The Astrophysical Implications of Dust Formation During The Eruptions of Hot, Massive Stars
aa r X i v : . [ a s t r o - ph . S R ] S e p The Astrophysical Implications of Dust FormationDuring The Eruptions of Hot, Massive Stars
C. S. Kochanek , ABSTRACT
Dust formation in the winds of hot stars is inextricably linked to the classic eruptivestate of luminous blue variables (LBVs) because it requires very high mass loss rates, ˙ M > ∼ - . M ⊙ /year, for grains to grow and for the non-dust optical depth of the wind to shield thedust formation region from the true stellar photosphere. Thus, dusty shells around hot starstrace the history of “great” eruptions, and the statistics of such shells in the Galaxy indicate thatthese eruptions are likely the dominant mass loss mechanism for evolved, M ZAMS > ∼ M ⊙ stars.Dust formation at such high ˙ M also explains why very large grains ( a max > ∼ µ m) are frequentlyfound in these shells, since a max ∝ ˙ M . The statistics of these shells (numbers, ages, masses, andgrain properties such as a max ) provide an archaeological record of this mass loss process. Inparticular, the velocities v shell , transient durations (where known) and ejected masses M shell ofthe Galactic shells and the supernova “impostors” proposed as their extragalactic counterpartsare very different. While much of the difference is a selection effect created by shell lifetimes ∝ ( v shell √ M shell ) - , more complete Galactic and extragalactic surveys are needed to demon-strate that the two phenomena share a common origin given that their observed properties areessentially disjoint. If even small fractions (1%) of SNe show interactions with such denseshells of ejecta, as is currently believed, then the driving mechanism of the eruptions must beassociated with the very final phases of stellar evolution, suggestive of some underlying nuclearburning instability. Subject headings: stars: evolution; stars: massive; stars: mass-loss; supernovae: general; stars:winds, outflows; dust; extinction
1. Introduction
The winds of hot stars do not form dust, as illustrated by the single uses of the word “dust” in thereviews of these winds by Kudritzki & Puls (2000) and Puls et al. (2008). The exceptions which provethe rule are the rare, dust forming WC stars, all of which appear to be binaries where dust forms due tothe collision of the two stellar winds (see the review by Crowther 2007), and the relatively rare B[e] starswhere dust is believed to form in a disk/dense equatorial wind surrounding the star (see the review by Department of Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus OH 43210 Center for Cosmology and AstroParticle Physics, The Ohio State University, 191 W. Woodruff Avenue, Columbus OH 43210 T ∗ > T ∗ ≃ ˙ M ∼ - to 10 - M ⊙ /year), and a similarly cool, “great” eruptive state of significantly higher bolometricluminosity and enormously enhanced mass loss rates ( ˙ M > ∼ - –10 - M ⊙ /year). We will call these the hot(or quiescent), cool (or S Doradus), and (great) eruptive states, and we will refer to the ejected material froman eruption as a shell since the low duty cycles of eruptions produce relatively thin dusty shells of ejecta. Intheir (great) eruptive state, these stars can expel enormous amounts of material under conditions favorableto the growth of dust grains, as illustrated by the massive ( ∼ M ⊙ , Smith et al. 2003), optically thick, dustyshell surrounding η Carinae (see the reviews by Davidson & Humphreys 1997, Smith 2009). Indeed, it islikely that such phases represent the bulk of the mass loss from higher mass stars ( M > ∼ M ⊙ ) becausenormal winds are inadequate to the task (Humphreys & Davidson 1984, Smith & Owocki 2006a). Therecent discovery of many 24 µ m shells surrounding hot stars by Wachter et al. (2010) and Gvaramadze et al.(2010) further suggests that the phenomenon is more common than previously thought.Table 1 summarizes the properties of Galactic LBVs and LBV candidates mainly drawn from Humphreys & Davidson(1994) and Smith & Owocki (2006a). All the stars have substantial, ˙ M ∼ - M ⊙ /year, relatively fast, v ∞ ≃
200 km/s, present day winds that are not forming dust, as expected for hot stellar winds. However,all but one system is surrounded by a relatively massive, M shell ∼ M ⊙ , slowly expanding, v shell ∼
100 km/s,shell of dusty material, and in at least four cases, models of the shell appear to require surprisingly largemaximum grain sizes, a max > ∼ µ m. We focused on these sources because most of these ancillary propertieshave been measured. Most of the mass estimates are based on assuming a dust-to-gas ratio X d = 0 .
01 and socould be underestimates.While Table 1 is certainly incomplete and subject to many selection effects, that it contains 13 objectsmeans that shell ejections are an important or even dominant mass loss process for massive stars, as hasbeen previously suggested in order to compensate for the steady downward revisions of the mass loss ratesin normal, hot stellar winds (e.g., Humphreys & Davidson 1984, Smith & Owocki 2006a). We can quantifythis by estimating the number of dusty shells that should exist in the Galaxy given the rate of Galacticsupernovae, r SN . For simplicity, we use a Salpeter initial mass function (IMF) and assume that supernovaearise from stars with initial masses M ∗ in the range M SN ≃ M ⊙ < ∼ M ∗ < ∼ M up , where for now we will let M up → ∞ . If eruptions occur in stars with M ∗ > M erupt and there are an average of N erupt occurrences perstar, then the eruption rate r erupt is of order r erupt ≃ . (cid:18) M ⊙ M erupt (cid:19) . N erupt r SN . (1)The optical depth of a dusty shell of mass M shell with visual opacity κ V ≃
100 cm /g expanding at velocity 3 – v shell is τ V = M shell κ V / π v shell t so the shell will be detectable for of order t shell ≃ (cid:18) M shell κ V π v shell τ V (cid:19) / ≃ (cid:18) . τ V (cid:19) / (cid:18) M shell M ⊙ (cid:19) / (cid:18)
70 km/s v shell (cid:19) years . (2)Most shells should be seen near their maximum size, v shell t shell ≃ . τ V because of theincreased dust opacity in the UV. The expected number of Galactic shells is the product of the rate and thelifetime, N shell = r erupt t shell = 6 N erupt (cid:18) . τ V (cid:19) / (cid:18) r SN century - (cid:19) (cid:18) M ⊙ M erupt (cid:19) . (cid:18) M shell M ⊙ (cid:19) / (cid:18) v shell (cid:19) . (3)As we see from Table 1, there are at least N shell ∼
10 LBV stars surrounded by massive, dusty shells in theGalaxy, which means that the number of eruptions per star is N erupt ≃ (cid:18) N shell (cid:19) (cid:16) τ V . (cid:17) / (cid:18) century - r SN (cid:19) (cid:18) M erupt M ⊙ (cid:19) . (cid:18) M ⊙ M shell (cid:19) / (cid:16) v shell (cid:17) . (4)and the amount of ejected mass per star due to the eruptions is M tot = N erupt M shell ≃ M ⊙ . A “normal”wind from a hot star with ˙ M normal ≃ - M ⊙ /year would have to operate continuously for over 10 years toequal the typical eruptive mass loss implied by the existence of even the well-studied Galactic shells. That N erupt > N erupt t shell ≃ years as compared to post-main-sequence lifetimes of order 10 years. Theseestimates are broadly consistent with earlier estimates (e.g. Humphreys & Davidson 1994, Lamers 1989)but based on a different approach.Given such a large contribution to the mass loss history of massive stars, we need to understand therelationship between mass loss and dust formation around hot stars. We consider a parcel of fluid ejectedin a wind of mass loss rate ˙ M and velocity v w ejected from a star of luminosity L ∗ and temperature T ∗ andexamine the conditions under which dust can form in §2. Not surprisingly, the key variable is the mass lossrate. First, for stellar winds with velocities of order the escape velocities of massive hot stars, very highmass loss rates are needed for particle growth. Second, the dust formation region must be shielded from thehot stellar photosphere, which these high density winds can achieve by forming a pseudo-photosphere in thewind with a characteristic temperature of roughly 7000 K. Dust formation around hot blue stars is neces-sarily tied to very high mass loss rates, the classic LBV eruptive state and the formation of shells of ejecta.
In §3 we discuss some implications of this model for dust formation, stellar evolution and supernovae.
2. The Physics of Dust Formation in Stellar Transient Ejecta
We consider the formation of dust grains of radius a comprised of N atoms of average mass m where4 π a ρ bulk / Nm and ρ bulk is the bulk density of the grain. We will use ρ bulk = 2 . (3 . ) and 4 – -6-4-20-6-4-20 3.6 3.8 4 4.2 4.4-6-4-20 3.6 3.8 4 4.2 4.4 3.6 3.8 4 4.2 4.4 Fig. 1.— Minimum mass loss rates ˙ M for silicate dust formation as a function of stellar temperature T ∗ .The results for graphitic dusts are very similar. The lower left panel labels the regions. Dust can formin the region labeled “dust forms”, above the photoionization (“ionized”), growth (“no growth”) and pho-toevaporation (“photoevaporated”) limits. The photoevaporation limits are for minimum photon energiesof E = 7 . T phot = T ∗ , in the S Doradus state (middle) the photospheric temperatureof T phot = min ( T ∗ , T phot = 7000 K.The upper, middle and lower panels show the changes for stellar luminosities of L ∗ ≡ L phot = 10 , 10 and10 L ⊙ , respectively. The filled points are the present day properties of the systems from Table 1. Otherparameters are set to M ∗ = 20 M ⊙ , v c = 1 km/s, X g = 0 . T d = 1500 K, N = 7, ˆ ρ = 3 . m = 20 m p . 5 – m = 12 m p (20 m p ) for graphitic (silicate) grains. The smallest possible grain is the point where the inter-particle bond strengths shift from strong molecular bonds to weaker intra-molecular interactions, and we willgenerally require N ≥ SiO . There are many prior treatments of dustformation in (generally cool) stellar winds (e.g. Salpeter 1977, Draine 1979, Gail et al. 1984), novae (see thereview by Gehrz 1988) and supernovae (e.g. Clayton 1979, Dwek 1988, Kozasa et al. 1991) which containmost of the basic physical picture we use here. For some standard results in dust physics we will refer toDraine (2011) as a reference source. For dust to grow, the medium must be largely neutral, sufficiently coolfor growth to occur, have a high enough density for there to be an appreciable particle collision rate, andthe grains must grow faster than they can be photo-evaporated by ultraviolet (UV) photons. We will assumethat the first stage of particle formation, nucleation to form the smallest grains, simply occurs once thetemperature is sufficiently low, and consider only the subsequent collisional growth of the grains. Makingnucleation an additional bottleneck to dust formation will only strengthen our conclusions.The dust forms in a (time varying) wind which we can characterize by the mass loss rate ˙ M , the windvelocity v w , and the mass fraction of condensible species X g . Not all the condensible mass need condenseonto grains, so the ultimate mass fraction of dust X d ≤ X g . The wind is produced by a star of luminosity L ∗ , photospheric radius R ∗ and effective temperature T ∗ where L ∗ = 4 π R ∗ σ T ∗ . At the time the dust is beingformed, the star has luminosity L phot , radius R phot and temperature T phot , with L phot = 4 π R phot σ T phot . Weconsider dust formation in three physical states. First, we have the hot star in quiescence with L phot = L ∗ and T phot = T ∗ . Second, we have the S Doradus state, where L phot = L ∗ and T phot ≡ = T ∗ . Finally, wehave the eruptive state where both L phot = L ∗ and T phot = T ∗ and we will use the optical depth of the wind toestimate T phot . In many cases we can assume that the spectral energy distribution is simply a black body, butthere are several areas where the differences between black bodies and stellar photospheres are importantbecause line opacities suppress the ultraviolet emission. Where this is important, we will use the models ofCastelli & Kurucz (2003).For the physics of dust formation, the most relevant velocity is that near the dust formation radius sinceit sets the particle density that determines the growth of the grains. For hot stars, dust formation occurssufficiently far from the star that the wind acceleration should be largely complete and we can view thevelocity as constant. We will scale the wind velocity by the surface escape velocity of the quiescent stars v e = (cid:18) GM ∗ R ∗ (cid:19) / = 151 (cid:18) M ∗ M ⊙ (cid:19) / (cid:18) L ⊙ L ∗ (cid:19) / (cid:18) T ∗ K (cid:19) km/s , (5)as this is the typical asymptotic velocity scale of radiatively accelerated winds (see the reviews by Kudritzki & Puls2000, Puls et al. 2008). Here we have scaled the stellar mass to M ∗ = 20 M ⊙ under the assumption that thestars have undergone significant mass loss by the time of the eruption. Since v e ∝ T ∗ , this introduces a strongstellar temperature dependence to dust formation and growth. In most of our results, the wind velocity isalways scaled by the escape velocity of the quiescent star!
Eqn. 5 yields an appropriate velocity scale for η Carinae but may be somewhat high for many of the sources in Table 1. For the more massive examples inTable 1 the low shell expansion velocities are almost certainly intrinsic because only a very high density in- 6 –terstellar medium (ISM) can significantly slow the expansion of these massive shells of material. Moreover,some of the systems have multiple shells where the inner shells have low velocities but cannot be interactingwith the ISM or an older slower wind (e.g. P Cyg, Meaburn et al. 1996; G79.29 + L ∗ and T ∗ by L phot and T phot , whichwill change the scaling of the wind velocity to use the escape velocity from the photosphere of the transient.Luminous hot stars produce large numbers of ionizing photons, and dust cannot form in such a hot,ionized medium. For a pure hydrogen wind, the wind can recombine if the rate of production of ionizingphotons is less than Q = ˙ M α B / π v w m p R ∗ (e.g. Fransson 1982). The production of ionizing photons is Q = L ∗ kT ∗ F ( G = x - , E / kT ∗ , ∞ ) (6)where E = 13 . F ( G , x , x ) = R x x GF ν dx R ∞ F ν dx → π Z x x Gx dx exp( x ) - , (7)where x = h ν/ kT and the limit is that of a black body. Thus, the minimum mass loss rate for the wind torecombine is ˙ M > ∼ π L ∗ GM ∗ m p FkT ∗ α B ! / (cid:18) v w v e (cid:19) ≃ . × - F / (cid:18) × - cm / s α B L ∗ L ⊙ K T ∗ M ∗ M ⊙ (cid:19) / (cid:18) v w v e (cid:19) M ⊙ / year . (8)For a black body, F ≃ (15 /π )(2 + x (2 + x ) exp( - x )) where x = 158000 / T ∗ , making it a small number unlessthe star is very hot ( F / = 0 . T ∗ = 10 K black body), and the line blanketing of stellar atmospheresreduces it still further. Fig. 1 shows this photoionization limit (“ionized”) on ˙ M for forming dust based onthe Castelli & Kurucz (2003) model atmospheres – unless the star is very hot, ionization cannot prevent dustformation in dense winds. Even then, the photoionization limit is only important for the quiescent star.If the gas is relatively neutral, and so can carry out chemical reactions to form molecules, the tem-perature must be low enough to aggregate the molecules into grains. In general, the dust temperature iscontrolled by the radiation field because collisional time scales are very much longer (see below). We candivide the effects of radiation into the equilibrium temperature and stochastic heating of small grains by Given the shell masses it is very hard to slow them down by large factors. Even slowing a 1 M ⊙ shell by a factor of two (from140 to 70 km/s) within an expansion radius of 0 . cm - . A dense ( ˙ M > ∼ - M ⊙ /year),pre-existing slow wind from a red supergiant phase is a more promising means of having this much mass present, but the timingmust be right and it still would not explain the slow multiple shell systems. Slowing a massive (10 M ⊙ ), fast (500 km/s) shell likethat of η Carinae down to 100 km/s requires 40 M ⊙ of material, and is essentially impossible. T d . Ifwe consider only the mean temperature of the grains, then dust can form outside radius (Draine 2011) R f orm = (cid:18) L phot Q P ( T phot , a min )16 πσ T d Q P ( T d , a min ) (cid:19) / = R ∗ (cid:18) L phot L ∗ (cid:19) / (cid:18) T ∗ T d (cid:19) (cid:18) Q P ( T phot , a min ) Q P ( T d , a min ) (cid:19) / (9)= 5 . × (cid:18) L phot L ⊙ (cid:19) / (cid:18) T d (cid:19) (cid:18) Q P ( T phot , a min ) Q P ( T d , a min ) (cid:19) / cm (10)where Q P ( T , a min ) is the Planck-averaged absorption efficiency for the smallest grains. To simplify manysubsequent results, we define Q rat ≡ Q P ( T d , a min ) Q P ( T phot , a min ) , (11)so R f orm = ( R ∗ / L phot / L ∗ ) / ( T ∗ / T d ) Q - / rat . Note that the Planck factor for the star is evaluated at theapparent photospheric temperature T phot which may not be the same as the temperature at the stellar surface T ∗ . Unless T d ≃ T phot , the corrections for the finite size of the star are unimportant and for sufficiently smallgrains the result is independent of the grain size because the Q ∝ a dependence cancels. If we use thegraphitic models of Draine & Lee (1984) and a min = 0 . µ m, the Planck average for small graphitic dustsis approximately a power law Q P ( T )( µ m / a min ) ≃ . (cid:18) T (cid:19) / (12)for 1000 < T < T phot ≃ R f orm ≃ . × (cid:18) L phot L ⊙ (cid:19) / (cid:18) T d (cid:19) / cm . (13)The Planck averages for small silicate dusts cannot be reasonably approximated as a simple power law, buta reasonable piecewise approximation islog (cid:2) Q P ( T )( µ m / a min ) (cid:3) ≃ - . - . t + . t for 10 < T < (14)where t = log ( T / (cid:2) Q P ( T )( µ m / a min ) (cid:3) ≃ . + . t - . t for 10 < T < × (15)where t = log ( T / < T d < T phot ≃ R f orm ≃ . × (cid:18) L phot L ⊙ (cid:19) / (cid:18) T d (cid:19) cm . (16) 8 –In general, including the Planck factors makes the formation radius roughly three times larger than if theyare ignored, and a reasonably general approximation is that R f orm ≃ ( L phot / L ⊙ ) / cm.If we assume that particle nucleation occurs rapidly once the ejecta are cool enough to form dust, thenthe subsequent properties are limited by the growth of the dust particles. The collisional growth rate of aparticle of radius a is dadt = v c X g ˙ M π v w r ρ bulk (17)where v c is an effective collisional velocity (e.g. Kwok 1975, Deguchi 1980). For thermal collisions,accreting particles of mass m a at gas temperature T , v c = (cid:18) kT π m a (cid:19) / = 4 . (cid:18) T (cid:19) / (cid:18) m p m a (cid:19) / km/s . (18)This means that growth cannot proceed by coagulation of large particles if the particle velocities are thermalbecause m a = Nm means that v c ∝ N - / and the growth rate freezes out at very tiny grain sizes. In this case,growth must be dominated by the accretion of monomers and very small clusters, so we can regard m a ≃ m as effectively constant. Coagulation will matter if v c is controlled by turbulent motions (e.g. Voelk et al.1980) with the net effect that particles can grow up to 4 times faster than by monomer accretion. The gaspresumably cools as it expands, so we will let v c = v c ( R f orm / R ) n where R f orm is the radius at which particlegrowth commences and n = 2 / n = 1 / v c or n . With these assumptions, wefind that the particles grow to a maximum size of a max = v cn X g ˙ M πρ bulk v w R f orm (19) ≃ . × - ˙ M ˆ ρ - bulk - M ⊙ / year ! (cid:18) X g . (cid:19) (cid:16) v cn km/s (cid:17) (cid:18)
500 km/s v w (cid:19) (cid:18) cm R f orm (cid:19) µ m= v cn X g ˙ M π GM ∗ ρ bulk (cid:18) L ∗ L phot (cid:19) / (cid:18) v e v w (cid:19) (cid:18) T d T ∗ (cid:19) Q / rat (20) ≃ . × - ˙ M ˆ ρ - bulk Q / rat - M ⊙ / year ! (cid:18) X g . (cid:19) (cid:18) M ⊙ M ∗ (cid:19) (cid:16) v cn km/s (cid:17) (cid:18) L ∗ L phot (cid:19) / (cid:18) v e v w (cid:19) (cid:18) T d K T ∗ (cid:19) µ mwhere v cn = v c / (1 + n ) absorbs the effects of the cooling model on a max and ˆ ρ bulk = ρ bulk / g/cm . The grainsize grows with radius as a = a max - R + nf orm R + n ! . (21)Because the density is already dropping rapidly, reasonable assumptions about the temperature scaling n have little effect on the results. Faster cooling leads to smaller particles, but the full range from a constanttemperature to adiabatic cooling reduces a max by less than a factor of two, and the particles are close to theirfinal sizes by the time R ≃ R f orm . 9 –If the mass loss rates are too low, then the particles cannot grow, which implies there is a minimummass loss rate for dust growth of ˙ M > π GM ∗ ρ bulk v c X g (cid:18) Nm πρ bulk (cid:19) / (cid:18) v w v e (cid:19) (cid:18) L phot L ∗ (cid:19) / (cid:18) T ∗ T d (cid:19) Q - / rat (22) > ∼ . × - (cid:18) km/s v c (cid:19) (cid:18) . X g (cid:19) (cid:18) v w v e (cid:19) (cid:18) M ∗ M ⊙ (cid:19) (cid:18) L phot L ∗ (cid:19) / (cid:18) T d T ∗ k (cid:19) (cid:0) ˆ ρ N ˆ m (cid:1) / Q / rat M ⊙ year(23)where we have phrased the limit in terms of the particle number N rather than the size a and used ˆ m = m / m p . Fig. 1 compares these limits from particle growth (“no growth”) to those from photoionization.The limits from particle growth are always the more stringent. When the wind velocity has v w ∼ v e , it is verydifficult for hot stars to form dust because of the rapid increase in the wind velocity with stellar temperature, v w ∝ T ∗ . We note that the limit at low temperatures appears high compared to typical AGB stars (e.g.van Loon et al. 2005, Matsuura et al. 2009) primarily because the mass has been scaled to M ∗ = 20 M ⊙ and ˙ M ∝ M ∗ .In the interstellar medium, the temperatures of the smallest dust grains are stochastic because the ab-sorption of individual photons can temporarily heat the grains to temperatures far higher than the equilib-rium temperature predicted by the ambient radiation density (e.g. Draine & Anderson 1985, Dwek 1986).This effect also plays a key role in the formation of dust by transients, but seems not to have been gener-ally considered outside estimates of dust formation in the colliding wind environments of WC stars (e.g.Cherchneff & Tielens 1995). Under the assumption that the ejecta must recombine in order to form dust, weare interested in photons with energies E < E < E ≃ . E using the modelsof Guhathakurta & Draine (1989) for stochastic dust heating as the energy at which the grain cooling timescale equals the time to lose an atom from the grain. This energy depends crucially on what we view asthe smallest number of particles in a grain because the peak temperature increases for smaller particle sizesand the probability of losing an atom rises exponentially with the peak temperature. If we consider a singleMg SiO molecule with N = 7 atoms as the smallest grain, then we find E ≃ . N = 14, then E ≃ . . Q = Q ′ ( a /λ ), the rate at which such photons are absorbed is t - γ = L phot a r hc F ( G = Q ′ , E / kT phot , E / kT phot ) (24)where the function F is the same as for the estimate of the number of ionizing photons in Eqn. 7 but with G = Q ′ rather than G = 1 / x . Dust can only grow once the evaporation rate is lower than the collisional growth 10 –rate (Eqn. 17), leading to a photoevaporation limit on the mass loss rate for dust formation of ˙ M > v w v c L phot am X g hc F = 3100 F v w v e km/s v c L phot L ⊙ (cid:18) L ⊙ L ∗ (cid:19) / (cid:18) T ∗ K (cid:19) (cid:18) M ∗ M ⊙ (cid:19) / (cid:18) . X g (cid:19) (cid:18) ˆ m N ˆ ρ (cid:19) / M ⊙ year . (25)The factor ( ˆ m N /ρ ) / ≃ N ≃
7. The enormous difference between the photon and particle densitiesmeans that the possibility of dust formation is entirely controlled by the spectral energy distribution of thetransient and the value of E . As with the recombination limits, the differences between black bodies andactual photospheres are crucial – the limits on ˙ M for black bodies are several orders of magnitude higherthan those for the Castelli & Kurucz (2003) models. As we see in Fig. 1, the photoevaporation limit on ˙ M is a wall blocking dust formation in the quiescent state independent of mass loss rate. Thus, dust can onlyform around hot stars if they do not appear to be hot when observed from the dust formation radius. Thesestars appear to have two means of achieving this – the S Doradus phase and (great) eruptions.In the S Doradus phase, the luminosity of the star is little changed, L phot ≃ T ∗ , but the stars have coolerphotospheric temperatures, T phot ≃ ˙ M ∼ - –10 - M ⊙ , and fast winds with v ∞ ≃ v e , but they cannotbe forming significant amounts of dust even though they satisfy the condition on photospheric temperature.With a dust optical depth of τ V = ˙ M κ V π v w R f orm ≃ (cid:18) ˙ M - M ⊙ / year (cid:19) (cid:18) κ V
100 cm / g (cid:19) (cid:18) cm R in (cid:19) v e v w (26)the stars would become bright, hot mid-IR sources and some would be heavily enshrouded by their owndust, yet neither phenomenon seems to be reported.As we show in the middle panels of Fig. 1, where we simply set the apparent photospheric temperatureto T phot = min ( T ∗ , E needed to photo-evaporate a grain, but for E = 10 eV the limit is close to the limit for any particle growth, at roughly ˙ M > ∼ - M ⊙ /year, If we lowerthe minimum energy to E = 7 . T phot and E , the combination of slow growth and photoevaporation mean that dustcannot form in the S Doradus phase.The final case we consider is a (giant) eruption where there is an increase in the bolometric luminosity L phot > L ∗ , the apparent temperature is cooler, T phot < T ∗ , as in the S Doradus phases, and the mass loss ratesare much higher, ˙ M > - M ⊙ /year. For sufficiently dense winds, the dust formation region sees a pseudo-photosphere created by the non-dust opacity of the wind rather than the hot stellar photosphere. Davidson 11 –(1987) explains this as a consequence of combining a dense wind with an opacity law that is falling rapidlywith temperature in this temperature range. Consider the Rosseland mean optical depth τ R ( R ) = Z R R ρκ R ( ρ, T ) dR (27)looking inwards from the radius R where the gas temperature is 1000 K and dust formation may be possibleat some interior radius R . If we combine the rapidly rising ρ ∝ r - density profile of the wind with a temper-ature regime where the opacity rises rapidly, then there will be a tendency to produce a pseudo-photospherewhere τ R ( R ) = 1 near that temperature. We computed the temperature T w ( R ( τ R = 1)) at the radius where τ R = 1using the solar composition opacity models of Helling & Lucas (2009), our standard wind density profileand assuming a temperature profile T w = T ∗ ( R / R ∗ ) - / . This is not a self-consistent wind model, but theresults are insensitive to the assumptions because the opacity and optical depth increase so rapidly towardssmaller radii in the wind. Fig. 1 shows the consequences of using this “pseudo-photospheric” temperature indetermining the photo-evaporation limit rather than T ∗ , as well as the contour where T w ( R ( τ R = 1)) = 7000 K.The limits now have two branches. For small ˙ M or low stellar temperatures, the wind is optically thin,the observed temperature is simply the photospheric temperature and the photo-evaporation limits are un-changed. For high ˙ M and high temperatures, the wind becomes optically thick and the observed temperatureis of order 7000 K with relatively weak dependencies on ˙ M and v w because of the steep slope of the opacity,as predicted by Davidson (1987). As expected from the arguments summarized by Vink (2009), the massloss rates needed to form a pseudo-photosphere are higher than are typically found for the S Doradus phase.However, once ˙ M > ∼ - . M ⊙ /year, the wind forms a pseudo-photosphere whose temperature slowly dropswith increasing mass loss rate, which makes the photevaporation limits less sensitive to E than in our SDoradus model. Note that in both the S Doradus and eruption models the photosphere must stay in its coolstate long enough for the ejecta to reach the dust formation radius ( ∼ κ λ = 3 X d ρ bulk h Q λ ( a max ) i a max , (28)where the dimensionless function h Q λ ( a max ) i = a max R a max Q λ ( a ) a dnda da R a max a dnda da (29)depends on the grain size distribution dn / da , the dimensionless (absorption or scattering) cross section Q λ ( a ), and the fraction of the gas mass in condensed dust X d ≤ X g . The function h Q λ ( a max ) i is proportionalto a max for very small grains, where Q λ ∝ a and becomes constant for very large grains where Q λ becomesconstant (e.g. Draine & Lee 1984). Thus, the ratio h Q λ ( a max ) i / a max appearing in the opacity becomesconstant for very small grains, decays as a - max for very large grains and has a maximum at an intermediatesize a peak . At V band for a Mathis et al. (1977) size distribution dn / da ∝ a - . with a range of a max / a min =50, we find a peak ≃ . µ m (0 . µ m) with (cid:10) Q λ (cid:0) a peak (cid:1)(cid:11) ≃ . ≃ .
5) and (cid:10) Q λ (cid:0) a peak (cid:1)(cid:11) / a peak = 15 . µ m -
12 –(3 . µ m - ) for graphitic (silicate) dust. These estimates were made for the effective absorption optical depth( τ abs ( τ abs + τ scat )) / and lead to maximum visual opacities of κ V , max ≃ X d .
005 cm / g graphitic κ V , max ≃ X d .
005 cm / g silicate . (30)For τ abs /τ scat the coefficients are 210/13 and 130/75 for graphitic/silicate dusts. In general, however, thesize dependence of the visual opacity is relatively weak. If a max > ∼ µ m the opacity begins to drop as a - max (Eqn. 21), but this requires ˙ M > ∼ M ⊙ /year, which no star seems to significantly exceed. For very small grains, h Q λ ( a max ) i / a max → . µ m - (0 . µ m - ) for graphitic (silicate) dust, so the opacity is only a factor of two(six) lower than the maximum opacity. If the grains cannot grow to moderate size, then the dust opacity willbe significantly reduced.
3. Discussion
To summarize, the growth of dust particles in the ejecta of massive stars is limited by particle growthrates and photo-evaporation by soft, non-ionizing UV photons from the star. The particle growth rate is thelimiting factor for lower temperature stars ( T ∗ < ∼ T ∗ > ∼ T phot ≃ ˙ M > ∼ - . M ⊙ /year do these stars have the proper conditions forforming dust. Moreover, when the mass loss rates are this high, the wind does form a “pseudo-photosphere”with a temperature T phot ∼ ˙ Mv ∞ ≃ τ L / c , where τ is the non-dust opacity source responsible foraccelerating the wind (see Kudritzki & Puls 2000, Puls et al. 2008). For an asymptotic velocity of v ∞ , theoptical depth must be τ ≃ (cid:18) ˙ MM ⊙ / year (cid:19) (cid:18) L ⊙ L ∗ (cid:19) v ∞ v esc . (31)In the S Doradus phase, having ˙ M < ∼ - M ⊙ /year and v ∞ ≃ v esc means that τ < Fig. 2.— Mass loss rates needed to grow dust particles to radius a max (Eqn. 21). In computing the ratio ofPlanck factors Q rat we have either assumed the existence of a cooler photosphere T phot = min ( T ∗ , T phot = T ∗ (dashed). The triangles show the estimated mass loss rates during (great) eruptionsbased on either the observed duration (filled triangles, η Car and P Cyg) or durations estimated from theshell widths (open triangles). We argue in the text that these latter estimates are gross underestimates of themass loss rates in eruption. The filled squares show the present day properties of the systems in Table 1.Objects in Table 1 noted as having exceptionally large grain sizes are circled.
The temperatures are left fixedat the present day temperature estimates – in reality they were cooler during the eruption but we lack directmeasurements.
14 –
Fig. 3.— Asymptotic expansion velocities of Type IIP SNe (top), SN “impostors” (middle) and GalacticLBV shells (bottom). The impostor velocities have been corrected for expansion out of the stellar potentialwell following Eqn. 33 in order to properly compare them to the LBV shells. The general pattern of theresults is not sensitive to the details of this correction. The SNe and the shells require no corrections becauseof their high velocities and ages, respectively. The curves in the impostor and Galactic LBV panels showthe expected distributions for the bins assuming a velocity-dependent rate r ( v shell ) ∝ v - / shell for 10 km/s < v shell < ∝ / v shell for the Galactic shells normalizedto the numbers of objects in each panel and excluding SN 1961V. 15 –a pseudo-photosphere. In the (great) eruptions with ˙ M > ∼ - M ⊙ /year, the optical depth must be τ ≫ Γ dust = κ rp L phot π cGM ∗ ≃ (cid:18) κ rp
100 cm / g (cid:19) (cid:18) L phot L ⊙ (cid:19) (cid:18) M ⊙ M ∗ (cid:19) , (32)is large. Dust formation, as a large additional source of continuum opacity, may help to address some of theproblems in accelerating these heavy winds (e.g. Owocki et al. 2004).The first important consequence of this close relationship between dust formation and the need for veryhigh mass loss rates is that the dust shells around luminous blue stars are formed exclusively in great erup-tions and so trace the history of these eruptions. Given a census of such dusty shells, their radii, expansionvelocities, (dust) masses and optical depths in the Milky Way or other galaxies, it should be possible toreconstruct this dominant mass loss mechanism for these massive stars. A particularly interesting diagnosticis the maximum grain size. Where the total dust mass or optical depth of a shell probes the total mass lost inthe eruption, the maximum grain size probes the mass loss rate because, as shown in Fig. 2, the maximumgrain size is proportional to the mass loss rate, a max ∝ ˙ M (Eqn. 21). Models of four of the Galactic shellsappear to require a max > ∼ µ m (see Table 1) which strongly suggests ˙ M > ∼ - M ⊙ /year or possibly evenhigher.A second consequence is that the mass loss rates associated with most of the shells in Table 1 are grosslyunderestimated – they are too low to make any dust let alone super-sized grains. These low estimates of ˙ M come from the assumption that the duration of the transient can be estimated from the radial thickness of theshell: ∆ t ≃ ∆ R / v shell ≃ years since ∆ R ≃ R shell ≃ v shell ≃
70 km/s. This leads to an estimateof ˙ M = 10 - M ⊙ /year for M shell = M ⊙ that is not very different from many of the present day winds whichare not making dust, as illustrated in Fig. 2. The flaw here is that the observed spread in radius probablycomes from temporal and azimuthal variations in velocity rather than the duration of the transient, just aswe see in η Carinae. This is proved by the simple geometric observation that all shells have comparablethickness ratios, which is the characteristic of a spread in velocity: ∆ R = ∆ vt and R = vt so ∆ R / R = ∆ v / v isindependent of time. If it were due to the duration of the transient then ∆ R = v ∆ t and R = vt so ∆ R / R = ∆ t / t and the shells only appear geometrically thin as they become old. Roughly speaking, for every shell witha 2:1 thickness ratio there should be one which is a filled sphere just finishing its eruption, and this is notobserved.It should be possible to determine the geometric structure of these shells in some detail because manyof the central stars are known to be significantly variable (e.g. η Carinae, see, e.g., Fernández-Lajús et al.(2009) for a full light curve, or, e.g., Martin et al. 2006 for spatially resolved data; AG Car, Groh et al.2009; IRAS 18576+3341, Clark et al. 2009). You can determine both the structure of the shell and obtaina geometric distance to the source by mapping the time delay between the variability of the star and theechoes of the variability across the shell, by essentially the same procedure as is used in reverberationmapping of quasars (see the review by Peterson 1993) or at a less involved level in studies of SN dust echoes(e.g. Patat 2005). This would complement the proper motion measurements possible for some systems (e.g. 16 –Smith et al. (2004a) for η Carinae). The optimal wavelength is probably on the blue side of the mid-IR peak,at 10-20 µ m to maximize the sensitivity to dust temperature variations while minimizing the direct radiationfrom the star, but scattered optical or near-IR emission is another possibility if the central star is faint enoughto allow imaging of the shell.With the exception of the Great Eruption of η Carinae (250-500 km/s), the typical expansion velocitiesof the Galactic shells are only 50-100 km/s (see Table 1, Fig. 3). As we argued earlier in §2, these expansionvelocities are unlikely to have been significantly slowed by decelerations due to sweeping up the surroundinginterstellar medium and so must be associated with the ejection mechanism. The relatively low velocities ofthe Galactic shells mean that comparisons of the so-called “SN impostors” to LBV eruptions require detailedexamination. Fig. 3 shows the expansion velocities of a sample of normal Type IIP SNe (Poznanski et al.2009), the Galactic eruptions from Table 1, and the SN “impostors” from Smith et al. (2011). The latterhave been conservatively corrected to an asymptotic expansion velocity at large radius by v ∞ = v - GM ∗ vt (33)where we used M ∗ = 40 M ⊙ and t = 14 days. While the corrections for some of the individual objectsare sensitive to the choice of these parameters, the overall results are not. In this recasting of the similarfigure from Smith et al. (2011), we see that almost none of the impostors have velocities similar to theGalactic shells. We must, however, exercise care in comparing the velocity distributions of impostors andGalactic shells in Fig. 3 because slowly expanding shells are detectable for longer periods of time, t shell ∝ / v shell (Eqn. 2). If the intrinsic rate of eruptions with asymptotic velocities v shell is r ( v shell ), the number ofobservable Galactic shells is ∝ r ( v shell ) / v shell , independent of any other consideration such as correlationsbetween v shell and M shell or completeness.Fig. 3 also shows a model for the velocity distributions that is consistent with both samples. Weassumed the intrinsic rate as a function of asymptotic velocity is a power law, r ( v shell ) ∝ v α shell with v min < v shell < v max . We excluded SN 1961V since it was probably an SN (see Kochanek et al. 2011, Smith et al.2011), but otherwise ignored other ambiguities as to the nature of the impostor sample (e.g. the very differentphysics of SN 2008S and the NGC 300-OT, see Kochanek 2011). The best fitting model has α ≃ - / v min ≃
10 km/s and v max ≃ - . <α < .
15 for an order of magnitude change in K-S test probabilities). If the true rate is r ( v shell ) ∝ v β shell , thenthe difference can be interpreted as a velocity-dependent completeness c ( v shell ) ∝ v α - β shell . For example, if thetrue rate is independent of v shell ( β = 0), then either the impostor sample is incomplete at low velocities orthe Galactic sample is incomplete at high velocities. There can be additional biases created by asymmetriesin the ejection velocities, since the early time velocities may represent the fastest expanding material whilethe late time shell emission may be dominated by the slowest moving material – however, only some 50%of the shells are strongly aspherical ( ? ) and the factor of ∼ η Carinae is not large enough torepresent a significant bias.We should note that velocity is not the only parameter in which there is essentially no overlap betweenthe Galactic and impostor samples. First, the eruption time scales of the only two Galactic systems wherethey are known, η Carinae and P Cyg, are an order of magnitude (or more) longer than those of almost all 17 –impostors (years to decades versus months, see Smith et al. 2011 for a summary), even though they are theonly Galactic systems with relatively high velocities. Second, the ejected masses of the impostors almostcertainly have to be far smaller than the typical Galactic shell. Assuming the impostors are radiativelydriven, energy conservation means that the upper bounds on their ejected masses are < ∼ . M ⊙ , while thetypical Galactic shell has a mass > ∼ M ⊙ . Like the velocity distribution, the mass differences can be drivenby the lifetimes of the Galactic shells, t shell ∝ M - / shell (Eqn. 2). Nonetheless, while the Galactic sourcesand the impostors may be produced by a single process with a broad parameter range (velocity, mass, timescale), the observed Galactic and extragalactic sources basically sample completely different regions of thatparameter space. If these two populations are to be unified, then the statistics and properties of the localsystems need to be systematically determined, and the completeness of the extragalactic surveys need to beimproved.In addition to (probably) being the dominant mass loss mechanism for massive stars, the eruptions mayhave a comparable importance to SNe as a source of dust, particularly as a source of large grains due totheir favorable conditions for particle growth. Suppose that the final pre-SN mass of the stars undergoingeruptions is M and that fraction f of the lost mass is in eruptions producing material with a dust-to-gas ratioof X d . For SN to dominate the dust production by massive stars, they must produce M d , SN > ∼ . f (cid:18) X d . (cid:19) (cid:18) M ⊙ M erupt (cid:19) . " - . M M erupt + . M M erupt (cid:18) M e M up (cid:19) . - (cid:18) M e M up (cid:19) . M ⊙ (34)of dust per SN. Unlike §1, we have allowed M up to be finite. We know either from the arguments ofHumphreys & Davidson (1984) and Smith & Owocki (2006a) or our estimates of shell statistics in §1 that f > ∼ .
5, so eruptions dominate the dust distribution unless M d , SN > ∼ . f M ⊙ if we are conservative ( M erupt =40 M ⊙ , M = 10 M ⊙ , M up = 100 M ⊙ ) or M d , SN > ∼ . f M ⊙ if we are more liberal ( M erupt = 20 M ⊙ , M = 5 M ⊙ , M up = 300 M ⊙ ). Moreover, the presence of a dense circumstellar medium may also enhance dust productionby SNe (Smith et al. 2008a). While SN dust production rates are uncertain, few SNe show evidence forproducing this amount of dust (see the discussion in Matsuura et al. 2009), although Matsuura et al. (2011)subsequently reported the detection of 0 . . M ⊙ of dust associated with SN 1987A.There is also increasing evidence that such high mass loss phases are crucial to understanding SNeon two levels. The first problem is simply one of rates. For the nominal parameters suggested by ourdiscussion of the abundance of shells in §1, the rate of eruptions must be roughly the same as the rate ofSNe. However, the rates of the faint Type IIn SNe generally believed to correspond to such transients aresignificantly lower (e.g. Li et al. 2011). This is consistent with the arguments in Thompson et al. (2009)and Horiuchi et al. (2011) that there is strong evidence for incompleteness in M V < ∼ -
16 mag transients.Many candidate eruptions are significantly fainter, with - . < M V < -
15 mag (Smith et al. 2011), so thecompleteness of these surveys for eruptions is presumably still worse. Local surveys need to find thesefainter transients in order to make a complete inventory. The second issue is that there appears to be amismatch between massive star formation rates and SN rates of almost a factor of two (Horiuchi et al. 2011).If the underlying estimates of massive star formation rates and SNe rates are correct, then either many ofthese fainter transients need to be SNe or there must be a significant population of failed SN (Horiuchi et al. 18 –2011). This is another facet of the need to correctly classify the impostors in order to understand theirstatistics.The final issue we consider is that some supernova show evidence in their evolution that they are inter-acting with the massive dense shells of material created by these eruptions. The most dramatic examples arethe hyperluminous Type IIn SNe (Smith & McCray 2007), although these seem to be related to low metal-licity environments (Kozłowski et al. 2010, Stoll et al. 2011). However, Fox et al. (2011) argue that manyType IIn SNe show evidence for CSM interactions requiring the dense shells produced by eruptions. Moregenerally, the existence of a dust echo from an SNe implies an eruption within 10 -10 years whenever theprogenitor is a hot star if our theory that dust only forms in eruptions is correct. In fact, some hyperlumi-nous SNe shows evidence for the presence of two shells – an inner one to boost the total luminosity andan outer dusty shell (e.g. Smith et al. 2008b, Kozłowski et al. 2010). The existence of any such correlationhas dramatic implications for the cause of stellar eruptions. In order to produce any strong SN interactionphenomena, the shell of material must have been produced within time t int = 10 . –10 . years of the SN.Suppose fraction f int ≃ - –10 - of SNe require the CSM densities of eruptions, then the time period ∆ t prior to the SNe over which the ejections can be occurring is ∆ t = 0 . N erupt t int f int (cid:18) M ⊙ M erupt (cid:19) . ≃ (cid:18) N erupt (cid:19) (cid:18) t int
300 years (cid:19) (cid:18) . f int (cid:19) (cid:18) M ⊙ M erupt (cid:19) . years . (35)The existence of dust echoes leads to a similar conclusion – while t int is larger, their incidence in SNe f int is higher. While this point has been made before in a qualitative sense (see, e.g. Smith et al. 2010,Smith et al. 2011), Eqn. 35 makes it quantitatively clear how strong a constraint results from the existenceof any such correlation. Moreover, the parameters chosen for the scaling in Eqn. 35 may be significantlyoverestimating ∆ t . If any such correlation exists, massive shell ejections are forced to be associated withthe very last phases of massive star evolution, roughly to the onset of carbon burning, and this suggests thatthe underlying driving mechanism is post-carbon ignition nuclear burning instabilities (see the discussionin Smith & McCray 2007). Unfortunately, the only known mechanism of this kind, the pair instability SN(Woosley et al. 2007), requires very high masses ( M ∗ > ∼ M ⊙ ) and should not function at the metallicitiesof any of these nearby examples. There could still be a strong metallicity effect because of the dependenceof line-driven stellar winds on metallicity (see Puls et al. 2008). As mass loss by normal winds becomesless efficient, stars may be more dependent on eruptions for mass loss, although this begs the question ofhow eruptions might become more efficient at lower metallicity. This question could be addressed by theinvestigating the statistics of LBV eruptions and shells as a function of environment.The author thanks K. Davidson, R. Humphreys, M. Pinsonneault, K. Sellgren, N. Smith, K.Z. Stanek,D.M. Szczygiel, T.A. Thompson and B.E. Wyslouzil for comments and discussions. C.S.K. is supported byNational Science Foundation (NSF) grant AST-0908816 This discussion does not apply to dust echoes from the SNe of red supergiants where it is feasible to produce the dust in a slow,steady wind.
19 –Table 1. Summary of Galactic LBVs With Dusty ShellsObject L ∗ T ∗ ˙ M now v w , now R shell v shell M shell ˙ M shell a max References L ⊙ K M ⊙ /yr km/s pc km/s M ⊙ M ⊙ /yr µ m η Car (1840) 10 . - .
500 0 .
08 250 /
500 15 10 . η Car (1890) 0 .
03 140 /
300 0 . - . Wray 17–96 10 . - .
100 1 . - . E02AG Car 10 . - .
110 0 .
80 70 25 10 - .
10 S91,L94,V00G79.29+0.46 10 . - .
110 1 . - . W96,V00b,J10G26.47+0.02 10 . - . ≡
200 2 . . - . C03P Cyg 10 . - .
190 0 .
07 136 0 . - . N01, S06Wra 751 10 . - . ≡
500 0 .
34 26 1 . - . . - .
160 0 .
15 70 10 10 - . U01,U05,C09,B10W 243 10 . - . C04Hen 3–519 10 . - .
365 1 . .
66 S94HR Car 10 . - .
145 0 . /
100 3 . . - . ≡
200 1 . . - . C03HD 168625 10 . - .
180 0 .
48 19 0 .
25 10 - . X d = 0 .
01. For η Carinae the dust mass estimate agrees with estimates of the ejected gas mass(Smith & Ferland 2007). The S06 estimate for P Cyg is a gas mass estimate. The dust content of P Cyg is uncertain,although there is a mid-IR excess (see S06). 20 –
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