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Evolution and pulsation period change in the Large Magellanic Cloud Cepheids

Theoretical estimates of the pulsation period change rates in LMC Cepheids are obtained from consistent calculation of stellar evolution and nonlinear stellar pulsation for stars with initial chemical composition X=0.7, Z=0.008, initial masses 5M_\odot<= Mzams<= 9M_\odot and pulsation periods ranged from 2.2 to 29 day. The Cepheid hydrodynamical models correspond to the evolutionary stage of thermonuclear core helium burning. During evolution across the instability strip in the HR diagram the pulsation period of Cepheids is the quadratic function of the evolution time for the both fundamental mode and first overtone. Cepheids with initial masses Mzams>= 7M_\odot pulsate in the fundamental mode and the period change rate \dot\Pi varies nearly by a factor of two for both crossings of the instability strip. In the period -- period change rate diagram the values of the period and \dot\Pi concentrate within the strips, their slope and half--width depending on both the direction of the movement in the HR--diagram and the pulsation mode. For oscillations in the fundamental mode the half-widths of the strip are \delta\log\dot\Pi = 0.35 and \delta\log\dot\Pi = 0.2 for the first and the second crossings of the instability strip, respectively. Results of computations are compared with observations of nearly 700 LMC Cepheids. Within existing observational uncertainties of \dot\Pi the theoretical dependences of the period change rate on the pulsation period are in a good agreement with observations.

term indicating the secular period change (Kukarkin and Florja 1932). Interest in such a property grew after works by Hofmeister et al. (1964) and Iben (1966) where the evolutionary state of Cepheids was determined and long-term period changes were thought to be due to evolutionary changes of the stellar structure during thermonuclear core helium burning (Fernie 1979; Mahmoud and Szabados 1980;Szabados 1983;Deasy and Wayman 1985).
In recent years a great deal of observational data on long-term pulsation period changes in the Large Magellanic Cloud (LMC) Cepheids was obtained in ASAS, MACHO and OGLE projects. Pietrukowicz (2001) considered data on 378 LMC Cepheids and concluded that all studied variables with periods longer 8 days show period changes. Later Poleski (2008) carried out an analysis of 655 LMC Cepheids and found that 18% of fundamental mode and 41% of first overtone pulsators have evolutionary period changes.
The estimation of the period change rateΠ from observations is of great interest since it provides with the direct test of the stellar evolution theory. Unfortunately, theoretical studies of pulsation period changes in Cepheids based on consistent solution of the equations of stellar evolution and stellar pulsation have not been done yet. Pietrukowicz (2001) compared his observational data with evolutionary and pulsation models studied by Alibert et al. (1999) and Bono et al. (2000). However the pulsation period change ratesΠ were not evaluated in these theoretical works, so that Pietrukowicz (2001) used rough estimates from presented tabular data. Moreover, Alibert et al. (1999) in their linear analysis of pulsational instability did not take into account effects of convection. Such a simplification might be responsible for a large disagreement between theoretical models and observational estimates ofΠ (Pietrukowicz 2001). Poleski (2008) compared his observational data with stellar evolution theory using the approach by Turner et al. (2006) which is also based on strong simplifications.
The goal of the present work is to obtain theoretical estimates of the pulsation period change rateΠ as a function of the age of the Cepheid using the consistent calculations of stellar evolution and nonlinear stellar pulsation. Initial relative mass abundances of hydrogen and elements heavier than helium correspond to the LMC chemical composition: X = 0.7, Z = 0.008. In hydrodynamical calculations of nonlinear stellar pulsations we take into account effects of turbulent convection, so that the hydrodynamical models occupy the whole interval of effective temperatures bounded in the Hertzsprung-Russel (HR) diagram by the blue and red edges of the instability strip. Methods of stellar evolution calculation and basic equations of radiation hydrodynamics and turbulent convection used for calculation of nonlinear stellar pulsation are given in our previous paper (Fadeyev 2013).

results of computations
Solution of the equations of hydrodynamics for nonlinear stellar oscillations as a function of time t was done with initial conditions taken in the form of stellar models of evolutionary sequences of stars with initial masses 5M ⊙ ≤ M ZAMS ≤ 9M ⊙ . Evolutionary tracks in the HR diagram of stars under consideration are shown in Fig. 1 where in dotted lines are shown parts of the track corresponding to the instability against radial oscillations. In the starting and in the ending track points the rate of the thermonuclear energy generation rate ε n,c and the rate of the gravitational energy production ε g,c in the stellar center are nearly the same: ε n,c ≈ ε g,c .
Therefore the tracks displayed in Fig. 1 represent the evolutionary stage when the only source of energy generation in the stellar center is thermonuclear helium burning.
For each evolutionary track the bounds of pulsational instability in the HR diagram were determined from hydrodynamical computations where as in our previous work (Fadeyev 2013) the kinetic energy of pulsation motions E K was calculated. The part of the evolutionary track with pulsational instability was determined from condition η > 0, where η = Π −1 d ln E Kmax /dt is the growth rate of the kinetic energy, E Kmax is the maximum value of the kinetic energy reached during one pulsational cycle. The pulsation period Π was evaluated from the discrete Fourier transform of the kinetic energy E K . It should be noted that the interval of time t within of which we integrate equations of hydrodynamics is comparable with the thermal scale of outer layers of the Cepheid and is much shorter in comparison with the nuclear evolution time scale.
Fot example, in the Cepheid with initial mass M ZAMS = 7M ⊙ the evolution time between the red and blue edges of the instability strip is ∼ 10 5 years, whereas hydrodynamic computations of the instability growth with subsequent limit cycle attainment are done on the time interval of ∼ 10 years.
In Fig. 2 we give the plots of the instability growth rate η versus effective temperature averaged over the pulsational cycle T eff for four Cepheid evolutionary sequences with initial masses form 5M ⊙ to 8M ⊙ . The evolutionary track crosses the instability strip twice in the HR diagram and therefore each evolutionary sequence is represented by two plots where the first one correponds to the movement across the HR diagram with increasing effective temperature (dotted lines) and the second plot corresponds to the movement in the opposite direction (dash-dotted lines). In Fig. 2 we use the averaged over the cycle effective temperature T eff as independent variable because in the hydrodynamical model the average radius of the photosphere r ph is smaller than the radius of the photosphere of the hydrostatically equilibrium model r ph,0 . For hydrodynamical models of Cepheids calculated in the present study the ratio of the photosphere radii ranges within 0.975 ≤ r ph /r ph,0 < 1 and edges of the instability strip shift to the blue in the HR diagram by 30 K < ∆T eff < 70 K.
The presence of two maxima in plots of η for M ZAMS ≤ 6M ⊙ is due to the fact that near the red edge of the instability strip radial pulsations are excited in the fundamental mode, whereas at higher effective temperatures pulsations are excited in the first overtone. Transition between oscillation modes takes place within the effective temperature range 6000 K < T eff < 6100 K.
In Cepheids with initial mass M ZAMS = 7M ⊙ evolving blueward across the instability strip radial oscillations are due to instability of the fundamental mode and transition to the first overtone takes place just near the blue edge at T eff ≈ 6200 K. During the second crossing of the instability strip radial oscillations exist in the form of the fundamental mode.
Pulsations of Cepheids with initial mass M ZAMS ≥ 8M ⊙ are always due to instability of the fundamental mode and the pulsation period Π gradually changes while the star moves in the HR diagram from one edge of the instability strip to another. The change of the pulsation period of the Cepheid with initial mass M ZAMS = 8M ⊙ is illustrated in Fig. 3 where for the sake of convenience we set the evolution time t ev to zero when the star crosses the edge of the instability strip and begins to oscillate. The plot with gradual decrease of the pulsation period corresponds to the first crossing of the instability strip and the plot with gradually increasing period corresponds to the second crossing. Hydrodynamical models with positive and negative growth rates η are shown in filled circles and opened circles, respectively. As is seen from shown plots the pulsation period Π is fitted by an algebraic polynomial Π(t ev ) = a 0 + a 1 t ev + a 2 t 2 ev for both evolutionary sequences with a good accuracy (i.e. with relative r.m.s. error less than one per cent). Polynomial approximation is shown in Fig. 3 by dotted and dash-dotted lines for the first crossing and the second crossing of the instability strip, respectively.
Expression of the pulsation period Π as a quadratic polynomial of t ev was found to be a good approximation for all Cepheid models considered in the present study. The only exception is a discontinuity of the period due to transition from one pulsation mode to other. This is illustrated in Fig. 4

comparison with observations
The period of light variations is the only quantity which can be determined from observations of the pulsating variable star with sufficiently high precision. Therefore for comparison of the results of theoretical computations with observational data we will consider the period change rateΠ as a function of the pulsation period Π.
In Fig. 5 where the period Π is expressed in days, whereas k = 0 and k = 1 correspond to the fundamental mode and to the first overtone, respectively. Due to the finite width of the instability strip the plots of evolutionary sequences shown in Fig. 5 by solid lines are confined within the bands with half-width δ logΠ = 0.035 for k = 0 and δ logΠ ≈ 0.1 for k = 1.
The diagram period -period change rate for Cepheids of the second crossing of the instability strip is shown in Fig. 6. Unfortunately, reliable estimates of the period change rate for Cepheid models with initial mass M ZAMS = 5M ⊙ evolving redward in the HR diagram were not obtained, so that in Fig. 6 the mean dependence of the period change rate is shown only for the fundamental mode: logΠ = −10.33 + 3.386 log Π.
The half-width of the band in the period -period change rate diagram is δ logΠ = 0.2.
To compare results of our theoretical computations with observations we used observational estimates of the period Π and the period change rateΠ from works by Pietrukowicz (2001) and Poleski (2008). Electronic tables 1 -3 supplementing the paper by Pietrukowicz (2001) give data on 369 LMC Cepheids, whereas the period change rates were evaluated using the Harvard photographic observations obtained in time interval from 1910 to 1950. Observational data on LMC Cepheids obtained from the OGLE survey were received from the author (Poleski 2008).
The diagram period -period change rate for Cepheids crossing the instability strip blueward with negativeΠ is presented in Fig. 7a  An interesting result of our calculations is that the dependence of the period change ratė Π on the pulsation period Π for the fundamental mode differs from that for the first overtone.
Unfortunately, at present observational confirmation of this feature seems to be impossible because of insufficiently high accuracy of observational evaluation ofΠ for the first overtone Cepheids. Significant scatter of points in Fig. 7 at short periods (Π < 7 day) is mainly due to the power law decrease ofΠ with decreasing pulsation period. Indeed, as is seen in Fig. 5 the period change rates in first overtone Cepheids are three orders of magnitude smaller in comparison with those in long period (Π ≈ 30 day) Cepheids. Therefore, to reduce the error of the observational estimate ofΠ in first overtone pulsators to the value comparable with that in long period Cepheids the time interval of the O − C diagram should be expanded by two orders of magnitude.
Thus, for comparison of the stellar evolution calculations with observations of most interest are fundamental mode Cepheids. Here among the important questions we should emphasize the role of chemical composition and convective overshooting in the period -period change rate diagram. It should also be noted that the crossing time of the Cepheid instability strip by core helium burning stars with initial masses M ZAMS > 9M ⊙ becomes comparable with that during the gravitational contraction of the helium core before the stage of the red supergiant.
Thefore the most massive long period Cepheids with gravitationally contracting core may play perceptible role in the period -period change rate diagram.
The author thanks Radek Poleski who kindly placed the Cepheid OGLE data at his disposal.
The study was supported by the Basic Research Program of the Russian Academy of Sciences "Nonstationary phenomena in the Universe".