From Solar-like to Mira stars: a unifying description of stellar pulsators in the presence of stochastic noise
MMNRAS , 1– ?? (2019) Preprint 22 September 2020 Compiled using MNRAS L A TEX style file v3.0
From Solar-like to Mira stars: a unifying description of stellarpulsators in the presence of stochastic noise
M. S. Cunha , , P. P. Avelino , , , W. J. Chaplin , Instituto de Astrof´ısica e Ciˆencias do Espac¸o, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal Departamento de Fsica e Astronomia, Faculdade de Ciłncias, Universidade do Porto, Rua do Campo Alegre 687, PT4169-007 Porto, Portugal School of Physics and Astronomy, University of Birmingham, Birmingham, B15 2TT, United Kingdom Stellar Astrophysics Centre (SAC), Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We discuss and characterise the power spectral density properties of a model aimed at de-scribing pulsations in stars from the main-sequence to the asymptotic giant branch. We showthat the predicted limit of the power spectral density for a pulsation mode in the presence ofstochastic noise is always well approximated by a Lorentzian function. While in stars pre-dominantly stochastically driven the width of the Lorentzian is defined by the mode lifetime,in stars where the driving is predominately coherent the width is defined by the amplitudeof the stochastic perturbations. In stars where both drivings are comparable, the width is de-fined by both these parameters and is smaller than that expected from pure stochastic driving.We illustrate our model through numerical simulations and propose a well defined classifica-tion of stars into predominantly stochastic (solar-like) and predominately coherent (classic)pulsators. We apply the model to the study of the Mira variable U Per, and the semiregularvariable L2 Pup and, following our classification, conclude that they are both classical pul-sators. Our model provides a natural explanation for the change in behaviour of the pulsationamplitude-period relation noted in several earlier works. Moreover, our study of L2 Pup en-ables us to test the scaling relation between the mode line width and effective temperature,confirming that an exponential scaling reproduces well the data all the way from the mainsequence to the asymptotic giant branch, down to temperatures about 1000 K below what hasbeen tested in previous studies.
Key words: stars: evolution – stars: interiors – stars: oscillations
Stochastic noise from near-surface convection is generally acceptedto be the main driver of the pulsations observed in solar-like pul-sators, from the main-sequence to the red-giant evolution phase(see, e.g Houdek & Dupret 2015, for a review). However, the sit-uation is not so clear for further evolved cool, luminous long-period pulsators located near the tip of the red-giant branch andin the asymptotic giant branch, that we shall collectively call theLong Period Variables (LPVs), comprising the semiregular andMira variables. In fact, the driving mechanism behind the pul-sations observed in LPVs, in particular in the semiregular vari-ables, has been a matter of significant debate. On the theoreti-cal side, stability studies of pulsations in LPVs face a significantchallenge related to the need to correctly account for the couplingbetween the pulsations and convection. Several efforts to accountfor this coupling have been made over the years, generally point-ing towards pulsations in LPVs being intrinsically unstable, withthe turbulent pressure playing an important role in the driving ofthe pulsations (e.g. Gough 1967; Munteanu et al. 2005; Xiong & Deng 2013; Xiong et al. 2018). However, on the observational side,the similarity between the pulsation properties in semiregular vari-ables and solar-like pulsators has led a number of authors to arguethat pulsations in semiregular variables are stochastically excited(Christensen-Dalsgaard et al. 2001; Bedding et al. 2005; Soszyn-ski et al. 2007; Dziembowski & Soszy´nski 2010; Mosser et al.2013; Yu et al. 2020), as opposed to being coherently-driven andintrinsically-unstable, as assumed for Mira stars.Independently of what is the main source of driving of pulsa-tions in LPVs and of whether their modes are intrinsically stableor unstable, it is clear that convection interacts and influences thepulsations they exhibit. Indeed, even the Mira stars often show pul-sation amplitude and phase variability which may be interpreted asresulting from random fluctuations (Eddington & Plakidis 1929),likely associated to the presence of convection (Percy & Colivas1999). This is in addition to secular variations observed in the pul-sation periods of some Mira stars, whose origin may be related withthe onset of thermal pulses (e.g. Wood & Zarro 1981; Moln´ar et al.2019; Templeton et al. 2005), as well as period variations observed c (cid:13) a r X i v : . [ a s t r o - ph . S R ] S e p Cunha al. on timescales of decades (e.g. Zijlstra & Bedding 2002; Temple-ton et al. 2005). Recently, the impact of stochasticity on otherwisecoherently-driven pulsators has been considered by Avelino et al.(2020) (hereafter Paper I), based on an internally driven dampedharmonic oscillator model.In the present paper we explore further the model introducedin Paper I, by considering a range of possibilities for the impact ofthe stochastic perturbations on pulsations, from the limit when thepulsations are essentially stochastically driven to the limit when thecoherent driving dominates. Since our model is phenomenological,we cannot draw conclusions on the exact physical sources of exci-tation and damping of the modes, as done in stability analyses suchas that discussed by Xiong & Deng (2013). However, our approachallows us to draw a bridge between theory and observations, byassuming that a pulsation mode is described by a model that incor-porates simultaneously a coherent and a stochastic driving source,in addition to a damping term. The focus is on the model predic-tions for the power spectral density and how they compare with theknown observational properties of pulsations in stars where thesepulsations may be influenced by convection.The paper is organised as follows: in Sec. 2 we briefly re-view the properties of the model introduced in Paper I along withthe main conclusions drawn in that work, which are of importanceto the discussion presented here. Section 3 discusses the proper-ties of the power spectral density for a pulsation mode in the limitcases when stochasticity has no significant influence on the pulsa-tion amplitude (classical limit) and when it dominates the driving(stochastic limit), as well as in the case when both drivings con-tribute significantly to the pulsation (intermediate case). In Sec. 4we perform numerical simulations based on the proposed model totest the analytical predictions for the power spectral density intro-duced in Sec. 3 and discuss further the impact of the main physicalmodel parameters, namely, the mode lifetime and the amplitude ofthe stochastic noise, on the properties of the power spectral den-sity. In Sec. 5 we apply our model to the study of the Mira starU Per and the semiregular star L2 Pup, and draw conclusions onthe main driving source for the pulsations observed in these stars.In Sec. 6 we argue that our model provides a natural explanationto the change in behaviour observed in the pulsation amplitude-period relation of LPVs and show how the results of our analysisfor L2 Pup allow us to test the mode line width-effective tempera-ture scaling relation at low effective temperatures. Finally, in Sec. 7we summarise our main conclusions.
In this work we adopt the internally driven damped harmonic oscil-lator model proposed in Paper I. In this model the displacement, x ,is described by the equation of motion ¨ x ( t ) + 2 η ˙ x ( t ) + ω x ( t ) = a f ( t ) + ξ ( t ) , (1)where a dot indicates a derivative with respect to time, t , η is thedamping constant, ω is the natural angular frequency of the os-cillator, a f is the acceleration associated to the internal coherentdriving mechanism, given by a sinusoidal function of frequency ω f and a varying phase (see their section 3 for details), and ξ isthe function that parameterizes the stochastic noise. The underly-ing assumption is that the stochastic perturbations induce a seriesof successive random velocity kicks separated by a time interval ∆ t . In that case, ξ ( t ) may be written as ξ ( t ) = 2 A N ω √ ω ∆ t K (cid:88) k =0 r ( k ) δ ( t − t k ) , (2)where A N (with dimensions of a length) parameterizes the am-plitude of the noise, r ( k ) are independent random variables witha normal distribution of mean zero and unit standard deviation( r ( k ) ∼ N (0 , ), and δ is the Dirac delta function. Moreover, t k = k ∆ t , and t obs = K ∆ t is the total observing time. The nor-malisation adopted in Eq. (2) is such that the average velocity vari-ation in a timescale equal to the oscillation period P is proportionalto A N ω .The model described by Eq. (1) will be used to study pul-sations where stochastic and coherent driving are simultaneouslypresent. There are two limits to this model, the first correspondingto setting a f = 0 , in which pulsations are driven solely by stochas-tic perturbations, and the second corresponding to setting ξ ( t ) = 0 ,in which the driving is fully coherent at all times. We shall referto pulsations described by these limits as stochastic (or solar-like)and purely classical, respectively. In addition we shall introducethe classical limit (Sec. 3.1), which is the limit when the impact ofstochasticity on the pulsation amplitude is small, but the inducedchange in the pulsation phase can still be significant, if consideredover a sufficiently long period of time.In this work we are interested in bridging between the classicaland stochastic limits, focusing, in particular, on how the pulsationpower spectrum properties change as the impact of stochasticityincreases. The classical limit has been studied in detail in Paper I.In particular, the authors have shown that in this limit the phasevariation displays a random walk behaviour, while the amplitudevariation remains small. Working under the assumption that the ob-served (angular) frequency equals ω f and is approximately equal to ω (requiring that η (cid:28) ω / ), the authors derived analytical ex-pressions for the root mean square (rms) of the relative pulsationamplitude variation ∆ A/A and of the phase variation ∆ ϕ after atotal observing time, t obs . These expressions, valid in the classicallimit, are given by σ ∆ A/A = A N A (cid:114) ω η , (3)and σ ∆ ϕ = A N A √ ω t obs , (4)respectively ( cf. their equations 25 and 27), where t obs is assumedto be much longer than η − . In the model ∆ A and ∆ ϕ are definedas the difference in amplitude and phase relative to the values inthe case of no stochastic noise ( ξ = 0 ). In practice, when deal-ing with observations the exact underlying coherent amplitude ofthe pulsation is unknown. In that case, the time average of the am-plitude, (cid:104) A (cid:105) , can be used as an estimator of A and ∆ A computedwith respect to that average. Moreover, ∆ ϕ can be computed withrespect to any fixed phase value (e.g., the phase at the start of theobservations). Observationally, phase variations in variable-star re-search are usually expressed in terms of O − C diagrams (e.g.,Sterken 2005; Catelan & Smith 2015, and references therein). Inthat context, ∆ ϕ is to be interpreted as the stochastic signature seenin these diagrams. While the O − C diagrams of some LPVs mayalso exhibit signatures of the decadal and secular pulsation periodvariations mentioned in Sec. 1, these long-term variations will havea negligible impact on σ ∆ ϕ , compared to the impact of the short-term stochastic variations. In addition to characterising the phase MNRAS , 1– ?? (2019) tochastic Signatures II and amplitude variations, the authors have shown that, unlike insolar-like pulsators, the power of the signal in the classical limit isindependent of η .Throughout the remainder of this paper, we will focus on thepower spectrum distribution that results from the model describedby Eq. (1), considering not only the classical and stochastic limits,but also the intermediate regime in which the two driving termshave a comparable impact on the pulsations. When the coherent driving dominates, the amplitude variations re-main small at all times, but the phase variation with respect to afixed time (e.g., the time t at the beginning of the observations)follows a random walk, its variance thus increasing with time. Inthis section we shall derive an analytical expression for the powerspectrum of the signal, x , under the classical limit, which we defineas being the limit when the amplitude variations may be neglectedin the computation of the power (see Hajimiri & Thomas (1999)for an analogous calculation in the context of circuits and commu-nications). In that limit, and under the assumption that η (cid:28) ω / (so that the observed frequency is ≈ ω ; Paper I), the solution toEq. (1) can be written as x = A sin [ ω t + ϕ ] , (5)where A is essentially a constant (equal to the amplitude of the pul-sation in the stochastic-free case) and ϕ describes a random walk.Let us start by defining the parameter Γ c ≡ (cid:18) A N A (cid:19) ω π = σ ϕ π | τ | , (6)which is a constant in our problem. Here ∆ ϕ [ τ ] ≡ ϕ [ t + τ ] − ϕ [ t ] , τ being any given positive or negative time interval whose absolutevalue is much longer than the time interval between the randomkicks ∆ t , and the equality follows directly from Eq. (4), which isvalid when the coherent driving dominates.The variance of the phase change over an observationtimescale τ is given by σ ϕ [ τ ] = (cid:104) ( ϕ [ t + τ ] − ϕ [ t ]) (cid:105) , (7)where the brackets represent an average over an infinite number ofrandom walk realizations at fixed time t (the time t being arbitrary).Next, consider the auto-correlation function of the signal givenby (cid:104) x [ t ] x [ t + τ ] (cid:105) == A (cid:104) sin [ ω t + ϕ [ t ]] sin [ ω ( t + τ ) + ϕ [ t + τ ]] (cid:105) = A (cid:104) cos [ ω τ + ∆ ϕ ] (cid:105)− A (cid:104) cos [ ω (2 t + τ ) + 2 ϕ [ t ] + ∆ ϕ ] (cid:105) . (8)If the time t is sufficiently long, the last term in Eq. (8) vanishes.Indeed, due to its random walk nature, the phase ϕ in the last termof Eq. (8) will follow a normal distribution. When the time t atwhich the random walk realisations are considered is sufficientlylong to ensure that the rms of ϕ is much greater than 2 π , ϕ mod 2 π will approach a uniform distribution in the interval [0,2 π ] and the average of the last term in Eq. (8) over many realisations will tendto zero. Hence, the auto-correlation function becomes simply (cid:104) x [ t ] x [ t + τ ] (cid:105) = A (cid:104) cos [ ω τ + ∆ ϕ ] (cid:105) = A ω τ ] (cid:104) cos ∆ ϕ (cid:105) (9) − A ω τ ] (cid:104) sin ∆ ϕ (cid:105) , (10)The probability density distribution of the phase variation ∆ ϕ [ τ ] on a timescale τ , is given by p (∆ ϕ ) = 1 √ πσ ∆ ϕ exp (cid:32) − ∆ ϕ σ ϕ (cid:33) . (11)Therefore (cid:104) cos ∆ ϕ (cid:105) = (cid:90) ∞−∞ p [∆ ϕ ] cos[∆ ϕ ] d ∆ ϕ (12) = exp (cid:20) − σ ϕ (cid:21) = exp [ − π Γ c | τ | ] (cid:104) sin ∆ ϕ (cid:105) = (cid:90) ∞−∞ p [∆ ϕ ] sin[∆ ϕ ] d ∆ ϕ = 0 , (13)where the last equality in Eq. (12) follows from Eq. (6).Substituting Eqs (12) and (13) into Eq. (10), the auto-correlation function may be finally written as ζ ( τ ) = (cid:104) x [ t ] x [ t + τ ] (cid:105) = A ω τ ] exp ( − π Γ c | τ | ) . (14)The limit power spectral density PSD ( i.e. the power spectral den-sity expected after an infinite number of realisations) may thenbe obtained by performing the Fourier transform of the auto-correlation function of the signal. Defining f = ω / π and f = ω/ π , we have PSD( f ) = (cid:90) + ∞−∞ e − iωτ ζ ( τ ) dτ == A π Γ c (cid:18)
11 + ( f − f ) / Γ c + 11 + ( f + f ) / Γ c (cid:19) , (15)In general Γ c (cid:28) f , thus implying that the limit power spectraldensity around the frequency f (for | f − f | (cid:28) f ) is wellapproximated by a Lorentzian function.This power is to be compared with that computed from thelight curves of pulsating stars. Hence we shall consider a single-sided power spectrum and, accordingly, multiply the power in eachpositive frequency by two, to ensure it is calibrated according toParsevals Theorem. Doing so, the power spectral density aroundthe frequency f in the classical limit becomes PSD( f ) = A π Γ c
11 + ( f − f ) / Γ c , (16)representing a Lorentzian function with a half width at half max-imum power equal to Γ c . Thus, so long as a classical pulsator inthe presence of stochastic noise is observed for a long enough pe-riod of time, we may expect the power spectral density of a singlepulsation mode to be a resolved Lorentzian function whose widthdepends on the amplitude of the stochastic noise (tending to zero inthe limit when ξ → ) but not on the damping constant. here and throughout this work we have adopted the following definitionfor the Fourier transform of a function g ( t ) : G ( ω ) = (cid:82) + ∞−∞ e − iωt g ( t ) dt MNRAS , 1– ????
11 + ( f − f ) / Γ c , (16)representing a Lorentzian function with a half width at half max-imum power equal to Γ c . Thus, so long as a classical pulsator inthe presence of stochastic noise is observed for a long enough pe-riod of time, we may expect the power spectral density of a singlepulsation mode to be a resolved Lorentzian function whose widthdepends on the amplitude of the stochastic noise (tending to zero inthe limit when ξ → ) but not on the damping constant. here and throughout this work we have adopted the following definitionfor the Fourier transform of a function g ( t ) : G ( ω ) = (cid:82) + ∞−∞ e − iωt g ( t ) dt MNRAS , 1– ???? (2019) Cunha al.
The opposite limit is the one in which the driving is dominated bythe stochastic source and the amplitude of the coherent oscillations A → . In this limit the equation of motion describing the evolutionof x with time approaches that of a damped harmonic oscillatorwith noise as given by ¨ x + 2 η ˙ x + ω x = ξ . (17)The Fourier transform of Eq. (17) is given by − ω ˜ x − iηω ˜ x + ω ˜ x = ˜ ξ , (18)where a tilde indicates the Fourier transform of the correspondingfunction.The function ξ is defined by Eq. (2). Its Fourier transform isthen ˜ ξ ( ω ) = 2 A N ω √ ω ∆ t K (cid:88) k =0 r ( k ) (cid:90) ∞−∞ δ ( t − t k ) e − iωt dt = 2 A N ω √ ω ∆ t K (cid:88) k =0 r ( k ) e − iωt k . (19)Hence, the limit power spectral density of ξ is given approximatelyby PSD ξ ( ω ) = (cid:104)| ˜ ξ | (cid:105) T = 4 A N ω , (20)where we have taken into account that K (cid:88) k =0 r ( k ) ∼ K = T ∆ t , (21)for K (cid:29) .The limit power spectral density of the signal, x , expressed interms of the angular frequency ω , can then be derived from Eqs (18)and (20), being given by PSD( ω ) = (cid:104)| ˜ x | (cid:105) T = 1 T (cid:104)| ˜ ξ | (cid:105) ( ω − ω ) + 4 η ω ≈ A N ω ( ω − ω ) + η (22)where we have taken into account that ( ω − ω ) ∼ ω ( ω − ω ) for ω ∼ ω , up to first order in ω − ω .Taking into account that ω = 2 πf , and binning the poweron negative frequencies onto their positive counterparts (as donein Sec. 3.1), one finally obtains the limit power spectral densityexpressed in terms of the cyclic frequency, PSD( f ) = A Γ c π Γ
11 + ( f − f ) / Γ s2 , (23)where Γ s = η/ (2 π ) , Γ c = (cid:18) A N A (cid:19) ω π . (24)In writing Eq. (23) we have assumed that A is small but non-zero.If A is exactly zero then the prefactor in Eq. (23) should be writtenas A N ω / (2 π Γ s ) . In the stochastic limit we therefore recover thewell known result that the limit power spectral density of a singlepulsation mode is described by a Lorentzian function whose widthdepends only on the damping constant and not on the amplitude ofthe stochastic noise. According to the model described by Eq. (1), the limit power spec-tral density for a single pulsation mode is well described by aLorentzian function in both the classical (Sec. 3.1) and stochastic(Sec. 3.2) limits. In the intermediate regime, where both the coher-ent and stochastic driving play a role, the assumptions underlyingthe analytical derivations break down and the problem must be ad-dressed numerically. Nevertheless, it would be of interest to have anapproximate analytical solution that could be used to estimate themain underlying physical parameters of the model, η and A N /A ,in the intermediate regime, in addition to the mode frequency.To achieve that, let us start by defining Γ = (cid:0) Γ − + Γ − (cid:1) − , (25)so that Γ = Γ c if Γ c (cid:28) Γ s (classical limit) and Γ = Γ s if Γ s (cid:28) Γ c (stochastic limit) (note that Γ ≤ Γ c , Γ s ). With this definitionin hand, it is possible to write a single expression for the powerspectral density that is valid both in the classical and stochasticlimits, namely, PSD( f ) = A π Γ (cid:18) c Γ s (cid:19)
11 + ( f − f ) / Γ . (26)As we will show in Sec. 4, this expression constitutes also a goodapproximation to the limit power spectral density of a pulsationmode in the intermediate regime where Γ s / Γ c ∼ . Consequently,Eq. (26) provides a unifying description of the limit power spectradensity of individual modes in stellar pulsators where stochasticperturbations play a role. Equation (26) reproduces exactly the limit power spectral densitypredicted in the classical (Eq.(16)) and stochastic (Eq.(23)) lim-its. To verify that it also provides a good representation of thelimit power spectral density for cases when both the classical andstochastic drivings impact the oscillations, we carried out a numberof numerical simulations based on Eq. (1). Specifically, we numer-ically solved the equation of motion using a fourth order Runge-Kutta algorithm, setting the initial conditions to be those appropri-ate for the purely classical pulsations ( i.e., the solution for ξ = 0 ).In the simulations, time is measured in units of the pulsation period P and length in units of the amplitude A . The time interval betweensuccessive kicks was assumed to be ∆ t = 0 . P . The fact thatit is much shorter than the pulsation period ensures that both theassumed regularity of the velocity kicks and the specific value ofthe parameter ∆ t have a negligible impact on our results.Each realisation of the simulations spans a total time equal to × pulsation cycles. We consider two series of simulations,one with η = 0 . P − and the other with η = 0 . P − . Since ω ≈ πP − , the condition that η (cid:28) ω / (Sec. 3.1) is satis-fied in all cases. For each series, we consider different values of A N /A such that the ratio Γ c / Γ s changes from a value of 10000to a value of 0.01. The aim is to illustrate the changing in regime,from a stochastic dominated case (where Γ c / Γ s > ) to a classicaldominated case (where Γ c / Γ s < ).The results of our simulations are displayed in Fig. 1, wherethe series with η = 0 . P − and η = 0 . P − are shown on theleft and right columns, respectively. For each simulation we showone single realisation (light grey line) and the average of one hun-dred realisations (pink line). In addition, we show the predicted MNRAS , 1– ?? (2019) tochastic Signatures II limit power spectral density, given by Eq. (26) (green line), and theLorentzian function derived from a fit to the average power spectraldensity (black-dashed line). All results are shown in the form of apower spectral density divided by the square of the underlying co-herent pulsation amplitude ( i.e. PSD/ A ). Besides the dependenceon f , the quantity PSD/ A (hereafter the amplitude-normalisedpower spectral density) depends only on the parameters Γ c and Γ s ,whose impact on the simulations we want to explore.As expected, when the driving is strongly dominated by eitherthe stochastic or the coherent source (upper and lower panels inFig. 1, respectively), Eq. (26) provides an excellent description ofthe average power spectral density obtained from our simulations.In the intermediate cases (four middle panels in Fig. 1), Eq. (26)predicts an amplitude that is somewhat smaller than that found inour simulations, but a half width at half maximum power alwaysvery close to that of the simulations. Comparing the parameterscharacterising the Lorentzian function derived from the fit to theaverage power spectral density and the theoretical Lorentzian givenby Eq. (26), we find a maximum relative difference in the amplitudeof 22 % (for the cases with Γ c / Γ s = 1 ). Moreover, for a fixed valueof the ratio Γ c / Γ s , we find that this difference is the same (to betterthan one per cent) for the two values of η considered. For the halfwidth at half maximum power, the agreement is always better than5 % . The dependence of the width of the Lorentzian describing thepower spectral density on the model parameters is also clear fromFig. 1. Comparison of the two top panels confirms the well knownresult that in the stochastically dominated case the width is propor-tional to η (note the factor of 10 difference in the range of the x-axisbetween the two panels). However, when comparing the three toppanels on either column, we can see that the width of the Lorentziandecreases (at fixed η ) as Γ c / Γ s decreases (i.e, as the impact of thecoherent driving becomes non-negligible). Finally, when the co-herent driving dominates (bottom panels), the width becomes inde-pendent of η (see figure 5 of Paper I, and Sec. 3.1 above). Here, thefactor of 10 change in width seen between the left and right bottompanels (note the scale on the x-axis) is due to the change of A N /A and the fact that at fixed pulsation frequency Γ c is proportional to ( A N /A ) . Finally, we note that the factor of 10 change in the am-plitude between the left and right panels is a natural consequence ofour choice of changing η by a factor of 10 at fixed Γ c / Γ s . Increas-ing η by one order of magnitude increases Γ s by the same factor,hence, at fixed Γ c / Γ s , also Γ c . This, in turn, increases Γ by an or-der of magnitude ( cf. Eq. (25)) reducing the amplitude by a factorof 10 ( cf.
Eq. (26)).
In this section we shall verify that the analytical expression for thepower spectral density of the signal derived in Sec. 3 (Eq. (26))represents well the observations of long-period variables. To thatend, we will consider two stars: the Mira variable U Per and thesemiregular variable L2 Pup. As we shall see, these stars providegood examples of the classical limit (Sec. 3.1) and of the interme-diate regime (Sec. 3.3), respectively. The stochastic limit (Sec. 3.2)has been extensively discussed in the literature, in connection to thestudy of solar-like pulsators (e.g. Houdek et al. 2019; Belkacemet al. 2019, and references therein), thus it shall not be of furtherconcern here.
We downloaded visual magnitudes for U Per and L2 Pup from theAmerican Association of Variable Star Observers (AAVSO) database . The corresponding light curves are shown on the top panelsof Figs 2 and 3, respectively. To identify the pulsation periodicity,we performed a Fourier analyses of the data using Period04 (Lenz& Breger 2005). The power spectra of the light curves are shownin the bottom panel of Fig. 2 and middle panel of Fig. 3, respec-tively. The pulsation frequency of U Per is clearly visible in thepower spectrum, along with side lobes at at ± − c/d. The powerspectrum of the residuals is shown in the bottom panel of Fig. 3,where the power is now dominated by the pulsation frequency andits ± The data considered for U Per extends for over 107 years (from JD2419771.74 to JD 2458909.522), covering nearly 123 pulsation cy-cles. A clear pulsation amplitude variability is observed in the star’slight curve shown on the top panel of Fig. 2. Here we propose thatthis variability is due to stochastic perturbations that can be well de-scribed by the phenomenological model introduced in Paper I andfurther discussed in this paper.In order to test the above proposition, we first derive from thedata the rms of the relative amplitude variation, σ ∆ A/A , and therms of the phase variation, σ ∆ ϕ after the total observation time t obs ≈ yr. To compute the amplitude and phase time varia-tions we subdivided the light curve in sections of roughly one pul-sation period ( ≈ d) and performed sine-wave least-square fitsto each subsection using the “Amplitude/Phase variations” func-tionality of Period04. The retrieved amplitude, normalised by theaverage value, (cid:104) A (cid:105) = 1 . mag, is shown on the top left panel ofFig. 4 and the corresponding phase measurements in the top rightpanel of the same figure.Since the phase variation follows a random walk, σ ∆ ϕ can beestimated from σ ∆ ϕ ∼ √ N × (cid:112) (cid:104) (∆ ϕ i ) (cid:105) = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) i =1 (∆ ϕ i ) , (27)where ∆ ϕ i is the phase difference computed between two consec-utive measurements, i and i − , N is the total number of phasedifference measurements, and the brackets (cid:104)(cid:105) indicate an averageover all values of ∆ ϕ . Unlike the phase, the amplitude loses mem-ory of past perturbations after a time ∼ η − , varying always closeto the mean. As a result, σ ∆ A/A can be estimated from σ ∆ A/A ∼ (cid:113) σ A i /A = (cid:113) σ A i / (cid:104) A (cid:105) , (28)where ∆ A i is the difference between the i th measurement of theamplitude and the amplitude average (cid:104) A (cid:105) .From the amplitude and phase shown in Fig. 4 and Eqs (27)-(28), we derive σ ∆ A/A = 0 . and σ ∆ ϕ = 2 . for U Per. To verify , 1– ????
We downloaded visual magnitudes for U Per and L2 Pup from theAmerican Association of Variable Star Observers (AAVSO) database . The corresponding light curves are shown on the top panelsof Figs 2 and 3, respectively. To identify the pulsation periodicity,we performed a Fourier analyses of the data using Period04 (Lenz& Breger 2005). The power spectra of the light curves are shownin the bottom panel of Fig. 2 and middle panel of Fig. 3, respec-tively. The pulsation frequency of U Per is clearly visible in thepower spectrum, along with side lobes at at ± − c/d. The powerspectrum of the residuals is shown in the bottom panel of Fig. 3,where the power is now dominated by the pulsation frequency andits ± The data considered for U Per extends for over 107 years (from JD2419771.74 to JD 2458909.522), covering nearly 123 pulsation cy-cles. A clear pulsation amplitude variability is observed in the star’slight curve shown on the top panel of Fig. 2. Here we propose thatthis variability is due to stochastic perturbations that can be well de-scribed by the phenomenological model introduced in Paper I andfurther discussed in this paper.In order to test the above proposition, we first derive from thedata the rms of the relative amplitude variation, σ ∆ A/A , and therms of the phase variation, σ ∆ ϕ after the total observation time t obs ≈ yr. To compute the amplitude and phase time varia-tions we subdivided the light curve in sections of roughly one pul-sation period ( ≈ d) and performed sine-wave least-square fitsto each subsection using the “Amplitude/Phase variations” func-tionality of Period04. The retrieved amplitude, normalised by theaverage value, (cid:104) A (cid:105) = 1 . mag, is shown on the top left panel ofFig. 4 and the corresponding phase measurements in the top rightpanel of the same figure.Since the phase variation follows a random walk, σ ∆ ϕ can beestimated from σ ∆ ϕ ∼ √ N × (cid:112) (cid:104) (∆ ϕ i ) (cid:105) = (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) i =1 (∆ ϕ i ) , (27)where ∆ ϕ i is the phase difference computed between two consec-utive measurements, i and i − , N is the total number of phasedifference measurements, and the brackets (cid:104)(cid:105) indicate an averageover all values of ∆ ϕ . Unlike the phase, the amplitude loses mem-ory of past perturbations after a time ∼ η − , varying always closeto the mean. As a result, σ ∆ A/A can be estimated from σ ∆ A/A ∼ (cid:113) σ A i /A = (cid:113) σ A i / (cid:104) A (cid:105) , (28)where ∆ A i is the difference between the i th measurement of theamplitude and the amplitude average (cid:104) A (cid:105) .From the amplitude and phase shown in Fig. 4 and Eqs (27)-(28), we derive σ ∆ A/A = 0 . and σ ∆ ϕ = 2 . for U Per. To verify , 1– ???? (2019) Cunha al.
Figure 1.
Amplitude-normalised power spectral density for numerical simulations based on the model described by Eq. (1). Panels in the left and rightcolumns display results for η = 0 . P − and η = 0 . P − , respectively. The value of A N /A adopted in each simulation is shown in the inset of thecorresponding panel, along with the value of η in units of P − . The ratio Γ c / Γ s derived from these parameters through Eq. (24) is also shown. Light greyline: one realisation of the simulations;
Pink line: the average of 100 realisations;
Dashed-black line: the Lorentzian function derived from fitting the averageshown in pink;
Green line: the Lorentzian function given by Eq. (26) computed with the parameters of the corresponding simulation. The transition froma stochastically-dominated model (top panels) gradually to a classically-dominated model (bottom panels), is evident from the change in the width of theLorentzian function at fixed η . the sensitivity of these results to the size of the light-curve sectionsconsidered in the fits, we repeat the calculation dividing the lightcurve in sections of 2, 3, and 4 pulsation periods, at a time. More-over, in the case of fits to light-curve sections extending for onepulsation period, we also determine σ ∆ A/A and σ ∆ ϕ from phaseand amplitude variations computed from non-consecutive measure-ments (specifically, taking values distanced by 2, 3, and 4 measure-ments). Changing the extent of the light-curve sections or the dis-tance between amplitude and phase measurements within the limits above results in values of σ ∆ A/A varying between 0.14 and 0.18and of σ ∆ ϕ varying between 1.9 and 2.3.According to our simulations, which we shall detail below,the values of σ ∆ A/A above imply that the star is near the classicallimit discussed in Sec. 3.1. Thus, the half width at half maximum ofthe Lorentzian is ∼ Γ c , which can be estimated from σ ∆ ϕ throughEq. (6). Doing so, we find Γ c = 8 . × − d − for the phase vari-ability shown in Fig. 4. Given that the resolution of the power spec-tral density is / (2458909 . − . ≈ . × − d − , MNRAS , 1– ?? (2019) tochastic Signatures II Figure 2.
Top: U Per light curve extracted from AAVSO. The data points areprovided rounded to the decimal point, which explains the discrete appear-ance of the light curve. Data runs from JD 2419771.74 to JD 2458909.522in a total of 26525 data points collected over 107 years. Bottom: Powerspectrum of the light curve showing the star’s pulsation frequency near0.0031 c/d along with side lobes at ± cf. windowfunction shown in the inset). Figure 3.
Top: L2 Pup light curve. Data runs from JD 2411839.4 to JD2458870.074 in a total of 21448 data points collected over nearly 129 years.The dotted line shows the observation period analysed by Bedding et al.(2005), running from JD 2425249 to JD 2453487, and considered for com-parison in Sec. 5.3. Middle: Power spectrum of the light curve dominated bylow frequency power associated to the long-period modulation of the lightcurve and the correspondent 1 cycle per year side lobe ( cf. window functionshown in the inset). Bottom: Power spectrum after pre-whitening by the lowfrequencies, showing the star’s pulsation frequency near 0.007 c/d and the ± the Lorentzian is not expected to be fully resolved for U Per. Infact, if we assume that the transition from an unresolved to a re-solved Lorentzian occurs when the total observing time reaches T res = 1 /π Γ (e.g. Basu & Chaplin 2017), our estimate of theLorentzian’s half width implies T res = 35526 d. For the Lorentzianto be fully resolved, the observing time would need to be about oneorder of magnitude longer than T res , but in the case of U Per theobservations run for 39138 d, so for a time only about longerthan the estimated T res . Unfortunately, this seems to be also thecase for the other Mira stars with long-term observations available in the AAVSO database. Thus, it seems that for the pulsation pe-riods typical of Mira stars and the level of stochasticity observed,the many decades-long observations available are not enough tofully resolve the Lorentzian in the power spectral density, despitethe stochastic impact being observed in the pulsation amplitude andphase of these stars.While the Lorentzian profile of U Per is not properly re-solved, there is no limitation to the length of simulated obser-vations. Hence, we can simulate a star with levels of amplitudeand phase variability similar to those inferred for U Per and ver-ify if the power spectral density from those simulations is consis-tent with the one derived from the light curve of U Per, once thesimulated light curve is limited to the true length of the observa-tions. To that end, we simulate 5000 pulsation cycles of a star withthe pulsation frequency observed in U Per, A N /A = 0 . , and η = 0 . P − . With this input the theoretical rms of the relativeamplitude and phase variation of the simulation are σ s ∆ A/A = 0 . and σ s ∆ ϕ = 2 . , respectively. We then cut the simulation is sec-tions of 123 pulsation cycles, each corresponding to one realisationwith the length of the observations, and interpolate each section onthe observation times, to guarantee that the effect of uneven timesampling and gaps is incorporated into the simulations. The ampli-tude and phase variability differ from realisation to realisation, asexpected for a stochastic process. We show an example of that vari-ability for one realisation that resembles the observations of U Perin the lower panels of Fig. 4. The values of σ ∆ A/A and σ ∆ ϕ forthis realisation computed in exactly the same manner as done forthe observations are σ ∆ A/A = 0 . and σ s ∆ ϕ = 2 . .The amplitude-normalised power spectral density for the fullsimulation (PSD/ A ) is shown in the upper panel of Fig. 5, alongwith the Lorentzian function computed with the parameters ofthe simulation. If U Per had been observed long enough for theLorentzian to be fully resolved, one would expect its PSD to looksomewhat similar to what is shown in this simulation. Next we in-spect how the amplitude-normalised PSD derived from the observa-tions of U Per compares with those obtained from the 123 pulsationcycle cuts from the simulation. The results are shown in the lowerpanel of Fig. 5. The thick black continuous line displays the ob-servational results, while the other thick lines, of various line stylesand colours, show the results obtained for 6 different realisations ofthe simulations (all with a length of 123 pulsation cycles). The dif-ferent realisations show that the PSD of the simulated data can takea variety of shapes when cuts of a length of 123 pulsation cyclesare considered. This is not surprising given the stochastic nature ofthe process and the fact that the resolution in frequency limits thesampling of the true power density to a few points. It is also evi-dent that the observed PSD is consistent with what the simulationspredict. Unfortunately, without a fully resolved Lorentzian profile,the PSD is of little use to estimate the parameters associated withthe stochastic impact on the pulsations. It thus appears that for Mirastars the best procedure to infer the impact of the stochasticity onthe pulsations consists in computing the rms of the relative am-plitude and phase variations from the light curve and using themto constrain the parameters A N /A and η from Eqs (3) and (4). Inaddition, a direct comparison with model simulations is a fruitfulapproach to test the robustness of the method used to infer the rmsof the relative pulsation amplitude variation and of the phase varia-tion. MNRAS , 1– ????
Top: L2 Pup light curve. Data runs from JD 2411839.4 to JD2458870.074 in a total of 21448 data points collected over nearly 129 years.The dotted line shows the observation period analysed by Bedding et al.(2005), running from JD 2425249 to JD 2453487, and considered for com-parison in Sec. 5.3. Middle: Power spectrum of the light curve dominated bylow frequency power associated to the long-period modulation of the lightcurve and the correspondent 1 cycle per year side lobe ( cf. window functionshown in the inset). Bottom: Power spectrum after pre-whitening by the lowfrequencies, showing the star’s pulsation frequency near 0.007 c/d and the ± the Lorentzian is not expected to be fully resolved for U Per. Infact, if we assume that the transition from an unresolved to a re-solved Lorentzian occurs when the total observing time reaches T res = 1 /π Γ (e.g. Basu & Chaplin 2017), our estimate of theLorentzian’s half width implies T res = 35526 d. For the Lorentzianto be fully resolved, the observing time would need to be about oneorder of magnitude longer than T res , but in the case of U Per theobservations run for 39138 d, so for a time only about longerthan the estimated T res . Unfortunately, this seems to be also thecase for the other Mira stars with long-term observations available in the AAVSO database. Thus, it seems that for the pulsation pe-riods typical of Mira stars and the level of stochasticity observed,the many decades-long observations available are not enough tofully resolve the Lorentzian in the power spectral density, despitethe stochastic impact being observed in the pulsation amplitude andphase of these stars.While the Lorentzian profile of U Per is not properly re-solved, there is no limitation to the length of simulated obser-vations. Hence, we can simulate a star with levels of amplitudeand phase variability similar to those inferred for U Per and ver-ify if the power spectral density from those simulations is consis-tent with the one derived from the light curve of U Per, once thesimulated light curve is limited to the true length of the observa-tions. To that end, we simulate 5000 pulsation cycles of a star withthe pulsation frequency observed in U Per, A N /A = 0 . , and η = 0 . P − . With this input the theoretical rms of the relativeamplitude and phase variation of the simulation are σ s ∆ A/A = 0 . and σ s ∆ ϕ = 2 . , respectively. We then cut the simulation is sec-tions of 123 pulsation cycles, each corresponding to one realisationwith the length of the observations, and interpolate each section onthe observation times, to guarantee that the effect of uneven timesampling and gaps is incorporated into the simulations. The ampli-tude and phase variability differ from realisation to realisation, asexpected for a stochastic process. We show an example of that vari-ability for one realisation that resembles the observations of U Perin the lower panels of Fig. 4. The values of σ ∆ A/A and σ ∆ ϕ forthis realisation computed in exactly the same manner as done forthe observations are σ ∆ A/A = 0 . and σ s ∆ ϕ = 2 . .The amplitude-normalised power spectral density for the fullsimulation (PSD/ A ) is shown in the upper panel of Fig. 5, alongwith the Lorentzian function computed with the parameters ofthe simulation. If U Per had been observed long enough for theLorentzian to be fully resolved, one would expect its PSD to looksomewhat similar to what is shown in this simulation. Next we in-spect how the amplitude-normalised PSD derived from the observa-tions of U Per compares with those obtained from the 123 pulsationcycle cuts from the simulation. The results are shown in the lowerpanel of Fig. 5. The thick black continuous line displays the ob-servational results, while the other thick lines, of various line stylesand colours, show the results obtained for 6 different realisations ofthe simulations (all with a length of 123 pulsation cycles). The dif-ferent realisations show that the PSD of the simulated data can takea variety of shapes when cuts of a length of 123 pulsation cyclesare considered. This is not surprising given the stochastic nature ofthe process and the fact that the resolution in frequency limits thesampling of the true power density to a few points. It is also evi-dent that the observed PSD is consistent with what the simulationspredict. Unfortunately, without a fully resolved Lorentzian profile,the PSD is of little use to estimate the parameters associated withthe stochastic impact on the pulsations. It thus appears that for Mirastars the best procedure to infer the impact of the stochasticity onthe pulsations consists in computing the rms of the relative am-plitude and phase variations from the light curve and using themto constrain the parameters A N /A and η from Eqs (3) and (4). Inaddition, a direct comparison with model simulations is a fruitfulapproach to test the robustness of the method used to infer the rmsof the relative pulsation amplitude variation and of the phase varia-tion. MNRAS , 1– ???? (2019) Cunha al.
Figure 4.
Top: amplitude (left) and phase (right) for U Per as a functionof time measured in units of the pulsation period, P = 1 /f , where f = 0 . c/d was taken to be the frequency at which the poweris maximum. The amplitude has been normalised by the average amplitudein the period considered. Bottom: the same as the top panels but for onerealisation of the simulation performed for U Per (see text for details). Figure 5.
Amplitude-normalised power spectral density (PSD/ A ) in unitsof the pulsation period. The lower horizontal axis shows the frequency inunits of the inverse of the pulsation period and the top horizontal axis in − cycles per day. Top panel: PSD/ A for the full simulation of 5000pulsation cycles (grey) and the Lorentzian described by Eq. (16) with theparameters adopted for the simulation (pink): f = 0 . c/d (=1 P − ), Γ c = 1 . × − d − (=0.036 P − ). Lower panel: comparison betweenthe observed PSD/ A (black, continuous, thick line) and the PSD/ A forsix different realisations of the simulation, all 123 pulsation cycles long(thick coloured lines, different line styles). In the case of the observed data,the time-averaged amplitude was used as an estimate of A in the normali-sation. The observed power spectral density is hardly distinguishable fromthe simulation representative set shown. The data considered for L2 Pup extends for nearly 129 years (fromJD 2411839.4 to JD 2458870.074), covering about 342 pulsationcycles. Significant dimming episodes, presumably due to extinc-tion by circumstellar dust (Bedding et al. 2002), can be seen in thelight curve. The long time scales associated with this phenomenon,which is external to the star, allow us to easily separate its signaturefrom that of the pulsations in the power spectrum. The amplitudeand phase derived from the data, following the same procedure asfor U Per are shown in Fig. 6, where, as before, the amplitude hasbeen normalised by its average value, (cid:104) A (cid:105) = 0 . mag. For the caseshown in the figure we have subdivided and fitted light curve sec-tions of roughly 3 pulsation cycles ( ≈
417 d). We did not considerfits to light curve sections of one pulsation cycle, as we did forU Per, because gaps of length comparable to half a pulsation cycleare common in the data of L2 Pup. The rms variance of the relativeamplitude variation computed from Eq. (28) for the case shown inFig. 6 is σ ∆ A/A = 0 . . From this value we may expect the star tobe a good example of the intermediate regime, where both Γ c and Γ s should contribute non-negligibly to the Lorentzian half width athalf maximum power, Γ .The amplitude-normalised power spectral density for L2 Pupis shown in Fig. 7, along with the Lorentzian function that bestfits the data. The fit was performed assuming that the power ata given frequency is described by a χ distribution with two de-grees of freedom as discussed byAnderson et al. (1990), where weminimised the expression given in their equation (11) consideringthe model to be a Lorentzian function plus a constant background.From the fit we find Γ = 7 . × − d − ( = 0 . P − ). This valueis comparable, but slightly smaller than the value of . × − d − reported by Bedding et al. (2005). Repeating the fit to the powerspectral density derived from the observation period considered intheir work we find Γ = 6 . × − d − , so the difference is notexplained by the extent of the data set considered. The explanationfor the difference found may stand, instead, on the fact that the dataused in Bedding et al. (2005) was from a different source.In the case of L2 Pup, the width of the Lorentzian is ∼ /t obs , or t obs ∼ T res . Thus, unlike the case of U Per, theLorentzian is resolved and we can apply Eqs (25)-(26), using (cid:104) A (cid:105) as an estimate of A , to estimate Γ c = 9 . × − d − and Γ s = 3 × − d − . While these values set the star closer to theclassical limit than to the stochastic limit, Γ c is not much smallerthan Γ s and, thus, the terms involving Γ s cannot be neglected inEq. (26). From these values, we can further estimate the model pa-rameters A N /A = 0 . and η = 0 . d − , from which we es-timate a mode lifetime, τ = η − , of 1.4 years. This is in contrastwith the value that would be derived if the star was assumed to bein the stochastic limit. In that case, Γ s would have been identicalto Γ and the mode lifetime would have been estimated to be about6 years, instead of the 1.4 years inferred from our model. This im-plies that the mode lifetimes and quality factors (proportional to theratio of the mode lifetime to the pulsation period) derived assumingthat semiregular variables are in the stochastic limit (e.g. Beddinget al. 2005; Stello et al. 2006) are likely overestimated and need tobe revised. The model presented in this work, along with the results from theanalysis of L2 Pup, can be used to interpret observational trends
MNRAS , 1– ?? (2019) tochastic Signatures II Figure 6.
Amplitude (left) and phase (right) for L2 Pup as a function oftime measured in units of the pulsation period, P = 1 /f , where f =0 . c/d. The amplitude has been normalised by the average amplitudein the time interval considered. Figure 7.
Amplitude-normalised power spectral density (PSD/ A ; black)in units of the pulsation period obtained from the full light curve of L2 Pup(about 342 pulsation cycles). The time-averaged amplitude was used as anestimate of A in the normalisation. The lower horizontal axis shows the fre-quency in units of the inverse of the pulsation period and the top horizontalaxis in − cycles per day. Also shown is the Lorentzian function (pink)derived from the fit to the amplitude-normalised power spectral density inthe frequency interval shown ([0.8,1.2] P − ). reported in the literature concerning both pulsation amplitudes andmode line widths. In this section we discuss two such trends inthe light of our model, namely, the observed pulsation amplitude-period relation and the mode line width-temperature scaling. An interesting question to address is where to seek evidence for atransition between the predominantly stochastic to predominantlycoherent regimes. The transition occurs where Γ c / Γ s = 1 , but de-termining this ratio from stellar data requires an estimate of theunderlying coherent amplitude, which may be difficult for stars ap-proaching the transition, as the amplitude variations become verylarge and stars tend to become multi-mode pulsators.A possible alternative to search for the transition between thepredominantly stochastic and the predominantly coherent regimesis to consider the behaviour of the pulsation amplitudes mea-sured from the power spectrum. Inspecting the amplitude of theLorentzian function in Eq. (26) ( A / π Γ × (1 + 2Γ c / Γ s ) ), wesee that as the transition is approached from the stochastic side,two additional terms are added to what would be the amplitudein the stochastic limit. The second of this terms, proportional to A /A N , will dominate after the transition, as the classical limit isapproached, but at some point before the transition ( i.e. at shorter periods) a change in the behaviour of the amplitude is expected, asthe impact of the coherent driving starts to be noticeable. Indeed,several studies have noted a change in behaviour in the pulsationamplitude-period relation (B´anyai et al. 2013; Mosser et al. 2013;Yu et al. 2020) at a pulsation period ranging from about 5 to 10 d.We conjecture that the change in behaviour reported in these worksresults from the increasing impact of the coherent driving on theoscillations. We can thus use these observations to estimate the pul-sation period at which the transition between the two regimes takesplace.According to our model, at Γ c / Γ s = 1 the amplitude of theLorentzian is predicted to be three times larger than what would beexpected in the stochastic limit ( cf. Eqs (23) and (26)). However,as we have seen in Sec. 4, the Lorentzian amplitude in Eq. (26)is somewhat underestimated, so a better estimate of this amplituderatio can be derived from numerical simulations, from which wefind a value close to 4. To estimate the pulsation period at which Γ c / Γ s = 1 we consider the pulsation amplitude-period relationpublished by Yu et al. (2020) (their figure 2(a)). Notice that theamplitudes reported in that work were not derived from fitting aLorentzian function to the power spectral density, as in our model.Nevertheless, we are only interested in amplitude ratios, so we canuse these results as a proxy to our test. The authors fitted a piece-wise linear model to the data and found a change in the slope of thepulsation amplitude-period relation at around 4.5 d. We extrapo-late the linear relation they found on the short-period side to derivethe pulsation amplitude that would be expected at larger pulsationperiods if the pulsations were driven only by the stochastic noise.Doing so, we find that the ratio between the measured and (extrap-olated) stochastic amplitudes is 4 for a pulsation period of about60 d. Thus, under the assumption that the change in the slope of thepulsation amplitude-period relation reported earlier by several au-thors results from the additional coherent driving, we conclude thatthe latter starts to have an impact on the amplitudes at around a pe-riod of 5 to 10 d, while the driving is still stochastically dominated,the actual transition between the two driving regimes ( Γ c / Γ s = 1 in our model) taking place at a pulsation period of about 60 d. The mode line widths in stochastically-driven pulsators ( = 2Γ s inthis work) have been shown to scale with effective temperature (e.g.Chaplin et al. 2009; Baudin et al. 2011; Appourchaux et al. 2012;Corsaro et al. 2012; Lund et al. 2017). The inference of Γ s for asample of long period variables, based on the model proposed inthis work, will enable this scaling relation to be tested up to theluminous asymptotic giant branch stars, thus, down to tempera-tures more than 1000 K lower than considered in previous studies.Since we have derived Γ s for one star only, we defer to future worka thorough test of the scaling relation. Still, we can verify whereL2 Pup stands in comparison to scaling relations published in theliterature. In order to do so, we adopt the effective temperature of T eff = 3500 ± K published by Kervella et al. (2014, 2016) forL2 Pup, which the authors derived from the star’s near infrared H and K band magnitudes and the angular diameter determined frominterferometry. Figure 8 shows the position of L2 Pup (black star)on the s − T eff diagram, along with the position of the sun (or-ange circle). Two scaling relations published by Lund et al. (2017)are also shown, one based on a power law (pink continuous line)and the other on an exponential function (pink dashed line) fittedto Kepler data. To derive these fits the authors used line widths andeffective temperatures from the
Kepler
Legacy sample, consisting
MNRAS , 1– ?? (2019) Cunha al.
Figure 8.
Comparison between the stochastic component of the line width(2 Γ s ) estimated for L2 Pup (black star) and the fits to the line widths at fre-quency of maximum power derived from solar-like stars at different (earlier)evolutionary status. The two pink curves show the fits performed by Lundet al. (2017) of a power law function and an exponential function, respec-tively, to the line widths of stars spanning from the main sequence to thered-giant phase (occupying roughly the position in the diagram marked bythe shaded light-grey areas). The orange circle marks the location of sun(Houdek et al. 2019). The value of Γ s inferred for L2 Pup is consistent witha single exponential dependence of Γ s on effective temperature. of main-sequence and subgiant stars, and from Kepler red giantsanalysed by Corsaro et al. (2012) and Handberg et al. (2017). Theresult for L2 Pup does not support the power law fit with the param-eters derived based on the data considered by Lund et al. (2017), butconfirms that a single exponential function represents well the de-pendence of Γ s on the effective temperature, down to temperaturesas low as 3500 K. This work provides a unifying description of the limit power spec-tral density for pulsations in the presence of stochastic noise,from main-sequence solar-like pulsators to the luminous AGB Mirastars. The model, first proposed by Avelino et al. (2020) (Paper I),is based on an internally driven damped harmonic oscillator andpredicts that in all cases, the limit power spectral density of a pul-sation mode in the presence of stochastic noise is described by aLorentzian function. Generally, the half width at half maximumpower of the Lorentzian function, Γ , depends both on the damp-ing constant (thus, on mode lifetime) and on the amplitude of thestochastic noise. Nevertheless, as the stochastic limit is approached, Γ tends to Γ s = η/ (2 π ) , thus becoming independent of the ampli-tude of the stochastic noise, and as the classical limit is approached, Γ tends to Γ c = ( A N /A ) × ω / (2 π ) , thus, becoming independentof the mode lifetime.The model allows us to introduce a quantitative classifica-tion of pulsating stars in the presence of stochastic noise. When Γ c / Γ s > , the driving is predominantly stochastic and the starsare classified as solar-like pulsators, while when Γ c / Γ s < , thedriving is predominantly coherent and the stars are classified asclassical pulsators. In this context, when no coherent driving ispresent, the star is a pure solar-like pulsator and when there is no source of stochastic noise, the star is a pure classical pulsator. Wenote, however, that the latter case was not considered in this study,as we have always assumed the presence of stochastic noise.The model has been applied in the analysis of the observa-tions of the Mira star U Per and the semiregular star L2 Pup. Inthe case of the Mira star, a classical pulsator according to our clas-sification scheme, we have shown that the Lorentzian is not fullyresolved. Unfortunately, that seems to be the case also for otherMira stars. Nevertheless, the model parameters for U Per could stillbe estimated through inspection of the amplitude and phase vari-ability, and used to perform model numerical simulations. Com-parison of the simulation results with the power spectral density forU Per corroborates our model predictions. In the case of L2 Pup, theLorentzian function is resolved and we could infer both Γ c and Γ s from the inspection of the power spectral density and the amplitudevariability. We could then derive the mode lifetime and show thatit is significantly shorter than what would have been derived if wehad wrongly assumed that the width of the Lorentzian dependedonly on η , as in the case of solar-like pulsators. In fact, we haveconfirmed that in this star Γ c / Γ s < , thus the driving is predomi-nantly coherent. Hence, according to our classification, L2 Pup is aclassical pulsator.Our model also provides a natural explanation for the changein behaviour of the observed pulsation amplitude-period relationreported in several earlier works. Based on our model, we arguethat the change in the amplitude behaviour seen at periods around5 to 10 d is associated to the beginning of a non-negligible contribu-tion of the coherent driving which increases towards longer periodswith the actual change between the stochastically- and coherently-dominated regimes occurring around pulsation periods of 60 d.Moreover, with the results inferred for L2 Pup we are able to ex-tend the test to the scaling relation between the mode line widthand the effective temperature, confirming that a single exponentialrelation provides a good representation of the scaling relation downto temperatures about 1000 K cooler than previously considered inthe literature.Finally, we emphasise that our model can be considered in thestudy of any stellar pulsator holding a source of stochastic noise, in-dependently of how important stochasticity is to the driving of thepulsations. Moreover, given our findings, we note that stochasticnoise should be added to the list of possible sources for the pulsa-tion phase variability observed in classical pulsators across the HRdiagram, including those in the main-sequence and in later stagesof evolution. DATA AVAILABILITY
The data underlying this article will be shared on reasonable re-quest to the corresponding author.
ACKNOWLEDGEMENTS
We acknowledge with thanks the variable star observations fromthe AAVSO International Database contributed by observers world-wide and used in this research. P. P. A. thanks the support fromFCT – Fundac¸˜ao para a Ciˆencia e a Tecnologia – through theSabbatical Grant No. SFRH/BSAB/150322/2019. M. S. Cunhais supported by national funds through FCT in the form ofa work contract. This work was supported by FCT throughthe research grants UIDB/04434/2020, UIDP/04434/2020 and
MNRAS , 1– ?? (2019) tochastic Signatures II PTDC/FIS-AST/30389/2017, and by FEDER - Fundo Europeu deDesenvolvimento Regional through COMPETE2020 - ProgramaOperacional Competitividade e Internacionalizao (grant: POCI-01-0145-FEDER-030389). W. J. C. acknowledges support from theUK Science and Technology Facilities Council (STFC). Fundingfor the Stellar Astrophysics Centre is provided by The Danish Na-tional Research Foundation (grant agreement no. DNRF106).
REFERENCES
Anderson E. R., Duvall Thomas L. J., Jefferies S. M., 1990, ApJ, 364, 699Appourchaux T., et al., 2012, A&A, 537, A134Avelino P. P., Cunha M. S., Chaplin W. J., 2020, MNRAS, 492, 4477B´anyai E., et al., 2013, MNRAS, 436, 1576Basu S., Chaplin W. J., 2017, Asteroseismic Data Analysis: Foundationsand Techniques. ISBN: 9780691162928. Princeton University Press,2017Baudin F., et al., 2011, A&A, 529, A84Bedding T. R., Zijlstra A. A., Jones A., Marang F., Matsuura M., Retter A.,Whitelock P. A., Yamamura I., 2002, MNRAS, 337, 79Bedding T. R., Kiss L. L., Kjeldsen H., Brewer B. J., Dind Z. E., KawalerS. D., Zijlstra A. A., 2005, MNRAS, 361, 1375Belkacem K., Kupka F., Samadi R., Grimm-Strele H., 2019, A&A, 625,A20Catelan M., Smith H. A., 2015, Pulsating StarsChaplin W. J., Houdek G., Karoff C., Elsworth Y., New R., 2009, A&A,500, L21Christensen-Dalsgaard J., Kjeldsen H., Mattei J. A., 2001, ApJ, 562, L141Corsaro E., et al., 2012, ApJ, 757, 190Dziembowski W. A., Soszy´nski I., 2010, A&A, 524, A88Eddington A. S., Plakidis S., 1929, MNRAS, 90, 65Gough D. O., 1967, AJ, 72, 799Hajimiri A., Thomas H. L., 1999, The Design of Low Noise Oscilla-tors. ISBN: 978-0-7923-8455-7. Kluwer Academic Publishers 1999,Boston, MAHandberg R., Brogaard K., Miglio A., Bossini D., Elsworth Y., SlumstrupD., Davies G. R., Chaplin W. J., 2017, MNRAS, 472, 979Houdek G., Dupret M.-A., 2015, Living Reviews in Solar Physics, 12, 8Houdek G., Lund M. N., Trampedach R., Christensen-Dalsgaard J., Hand-berg R., Appourchaux T., 2019, MNRAS, 487, 595Kervella P., et al., 2014, A&A, 564, A88Kervella P., Homan W., Richards A. M. S., Decin L., McDonald I., Mon-targ`es M., Ohnaka K., 2016, A&A, 596, A92Lenz P., Breger M., 2005, Communications in Asteroseismology, 146, 53Lund M. N., et al., 2017, ApJ, 835, 172Moln´ar L., Joyce M., Kiss L. L., 2019, ApJ, 879, 62Mosser B., et al., 2013, A&A, 559, A137Munteanu A., Bono G., Jos´e J., Garc´ıa-Berro E., Stellingwerf R. F., 2005,ApJ, 627, 454Percy J. R., Colivas T., 1999, PASP, 111, 94Soszynski I., et al., 2007, Acta Astron., 57, 201Stello D., Kjeldsen H., Bedding T. R., Buzasi D., 2006, A&A, 448, 709Sterken C., 2005, The O-C Diagram: Basic Procedures. p. 3Templeton M. R., Mattei J. A., Willson L. A., 2005, AJ, 130, 776Wood P. R., Zarro D. M., 1981, ApJ, 247, 247Xiong D.-R., Deng L.-C., 2013, Research in Astronomy and Astrophysics,13, 1269Xiong D. R., Deng L., Zhang C., 2018, MNRAS, 480, 2698Yu J., Bedding T. R., Stello D., Huber D., Compton D. L., Gizon L., HekkerS., 2020, MNRAS, 493, 1388Zijlstra A. A., Bedding T. R., 2002, Journal of the American Association ofVariable Star Observers (JAAVSO), 31, 2MNRAS , 1– ????