Theory of Dust Formation

Transcription

1 Theory of Dust Formation From Giant Stars to Brown Dwarfs How do dust grains form and what do they do in brown dwarf atmospheres? Peter Woitke, Christiane Helling The Scottish Universities Physics Alliance (SUPA), School of Physics and Astronomy, University of St Andrews, Scotland, UK

2 Contents 1. Introduction to Dust in Stellar Atmospheres 2. Dust Processes 2.1 Gravitational Settling 2.2 Growth & Evaporation 2.3 Particle Energetics 2.4 Coagulation 2.5 Nucleation 3. Solution Method 3.1 Moment Method for Nucl., Growth, Evap. & Drift 3.2 Application to Static Stellar Atmospheres 3.3 Mixing by Convection & Overshoot 4. Basic Results 5. Summary

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5 Effects of Dust Particles dust provides strong continuum opacity reduces the line blanketing effect causes a backwarming of deeper layer dominates surface albedo changes the convective stability dust consumes condensable elements strongly metal-deficient gas component spectral appearance of molecular features spectral classification (M, L, T), T eff -scale fundamental parameters, evolutionary state

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7 Little Dust Dictionary stellar community planetary community nucleation growth } nucleation, condensation coagulation { growth coagulation (thermal induced) coalescence (drift-induced) equilibrium drift velocity terminal fall speed

8 2. Basic Processes 2.1 Gravitational Settling equation of motion m s v dr = g m s + F fric (a, ρ, T, v dr ) equilibrium drift g ρ s 4π 3 a3 = F fric (a, ρ, T, v dr) (quick relaxation) Knudsen number Kn = l/(2a) Reynolds number Re = 2a ρ v dr /µ kin F fric = 8 π a 2 ρ c 3 T v dr (Kn 1, v dr c T ) subsonic free molecular flow = πa 2 ρ vdr 2 (Kn 1, v dr c T ) supersonic free molecular flow = 6π a µ kin v dr (Kn 1, Re 1000) laminar viscous flow (Stokes) = 1.3πa 2 ρ v 2 dr (Kn 1, Re 1000) turbulent flow (Newton) implicit equation for v dr = v dr (g, a, ρ s, ρ, T ) First Prev Next Last Go Back Full Screen Close Quit

9 different regimes e. g. at ρ=10 5 g cm 3 : τ sink = H p /v dr =... 8 months (a=0.1 µm) 1/4 hour (a=100 µm) effects of ρ s, T, porosity, non-spherical shapes... reliable results! log(g)=5, T g =1500 K, SiO 2 grains (ρ s =2.65 g cm 3 ) (Woitke & Helling 2003, A&A 399, 297)

10 2.2 Dust Growth & Evaporation for example SiO + H 2 O SiO 2 [s] + H 2 Kn 1, v dr c S : Kn 1, Re 1000 : dv dt = 4π a2 r dv dt = 4π a r V r n r v rel r α r ( 1 1 S r ) V r D r n r ( 1 1 S r )

11 Growth & Evaporation in the Diffusion Limit Consider diffusion of species i with concentration c i = n i /ρ to grain surface t (ρ c i) + ( v gas ρ c i ) = (ρ D i c i ) (1) stationarity (. t = 0), spherical symmetry, v gas = 0, constant ρ and D i : r 2 c i ρ D i r = const i. (2) ( 1 S i = n ) ikt n ikt boundary conditions c i ( ) = n i ρ and c i(a) = n i ρ = n i ρ c i (r) = n i ρ ( 1 a ( 1 1 ) ) (3) r S i particle flux through any sphere = 4πr 2 ρ D i c i r = 4πa D i n i ( 1 1 S i ). volume accretion rate dv dt = 4πa V i D i n i ( 1 1 S i ) S i

12 SiO + H 2 O SiO 2 [s] + H 2 log(g)=5, T g =1500 K, solar ɛ k in gas phase (S ) (Woitke & Helling 2003, A&A 399, 297) τ gr = 4π 3 a3 / dv dt τ sink < τ gr only for a < 100µm (deeper layers) a < 1µm (upper layers) maximum particle size a max in BD atmospheres a < a max, but not a a max! supersonic and turbulent regimes not relevant

13 Solid s Surface reaction TiO 2 [s] TiO 2 TiO 2 [s] rutile TiO + H 2 O TiO 2 [s] + H 2 Ti + 2 H 2 O TiO 2 [s] + 2 H 2 TiS + 2 H 2 O TiO 2 [s] + H 2 S + H 2 SiO 2 [s] SiO 2 SiO 2 [s] silica SiO + H 2 O SiO 2 [s] + H 2 SiS + 2 H 2 O SiO 2 [s] + H 2 S + H 2 SiO[s] SiO SiO[s] SiO 2 + H 2 SiO[s] + H 2 O SiS + H 2 O SiO[s] + H 2 S Fe[s] Fe Fe[s] solid iron FeO + H 2 Fe[s] + H 2 O FeS + H 2 Fe[s] + H 2 S Fe(OH) 2 + H 2 Fe[s] + 2 H 2 O FeO[s] FeO FeO[s] Fe + H 2 O FeO[s] + H 2 FeS + H 2 O FeO[s] + H 2 S Fe(OH) 2 FeO[s] + H 2 O FeS[s] FeS FeS[s] Fe + H 2 S FeS[s] + H 2 FeO + H 2 S FeS[s] + H 2 O Fe(OH) 2 + H 2 S FeS[s] + 2 H 2 O Fe 2 SiO 4 [s] 2 Fe + SiO + 3 H 2 O Fe 2 SiO 4 [s] + 3 H 2 fayalite 2 Fe + SiS + 4 H 2 O Fe 2 SiO 4 [s] + H 2 S + 3 H 2 2 FeO + SiO + H 2 O Fe 2 SiO 4 [s] + H 2 2 FeO + SiS + 2 H 2 O Fe 2 SiO 4 [s] + H 2 S + H 2 2 FeS + SiO + 3 H 2 O Fe 2 SiO 4 [s] + H H 2 S 2 FeS + SiS + 4 H 2 O Fe 2 SiO 4 [s] + H 2 S + 3 H 2 2 Fe(OH) 2 + SiO Fe 2 SiO 4 [s] + H 2 O + H 2 2 Fe(OH) 2 + SiS Fe 2 SiO 4 [s] + H 2 S + H 2 Surface Reactions Solid s Surface reaction MgO[s] MgO MgO[s] periclase Mg + H 2 O MgO[s] + H 2 2 MgOH 2 MgO[s] + H 2 Mg(OH) 2 MgO[s] + H 2 O MgSiO 3 [s] Mg + SiO + 2 H 2 O MgSiO 3 [s] + H 2 enstatite Mg + SiS + 3 H 2 O MgSiO 3 [s] + H 2 S + 2 H 2 2 MgOH + 2 SiS + 4 H 2 O 2 MgSiO 3 [s] + 2 H 2 S + 3 H 2 2 MgOH + 2 SiO + 2 H 2 O 2 MgSiO 3 [s] + 3 H 2 Mg(OH) 2 + SiO MgSiO 3 [s] + H 2 Mg(OH) 2 + SiS + H 2 O MgSiO 3 [s] + H 2 S+ H 2 Mg 2 SiO 4 [s] 2 Mg + SiO + 3 H 2 O Mg 2 SiO 4 [s] + 3 H 2 forsterite 2 Mg + SiS + H 2 O Mg 2 SiO 4 [s] + H 2 S + 3 H 2 2 MgOH + SiO + H 2 O Mg 2 SiO 4 [s] + 2 H 2 2 MgOH + SiS + 2 H 2 O Mg 2 SiO 4 [s] + H 2 S + 2 H 2 2 Mg(OH) 2 + SiO Mg 2 SiO 4 [s] + H 2 O + H 2 2 Mg(OH) 2 + SiS Mg 2 SiO 4 [s] + H 2 + H 2 S Al 2 O 3 [s] 2 AlOH + H 2 O Al 2 O 3 [s] + 2 H 2 aluminia 2 AlH + 3 H 2 O Al 2 O 3 [s] + 4 H 2 Al 2 O + 2 H 2 O Al 2 O 3 [s] + 2 H 2 2 AlS + 3 H 2 O Al 2 O 3 [s] + 2 H 2 S + H 2 2 AlO 2 H Al 2 O 3 [s] + H 2 O CaTiO 3 [s] Ca + TiO + 2 H 2 O CaTiO 3 [s] + 2 H 2 Ca + TiO 2 + H 2 O CaTiO 3 [s] + H 2 CaO + TiO + H 2 O CaTiO 3 [s] + H 2 CaO + TiO 2 CaTiO 3 [s] CaS + TiO + 2 H 2 O CaTiO 3 [s] + H 2 S + H 2 CaS + TiO 2 + H 2 O CaTiO 3 [s] + H 2 S Ca(OH) 2 + TiO CaTiO 3 [s] + H 2 Ca(OH) 2 + TiO 2 CaTiO 3 [s] + H 2 O

14 The Stability Sequence for Solar Composition Gas T S = sublimation temperatures (S = 1) of different solid materials in a dust-free (undepleted) solar composition gas S i = n ikt n i kt = p ikt p vap (T ) supersaturation ratio First Prev Next Last Go Back Full Screen Close Quit

15 2.3 Dust Particle Energetics Q cond + Q fric = Q rad + Q coll implicit equation for T = T d T g (Woitke & Helling 2003, A&A 399, 297) Q cond = dv dt ρ s f H Q fric = α fric F fric v dr Q rad = 4π πa 2 Q abs (a, λ) [ ] B λ (T d ) J λ dλ πa 2 n v th α acc 2k(T d T g ), Kn 1 Q coll = 4π κ a (T d T g ), Kn 1 a < a max : dust temperature increase T < 3 K negligible: T d T g growth not limited by the need to eliminate the heat of condensation log(g)=5, T g =1500 K, SiO 2 grains, J λ =B λ (T g )

16 2.4 Coagulation df(v ) dt = f(v ) V V l f(v ) Ω(V, V ) dv V l f(v )f( V ) Ω( V, V ) dv V = V V, Ω(V, V ) = σ(v, V ) v rel (V, V ) thermal motion (Brownian motion) turbulence (large grains don t follow small eddy motion) drift-induced ( coalescence ) τ coag τ gr, but last active process when S 1. relevant for growth of rain droplets in Earth atmosphere requires detailed knowledge about the number and size of intial particles

17 2.5 Nucleation The Necessity of Nucleation: Earth atmosphere: seed particles aerosols formed by volcano eruptions, in particular H 2 SO 4 -water droplets ( µm) coal burning, e. g. tropic fires, human activities salt particles from ocean decomposition of organic matter tire particles brown dwarfs, giant gas planets: no such mechanism exists, missing surface destiny of dust particles: complete thermal evaporation (including seeds) renewal of seed particles necessary

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20 linear chain of reactions Basics of nucleation theory τgr N 1 A 1 A 2 A N 1 τev N detailed balance 1 τ N ev = 1 τ N 1 gr f(n 1) f(n) A N τ N gr τ N+1 ev A N+1 A Nmax contrained equilibrium ( f(n) = f(1) (N 1) S N 1 f exp G S f G(N) + f G(1) ) RT } ( {{} exp G(N) ) RT f(1)kt where S = is the supersaturation ratio p vap (T )

21 J = Becker-Döring method ( Nmax N=1 τ N gr f(n) ) 1 f(n ) τ N gr where N = critical cluster modifications heterogeneous nucleation Gail & Sedlmayr (1988, A&A 206, 153) thermal non-eq. T d (N) T g Gauger et al. (1990, A&A 235, 345) chemical non-eq. f(1) f(1), etc. Patzer et al. (1998, A&A 337, 847) reliable thermodynamical cluster data f G(N) classical nucleation theory N f G S f G(N) σa N

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23 Stability and Nucleation Within a few 100 K, many refractory solid materials become thermodynamically stable. However, nucleation requires strongly supersaturated gas efficient nucleation occurs at much lower temperatures, where different solid species can grow simultaneously on the surface of first seed particles. formation of heterogeneous ( dirty ) grains can be expected

24 3. Solution Method 3.1 Moment Method for Nucl., Growth, Evap. and Drift assumptions: spherical grains with macroscopic properties (e. g. reactive surface a 2 ), neglection of coagulation ( V r V ), T d (a)=const, S r (a)=const ρl j ( x, t) = f(v, x, t) V j/3 dv V l ( ) ( [ vgas f(v ) dv + + v t dr(v ) ] f(v ) dv) = ( R R + R R ) dv (1) ( ) ( ) ρlj + vgas ρl j = R k V j/3 dv t V k } l {{} effect of surface reactions V l f(v ) V j/3 v dr(v ) dv } {{ } effect of size-dependent particle drift (2)

25 a) subsonic free molecular flow (Kn 1 v dr c T) R dv R dv = r = r f(v ) dv 4π[a(V )] 2 n r v rel r α r (3) f(v V r ) dv 4π[a(V V r )] 2 n r v rel r α r (4) R dv = r f(v + V r ) dv 4π[a(V )] 2 n r v rel r α r 1 S r (5) R dv = f(v ) dv 4π[a(V V r )] 2 n r v rel 1 r α r (6) S r r subsonic equilibrium drift v πgρs a dr = e r (7) 2ρ c T ( ) ( ) ρlj + vgas ρl j = j/3 Vl J(V l ) + j ( ) Lj+1 t 3 χnet lkn ρl j 1 + ξ lkn e r (8) c T with χ net lkn = 3 36π r V r n r v rel r α r ( 1 1 S r ) ξ lkn = π 2 ( ) 1/3 3 g ρ s 4π

26 b) laminar viscous flow (Kn 1 Re d <1000) R dv R dv = r = r f(v ) dv 4πa(V ) n r D r (9) f(v V r ) dv 4πa(V V r ) n r D r (10) R dv = r f(v + V r ) dv 4πa(V ) n r D r 1 S r (11) R dv = 1 f(v ) dv 4πa(V V r ) n r D r (12) S r r equilibrium drift according to Stokes friction v dr = 2 g ρ s a 2 e r (13) 9 µ kin ( ) ( ) ρlj + vgas ρl j = j/3 Vl J(V l ) + j ( ) ρlj+2 t 3 χnet skn ρl j 2 + ξ skn e r µ kin with χ net skn = 3 48π 2 r V r n r D r ( 1 1 S r ) ξ skn = 2 9 ( ) 2/3 3 g ρ s 4π (14)

27 3.2 Application to Static Stellar Atmospheres subsonic free molecular flow ( Lj+1 ( ) ) ρlj + ( vgas ρl j = j/3 Vl J + j ) t 3 χnet lkn ρl j 1 + ξ lkn e r (1) c T ( ) ) R ( n H ɛ i + ( vgas n H ɛ i = νi,0 N l J 3 36π ρl2 ν i,r n r vr rel α r 1 1 )(2) t S r static ( v gas = 0), plane-parallel ( ( e r ) = d dz ( )), and stationary ( t r=1 ( ) = 0) d ( ) Lj+1 = 1 ( V j/3 l J + j ) dz c T ξ lkn 3 χnet lkn ρl j 1 R ( 0 = ν i,0 N l J π ρl2 ν i,r n r vr rel α r 1 1 ) (3) S r only trivial solution, namely the dust-free and saturated atmosphere: r=1 L j = 0 S r 1

28 The Convective Nature of Dust Formation in Ultra-Cool, Compact Atmospheres The Life Cycle of Dust (Helling 2003, Review in Modern Astronomy 19, 114)

29 3.3 Mixing by Convection & Overshoot Numerical 3D simulations of convection in late M-type dwarfs (Ludwig, Allard & Hauschildt, 2002, AA 395, 99): mixing ( mass exchange ) timescale: log τ mix (z) log τ min mix + β max τ min mix αh p/v max MLT, β 2.2 { 0, log p 0 log p(z) }

30 Assumption: convective mixing with uncondensed matter from the deep interior of the BD with abundances ɛ 0 i on time-scale τ mix(z) system of ordinary differential equations for j = 0, 1, 2,... d dz ( Lj+1 c T ) = 1 ( ρl j ξ lkn τ mix (z) + V l j/3 J + j ) 3 χnet lkn ρl j 1 (4) with one algebraic auxiliary condition for each involved element i n H (ɛ 0 i ɛ i) τ mix (z) = ν i,0 N l J π ρl2 R r=1 ν i,r n r v rel r α r ( 1 1 S r ) (5) which can be included e. g. into classical stellar atmosphere codes. closure condition L 0 = F(L 1, L 2, L 3,...) required inward drift motion inward integration natural outer boundary condition required (here L j (z max ) = 0, i. e. dust-free) First Prev Next Last Go Back Full Screen Close Quit

31 4. Simple Application to Static Cloud Structure one example refractory solid species TiO 2 (rutile) with homogeneous classical nucleation theory and surface growth reactions TiO 2 TiO 2 [s] (r =1) TiO + H 2 O TiO 2 [s] + H 2 (r =2) prescribed atmospheric gas stratification (Tsuji 2002, ApJ 575, 264) for T eff =1400 K, log(g)=5, solar abundances First Prev Next Last Go Back Full Screen Close Quit

32 Structure of a Quasi-Static Cloud Layer T eff =1400 K, log(g)=5, τmix min =300 s, β = dust-poor, depleted gas few dust particles (τ sink < τ mix ), strongly depleted gas phase (ɛ gas Ti 10 6 ɛ 0 Ti ) (Woitke & Helling 2003, A&A, 414, 335) First Prev Next Last Go Back Full Screen Close Quit

33 Structure of a Quasi-Static Cloud Layer T eff =1400 K, log(g)=5, τmix min =300 s, β = dust-poor, depleted gas few dust particles (τ sink < τ mix ), strongly depleted gas phase (ɛ gas Ti 10 6 ɛ 0 Ti ) I. region of efficient nucleation maximum nucleation rate (S 1), very small particles ( a 0.01µm) (Woitke & Helling 2003, A&A, 414, 335) First Prev Next Last Go Back Full Screen Close Quit

34 Structure of a Quasi-Static Cloud Layer T eff =1400 K, log(g)=5, τmix min =300 s, β = dust-poor, depleted gas few dust particles (τ sink < τ mix ), strongly depleted gas phase (ɛ gas Ti 10 6 ɛ 0 Ti ) I. region of efficient nucleation maximum nucleation rate (S 1), very small particles ( a 0.01µm) II. dust growth region inward increase of mean particle size, increase of molecular concentrations, vanishing nucleation rate (Woitke & Helling 2003, A&A, 414, 335) First Prev Next Last Go Back Full Screen Close Quit

35 Structure of a Quasi-Static Cloud Layer T eff =1400 K, log(g)=5, τmix min =300 s, β = dust-poor, depleted gas few dust particles (τ sink < τ mix ), strongly depleted gas phase (ɛ gas Ti 10 6 ɛ 0 Ti ) I. region of efficient nucleation maximum nucleation rate (S 1), very small particles ( a 0.01µm) II. dust growth region inward increase of mean particle size, increase of molecular concentrations, vanishing nucleation rate III. drift dominated region a a crit const ( 100µm), ρ d /ρ g decreases (dynamic dilution), dust growth not exhaustive (S > 1 ɛ gas Ti ɛ 0 Ti ) (Woitke & Helling 2003, A&A, 414, 335) First Prev Next Last Go Back Full Screen Close Quit

36 Structure of a Quasi-Static Cloud Layer T eff =1400 K, log(g)=5, τmix min =300 s, β = dust-poor, depleted gas few dust particles (τ sink < τ mix ), strongly depleted gas phase (ɛ gas Ti 10 6 ɛ 0 Ti ) I. region of efficient nucleation maximum nucleation rate (S 1), very small particles ( a 0.01µm) II. dust growth region inward increase of mean particle size, increase of molecular concentrations, vanishing nucleation rate III. drift dominated region a a crit const ( 100µm), ρ d /ρ g decreases (dynamic dilution), dust growth not exhaustive (S > 1 ɛ gas Ti ɛ 0 Ti ) IV. evaporation region S < 1 χ < 0 i. e. particles shrink (thermal evaporation), finite extension of cloud layer below cloud base (Woitke & Helling 2003, A&A, 414, 335) First Prev Next Last Go Back Full Screen Close Quit

37 Sketch of Cloud Structure First Prev Next Last Go Back Full Screen Close Quit

38 5. Summary 5.1 Physical Processes important processes are: nucleation, growth, evaporation, coagulation, drift, mixing T d = T g is a reasonable assumption quasi-static atmospheres: drift differential equations convection dust formation, in particular: no dust without mixing, dust forms in upwinds/convective overshoots nucleation must take place, requires S 1, phase equilibrium not valid

39 5.2 Summary Cloud Structure cloud deck insufficient mixing (Earth: transition troposphere to stratosphere) cloud base thermal stability, but big evaporating grains exist even below the cloud base hierarchical structure of cloud layer nucleation growth drift evaporation inward increasing metal abundances in the gas phase (-7 to -5) orders of mag.... (0 to +0.5) orders of mag. inward increasing mean dust particle size a (z) < 0.01µm... > 100µm

40 5.3 Conclusions and Outlook Need for more detailed brown dwarf atmosphere models inclusion of kinetic treatment of dust formation (static 1D) better treatment of convective mixing (static 1D) influence of turbulence on dust formation (effect of small-scale perturbations, both 1D and 3D) time-dependent HD simulations (3D)...

41 5.4 Differences between AGB winds and BD atmospheres red giants brown dwarfs Knudsen numbers very small small large free molecular flow free viscous case important force radiation pressure gravity direction outward inward temperatures T d T g T d T g formation hierarchy warm cold cold warm

42 Possible closure condition simple: power law of the dust moments: y(j) = αj β with y(j) = L j L j 1 more general: size distribution function f(v ) is approximated by a weight function Ψ α (V ), which describes the basic shape of f(v ), modified by a sum of orthogonal polynomials {p k (V )} (k = 0, 1, 2,..., n): f(v, x, t) = Ψ α (V ) n a k ( x, t) p α k(v ). k=0 a k polynomial coefficients α = α( x, t) additional parameter of Ψ α and a k (k = 0, 1, 2,..., n) are stepwise adjusted in order to fit the known moments L j (j =0, 1, 2,..., n) exactly, using the orthogonality relation Ψ α (V ) p α i (V ) p α k(v ) dv = δ ik (Deuflhard & Wulkow 1989, Wulkow 1992) [back]

43 Variation of Convection Parameters T eff =1400 K, log(g)=5, τmix min =2000 s, β =2.2 T 1. Minimum mixing time-scale τ min mix eff =1400 K, log(g)=5, τ min mix =10 7 s, β =2.2

44 Variation of Convection Parameters 1. Minimum mixing time-scale τ min mix

45 Variation of Convection Parameters 2. Slope of τ mix (z)