:? White dwarf:? 46

47 Recap from last lecture: Δr/R << 1: Plane-parallel symmetry Solar photo/chromosphere, dwarfs, (giants) Δr/R ~ 1: Spherical symmetry Solar corona, supergiants, expanding envelopes (OBA stars), novae, SN for geometrically thin atmosphere: Δ M phot ρ 4π R 2 Δr Example: Solar Photosphere: Δr/Rsun ~ 7 x 10-4 ΔM/Msun ~ Surface Gravity: g(r) = const g = G M * /R * 2 Star log(g) A dwarf: ~4.0 K subgiant: ~3.5 B supergiant: ~0.0 White dwarf: ~8.0 47

48 Recap from last lecture: dp dr = ρ( r)[ g( r) g rad ] Hydrostatic equilibrium 48

49 Recap from last lecture: dp dr = ρ( r)[ g( r) g rad ] Hydrostatic equilibrium ρ( r) = ρ( r ) e 0 r r H 0 Barometric Formula 49

50 Recap from last lecture: dp dr = ρ( r)[ g( r) g rad ] Hydrostatic equilibrium ρ( r) = ρ( r ) e 0 r r H 0 Barometric Formula g ( v ) ( )[1 4 ] dρ rad v sound v = ρg r 2 dr g( r) vesc ( r) Hydrodynamic equation of motion 50

51 3. Conservation of energy Stellar interior: production of energy via nuclear reactions Stellar atmosphere: negligible production of energy the energy flux is conserved at any given radius 51

52 3. Conservation of energy Stellar interior: production of energy via nuclear reactions Stellar atmosphere: negligible production of energy the energy flux is conserved at any given radius In spherical symmetry: r 2 F(r) = const F ( r) = energy area time F(r) ~ 1/r π ( ) r F r = const = luminosity L Plane-parallel: r 2 ~ R 2 ~ const F(r) ~ const 52

53 4. Concepts of Thermodynamics Radiation field Consider a closed cavity in thermodynamic equilibrium TE (photons and particles in equilibrium at some temperature T).! The specific intensity emitted (through a small hole) is (energy per area, per unit time, per unit frequency, per unit solid angle) (Planck function): 3 2hν ν ν = ν ( ) ν = ν (erg cm s Hz sr ) 2 hν / kt I d B T d d c e 1 53

54 4. Concepts of Thermodynamics Radiation field Consider a closed cavity in thermodynamic equilibrium TE (photons and particles in equilibrium at some temperature T).! The specific intensity emitted (through a small hole) is (energy per area, per unit time, per unit frequency, per unit solid angle) (Planck function): 3 2hν ν ν = ν ( ) ν = ν (erg cm s Hz sr ) 2 hν / kt I d B T d d c e 1 black body radiation: universal function dependent only on T not on chemical composition, direction, place, etc. Note: stars don t radiate as blackbodies, since they are not closed systems!!! But their radiation follows the Planck function qualitatively 54

55 Properties of Planck function (a) B ν (T 1 ) > B ν (T 2 ) (monotonic) for every T 1 > T 2 (no function crossing) Rybicki & Lightman, 79 55

56 Properties of Planck function (a) B ν (T 1 ) > B ν (T 2 ) (b) ν max /T = const (monotonic) for every T 1 > T 2 (no function crossing) Rybicki & Lightman, 79 56

57 Properties of Planck function (a) B ν (T 1 ) > B ν (T 2 ) (monotonic) for every T 1 > T 2 (no function crossing) (b) ν max /T = const (c) Stefan-Boltzmann law: F = π B ( T ) dν = σ T 0 ν 4 σ = ergcm 2 s 1 K 4 Rybicki & Lightman, 79 57

58 Concepts of Thermodynamics Gas particles 1. Velocity distribution In complete equilibrium, the velocity distribution of particles is given by MaxwellBoltzmann distribution (needed, e.g., for collisional rates): m f (v)dv = 4π v 2π kt 2 3/ 2 e mv2 2 kt dv 1/ 2 most probable speed: vp = (2kT m ) 1/ 2 average speed: v = (8kT π m ) 1/ 2 r.m.s. speed: vrms = (3kT m ) Gray

59 2. Energy level distribution upper level: u lower level: l 59

60 2. Energy level distribution Boltzmann s equation for the population density of excited states in thermodynamic equilibrium (TE) upper level: u n n ( Eu El ) u u kt l = g e g l g u, g l : statistical weights = 2J+1: number of degenerate states (J=inner quantum number); g u = 2n 2 for Hydrogen lower level: l n n En n n kt tot = i g e u u = gie Ei kt partition function 60

61 3. Kirchhoff s laws For a thermal emitter in TE εν emission coefficient ν ( ) κ = absorption coefficient = B T ν Direct consequence of Boltzmann s equation for excitation and Maxwell s velocity law. 61

62 3. Kirchhoff s laws For a thermal emitter in TE εν emission coefficient ν ( ) κ = absorption coefficient = B T ν Direct consequence of Boltzmann s equation for excitation and Maxwell s velocity law. 62

63 Local Thermodynamic Equilibrium A star as a whole (or a stellar atmosphere) is far from being in thermodynamic equilibrium: energy is transported from the center to the surface, driven by a temperature gradient. But for sufficiently small volume elements dv, we can assume TE to hold at a certain temperature T(r) Local Thermodynamic Equilibrium (LTE) 63

64 Local Thermodynamic Equilibrium A star as a whole (or a stellar atmosphere) is far from being in thermodynamic equilibrium: energy is transported from the center to the surface, driven by a temperature gradient. But for sufficiently small volume elements dv, we can assume TE to hold at a certain temperature T(r) Local Thermodynamic Equilibrium (LTE) good approx: stellar interior (high density, small distance travelled by photons, nearly-isotropic, thermalized radiation field) bad approx: gaseous nebulae (low density, non-local radiation field, optically thin) stellar atmospheres: optically thick but moderate density 64

65 Local Thermodynamic Equilibrium 1. Energy distribution of the gas determined by local temperature T(r) appearing in Maxwell s/ Boltzmann s equations, and Kirchhoff s law 65

66 Local Thermodynamic Equilibrium 1. Energy distribution of the gas determined by local temperature T(r) appearing in Maxwell s/ Boltzmann s equations, and Kirchhoff s law 2. Energy distribution of the photons photons are carriers of non-local energy through the atmosphere! I ν (r) B ν [T(r)] I ν is a superposition of Planck functions originating at different depths in the atmosphere! radiation transfer 66

67 LTE vs NLTE LTE each volume element separately in thermodynamic equilibrium at temperature T(r) 1. f(v) dv = Maxwellian with T = T(r) 2. Saha: (n p n e )/n 1 ~ T 3/2 exp(-hν 1 /kt) 3. Boltzmann: n i / n 1 = g i / g 1 exp(-hν 1i /kt) 67

68 LTE vs NLTE LTE each volume element separately in thermodynamic equilibrium at temperature T(r) 1. f(v) dv = Maxwellian with T = T(r) 2. Saha: (n p n e )/n 1 ~ T 3/2 exp(-hν 1 /kt) 3. Boltzmann: n i / n 1 = g i / g 1 exp(-hν 1i /kt) However: volume elements not closed systems, interactions by photons LTE non-valid if absorption of photons disrupts equilibrium 68

69 LTE vs NLTE NLTE if rate of photon absorptions?? rate of electron collisions I ν (T) ~ T α, α > 1?? n e T 1/2 69

70 LTE vs NLTE NLTE if rate of photon absorptions >> rate of electron collisions I ν (T) ~ T α, α > 1 >> n e T 1/2 LTE valid: low temperatures & high densities non-valid: high temperatures & low densities 70

71 LTE vs NLTE for hot stars Kudritzki