Big, Cool, and Losing mass: Dependence of mass loss rates on L, R, M and Z Examples of mass loss formulae 4x0-3 L R M - Reimers 975 for red giants 4x0-2 0^(0.023P) Vassiliadis & Wood 993 for Miras 2.2x0-6 (L/0 4 ).05 (T eff /3500) -6.3 van Loon et al. 2005 for LMC high-l sources L. A. Willson Iowa State University 7.2x0 8 T eff 6.8 L 2.47 M -.95 Wachter et al. 2002 from models for carbon stars 8x0-4 LR/M (T eff /4000) 3.5 (+g Sun /4300g * ) a modified Reimers relation Schröder & Cuntz 2005 More modified Reimers formulae Mdot =.8e-2 (M Zams /8) LR/M Volk & Kwok 988 Mdot =.5e-3! BVK LR/M Bryan, Volk & Kwok 990 with! BVK = M ZAMS 2-0.6M ZAMS + 0.2 Mdot = 4e-3 (M e,o /M e ) LR/M Baud & Habing 983 Mdot Bi = 4.8x0-9 M Zams -2. L 2.7 MdotR Blöcker 995 Mdot Bi = 4.8x0-9 M -2. L 2.7 MdotR (Blöcker gave two versions) Compare formulae for Mdot = -dm/dt: Reduce to Mdot(L,M) or Mdot(L,M,Z) using (as needed) P(M,R), T eff (L,R), R(L,M,Z)*. Find the deathline Mdot/M = Ldot/L* Near the deathline, find dlogmdot/dlogl and dlogmdot/dlogm *from evolutionary models. First iteration: Steady (mean) dlogl/dt. Later: Shell flash modulations included.
Reimers VW vanloon SC Wachter BW logmdot 0-2 -4-6 -8-0 3 3.5 logl 4 4.5 5 What models and observations tell us about Mdot. Some observations, most models => which stars lose mass, i.e. P, L, R, M or a combination of these for which Mdot/M! Ldot/L or t Mdot! t nuclear 2. Different observations, most models => dlogmdot/dlogl holding M fixed and constraining R; dlogmdot/dlogm holding L and R fixed. Where the mass-losing AGB stars are found For the superwind phase, near L crit nearly all the action takes place in the Death Zone, where 0. < (Mdot/M)/(Ldot/L) < 0 There are 3 important numbers to compare a mass loss formula with observations The L crit (M, Z) where Mdot/M = Ldot/L for stars evolving up an evolutionary track R(L, M, Z) The slope dlogmdot/dlogl at L crit Therefore. The slope dlogmdot/dlogm at L crit
Constraining those numbers L crit : Relatively easy to constrain, nearly all observational relations give this dlogmdot/dlogl at L c : Look to distributions N(L), N(P), N(Mdot) to constrain this one; equivalent = duration of mass loss epoch (time from 0. to 0 x Mdot c or from 0-7 to 0-5 solar masses/yr for AGB stars) dlogmdot/dlogm: Also affects the duration and distributions L crit for Reimers, VW and BW loglcrit 4. 4 3.9 3.8 3.7 3.6 3.5 solid: BW dashed: VW dotted: R 3.4 0.6 0.8.2.4.6.8 2 2.2 Mass The deathzone: 0. to 0 x Mdot crit = M/t nuc Adding lower-metallicity models 3lawsLcritAndSlope LogL 4.2 4 3.8 3.6 3.4 " = (dlogmdot/dlogl) - VW loglcrit VW loglcrit+! VW loglcrit-! BW loglcrit BW loglcrit+! BW loglcrit-! RR Lcrit logl 4.8 4.6 4.4 4.2 4 3.8 3.6 3.4 Z /Sun = 0.00 VW loglcrit VW loglcrit+! VW loglcrit-! BW loglcrit BW loglcrit+! BW loglcrit-! Z /Sun = 0.0 0. 0.3 Short-P Miras have lower mass, lower Z progenitors 3.2 0.6 0.8.2.4.6.8 2 2.2 Mass 3.2 0.5.5 2 2.5 3 Mass
Reverse the axes and add tracks We know where stars die Cliff or DeathLine 0.6 0.6 4 log"= -0-8 -6-4 logm 0.4 0.2 0 2 M Sun M Sun logm 0.4 0.2 0.0 2.8 2.4 Chandrasekhar limit -0.2 core mass needed -0.2 0.7 upper limit to core mass -0.4 3 3.5 4 4.5 5 logl 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 logl What observations tell us Observations of Mdot, fitted to give Mdot as a function of L, R, M, P and/or T, generally yield the location of the death line, L crit or R crit or P crit from setting log(m/t nuc ) = Mdot(L, R, M, P and/or T eff ) using an evolutionary track R(L, M, Z, ML) and L(R, T eff ) or P(M,R) as needed The extent of the death zone in L How much does L need to change to get Mdot to increase by 2 orders of magnitude? That will be roughly* from L crit + /(dlogmdot/dlogl) to L crit - /(dlogmdot/dlogl) using M = const* Or (observationally simpler) from 0-7 to 0-5 M sun /yr: Time = 2/(dlogMdot/dlogL)* * After L crit the mass will be significantly smaller than it was, so we also need dlogmdot/dlogl to do this right.
We would really like to know how Mdot depends on L,R,M and Z Locally, we may study the slopes dlogmdot/dlogl and dlogmdot/dlogm The initial-final mass relation is set by what is happening in the range 0. # (Mdot/M)/(Ldot/L) # 0 or 0. # t Mdot /t nuc # 0 and is close to M core (L crit ) for a steep Mdot formula slope at Lcrit slope dlogmdot/dlogl at L c 7 6 5 4 3 2 0 0.6 0.8.2.4.6.8 2 2.2 Mass solid: BW Z=solar dashed: VW compare Reimers =.68 Blöcker= 4.38 vanloon = 0.77 SC = 2.65 Wachter = 3.08 If the slope is steep enough it affects only the shape of the corner in logmdot vs time or logl, not M i vs M f Duration of terminal mass loss for M = formula Reimers VW vanloon SC Wachter BW Simple dlog(dm/dt)/dlogl.68 5.75 0.77 2.65 3.08 4.3 duration, 0 6 yrs # 3.6 # 0.4 # 7.8 # 2.3 # 2.0 # 0.4 Observed: ~0.2e6 years
Conclusion, part I: Published mass loss laws lead to very different dlogmdot/dlogl and dlogmdot/dlogm Duration and distribution of mass-losing stars => these slopes should be very steep, or for Mdot = AL B along an evolutionary track: 0 # B # 20 This eliminates most published mass loss formulae Pulsation Models for mass loss* Essential Shocks transfer outward momentum to the gas Dust Makes a big difference in some stars Molecule & grain chemistry is complex! Non-LTE gas reheating, cooling Also essential, particularly in low Z stars *In case you missed Don s talk, and a little ancient history Periodic structure Pulsation extended atmosphere Shocks compression heating expansion cooling Multiple shocks per cycle when P/P acoustic > Periodic Structure for one shock per cycle Set r(t S +P) = r(t S ) where t S = time of shock passage During most of the cycle the motion is ballistic For very short periods (Q < 0.0 d ) this gives shock "v # gp For most pulsating stars, 0.02 < Q < 0.2 => periodicity condition v esc P/2r = 38.3 Q d! (#/(-# 2 ))+ (- # 2 )- 3 / 2 arcsin(#) where #=v out!"v/2 From Hill & Willson 978 and Willson & Hill 978; also Willson 2000
Periodicity constraint on "v Periodic Pulsation Model 2.0!v=gP Photosphere is the red line 6!v/v esc.5.0 No periodic solution Two shocks per cycle when P>P acoustic 5 R/R* 4 3 0.5 Allowed Refrigerated region 2 0.0.000.00.0. Time -> Q(r), days Part I: (Some) Models match observations: Are we done? Non-LTE cooling/heating of compressed/expanding gas is important in determining atmospheric structure Means that dynamics and radiative transfer should be treated together May be approximately described in terms of critical density: Q/Q LTE = /(+$ x /$) Non-LTE Detailed calculations so far only for some of the constituents and some conditions Temperature versus radius from two models that are identical, except that in the top panel the critical density $ x =0-0 gm cm 3, while in the bottom panel $ x = 0-8 gm cm 3. Lower critical density results in a more extended region where T ~T RE between shocks. (Figure courtesy of GH Bowen.) From Willson 2000
Part I: (Some) Models match observations: Are we done? Can non-grey silicate grains drive/enhance mass loss? They are too transparent in the optical/ir Add iron for opacity: they get too hot Insert here. What is the problem? Contrasting carbon grains and silicates 2. Why might non-lte chemistry solve the problem? Noted independently by S. Höfner, P. Woitke, 2006 Possible solution (Höfner & Anderson 2007): Non-LTE chemistry => both silicate and carbon grains form in oxygen-rich stars. For carbon grains Carbon grains Silicate grain materials High opacity near micron is needed to drive mass loss, but adding Fe also => hot grains illustration from Woitke 2006
Radiative acceleration of a grain Radiative equilibrium for an opaque grain or a planet! dust (r) = c # k dust F " (r)d" GM (r) r 2!T 4 " eq = xarea % " # $ rarea & ' # $ 4(d 2 " $ % & ' $ $ $ # * + 0 * + 0 % A())L star ())d) ' ' ' A())L grain ())d) ' & Non-LTE chemistry When conditions change on a time scale (dlogx/dt) - < destruction time scales, then the abundances of various species depends more on the rate of formation than on the equilibrium value. Thus one may not have all the C or all the O in CO, but some of each left to form grains. May also be relevant In a range of density where collisional excitation is followed by radiative deexcitation, under typical conditions the excitation T is < LTE T excitation kt is ~ 0. ev and the excitation energies of molecules and atoms include 0. - ev. Under-excited atoms are more likely to stick to a growing grain.
Wish List Questions? Successful models for mass loss from silicate-rich stars with non-gray grains. An appropriate Q/Q LTE to use, or an efficient method for treating the coupling between the gas and the radiation field. Better observational constraints on the slopes (dlogmdot/dlogl and dlogmdot/dlogm).