Testing a Predictive Theoretical Model for the Mass Loss Rates of Cool Stars
aa r X i v : . [ a s t r o - ph . S R ] A ug T HE A STROPHYSICAL J OURNAL , 2011,
IN PRESS
Preprint typeset using L A TEX style emulateapj v. 03/07/07
TESTING A PREDICTIVE THEORETICAL MODEL FOR THE MASS LOSS RATES OF COOL STARS S TEVEN
R. C
RANMER AND S TEVEN
H. S
AAR
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138
Draft version February 7, 2018
ABSTRACTThe basic mechanisms responsible for producing winds from cool, late-type stars are still largely unknown.We take inspiration from recent progress in understanding solar wind acceleration to develop a physicallymotivated model of the time-steady mass loss rates of cool main-sequence stars and evolved giants. Thismodel follows the energy flux of magnetohydrodynamic turbulence from a subsurface convection zone to itseventual dissipation and escape through open magnetic flux tubes. We show how Alfvén waves and turbulencecan produce winds in either a hot corona or a cool extended chromosphere, and we specify the conditionsthat determine whether or not coronal heating occurs. These models do not utilize arbitrary normalizationfactors, but instead predict the mass loss rate directly from a star’s fundamental properties. We take account ofstellar magnetic activity by extending standard age-activity-rotation indicators to include the evolution of thefilling factor of strong photospheric magnetic fields. We compared the predicted mass loss rates with observedvalues for 47 stars and found significantly better agreement than was obtained from the popular scaling lawsof Reimers, Schröder, and Cuntz. The algorithm used to compute cool-star mass loss rates is provided asa self-contained and efficient computer code. We anticipate that the results from this kind of model can beincorporated straightforwardly into stellar evolution calculations and population synthesis techniques.
Subject headings: stars: coronae — stars: late-type — stars: magnetic field — stars: mass loss — stars: winds,outflows — turbulence INTRODUCTION
All stars are believed to possess expanding outer atmo-spheres known as stellar winds. Continual mass loss hasa significant impact on the evolution of the stars them-selves, on surrounding planetary systems, and on the evolu-tion of gas and dust in galaxies (see reviews by Dupree 1986;Lamers & Cassinelli 1999; Puls et al. 2008). For example, theSun’s own mass loss was probably an important factor in theearly erosion of atmospheres from the inner planets of our so-lar system (e.g., Wood 2006; Güdel 2007). On the oppositeend of the distance scale, a better understanding of the windsfrom supergiant stars is leading to new ways of using them as“standard candles” to measure the distances to other galaxies(Kudritzki 2010). By studying the physical mechanisms thatdrive stellar winds, as well as their interaction with processesoccurring inside the stars (convection, pulsation, rotation, andmagnetic fields), we are able to make better quantitative pre-dictions about a wide range of astrophysical environments.Over the last half-century, there has been a great dealof research into possible mechanisms for driving stellarwinds on the “cool side” of the Hertzsprung-Russell di-agram; i.e., effective temperatures less than about 8000K (Holzer & Axford 1970; Hartmann & MacGregor 1980;Hearn 1988; Lafon & Berruyer 1991; Mullan 1996; Willson2000; Holzwarth & Jardine 2007). Despite this work, thereis still no agreement about the fundamental mechanisms re-sponsible for producing these winds. Many studies of stellarevolution use approximate prescriptions for mass loss that donot depend on a true physical model of how the outflow isproduced (Reimers 1975; Leitherer 2010). Observational val-idation of models is made difficult because mass loss ratessimilar to that of the solar wind ( ˙ M ∼ - M ⊙ yr - ) tend tobe too low to be detectable in most observational diagnostics.Fortunately, there has been a great deal of recent Electronic address: [email protected],[email protected] progress toward identifying and characterizing the pro-cesses that produce our own Sun’s wind. Self-consistentmodels of turbulence-driven coronal heating and solarwind acceleration have begun to succeed in reproducing awide range of observations without the need for ad hocfree parameters (e.g., Suzuki 2006; Cranmer et al. 2007;Rappazzo et al. 2008; Verdini et al. 2010; Bingert & Peter2011; van Ballegooijen et al. 2011). This progress on the so-lar front provides a fruitful opportunity to better understandthe fundamental physics of coronal heating and wind acceler-ation in other kinds of stars.The goal of this paper is to construct self-consistent physi-cal models of cool-star wind acceleration. These models pre-dict stellar mass loss rates without the need for observation-ally constrained normalization parameters, artificial heatingfunctions, or imposed damping lengths for waves. We aim todescribe time-steady mass outflows from main-sequence starswith solar-type coronae and from giants with cooler outer at-mospheres. In principle, then, these models cross the well-known dividing line (Linsky & Haisch 1979) between starswith and without X-ray emission. However, there are sev-eral types of late-type stellar winds that our models do not at-tempt to explain: (1) Highly evolved supergiants and asymp-totic giant branch (AGB) stars presumably have winds drivenby radiation pressure on dust grains (Lafon & Berruyer 1991;Höfner 2011) and/or strong radial pulsations (Willson 2000).(2) T Tauri stars have polar outflows that may be energized bymagnetospheric streams of infalling gas from their accretiondisks (e.g., Cranmer 2008). (3) Blue horizontal branch starsmay have line-driven stellar winds similar to those of O, B,and A type stars (Vink & Cassisi 2002).The remainder of this paper is organized as follows. InSection 2 we outline the relevant properties of Alfvén wavesand magnetohydrodynamic (MHD) turbulence that we expectto find in cool-star atmospheres. Section 3 presents deriva-tions of two complementary models of mass loss for stars with CRANMER AND SAARand without hot coronae, and also describes how we estimatethe total mass loss due to both gas pressure and wave pres-sure gradients. Section 4 summarizes how we determine thelevel of magnetic activity in a star based on its rotation rateand other fundamental parameters. We then give the result-ing predictions for mass loss rates of cool stars in Section 5and compare the predictions with existing observational con-straints. Finally, Section 6 concludes this paper with a briefsummary of the major results, a discussion of some of thebroader implications of this work, and suggestions for futureimprovements. ALFVÉN WAVES IN STELLAR ATMOSPHERES
For several decades, MHD fluctuations have beenstudied as likely sources of energy and momen-tum for accelerating winds from cool stars (see, e.g.,Hollweg 1978; Hartmann & MacGregor 1980; DeCampli1981; Wang & Sheeley 1991; Airapetian et al. 2000;Falceta-Gonçalves et al. 2006; Suzuki 2007). Specifically,the dissipation of MHD turbulence as a potential source ofheating for the solar wind goes back to Coleman (1968) andJokipii & Davis (1969). Despite the fact that other sources ofheating and acceleration may exist, we choose to explore howmuch can be explained by restricting ourselves to just thisone set of processes. The ideas outlined here will be appliedto both the “hot” and “cold” models for mass loss describedin Section 3.
Setting the Photospheric Properties
We begin with five fundamental parameters that are as-sumed to determine (nearly) all of the other relevant proper-ties of a star: mass M ∗ , radius R ∗ , bolometric luminosity L ∗ ,rotation period P rot , and metallicity. We also assume that thestar’s iron abundance, expressed logarithmically with respectto hydrogen, is a good enough proxy for the abundances ofother elements heavier than helium; i.e., [Fe/H] ≈ log( Z / Z ⊙ ).For spherical stars, the effective temperature T eff and surfacegravity g are calculated straightforwardly from σ T = L ∗ π R ∗ , g = GM ∗ R ∗ , (1)where σ is the Stefan-Boltzmann constant and G is the New-tonian gravitation constant.We need to know the mass density in the stellar photo-sphere ρ ∗ in order to specify the properties of MHD wavesat that height. The density was computed from the criterionthat the Rosseland mean optical depth should have a valueof 2 / which is a tabulation of the Rosseland mean opac-ity κ R as a function of temperature, density, and metallicity(see Marigo & Aringer 2009). An approximate expression forthe Rosseland optical depth τ R in the photosphere, τ R = κ R ρ ∗ H ∗ = 2 / , (2)was solved for ρ ∗ , where H ∗ is the photospheric value of thedensity scale height (see below). We used straightforward lin-ear interpolation to locate the relevant solutions for ρ ∗ as afunction of T eff , g , and [Fe/H].Figure 1(a) shows how the photospheric density varies as afunction of T eff and log g under the assumption of solar metal-licity ([Fe/H] = 0). For context we also show the location http://stev.oapd.inaf.it/cgi-bin/aesopus F IG . 1.— Derived photospheric parameters shown as a function of both T eff and log g . Contour labels denote (a) base-10 logarithm of mass density ρ ∗ ing cm - , (b) magnetic field strength B ∗ in G, and (c) Alfvén wave amplitude v ⊥∗ in km s - . Also shown is a post-main-sequence evolutionary track for a1 M ⊙ star (red curves) and the location of the ZAMS (white and gray curves). of the zero-age main sequence (ZAMS) from the models ofGirardi et al. (2000), as well as a post-main-sequence evolu-tionary track for a 1 M ⊙ star from the BaSTI model database(Pietrinferni et al. 2004).We used the 2005 release of OPAL plasma equations ofstate (see also Rogers & Nayfonov 2002) to estimate themean atomic weight µ in a partially ionized photosphere. Forthe range of parameters appropriate for cool stars, we foundthat µ is primarily sensitive to T eff , and not to gravity or metal-licity, so we produced a single parameter fit, µ ≈ +
12 tanh (cid:18) - T eff (cid:19) (3)where T eff is expressed in K. Other quantities that will beneeded later include the photospheric density scale height, http://albione.oa-teramo.inaf.it/main.php http://opalopacity.llnl.gov/EOS_2005/ HEORETICAL MASS LOSS RATES OF COOL STARS 3which is given by H ∗ = k B T eff µ m H g (4)where k B is Boltzmann’s constant and m H is the mass of ahydrogen atom. We also need to compute the equipartitionmagnetic field strength, B eq = p π P ∗ = s πρ ∗ k B T eff µ m H , (5)where P ∗ is the photospheric gas pressure. Because T eff and µ do not vary over many orders of magnitude, it is roughlythe case that B eq ∝ ρ / ∗ . However, in all calculations belowwe compute B eq fully from Equation (5). In Section 4 we de-scribe observations that show the photospheric magnetic fieldstrength B ∗ is roughly linearly proportional to B eq for manystars. The measurements determine the constant of propor-tionality, and we use B ∗ = 1 . B eq (6)in the remainder of this paper. Figure 1(b) shows B ∗ as afunction of T eff and log g .We consider MHD waves that are driven by turbulent con-vective motions in the stellar interior. The original mod-els of wave generation from turbulence (e.g., Lighthill 1952;Proudman 1952; Stein 1967) dealt mainly with acousticwaves in an unmagnetized medium. More recently, how-ever, it has been shown that when a stellar atmosphereis filled with magnetic flux tubes, the dominant carrier ofwave energy should be transverse kink-mode oscillations(Musielak & Ulmschneider 2002a). When the magnetic fluxtubes extend above the stellar surface and expand to fill thevolume, the kink-mode waves become shear Alfvén waves(see Cranmer & van Ballegooijen 2005).We utilize the model results of Musielak & Ulmschneider(2002a) to estimate the flux of energy in kink/Alfvén wavesin stellar photospheres. For simplicity, we used only the sim-ulations of Musielak & Ulmschneider (2002a) with their stan-dard parameter choices: a mixing length parameter of α = 2and a constant magnetic field strength that is 0.85 times theequipartition field strength. Our analytic fit to the resultsshown in their Figure 8 is F A ∗ = F (cid:18) T eff T (cid:19) ε exp " - (cid:18) T eff T (cid:19) (7)where the dependence on ˜ g = log g is given by F erg cm - s - = 5 .
724 exp (cid:18) - ˜ g . (cid:19) , (8) T . + . g , (9) ε = 6 . + . g . (10)These fits are similar in form to those given byFawzy & Cuntz (2011) for longitudinal MHD waves.Figure 2 shows a comparison between the above fittingformula and the plotted results of Musielak & Ulmschneider(2002a) for log g = 3, 4, and 5. The behavior of F A ∗ for lowervalues of log g was not given by Musielak & Ulmschneider(2002a), but similar results were found for a wider range ofgravities by Ulmschneider et al. (1996) for acoustic waves. F IG . 2.— Comparison between the Musielak & Ulmschneider (2002a) nu-merical models (dot-dashed curves) and analytic fits (solid curves) for photo-spheric transverse wave energy fluxes F A ∗ as a function of effective tempera-ture and photospheric gravity. Numerical labels denote log g for each curve. We used the kink-mode energy flux to determine the trans-verse velocity amplitude v ⊥ of Alfvén waves in the photo-sphere. The flux is defined as F A ∗ = ρ ∗ v ⊥∗ V A ∗ (11)with V A ∗ = B ∗ / (4 πρ ∗ ) / being the photospheric Alfvénspeed. The above expression is not exact for waves under-going strong reflection (see, e.g., Heinemann & Olbert 1980),but it ends up giving a similar prediction for the height vari-ation of v ⊥ in the corona that would come from a more ac-curate non-WKB model (Cranmer & van Ballegooijen 2005).Figure 1(c) shows how v ⊥∗ varies as a function of T eff andlog g for solar metallicity stars.For the well-observed case of the Sun, we know that mostof the photospheric magnetic field is concentrated into small(100–200 km diameter) flux tubes concentrated in the in-tergranular downflow lanes (Solanki 1993; Berger & Title2001). The field strength in these tubes is close to equipar-tition, with B ∗ ≈ filling factor f ∗ in the photosphere of about 0.1% to 1%, sothe spatially averaged magnetic flux density B ∗ f ∗ is only oforder 1–10 G (Schrijver & Harvey 1989). Radial Evolution of Waves and Turbulence
Figure 3 illustrates the stellar magnetic field geometry thatwe assume to exist above the surface of a cool star. Flux tubesthat are open to the stellar wind have a cross-sectional area A ( r ) that expands monotonically with increasing radial dis-tance r from the star. The condition ∇· B = 0 demands that theproduct of A and the magnetic field strength B remains con-stant. Thus, B ( r ) inside a flux tube decreases monotonically,from its photospheric value of B ∗ , with increasing distance.We normalize A such that at a given distance the total stellarsurface area covered by open flux tubes is defined to be A = 4 π r f . (12) The presumed non-existence of magnetic monopoles implies that “open”field lines must eventually be closed far from the star, presumably via inter-actions with the larger-scale interstellar field (Davis 1955).
CRANMER AND SAAR f ∗ f TR f ∞ → F IG . 3.— Summary illustration of flux tube expansion on a representativecool star. The filling factor (for open magnetic flux tubes) grows from f ∗ inthe photosphere to f TR at the transition region, and to an asymptotic value f ∞ → The dimensionless filling factor f tends to increase withheight to an asymptotic value of 1 as r → ∞ (see alsoCuntz et al. 1999), but its increase is not necessarily mono-tonic. We do not explicitly consider the properties of closedmagnetic “loops” on the stellar surface, but the radial varia-tion of f ( r ) takes into account their presence.For each star, we intend to specify f ∗ on the basis of ei-ther direct measurements or empirical scaling relations. Themodel described in Section 3.1 also requires specifying thevalue of f at the sharp transition region (TR) between thecool chromosphere and hot corona. We generally know that f ∗ < f TR <
1, so in the absence of better information we willapply the assumption that that f TR = f θ ∗ , where θ is a dimen-sionless constant between 0 and 1. For the solar wind modelsof Cranmer et al. (2007) the exponent θ ranges between about0.3 and 0.5.Alfvén waves propagate up from the stellar photosphere,partially reflect back down toward the Sun, develop intostrong MHD turbulence, and dissipate gradually (Velli et al.1991; Matthaeus et al. 1999; Cranmer & van Ballegooijen2005). Temporarily ignoring the reflection and turbulent cas-cade, the overall energy balance of an Alfvén wave train isgoverned by the conservation of wave action. We define theflux of wave action ˜ S as˜ S ≡ ρ v ⊥ V A (1 + M A ) A = constant (13)where M A = u / V A is the Alfvén Mach number and u is theradial outflow speed of the wind (see, e.g., Jacques 1977;Tu & Marsch 1995). Close to the stellar surface, where M A ≪
1, this condition is equivalent to energy flux conser-vation ( F A A = constant). In any case, the constant value of ˜ S in Equation (13) is known for each star because the conditionsat the photosphere are known (and it is also valid to assume M A → v ⊥ ∝ ρ - / close to thestar to v ⊥ ∝ ρ + / at larger distances.The waves gradually lose energy due to turbulent dis-sipation, but for locations reasonably close to the stellarsurface—e.g., the region shown in Figure 3—it is not a badapproximation to use the undamped form of wave actionconservation to compute the radial dependence of v ⊥ (seeCranmer & van Ballegooijen 2005). Wave damping gives rise to plasma heating, and we adopt a phenomenological heatingrate that is consistent with the total energy flux that cascadesfrom large to small eddies. This rate is constrained by theproperties of the Alfvénic fluctuations at the largest scales,and it does not specify the exact kinetic means of dissipationonce the energy reaches the smallest scales. Dimensionally,it is similar to the rate of cascading energy flux derived byvon Kármán & Howarth (1938) for isotropic hydrodynamicturbulence. The volumetric heating rate is given by Q = ˜ αρ v ⊥ λ ⊥ (14)(Hollweg 1986; Hossain et al. 1995; Zhou & Matthaeus 1990;Matthaeus et al. 1999; Dmitruk et al. 2002). The dimension-less efficiency factor ˜ α depends on the local degree of wavereflection and is discussed further below. The perpendicu-lar length scale λ ⊥ is an effective correlation length for thelargest eddies in the turbulent cascade.MHD turbulence occurs only when there exist counter-propagating Alfvén wave packets along a flux tube. The starnaturally creates upward waves, and we assume that linear re-flection gives rise to downward waves (Ferraro & Plumpton1958). We specify the ratio of downward to upward waveamplitudes by the effective reflection coefficient R , and theefficiency factor ˜ α is given by˜ α = α R (1 + R ) √ + R ) / (15)(see, e.g., Cranmer et al. 2007). At the photospheric lowerboundary, we assume total reflection with R = 1 and thus˜ α = α . Higher in the stellar atmosphere, we use the low-frequency limiting expression of Cranmer (2010), R ≈ V A - u ∞ V A + u ∞ , (16)where the wind’s terminal speed is u ∞ and we also assumethat M A ≪ M A = 1 (pre-sumably far from the stellar surface) is roughly equal to u ∞ .We also set α = 0 . MODELS FOR MASS LOSS
In this section we present two complementary descriptionsof cool-star mass loss that make use of the Alfvén wave prop-erties discussed above. Supersonic winds can be driven byeither gas pressure in a hot corona (Section 3.1) or wave pres-sure in a cool, extended chromosphere (Section 3.2). We firstinvestigate each idea by assuming the other one is negligi-ble, and then we explore how to incorporate both processestogether (Section 3.3).
Hot Coronal Mass Loss
If the turbulent heating given by Equation (14) is sufficientto produce a hot ( T & K) corona, then the plasma’s highgas pressure gradient may provide enough outward accelera-tion to produce a transition from a subsonic (bound) state nearthe star to a supersonic (outflowing) state at larger distances(Parker 1958). In this section we estimate the mass loss rate ˙ M of such a gas-pressure-driven stellar wind.We begin by computing Q ∗ in the photosphere using ρ ∗ and v ⊥∗ in Equation (14). For the Sun, we have the observationalconstraint that λ ⊥∗ must be about the size of the granular mo-tions that jostle the flux tubes (i.e., roughly 100–1000 km).HEORETICAL MASS LOSS RATES OF COOL STARS 5For other stars we can assume that the horizontal scale ofgranulation remains proportional to the photospheric pressurescale height (Robinson et al. 2004). Thus, we use λ ⊥∗ = λ ⊥⊙ H ∗ H ⊙ , (17)where H ⊙ = 139 km and the models ofCranmer & van Ballegooijen (2005) were used to setthe solar normalization of the correlation length to λ ⊥⊙ = 300km.In the photosphere, we assume the turbulent heating isswamped by radiative gains and losses that are determinedby the conditions of local thermodynamic equilibrium (LTE),and the temperature is set by those processes alone. At largerheights in the flux tube, the turbulent heating Q begins to havean effect. We define the chromosphere as the region in which Q is balanced by radiative losses. As one increases in height,however, the density drops to the point where radiative lossesalone can no longer balance the imposed heating rate; this oc-curs at the sharp TR between chromosphere and corona. (SeeSection 3.2 for cases where this transition does not occur atall.)In the region between the photosphere and the TR, we as-sume that the wind flow speed is sufficiently sub-Alfvénicsuch that v ⊥ ∝ ρ - / . We also assume that λ ⊥ scales with thetransverse size of the magnetic flux tube, so that λ ⊥ ∝ A / ∝ B - / (Hollweg 1986). Thus, Equation (14) can be rewrittenas Q TR Q ∗ = ˜ α TR ˜ α ∗ (cid:18) ρ TR ρ ∗ (cid:19) / (cid:18) B TR B ∗ (cid:19) / (18)where ˜ α ∗ = 0 . B TR B ∗ = f ∗ f TR ≈ f - θ ∗ (19)where the last approximation holds if there is a universal re-lationship between f ∗ and f TR as speculated in Section 2.2above.Just below the TR, the heating is just barely balanced byradiative cooling. In the optically thin limit, radiative coolingbehaves as Q cool = - n Λ ( T ), where n is the number density inthe fully ionized TR region. Let us then assume that Q TR =max | Q cool | , where max | Q cool | = ρ Λ max m . (20)The quantity Λ max is the absolute maximum of the radiativeloss curve Λ ( T ), and it occurs roughly at T TR = 2 × K.The value of Λ max depends on metallicity. To work out itsdependence on Z , we computed a number of radiative losscurves for different metal abundances using version 4.2 ofthe CHIANTI atomic database (Young et al. 2003) with col-lisional ionization balance (Mazzotta et al. 1998). We startedwith a traditional (Grevesse & Sauval 1998) solar abundancemixture ( Z / Z ⊙ = 1) and then recomputed Λ ( T ) by varying themetal abundance ratio Z / Z ⊙ between 0 and 10. We foundthat the maxima of the curves were fit well by the followingparameterized function, Λ max - erg cm s - ≈ . + (cid:18) ZZ ⊙ (cid:19) . . (21)Other examples of the metallicity dependence of Λ ( T )have been given by, e.g., Boehringer & Hensler (1989) and Gnat & Sternberg (2007). We have ignored any possible dif-ferences between a star’s photospheric metal abundances andthose in the low corona, although such differences have beenmeasured in some cases (Testa 2010).With the above assumptions, we solve for the TR density, ρ TR = " ˜ α TR Q ∗ m ˜ α ∗ ρ / ∗ Λ max / f ∗ - θ ) / (22)and we also derive the heating rate at the TR to be Q TR = (cid:18) ˜ α TR Q ∗ ˜ α ∗ (cid:19) / (cid:18) m ρ ∗ Λ max (cid:19) / f ∗ - θ ) / . (23)A potential roadblock to solving Equations (22–23) is that wedo not initially know the value of ˜ α TR . This quantity dependson the reflection coefficient R , which depends on the Alfvénspeed V A at the TR (see Equation (16)), which in turn dependson the unknown value of ρ TR . In practice, we solve theseequations iteratively. We start with an initial estimate of R =0 .
5, we compute ˜ α TR , ρ TR , and V A at the TR, and then werecompute R for the next iteration. In all cases the processconverges to a self-consistent set of values (with a relativeaccuracy of ∼ - ) in no more than 20 iterations.The mass loss rate of the stellar wind is determined by theheating rate Q TR as well as other sources and sinks of energyat the TR. The general idea that the solar wind’s mass fluxis set by the energy balance at the TR was first discussed byHammer (1982). Hansteen & Leer (1995) worked out the ba-sic scaling argument that is used below (see also Leer et al.1982; Withbroe 1988; Schwadron & McComas 2003). In thelow corona and wind, the time-steady equation of internal en-ergy conservation is1 A ∂∂ r (cid:26) A (cid:20) F H - F cond + ρ u (cid:18) u - GM ∗ r (cid:19)(cid:21)(cid:27) = 0 , (24)where F H is the energy flux associated with the heating, F cond is the energy flux transported by heat conduction along thefield, and u is the outflow speed. The term in braces is con-stant as a function of radius, so it is straightforward to equateits value at the TR to its asymptotic value at r → ∞ . Thekinetic energy term proportional to u is assumed to be negli-gibly small at the TR, but we assume it dominates the energybalance at large distances. Thus, A TR (cid:0) F H , TR - F cond (cid:1) - ( ρ uA ) TR GM ∗ R ∗ = ( ρ uA ) ∞ u ∞ , (25)where F H , TR is the heat flux F H at the TR, and we realizethat the product ρ uA is also constant via mass flux conser-vation. We also make the key assumption that u ∞ = V esc =(2 GM ∗ / R ∗ ) / , and thus we can write ˙ M ≡ ρ uA = 4 π R ∗ f TR V (cid:0) F H , TR - F cond (cid:1) . (26)To evaluate Equation (26) we need to estimate the value of F H , TR . Formally, Q = |∇ · F H | , so to determine the magnitude F H , TR one would have to integrate Q ( r ) along the flux tube.Taking account of the expanding flux tube area A ∝ B - , andalso assuming that r TR ≈ R ∗ , F H , TR = 1 A TR Z ∞ R ∗ dr Q ( r ) A ( r ) . (27) CRANMER AND SAARSpecifically, if Q ∝ r - β and A ∝ r γ , then F H , TR = Q TR R ∗ | β - γ - | ≡ Q TR R ∗ h . (28)Rather than specifying β and γ , we estimate the dimension-less scaling factor h by extracting both Q TR and F H , TR from theself-consistent solar wind models of Cranmer et al. (2007).For a range of fast and slow solar wind solutions, we foundthat Q TR is typically between 1 . × - and 4 × - erg cm - s - , and F H , TR is typically between 8 × and 3 × ergcm - s - . This results in h usually being between 0.5 and 1.5.It is important to also verify that F H , TR is less than the en-ergy flux carried “passively” by the Alfvén waves as theypropagate up from the photosphere. The latter quantity, whichwe call F A , TR , represents the upper limit of available energyin the waves (at the TR) that can be extracted by the turbu-lent heating. Assuming that wave flux is conserved (i.e., that M A ≪ F A , TR = f ∗ F A ∗ / f TR . For the cool-star models discussed in Section 5, we found that the ratio F H , TR / F A , TR is usually around 0.1 to 0.5. Only in two casesdid it exceed 1 (albeit with values no larger than 1.5), and inthose cases we capped F H , TR to be equal to F A , TR to maintainenergy conservation.To evaluate Equation (26), we also need to estimate themagnitude of the downward conductive flux F cond from the hotcorona. For the solar TR and low corona, Withbroe (1988)found there to be an approximate balance between conduc-tion and radiation losses. Withbroe (1988) determined that F cond ≈ c rad P TR , where P TR is the gas pressure at the TR, andthe constant of proportionality is c rad = s κ e k Z T TR T Λ ( T ) T / dT (29)where κ e is the electron thermal conductivity, T ≈ K isa representative chromospheric temperature, T TR = 2 × K,and c rad has units of speed. We evaluated the above integral tobe able to scale out the metallicity-dependent factor given inEquation (21) above, and found that c rad ≈ s Λ max ( Z ) Λ max ( Z ⊙ ) km s - . (30)We used this expression to estimate F cond = c rad P TR . For thespecific case of the Sun, conduction is relatively unimportantin open flux tubes, since F cond . . F H , TR . For the other starsmodeled in this paper, the ratio F cond / F H , TR spanned severalorders of magnitude from 10 - to 10 - . In the eventualitythat the estimated value of F cond may exceed F H , TR , one wouldneed an improved description of the coronal temperature T ( r )to compute a more accurate value of the conduction flux. Inour numerical code, however, we do not allow F cond to exceeda value of ξ F H , TR , where ξ is an arbitrary constant that wefixed to a value of 0.9. This condition was not met for any ofthe stars in the database of Section 5.Once these energy fluxes are computed, we then compute ˙ M using Equations (23), (26), and (28), as well as the otherdefinitions given above. Interestingly, this can be done with-out needing to know the temperature profile T ( r ). From a cer-tain perspective, the corona’s thermal response to the heatingrate Q may be considered to be just an intermediate step to-ward the “final” outcome of a kinetic-energy-dominated out- flow far from the star. However, it should be possible toestimate the maximum coronal temperature T max by invert-ing scaling laws given by, e.g., Hammer et al. (1996) andSchwadron & McComas (2003).The mass loss rate given by Equation (26) depends on ourassumption that u ∞ = V esc . Equation (25) shows that larger as-sumed values of u ∞ would give rise to lower mass loss rates,and smaller values of u ∞ would give larger mass loss rates.For the solar wind there is roughly a factor of three variationin u ∞ , from about 0 . V esc to 1 . V esc . For other stars, it is rareto see observations where u ∞ exceeds V esc , and Judge (1992)found generally that u ∞ < V esc for luminous evolved stars.Even in the extreme case of u ∞ = 0, however, Equation (25)would give only two times the mass loss assumed by Equa-tion (26). When compared to the larger typical observationaluncertainties in ˙ M , factors of two are not a major concern.The Sun’s mass loss rate of 2 × - to 3 × - M ⊙ yr - ismodeled reasonably well with the model described here. Thephotospheric energy flux of Alfvén waves is F A ∗ ≈ . × erg cm - s - , and the photospheric wave amplitude is v ⊥∗ ≈ .
28 km s - (Cranmer et al. 2007). Although magnetogramobservations sometimes give filling factors f ∗ as large as 1%at solar maximum, values of 0.1% tend to better representthe coronal holes that are connected to the largest volume ofopen flux tubes (see Figure 3 of Cranmer & van Ballegooijen2005). Assuming f ∗ = 0 .
001 and values for the other constantsof α = 0 . θ = 1 /
3, Equations (22–23) give ρ TR ≈ × - g cm - and Q TR ≈ × - erg cm - s - , which are inagreement with the models of Cranmer et al. (2007) and oth-ers. Thus, with h = 0 .
5, Equation (26) gives ˙ M ≈ . × - M ⊙ yr - .Although the above calculation of ˙ M is relatively straight-forward, it has not been boiled down to a simple scalinglaw such as that of Reimers (1975, 1977), Mullan (1978),or Schröder & Cuntz (2005). However, if we make the fur-ther assumptions that ˜ α and h are fixed constants, and that F H , TR ≫ F cond , we can isolate several interesting scalings:1. The ultimate driving of the wind comes from the basalflux of Alfvén wave energy F A ∗ . Schröder & Cuntz(2005) assumed that ˙ M scales linearly with F A ∗ , but inour case we can combine the above equations with thedefinition of Q ∗ to find ˙ M ∝ F / ∗ , which is noticeablysteeper than a pure linear dependence. This positivefeedback is qualitatively similar to what occurs in ra-diatively driven winds of more massive stars, for whichthe mass loss rate is proportional to the radiative flux (orluminosity) to a power larger than one (i.e., ˙ M ∝ L . ∗ ;Castor et al. 1975; Owocki 2004).2. Extracting the dependence on magnetic filling factor,we found that ˙ M ∝ f (4 + θ ) / ∗ . Using the range of θ fromsolar models (0.3–0.5), this gives a relatively narrowrange of exponents, ˙ M ∝ f . ∗ to f . ∗ . Saar (1996a) esti-mated that f ∗ ∝ P - . for rotation periods P rot > Our approach, which ignores the details of this intermediate step, is anapproximation that also sidesteps some other important issues. For example,a time-steady wind solution should pass through one or more critical points,and it and should also satisfy physical boundary conditions at r = R ∗ and r → ∞ . In Section 6 we summarize the necessary steps to producing moreself-consistent versions of this model. HEORETICAL MASS LOSS RATES OF COOL STARS 7 F IG . 4.— Density dependence of the heating rate Q defined in Equation (14)compared to the density dependence of the maximum radiative cooling ratefrom Equation (20) (dashed curve). For the Sun, a numerical model (blacksolid curve) compares favorably with a simple analytic “bridging” betweenthe near-star ( Q ∝ ρ / ) and distant ( Q ∝ ρ / ) scalings discussed in the text(black dotted curve). For an example evolved giant star (blue dot-dashedcurve), the transition to the steeper density dependence occurs to the right ofthe cooling boundary. possible to estimate ˙ M ∝ P - . .These simple scaling relations are given only for illustrativepurposes (see also Equation (45) below). The predictions ofour “hot” coronal mass loss model should be considered to bethe solutions of the full set of Equations (17–30). Cold Wave-Driven Mass Loss
In a high-density stellar atmosphere, it is possible that theturbulent heating described by Equation (14) could be bal-anced by radiative cooling even very far from the star. Inthat case, hot coronal temperatures may never occur (see, e.g.,Suzuki 2007; Cranmer 2008). The density dependence of theheating rate Q determines whether radiative cooling remainsimportant at large distances, and Figure 4 shows two exam-ples of how Q may vary as a function of ρ . Near the stellarsurface, where v ⊥ ∝ ρ - / and B ∝ ρ / , we can combine var-ious assumptions to estimate that Q ∝ ρ / . However, furtherfrom the star, where v ⊥ ∝ ρ + / and B ∝ ρ , the density depen-dence becomes steeper, with Q ∝ ρ / .Figure 4 compares the modeled heating rates with the“maximum cooling boundary” implied by Equation (20). Thesolar model crosses the boundary, and thus undergoes a tran-sition to a hot corona. The solid curve was taken from a nu-merical model of fast solar wind from a polar coronal hole(Cranmer et al. 2007). On the other hand, a model for arepresentative late-type giant sits to the right of the coolingboundary, which implies that radiative losses can maintain thecircumstellar temperature at chromospheric values of ∼ K even at large distances. The two models differ becausethe density at which M A ≈ Q curves undergoa change in slope) for the giant is several orders of magnitudelarger than the corresponding density for the solar case. Thisdensity is an output of a given mass loss model and cannot bespecified a priori.In this section we develop a model for cool-star mass lossunder the assumption of strong radiative cooling. In this case the Parker (1958) gas pressure driving mechanism cannotdrive a significant outflow. However, when the flux of Alfvénwaves is large, they can impart a strong bulk acceleration tothe plasma due to wave pressure, which is a nondissipative netponderomotive force exerted by virtue of wave propagationthrough an inhomogeneous medium (Bretherton & Garrett1968; Jacques 1977). The subsequent calculation of ˙ M fora “cold wave-driven” stellar wind largely follows the devel-opment of Holzer et al. (1983) (see also Cranmer 2009).Three key assumptions are: (1) that the Alfvén wave ampli-tudes in the wind are larger than the local sound speeds, (2)that there is negligible wave damping between the stellar sur-face and the wave-modified critical point of the flow, and (3)that the critical point occurs far enough from the stellar sur-face that f ≈ M A ≪ r crit R ∗ ≈ / + ( v ⊥∗ / V esc ) . (31)Once the critical point radius is known, it becomes possibleto use the known properties of the Alfvén waves to determinethe wind velocity and density at the critical point. Holzer et al.(1983) found analytic solutions for these quantities in the lim-iting case of M A ≪ M A and themass loss rate, but we also continue to use Equation (31) thatwas derived in the limit of M A ≪
1. There are three unknownquantities and three equations to constrain them. The threeunknowns are the critical point values of the wind speed u ,density ρ , and wave amplitude v ⊥ . The first equation is theconstraint that the right-hand side of the time-steady momen-tum equation must sum to zero at the critical point of the flow(e.g., Parker 1958). For the conditions described above, thisgives 2 u r crit - GM ∗ r = 0 , (32)and it is solved straightforwardly for u crit . The second andthird equations are, respectively, the definition of the criticalpoint velocity in the “cold” limit of zero gas pressure, u = v ⊥ (cid:18) + M A + M A (cid:19) (33)and the conservation of wave action as given by Equation(13). The fact that V A appears in these equations and dependson the (still unknown) density makes it difficult to find an ex-plicit analytic solution for ρ crit . We again use iteration froman initial guess to reach a self-consistent solution for u , ρ , and v ⊥ at the critical point. The stellar wind’s mass loss rate isthus determined from ˙ M = 4 π r u crit ρ crit .Because the mass loss rate is set at the critical point, wedo not need to specify the terminal speed u ∞ . For most im-plementations of the above model, the denominator in Equa-tion (31) is close to unity and thus we have u ≈ V /
7, CRANMER AND SAARor that u crit is about 38% of the presumed value of u ∞ .Of course, there have been stellar wind models with non-monotonic radial variations of u ( r ), with u crit > u ∞ (e.g.,Falceta-Gonçalves et al. 2006). It is also possible for “toomuch” mass to be driven past the critical point, such thatparcels of gas may be decelerated to stagnation at some heightabove r crit and thus would want to fall back down towards thestar. In reality, this parcel would collide with other parcels thatare still accelerating, and a stochastic collection of shockedclumps is likely to result. Interactions between these parcelsmay result in an extra degree of collisional heating that couldact as an extended source of gas pressure to help maintain amean net outward flow. Situations similar to this have beensuggested to occur in the outflows of pulsating cool stars(Bowen 1988; Struck et al. 2004), T Tauri stars (Cranmer2008), and luminous blue variables (van Marle et al. 2009). Combining Hot and Cold Models
A proper treatment of a stellar wind powered by MHDturbulence—and accelerated by a combination of gas pressureand wave pressure effects—requires a self-consistent numeri-cal solution to the conservation equations (e.g., Cranmer et al.2007; Suzuki 2007; Cohen et al. 2009; Airapetian et al.2010). However, in this paper, we explore simpler ways ofestimating the combined effects of both processes.Sections 3.1 and 3.2 gave us independent estimates for themass loss rate assuming only gas pressure or wave pressurewere active in the flux tube of interest. We refer to these twomass loss rates as ˙ M hot and ˙ M cold , respectively. It seems clearthat when one of these values is much larger than the other,then one process is dominant and the actual mass loss rateshould be close to that larger value. For the manifestly “hot”example of the Sun, we found that ˙ M hot / ˙ M cold ≈
20, whichcorrectly implies that gas pressure driving is dominant. Formost examples of late-type giants with L ∗ > L ⊙ , the ratio ˙ M hot / ˙ M cold was found to decrease to values between about 0.1and 3. This could mean that gas and wave pressure gradientsare of the same order of magnitude for these stars.One of the most straightforward things that can be doneis to assume the combined effect of gas and wave pressureproduces a mass loss rate equal to the sum of the two in-dividual components, ˙ M hot + ˙ M cold . This preserves the ideathat one dominant mechanism should determine ˙ M when theother would predict a negligibly small effect. It also makessense based on Equation (24), which shows how the energyfluxes sum together linearly in the internal energy equation.If there were multiple sources of input energy flux, Equation(26) would show that the resulting mass loss rate should beproportional to their sum.However, there is one complication that hinders us fromsimply adding together ˙ M hot and ˙ M cold . The calculation of ˙ M hot from Section 3.1 contains the assumption that the TRturbulent heating always obeys the near-star density scaling Q ∝ ρ / . It therefore predicts that all stars eventually undergoa transition to a hot corona. For some stars (like the late-typegiant in Figure 4), however, we know that there should be nocorona and it is erroneous to assume that ˙ M hot has any realmeaning. Thus, for each model we compute the wind speedat the TR from mass flux conservation, u TR = ˙ M hot π R ∗ f TR ρ TR , (34)and we demand that for ˙ M hot to have a consistent interpreta- tion, the TR Mach number M A , TR = u TR / V A , TR should be muchsmaller than one. As expected, this condition was found tobe violated for late-type giants having L ∗ & L ⊙ . Thus, inthese cases we should replace ˙ M hot with either a drastically re-duced value or zero—the latter in cases where the Q ( ρ ) curvealways falls to the right of the maximum cooling boundaryin Figure 4. After some experimentation, we found that re-ducing the initially computed value of Q TR by a factor ofexp( - M , TR ) does a reasonably good job of reproducing theresult of using a more consistent Q ( ρ ) function. Thus, we pro-pose that the summing of the “hot” and “cold” mass loss ratesbe done with the following approximate expression, ˙ M ≈ ˙ M cold + ˙ M hot exp (cid:0) - M , TR (cid:1) (35)where ˙ M hot and M A , TR are computed using the assumptions ofSection 3.1 and ˙ M cold is computed using the model given inSection 3.2. MAGNETIC ACTIVITY AND ROTATION
An important ingredient in the above models—which re-mains unspecified for most stars—is the photospheric fillingfactor f ∗ . It is now well-known that both f ∗ and the mag-netic flux density B ∗ f ∗ exhibit significant correlations withstellar rotation speed (Saar & Linsky 1986; Marcy & Basri1989; Montesinos & Jordan 1993; Saar 2001). For many starsthe rotation rate also scales with age, chromospheric activity,and coronal X-ray emission (Skumanich 1972; Noyes et al.1984; Pizzolato et al. 2003; Mamajek & Hillenbrand 2008).A prevalent explanation for these correlations is that anMHD dynamo amplifies the magnetic flux in proportionto the large-scale energy input from differential rotation(e.g., Parker 1979; Montesinos et al. 2001; Bushby 2003;Moss & Sokoloff 2009; Christensen et al. 2009; I¸sık et al.2011).In this section we construct an empirical scaling relationthat will allow a reasonably accurate determination of f ∗ asa function of P rot and the other stellar parameters. Otherestimates of this relationship have been made in the past(Montesinos & Jordan 1993; Ste¸pien 1994; Saar 1996a, 2001;Cuntz et al. 1998; Fawzy et al. 2002). However, since ourgoal is to apply this relation to stellar wind acceleration (inopen flux tubes that cover a subset of the inferred f ∗ area) andto evolved giants (which are greatly undersampled in obser-vational studies of f ∗ ), we aim to reanalyze the existing datarather than rely on other published scalings.Table 1 lists the properties of 29 stars that have reliablemeasurements of their fundamental parameters, rotation rates,and either independent or combined values of B ∗ and f ∗ . Thesources for these values are given as numbered referencesin the final column. In many cases the available sourcesgave only a subset of the basic stellar parameters. Whennecessary, we used Equation (1) and information from theNASA/IPAC/NExScI Star and Exoplanet Database (NStED) to fill in missing values (see Berriman et al. 2010). Table 1also gives approximate “quality factors” q that describe therelative accuracy of the measurements, and in the Appendixwe describe these factors in more detail.It has been known for some time that, for dwarf stars, B ∗ never appears to be very far from the equipartition fieldstrength B eq (e.g., Saar & Linsky 1986). Figure 5 plots theratio B ∗ / B eq for the measurements in Table 1 that have sepa-rate determinations of B ∗ and f ∗ . The sizes of the symbols are http://nsted.ipac.caltech.edu/ HEORETICAL MASS LOSS RATES OF COOL STARS 9
Table 1. M
AGNETIC A CTIVITY D ATA FOR
G/K/M S
TARS
Name T eff (K) log g M ∗ / M ⊙ R ∗ / R ⊙ L ∗ / L ⊙ P rot (d) [Fe/H] B ∗ × f ∗ (G) Qual. Ref.Sun 5770 4.44 1 1 1 25.3 0 1400 × (0.001–0.01) — —59 Vir (G0 V) 6234 4.60 1.17 0.897 1.10 3.3 + .
280 1000 × χ Ori (G0 V) 5955 4.30 0.67 0.962 1.05 5.2 - .
039 1000 × + .
024 1800 × - .
013 330 3 9, 10, 79 Cet (G2 V) 5790 4.40 1.0 1.04 1.11 7.7 + .
159 1400 × + .
137 1700 × κ Cet (G5 V) 5771 4.56 1.02 0.877 0.770 9.4 + .
056 321 2 9, 5, 7392 2 9, 5, 7406 2 9, 5, 7480 2 9, 5, 71500 × ξ Boo A (G8 V) 5551 4.57 0.86 0.801 0.550 6.2 - .
122 1600 × × × × - .
049 1700 × + .
040 1200 × + .
108 1700 × + .
067 1500 × + .
330 3500 × - .
193 2100 × - .
27 1600 × - .
170 2400 × ǫ Eri (K4.5 V) 5094 4.60 0.83 0.754 0.345 11.7 - .
097 165 3 18, 5, 10, 71000 × × × - .
206 1500 × - .
195 60 2 19, 5, 10, 7V833 Tau (K5e V) 4450 4.57 0.80 0.77 0.209 1.85 + .
340 2600 × - .
193 1200 × - .
075 2500 × × + .
050 2800 × .
00 3000 × - .
75 3300 3 25, 104000 × - .
200 3900 3 26, 27, 103400 × - .
238 2000 3 28, 6, 29, 30, 102400 × × + .
07 3300 3 31, 10, 32, 6(1) Anderson et al. (2010), (2) Pizzolato et al. (2003), (3) Saar (1996a), (4) Kovtyukh et al. (2004), (5) Montesinos & Jordan (1993), (6) NStED, (7) Soubiran et al.(2010), (8) Mishenina et al. (2008), (9) Baumann et al. (2010), (10) Saar (2001), (11) Masana et al. (2006), (12) Fernandes et al. (1998), (13) Saar (1996b),(14) Eggenberger et al. (2008), (15) Kovári et al. (2004), (16) Taylor (2003), (17) Marcy & Basri (1989), (18) Gai et al. (2008), (19) Wood & Linsky (2006),(20) Pettersen (1989), (21) Vogt et al. (1983), (22) Kervella et al. (2008), (23) Wood et al. (2005b), (24) Alonso et al. (1996), (25) Favata et al. (2000), (26) Jenkins et al.(2009), (27) Reid et al. (1995), (28) Morales et al. (2008), (29) Kiraga & Ste¸pie´n (2007), (30) Eggen (1996), (31) Veeder (1974), (32) Bonfils et al. (2005). proportional to the observational quality factors, and all statis-tical fits and moments discussed below were weighted linearlywith q . Figure 5(a) shows that there is no strong correlationof B ∗ / B eq with T eff . Saar (1996a) found a slight increase in B ∗ / B eq for the most rapid rotators ( P rot < B ∗ ∝ P - . , but we do not consider it significantenough to apply it below or to extrapolate it to longer rotationperiods.We found that the q -weighted mean value of B ∗ / B eq for theentire sample (1.16, with standard deviation ± .
38) is onlymarginally higher than the mean value for the subset of slowerrotating, non-saturated stars with P rot > ± . f ∗ for the caseswhere only the product B ∗ f ∗ has been measured.A primary indicator of stellar magnetic activity appears tobe the photospheric filling factor f ∗ . There have been a num-ber of different proposed ways to express the general anti-correlation between activity and rotation period. Noyes et al.(1984) found that indices of chromospheric activity corre-late better with the so-called Rossby number Ro ≡ P rot /τ c ,where τ c is a measure of the convective turnover time, thanwith P rot alone. For other data sets, however, the useful-ness of the Rossby number has been called into question(Basri 1986; Ste¸pien 1994). Saar (1991) postulated that B ∗ f ∗ (and presumably also f ∗ itself) is proportional to Ro - (see also Montesinos & Jordan 1993; Cuntz et al. 1998; Saar2001; Fawzy et al. 2002).0 CRANMER AND SAAR F IG . 5.— Observational data for B ∗ / B eq as a function of (a) effectivetemperature and (b) rotation rate. Solid lines in (a) show mean values for alldata (red) and for only stars having P rot > ± σ of the means. Quality factors are denoted bycrosses ( q = 1), triangles ( q = 2), squares ( q = 3) and circles ( q = 4). To compute the Rossby number for a given star, we needto know the convective turnover time τ c . Figure 6 comparesseveral past calculations of τ c with one another. For most starswe will utilize a parameterized fit to the set of ZAMS stellarmodels given by Gunn et al. (1998), τ c = 314 .
24 exp " - (cid:18) T eff . (cid:19) - (cid:18) T eff (cid:19) + . , (36)where τ c is expressed in units of days and the fit is valid for theapproximate range 3300 . T eff . τ c may depend on other stellar parameters besides effec-tive temperature, but more recent sets of models (Landin et al.2010; Barnes & Kim 2010; Kitchatinov & Olemskoy 2011)also found reasonably monotonic behavior as a function of T eff for a broad range of stellar ages and masses.There are indications that the simple relationship between τ c and T eff seen for main-sequence stars is not universal. Forexample,1. Low-mass M dwarfs (with M ∗ . . M ⊙ ) are likelyto be fully convective, and thus their dynamos arelikely to be driven by fundamentally different processesthan exist in more massive stars (Mullan & MacDonald2001; Reiners & Basri 2007; Irwin et al. 2011). There F IG . 6.— Estimates of the convective turnover time τ c as a function of T eff .ZAMS models of Gunn et al. (1998) were fit by Equation (36) (black solidcurve). The Gunn et al. (1998) evolutionary track for a 2 . M ⊙ star (bluedashed curve) outlines an upper limit for τ c at intermediate temperatures. Lo-cal values of τ c from Landin et al. (2010) (green symbols) and Barnes & Kim(2010) (orange dot-dashed curve), the parameterization given by Noyes et al.(1984) (black dotted curve), and the M dwarf estimate of τ c ≈
70 d used byReiners et al. (2009) (red box labeled by “dM”) are also shown. is also some disagreement about the relevant τ c val-ues for these stars. Figure 6 shows that the models ofBarnes & Kim (2010) exhibit a slight discontinuity atthe fully convective boundary. The Reiners et al. (2009)semi-empirical estimate of τ c ≈
70 days for M dwarfs isabout a factor of 2–3 lower than that of Barnes & Kim(2010). However, because the Reiners et al. (2009)value is in reasonable agreement with an extrapolationof Equation (36) to lower effective temperatures, wewill just use this expression and not make any specialadjustments to the Rossby numbers of fully convectiveM dwarfs.2. Luminous evolved giants exhibit qualitatively differentinterior properties than do main sequence stars of sim-ilar T eff . Despite not having firm measurements of themagnetic activities of evolved giants, we will want toestimate f ∗ for such stars in order to compute their massloss rates. Gondoin (2005, 2007) found that the cor-relation between X-ray activity and rotation in G andK giants is consistent with that of main-sequence starsif the larger values of τ c from the evolved models ofGunn et al. (1998) were used instead of the ZAMS val-ues (see the blue dashed curve in Figure 6). Similarly,Hall (1994) calculated luminosity-dependent scalingfactors that can be used to multiply the ZAMS valueof τ c to obtain a consistent relation between rotationand photometric activity (see also Choi et al. 1995). Wefound that the above results can be generally repro-duced by multiplying the ZAMS value of τ c by a fac-tor ( g ⊙ / g ) . , which applies only for low-gravity sub-giants and giants (i.e., only for g < g ⊙ ). In Section 5we explore the extent to which this kind of approximatecorrection factor helps to explain the activity and massloss of evolved stars.A slightly different way of estimating the magnetic flux ofa rotating star is to take advantage of a proposed “magneticHEORETICAL MASS LOSS RATES OF COOL STARS 11Bode’s law;” i.e., the conjecture that the star’s magnetic mo-ment scales linearly with its angular momentum (Arge et al.1995; Baliunas et al. 1996). Using the stellar parameters de-fined above, this corresponds approximately to B ∗ f ∗ R ∗ ∝ M ∗ R ∗ P rot . (37)The above relationship does not take into account variationsof the moment of inertia (for different stars) away from anidealized scaling of I ∼ M ∗ R ∗ , and it assumes the magneticmoment is dominated by a large-scale dipole component. Itis possible to test this idea with the data given in Table 1 byevaluating the correlation between f ∗ and M ∗ ( R ∗ P rot B ∗ ) - .Figure 7 shows how the empirical set of f ∗ values corre-lates with rotation period, Rossby number, and the proposedmagnetic Bode’s law. Rather than use the Rossby number it-self, we instead plot the data in Figure 7(b) as a function of anormalized ratio Ro / Ro ⊙ , where according to Equation (36),the Sun’s Rossby number Ro ⊙ = 1 .
96. Such a normalizationallows us to neglect any scaling discrepancies between “lo-cal” and “global” definitions of τ c (e.g., Pizzolato et al. 2001;Landin et al. 2010). The Sun’s large range of measured f ∗ values (10 - to 10 - ) is indicated with a vertical bar, and itis likely that all other stars exhibit such a range on both rota-tional and dynamo-cycle time scales.Figures 7(a) and 7(c) show that the correlations with P rot and the proposed magnetic Bode’s law are not especiallystrong. However, if all of the lowest quality ( q = 1) mea-surements were removed, the correlation with P rot would beimproved significantly. Figure 7(b) shows that the Rossbynumber seems to be a slightly better ordering parameter, andit compares the individual data points with several functionalrelationships. The blue and red solid curves are subjective fitsto the minimum and maximum bounds on the envelope of datapoints, with f min = 0 . + ( x / . . ] . , (38) f max = 11 + ( x / . . (39)where x = Ro / Ro ⊙ . We also show empirical and theoreti-cal fitting formulae from Montesinos & Jordan (1993). Othercomparisons could also be made with relationships given byCuntz et al. (1998), Fawzy et al. (2002), and others, but theyall appear to fall near the green and red curves.Note that for slow rotation rates (i.e., large Rossby num-bers) the scaling laws shown in Figure 7(b) imply a sig-nificantly steeper decline of f ∗ than has been suggested inthe past. For example, Saar (1991) estimated f ∗ ∝ Ro - ,and Saar (1996a) estimated f ∗ ∝ P - . . On the other hand,our empirical upper and lower bounds suggest f max ∝ Ro - . and f min ∝ Ro - . respectively. This is similar to the ob-served relationship between Rossby number and X-ray activ-ity. Mamajek & Hillenbrand (2008) found that the ratio ofX-ray to bolometric luminosity L X / L bol drops by about a fac-tor of 700 as the Rossby number increases by a factor of tenfrom 0.25 to 2.5 (see also Wright et al. 2011). This corre-sponds very roughly to a power-law decrease of Ro - . . Itsagreement with the behavior of f ∗ shown above is also consis-tent with existing empirical correlations between X-rays andmagnetic activity (Pevtsov et al. 2003).In addition to the rotational scaling of f ∗ with Ro,there is also likely to be a “basal” lower limit on the F IG . 7.— Comparison of possible correlations between measured f ∗ fillingfactors with (a) rotation rate, (b) Rossby number, and (c) a magnetic Bode’slaw parameter (see Equation (37)). Solid curves in (b) denote the lower (blue)and upper (red) envelopes surrounding the data, and green curves show fittingformulae from Equations 2.3 (dotted; empirical) and 7.3 (dashed; theoretical)of Montesinos & Jordan (1993). Quality factors are denoted by the samesymbols used in Figure 5, and the Sun’s range of f ∗ is shown with a verticalbar. outer atmospheric activity of a star (e.g., Schrijver 1987;Cuntz et al. 1999; Bercik et al. 2005; Takeda & Takada-Hidai2011; Pérez Martínez et al. 2011). Whether this lower limitis the result of acoustic waves, a turbulent dynamo, or someother physical process, there is probably a minimum valueof f ∗ that is independent of rotation rate. For example,Bercik et al. (2005) found that turbulent dynamos in main se-quence stars can generate a flux density B ∗ f ∗ ≈ B ∗ , we can estimate a basal filling factor for thesestars of f ∗ ≈ . f basal = 10 - to be used in the mass loss models below.Before moving on to apply the empirical values of f ∗ to ourmodel of mass loss, we emphasize that the measurements donot directly provide the filling factor of open magnetic fluxtubes. Ideally, Zeeman broadening measurements should besensitive to the total flux in strong magnetic elements on thestellar surface, no matter whether the field lines are closedor open. In many cases, however, the closed-loop active re-gions have significantly stronger local field strengths than theopen regions. Therefore the closed-field regions are likely todominate the spectral line broadening that gives rise to theobservational determinations of f ∗ (see the Appendix). With-out spatially resolved magnetic field measurements, we do notyet have a definitive way to predict how a given star dividesup its flux tubes between open and closed. Mestel & Spruit(1987) claimed that as the rotation rate increases (from slowvalues similar to the Sun’s), the relative fraction of closedfield regions should first increase, then eventually it shoulddecrease as centrifugal forces strip the field lines open. Wecan speculate that the spread in the measured f ∗ data may tellus something about the closed and open fractions. Becauseclosed-loop active regions tend to have stronger fields thanopen coronal holes, the lower and upper envelopes that sur-round the data in Figure 7(b) could be good proxies for thefilling factors of open and closed regions, respectively. Morespecifically, we hypothesize that f min is seen when no activeregions are present on the visible surface (i.e., f min ≈ f open )and that f max is seen when active regions dominate the ob-served magnetic flux. This idea is tested, in a limited way, inSection 5.2. RESULTS
Here we present the results of solving the mass loss equa-tions derived in Section 3 using the empirical estimates forthe rotational dependence of the magnetic filling factor de-rived in Section 4. For hot coronal mass loss, we assumedvalues for the dimensionless parameters α = 0 . h = 0 .
5, and θ = 1 /
3. As discussed above, these values were “calibrated”from our more detailed knowledge of the Sun’s coronal heat-ing and wind acceleration. Our use of these values for otherstars is an extrapolation that can be tested by comparison withobserved mass loss rates.
Database of Stellar Mass Loss Rates
Figure 8 is a broad overview of observed stellar mass loss.It plots the locations of individual stars in a Hertzsprung-Russell type diagram with their mass loss rates shown as sym-bol color (see also de Jager et al. 1988). A box illustrates theapproximate regime of parameter space covered by the mod-els developed in this paper; it extends from the main sequenceup through the regime of the so-called “hybrid chromosphere”stars (Hartmann et al. 1980), and possibly also into the pa-rameter space of cool luminous supergiants. In addition to thecool-star data discussed below, we also include in Figure 8measured mass loss rates of hot, massive stars (Waters et al.1987; Lamers et al. 1999; Mokiem et al. 2007; Searle et al.2008), FGK supergiants (de Jager et al. 1988), AGB stars(Bergeat & Chevallier 2005; Guandalini 2010), red giants inglobular clusters (Mészáros et al. 2009), and M dwarfs in pre-cataclysmic variable binaries (Debes 2006). Many of thesestars are not included in the subsequent analysis because wehave no firm rotation periods or magnetic activity indices forthem. Table 2 lists the properties of 47 stars for which our knowl-edge appears to be complete enough to be able to comparetheoretical and observed values of ˙ M . For the Sun, therange of volume-integrated mass loss rates comes from Wang(1998). The sources for all listed values are given as num-bered references that continue the sequence started in Table1; the citations corresponding to numbers 1–32 are given inTable 1. In cases where T eff , log g , or [Fe/H] were estimatedfrom the PASTEL database (Soubiran et al. 2010), we aver-aged together multiple measurements when more than onewas given. In the few cases where the same star appears inboth Table 1 and Table 2, for consistency’s sake we will re-compute f ∗ from the star’s rotation period when calculatingtheoretical mass loss rates (see Section 5.2).For binary systems with astrospheric measurements of ˙ M (see, e.g., Wood et al. 2002), the numbers given are assumedto be the sum of both stars’ mass loss rates. We list that samevalue for both components and denote it with “(A + B).” We didnot utilize the published astrospheric measurements of Prox-ima Cen and 40 Eri A, which gave only upper limits on ˙ M ,and λ And and DK UMa, which had uncertain detections ofastrospheric H I Ly α absorption (Wood et al. 2005a,b).At the bottom of Table 2 we list three stars that have pa-rameters at the outer bounds of what we intend to model.They are test cases for the limits of applicability of thephysical processes summarized in Section 3. EV Lac isan active M dwarf and flare star that probably has a fullyconvective interior (e.g., Osten et al. 2010). Such starsmay exhibit qualitatively different mechanisms of mass lossand rotation-activity correlation than do stars higher upthe main sequence (Mullan 1996; Reiners & Basri 2007;Irwin et al. 2011; Martínez-Arnáiz et al. 2011; Vidotto et al.2011). V Hya is an N-type carbon star with an extendedand asymmetric AGB envelope and evidence for rapid ro-tation (Barnbaum et al. 1995; Knapp et al. 1999). 89 Heris a post-AGB yellow supergiant with multiple detectionsof circumstellar nebular material (Sargent & Osmer 1969;Bujarrabal et al. 2007). It is worthwhile to investigate to whatextent the mass loss mechanisms proposed in this paper couldbe applicable to these kinds of stars.Not all stars in Table 2 have precise measurements for theirrotation period. For 61 Vir and 70 Oph B, we used publishedestimates of the rotation period that were obtained from theknown correlation between rotation and chromospheric Ca IIactivity (Baliunas et al. 1996). For essentially all stars moreluminous than ∼ L ⊙ (with the exception of HR 6902; seeGriffin 1988) we estimated P rot via spectroscopic determina-tions of v sin i from the rotational broadening of photosphericabsorption lines. The inclination angle i is the main unknownquantity. Chandrasekhar & Münch (1950) found that for anisotropically distributed set of inclination vectors, the meanvalue of sin i is π/
4. Thus, we estimate a mean rotation pe-riod h P rot i = 2 π R ∗ (4 /π ) v sin i . (40)Note that the median of sin i for an isotropic distribution isnot equal to the mean; the former is given by √ /
2. In orderto encompass both values, as well as the majority of “mostlikely” values of P rot , we can adopt generous uncertainty lim-its for which we will estimate f ∗ and the other derived quan-tities for mass loss. For the isotropic distribution of direc-tion vectors, the quantity sin i falls between 0.5 and 1 approx-imately 87% of the time. This is a reasonably good definitionHEORETICAL MASS LOSS RATES OF COOL STARS 13 F IG . 8.— Hertzsprung-Russell diagram showing observed mass loss rates. The color scale at lower-left specifies log ˙ M , where ˙ M is measured in M ⊙ yr - . Anestimate for the ZAMS is shown in gray, and the approximate domain of parameter space covered by the models of this paper is outlined by a black dotted box. for uncertainty bounds that would correspond to ± . σ if thedistribution were Gaussian. Thus, for stars with only v sin i measurements, we use the following values as error bars onthe derived rotation period:2 π . P rot h P rot i . π . (41) Comparing Predictions with Observations
We applied the combined model for mass loss that culmi-nated in Equation (35) to the stars listed in Table 2. Belowwe show results of direct forward modeling; i.e., utilizing aknown relationship for f ∗ as a function of Rossby number.First, however, we wanted to investigate whether or not asingle monotonic relationship for f ∗ (Ro) could produce massloss rates that were even remotely close to the measured val-ues. Thus, we produced trial grids of models in which f ∗ wastreated as a free parameter. For each star, we varied f ∗ from10 - to 1 and found the empirical value of the filling factor( f emp ) for which the modeled value of ˙ M matched the ob-served value given in Table 2. For the four binaries that haveonly systemic measurements of ˙ M we summed the model pre-dictions for each component and made a single comparisonwith the observations.Figure 9 shows the result of this process of “working back-wards” from the measured mass loss rates. The empiricallyconstrained f emp values are plotted against Rossby number,which is defined with (a) the simple Gunn et al. (1998) ZAMSvalue for τ c (Equation (36)) and (b) a gravity-modified ver-sion of τ c that gives giants larger convective overturn times. We do not show f emp for the test-case stars EV Lac or 89 Her, sinceno values in the range 10 - –1 produced agreement with their observed massloss rates. Extrapolating from the grid of modeled ˙ M values to the observedvalue would have required impossible values of f emp & f ∗ (Ro) diagram just like the F6 main sequence star HD 68456(Anderson et al. 2010). This may be relevant for deducing the relevant phys-ical processes in other F-type stars with T eff ≈ We varied the exponent in the gravity modification term andfound that multiplying the ZAMS τ c by ( g ⊙ / g ) . gives thenarrowest distribution of f ∗ versus Ro. The optimal exponentof 0.18 is very close to the value of 0.23 that we found repro-duced the results of Hall (1994) and Gondoin (2005, 2007).In Figure 9 we also show the same curves from Figure 7(b)that outline the measured range of filling factors. The lowerenvelope curve f min , defined in Equation (38), appears to bea good match to the gravity-modified empirical values f emp .This provides circumstantial evidence that f min is indeed anappropriate proxy for the filling factor of open flux tubes as afunction of Rossby number.We now put aside the empirical estimates for the filling fac-tor and use only Equation (38) for f ∗ = f min in the remainderof this paper. Table 3 shows some of the predicted propertiesof stellar coronae and winds for the 47 stars in our database.There were only seven stars for which Equation (38) gave afilling factor below the adopted “floor” value of f basal = 10 - ;we replaced f min with f basal in those cases. Table 3 also gives F H , TR , the coronal heat flux deposited at the TR for each star.It may be useful to use this to predict the X-ray flux associatedwith open-field regions on these stars, but we should note thatthe closed-field regions (which we do not model) are likely todominate the observed X-ray emission. We also list the vari-ous components of Equation (35) so that the contributions ofgas pressure and wave pressure can be assessed (see below).Figure 10 compares the theoretical and measured mass lossrates with one another as a function of L ∗ . For the four bi-nary systems listed in Table 2 with combined A + B mass lossrates, we separated the measured value into two pieces us-ing the modeled ˙ M ratio for the two components. (This wasdone for this figure only because the measured rates are shownas a function of a single star’s luminosity.) For stars withonly v sin i rotation period estimates, we used Equation (40) tocompute ˙ M for the central plotting symbol and the entries inTable 3, and we recomputed ˙ M for the lower and upper limits4 CRANMER AND SAAR Table 2. C
OOL S TAR M ASS L OSS D ATA
Name T eff (K) log g M ∗ / M ⊙ R ∗ / R ⊙ L ∗ / L ⊙ P rot (d) [Fe/H] - log( ˙ M / [ M ⊙ yr - ]) Ref.Sun 5770 4.44 1 1 1 25.3 0 13.5–13.7 — α Cen A (G2 V) 5886 4.31 1.105 1.224 1.622 29 + .
197 13.40 (A + B) 33, 34, 23, 7 α Cen B (K0 V) 5473 4.54 0.934 0.863 0.603 36.2 + .
230 13.40 (A + B) 33, 34, 35, 770 Oph A (K0 V) 5300 4.52 0.89 0.86 0.53 19.7 + .
040 11.70 (A + B) 14, 34, 2370 Oph B (K5 V) 4390 4.65 0.73 0.67 0.15 34 + .
040 11.70 (A + B) 14, 34, 36 ǫ Eri (K4.5 V) 5094 4.60 0.83 0.754 0.345 11.7 - .
097 12.22 18, 34, 23, 761 Cyg A (K5 V) 4425 4.63 0.69 0.665 0.153 35.4 - .
193 14.00 22, 34, 23, 7 ǫ Ind (K5 V) 4635 4.54 0.70 0.745 0.231 22 - .
088 14.00 37, 34, 23, 736 Oph A (K5 V) 5135 4.54 0.602 0.69 0.299 20.3 - .
206 12.52 (A + B) 19, 34, 736 Oph B (K5 V) 5103 4.58 0.486 0.59 0.213 22.9 - .
195 12.52 (A + B) 19, 34, 7 ξ Boo A (G8 V) 5551 4.57 0.86 0.801 0.550 6.2 - .
122 13.00 (A + B) 12, 34, 38, 7 ξ Boo B (K4 V) 4350 4.80 0.70 0.550 0.0977 11.5 - .
122 13.00 (A + B) 12, 34, 38, 761 Vir (G5 V) 5560 4.39 0.946 0.972 0.804 29 - .
002 14.22 39, 34, 36, 7 δ Eri (K0 IV) 5025 3.75 1.122 2.33 3.185 55.3 + .
069 13.10 40, 34, 23, 7 α Boo (K1.5 III) 4290 1.76 1.10 23 170 447 - .
53 9.60 41, 42, 7 α Tau (K5 III) 3898 1.33 1.5 44 394 648 - .
180 10.83 43, 44 γ Dra (K5 III) 3985 1.53 3.0 49 535 557 - .
150 11.06 43, 44HR 6902 (G9 IIb) 4900 1.99 3.86 33 566 220 + .
430 10.68 45, 46, 47 β And (M0 III) 3742 1.55 1.5 34 204 188 - .
04 10.19 48, 49, 7 β UMi (K4 III) 4040 1.27 1.3 44 475 1030 - .
26 10.01 48, 50, 51, 7 µ UMa (M0 III) 3700 0.69 1.5 92 1430 488 0.00 8.92 48, 49, 7 α TrA (K2 II-III) 4150 1.50 23.7 143 5500 888 - .
06 9.77 52, 53, 7 λ Vel (K4 Ib-II) 3820 0.64 7.0 210 8510 1250 + .
23 8.52 54, 55, 7BD +
01 3070 (RGB) 5130 2.70 0.749 6.4 25.6 50.9 - .
85 8.76 56, 57BD +
05 3098 (RGB) 4930 2.00 0.746 14.3 109 109 - .
40 8.91 56, 57BD +
09 2574 (RGB) 4860 2.10 0.753 12.8 82.5 204 - .
95 8.94 56, 57BD +
09 2870 (RGB) 4600 1.40 0.864 30.7 381 235 - .
37 8.43 56, 57BD +
10 2495 (RGB) 4920 2.12 0.723 12.3 80.0 156 - .
83 9.06 56, 57BD +
12 2547 (AGB) 4610 1.50 0.780 26.0 275 265 - .
72 8.15 56, 57BD +
17 3248 (RHB) 5250 2.21 0.458 8.80 53.1 64.8 - .
02 8.95 56, 57BD +
18 2757 (AGB) 4840 1.43 0.446 21.3 225 127 - .
19 8.24 56, 57BD +
18 2976 (RGB) 4550 1.30 0.769 32.5 408 248 - .
40 7.65 56, 57BD -
03 5215 (RHB) 5420 2.60 0.884 7.80 47.4 42.5 - .
66 8.68 56, 57HD 083212 (RGB) 4550 1.40 0.663 26.9 280 146 - .
49 8.14 56, 57HD 101063 (SGB) 5070 3.40 1.19 3.60 7.73 28.6 - .
13 9.56 56, 57HD 107752 (AGB) 4750 1.70 0.838 21.4 210 185 - .
88 8.24 56, 57HD 110885 (RHB) 5330 2.50 0.757 8.10 47.8 39.3 - .
44 8.66 56, 57HD 111721 (RGB) 5080 2.35 0.460 7.50 33.8 74.5 - .
26 9.20 56, 57HD 115444 (RGB) 4750 1.62 0.584 19.6 176 169 - .
77 8.56 56, 57HD 119516 (RHB) 5440 2.37 0.626 8.60 58.4 39.8 - .
50 8.38 56, 57HD 121135 (AGB) 4925 1.90 0.789 16.5 144 76.3 - .
57 8.23 56, 57HD 122956 (RGB) 4600 1.35 0.447 23.4 221 130 - .
78 8.03 56, 57HD 135148 (RGB) 4275 0.80 0.239 32.2 312 164 - .
90 7.83 56, 57HD 195636 (RHB) 5370 2.17 0.442 9.10 62.1 16.9 - .
83 8.05 56, 57EV Lac (M3.5 V) 3168 4.80 0.315 0.369 0.0124 4.38 - .
200 13.70 26, 27, 34 23V Hya (N6, AGB) 2160 - .
89 4.20 945 17540 576 + .
10 5.12 58, 59, 6089 Her (F2 Ib) 6550 0 .
60 0.61 64.8 6970 143 - .
41 8.00 61, 62, 63, 7(1)–(32) See Table 1, (33) Porto de Mello et al. (2008), (34) Wood et al. (2005a), (35) DeWarf et al. (2010), (36) Baliunas et al. (1996), (37) Janson et al. (2009),(38) Wood & Linsky (2010), (39) Vogt et al. (2010), (40) Hekker & Aerts (2010), (41) Schröder & Cuntz (2007), (42) Carney et al. (2008), (43) Robinson et al. (1998),(44) Cayrel de Strobel et al. (2001), (45) Kirsch et al. (2001), (46) Griffin (1988), (47) Marshall (1996), (48) Judge & Stencel (1991), (49) Massarotti et al. (2008),(50) Tarrant et al. (2008), (51) de Medeiros & Mayor (1999), (52) Ayres et al. (2007), (53) Harper et al. (1995), (54) Carpenter et al. (1999), (55) Setiawan et al. (2004),(56) Dupree et al. (2009), (57) Cortés et al. (2009), (58) Bergeat & Chevallier (2005), (59) Lambert et al. (1986), (60) Barnbaum et al. (1995), (61) Sargent & Osmer(1969), (62) Danziger & Faber (1972), (63) Stasi´nka et al. (2006). given by Equation (41) to obtain the error bars. Figure 10(b)shows a comparison with the semi-empirical scaling law pro-posed by Schröder & Cuntz (2005), with ˙ M SC = η L ∗ R ∗ M ∗ (cid:18) T eff (cid:19) . (cid:18) + g ⊙ g (cid:19) (42)where L ∗ , R ∗ , and M ∗ are assumed to be in solar units. Forthis plot we computed the normalization constant η such thatthe average modeled mass loss rate would equal the averagemeasured mass loss rate for all 47 stars. Averages were takenusing the logarithm of ˙ M so that all stars would contribute tothe average comparably. We found η = 8 . × - M ⊙ yr - ,which is within the error bars of the Schröder & Cuntz (2005)value. Overall, our “standard model” (i.e., Equation (35) with α = 0 . h = 0 .
5, and θ = 1 /
3) appears to match the mea-sured mass loss rates reasonably well. We emphasize that thismodel does not contain any arbitrary η normalization factors.For the three test-case stars at the bottom of Table 2, however,our model does not do as well. The model underpredicts themass loss from the dM flare star EV Lac by at least four ordersof magnitude, and it also fails for the F supergiant 89 Her bya slightly smaller amount. For EV Lac and other flare-activeM dwarfs, it is possible that coronal mass ejections and otherepisodic sources of energy (Mullan 1996) could be responsi-ble for the bulk of the observed mass loss. For the carbon starV Hya, the reasonably good agreement between the modelHEORETICAL MASS LOSS RATES OF COOL STARS 15 TABLE 3T
HEORETICAL W IND P ROPERTIES OF C OOL S TARS
Name Ro log f min B ∗ (G) log F H , TR ˙ M hot / ˙ M cold log M A , TR - log( ˙ M / [ M ⊙ yr - ])Sun 1.960 -2.996 1513.05 6.14 19.73 -2.42 13.44 α Cen A (G2 V) 2.074 -3.078 1308.79 6.17 11.43 -2.17 13.21 α Cen B (K0 V) 1.755 -2.835 1545.97 5.62 51.73 -2.93 14.0970 Oph A (K0 V) 0.996 -2.025 1666.45 5.99 194.1 -3.25 13.4370 Oph B (K5 V) 1.027 -2.067 2130.57 4.55 417.3 -4.50 15.17 ǫ Eri (K4.5 V) 0.519 -1.174 1832.80 5.97 977.4 -3.94 13.3161 Cyg A (K5 V) 1.107 -2.173 2145.92 4.59 246.1 -4.43 15.15 ǫ Ind (K5 V) 0.755 -1.646 1941.88 5.15 551.7 -4.22 14.2536 Oph A (K5 V) 0.923 -1.918 1927.79 5.67 228.1 -3.63 13.8436 Oph B (K5 V) 1.021 -2.059 1973.54 5.51 173.1 -3.66 14.16 ξ Boo A (G8 V) 0.381 -0.853 1788.75 6.69 1335 -3.59 12.42 ξ Boo B (K4 V) 0.340 -0.755 2620.92 4.83 5964 -5.37 14.6861 Vir (G5 V) 1.765 -2.843 1524.08 5.99 27.96 -2.63 13.56 δ Eri (K0 IV) 1.844 -2.907 1077.96 5.77 12.04 -2.18 12.72 α Boo (K1.5 III) 4.217 -4.000 411.54 5.23 0.57 -0.31 10.33 α Tau (K5 III) 4.183 -4.000 270.35 5.17 0.75 0.11 9.88 γ Dra (K5 III) 4.097 -4.000 301.38 5.24 0.85 -0.12 9.99HR 6902 (G9 IIb) 3.157 -3.692 287.82 6.11 1.28 0.13 9.86 β And (M0 III) 1.229 -2.321 311.31 5.61 0.47 -0.85 8.86 β UMi (K4 III) 6.960 -4.000 262.45 5.32 0.85 0.28 9.72 µ UMa (M0 III) 2.182 -3.152 165.83 5.81 0.63 0.68 7.93 α TrA (K2 II-III) 7.010 -4.000 270.87 5.55 1.25 -0.13 9.32 λ Vel (K4 Ib-II) 5.808 -4.000 136.66 5.67 2.15 1.05 8.47BD +
01 3070 (RGB) 1.122 -2.192 792.36 6.49 2.36 -1.48 10.14BD +
05 3098 (RGB) 1.600 -2.700 553.26 6.37 0.50 -0.63 9.08BD +
09 2574 (RGB) 3.001 -3.618 632.02 5.74 0.49 -0.62 10.17BD +
09 2870 (RGB) 2.249 -3.196 476.21 6.00 0.37 -0.17 8.68BD +
10 2495 (RGB) 2.390 -3.284 602.67 6.01 0.51 -0.62 9.83BD +
12 2547 (AGB) 2.657 -3.439 378.07 5.98 0.60 0.10 9.20BD +
17 3248 (RHB) 1.259 -2.355 493.93 6.81 0.65 -0.53 9.05BD +
18 2757 (AGB) 1.400 -2.508 384.43 6.55 0.32 -0.03 8.05BD +
18 2976 (RGB) 2.218 -3.175 464.10 5.96 0.32 -0.12 8.54BD -
03 5215 (RHB) 1.095 -2.157 583.84 7.06 1.91 -0.90 9.36HD 083212 (RGB) 1.361 -2.467 464.86 6.20 0.23 -0.44 8.07HD 101063 (SGB) 0.813 -1.744 1312.86 6.26 40.43 -2.65 11.31HD 107752 (AGB) 2.171 -3.145 524.60 6.08 0.42 -0.37 9.05HD 110885 (RHB) 0.909 -1.897 575.05 7.06 2.02 -0.97 9.17HD 111721 (RGB) 1.379 -2.486 609.98 6.42 0.61 -0.91 9.64HD 115444 (RGB) 1.919 -2.965 491.45 6.14 0.34 -0.31 8.82HD 119516 (RHB) 0.945 -1.950 483.08 7.24 1.29 -0.62 8.80HD 121135 (AGB) 1.071 -2.126 502.44 6.68 0.60 -0.68 8.47HD 122956 (RGB) 1.218 -2.309 444.03 6.27 0.19 -0.38 7.88HD 135148 (RGB) 1.033 -2.075 394.38 5.99 0.07 -0.25 6.98HD 195636 (RHB) 0.350 -0.779 435.75 7.64 3.38 -0.88 7.90EV Lac (M3.5 V) 0.0706 -0.313 3005.25 2.72 2641 -8.15 17.78V Hya (N6, AGB) 0.609 -1.367 12.39 6.53 1.32 1.99 4.3489 Her (F2 Ib) 27.12 -4.000 43.59 1.92 0.05 -1.11 11.15 and measurements is probably a coincidence, since our modeldoes not include the dusty radiative transfer or strong radialpulsations that are likely to be important for AGB stars.It is interesting to highlight the case of the moderatelyrotating K dwarfs ǫ Eri, 70 Oph, and 36 Oph, whichHolzwarth & Jardine (2007) found to have anomalously highmass loss rates. They concluded that the observed magneticfluxes for these stars were insufficient to produce their denseoutflows. For these stars, our modeled mass loss rates tendedto be about a factor of 10–20 below the measured values.However, these models were computed using f min from Equa-tion (38). The measured values of f ∗ given in Table 1 for thesestars are larger than their corresponding f min values by factorsranging from 3 to 20. If instead these values were used, ourmodeled mass loss rates would be in better agreement withthe astrospheric observations of Wood et al. (2005a).We also developed a statistical measure of how well a givenmodel agrees with the measured database of mass loss rates. We defined a straightforward least-squares parameter χ = 1 N N X i =1 (cid:2) log ˙ M i (model) - log ˙ M i (obs) (cid:3) (43)where the total number of comparisons ( N = 40) excludes thefinal three test cases in Table 2 and counts each of the fourA + B binaries as one. When χ <
1, then (on average) themodeled and measured mass loss rates are within an order ofmagnitude of one another. Table 4 summarizes the results, in-cluding comparisons with other published empirical prescrip-tions. The η normalization factors for each of these scalinglaws were computed similarly as the factor in Equation (42)above. Note that our standard model appears to be a sig-nificant improvement over both the popular Reimers (1975,1977) and Schröder & Cuntz (2005) scalings.To further explore the proposed model, we varied some ofthe modeling parameters described in Section 3. Varying theTR filling factor exponent θ did not have much of an effecton χ . However, varying the flux height scaling factor h didchange χ significantly. We found that a larger value of h ≈ F IG . 9.— Empirical f emp filling factors computed to match measured massloss rates. Rossby numbers were computed in two ways: (a) directly fromEquation (36), and (b) multiplying τ c from Equation (36) by ( g ⊙ / g ) . .Symbol shading is proportional to log( L ∗ / L ⊙ ), and the curves are the sameas those in Figure 7(b). The thick orange bar denotes the Sun’s empiricalrange of values (computed from the small variation in ˙ M ). The open orangecircle denotes V Hya, the only one of the three test cases (at bottom of Table2) that gave a realistic solution for f emp .TABLE 4M ASS L OSS “G OODNESS OF F IT ”Model χ This paper (standard model) 0.650
This paper (ZAMS τ c ) 1.575This paper ( h = 0 .
25) 0.794This paper ( h = 1) 0.564This paper ( h = 3) 0.504This paper ( θ = 0 .
2) 0.620This paper ( θ = 0 .
5) 0.707This paper (all [Fe/H] = 0) 0.647This paper ( ˙ M = ˙ M hot + ˙ M cold ) 0.703This paper ( α from Trampedach & Stein 2011) 0.770Reimers (1975, 1977) 1.260Mullan (1978), Equation (4a) 3.768Nieuwenhuijzen & de Jager (1990) 2.356Catelan (2000), Equation (A1) 1.924Schröder & Cuntz (2005) 1.131 F IG . 10.— Comparison of modeled (open circles) and measured (bluecrosses) mass loss rates for the stars in Table 2, plotted as log ˙ M versus stellarluminosity. Panels show (a) the standard model developed in this paper, and(b) the empirical scaling relation of Schröder & Cuntz (2005). The Sun isshown as a filled black circle, and the three test-case stars from the bottom ofTable 2 are shown in orange. Vertical error bars in (a) correspond to modelscomputed for the P rot range of Equation (41). h = 0 . h values forthe Sun’s corona.We also tried removing some of the imposed complexity ofthe standard model to see if simpler assumptions would giveadequate results. Removing the gravity-dependent modifica-tion factor of ( g ⊙ / g ) . from the definition of the convectiveoverturn time resulted in significantly poorer agreement withthe data (i.e., more than double the χ of the standard model).We explored the importance of metallicity by replacing thepublished [Fe/H] by purely solar values ([Fe/H] = 0). This ac-tually improved the value of χ from the standard model, butonly by < χ (8% larger than thestandard model).We also noted that the theoretical photospheric Alfvén wavefluxes from Musielak & Ulmschneider (2002a) exhibited astrong dependence on the convective mixing length parameterHEORETICAL MASS LOSS RATES OF COOL STARS 17 F IG . 11.— Illustrations of the relative importance of “hot” versus “cold”mass loss mechanisms. (a) Ratio of hot to cold modeled values for ˙ M (open circles) compared with the modified ratio ˙ M hot exp( - M , TR ) / ˙ M cold (red filled circles). (b) Mach number at the TR, M A , TR = u TR / V A , TR . The Sunis shown as a thicker black circle in (a) and a filled black circle in (b). (i.e., F A ∗ ∝ α . ). Thus, instead of simply assuming α = 2 asin the standard model, we created a linear regression fit to thetabulated simulation results of Trampedach & Stein (2011),who found empirical values of α between 1.6 and 2.2 depend-ing on T eff , log g , and M ∗ . We used the following approximatefit α TS ≈ . - T eff + log g . + M ∗ . M ⊙ (44)and did not allow α TS to be less than 1.6 or greater than 2.2.Thus, we multiplied the value of F A ∗ from Equation (7) bya factor of ( α TS / . . The predicted mass loss rates for theTable 2 stars had about an 18% higher value of χ than thestandard model, so we did not pursue this mixing length pre-scription any further.For additional context about the hot and cold mass lossmodels described in Section 3.3, Figure 11 shows the ratio ˙ M hot / ˙ M cold for the 47 modeled stars as well as the TR Machnumber M A , TR = u TR / V A , TR . It is clear that the dwarf starsare dominated by hot coronae, and the stellar wind outflow isstill negligibly small at the coronal base. However, as the lu-minosity exceeds ∼ L ⊙ for the giant stars, the hot coronal F IG . 12.— Plasma number density at the TR plotted as a function of stellarrotation period. Symbols are the same as in Figure 10. contribution goes away and the acceleration becomes domi-nated by wave pressure.Figure 12 examines how the plasma number density at thetransition region, n TR = ρ TR / m H , depends on stellar rotation.Holzwarth & Jardine (2007) assumed n TR ∝ Ω . ∝ P - . (seealso Ivanova & Taam 2003). Although our models do not fol-low a single universal relation for both giants and dwarfs, theproposed scaling (or one slightly steeper) may be appropri-ate for certain sub-populations of stars. For the dwarf starswith well-determined rotation periods, it is interesting that theSun’s computed value of n TR is larger than that of stars havinghigher magnetic activity. Equation (22) shows that the depen-dence on filling factor is weak (i.e., about f . ∗ for θ = 1 / n tends toincrease with activity (Güdel 2004). However, X-ray determi-nations of number density are probably dominated by closed-field active regions, which are not necessarily correlated withthe regions driving the stellar wind. Predictions for Idealized Stellar Parameters
In addition to the above comparisons with the individualstars of Table 2, we also created some purely theoretical setsof stellar models and computed ˙ M for them. We began withthe ZAMS model parameters given by Girardi et al. (2000),and we assumed solar metallicity for a range of constant rota-tion rates. This gave rise to a two-dimensional grid of models(varying T eff and P rot ) for main sequence stars. Because themodeled stars are all high-gravity dwarfs, we used only Equa-tion (36) for τ c , in combination with Equation (38) for f min asa function of Rossby number.Figure 13 shows the resulting mass loss rates as a func-tion of T eff and P rot . If we had not utilized a basal “floor” on f ∗ , we would have predicted a steep drop-off in mass lossfor T eff & τ c decreases rapidly. However, because of the floor,there appear to be reasonably strong mass loss rates up tothe point at which subsurface convection zones disappear at T eff & T eff ≈ F IG . 13.— Theoretical predictions of ˙ M for main sequence stars, plottedas a function of T eff . Differently colored curves show a range of assumedrotation periods (see labels for values). The Sun is indicated by an opencircle. cause of the iteration for R , ρ TR , and Q TR . If the calculationof these quantities is halted after only one iteration, the finalvalue of ˙ M varies more smoothly as a function of T eff . We planto utilize a more self-consistent non-WKB model of Alfvénwave reflection in future versions of this work.The mass loss rates shown in Figure 13 are almost all due tothe hot coronal processes discussed in Section 3.1. Thus, it ispossible to simplify the components of Equation (26) in orderto obtain an approximate scaling relation for ˙ M that is rea-sonable for these main sequence stellar models. Ignoring theweakest dependences on some stellar parameters (i.e., factorswith exponents less than or equal to 1 / ˙ M - M ⊙ / yr ∼ (cid:18) R ∗ R ⊙ (cid:19) / (cid:18) L ∗ L ⊙ (cid:19) - / × (cid:18) F A ∗ erg cm - s - (cid:19) / f (4 + θ ) / ∗ , (45)which reproduces the curves in Figure 13 to within about anorder of magnitude. Despite the fact that this scaling formulais relatively easy to apply, we do not recommend its use instellar evolution or population synthesis calculations. Oncestars leave the main sequence, Equation (45) is no longer agood approximation.We also computed a time-dependent mass loss rate for theevolutionary track of a star having M ∗ = 1 M ⊙ . There isevidence that the wind from the “young Sun” was signifi-cantly denser than it is today, and this more energetic out-flow may have been important to early planetary evolution(e.g., Wood 2006; Güdel 2007; Sterenborg et al. 2011; Suzuki2011). We used the BaSTI evolutionary track plotted in Fig-ure 1 (Pietrinferni et al. 2004) for the time variation of R ∗ and L ∗ . We grafted on a model of rotational evolution for a solar-mass star from Figure 6(a) of Denissenkov et al. (2010). Forlate ages ( t &
100 Myr, or log t & P rot ∝ t . . Such an age scaling is well within therange of empirically determined power laws ( t . to t . ) ob-tained from young solar analogs (e.g., Barnes 2003; Güdel2007). F IG . 14.— Theoretical predictions of stellar wind properties for an evolv-ing solar-mass star. (a) Base-10 logarithms of luminosity (green dot-dashedcurve), Rossby number (black solid curve), and f min (blue dashed curve) plot-ted as a function of age in years. (b) Our standard model for ˙ M (black solidcurve), compared with the Schröder & Cuntz (2005) scaling (blue dashedcurve) and an ideal power-law t - . decline with increasing age (green dottedcurve). A model and observations of CTTS are shown for comparison (rederror bars and dot-dashed curve). Figure 14(a) shows how the luminosity and two dimension-less parameters related to the rotational dynamo (Ro and f min )vary as a function of age for this model. Note that prior toabout t ≈
70 Myr the Rossby number is small enough thatthe filling factor appears to be saturated near its maximumassumed value of 0.5. At very late times, when the star be-gins to ascend the red giant branch, the Rossby number de-creases again because of the increase in τ c with decreasing T eff and gravity. We utilized the ( g ⊙ / g ) . correction factorwhen computing τ c , but it was relatively unimportant until thestar left the main sequence.Figure 14(b) gives our prediction for the age variation ofthe a solar-type star’s mass loss rate. For ages between about t ≈ . ˙ M ∝ t - . . This is asignificantly shallower age dependence than the t - declinesuggested by Wood et al. (2002) on the basis of astrospheremeasurements. We note that if the rotation period was theonly variable to change with time, Equation (45) would giveHEORETICAL MASS LOSS RATES OF COOL STARS 19something like ˙ M ∝ P - . (for f ∗ = f min and θ = 1 / P rot with age—such as the t . de-pendence in the solar-mass rotational model of Landin et al.(2010)—would give rise to a steeper age- ˙ M relationship moresimilar to that of Wood et al. (2002).For comparison, Figure 14(b) also shows that theSchröder & Cuntz (2005) scaling law predicts a much smallerrange of mass loss variation for the young Sun than doesthe present model. We also show a model (Cranmer 2008,2009) and measurements (Hartigan et al. 1995) for classicalT Tauri stars (CTTS) at the youngest ages. It is clear thatfor t .
10 Myr some additional physical processes must beincluded (e.g., accretion-driven turbulence on the stellar sur-face) to successfully predict mass loss rates. DISCUSSION AND CONCLUSIONS
The primary aim of this paper was to develop a new genera-tion of physically motivated models of the winds of cool mainsequence stars and evolved giants. These models follow theproduction of MHD turbulent motions from subsurface con-vection zones to their eventual dissipation and escape throughthe stellar wind. The magnetic activity of these stars is takeninto account by extending standard age-activity-rotation indi-cators to include the evolution of the filling factor of strongmagnetic fields in stellar photospheres. The winds of G andK dwarf stars tend to be driven by gas pressure from hotcoronae, whereas the cooler outflows of red giants are sup-ported mainly by Alfvén wave pressure. We tested our modelof combined “hot” and “cold” winds by comparing with theobserved mass loss rates of 47 stars, and we found that thismodel produces better agreement with the data than do pub-lished scaling laws. We also made predictions for the para-metric dependence of ˙ M on T eff and rotation period for mainsequence stars, and on age for a one solar mass evolutionarytrack.The eventual goal of this project is to provide a straightfor-ward algorithm for predicting the mass loss rates of cool starsfor use in calculations of stellar evolution and population syn-thesis. A brief stand-alone subroutine called BOREAS hasbeen developed to implement the model described in this pa-per. This code is written in the Interactive Data Language(IDL) and it is included with this paper as online-only mate-rial. This code is also provided, with updates as needed, onthe first author’s web page. Packaged with the code itself aredata files that allow the user to reproduce many of the resultsshown in Section 5.In order to further test the conjecture that Alfvén wavesand turbulence drive cool-star winds, the models need tobe expanded from the simple scaling laws of Section 3to fully self-consistent solutions of the mass, momentum,and energy conservation equations along open flux tubes.Modeling the full radial dependence of density, temper-ature, magnetic field strength, and outflow speed wouldeliminate our reliance on approximate factors like h and θ . We believe that the models of Cranmer et al. (2007)for the solar wind, and Cranmer (2008) for T Tauri stars,can be extended straightforwardly and applied to othertypes of stars. However, there are many other approachesto producing self-consistent and/or three-dimensional mod- IDL is published by ITT Visual Information Solutions. There are alsoseveral free implementations with compatible syntax, including the GNUData Language (GDL) and the Perl Data Language (PDL). ∼ scranmer/ els that should be explored (e.g., Airapetian et al. 2000,2010; Holzwarth & Jardine 2005; Schrijver & Title 2005;Falceta-Gonçalves et al. 2006; Suzuki 2007; Vidotto et al.2009; Cohen et al. 2009; Cohen 2011).There are additional ways that our simplified models ofcoronal energy balance (Section 3.1) and wave-pressure driv-ing (Section 3.2) may be improved:1. Our standard assumption for the outflow speed in acoronal wind was u ∞ = V esc . However, Judge (1992)found that many stars have significantly smaller termi-nal speeds. It should be possible to use something likethe Schwadron & McComas (2003) solar wind scalinglaw to estimate the peak temperature in the corona, andthus apply the Parker (1958) theory of gas pressure ac-celeration to compute the wind speed.2. We assumed in Section 3.1 that r TR ≈ R ∗ . However,Schröder & Cuntz (2005) estimated that some low-gravity stars should exhibit “puffed up” chromosphereswith a fractional extent given by the final term in paren-theses in Equation (42). We applied this correction fac-tor to the modeled values of r TR and ˙ M hot for the stars inTable 2. Doing so yielded significant differences fromthe standard model only for stars having ˙ M cold ≫ ˙ M hot ,i.e., the combined model value of ˙ M was relatively un-changed in those cases. However, in general there maybe other stars for which this kind of correction factorneeds to be considered in more detail.3. We also assumed that the flux height scaling factor h took on a single constant value for all stars. It maybe useful to explore extending the Schröder & Cuntz(2005) idea of gravity-dependent spatial expansion tothis parameter as well. The χ results shown in Table 4suggest that a larger value of h ≈ h ≈ . F A ∗ in kink/Alfvén waves in the pho-tosphere may depend on other parameters that we havenot considered. Musielak & Ulmschneider (2002a)found that the flux is rather sensitive to B ∗ / B eq in thephotosphere, so departures from our assumed value of1.13 may give rise to significantly different predictions.Also, Musielak & Ulmschneider (2002b) examined thesensitivity to metallicity and found that lower Z / Z ⊙ tends to give lower values of F A ∗ for T eff . α parameter—thus possibly making thiseffect more important.5. Instead of assuming a simple monotonic dependenceof the open-field filling factor on Rossby number, itmay be possible to construct realistic surface distri-butions of active regions for a given activity leveland rotation period, and model the opening up offlux tubes by both stellar winds and centrifugal forces(Mullan & Steinolfson 1983; Mestel & Spruit 1987;Jardine 2004; Holzwarth & Jardine 2005; Cohen et al.2009).0 CRANMER AND SAARTo continue testing and refining these models, it is also im-portant to utilize the newest and most accurate measure-ments of stellar mass loss rates (see, e.g., Schröder & Cuntz2007; Willson 2009; Catelan 2009; Mauron & Josselin 2011;Vieytes et al. 2011) and magnetic fields (Donati & Landstreet2009; Vlemmings et al. 2011).Finally, we emphasize that a complete description of late-type stellar winds requires the incorporation of other phys-ical processes besides Alfvén waves and turbulence. Theouter atmospheres of cool stars are also likely to be pow-ered by acoustic or longitudinal MHD waves (Cuntz 1990;Buchholz et al. 1998), episodic flares or coronal mass ejec-tions (Mullan 1996; Aarnio et al. 2009), and large-amplitudepulsations (Bowen 1988; de Jager et al. 1997; Willson 2000).It is well known that radiative driving should not be neglectedfor AGB stars and red supergiants, and it may be importantfor Cepheids (Neilson & Lester 2008) and horizontal branch stars (Vink & Cassisi 2002) as well.The authors gratefully acknowledge Nancy Brickhouse,Andrea Dupree, Adriaan van Ballegooijen, Stan Owocki, OferCohen, and the anonymous referee for many valuable discus-sions. This work was supported by the Sprague Fund of theSmithsonian Institution Research Endowment, and by the Na-tional Aeronautics and Space Administration (NASA) undergrants NNX09AB27G and NNX10AC11G to the Smithso-nian Astrophysical Observatory. This research made exten-sive use of NASA’s Astrophysics Data System and the SIM-BAD database operated at CDS, Strasbourg, France. Thisresearch has also made use of the NASA/IPAC/NExScI Starand Exoplanet Database (NStED), which is operated by theJet Propulsion Laboratory, California Institute of Technology,under contract with NASA. APPENDIXNOTES ON STELLAR MAGNETIC FIELD MEASUREMENTS
In this work we focus on observations of unpolarized spectral lines sensitive to Zeeman splitting by stellar magnetic fields(e.g., Robinson 1980). The resulting “Zeeman broadened” line profiles are valuable probes of both the intensity-weighted meanabsolute value of the field strength (i.e., B ∗ = h I B | B |i / h I B i , where I B is the continuum intensity in the magnetic regions) andthe intensity-weighted fraction of the visible stellar hemisphere that is covered by these fields (i.e., the filling factor f ∗ ). Thesedetections are thus weighted towards the brightest regions of the stellar surface; i.e., plage or network regions. However, thistechnique allows detection of more topologically complex fields, and thus more comprehensive values of f ∗ and B ∗ , than does theuse of circular polarization. The latter exhibits significant signal cancellation when there are multiple oppositely directed patchesof magnetic field in the same resolution element. In many cases, however, only the disk-averaged magnetic flux density ( B ∗ f ∗ )can be determined reliably from Zeeman broadened spectra and not the separate values of B ∗ and f ∗ (see also Rüedi et al. 1997;Anderson et al. 2010).Many details of the observations of the stars discussed in Section 4 were given by Saar & Linsky (1985, 1986) and Saar (1990,1991, 1996a,b, 2001). The approximate quality factors listed in Table 1 span the range from low ( q = 1) to high ( q = 4) relativeconfidence in the derived magnetic parameters. The values for q were assigned based on a combination of the following propertiesof the spectroscopic data and its magnetic analysis:1. Detections using lines with longer wavelengths and higher Landé g eff factors are given higher q values. The ratio of thestrength of the Zeeman effect to the nonmagnetic Doppler width is ∆ λ B / ∆ λ D , where ∆ λ B ∝ g eff λ is the Zeeman splittingamplitude and ∆ λ D ∝ λ is the nonmagnetic Doppler width. Thus, the relative detectability of the effect increases as g eff λ .2. Spectra with higher signal-to-noise (S/N) ratios and higher spectral resolution ( R = λ/ ∆ λ ) tend to have higher quality(see, e.g., Saar 1988), although a longer wavelength measurement can trump better R . For example, the greater magneticsensitivity at large λ can lead to partial resolution of the individual Zeeman components (Saar & Linsky 1985).3. The simultaneous analysis of larger numbers of lines (especially with higher g eff ) contributes to a good quality measurement(Rüedi et al. 1997). It is additionally helpful for the lines to be free of blends and for any rotational broadening to be small(i.e., v sin i .
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