Stellar granulation as seen in disk-integrated intensity. I. Simplified theoretical modeling
aa r X i v : . [ a s t r o - ph . S R ] S e p Astronomy&Astrophysicsmanuscript no. 20816 c (cid:13)
ESO 2018November 24, 2018
Stellar granulation as seen in disk-integrated intensity
I. Simplified theoretical modeling
R. Samadi , K. Belkacem , H.-G. Ludwig , LESIA, Observatoire de Paris, CNRS UMR 8109, UPMC, Universit´e Denis Diderot, 5 Place Jules Janssen, 92195 Meudon Cedex,France Zentrum f¨ur Astronomie der Universit¨at Heidelberg, Landessternwarte, Knigstuhl 12, D-69117 Heidelberg, Germany GEPI, Observatoire de Paris, CNRS UMR 8111, Universit´e Denis Diderot, 5 Place Jules Janssen, 92195 Meudon Cedex, FranceNovember 24, 2018
ABSTRACT
Context.
Solar granulation has been known for a long time to be a surface manifestation of convection. The space-borne missionsCoRoT and
Kepler enable us to observe the signature of this phenomena in disk-integrated intensity on a large number of stars.
Aims.
The space-based photometric measurements show that the global brightness fluctuations and the lifetime associated with gran-ulation obeys characteristic scaling relations. We thus aimed at providing simple theoretical modeling to reproduce these scalingrelations, and subsequently at inferring the physical properties of granulation across the HR diagram.
Methods.
We developed a simple 1D theoretical model. The input parameters were extracted from 3D hydrodynamical models ofthe surface layers of stars, and the free parameters involved in the model were calibrated with solar observations. Two di ff erent pre-scriptions for representing the Fourier transform of the time-correlation of the eddy velocity were compared: a Lorentzian and anexponential form. Finally, we compared our theoretical prediction with 3D radiative hydrodynamical (RHD) numerical modeling ofstellar granulation (hereafter ab initio approach). Results.
Provided that the free parameters are appropriately adjusted, our theoretical model reproduces the observed solar granulationspectrum quite satisfactorily ; the best agreement is obtained for an exponential form. Furthermore, our model results in granulationspectra that agree well with the ab initio approach using two 3D RHD models that are representative of the surface layers of anF-dwarf and a red-giant star.
Conclusions.
We have developed a theoretical model that satisfactory reproduces the solar granulation spectrum and gives resultsconsistent with the ab initio approach. The model is used in a companion paper as theoretical framework for interpretating the ob-served scaling relations.
Key words.
Convection – Turbulence – Stars: granulation – Sun: granulation
1. Introduction
The solar surface reveals irregular cellular patterns commonlycalled granules. These structures were first discovered and dis-cussed during the 19th century (see an historical review inBray & Loughhead 1967). Their origin was first attributed byUns¨old (1930) to convective currents occuring beneath the vis-ible photospheric layers, while their turbulent nature was firsthighlighted by Siedentopf (1933). Several decades later, with theadvances in numerical hydrodynamical simulations, the proper-ties of solar granulation are well explained (e.g. Muller 1999,and references therein).Stars with e ff ective temperature lower than about 7 000 Khave an appreciable upper convective envelope and are thusexpected to show – as in the Sun – granules on their sur-face. Because the granules evolve with time, their evolutionproduces small brightness fluctuations that can now be accu-rately monitored and measured with space-based high-precisionphotometry measurements performed with the MOST, CoRoTand Kepler missions (Matthews et al. 2004; Michel et al. 2008;Kallinger & Matthews 2010; Mathur et al. 2011; Chaplin et al.2011b). These observations reveal that the characteristic time
Send o ff print requests to : R. Samadi Correspondence to : [email protected] τ e ff of the granules and the total brightness fluctuations σ as-sociated with them scales as a function of the frequency ν max at which the solar-like oscillations peak (e.g., Huber et al. 2009;Kallinger & Matthews 2010; Chaplin et al. 2011b; Mathur et al.2011; Kjeldsen & Bedding 2011). In turn, ν max is shown to scalewith the cut-o ff frequency ν c of the atmosphere, therefore mainlythe pressure scale-height near the photosphere (Brown et al.1991; Kjeldsen & Bedding 1995; Stello et al. 2009; Huber et al.2009; Mosser et al. 2010; Belkacem et al. 2011). The observedrelation between the properties of the stellar granulation and ν c were explained on the basis of some simplified physicalconsiderations (Huber et al. 2009; Kjeldsen & Bedding 2011;Mathur et al. 2011). A detailed theoretical study of the observedscaling relations is, however, lacking.A possible theoretical approach is the one proposed byLudwig (2006). This ab initio approach consists of model-ing the stellar granulation as seen in disk-integrated intensityfrom the intensity emerging directly from given 3D RHD mod-els. This numerical approach was applied by several authors(Svensson & Ludwig 2005; Ludwig et al. 2009; Mathur et al.2011). It is very time-consuming and hence does not easily al-low envisaging a large set of calculations with di ff erent surfacemetal abundance, for example. Furthermore, interpreting the re-sults is not trivial, and the systematic di ff erences obtained by Mathur et al. (2011) between observed and modeled spectra ofred giants are not well understood. On the other hand, a theoret-ical model based on a more simplified physical approach o ff ersthe advantage of separately testing several properties of turbulentconvection. In addition, it allows one to compute the granulationspectrum for a variety of stars on a large scale. Furthermore, itis possible to derive scaling relations from such a simplied the-oretical model that could provide additional theoretical supportfor the observed scaling relations and extend the current theo-retical scaling relations. We here present such a simple theoret-ical model of the stellar granulation as seen in disk-integratedintensity. In the companion paper (Samadi et al. 2013, hereafterpaper II), we derive theoretical scaling relations for σ and τ e ff from this model. Comparisons with a large sample of observedstars as well as with previously published scaling relations arereported in paper II.This paper I is organized as follows: In Sect. 2, we out-line our theoretical model for the stellar granulation spectrum indisk-integrated intensity and the di ff erent prescriptions adoptedfor modeling the properties of turbulent granules. The free pa-rameters introduced in the theoretical model are next calibratedin Sect. 3 such that our theoretical calculations match at best thesolar observations. In Sect. 4, we compare our calculations withthose obtained with ab initio of Ludwig (2006) approach usingtwo 3D RHD models of the surface layers of an F-dwarf star anda red giant star. Finally, Sect. 5 is devoted to the conclusion.
2. Modeling of the power density spectrum
We here outline our theoretical model for the power spectrumassociated with the relative variations of the bolometric fluxemerging from the star in the direction of an observer, whowould measure it continuously during a given duration (typi-cally much longer than the timescale of the granulation). The de-tailed derivation of the model is presented in Appendix A, whilethe underlying approximations and assumptions are discussed inAppendix B. F ( t ) is the bolometric flux toward the observer at the instant t ,and we define ∆ F ( t ) = F ( t ) − h F i t as the instantaneous variationof the flux with respect to its time average, h F i t . From now on hi t stands for a time average. The power density spectrum (PDS)associated with the relative variation of the flux ( ∆ F ( t ) / h F i t ) ishence defined by F ( ν ) = T (cid:12)(cid:12)(cid:12)(cid:12) c ∆ F (cid:12)(cid:12)(cid:12)(cid:12) h F i t , (1)where T is the duration of the observation, and ν a given fre-quency. The operator d ( . ) is b f ( ν ) = Z T / − T / d t f ( t ) e i π ν t . (2)We point out that with CoRoT or Kepler observations T is typi-cally much longer than the granule lifetime such that the operator d ( . ) tends to the classical Fourier transform.We consider a gray atmosphere . This is a necessary condi-tion to obtain an analytical formulation for the granulation spec-trum. We adopt a spherical coordinate system ( r , θ, φ ) with the z -axis pointing toward the observer. Accordingly, the bolometricflux received from the star at the instant t is given by F ( t ) = R s Z πφ = d φ Z µ = d µ µ I ( t , τ = , µ, φ ) , (3) where µ = cos( θ ), I ( t , τ, µ, φ ) is the specific intensity in the di-rection ( µ, φ ), R s the radius of the star, and τ ( r ) the mean opticaldepth τ ( r ) = Z + ∞ r ′ = r d r ′ κ ( r ′ ) ρ ( r ′ ) , (4)where κ is the mean opacity, ρ is the mean density (i.e., averagedin time and over a sphere of radius r ) and the specific intensity I ( τ, µ, φ ) is related to the source function according to (e.g., Gray1992, p. 114) I ( τ, µ, φ ) = Z + ∞ τ d τ ′ e − τ ′ ( r ) /µ µ S ( t , τ ′ ( r ) , µ, φ ) , (5)where S is the source function.To proceed, we assume the local thermodynamic equilibrium(LTE) so that S = B , where B is the Planck function at the instant t and the position ( τ ( r ) , µ, φ ). Note that LTE is fully justified be-cause the region where the granules are seen most often extendsto the small region around the optical depth τ ∼ ∆ F ( t ) isrewritten such as ∆ F ( t ) = R s Z πφ = d φ Z µ = d µ Z + ∞ d τ e − τ ( r ) /µ ∆ B , (6)where we have defined ∆ B ≡ B − h B i t .We neglect length-scales longer than the granulation length-scales. In that case ∆ B represents the instantaneous di ff erencebetween the brightness of the granules situated at position( τ ( r ) , µ, φ ) and the brightness of the material in the steady state( < B > t ). Finally, we assume that κρ varies at a length-scalelonger than the characteristic size of the granule. From this setof assumptions, and after some calculations, Eq. (1) can be writ-ten (see details in Appendix A) F ( ν ) = Z d µ Z + ∞ d τ e − τ/µ h B i t F ! F τ ( τ, ν ) (7)with F τ ( τ, ν ) = (2 π ) κρ R s h B i t ^ h ∆ B ∆ B i ( ν, k = , τ ) , (8)where the constant F is given by Eq. (A.15), ^ h ∆ B ∆ B i ( ν, k , τ )is the space and time Fourier transform of h ∆ B ∆ B i (seeEq. (A.13)), where k is a wavenumber, ν a frequency, and thesubscripts 1 and 2 refer to two di ff erent spatial and temporal po-sitions.To continue we need to derive an expression for the correla-tion product h ∆ B ∆ B i . To this end, we recall that B = σ T /π ,where σ is the Stefan-Boltzmann constant, and introduce ∆ T as the di ff erence between the temperature of the granule andthat of the surrounding medium. We further more use the quasi-normal approximation (Millionshchikov 1941) and assume thatthe scalar Θ ≡ ∆ T / h T i t is isotropic and behaves as a passivescalar. E Θ ( k , ν ) is then introduced as its associated spectrum(Lesieur 1997, Chap V-10) and factorized into a spatial spectrum E Θ ( k ) and a frequency-dependent factor χ k ( ν ) (see Eq. (A.26)).The above-mentionned set of approximations, after tedious cal-culations, leads us to (see details in Appendix A) F τ ( τ, ν ) = σ τ ν ! S Θ ( τ, ν ) , (9) where σ τ = √ r τ g N g Θ , τ g = κ ρ Λ , N g = π R s Λ , (10) Θ rms the root-mean-square of Θ (Eq. (A.25)), and Λ a charac-teristic length (see below). Note that in Eq. (10), τ g correspondsto the characteristic optical thickness of the granules, N g to theaverage number of granules distributed over half of the photo-sphere (i.e., at r = R s ), and σ τ to the global brightness fluctu-ations associated with the granulation spectrum that one wouldsee at the optical depth τ . The RHS of Eq. (9) involves the di-mensionless source function S Θ ( τ, ν ), whose expression is givenin Eq. (A.32). The latter depends on E Θ ( k ) and χ k ( ν ).The adopted expression for the spatial spectrum E Θ ( k ) (seeEq. (A.38)) involves the length-scale Λ as well as the charac-teristic wavenumber k c , which separates the inertial-convectiverange from the inertial-conductive range. Some prescriptions arerequired for Λ and k c , however. We hence assume that Λ = β H p ,where H p is the pressure scale height and β a free parameter. For k c , we adopt the prescription k c = ζ (cid:18) ǫχ (cid:19) / , where ζ is a freeparameter introduced to exert some control on this prescription, χ rad is the radiative di ff usivity coe ffi cient, and ǫ is the rate ofinjection of kinetic energy into the turbulent cascade.We turn now to the frequency component χ k ( ν ). In a stronglyturbulent medium, χ k is well described by a Lorentzian function χ k ( ν ) = πν k + ( ν/ν k ) , (11)where ν k is by definition the half-width at half-maximum of χ k ( ν ). The expression for ν k is given in Eq. (A.45), which in-volves the free parameter λ . This was introduced to have somecontrol on this definition. As an alternative for a Lorenztian func-tion (see Appendix A.4.2), we also consider an exponential form χ k ( ν ) = ln 22 ν k exp " − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ln 2 νν k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (12)
3. Calibration using solar observations
The theoretical granulation representative of the Sun were com-puted on the basis of the present model using inputs extractedfrom a 3D hydrodynamical model of the solar surface layers. Toreproduce the solar observations, we calibrated the free parame-ters involved in the theory.
We computed the theoretical PDS of the granulation ( F ) ac-cording to Esq. (7), (9), and (10) together with Eq. (A.15)and the set of Eqs. (A.32)–(A.36). The di ff erent quantities in-volved in the theoretical model (stratification, convective veloc-ity, etc) were obtained from a 3D model in a similar way as inSamadi et al. (2003b). However, while Samadi et al. (2003b) ex-tracted these quantities from horizontal averages at constant ge-ometrical depth, we here performed the averages at a constantoptical depth τ . This is justified because the RHS of Eq. (7) isintegrated over the optical depth.For later use, we define σ as the root-mean-square (rms)brightness fluctuations associated with a given PDS. The lattersatisfies the following relation σ = Z + ∞−∞ d ν F ( ν ) , (13) where F ( ν ) refers to a given PDS. Another important character-istic of the granulation spectrum is its associated timescale τ e ff ,which is here defined, following Mathur et al. (2011), as the e-folding time associated with the auto-correlation function (ACF)of the relative flux variations caused by the granulation. Di ff erentPDS are next compared, in terms of their shape (i.e., ν variation), σ and τ e ff .We considered a 3D model representative of the surface lay-ers of the Sun (see details in Ludwig et al. 2009) and computedthe associated theoretical PDS. This was then compared withthe PDS obtained from the green channel of the SOHO / VIRGOthree-channel sun-photometer (SPM, see Frohlich et al. 1997).We multiplied the original data by the instrumental function re-sponse function as derived by Michel et al. (2009) to convert theobserved photometric fluctuations in terms of bolometric ones.To compare theoretical PDS with the observations, we system-atically added the instrumental whited noise component, whichwe measured at high frequency on the observed solar spectrum. χ k We first assumed by default λ = β = ζ = χ k (Eq. (11)). The theoretical PDS un-derestimates σ by a factor of about 30 and the width of the solargranulation by a factor about five (not shown). Part of this signif-icant discrepancy is due to our prescription for Λ (Eq. (A.42)), k c (Eq. (A.43)), and ν k (Eq. (A.45)). The estimation of these quan-tities can been controlled by the parameters β and ζ , and theproduct β λ , respectively. However, there is some degeneracybetween ζ and β (see below). At fixed value of ζ , we then si-multaneously adjusted β and β λ such that σ and the width of thespectrum best matches the observations. This led to β = . λβ = .
7. The resulting theoretical PDS is shown in Fig. 1. Weobtained an overall satisfactory agreement between the theoreti-cal PDS and the observations, except for the frequency variationsat high frequencies.
Fig. 1.
Power density spectrum in ppm /µ Hz as a function of fre-quency ν . In black: PDS obtained from the green channel of theSOHO / VIRGO three-channel sun-photometer. In red and blue:theoretical PDS of the granulation background ( F ) computedusing the quantities obtained from the solar 3D model. The redcurve assumes a Lorentzian function for χ k , the blue curve anexponential functional. There are several pieces of observational evidence thatthe granulation spectrum significantly departs at high frequen-cies from a Lorentzian function (e.g., Mathur et al. 2011, andreferences therein). Furthermore, calculations performed withthe Ludwig (2006) method also confirm this general trend(Ludwig et al. 2009; Ludwig & Ste ff en 2012). Accordingly, wenow considered the exponential function given by Eq. (12). Thefree parameter β and the product β λ were adjusted such that thetheoretical PDS reproduces the observations best. For this wefitted the observed solar spectrum by means of the maximum-likelihood estimator (see e.g. Toutain & Appourchaux 1994;Appourchaux et al. 1998). For ζ =
1, the fit leads to β = . β λ = .
0. As seen in Fig. 1, we obtained an overall satisfactoryagreement with the observations. In particular, the observationsare much better fitted at high frequencies than when a Lorentzianfunction is adopted. Consequently, unless mentioned otherwise,we adopted the exponential function from now on.Other values of the parameter ζ were also tested. For eachadopted value of ζ , we have adjusted the parameter β and theproduct β λ such as to reproduce the solar data best. They all re-sult in almost the same agreement with the solar observations.As for the Lorentzian χ k (see Sect. 3.2), there is thus a degen-eracy between the parameters β and ζ . Therefore, the observedsolar granulation background does not permit one to constrainthese parameters independently. However, the granules observedon the solar surface have a typical size of about 2 Mm (seee.g. Muller 1989; Roudier et al. 1991). Furthermore, the obser-vations reported by Espagnet et al. (1993) and Hirzberger et al.(1997) suggest that k c / k ≈
2. These observations accordinglyfavor a value of ζ of about five since this value results in k c / k = . Λ = . ζ =
5. The corresponding value of β is 12.3, which valueis within the range found by Trampedach et al. (2013), that is,9-13.
4. Comparison with the Ludwig (2006) ab initio modeling
We compare here theoretical PDS computed with calculationsperformed on the basis of the Ludwig (2006) ab initio modelingassuming the same 3D hydrodynamical models in both cases.Two di ff erent 3D models were considered: a 3D model repre-sentative of the surface layers of an F-type main-sequence star(Sect. 4.1) and a second representative of the surface layers ofa red giant star (Sect. 4.2). These 3D models constitute two ex-treme cases in the H-R diagram.Because calculation of the PDS requires knowing the stellarradius R s , this was obtained by matching a complete 1D stan-dard model to the stratification of the 3D model (see details inTrampedach 1997; Samadi et al. 2008). The standard 1D modelwas computed using the CESAM2K code (Morel & Lebreton2008). Ludwig et al. (2009) have computed two F-dwarf 3D modelsthat included the CoRoT target HD 49933 in terms of surfacemetal abundance. Of these two 3D models, we considered herethe one with the solar surface abundance. This model has T e ff = g = .
25. We computed the theoretical PDS asdetailed in Sect. 3.1. We adopted the values of the free parame-ters that give the best agreement between the model and the ob- servations for the Sun (see Sect. 3). The result of the calculationis presented in Fig. 2 (top panel).We compared the theoretical PDS with the spectrum ob-tained by Ludwig et al. (2009) on the basis of the ab initio method. Their calculation was based on a radius of R S , Ludwig = . R ⊙ , while our associated global 1D model has a radius of R S = . R ⊙ . To compare ours with their results, we multipliedtheir spectrum by ( R S , Ludwig / R S ) since the PDS is inversely pro-portional to the square of the radius (Ludwig 2006). The result-ing PDS is shown in Fig. 2 (top panel). Fig. 2. Top:
Power density spectrum in ppm /µ Hz as a functionof frequency ν obtained for a 3D RHD model representative ofan F-dwarf star (see Sect. 4.1). The black curve corresponds tothe theoretical PDS obtained by Ludwig et al. (2009) on the ba-sis of the ab initio approach, the red curve to the PDS obtainedwith our theoretical model using inputs extracted from the same3D model (see Sect. 3.1). Bottom:
Same as the top panel for the3D model representative of a red-giant star (see Sect. 4.2).The ν -variation of the PDS obtained with the ab initio ap-proach is well reproduced. For the characteristic time τ e ff , ourtheoretical calculations result in a value of τ e ff σ = ±
17 ppm.Our theoretical model results in an rms brightness fluctuationof σ = σ for this F-dwarfstar. Furthermore, we show in paper II that our theoretical modelresults for F-dwarf stars in a systematic overestimation of σ . Asdiscussed in paper II, this trend is very likely linked to the impactof the high level of magneticactivities on the surface convection(see also Chaplin et al. 2011a,b). We therefore emphasize that all our calculations must be rigorously considered to be valid onlyfor stars with a low activity level. We considered a 3D model of the surface layers of a red-giantstar characterised by log g = . T e ff = ±
22 K (seedetails in Ludwig & Ste ff en 2012). The associated theoreticalPDS is compared in Fig. 2 (bottom panel) with the PDS com-puted by Ludwig & Ste ff en (2012) on the basis on the Ludwig(2006) ab initio approach. Their calculation assumed a radius R S , Ludwig = R ⊙ , while our associated global 1D model has aradius of R s = . R ⊙ . Accordingly, we multiplied their PDS by( R S , Ludwig / R s ) .Except at very high frequency ( ν & µ Hz), we obtaina good match between the two calculations, both in terms ofamplitude and ν -variation. Our theoretical spectrum results in σ =
352 ppm, which is only 9 % higher than the one ob-tained by Ludwig & Ste ff en (2012). For τ e ff , we obtain τ e ff = . × s. This is 29 % higher than the value derived from theLudwig & Ste ff en (2012) PDS.We note that assuming di ff erent sets of the calibrated param-eters (see Sect. 3.2) has a negligible impact on the ν -variation ofthe PDS and a moderate e ff ect on σ , for both the red giant andF-dwarf models. Furthermore, for both 3D models, assuming aLorenztian function for χ k ( ν ) instead of an exponential function,results in a considerable discrepancy at high frequency, as in thecase of the Sun (not shown).
5. Conclusion
We have developed a simple 1D theoretical model for the gran-ulation spectrum. One advantage of this model is that it can beused to compute theoretical granulation spectra on a large scale.Another benefit is that any prescription for the turbulent spec-trum can been considered, in particular, di ff erent prescriptionscan be tested for χ k , the Fourier transform of the time correla-tion function of the eddy velocity at a fixed k wave-number. Inthis way, we have established a link between the ν -variation ofthe granulation spectrum in intensity with the frequency compo-nent χ k .The theoretical model was first applied to the solar case. Thestratification of the surface layers of the Sun and the propertiesof its surface convection were obtained from a 3D hydrodynam-ical model of the surface layers of the Sun. Assuming defaultvalues for the free parameters involved in the theoretical model,our theoretical model strongly underestimates the quantities σ and τ e ff derived from the observed solar granulation spectrum.Nevertheless, the solar granulation spectrum can be well repro-duced by adjusting two of the three free parameters involved inthe theoretical model, while the degeneracy between the free pa-rameters was removed by using constraints from solar images.Two di ff erent functions for χ k ( ν ) were tested: a Lorentzianfunction and an exponential one. Whatever the choice of the freeparameters, the ν -variation of the solar spectrum is not repro-duced at high frequencies with the Lorentzian function. Indeed,at high frequencies the theoretical spectrum decreases as ν − ,while the observations decrease much more rapidly with ν . Onthe other hand, adopting an exponential χ k results in a much bet-ter agreement at high frequency. As discussed in Appendix C,this result confirms the previous claim by Nordlund et al. (1997)that the granules seen in the visible part of the atmosphere havea low level of turbulence. We then compared theoretical PDS with PDS calculated onthe basis of the Ludwig (2006) ab inito approach, that is from thetimeseries of the intensity emerging directly from 3D hydrody-namical models of the surface layers of stars. The time-averagedproperties of the surface layers were obtained from two 3D hdy-rodynamical models, one representative of an F-type star and thesecond one for a red giant star. The theoretical PDS reproducesthe main characteristics of the PDS obtained with the ab initio modeling. As in the solar case, the best agreement was obtainedwith an exponential χ k ( ν ).However, some residual di ff erences remain between our the-oretical calculations and the solar granulation spectrum as wellas between our calculations and those obtained with the ab ini-tio approach. As discussed in Appendix D, all these di ff erencesdo not exhibit a severe defect of our theoretical model, however,and we conclude that the main physical assumptions of the theo-retical model are validated by our di ff erent comparisons. Furthertheoretical developments are required to improve the modelingof the granulation background, however.We compute in the companion paper on the basis of this the-oretical model the global brightness fluctuations ( σ ) and the life-time ( τ e ff ) associated with the stellar granulation for a set of starswith di ff erent spectral types and luminosity classes. This willthen allow us to derive theoretical scaling laws for σ and τ e ff ,which we then compare with Kepler observations.
Acknowledgements.
SOHO is a mission of international collaboration betweenESA and NASA. RS and KB acknowledge financial support from the ProgrammeNational de Physique Stellaire (PNPS) of CNRS / INSU and from AgenceNationale de la Recherche (ANR, France) program “Interaction Des ´Etoileset des Exoplan`etes” (IDEE, ANR-12-BS05-0008). HGL acknowledges finan-cial support by the Sonderforschungsbereich SFB 881 “The Milky Way System”(subproject A4) of the German Research Foundation (DFG). RS thanks Fr´ed´ericBaudin for useful discussions about statistical issues.
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Appendix A: Theoretical developments
A.1. General expressionof the PDS
We establish here our general expression of the PDS, that it isEqs. (7) and (8). We start from the term of Eq. (6), which entersin the RHS of Eq. (1). We apply the operator given by Eq. (2) onEq. (6), it gives c ∆ F ( ν ) = R s Z d r r Z T / − T / d t ρκ e − τ ( r ) /µ e i π ν t ∆ B ( t , r ) , (A.1)where the space integration is performed over half of the (stellar)sphere, d r = − d µ r d r d φ , and d τ = κ ρ .The extent of the region where the stellar granulation is seenis very small compared to the stellar radius R s , such that r isalmost constant over the optical depth range relevant for calcu-lating the flux. Therefore, in very good approximation we have( R s / r ) ≃ c ∆ F ( ν ) = Z d r Z T / − T / d t ρκ e − τ ( r ) /µ e i π ν t ∆ B ( t , r ) . (A.2)Since c ∆ F ( ν ) cannot be evaluated in a deterministic way but a sta-tistical one only, we consider the square of Eq. (A.2) and averageit over a large number of independent realizations. This gives h| c ∆ F | i ( ν ) = Z d r κ ( r ) ρ ( r ) e − τ ( r ) /µ ( r ) Z T / t = − T / d t e i π ν t × Z d r κ ( r ) ρ ( r ) e − τ ( r ) /µ ( r ) Z T / t = − T / d t e − i π ν t × h ∆ B ∆ B i , (A.3)where ∆ B (resp. ∆ B ) corresponds to the quantity ∆ B evaluatedat the time-space position ( t , r ) (resp. ( t , r )).We then define the following new coordinates t = ( t + t ) / t ′ = t − t , (A.4) r = ( r + r ) / r = r − r . (A.5)In these coordinates, r and t ′ are the spatial correlation and tem-poral correlation lengths associated with the local properties ofthe turbulence, while r and t are mean space and time posi-tions. Using the new coordinates given by Eqs.(A.4) and (A.5)in Eq. (A.6) leads to h| c ∆ F | i ( ν ) = π Z Z T / − T / d t d r (A.6) × κ ( r ) ρ ( r ) e − τ /µ Γ ( t , r , ν ) , where Γ ( t , r , ν ) = Z d r γ ( r , r )2 π Z T / T / d t ′ e − i π ν t ′ h ∆ B ∆ B i (A.7) γ ( r , r ) = κ ( r ) ρ ( r ) κ ( r ) ρ ( r ) κ ( r ) ρ ( r ) e τ /µ − ( τ /µ + τ /µ ) (A.8)and τ i = τ ( r i ) , µ i = µ ( r i ) with i = { , , } .At this stage, a tractable expression for Eq. (A.6) calls formore assumptions. We assume that T is much longer than thegranule lifetime. We neglect length-scales longer than the gran-ulation length-scales. In that case ∆ B represents the instanta-neous di ff erence between the brightness of the granules situated at the position ( τ ( r ) , µ, φ ) and the brightness of the material inthe steady state ( < B > t ). Accordingly, Eq. (A.6) reduces to h| c ∆ F | i = (2 π ) R s T Z d µ Z + ∞ d τ e − τ /µ Γ ( τ , ν )(A.9) Γ ( τ , ν ) = Z d r γ ( r , r )2 π Z + ∞−∞ e − i π ν t ′ h ∆ B ∆ B i d t ′ (A.10) γ ( r , r ) = κ ( r ) ρ ( r ) κ ( r ) ρ ( r ) κ ( r ) ρ ( r ) . (A.11)We now assume that κρ varies at a length-scale longer thanthe characteristic size of the granules. This assumption is dis-cussed in Appendix B. Accordingly, γ ≃ κ ( r ) ρ ( r ) and Γ re-duces to Γ ( τ , ν ) = (2 π ) κ ( τ ) ρ ( τ ) ^ h ∆ B ∆ B i ( ν, k = ) , (A.12)where ^ h ∆ B ∆ B i is the space and time Fourier transform of h ∆ B ∆ B i , defined as ^ h ∆ B ∆ B i ( ν, k ) ≡ π ) Z d t ′ Z d r e − i πν t ′ − i k . r h ∆ B ∆ B i . (A.13)Note that, from now on, we substitute the notation τ by τ .We now turn to the time-averaged flux h F i t . From Eqs. (3),(4), and (5) one derives h F i t = π R s F , (A.14)where we have defined F ≡ Z d µ Z + ∞ d τ e − τ/µ h B i t ( τ ) . (A.15)Finally, using Eq. (A.9), Eq. (A.12) and Eq. (A.14) we derive thegeneral expressions of Eqs. (7) and (8). The term F τ (Eq. (8))stands for the PDS of the granulation as it would be seen at theoptical depth τ . However, the observed PDS associated with thegranulation background is given by Eq. (7) and corresponds tothe sum of the spectra seen at di ff erent layers in the atmosphere,but weighted by the term e − τ/µ ( h B i t / F ) . A.2. Source function( h ∆ B ∆ B i ) To proceed we need to derive an expression for the correlationproduct h ∆ B ∆ B i . To this end, we recall that B = σ T /π , where σ is the Stefan-Boltzmann constant, and introduce ∆ T as thedi ff erence between the temperature of the granule and that of thesurrounding medium. We thus have ∆ B = (cid:16) (1 + Θ ) − (cid:17) h B i t , (A.16)where Θ ≡ ∆ T / h T i t . The second-order Taylor expansion of theRHS of Eq. (A.16) gives ∆ B = (cid:16) Θ + Θ (cid:17) h B i t . (A.17)We have neglected the third and fourth order terms in Θ . Indeed,a 3D hydrodynamical simulation of the solar surface shows thatterms higher than second order contribute less than about 15 %of Eq. (A.16). Accordingly, h ∆ B ∆ B i = (A.18)4 (cid:16) h Θ Θ i + h Θ Θ i + h Θ Θ i + h Θ Θ i (cid:17) h B i t , amadi et al.: Stellar granulation as seen in disk-integrated intensity , Online Material p 2 where Θ ≡ Θ ( r , t )) and Θ ≡ Θ ( r , t ).We now adopt the quasi-normal approximation (QNA).This approximation is rigorously valid for normally distributedquantities. Departure from this approximation is discussed inAppendix B. Normally distributed quantities are necessarilysymmetric, such that h Θ Θ i = h Θ Θ i =
0. This ap-proximation also implies (e.g., Lesieur 1997, Chap. VII-2) h Θ Θ i = h Θ i h Θ i + h Θ Θ i . (A.19)The first term in the RHS of Eq. (A.19) does not contribute in thetime Fourier domain, except at the null frequency. Accordingly, h ∆ B ∆ B i = h h Θ Θ i + h Θ Θ i i h B i t . (A.20)Now using the Parseval-Plancherel relation, Eq. (A.20) becomes ^ h ∆ B ∆ B i ( ν, k ) = h B i t (cid:20) ^ h Θ Θ i ( ν, k ) + e B Θ ( ν, k ) (cid:21) , (A.21)with˜ B Θ ( ν, k ) ≡ R d ν ′ R d k ′ ^ h Θ Θ i ( ν ′ , k ′ ) ^ h Θ Θ i ( ν ′′ , k ′′ ) , (A.22)where ν ′′ = ν + ν ′ and k ′′ = k + k ′ .We assume that the scalar Θ is isotropic. Accordingly, thespatial Fourier transform of h Θ Θ i ( r , ν ) is given by (see e.g.Lesieur 1997, Chap V-10) ^ h Θ Θ i ( k , ν ) = E Θ ( k , ν )2 π k . (A.23)The scalar spectrum E Θ ( k , ω ) is related to the scalar variance as(Lesieur 1997, Chap V-10)12 h Θ i ( τ ) = Θ = Z ∞−∞ d ν Z ∞ d k E Θ ( k , ν ) , (A.24)where Θ rms is by definition the rms of Θ , which is related to ∆ T rms (the rms of ∆ T ) according to Θ = ∆ T h T i t . (A.25)Following Stein (1967), the scalar energy spectrum E Θ ( k , ν ) canbe factorized into a spatial spectrum E Θ ( k ) and a frequency-dependent factor χ k ( ν ) according to E Θ ( k , ν ) = E Θ ( k ) χ k ( ν ) , (A.26)where the frequency-dependent factor χ k ( ν ) is normalized suchthat Z + ∞−∞ d ν χ k ( ν ) = . (A.27) A.3. Finalexpression ofrelativeflux variations
The final expression for the term F τ (Eq. (8)) that appears in theRHS of Eq. (7) is obtained by inserting Eq. (A.21) to (A.23) intoEq. (8), so that F τ ( τ, ν ) = (2 π ) κρ R s (cid:20) ^ h Θ Θ i +
72 ˜ B Θ (cid:21) , (A.28)where the two terms in the RHS of Eq. (A.28) are considered for( τ, ν, k = F τ ( τ, ν ) =
72 (2 π ) κρ R s ˜ B Θ ( τ, ν, k = , (A.29)with ˜ B Θ ( τ, ν, = π Z d k E Θ ( k ) k ! ψ k ( τ, ν ) , (A.30) ψ k ( τ, ν ) = Z d ν ′ χ k ( ν ′ ) χ k ( ν ′ + ν ) . (A.31)The final expression, Eq. (A.29), is recast to a more suitableform. To this end, we define the characteristic wave-number k = π/ Λ where Λ is a characteristic length. We also definethe characteristic frequency ν = / (2 π τ c ) where τ c is a charac-teristic time. Eventually, Eq. (A.29) leads to Eqs. (9)-(10) wherewe have defined the dimensionless source function S Θ ( τ, ν ) = π Z d KK ˜ E Θ ( K ) Ψ K ( τ, ξ ) , (A.32)as well as the following dimensionless quantities: Ψ K ( τ, ξ ) = Z d ξ ′ ˜ χ K ( ξ ′ ) ˜ χ K ( ξ ′ + ξ ) , (A.33)˜ E Θ ( K ) = k Θ − E Θ ( k ) , (A.34)˜ χ K ( ξ ) = ν χ k ( ν ) , (A.35) K = kk , ξ = νν , (A.36)where we have defined the characteristic wavenumber k = π / Λ , and the characteristic frequency ν = / (2 π τ c ), where τ c ≡ / ( k u ) is a characteristic time and u a characteristic ve-locity (see Eq. (A.40) below). Note that the dimensionless quan-tities ˜ E ( K ) and ˜ χ K ( ξ ) verify the following normalization condi-tions: Z + ∞ d K ˜ E Θ ( K ) = , and Z + ∞−∞ d ξ ˜ χ K ( ξ ) = . (A.37) A.4. TurbulencemodelingA.4.1. Spatial spectrum
As already mentioned, it is assumed that the temperature fluc-tuations behave as a passive scalar (see the discussion inAppendix B). Therefore E Θ ( k ) is given according to (see Lesieur1997, Chap VI-10) E Θ ( k ) = a Θ u E ( k ) , for k ≤ k c a Θ u k c k ! E ( k ) , for k > k c , (A.38)where a is a normalization factor, E ( k ) the kinetic energy spec-trum, u a characteristic velocity (see Eq. (A.40) below), and k c the wavenumber, which separates two characteristic ranges: amadi et al.: Stellar granulation as seen in disk-integrated intensity , Online Material p 3 – the inertial-convective range ( k < k c ). In this domain, advec-tion of the temperature fluctuations by the turbulent velocityfield dominates over the di ff usion. – the inertial-conductive range ( k > k c ). In this domain, thedi ff usion of the temperature fluctuations dominates over ad-vection.We consider for E ( k ) the Kolmogorov spectrum, and thecharacteristic velocity u is defined such that E ( k ) = k < u k K − / for k ≥ k , (A.39)where the characteristic velocity u is defined such that32 u ≡ Z + ∞ d k E ( k ) . (A.40)From Eq. (A.38) and Eq. (A.39), the spacial spectrum finallybecomes˜ E Θ ( K ) = K < a K − / for 1 ≤ K ≤ K C a K C K − / for K C < K , (A.41)where we have defined K C = k c / k .Some prescriptions are required for k c and k . 3D RHD showthat from one stellar 3D model to another, the granule size Λ scales approximately as the pressure scale-height ( H p ) at thephotosphere (Freytag et al. 1997). Accordingly, we assume that k = π Λ with Λ = β H p , (A.42)where Λ is a characteristic length-scale, H p the pressure scaleheight and β is a free parameter. Concerning the characteris-tic wavenumber k c , in a medium with very low Prandtl number(which is the case in the stellar medium), k c is given by (seeLesieur 1997, Chap. VI) k c = ζ ǫχ / , (A.43)where ζ is a free parameters introduced to exercise some con-trol on this prescription, χ rad is the radiative di ff usivity coef-ficient, and ǫ is the rate of injection of kinetic energy intothe turbulent cascade. The latter is estimated according to ǫ = Φ w / Λ , where we have introduced the anisotropy factor Φ ≡ (cid:16) u + v + w (cid:17) / w , where u rms , v rms , and w rms are the rmsof the three components of the velocity (horizontal and verticalones). A.4.2. Frequency spectrum
Since in the inertial-convective range ( i.e , k < k c ) temperaturefluctuations are dominated by advection, we assume that – at thisscale range – χ k ( ν ) is the same as the frequency spectrum associ-ated with the velocity field, χ vk . This hypothesis tends to be sup-ported by large eddies simulations (see e.g. Samadi et al. 2003a).Indeed, using a solar hydrodynamic simulation, Samadi et al.(2003a) have found that this property is rather well verified bythe entropy fluctuations. Since entropy fluctuations are mostlydominated by temperature fluctuations, this must be the same forthe temperature fluctuations. For the inertial-conductive range( k > k c ): temperature fluctuations are no longer dominated by advection. However, to our knowledge, no study has been con-ducted yet about the properties of χ k in this range. Therefore, weassume by default that in this range χ k varies with ν , as do χ vk .In a strongly turbulent medium, χ vk is well described bya Lorentzian function (Sawford 1991; Samadi et al. 2003a;Belkacem et al. 2010), i.e. , χ k ( ν ) = πν k + ( ν/ν k ) , (A.44)where ν k is by definition the half-width at half-maximum of χ k ( ν ). In the framework of the Samadi & Goupil (2001) formal-ism, this latter quantity is evaluated as ν k = k u k πλ , (A.45)with u k = Z kk E ( k ) d k , (A.46)where λ is a free parameter introduced, following Balmforth(1992), to have some control on the adopted definition for ν k . Wedefine the characteristic time τ c ≡ / ( k u ), which correspondsto an estimate of the lifetime of the largest eddies. Accordingly,we have ν = (2 πτ c ) − .The Lorentzian χ k (Eq. (A.44)) has a justification for astrongly turbulent medium. However, Georgobiani et al. (2006)have found on the basis of a 3D RHD solar model that χ k de-creases more rapidly with ν near the photosphere than it doesin deeper layers. Accordingly, as an alternative for a Lorenztianfunction and following Musielak et al. (1994), we also considerfor χ k an exponential form χ k ( ν ) = ln 22 ν k exp " − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ln 2 νν k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (A.47)where ν k is the half-width at half-maximum. We alternativelyadopted here the two di ff erent prescriptions for χ k . Appendix B: Approximations and assumptions
Our theoretical model is based on three major approximationsand assumptions. They are discussed below.
Passive scalar assumption:
It was assumed that the tempera-ture fluctuations behave as a passive scalar. We recall that apassive scalar f is a quantity that obeys an equation of di ff u-sion (e.g., Lesieur 1997). For the temperature fluctuations, thisequation of di ff usion is rigorously valid when the di ff usion andthe Boussinesq approximations are verified and the stratificationis negligible compared with the eddy size. However, all theseconditions are not fulfilled in the vicinity of the photospherewhere the granules are the more visible. Indeed, in this region themedium is optically thin such that the di ff usion approximationdoes no longer hold. Furthermore, this region is characterizedby a non-negligible turbulent Mach number that prevents theBoussinesq approximation from being valid. Finally, the gran-ule sizes are typically of the order of the pressure-scale height(see below). Nevertheless, as shown by Espagnet et al. (1993)and Hirzberger et al. (1997)s the spectrum, E Θ , associated withthe temperature fluctuations at the surface of the Sun scales with k as predicted by Eq. (A.38), which was derived assuming thatthe temperature fluctuations behave as a passive scalar. This sug-gests that somehow the temperature fluctuations obey an equa-tion of di ff usion. amadi et al.: Stellar granulation as seen in disk-integrated intensity , Online Material p 4Quasi-normal approximation:
The approximation of Eq. (A.19)assumes that fluctuating quantities are distributed according toa normal distribution. However, it is well known that the depar-ture from the QNA is important in a strongly turbulent medium(Ogura 1963). Furthermore, the upper-most part of the convec-tion zone is a turbulent convective medium composed of essen-tially two flows (the granules and the downdraft plumes) that areasymmetric with respect to each other. Therefore, we obviouslydo not deal with symmetric distribution, as it is the case for anormal distribution. With the help of a solar 3D hydrodynamicalsimulation, Belkacem et al. (2006) have quantified the departurefrom the QNA (seee also Kupka & Robinson 2007). However,by comparing 3D hydrodynamical models representative of dif-ferent main-sequence stars, we have found that this departuredoes not vary significantly across the main sequence.
Length-scale separation:
The derivation of Eq. (7) is based onthe assumption that the product κρ varies at a length-scale sig-nificantly longer than the granule size. However, in the case ofthe Sun, for instance, the granules have a size of about 2 Mm(see e.g. Muller 1989; Roudier et al. 1991) while the pressure-scale height, H p , is of the order of few hundred kilometers at thephotosphere. Because near the photosphere the density scale-height H ρ is of the same order as H p and even lower, our as-sumption does not hold near the photosphere. Nevertheless, 3Dhydrodynamic models show that from a stellar model to another,the granule size scales as the pressure scale-height at the pho-tosphere (Freytag et al. 1997; Samadi et al. 2008). Therefore, asfor the QNA, we expect that the departure from our hypothe-sis introduces a bias that remains almost constant across the HRdiagram.The major approximations and assumptions adopted in ourmodel are expected to be in default near the photosphere.However, avoiding these approximations and assumptions wouldrequire additional theoretical improvements, and they constitute– at the present time – the only way for deriving an analyti-cal model of the granulation spectrum. We also recall that ourobjective is to derive a simple analytical model for the interpre-tation of the observed scaling relations. Furthermore, providedthat the three free parameters involved in the model are appro-priately tuned, the theoretical model agrees reasonably well withthe Ludwig (2006) 3D hydrodynamical approach (see Sect. 4). Appendix C: Eddy-time correlation
Two di ff erent functions for χ k ( ν ) were tested: a Lorentzian func-tion and an exponential one. For a strongly turbulent medium,one expects a Lorentzian function (Sawford 1991; Samadi et al.2003a; Belkacem et al. 2010). However, with this function, thetheoretical granulation spectrum decreases as ν − , while the ob-servations decreases much more rapidly with ν . On the otherhand, adopting an exponential χ k results in a much better agree-ment at high frequency. This is because an exponential χ k de-creases more rapidly with frequency than does a Lorenztian χ k .Finally, as in the Sun, adopting an exponential χ k results in amuch better match with the theoretical PDS computed with theab initio approach for two 3D hdyrodynamical models, one rep-resentative for an F-type star and the other one for a red giantstar (see Sect. 4).According to Sawford (1991, see also Appourchaux et al.(2010)), the time-Fourier transform of the Lagrangian eddy-timecorrelation function is expected to tend to a Lorentzian functionwhen the Reynolds number tends to infinity. In contrast, the lessturbulent the medium (i.e., in general the lower Reynolds num-ber), the more rapid the decrease of χ k with increasing ν . This behavior is also supported for the Eulerian eddy-time correla-tion ( χ k ) by hydrodynamical numerical models (see e.g. Samadi2011). In other words, the ν variation of χ k is expected to dependon the degree of turbulence. Accordingly, our result confirmsthat the granules, which are mainly visible near the photosphere,are less turbulent than the super-adiabatic layers situated a fewhundred kilometers below the photosphere. We therefore con-firm the previous claim by Nordlund et al. (1997) that the gran-ules have a low level of turbulence. This is also consistent withthe results by Georgobiani et al. (2006, see also Nordlund et al.(2009)). Indeed, these authors showed that χ k decreases morerapidly with ν near the surface of a solar 3D model than itdoes in deeper layers, where χ k is close to a Lorentzian func-tion (Samadi et al. 2003a).Finally, the fact that the granules have a relatively low levelof turbulence is probably not specific to the Sun. Indeed, the 3DRHD models of stars show that the granules have similar prop-erties as in the Sun (e.g., Trampedach et al. 2013). Therefore,it is not surprising that the theoretical PDS computed with theab initio approach varies with ν in a similar way as in the Sun. Appendix D: Remaining discrepancies
Although globally satisfactory, the theoretical model does notreproduce the solar granulation spectrum perfectly. Indeed, theobserved spectrum shows a kink at ν ∼ ff ab initio approach. This indicates that pure hydrodynamical ap-proaches cannot fully account for the observed solar granulationspectrum. Nevertheless, the remaining discrepancies representonly a small fraction of the total brightness fluctuations producedby the granulation phenomenon.The theoretical model does not perfectly reproduced the PDSobtained with the ab initio approach for the two 3D models con-sidered in this work. For instance, a di ff erence of less than about15 % is obtained for σ with the 3D model for an F-dwarf starand a di ff erence less than about 30 % are obtained for τ e ff withthe 3D model for a red giant star. These di ff erences must be at-tributed to the di ff erent approximations and assumptions adopted(see Appendix B above). Nevertheless, they remain of the or-der of the dispersion obtained between the di ff erent methods ofanalysis investigated by Mathur et al. (2011, see also paper II).Because the ab initioab initio