aa r X i v : . [ a s t r o - ph . S R ] D ec NONLINEAR PULSATIONS OF RED SUPERGIANTSYu. A. Fadeyev Institute of Astronomy, Russian Academy of Sciences, Pyatnitskaya ul. 48, Moscow, 109017Russia
Received
Abstract — Excitation of radial oscillations in population I ( X = 0 . Z = 0 .
02) redsupergiants is investigated using the solution of the equations of radiation hydrodynamics andturbulent convection. The core helium burning stars with masses 8 M ⊙ ≤ M ≤ M ⊙ andeffective temperatures T eff < κ –mechanism in the hydrogen and helium ionization zones. Radialpulsations of stars with mass M < M ⊙ are strictly periodic with the light amplitude ∆ M bol ≤ . m
5. The pulsation amplitude increases with increasing stellar mass and for
M > M ⊙ themaximum expansion velocity of outer layers is as high as one third of the escape velocity. Themean radii of outer Lagrangean mass zones increase due to nonlinear oscillations by ≤
30% incomparison with the initial equilibrium. The approximate method (with uncertainty of a factorof 1.5) to evaluate the mass of the pulsating red supergiant with the known period of radialoscillations is proposed. The approximation of the pulsation constant Q as a function of themass–to–radius ratio is given. Masses of seven galactic red supergiants are evaluated using theperiod–mean density relation.Keywords: stars: variable and peculiar . E–mail: [email protected] INTRODUCTION
Red supergiants are long–period variables with semiregular light variations on a timescale & day. The period–luminosity relation (Glass 1979; Feast et al. 1980) and the lineartheory of adiabatic oscillations (Stothers 1969, 1972) allow us to suppose that such a typeof variability is due to radial stellar pulsations. At the same time together with semiregularvariability some red supergiants exhibit superimposed irregular light variations (Kiss et al.2006). The secondary stochastic variability is thought to be due to the large–scale convectionin the outer subphotospheric layers (Stothers and Leung 1971; Schwarzschild 1975; Stothers2010). Red supergiants are also remarkable due to intensive mass loss revealed through a largeinfrared excess indicating dust production in the stellar wind (Verhoelst et al. 2009).The period–luminosity relation of radially pulsating red supergiants is used for determina-tion of extra–galactic distances and in comparison with Cepheids the red supergiants allow usto substantially extend the distance scale due to their higher luminosities (Pierce et al. 2000;Jurcevic et al. 2000). Application of the theory of stellar pulsation to the analysis of observedvariability of red supergiants allows us to verify some conclusions of the stellar evolution the-ory in a way similar to that employed earlier for Cepheids. It should be noted also that thegrowing bulk of recent observations indicate that the strong stellar wind of massive late–typesupergiants is due to nonlinear stellar oscillations (van Loon et al. 2008)The nature of radial oscillations in red supergiants is still not completely clear yet. Thelinear analysis of pulsational instability of red supergiants with masses 15 M ⊙ ≤ M ≤ M ⊙ wasperformed by Li and Gong (1994) and Guo and Li (2002). According to their calculations radialoscillations of red supergiants are due to instability of the fundamental mode and, perhaps, thefirst overtone. However the theoretical period–luminosity relation agrees only with fundamentalmode oscillations.Nonlinear radial oscillations of red supergiants were considered only in two studies. In thefirst one (Heger et al. 1997) the authors investigated radial oscillations of the red supergiantwith mass M = 15 M ⊙ at the final stage of the core helium burning. In the second work(Yoon and Cantiello 2010) the authors investigated pulsational instability of red supergiantswith masses 15 M ⊙ ≤ M ≤ M ⊙ . It should be noted that in both these studies the self–exciting stellar oscillations were treated with modified methods of stellar evolution calculationand effects of interaction between pulsation motions and turbulent convection were not takeninto account.Below we present results of investigation of nonlinear pulsations of red supergiants obtainedfrom the self–consistent solution of the equations of radiation hydrodynamics and turbulentconvection. The need for such an approach is due to the significant length and mass of the2uter convection zone involved in pulsation motions. The treatment of convective heat transportuses the solution of the diffusion–type equations for the enthalpy and the mean turbulentenergy obtained by Kuhfuß(1986) for spherically–symmetric gas flows from the Navier–Stokesequation. Thus, the results presented below deal with modelling the semiregular variabilityand the secondary stochastic variability is not considered because this problem is beyond theapproximation of spherical geometry. We consider the stars with masses at the zero–age mainsequence 8 M ⊙ ≤ M ZAMS ≤ M ⊙ and initial fractional mass abundunces of hydrogen andelements heavier than helium X = 0 . Z = 0 . The problem of self–exciting stellar oscillations is the Cauchy problem for equations of hy-drodynamics with initial conditions corresponding to the hydrostatic and thermal equilibrium.In the present study the initial conditions were taken from the evolutionary models of stars atthe core helium burning. Methods of stellar evolution calculations are discussed in our previouspapers (Fadeyev 2011a, b).The evolutionary tracks in the Hertzsprung–Russel (HR) diagram of population I stars( X = 0 . Z = 0 .
02) with initial masses M ZAMS = 10, 15 and 20 M ⊙ are shown in Fig. 1. Finalpoints of tracks correspond to helium exhaustion in the stellar core ( Y c ≈ − ). Solid linesindicate parts of tracks when the star is the core helium burning red supergiant with effectivetemperature T eff ≤ M ZAMS = 10 M ⊙ the duration of gravitational contraction of the helium core is ≈ years.The luminosity ceases to increase when the triple alpha process becomes the main energysource in the stellar center. For the star witn initial mass M ZAMS = 10 M ⊙ the total duration ofthermonuclear helium burning is ≈ . · years and in the beginning the star remains the redsupergiant with luminosity decreasing by a factor of two during ≈ . · years. The star leavesthe red supergiant domain when the central helium abundance decreases below Y c ≈ .
52 andits evolutionary track loops the HR disgram to effective temperatures as high as T eff ≈ K.The star becomes again the red supergiant when its central helium abundance decreases to Y c ≈ .
05 and the time of helium exhaustion does not exceed 2 · years. Therefore, ofmost interest is the initial stage of helium burning during of which the luminosity of the red3upergiant decreases. Evolution of stars with initial masses M ZAMS < M ⊙ is nearly the samebut proceeds in different time scales.Evolutionary tracks of the core helium burning stars M ZAMS ≥ M ⊙ do not loop in theHR diagram and all the time remain in the red supergiant domain. For M ZAMS = 20 M ⊙ thecore helium burning proceeds during ≈ . · years, in the beginning during ≈ . · yearsthe stellar luminosity decreasing and then increasing to L ≈ L ⊙ .All red supergiant evolutionary models used as initial conditions for hydrodynamic compu-tations are chemically homogeneous between the inner boundary to the stellar surface. More-over,for the given value of M ZAMS the abundances in the stellar envelope do not change duringthe core helium burning. This is due to the fact that the size of the outer convective zone ismaximum at the final stage of gravitational contraction of the helium core just before ignitionof the triple alpha process. Depending on the mass and luminosity of the star the radius ofthe inner boundary r ranges within 0 . . r /R eq < .
1, where R eq is the radius of the upperboundary of the equilibrium model. The method for the self–consistent solution of the equations of radiation hydrodynamics andturbulent convection is described in our previous paper (Fadeyev, 2011b), so that below we onlydiscuss the results obtained. Computations were carried out with the number of Lagrangeanmass zones 500 ≤ N ≤ . To be confident that the solution is independent of the innerboundary radius r and the number of Lagrangean zones N some hydrodynamic models werecomputed with several different values of these quantities.Our hydrodynamic computations show that red supergiants with initial masses 8 M ⊙ ≤ M ZAMS ≤ M ⊙ are unstable against radial oscillations in the fundamental mode. Howeverdepending on the value of M ZAMS hydrodynamic models demonstrate different behaviour bothduring the growth of instability and after the limit amplitude attainment. In particular, in starswith initial mass M ZAMS < M ⊙ the oscillation amplitude is always enough small, whereas for M ZAMS > M ⊙ nonlinear effects play a substantial role.The main properties of hydrodynamical models are summarized in Table 1, where thebolometric luminosity L and the effective temperature T eff correspond to the initial equilibriummodel, Y c is the fractional helium abundance in the stellar center, Π and Q are the pulsationperiod and the pulsation constant in days, η = Π d ln E K /dt is the growth rate of the pulsationkinetic energy E K . Reciprocal of this quantity equals the number of pulsation periods duringwhich the kinetic energy increases by a factor of e = 2 . . . . . Ratios of the outer boundarymean radius h R i to the equilibrium radius of the model R eq are given in the last column of4able 1 and show the role of nonlinear effects after the limit amplitude attainment.The oscillation amplitude of red supergiants M ZAMS = 10 M ⊙ at the top of the evolutionarytrack is enough small and the relative radial displacement of the upper boundary is ∆ R/R eq =0 .
06. The oscillation amplitude gradually increases with decreasing luminosity and in the pointwith minimum luminosity before the loop in the HR diagram ∆
R/R eq = 0 .
13. Enhancementof the radial oscillation amplitude with decreasung luminosity is illustrated in Fig. 2 wherevariations of the velocity at the upper boundary U and the bolometric magnitude M bol areshown for L = 2 · , 1 . · and 10 L ⊙ .Fairly good repetition of small amplitude oscillations allows us to calculate the mechanicalwork done by Lagrangean mass zones and thereby to evaluate their contribution into excitationor damping of instability. The radial dependence of the mechanical work H P dV , where V is thespecific volume and P is the sum of gas, radiation and turbulent pressure, is shown in Fig. 3for the hydrodynamical model M ZAMS = 10 M ⊙ , L = 1 . · L ⊙ . The region of instabilityexcitation ( H P dV >
0) encompasses the layers with temperature 1 . · . T . · Kcorresponding to the hydrogen and helium ionization zones. In deeper layers (
T > · K)with fully ionized helium the pulsational instability is damped ( H P dV < ρ , temperature T , opacity κ and luminosity L rad = 4 πr F rad , where F rad is radiative flux, in Lagrangean mass zones of the hydrodynamical model. In Fig. 4(a) wegive the plots of relative variations δρ/ρ , δT /T and δκ/κ in the layers of fully ionized heliumwith temperature ranging within 4 . · K ≤ T ≤ . · K. For the sake of convenience, theplots are arbitrarily shifted along the vertical axis. Coincidence of the maxima of density andtemperature variations indicates that oscillations are nearly adiabatic. Decrease of opacity atmaximum compression damps the instability because, as seen in Fig. 4(b), heat losses due toradiation reach their maximum.Variations of same quantities for the layer with temperature 1 . · K ≤ T ≤ . · Kcorresponding to partial helium ionization are plotted in Fig. 5. Substantial phase shifts betweenmaxima of density and temperature indicate large nonadiabaticity of pulsation motions, whereasthe delay of the maximum temperature with respect to maximum compression is the cause ofthe positive mechanical work. Absorption of heat during compression is due to increase ofopacity and it is accompanied by decrease of the radiative flux.Thus, damping of oscillations in the layers of fully ionized helium and driving of pulsationalinstability in the hydrogen and helium ionization zones are due to the κ –mechanism, becauseeffects of heat gains and losses are connected with absorption and emission of radiation. Lowrates of the instability growth ( η ∼ − ) and the small limit cycle amplitude are due to the5mall fraction of radiation in the total energy flux ( L rad . − L r ). Driving of pulsational in-stability at so small radiation fluxes is due to the large amplitude of total luminosity variations.As seen in Fig. 6, the amplitude of luminosity variations is largest in vicinity of the heliumionization zone.Pulsational instability of red supergiants increases with increasing initial stellar mass andfor M ZAMS > M ⊙ nonlinear effects become significant. In Fig. 7 we give the temporaldependences of the kinetic energy E K of the pulsating envelope and the radius of the upperboundary R in units of the equilibrium radius R eq for the red supergiant model with initial mass M ZAMS = 16 M ⊙ and luminosity L = 8 . · L ⊙ . Compared to less massive supergiants thismodel demonstrates the growth rate which is higher by an order of magnitude, whereas afterthe attainment of the limiting amplitude the mean radii of outer Lagrangean mass zones exceedtheir equilibrium values by nearly one third. The amplitude ceases to grow in a transitionalprocess encompassing roughly a dozen of oscillation periods (20 < t/ Π < In the HR diagram red supergiants occupy the domain with relatively narrow effectivetemperature range (3000 K . T eff . L and the period of radial pulsations Π. Fromobservations such a correlation is known as the period–luminosity relation. The theoreticalperiod–luminosity relation obtained in the present study is shown in Fig. 8 for models of threeevolutionary sequences with initial masses M ZAMS = 10, 15 and 20 M ⊙ . Hydrodynamical mod-els of red giants at the evolutionary stage of decreasing luminosity are shown by filled circlesand the dashed lines show the linear fits for each evolutionary sequence.For stars with initial mass M ZAMS = 10 M ⊙ the evolutionary track loops the HR diagramduring helium burning and three models shown in Fig. 8 by triangles correspond to the initialpart of the loop with effective temperatures T eff < M < M ⊙ leave the red supergiant domain and others return to it.The chemical composition of outer layers involved in pulsation motions does not change,whereas effects of mass loss for M < M ⊙ are insignificant. Therefore, red supergiants withalmost exhausted helium in the stellar center obey the same period–luminosity relation as starsin earlier phases of helium burning. This is illustrated in Fig. 8 where two models of stars6 ZAMS = 10 M ⊙ with central helium abundances Y c = 2 . · − and 1 . · − are shown byopen circles.As can be seen in Fig. 8, the dispersion of the common correlation between the periodand luminosity of red supergiants is due to dependence of the both luminosity and period onthe stellar mass. Therefore, one of the causes of dispersion on the empirical period–luminositydiagram is due to different masses of observed stars. It should also be noted that the mass–luminosity relation of red supergiants and, therefore, the period–luminosity relation, depend onconvective overshooting and mass loss during the preceeding evolution. An important role inthe both mass–luminosity and period–luminosity relations belongs also to abundances of heavyelements Z . These effects, however, were beyond the scope of the present study. The equlibrium luminosity of the red supergiant during helium burning changes within theranges that depend on the initial stellar mass. For example, in stars with M ZAMS ≈ M ⊙ theluminosity decreases by a half, whereas in red supergiants with initial mass M ZAMS = 20 M ⊙ themaximum to minimum luminosity ratio decreases to ≈ .
6. The period of radial pulsations Πchanges simultaneosly with equilibrium luminosity L . Evolution of red supergiants M ZAMS ≤ M ⊙ between the upper and lower luminosity limits is accompanied by the change of thepulsation period by a factor of three. The maximum to minimum period ratio decreases to afactor of two for M ZAMS = 20 M ⊙ .This property of red supergiants is illustarted in Fig. 9 where for stars with initial masses8 M ⊙ ≤ M ZAMS ≤ M ⊙ we show the period–mass diagram. Hydrodynamical models of starsat the top of the evolutionary track are shown by filled circles. Open circles indicate the redsupergiant models with lower luminosity. The diagram in Fig. 9 takes into account effects ofmass loss and along the vertical axis we give the mass values M of evolving stars. Evolution ofthe red supergiant corresponds to the displacement on the diagram from right to left and thenin the opposite direction. For models M ZAMS = 10 M ⊙ and 20 M ⊙ this displacement is shownby arrows.The period–mass diagram in Fig. 9 demonstrates the existence of the limited area of massand period values. The borders of allowed masses and periods of radially pulsating red super-giants can be approximately fitted aslog( M/M ⊙ ) = ( .
153 + 0 .
365 log Π0 .
488 + 0 .
273 log Π (1)and in Fig. 9 they are shown by dashed lines. Thus, from the observational estimate of the7eriod Π relations (1) allow us to evaluate the upper and lower mass limits of the red supergiant.For periods Π ≤
300 the uncertainty of such an estimate is about a factor of ≈ . The more exact value of the red supergiant mass can be obtained from the period–meandensity relation Π = Q ( R/R ⊙ ) / ( M/M ⊙ ) − / (2)provided that the pulsation period Π and the mean stellar radius R are known from observations.The pulsation constant Q is obtained from the theory of stellar pulsation and in some casescan be expressed as a function of the stellar mass M and stellar radius R . Substitution of thisexpression into the period–mean density relation (2) allows us to obtain the explicit expressionfor the mass of the pulsating star.The pulsation constants obtained in our hydrodynamical calculations of red supergiantswith initial masses 8 M ⊙ ≤ M ZAMS ≤ M ⊙ and pulsation periods 45 ≤ Π ≤ f = ( M/M ⊙ ) / ( R/R ⊙ ). The linear fit of thepulsation constant is written as log Q = − . − .
778 log f (3)and is shown in Fig. 10 by the dashed line.Masses M of seven galactic red supergiants evaluated from substitution of (3) into theperiod–mean density relation (2) are given in Table 2. The periods Π are taken from theGeneral Catalogue of Variable Stars (Samus et al. 2011). The mean stellar radii were evaluatedby Levesque et al. (2005) and Josselin and Plez (2007). In last two columns of Table 2 we givethe lower M a and upper M b mass limits derived from (1).Unfortunately, the existing estimates of mean radii of red supergiants are still highly un-certain. For example, the uncertainty in the effective temperature is ≈
25% (Josselin and Plez2007), so that the uncertainty in the mean radius is as high as ≈ M a , M b ] should not be considered as a contradiction. For example, if we adopt that the radiusof AD Per is larger by 20% ( R = 548 R ⊙ ) then the stellar mass is M = 12 . M ⊙ , that is withinranges given by (1). 8 CONCLUSION
Given in the previous section estimates of masses of seven galactic red supergiants allow usto conclude that the theory of stellar evolution is in an agreement with observational estimatesof stellar radii. To compare more stars with the theoretical period–luminosity relation oneshould consider pulsational instability of red supergiants in the wider interval of initial masses M ZAMS .For more detailed theoretical period–luminosity relation one should consider the role of someparameters used in evolutionary computations. One of them is the overshooting parameter. Inthe present study the evolutionary computations were done for the ratio of the overshootingdistance to the pressure scale height l ov /H P = 0 .
15. The need to know the role of this parameteris due to the dependence of the mass–luminosity relation of helium burning stars on convectiveovershooting.In stars with masses M ≥ M ⊙ effects of mass loss during the red supergiant evolutionarystage become significant. In the present study the evolutionary calculations were done withmass loss rates by Nieuwenhuijzen and de Jager (1990) however determination of the massloss rate ˙ M as a function of fundamental stellar parameters remains disputable (Mauron andJosselin, 2011) Therefore, one should employ parametrization of the expression for ˙ M andconsider the mass–luminosity and period–luminosity relations as a function of this parameter.Another parameter which significantly affects the period–luminosity relation of red super-giants is the mass fraction abundance of heavy elements Z . Of special interest is the period–luminosity relation for Z = 0 .
008 which is typical for the Large and Small Magellanic Clouds.9
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R/R ⊙ M/M ⊙ M a /M ⊙ M b /M ⊙ SU Per 533 780 17.1 14.1 17.1W Per 485 620 12.2 13.6 16.6V602 Car 635 860 17.7 15.0 17.9AD Per 362.5 457 8.9 12.2 15.4FZ Per 184 324 8.2 9.5 12.8RW Cyg 550 676 12.9 14.2 17.2SW Cep 70 234 9.8 6.7 9.813 igure captions
Fig. 1. Evolutionary tracls of Population I stars with initial masses M ZAMS = 10, 15 and 20 M ⊙ in the HR diagram. Parts of tracks corresponding to the stage of the red supergiant areshown in solid lines.Fig. 2. Variations of the gas flow velocity at the upper boundary U (a) and bolometric magnitude M bol (b) in red supergiants with initial mass M ZAMS = 10 M ⊙ and luminosity L = 2 · (solid lines), 1 . · L ⊙ (dashed lines) and 10 L ⊙ (dot–dashed lines).Fig. 3. The radial dependence of the mechanical work done by a Lagrangean mass zone over thepulsation period Π.Fig. 4. () – Variations of gas density ρ (solid line), temperature T (dashed line) and opacity κ (dotted line) in the layer of fully ionized helium; (b) – variations of radiative luminosityin units of the total equilibrium luminosity L .Fig. 5. Same as Fig. 4 but for the layer with partial helium ionization.Fig. 6. The amplitude of variations of the total luminosity L r expressed in units of the equilibriumluminosity L versus the radial distance from the stellar center.Fig. 7. Kinetic energy of pulsation motions E K (a) and the upper boundary radius R (b) as afunction of time t for the hydrodynamical model M ZAMS = 16 M ⊙ , L = 8 . · L ⊙ .Fig. 8. The period–luminosity diagram for red supergiants with initial masses M ZAMS = 10, 15and 20 M ⊙ . Hydrodynamical models are represented by filled cicrles. In triangles areshown hydrodynamical models of stars that leave the red supergiant domain. In opencircles are represented the hydrodynamical models of stars with central helium abundance Y c ≤ . · − .Fig. 9. The period–mass diagram of red supergiants 8 M ⊙ ≤ M ZAMS ≤ M ⊙ . Hydrodynamicalmodels of stars with maximum and minimum equilibrium luminosity are shown by filledand open circles, respectively. Arrows indicate the direction of evolutionary change of thepulsation period Π of stars M ZAMS = 10 M ⊙ and 20 M ⊙ . The region of allowed values ofthe radial pulsation periods is limited by dashed lines.Fig. 10. The pulsation constant Q of red supergiants 8 M ⊙ ≤ M ZAMS ≤ M ⊙ as a function ofmass–to–radius ratio f = ( M/M ⊙ ) / ( R/R ⊙ ). Hydrodynamical models of red supergiantsare shown by filled circles. 14igure 1: Evolutionary tracls of Population I stars with initial masses M ZAMS = 10, 15 and20 M ⊙ in the HR diagram. Parts of tracks corresponding to the stage of the red supergiant areshown in solid lines. 15igure 2: Variations of the gas flow velocity at the upper boundary U (a) and bolometricmagnitude M bol (b) in red supergiants with initial mass M ZAMS = 10 M ⊙ and luminosity L =2 · (solid lines), 1 . · L ⊙ (dashed lines) and 10 L ⊙ (dot–dashed lines).16igure 3: The radial dependence of the mechanical work done by a Lagrangean mass zone overthe pulsation period Π. 17igure 4: () – Variations of gas density ρ (solid line), temperature T (dashed line) and opacity κ (dotted line) in the layer of fully ionized helium; (b) – variations of radiative luminosity inunits of the total equilibrium luminosity L . 18igure 5: Same as Fig. 4 but for the layer with partial helium ionization.19igure 6: The amplitude of variations of the total luminosity L r expressed in units of theequilibrium luminosity L versus the radial distance from the stellar center.20igure 7: Kinetic energy of pulsation motions E K (a) and the upper boundary radius R (b) asa function of time t for the hydrodynamical model M ZAMS = 16 M ⊙ , L = 8 . · L ⊙ .21igure 8: The period–luminosity diagram for red supergiants with initial masses M ZAMS = 10,15 and 20 M ⊙ . Hydrodynamical models are represented by filled cicrles. In triangles are shownhydrodynamical models of stars that leave the red supergiant domain. In open circles arerepresented the hydrodynamical models of stars with central helium abundance Y c ≤ . · − .22igure 9: The period–mass diagram of red supergiants 8 M ⊙ ≤ M ZAMS ≤ M ⊙ . Hydrody-namical models of stars with maximum and minimum equilibrium luminosity are shown byfilled and open circles, respectively. Arrows indicate the direction of evolutionary change of thepulsation period Π of stars M ZAMS = 10 M ⊙ and 20 M ⊙ . The region of allowed values of theradial pulsation periods is limited by dashed lines.23igure 10: The pulsation constant Q of red supergiants 8 M ⊙ ≤ M ZAMS ≤ M ⊙ as a functionof mass–to–radius ratio f = ( M/M ⊙ ) / ( R/R ⊙⊙