L55 NONLINEAR BEAT CEPHEID MODELS Z. KOLLÁTH J. P. BEAULIEU and J. R. BUCHLER and P. Yecko ABSTRACT the Numerical Hydrodynamic

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NONLINEAR BEAT CEPHEID MODELS Z. Kolla´th Konkoly Observatory, P.O. Box 67, Budapest XII, 1525 Hungary J. P. Beaulieu Kapteyn Institute, P.O. Box 800, 9700 AV Groningen, The Netherlands and J. R. Buchler and P. Yecko Physics Department, University of Florida, Gainesville, FL 32611 Received 1998 April 10; accepted 1998 May 18; published 1998 July 1

ABSTRACT The numerical hydrodynamic modeling of beat Cepheid behavior has been a long-standing quest in which purely radiative models have failed miserably. We ®nd that beat pulsations occur naturally when turbulent convection is accounted for in our hydrodynamics codes. The development of a relaxation code and of a Floquet stability analysis greatly facilitates the search for and analysis of beat Cepheid models. The conditions for the occurrence of beat behavior can be understood easily and at a fundamental level with the help of amplitude equations. Here a discriminantD arises whose sign decides whether single-mode or double-mode pulsations can occur in a model, and thisD depends only on the values of the nonlinear coupling coef®cients between the fundamental and the ®rst overtone modes. For radiative modelsD is always found to be negative, but with suf®ciently strong turbulent convection its sign reverses. Subject headings: Cepheids Ð convection Ð hydrodynamics Ð Magellanic Clouds Ð stars: oscillations Ð turbulence

The Fourier analysis of the observational data of the beat 1987) to obtain nonlinear periodic pulsations (limit cycles) Cepheid light curves and radial velocities shows constant power when they exist; (c) a stability analysis of the limit cycles that in two basic frequencies and in their linear combinations, which gives their (Floquet) stability exponents. indicates that the stars pulsate in two modes (or more if res- The one-dimensional turbulent diffusion equation and the onances are involved). Since the beginning of theoretical Ce- concomitant eddy viscosity, the turbulent pressure, and the con- pheid studies in the early 1960s, numerical hydrodynamical vective and turbulent ¯uxes contain (seven) order unity param- attempts at modeling the phenomenon of beat pulsation have eters that need to be calibrated through a comparison to ob- failed, and beat Cepheids have been a bane of stellar pulsation servations. In the ®rst paper (Yecko et al. 1998) in which we theory. performed a broad survey of the linear properties of TC Ce- In Cepheids energy is carried through the pulsating envelope pheid models, we found that of these the mixing length and to the surface by radiation transport as well as by turbulent the strengths of the convective ¯ux and of the eddy viscosity convection (TC). Even though convection can transport almost play a dominant role and that broad combinations of these three all the energy in the hydrogen partial ionization region, this parameters exist that give agreement with the observed widths convection is inef®cient in the sense that it only mildly affects of both the fundamental and ®rst overtone instability strips. In the structure of the envelope. It was thus generally thought that this Letter we show that the inclusion of TC produces pulsating convection, while important for providing a red edge to the beat Cepheid models that satisfy the observational constraints, instability strip, would play a minor role the appearance of the in particular those of period ratios, of modal pulsation ampli- nonlinear pulsation. Purely radiative models did indeed give tudes and of their ratios. Furthermore, the models are very good overall agreement with the observed light and radial ve- robust with respect to the numerical and physical parameters. locities. However, recently it has become increasingly clear that Our discovery of beat Cepheid models has been partially there are a number of severe problems with radiative models serendipitous. When we started to investigate the nonlinear (Buchler 1998), in addition to their inability to account for beat pulsations of a typical Small Magellanic Cloud Cepheid model behavior. (M ϭ 4.0 M,, L ϭ 1100 L,,Teff ϭ 5900 K,X ϭ 0.73 and We have recently implemented in our hydrodynamics codes Z ϭ 0.004) with the turbulent convective hydro code, we en- a one-dimensional model diffusion equation for turbulent en- countered beat pulsations that appeared steady. We use the ergy (Yecko, KollaÂth, & Buchler 1998) similar to those ad- OPAL opacities of Iglesias & Rogers (1996) combined with vocated by Stellingwerf (1982), Kuhfuss (1986), Gehmeyr & those of Alexander & Ferguson (1994). The values of the TC Winkler (1992), and Bono & Stellingwerf (1994). In contrast parametersÐfor a de®nition, see Yecko et al. (1998)Ðare to these authors, however, we have developed additional tools ac ϭ3,,aL ϭ0.41 a ptDsϭ 0.667 ,,,,a ϭ 1 a ϭ 4 a ϭ 0.75 that allow us to ®nd beat behavior without having to rely on an ϭ 1.2. The steadiness of these beat pulsations was con®rmed very time-consuming and sometimes inconclusive hydrody- when several nonlinear hydrodynamics calculations, each ini- namic integrations to determine if a model undergoes stable, tiated with a different admixture of fundamental and ®rst over- or steady beat pulsations. These are (a) a linear stability anal- tone eigenvectors, converged towards the same ®nal steady beat ysis, which yields the frequencies and growth rates of all pulsational state. This convergence could be corroborated when modes; (b) a relaxation method (based on the general algorithm we extracted the slowly varying amplitudes with the help of a of Stellingwerf with the modi®cations of KovaÂcs & Buchler time-dependent Fourier decomposition and plotted the resulting L55 L56 KOLLAÂ TH ET AL. Vol. 502

phase portraits [A0 (t) vs. A1(t)] that are shown in Figure 1, where all initializations are seen to converge toward a ®xed point located at A0 ϭ 0.0104 and A1 ϭ 0.0200 (These radial displacement amplitudes assume the eigenvectors to be nor- malized to unity at the stellar surface, dr/r ϭ 1). While the observed transient behavior of ∗the models provides a conclusive proof of the presence of steady beat pulsations, it is important to explain and describe the behavior on a more fundamental level. The phase portrait of Figure 1 is very similar to those found for nonresonant mode interaction on the basis of amplitude equations. (Buchler & KovaÂcs 1986, 1987, here- after BK86 and BK87). We show here that indeed the nonlinear behavior of the hydrodynamical model pulsations can be understood very simply that way. The amplitudes of the two nonresonantly interacting modes obey remarkably simple equations:

dA 0 2 2 ϭ A0 (k 0 Ϫ q00 A0 Ϫ q01 A1); (1a) Fig. 1.ÐEvolution of the modal amplitudes for different initial conditions dt (crosses). Equal time intervals between dots. The open squares denote the unstable fundamental (F), the unstable ®rst overtone (O), and the stable double- dA 1 ϭ A (k Ϫ q A2 Ϫ q A2). (1b) mode (DM) ®xed points. dt 1 1 10 0 11 1 (BK86). One can show that these conditions are equivalent to

These amplitude equations are ªnormal formsº and are there- requiring kÅ0(1) 1 0 and kÅ1(0) 1 0, conditions which also imply fore generic for any dynamical system in which two modes that both single-mode limit cycles (fundamental and ®rst over- interact nonresonantly. The assumptions underlying these am- tone) are individually unstable. This validates Stellingwerf's plitude equations are satis®ed for Cepheids: (a) The lowest (1975) suggestion that the simultaneous instability of the fun- modes (fundamental and ®rst overtone here) are weakly non- damental and the ®rst overtone leads to steady beat pulsations adiabatic, i.e., the ratios of linear growth rates k to periods are (in the absence of a resonance). It provides an economical tool small, a condition that is readily con®rmed by our linear sta- to search for double-mode behavior, because we can now rel- bility analysis. (b) The pulsations are weakly nonlinear, which atively easily compute single-mode limit cycles and their allows a truncation of the amplitude equations in the lowest stability. permissible (third) order; weak nonlinearity can be established As a further con®rmation that the nonresonant scenario ap- by comparing the linear and nonlinear periods, which differ plies to the pulsating Cepheid model, we have determined the less than a tenth of a percent. Furthermore, both the nonlinear coef®cients of equations (1a) and (1b) as in BK87 by ®tting self-saturation coef®cients q00 and q11 as well as the cross- time-dependent solutions of these equations to the temporal coupling coef®cients q01 and q10 have always been found to be variation of the amplitudes in their approach to the limit cycle positive in Cepheid models so that amplitude saturation can as shown in Figure 1. The ®tted trajectories in the phase portrait occur in third order, and it is suf®cient to keep terms up to are practically undistinguishable from the hydro results, con- cubic in the amplitudes. (c) In the range of interest there is no ®rming the applicability and accuracy of the amplitude equation important low-order resonance between the fundamental and formalism and the absence of any relevant resonances. the ®rst overtone modes, and possibly a higher mode. We mention in passing that the expression ªdouble-mode In the following discussion we look at the regime where both Cepheidsº is often used cavalierly for beat Cepheids. Since no modes are linearly unstable, k 0 1 0 and k1 1 0. Equations (1a) additional, resonant overtone is involved in the beat pulsations, and (1b) then have two single-mode ®xed points. The amplitude the latter are thus truly double-mode pulsations. of the single-mode fundamental (0) ®xed point is A0 ϭ With the relaxation code, we are able to compute both the 1/2 (k 0 /q00 ) , and its coef®cient for stability to ®rst overtone per- fundamental and the ®rst overtone limit cycles with their re- 2 turbations is kÅ1(0) ϭ k1 Ϫ q10 A0. In our notation a positive co- spective amplitudes and Floquet stability exponents l 1(0) ϭ ef®cient implies growth and thus instability. The corresponding P0kÅ1(0) and l 0(1) ϭ P1kÅ0(1). The above discussion then shows that 1/2 ®rst overtone (1) limit cycle amplitude is A1 ϭ (k1 /q11 ) , and from these four quantities we can extract the four nonlinear its coef®cient of stability to fundamental perturbations is qjk coef®cients when we have already computed the linear pe- 2 kÅ0(1) ϭ k 0 Ϫ q01 A1. The kÅ-values, when multiplied by the periods and growth rates. The values we obtain this way for this riods Pk of their limit cycles, are equal to the corresponding beat Cepheid model agree quite well with those that we obtain Floquet exponents (Buchler, Moskalik, & KovaÂcs 1991). from the ®t described in the previous paragraph. Note that these Equations (1a) and (1b) can also have a double-mode ®xed two determinations rely on independent numerical hydrody- 2 2 point whose amplitudes satisfy A0DM ϭ kÅ0(1) q11 /D ! A0, namical input, the ®rst on two periodic limit cycles (that are 2 2 A1DM ϭ kÅ1(0) q00 /D ! A1, where D ϭ q00 q11 Ϫ q01 q10. This ®xed linearly unstable), the second on transient evolution toward the 2 2 point exists provided A0DM 1 0 and A1DM 1 0. Then, if D ! 0, stable double-mode pulsation. the double-mode limit cycle is unstable. Stable pulsations occur In order to investigate the robustness of the observed beat either in the fundamental or ®rst overtone, and the pulsational behavior we now explore the pulsational behavior of a sequence mode is determined by the evolutionary history of the model of Cepheid models in which the effective temperature of the

(hysteresis). On the other hand, if D 1 0, the double-mode ®xed equilibrium modes of the sequence varies from Teff ϭ 6200 to point is stable and steady double-mode pulsations occur 5800 K. Such a sequence is approximately along an evolu- No. 1, 1998 NONLINEAR BEAT CEPHEID MODELS L57

Fig. 2.ÐThe limit cycle stability coef®cients (dayϪ1) along a Cepheid sequence (i.e., as a function of Teff) for various eddy viscosity strengths an; fundamental limit cycles kÅ1(0) (®lled circles) and ®rst overtone limit cycles kÅ0(1) (open circles).

Fig. 3.ÐModal selection in the eddy viscosity-Teff plane tionary path. The eddy viscosity parameter an is treated as an additional variable parameter to explore the effect of TC on the behavior. the sign of D that makes double-mode behavior possible for

In Figure 2 the stability coef®cients of the sequence are suf®ciently large an. plotted versus Teff, with open/®lled circles for those of the fun- The condition for beat behavior is thus seen to be rather damental/overtone single-mode cycles. The curves are labeled subtle in that it involves the effects of convection beyond the with the corresponding strengths an of the eddy viscosity. As linear regime for which it seems dif®cult to give a ªsimpleº discussed above we expect double-mode behavior where both physical explanation. Floquet exponents are positive. (The stability exponents re- Figure 3 gives the overall modal selection picture in the sulting from perturbations with other modes are always smaller an-Teff plane. The linear edges of the instability region (k 0 ϭ in this sequence and are therefore irrelevant here). For the low 0 and k1 ϭ 0) are shown as dashed lines. By computing the value of an ϭ 0.5 (dotted lines) the two stability coef®cients fundamental and ®rst overtone limit cycles for a number of are never positive simultaneously, thus excluding double-mode an and Teff-values, by interpolation, we can obtain kÅ0(1) or behavior. On the other hand, for an ϭ 1.2 a double-mode region kÅ1(0) as a function of an and Teff, and in particular the loci where appears between T 5875±5915 K, and for a ϭ 2.0 this they vanish. The solid curves give the nonlinear pulsation edges eff ∼ n broadens to Teff 5965±6050 K. and are marked ªOREº and ªFBE.º How does tur∼bulent convection bring about double-mode It is straightforward to show that if the two linear growth behavior? Figure 2 shows that, in the region of interest, an rates vanish at the same point, the four curves will intersect in increase in the turbulent eddy viscosity causes a rapid decrease a single point on this diagram, which we label critical point. in the stability of the fundamental limit cycle (kÅ1(0), ®lled cir- The curve marked ªOBEº is the linear blue edge of the ®rst cles), but an increase in that of the ®rst overtone limit cycle overtone mode, and it coincides with the overtone nonlinear

(kÅ0(1), open circles). This description, though, does not tell us blue edge up to and on the left of the critical point. The linear whether it is the effect of TC on the linear k's or on the non- fundamental blue edge becomes also the fundamental blue edge linear q's, or on both, that is responsible for the beat pulsations. above the critical point. Above the line ORE we have kÅ0(1) 1 Table 1 shows the variation with an of the relevant model 0, and the ®rst overtone limit cycle is unstable. Below the line quantities, viz. the linear growth rates, the nonlinear coupling FBE the quantity kÅ1(0) 1 0, and the fundamental limit cycle is coef®cients, the discriminant D ϭ q00 q11 Ϫ q01 q10, the ampli- unstable. Thus, in the region marked ªdmº both single-mode tude of the fundamental cycle and its stability coef®cient with limit cycles are unstable, and this is the region of double-mode respect to overtone perturbations, and the same for the ®rst pulsation. In the small triangular region at the bottom, on the overtone. other hand, both limit cycles are stable, and either fundamental The necessary condition for stable double-mode pulsations, or ®rst overtone limit cycles can occur. viz. D 1 0, is never found to be satis®ed in radiative models. In summary, stable ®rst overtone pulsations occur in the In these models the cross-coupling always dominates over the dotted region, delineated by the lines OBE and ORE. The fun- self-saturation coef®cients. Table 1 shows that an increase in damental limit cycle is stable in the region marked by open the strength of the eddy viscosity causes q00 and q11 to increase squares, delineated by FBE and FRE (not shown on the far faster than q01 and q10. It is therefore the resultant change in right). This ®gure makes it particularly evident how TC favors

TABLE 1 Nonlinear Coupling Coefﬁcients (5900 K Model)

an k0 k1 q00 q01 q10 q11 D A0 kÅ0(1) A1 kÅ1(0) 0.5 ...... 2.382EϪ3 1.124EϪ2 1.199 3.532 4.898 13.311 Ϫ1.343 4.458EϪ2 1.510EϪ3 2.906EϪ2 Ϫ6.012EϪ4 1.0 ...... 2.169EϪ3 8.636EϪ3 1.815 3.865 6.645 14.924 1.407 3.457EϪ2 6.959EϪ4 2.406EϪ2 Ϫ6.752EϪ5 1.2 ...... 2.082EϪ3 7.582EϪ3 2.179 4.200 7.554 16.001 3.140 3.091EϪ2 3.650EϪ4 2.177EϪ2 9.169EϪ5 1.5 ...... 1.947EϪ3 5.983EϪ3 2.893 4.917 9.212 18.172 7.269 2.595EϪ2 Ϫ2.188EϪ4 1.814EϪ2 3.285EϪ4 2.0 ...... 1.720EϪ3 3.282EϪ3 4.685 7.194 13.039 24.229 19.704 1.916EϪ2 Ϫ1.506EϪ3 1.164EϪ2 7.458EϪ4 L58 KOLLAÂ TH ET AL. Vol. 502 double-mode pulsations and why all efforts with radiative codes have failed in modeling beat Cepheids. We have seen that when TC effects are suf®ciently large then the Cepheids should run into the double-mode regime in both their crossings of the instability strip. Furthermore, as a Cepheid crosses the double-mode regime redward, say, the ®rst overtone amplitude should gradually go to zero, while the fundamental amplitude increases from zero to the value it attains as a fundamental mode Cepheid (BK86). The question arises whether this nonresonant scenario that is derived from the amplitude equations is in agreement with the observations. The four SMC beat Cepheids from the EROS survey (an- alyzed by Beaulieu and reproduced in Buchler 1998) all have the same amplitude ratios, A /A 0.45, a priori in disagree- Fig. 4.ÐSequence of models (same as in top of Fig. 2). Amplitudes of the 0 1 ∼ fundamental and of the ®rst overtone limit cycles (solid lines, stable; dashed ment with the nonresonant scenario shown in Figure 1 of BK86, lines, unstable). Double-mode component amplitudes of the double mode (®lled which suggests that Cepheids with all amplitudes ratios should squares, fundamental; open diamonds, ®rst overtone). occur. In Figure 4 we display the behavior of the component modal amplitudes of the beat Cepheid models for the an ϭ 1.2 se- We have demonstrated that turbulent convection leads nat- quence of Figure 2. The amplitudes of the stable single-mode urally to beat behavior in Cepheids, which does not occur with limit cycles are shown as solid lines with solid dots for the purely radiative models. The reason is that the nonlinear effects fundamental and open dots for the ®rst overtone, and as dashed of TC dissipation can create a region in which both the fun- lines where they are unstable. The fundamental and ®rst over- damental and the ®rst overtone cycles are unstable, and the tone component amplitudes of the stable double-mode pulsators model undergoes stable double-mode pulsations. At a more are shown as ®lled squares and open diamonds, respectively. basic level the amplitude equation formalism shows that tur- It is seen that although the modal amplitudes do indeed vary bulent convection modi®es the nonlinear coupling between the continuously throughout as the double-mode regime is tra- fundamental and ®rst overtone modes in such as way as to versed, the variation is very rapid near the cooler side. The allow beat behavior. reason for this unexpected behavior is that the q's are not The development of a relaxation code (TC) to ®nd periodic constant in this sequence, and what is more, they vary in such pulsations, and a Floquet stability analysis of these limit cycles a way that D happens go through zero around 5850 K. It is has made the search quite ef®cient, and a broader survey of the presence of this nearby pole that causes a change in the beat Cepheids, with wide ranges of metallicities is in progress. curvature of A0. This will also search for beat Cepheid models that pulsate in According to Figure 4 it is therefore much more likely to the ®rst and second overtones. ®nd beat Cepheids in the slowly varying regime in which the ratio A0 /A1 Շ 0.5. The computed behavior of the modal am- This research has been supported by the Hungarian OTKA plitudes is thus in agreement with the observed SMC Cepheids, (T-026031), AKP (96/2-412 2,1), and by the NSF and the nonresonant scenario is consistent with the (AST95±28338) at UF. Two of us (J. P. B. and Z. K.) thank observations. the French AcadeÂmie des Sciences for ®nancial support.

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