The relationship between photometric and spectroscopic oscillation amplitudes from 3D stellar atmosphere simulations
Yixiao Zhou, Thomas Nordlander, Luca Casagrande, Meridith Joyce, Yaguang Li, Anish M. Amarsi, Henrique Reggiani, Martin Asplund
MMNRAS , 1–16 (2021) Preprint 4 February 2021 Compiled using MNRAS L A TEX style file v3.0
The relationship between photometric and spectroscopic oscillationamplitudes from 3D stellar atmosphere simulations
Yixiao Zhou, Thomas Nordlander, , Luca Casagrande, , Meridith Joyce, , Yaguang Li, , Anish M. Amarsi, Henrique Reggiani and Martin Asplund Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), Australia Sydney Institute for Astronomy (SIfA), School of Physics, University of Sydney, NSW 2006, Australia Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark Theoretical Astrophysics, Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden Department of Physics and Astronomy, Johns Hopkins University, 3400 N Charles St., Baltimore, MD 21218, USA Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, D-85741 Garching, Germany
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We establish a quantitative relationship between photometric and spectroscopic detectionsof solar-like oscillations using ab initio , three-dimensional (3D), hydrodynamical numericalsimulations of stellar atmospheres. We present a theoretical derivation as proof of conceptfor our method. We perform realistic spectral line formation calculations to quantify the ratiobetween luminosity and radial velocity amplitude for two case studies: the Sun and the redgiant 𝜖 Tau. Luminosity amplitudes are computed based on the bolometric flux predicted by3D simulations with granulation background modelled the same way as asteroseismic obser-vations. Radial velocity amplitudes are determined from the wavelength shift of synthesizedspectral lines with methods closely resembling those used in BiSON and SONG observations.Consequently, the theoretical luminosity to radial velocity amplitude ratios are directly com-parable with corresponding observations. For the Sun, we predict theoretical ratios of 21.0 and23.7 ppm/[m s − ] from BiSON and SONG respectively, in good agreement with observations19.1 and 21.6 ppm/[m s − ]. For 𝜖 Tau, we predict K2 and SONG ratios of 48.4 ppm/[m s − ],again in good agreement with observations 42.2 ppm/[m s − ], and much improved over theresult from conventional empirical scaling relations which gives 23.2 ppm/[m s − ]. This studythus opens the path towards a quantitative understanding of solar-like oscillations, via detailedmodelling of 3D stellar atmospheres. Key words: convection – hydrodynamics – methods: numerical – stars: oscillations – stars:atmospheres – line: profiles
Solar-like oscillations can be observed via photometry and spec-troscopy. The photometric method allows us to detect stellar oscil-lations by measuring variations in the brightness of stars, whereasthe spectroscopic method exploits the Doppler shifts of spectrallines to detect stellar oscillations. The spectroscopic method (alsocalled the radial velocity method) is employed by ground-basedtelescopes such as Birmingham Solar Oscillations Network (Bi-SON, Chaplin et al. 1996) and Stellar Oscillations Network Group(SONG, Grundahl et al. 2006). These instrumments have laid thegroundwork for helioseismoloy and asteroseismology through theirdetailed observations of solar oscillations and their detection of thefirst solar-like oscillating stars (Claverie et al. 1979; Brown et al.1991; Bedding et al. 2001; Kjeldsen et al. 2003).Built upon these pioneering works, the field of asteroseismol- ogy has thrived in the last decade thanks to the CoRoT (Michel et al.2008),
Kepler (Borucki et al. 2010) and TESS (Ricker et al. 2015)missions that detect stellar oscillations by measuring variations instellar luminosity. The high-quality, long time-series, extensive pho-tometric data provided by these space-based telescopes enable ac-curate determination of oscillation frequencies and amplitudes forthousands of solar-like stars, thus ushering in the era of ensembleasteroseismology. However, in the low-frequency regime the photo-metric method is complicated by signals due to stellar atmosphericconvection (stellar granulation), which impedes the characterisa-tion of low-frequency oscillations. This difficulty can be avoided byobserving the star using the radial velocity method, as stellar gran-ulation noise is significantly less pronounced in velocity signals.Moreover, the radial velocity method has demonstrated great po-tential for measuring oscillations in cool dwarf stars (e.g. Kjeldsenet al. 2005), which are important in exoplanet science but difficult to © a r X i v : . [ a s t r o - ph . S R ] F e b Y. Zhou et al. detect with space-photometry due to their low intrinsic luminosityand small oscillation amplitude (Huber et al. 2019). Nonetheless,ground-based spectroscopy is limited by target brightness and theEarth’s atmosphere.It follows that the photometric and spectroscopic methods ofmeasuring stellar oscillations are highly complementary and thatcombining the two methods will yield extra information that canfurther constrain the properties of stars. Recently, solar-like oscil-lations in several stars, such as Procyon A and 𝜖 Tauri (hereafter 𝜖 Tau), have been observed in both photometry and spectroscopy(Huber et al. 2011; Arentoft et al. 2019). With the commencementof the TESS mission and SONG observations, many stars will soonhave both luminosity and radial velocity data available. Therefore,investigating the relationship between luminosity and radial velocityamplitude is of increasing importance.This topic was first explored in the pioneering study of solar-like oscillations by Kjeldsen & Bedding (1995), who proposed aquantitative relationship between luminosity and radial velocity os-cillation amplitudes for solar-like stars by scaling from the Sun. TheKjeldsen & Bedding (1995) amplitude ratio scaling relation hasbeen the industry standard in asteroseismology until now, provid-ing valuable guidance for many years. However, their relationshipis based on empirical arguments, and it is unable to reproduce theobserved amplitude ratio for some stars (Huber et al. 2011; Arentoftet al. 2019). It is therefore prudent and timely to refine the relation-ship between luminosity and radial velocity based on detailed stellarmodelling. As a first attempt to solve this problem from a modellingperspective, Houdek et al. (1999) and Houdek (2010) computed thetheoretical ratio between luminosity and velocity amplitudes. Thecalculations are based on their one-dimensional, non-local, timedependent convection model for the Sun (Houdek et al. 1999) andthe scaled VAL-C atmosphere for Procyon A (Vernazza et al. 1981;Houdek 2010). Their amplitude ratio results are in reasonable agree-ment with observations. Nevertheless, it is worth noting that thepredicted amplitude ratio depends on at which atmospheric heightthe velocity amplitude is evaluated (see Fig. 1 and 2 of Houdek2010).In this paper, we investigate the relationship between photo-metric and spectroscopic measurements of stellar oscillations. Wequantify the amplitude ratio in an essentially parameter-free man-ner, by carrying out detailed ab initio three-dimensional (3D) hy-drodynamical simulations of stellar surface convection. We base ouranalysis on realistic synthetic spectra, calculated using 3D radiativetransfer and taking into account departures from local thermody-namic equilibrium (LTE) where necessary.
In this pilot study, we focus on the Sun and on the G-type red giantstar 𝜖 Tau (HD 28305). As a bright star residing in the nearest openCluster, Hyades, and known exoplanet host, 𝜖 Tau is of great interestto stellar physics for a variety of reasons (Sato et al. 2007).We adopt the stellar parameters provided in Arentoft et al.(2019): 𝑇 eff = 𝑔 = .
67 dex, [ Fe / H ] = .
15 dex, asreference values. This effective temperature was determined via thebolometric flux measured by Baines et al. (2018) and the angulardiameter measured interferometrically from the CHARA array (Ar-entoft et al. 2019). The surface gravity was determined from theobserved frequency of maximum power, 𝜈 max , for this star (Stelloet al. 2017; Arentoft et al. 2019) through the 𝜈 max scaling relation(Brown et al. 1991; Kjeldsen & Bedding 1995). Moreover, detailed Table 1.
Fundamental parameters and basic information about the simula-tion of the Sun and 𝜖 Tau. Reference values are adopted from Prša et al.(2016) and Arentoft et al. (2019), respectively. We note that the effectivetemperature fluctuates over time in 3D models, therefore both mean effec-tive temperature and its standard deviation are given. Also, both minimumand maximum vertical grid spacing are provided, as mesh points are notuniformly distributed vertically. Sun 𝜖 Tau 𝑇 eff (K) Reference 5772 . ± . ± ±
16 4979 ± 𝑔 (cgs) Reference 4.438 2.67Modelling 4.438 2.67[Fe/H] (dex) Reference 0.00 0 . ± . Time duration (hour) 24 1205.8Sampling interval (s) 30 1447Vertical size (Mm) 3.6 250Vertical grid spacing (km) 7–33 562–2650Horizontal grid spacing (km) 25 2165 asteroseismic observations for 𝜖 Tau using both K2 (the successorof Kepler ; Howell et al. 2014) and SONG yield individual oscilla-tion frequencies for more than 20 modes as well as the amplituderatio between K2 and SONG, which makes 𝜖 Tau an ideal target toinvestigate in this work. Analogous parameters for the Sun are, ofcourse, known to the highest degrees of precision and accuracy ofany star. Observational parameters are included in Table 1.
In this section, we introduce the 3D hydrodynamic stellar atmo-sphere models that are the basis of our analysis. All 3D modelsare computed with a customized version of the Stagger code(Nordlund & Galsgaard 1995; Collet et al. 2018), a radiative-magnetohydrodynamic code that solves the time-dependent equa-tions of mass, momentum and energy conservation, as well as themagnetic-field induction equation and the radiative transfer equationon a 3D staggered Eulerian mesh. The stellar models in the presentstudy have been constructed without magnetic fields. All scalarsare evaluated at cell centres, whereas vectors, such as velocity, arestaggered at the centres of cell faces in order to improve numericalaccuracy. The code incorporates realistic microphysics and a de-tailed radiative transfer scheme. An updated version of the Mihalaset al. (1988) equation of state (Trampedach et al. 2013) is adopted,which accounts for all ionization stages of the 17 most abundant ele-ments in the Sun plus the H molecule. A comprehensive collectionof relevant continuous absorption and scattering sources is includedas described in Hayek et al. (2010). The pre-computed, sampled lineopacities are taken from the MARCS model atmosphere package(Gustafsson et al. 2008). Radiative energy transport is modelled bysolving the equation of radiative transfer at every time step of thesimulation for all mesh points above a certain Rosseland mean opti-cal depth ( 𝜏 Ross ≤
500 throughout this work) under the assumptionof LTE. The frequency dependence of the radiative transfer equationis approximated via the opacity binning method (Nordlund 1982;Collet et al. 2018), in which 12 opacity bins are divided based onwavelength and strength of opacities. Consequently, the integra-tion over wavelength reduces to the summation over 12 selectedbins. The spatial dependence of the radiative transfer equation is
MNRAS000
MNRAS000 , 1–16 (2021) uminosity and velocity oscillation amplitudes P S [ L / L ] [ pp m ] P S [ V y ] [ m / s ] L / L V y (a) P S [ L / L ] [ pp m ] P S [ V y ] [ m / s ] L / L V y (b) Figure 1. 𝜖 Tau. Three simulation modesare visible in both the velocity and luminosity spectra. represented by solving along a set of inclined rays in space. Ninedirections – one vertical and eight inclined directions representingcombinations of two polar and four azimuthal angles – are consid-ered for all models presented in this work. The integration over polarangle is carried out using the Gauss-Radau quadrature scheme. Thethus evaluated radiative heating rates can be used to calculate thesurface flux (i.e., the emergent radiative flux at the top boundary ofsimulation domain) and subsequently the effective temperature viathe Stefan-Boltzmann law.The basic configurations of our 3D models are summarised inTable 1. For both the Sun and the red giant 𝜖 Tau, the Staggermodel atmospheres are constructed based on the reference effectivetemperatures and surface gravities (Table 1). The Asplund et al.(2009) solar chemical composition is adopted in both cases. Thoughwe do not expect stellar metallicity to introduce any significantdifferences for the purposes of this study, we intend to consider theeffects of stellar metallicity in detail in a later investigation.Spatially, the simulation domain is discretized in a box locatedaround the stellar photosphere. Horizontally, the simulation domainis a square with 240 ×
240 evenly distributed mesh points. Thehorizontal size of the box is large enough to enclose at least tengranules at any time of the simulation (Magic et al. 2013a). Thereare 240 mesh points in the vertical direction covering roughly theouter 1% of the stellar radius, extending from the upper part ofthe surface convection zone, including the entire optical surface,and reaching the lower part of the chromosphere; we note that theouter-most layers are likely the least realistic given our neglect ofmagnetic fields in these simulations. Because the vertical scale ofthe simulation is very small compared to the total stellar radius, thespherical effects are negligible and gravitational acceleration canbe regarded as a constant (i.e. the surface gravity). Mesh points arenot evenly distributed vertically: the highest numerical resolution isapplied around the optical surface to resolve the transition betweenthe optically thick and thin regimes. Furthermore, in the case of 𝜖 Tau, a separate vertical mesh structure is employed for the radiativetransfer calculation in order to resolve the extremely steep temper-ature and opacity gradients near the optical surface of red giantsadequately (see e.g., Fig. 3 of Collet et al. 2018). Adaptive mesh refinement was used when constructing the vertical radiative mesh;the radiative mesh of each vertical sub-domain within the simulationdomain is arranged based on the distribution of Rosseland opticaldepth in this sub-domain, resulting in highest numerical resolutionnear the photosphere (see Fig. 6 in Collet et al. 2018 for an illustra-tion). At each simulation time step, radiative transfer calculationsare performed on the radiative mesh and then interpolated back tothe aforementioned hydrodynamical mesh. We refer the reader toSection 2.7 in Collet et al. (2018) for a detailed introduction to thistechnique.Boundaries are periodic in the horizontal direction while openin the vertical (Collet et al. 2018). At the bottom boundary, outgo-ing flows (vertical velocities towards stellar centre) are free to carrytheir entropy fluctuations out of the simulation domain, whereasincoming flows have invariant entropy and thermal (gas plus radi-ation) pressure. Temporally, the duration of the simulation is oneday for the Sun, and about 50 days for 𝜖 Tau. Simulation data isstored every 30 seconds in the solar simulation while every 1447seconds for the red giant case. A long stellar time coverage like thisis necessary for an accurate analysis of stellar oscillations.Sound waves and the resulting 𝑝 -modes are natural phenomenain surface convection simulations, which can be directly identifiedby looking at the power spectrum of the vertical velocity of thesimulations. Because 𝑝 -mode oscillations in the simulation domainperiodically shift the optical surface up and down, causing coher-ent changes in surface temperature, simulation modes can also beidentified indirectly from the power spectrum of the bolometric fluxvariation. The relative variation of the bolometric flux, in parts permillion (ppm), is defined as 𝛿𝐹 bol 𝐹 bol , = 𝐹 bol − 𝐹 bol , 𝐹 bol , × = 𝛿𝐿𝐿 . (1)This is essentially equivalent to the relative variation in luminosity 𝛿𝐿 / 𝐿 (in ppm) because oscillations hardly change the total stellarradius. The subscript “0” indicates time-averaged quantities, i.e. theequilibrium state.The power spectra (PS) of the vertical velocity variation and therelative luminosity variation (luminosity spectrum for short here- MNRAS , 1–16 (2021)
Y. Zhou et al. inafter) are computed viaPS [ 𝑓 ]( 𝜔 ) = 𝑁 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 𝑁 − ∑︁ 𝑠 = 𝑓 ( 𝑡 𝑠 ) 𝑒 𝑖𝜔𝑠 Δ 𝑡 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) 𝜔 = 𝜋𝑁 Δ 𝑡 (cid:18) , , ..., 𝑁 − (cid:19) 𝑓 = ¯ 𝑉 𝑦 or 𝛿𝐿𝐿 . (2)Here, ¯ 𝑉 𝑦 is the horizontally averaged vertical velocity, and the terms 𝑡 , Δ 𝑡 and 𝜔 are time, time interval between two consecutive snap-shots and angular frequency, respectively. The symbol 𝑠 denotesindividual simulation snapshots, and 𝑁 is the total number of snap-shots.Fig. 1 shows the results for the solar and the red giant sim-ulations. Three radial simulation modes with frequencies of ap-proximately 2.1, 3.3, and 4.7 mHz are seen in the vertical velocityspectrum of the solar simulation. These are the fundamental, firstovertone, and second overtone radial modes in the simulation box,respectively. Among the three, only the intermediate-frequency, firstovertone simulation mode is clearly recognizable from the luminos-ity spectrum. The reason is that at low frequencies, the granulationsignal is relatively strong, causing the signature of the low-frequencysimulation mode to be overwhelmed by “convective noise.” On theother hand, the amplitude of the high-frequency simulation mode istoo small to be clearly identified in the luminosity spectrum (blackline in Fig. 1(a)). We therefore refer to the first overtone radial modeas the dominant simulation mode, as it is the only one that is iden-tifiable in both the vertical velocity and luminosity spectra in thesolar case. The situation for the 𝜖 Tau simulation is different. Thethree simulation modes are visible in both the velocity and luminos-ity spectra owing to their large amplitude. We note that for both theSun and 𝜖 Tau, the duration of the simulation is long enough to coverat least 200 periods of the dominant simulation mode. Likewise, thesampling interval is short enough in both cases such that at least10 snapshots are stored within one pulsation cycle of the dominantsimulation mode. These two factors together ensure that the domi-nant simulation mode is well resolved in the frequency domain. Theexact frequency of the dominant simulation mode is important forthe analysis below (Sect. 5 and 6). It is determined by looking for thelocal maximum ¯ 𝑉 𝑦 for all vertical layers in the simulation domain,which is similar to the method used in Belkacem et al. (2019). Theexact frequency values are 3.299 mHz and 0.051 mHz for the solarand 𝜖 Tau simulation respectively, which are also highlighted in reddotted lines in Fig. 1.It is worth noting that the amplitude of simulation mode ison the order of 100 m s − . This is much greater than the observedamplitude of radial 𝑝 -modes as measured in the solar flux spectrum,which is around 0.2 m s − . This difference in amplitude between thesimulation mode and the observed stellar 𝑝 -mode was explained indetail in Belkacem et al. (2019) and Zhou et al. (2019). In short,the discrepancy emerges from the difference between the volumeof the simulation box and the volume of the real star. Stellar 𝑝 -modes propagate throughout the entire stellar surface and interior,whereas the simulation modes are confined to the simulation boxwhose horizontal and vertical extents are significantly smaller thanthe dimensions of a star. Therefore, the luminosity or velocity ampli-tudes from 3D atmosphere simulations are not directly comparableto the corresponding asteroseismic observations. A natural ques-tion is then whether realistic ratio between luminosity and velocityamplitude can be predicted from our simulations? We address thisquestion in detail in the subsequent section. We demonstrate in this section that, in principle, 3D surface convec-tion simulations are able to reliably predict the relationship betweenthe luminosity and velocity amplitudes (the amplitude ratio) despitetheir individual values not being comparable with observations. Webegin with the relative luminosity variation defined in Eq. (1). As-suming the source function in the radiative transfer equation 𝑆 𝜈 ( 𝜈 is the radiation frequency) is a linear function of optical depth 𝜏 𝜈 (i.e. equivalent to the Eddington-Barbier approximation), thesurface flux at a given frequency is given by: 𝐹 𝜈 ( 𝜏 𝜈 = ) = 𝜋𝑆 𝜈 ( 𝜏 𝜈 = / ) . (3)Further assuming LTE gives 𝐹 𝜈 ( 𝜏 𝜈 = ) = 𝜋𝐵 𝜈 ( 𝜏 𝜈 = / ) = 𝜋 ℎ𝜈 𝑐 (cid:104) ℎ𝜈𝑘 𝐵 𝑇 ( 𝜏 𝜈 = / ) (cid:105) − . (4)Here, 𝐵 𝜈 is the Planck function, 𝑐 , ℎ and 𝑘 𝐵 are speed of light,Planck constant and Boltzmann constant, respectively. The term 𝑇 ( 𝜏 𝜈 = / ) is the temperature at optical depth 𝜏 𝜈 = /
3. Becauseat different frequencies, the 𝜏 𝜈 = / 𝑇 ( 𝜏 𝜈 = / ) depends on frequency in general. In the case of a greyatmosphere where optical depth has no frequency dependence, theintegration of 𝐹 𝜈 over frequency gives the Stefan-Boltzmann law.The bolometric flux is hence 𝐹 bol = 𝜋 ∫ ∞ 𝐵 𝜈 ( 𝜏 = / ) 𝑑𝜈 = 𝜎𝑇 ( 𝜏 = / ) , (5)where 𝜎 is the Stefan-Boltzmann constant. Combining Eqs. (1) and(5) yields 𝛿𝐿𝐿 = 𝛿𝑇 ( 𝜏 = / ) 𝑇 ( 𝜏 = / ) × , (6)where 𝛿𝑇 denotes temperature fluctuation at constant optical depth.From Eq. (6) we can then recognise that the luminosity variationessentially captures the fluctuation in temperature at the optical sur-face. For solar-type stars without strong stellar activity, such fluc-tuation is due primarily to surface convection and secondarily dueto acoustic oscillations. The contribution due to surface convectionwill be separated from the acoustic oscillations in Sect. 5.1.Next we connect fluid velocity 𝑉 with the fluctuations of ther-modynamical quantities. Following the discussion in Aerts et al.(2010) (see Chapter 3.1.4), we assume 𝑉 is caused solely by soundwaves and is small compared to the sound speed. It is worth notingthat convective velocities are non-negligible in stellar convectionzones; their magnitude can even be comparable to the local soundspeed in the near-surface region. Nevertheless, convective velocitiesare effectively regarded as “equilibrium state,” since oscillation isthe focus here. Under this assumption, density 𝜌 , pressure 𝑃 andtemperature 𝑇 can be written as 𝑓 = 𝑓 + 𝑓 (cid:48) , where 𝑓 (cid:48) is the smallEulerian perturbation (second and higher order terms are ignored).After further assuming that the medium is spatially homogeneous,all derivatives of equilibrium quantities vanish. The fluid continuityequation 𝜕𝜌𝜕𝑡 + ∇ · ( 𝜌 (cid:174) 𝑉 ) = Perturbations at constant geometric depth (or radius)MNRAS000
3. Becauseat different frequencies, the 𝜏 𝜈 = / 𝑇 ( 𝜏 𝜈 = / ) depends on frequency in general. In the case of a greyatmosphere where optical depth has no frequency dependence, theintegration of 𝐹 𝜈 over frequency gives the Stefan-Boltzmann law.The bolometric flux is hence 𝐹 bol = 𝜋 ∫ ∞ 𝐵 𝜈 ( 𝜏 = / ) 𝑑𝜈 = 𝜎𝑇 ( 𝜏 = / ) , (5)where 𝜎 is the Stefan-Boltzmann constant. Combining Eqs. (1) and(5) yields 𝛿𝐿𝐿 = 𝛿𝑇 ( 𝜏 = / ) 𝑇 ( 𝜏 = / ) × , (6)where 𝛿𝑇 denotes temperature fluctuation at constant optical depth.From Eq. (6) we can then recognise that the luminosity variationessentially captures the fluctuation in temperature at the optical sur-face. For solar-type stars without strong stellar activity, such fluc-tuation is due primarily to surface convection and secondarily dueto acoustic oscillations. The contribution due to surface convectionwill be separated from the acoustic oscillations in Sect. 5.1.Next we connect fluid velocity 𝑉 with the fluctuations of ther-modynamical quantities. Following the discussion in Aerts et al.(2010) (see Chapter 3.1.4), we assume 𝑉 is caused solely by soundwaves and is small compared to the sound speed. It is worth notingthat convective velocities are non-negligible in stellar convectionzones; their magnitude can even be comparable to the local soundspeed in the near-surface region. Nevertheless, convective velocitiesare effectively regarded as “equilibrium state,” since oscillation isthe focus here. Under this assumption, density 𝜌 , pressure 𝑃 andtemperature 𝑇 can be written as 𝑓 = 𝑓 + 𝑓 (cid:48) , where 𝑓 (cid:48) is the smallEulerian perturbation (second and higher order terms are ignored).After further assuming that the medium is spatially homogeneous,all derivatives of equilibrium quantities vanish. The fluid continuityequation 𝜕𝜌𝜕𝑡 + ∇ · ( 𝜌 (cid:174) 𝑉 ) = Perturbations at constant geometric depth (or radius)MNRAS000 , 1–16 (2021) uminosity and velocity oscillation amplitudes then becomes 𝜕𝜌 (cid:48) 𝜕𝑡 + 𝜌 ∇ · (cid:174) 𝑉 = , (8)while the equation of motion 𝜌 𝜕 (cid:174) 𝑉𝜕𝑡 + 𝜌 (cid:174) 𝑉 · ∇ (cid:174) 𝑉 = −∇ 𝑃 + 𝜌 (cid:174) 𝑔 (9)becomes 𝜌 𝜕 (cid:174) 𝑉𝜕𝑡 = −∇ 𝑃 (cid:48) + 𝜌 (cid:174) 𝑔 (cid:48) , (10)where 𝑔 is gravitational acceleration. If we ignore the perturbationto gravitational acceleration (cid:174) 𝑔 (cid:48) (i.e. Cowling approximation), thissimplifies to 𝜌 𝜕 (cid:174) 𝑉𝜕𝑡 + ∇ 𝑃 (cid:48) = . (11)Now taking the time derivative of Eq. (8) and making use of Eq. (11),we have 𝜕 𝜌 (cid:48) 𝜕𝑡 − ∇ 𝑃 (cid:48) = . (12)In the case of adiabatic oscillation, pressure and density fluctuationsare connected by 𝑃 (cid:48) = 𝑐 𝑠, 𝜌 (cid:48) , (13)where 𝑐 𝑠 = √︁ ( 𝜕𝑃 / 𝜕𝜌 ) ad is the adiabatic sound speed. SubstitutingEq. (13) into Eq. (12) gives the wave equation (Aerts et al. 2010Eq. 3.51): 𝜕 𝜌 (cid:48) 𝜕𝑡 − 𝑐 𝑠, ∇ 𝜌 (cid:48) = . (14)If we now consider a pure radial sound wave in which all quan-tities depend only on the 𝑦 -coordinate, then density and pressurefluctuation can be written as 𝜌 (cid:48) = 𝑎 cos ( 𝑘 𝑦 − 𝜔𝑡 ) ,𝑃 (cid:48) = 𝑐 𝑠, 𝑎 cos ( 𝑘 𝑦 − 𝜔𝑡 ) , (15)where 𝑎 and 𝑘 denote amplitude and wave number, respectively.The dispersion relation is therefore 𝜔 = 𝑐 𝑠, 𝑘 . (16)Based on Eqs. (11), (15) and the dispersion relation (16), the ex-pression of fluid velocity can be written as 𝑉 = 𝑐 𝑠, 𝜌 𝑎 cos ( 𝑘 𝑦 − 𝜔𝑡 ) . (17)Comparing Eq. (15) and Eq. (17) gives the relation between pressurefluctuation and fluid velocity: 𝑃 (cid:48) = 𝜌 𝑐 𝑠, 𝑉. (18)We recall that, for adiabatic oscillations, pressure and temperaturefluctuations are related via 𝑇 (cid:48) 𝑇 = ∇ ad , 𝑃 (cid:48) 𝑃 , (19)with ∇ ad = ( 𝜕 ln 𝑇 / 𝜕 ln 𝑃 ) ad being the adiabatic temperature gradi-ent. The relation between the temperature fluctuation and the fluidvelocity is given by 𝑇 (cid:48) = 𝜌 𝑐 𝑠, 𝑇 ∇ ad , 𝑃 𝑉. (20) For additional information about relevant discussion and deriva-tions, see Landau & Lifshitz (1987) chapter 64 and Aerts et al.(2010) chapter 3.1.4.We have now obtained the relationship between 𝛿𝐿 / 𝐿 and 𝛿𝑇 (the temperature fluctuation at constant optical depth), as wellas the relationship between 𝑉 and 𝑇 (cid:48) (the temperature fluctuationat constant geometric depth). The next step is to link these twotemperature fluctuations. Considering only first order perturbations,the temperature at given optical depth 𝜏 at any given time can beseparated as 𝑇 ( 𝜏, 𝑡 ) = 𝑇 ( 𝜏 ) + 𝛿𝑇 ( 𝜏, 𝑡 ) . (21)At fixed geometric depth near the photosphere, the optical depthvaries with time because of the time-dependent nature of convection.Therefore, the temperature at fixed geometric depth, if expressed asa function of 𝜏 , reads 𝑇 ( 𝜏 + 𝑑𝜏, 𝑡 ) = 𝑇 ( 𝜏, 𝑡 ) + 𝜕𝑇𝜕𝜏 𝑑𝜏 = 𝑇 ( 𝜏 ) + 𝛿𝑇 ( 𝜏, 𝑡 ) + 𝜕𝑇𝜕𝜏 𝑑𝜏. (22)Because 𝑇 ( 𝜏 ) represents the equilibrium state, the Eulerian pertur-bation to temperature is therefore 𝑇 (cid:48) ( 𝜏, 𝑡 ) = 𝑇 ( 𝜏 + 𝑑𝜏, 𝑡 ) − 𝑇 ( 𝜏 ) = 𝛿𝑇 ( 𝜏, 𝑡 ) + 𝜕𝑇𝜕𝜏 𝑑𝜏. (23)This equation demonstrates the relationship between two differentkinds of perturbation, it can be expanded further by analysing theterm 𝑑𝜏 . In the equilibrium state, the optical depth is, by definition, 𝜏 = ∫ 𝑦 −∞ 𝛼 𝑑𝑦, (24)where 𝑦 is the geometric depth as before and 𝛼 is the mean absorp-tion coefficient. Recalling that at a given time 𝑡 , 𝜏 + 𝑑𝜏 correspondsto the same geometric depth 𝑦 , we have 𝜏 + 𝑑𝜏 = ∫ 𝑦 −∞ 𝛼 ( 𝑡 ) 𝑑𝑦. (25)Subtracting Eq. (24) from Eq. (25) gives 𝑑𝜏 = ∫ 𝑦 −∞ 𝛿𝛼 ( 𝑡 ) 𝑑𝑦, (26)which relates the perturbation of the absorption coefficient at con-stant 𝜏 to the change in optical depth at fixed geometric depth. Thevalue of the absorption coefficient, however, depends on the opac-ity source of the plasma in a complex way. As such, there is nosimple analytical function to describe the relationship between 𝛼 and the thermodynamical quantities. Nevertheless, given the factthat the H − opacity is the dominant source of opacity near the solarphotosphere, we adopt the simplification that the mass absorptioncoefficient consists only of H − opacity: 𝜅 H − (units cm g − ). Herewe adopt a power-law fit of 𝜅 H − (Hansen et al. 2004 Eq. 4.65), whichgives reasonable results in our range of interest (3000 (cid:46) 𝑇 (cid:46) − (cid:46) 𝜌 (cid:46) − g / cm ; hydrogen mass fraction of around0.7; metal mass fraction 0 . (cid:46) 𝑍 (cid:46) . 𝜅 H − (cid:39) . × − ( 𝑍 / . ) 𝜌 / 𝑇 cm / g ,𝛼 (cid:39) 𝜌𝜅 H − (cid:39) . × − ( 𝑍 / . ) 𝜌 / 𝑇 cm − . (27)The perturbation of the absorption coefficient is then 𝛿𝛼 (cid:39) 𝛼 (cid:18) 𝛿𝜌 𝜌 + 𝛿𝑇𝑇 (cid:19) , (28)where 𝛿𝜌 is the perturbation of density at fixed optical depth. Asindicated by 3D surface convection simulations, the magnitude of MNRAS , 1–16 (2021)
Y. Zhou et al. 𝛿𝜌 and 𝜌 (cid:48) are similar around photosphere (Fig. 3 of Magic et al.2013b). Hence, using also Eqs. (13), (18) and (20), we have 𝛿𝜌 (cid:39) 𝜌 (cid:48) = 𝑃 𝑐 𝑠, ∇ ad , 𝑇 𝑇 (cid:48) . (29)Substituting Eqs. (28) and (29) into Eq. (26) yields 𝑑𝜏 (cid:39) ∫ 𝑦 −∞ 𝛼 (cid:32) 𝑃 𝜌 𝑐 𝑠, ∇ ad , 𝑇 𝑇 (cid:48) + 𝑇 𝛿𝑇 (cid:33) 𝑑𝑦. (30)As the absorption coefficient 𝛼 increases rapidly when movingfrom the upper atmosphere to photosphere (Eq. (27)), the maincontribution to the right hand side of Eq. (30) comes from a thinlayer just above 𝑦 . Therefore, Eq. (30) can be approximated by 𝑑𝜏 (cid:39) 𝜏 (cid:32) 𝑃 𝜌 𝑐 𝑠, ∇ ad , 𝑇 𝑇 (cid:48) + 𝑇 𝛿𝑇 (cid:33) 𝜏 . (31)Under the assumption of the Eddington grey atmosphere and LTE,the temperature stratification is 𝑇 ( 𝜏 ) = 𝑇 eff (cid:18) 𝜏 + (cid:19) , (32)which is the so-called Eddington 𝑇 − 𝜏 relation. Making use of theEddington 𝑇 − 𝜏 relation and plugging Eq. (31) into Eq. (23), wehave 𝑇 (cid:48) (cid:39) 𝛿𝑇 + 𝑇 eff , (cid:18) 𝜏 + (cid:19) − 𝜏 (cid:32) 𝑃 𝜌 𝑐 𝑠, ∇ ad , 𝑇 𝑇 (cid:48) + 𝑇 𝛿𝑇 (cid:33) 𝜏 . (33)Evaluating the equation above at 𝜏 = / 𝑇 (cid:48) and 𝛿𝑇 at the optical surface: (cid:32) − 𝑃 𝜌 𝑐 𝑠, ∇ ad , (cid:33) 𝜏 = 𝑇 (cid:48) ( 𝜏 = / , 𝑡 ) (cid:39) 𝛿𝑇 ( 𝜏 = / , 𝑡 ) . (34)The ratio between 𝑇 (cid:48) and 𝛿𝑇 at 𝜏 = / 𝛿𝐿𝐿 (cid:39) × (cid:18) 𝜌 𝑐 𝑠, ∇ ad , 𝑃 − 𝑐 𝑠, (cid:19) 𝜏 = 𝑉 ( 𝜏 = / ) . (35)Eq. (35) demonstrates that, to first order, the ratio between rela-tive luminosity variation and photosphere velocity depends only onthe equilibrium state of the thermodynamic quantities. We are nowequipped to address the question put forward at the end of Sect. 3:it is thus demonstrated that 3D surface convection simulations dohave the potential to reliably predict the luminosity and velocityamplitude ratio, because the ratio does not depend on the luminos-ity or velocity amplitude nor on any other term that is subject tooverestimation by our box-in-a-star models.We note that a number of approximations and simplificationshave been employed when deriving Eq. (35). Itemized, these as-sumptions are: • The Eddington-Barbier approximation • Local thermodynamic equilibrium • Grey atmosphere • Convective velocities are regarded as the “equilibrium state”,such that fluid velocity 𝑉 consists only of an oscillation componentand is small compared to the sound speed • Spatially homogeneous medium • The Cowling approximation • Adiabatic oscillations • H − is the only source of opacity in the stellar photosphere,such that it can be represented by a power law 𝜅 H − ∝ 𝜌 / 𝑇 • The magnitude of 𝛿𝜌 and 𝜌 (cid:48) are similar around photosphereSome of these assumption, such as the grey atmosphere and spatiallyhomogeneous medium assumptions, are obviously not correct inthe near-surface regions. Therefore, the analysis above is only toillustrate that 3D simulations are capable of providing a reliableluminosity and velocity amplitude ratio. We emphasise, however,that Eq. (35) is not used to calculate the ratio between luminosityand velocity amplitude; rather, we evaluate luminosity variation andradial velocity directly from 3D simulations that do not rely on theseassumptions. Intrinsic bolometric flux is an output quantity from the 3D stellarsurface convection simulations. It is computed from the radiativetransfer calculations performed at each time step of the simulation(Sect. 3). The theoretical bolometric flux as a function of time, whichis analogous to the intrinsic light curve of star, is converted to 𝛿𝐿 / 𝐿 according to Eq. (1). This is then transformed to the (oscillation)frequency domain using a Lomb-Scargle Periodogram algorithm(Lomb 1976; Scargle 1982) to obtain the luminosity power spec-trum shown in Fig. 2 (grey lines). A general trend of the luminositypower spectra is that the luminosity power is higher at low frequen-cies and decreases with increasing frequency. In the solar case, apeak located around 3.3 mHz is clearly seen in the spectrum. Thisfeature is associated with surface convection (granulation), whereup- and downflows shift the location of the optical surface, produc-ing fluctuations in bolometric flux. The peak around 3.3 mHz iscaused by the main oscillation mode of the simulation box. Acous-tic waves naturally excited in the simulation domain periodicallychange the location of optical surface, leading to coherent varia-tions in bolometric flux. The variation due to granulation happenson all time-scales and thus provides the background signal in thepower spectrum; for an even longer time-sequence this granulationsignal becomes more and more smooth, making it easier to discernthe frequencies of the oscillation modes.In order to obtain the luminosity amplitude for the simulationmode, it is necessary to filter out the contribution from granula-tion. At a given spatial position near the photosphere, granulationemerges, evolves, and disappears with a typical time scale 𝑡 gran . Hav-ing the insight that granulation (essentially surface velocity field) isconstantly evolving, Harvey (1985) proposed that the autocorrela-tion function of stellar granulation can be described by exponentialfunction exp (− 𝑡 / 𝑡 gran ) . That is, the correlation between granulationat moments 𝑡 and 𝑡 + 𝑡 decreases exponentially with increasingtime interval. Because the autocorrelation of a signal correspondsto the Fourier transform of its power spectrum, the power spectrum MNRAS000
Y. Zhou et al. 𝛿𝜌 and 𝜌 (cid:48) are similar around photosphere (Fig. 3 of Magic et al.2013b). Hence, using also Eqs. (13), (18) and (20), we have 𝛿𝜌 (cid:39) 𝜌 (cid:48) = 𝑃 𝑐 𝑠, ∇ ad , 𝑇 𝑇 (cid:48) . (29)Substituting Eqs. (28) and (29) into Eq. (26) yields 𝑑𝜏 (cid:39) ∫ 𝑦 −∞ 𝛼 (cid:32) 𝑃 𝜌 𝑐 𝑠, ∇ ad , 𝑇 𝑇 (cid:48) + 𝑇 𝛿𝑇 (cid:33) 𝑑𝑦. (30)As the absorption coefficient 𝛼 increases rapidly when movingfrom the upper atmosphere to photosphere (Eq. (27)), the maincontribution to the right hand side of Eq. (30) comes from a thinlayer just above 𝑦 . Therefore, Eq. (30) can be approximated by 𝑑𝜏 (cid:39) 𝜏 (cid:32) 𝑃 𝜌 𝑐 𝑠, ∇ ad , 𝑇 𝑇 (cid:48) + 𝑇 𝛿𝑇 (cid:33) 𝜏 . (31)Under the assumption of the Eddington grey atmosphere and LTE,the temperature stratification is 𝑇 ( 𝜏 ) = 𝑇 eff (cid:18) 𝜏 + (cid:19) , (32)which is the so-called Eddington 𝑇 − 𝜏 relation. Making use of theEddington 𝑇 − 𝜏 relation and plugging Eq. (31) into Eq. (23), wehave 𝑇 (cid:48) (cid:39) 𝛿𝑇 + 𝑇 eff , (cid:18) 𝜏 + (cid:19) − 𝜏 (cid:32) 𝑃 𝜌 𝑐 𝑠, ∇ ad , 𝑇 𝑇 (cid:48) + 𝑇 𝛿𝑇 (cid:33) 𝜏 . (33)Evaluating the equation above at 𝜏 = / 𝑇 (cid:48) and 𝛿𝑇 at the optical surface: (cid:32) − 𝑃 𝜌 𝑐 𝑠, ∇ ad , (cid:33) 𝜏 = 𝑇 (cid:48) ( 𝜏 = / , 𝑡 ) (cid:39) 𝛿𝑇 ( 𝜏 = / , 𝑡 ) . (34)The ratio between 𝑇 (cid:48) and 𝛿𝑇 at 𝜏 = / 𝛿𝐿𝐿 (cid:39) × (cid:18) 𝜌 𝑐 𝑠, ∇ ad , 𝑃 − 𝑐 𝑠, (cid:19) 𝜏 = 𝑉 ( 𝜏 = / ) . (35)Eq. (35) demonstrates that, to first order, the ratio between rela-tive luminosity variation and photosphere velocity depends only onthe equilibrium state of the thermodynamic quantities. We are nowequipped to address the question put forward at the end of Sect. 3:it is thus demonstrated that 3D surface convection simulations dohave the potential to reliably predict the luminosity and velocityamplitude ratio, because the ratio does not depend on the luminos-ity or velocity amplitude nor on any other term that is subject tooverestimation by our box-in-a-star models.We note that a number of approximations and simplificationshave been employed when deriving Eq. (35). Itemized, these as-sumptions are: • The Eddington-Barbier approximation • Local thermodynamic equilibrium • Grey atmosphere • Convective velocities are regarded as the “equilibrium state”,such that fluid velocity 𝑉 consists only of an oscillation componentand is small compared to the sound speed • Spatially homogeneous medium • The Cowling approximation • Adiabatic oscillations • H − is the only source of opacity in the stellar photosphere,such that it can be represented by a power law 𝜅 H − ∝ 𝜌 / 𝑇 • The magnitude of 𝛿𝜌 and 𝜌 (cid:48) are similar around photosphereSome of these assumption, such as the grey atmosphere and spatiallyhomogeneous medium assumptions, are obviously not correct inthe near-surface regions. Therefore, the analysis above is only toillustrate that 3D simulations are capable of providing a reliableluminosity and velocity amplitude ratio. We emphasise, however,that Eq. (35) is not used to calculate the ratio between luminosityand velocity amplitude; rather, we evaluate luminosity variation andradial velocity directly from 3D simulations that do not rely on theseassumptions. Intrinsic bolometric flux is an output quantity from the 3D stellarsurface convection simulations. It is computed from the radiativetransfer calculations performed at each time step of the simulation(Sect. 3). The theoretical bolometric flux as a function of time, whichis analogous to the intrinsic light curve of star, is converted to 𝛿𝐿 / 𝐿 according to Eq. (1). This is then transformed to the (oscillation)frequency domain using a Lomb-Scargle Periodogram algorithm(Lomb 1976; Scargle 1982) to obtain the luminosity power spec-trum shown in Fig. 2 (grey lines). A general trend of the luminositypower spectra is that the luminosity power is higher at low frequen-cies and decreases with increasing frequency. In the solar case, apeak located around 3.3 mHz is clearly seen in the spectrum. Thisfeature is associated with surface convection (granulation), whereup- and downflows shift the location of the optical surface, produc-ing fluctuations in bolometric flux. The peak around 3.3 mHz iscaused by the main oscillation mode of the simulation box. Acous-tic waves naturally excited in the simulation domain periodicallychange the location of optical surface, leading to coherent varia-tions in bolometric flux. The variation due to granulation happenson all time-scales and thus provides the background signal in thepower spectrum; for an even longer time-sequence this granulationsignal becomes more and more smooth, making it easier to discernthe frequencies of the oscillation modes.In order to obtain the luminosity amplitude for the simulationmode, it is necessary to filter out the contribution from granula-tion. At a given spatial position near the photosphere, granulationemerges, evolves, and disappears with a typical time scale 𝑡 gran . Hav-ing the insight that granulation (essentially surface velocity field) isconstantly evolving, Harvey (1985) proposed that the autocorrela-tion function of stellar granulation can be described by exponentialfunction exp (− 𝑡 / 𝑡 gran ) . That is, the correlation between granulationat moments 𝑡 and 𝑡 + 𝑡 decreases exponentially with increasingtime interval. Because the autocorrelation of a signal correspondsto the Fourier transform of its power spectrum, the power spectrum MNRAS000 , 1–16 (2021) uminosity and velocity oscillation amplitudes × × × × Frequency ( µ Hz)10 P o w e r( pp m ) (a) Frequency ( µ Hz)10 P o w e r( pp m ) (b) Figure 2. 𝜇 Hz. Red solid line represents the granulation background of the solar simulation, as modelled based on Eq. (38). A Lorentzianfit to the dominant simulation mode is also shown in red dashed line. 2(b): Luminosity power spectrum and granulation background predicted from our 𝜖 Tausimulation. The black curve is obtained by smoothing the luminosity power spectrum with a running mean with width equals to 10 𝜇 Hz. of granulation background is the Fourier transform of
V ( 𝑡 ) = (cid:40) V 𝑒 − 𝑡 / 𝑡 gran ( 𝑡 ≥ ) ( 𝑡 < ) , (36)which is B( 𝜈 ) = 𝐶 V 𝑡 gran + ( 𝜋𝜈𝑡 gran ) . (37)This is a Lorentzian function. The term V is the velocity amplitudeassociated with granulation, 𝜈 is cyclic frequency and 𝐶 is a normal-ization constant such that the power spectrum satisfies the Parsevaltheorem (see also Eq. 5 of Lund et al. 2017). In recognition of this,we modelled the oscillation background caused by granulation withthe sum of one or more (generalized) Lorentzian profiles: B( 𝜈 ) = 𝑁 ∑︁ 𝑖 = √ 𝜋 𝑎 ,𝑖 / 𝑎 ,𝑖 + ( 𝜈 / 𝑎 ,𝑖 ) 𝑎 ,𝑖 , (38)which is similar to the functional forms commonly applied to realobservational data (e.g. Lund et al. 2017; Li et al. 2020). Here,the free parameters are { 𝑎 ,𝑖 , 𝑎 ,𝑖 , 𝑎 ,𝑖 } and 𝑁 is the number ofLorentzian components. As multiple Lorentzian components areoften employed to achieve a better fit to the observed stellar back-ground, the interpretation is that in real stars, there exists more thanone granulation scale as well as contributions from stellar activity.Nevertheless, the true analytical form of stellar granulation back-ground remains elusive (Lundkvist et al. 2020). Therefore, we testgranulation background models with one, two and three compo-nents ( 𝑁 = , ,
3) by computing the Bayesian evidence (marginallikelihood) for each model. The Bayesian evidence 𝑝 ( 𝐷 |M) , whichis the probability of the power spectrum data 𝐷 given a granulationmodel M , is frequently used to evaluate the relative probability ofthe models for given data. We find that for both the Sun and 𝜖 Tau, theBayesian evidence of the single-component background model issignificantly smaller than multi-component models while 𝑝 ( 𝐷 |M) of 𝑁 = 𝑁 = Strictly speaking, Eq. (38) is the sum of generalized Lorentzian profilesbecause 𝑎 ,𝑖 is a free parameter rather than fixed to 2. Here we refer themas Lorentzian profiles for simplicity. Bayesian approach therefore shows that given the theoretical lumi-nosity spectrum, the granulation background is better described bymulti-component Lorentzian profiles. In this study, we choose thetwo-component ( 𝑁 =
2) model, because it performs equally well asthe three-component one but involves fewer free parameters. We re-fer the readers to Lundkvist et al. (2020) for a detailed examinationof different granulation background models against results from 3Dsurface convection simulations.The granulation background fitting is performed with the par-allel tempering MCMC (Markov chain Monte Carlo) algorithm ofVousden et al. (2016). The best-fitting results are demonstratedin red solid lines in Fig. 2. The amplitude of the granulationbackground at the frequency of the dominant simulation mode is343 . ± . . ± . 𝜖 Tau case. Uncertainties presented here are returned from theMCMC samples, representing the statistical errors associated withthe background fitting. We then subtracted the power spectrum withthe best fitting background. The bolometric oscillation amplitudeis determined by taking the square root of the peak value (corre-sponds to the value at the frequency of the dominant simulationmode) of the subtracted spectrum, which is 1714 . ± . . ± . 𝜖 Tau (also tabulatedin Table 4). We note that another frequently used method to extractthe oscillation amplitude is to fit a parametric model to the peakregion, then take the peak value of the fitted curve. However, inour solar simulation, the width of the simulation mode, which isconnected to the mode damping rate, is on the order of 10 − 𝜇 Hz (see also Table 1 of Belkacem et al. 2019), being much largerthan the width of solar 𝑝 -modes. The reason is that the simulationsmodes have much less mode mass than stellar 𝑝 -modes. Therefore,to avoid further complications about the reliability of the width, wemeasure only the peak amplitude at the frequency of the dominantsimulation mode in our analysis. Space-based missions such as CoRoT,
Kepler and TESS measurestarlight in a certain band-pass. What is measured from these space-based observations is stellar flux in certain wavelength ranges, rather
MNRAS , 1–16 (2021)
Y. Zhou et al. F l u x [ e r g s c m Å ] Sun Binned observationBinned simulationRaw simulationSpectral response function3000 4000 5000 6000 7000 8000 9000 10000Wavelength [Å]0.00.10.20.30.40.50.60.7 F l u x [ e r g s c m Å ] Tau Binned observationBinned simulationRaw simulationSpectral response function
Figure 3.
Upper panel : Thin grey line is the emergent flux between 3500 and 9500 Å computed based on one example snapshot of our 3D solar modelatmosphere (at a resolution of 15 km s − ). The re-sampled synthetic spectrum using a 5 Å wavelength bin is depicted with the black line. The observed highresolution solar flux spectrum of Kurucz (2005) is also binned every 5 Å (blue line) in order to facilitate comparison between simulation and observation. Reddash-dotted line is the mean spectral response function of Kepler averaged from its 84 channels (Thompson et al. 2016).
Lower panel : Similar to the upperpanel, but the result for 𝜖 Tau. Because the observed 𝜖 Tau spectrum of Valdes et al. (2004) lacks an absolute flux level, we normalise the original observationdata such that its magnitude generally matches the 4976 K (the reference effective temperature of 𝜖 Tau) black body spectrum between 7000 and 9000 Å. Wenote that data shown in this figure is not used to calculate 𝑐 𝑃 − bol . Instead, the synthetic spectra that enter into Eq. (40) have a lower wavelength resolution(1 nm, see text). Table 2.
Conversion factor 𝑐 𝑃 − bol computed for Kepler for the Sun and 𝜖 Tau. “3D” , “ATLAS9” and “Planck” represent the choice of model atmo-sphere (3D model atmosphere in this work, ATLAS9 model atmosphere,and black body respectively) applied in the computation. The “ATLAS9”and “Planck” results are obtained by interpolating the data provided in Lund(2019). Star 3D ATLAS9 PlanckSun 0.92 0.94 0.98 𝜖 Tau 0.68 0.79 0.87 than the intrinsic bolometric stellar flux. In order to connect theluminosity amplitude provided by 3D simulations with observables,it is necessary to quantify the conversion factor between intrinsic andmeasured luminosity amplitude (also called bolometric correction factor for luminosity amplitude) which is defined as 𝐴 bol = 𝑐 𝑃 − bol 𝐴 𝑃 , (39)where 𝑐 𝑃 − bol is the conversion factor. The term 𝐴 bol and 𝐴 𝑃 areintrinsic luminosity amplitude and luminosity amplitude measuredin a certain band-pass respectively. The conversion factor was firstinvestigated by Michel et al. (2009) for CoRoT and Ballot et al.(2011) for Kepler . Recently, Lund (2019) quantified 𝑐 𝑃 − bol valuesfor CoRoT, Kepler and TESS across the HR diagram based on a gridof ATLAS9 model fluxes (Castelli & Kurucz 2003). Here we followtheir theoretical formulation, but we calculate the flux spectrumfrom our 3D models to obtain self-consistent conversion factors forthe two stars investigated in this work. For radial oscillations, theexpression of 𝑐 𝑃 − bol , as given in Lund (2019), is 𝑐 𝑃 − bol = ∫ T 𝑃 ( 𝜆 ) 𝐹 ( 𝜆 ) 𝑑𝜆𝑇 eff ∫ T 𝑃 ( 𝜆 ) 𝜕𝐹 ( 𝜆 ) 𝜕𝑇 eff 𝑑𝜆 , (40) MNRAS000
Conversion factor 𝑐 𝑃 − bol computed for Kepler for the Sun and 𝜖 Tau. “3D” , “ATLAS9” and “Planck” represent the choice of model atmo-sphere (3D model atmosphere in this work, ATLAS9 model atmosphere,and black body respectively) applied in the computation. The “ATLAS9”and “Planck” results are obtained by interpolating the data provided in Lund(2019). Star 3D ATLAS9 PlanckSun 0.92 0.94 0.98 𝜖 Tau 0.68 0.79 0.87 than the intrinsic bolometric stellar flux. In order to connect theluminosity amplitude provided by 3D simulations with observables,it is necessary to quantify the conversion factor between intrinsic andmeasured luminosity amplitude (also called bolometric correction factor for luminosity amplitude) which is defined as 𝐴 bol = 𝑐 𝑃 − bol 𝐴 𝑃 , (39)where 𝑐 𝑃 − bol is the conversion factor. The term 𝐴 bol and 𝐴 𝑃 areintrinsic luminosity amplitude and luminosity amplitude measuredin a certain band-pass respectively. The conversion factor was firstinvestigated by Michel et al. (2009) for CoRoT and Ballot et al.(2011) for Kepler . Recently, Lund (2019) quantified 𝑐 𝑃 − bol valuesfor CoRoT, Kepler and TESS across the HR diagram based on a gridof ATLAS9 model fluxes (Castelli & Kurucz 2003). Here we followtheir theoretical formulation, but we calculate the flux spectrumfrom our 3D models to obtain self-consistent conversion factors forthe two stars investigated in this work. For radial oscillations, theexpression of 𝑐 𝑃 − bol , as given in Lund (2019), is 𝑐 𝑃 − bol = ∫ T 𝑃 ( 𝜆 ) 𝐹 ( 𝜆 ) 𝑑𝜆𝑇 eff ∫ T 𝑃 ( 𝜆 ) 𝜕𝐹 ( 𝜆 ) 𝜕𝑇 eff 𝑑𝜆 , (40) MNRAS000 , 1–16 (2021) uminosity and velocity oscillation amplitudes where 𝜆 is wavelength. Note that the stellar spectral flux 𝐹 ( 𝜆 ) alsodepends on basic stellar parameters i.e. 𝑇 eff , log 𝑔 and [Fe/H]. Theinstrumental transfer function T 𝑃 ( 𝜆 ) is connected with the spectralresponse function S 𝜆 via T 𝑃 ( 𝜆 ) = S 𝜆 /( ℎ𝑐 / 𝜆 ) , where the latterrepresents the band-pass of the instrument. In this study we choosethe Kepler spectral response function as an example.The stellar spectral flux 𝐹 ( 𝜆 ) in Eq. (40) is computed usingthe 3D radiative transfer code scate (Hayek et al. 2011). scatesolves the 3D, time-dependent radiative transfer problem for bothspectral lines and the background continuum. The computation iscarried out under the LTE assumption, but with the ability to includeisotropic continuum scattering in opacities and source functions.Simulation snapshots generated from the Stagger code are inputmodels in scate. The equation-of-state, continuum absorption andscattering coefficients, and the pre-tabulated line opacities adoptedin scate are identical to those used in Stagger, thereby ensuringfull consistency between the 3D surface convection simulations and3D LTE line formation calculations. For more information aboutthe code and the numerical method therein, we refer the readers toHayek et al. (2010, 2011).Here we use the spectrum synthesis mode of scate. In this sce-nario, the code delivers the angle-resolved surface fluxes for manywavelength points while treating scattering as pure absorption; testcalculations reveal that for our target stars and for the wavelengthsof interest here, this is an excellent approximation. At a given wave-length, continuum opacities are computed on-the-fly based on themicrophysics and 3D atmosphere models mentioned above; lineopacities are read from the pre-tabulated opacity sampling data.Specific intensities are calculated by tilting the simulation domainto represent five different polar angles 𝜃 , and rotated to yield fourequidistant azimuthal angles 𝜙 , for a total of 20 rays. The emergentflux is finally computed through integration of the specific inten-sities on a Gauss-Legendre quadrature in the polar direction, andtrapezoidal integration in the azimuthal direction. We compute theemergent flux between 350 and 950 nm, covering the entire spectralresponse function of Kepler , in steps of 1 nm. In order to reducethe computational cost, we compute the spectral energy distributiononly for a subset of the simulation sequence covering four periodsof the dominant simulation mode. This corresponds to 40 snapshotsin the solar simulation and 55 snapshots in the red giant simulation.The spectral flux distributions computed based on one examplesnapshot of our 3D solar and 𝜖 Tau model atmosphere are depictedin Fig. 3, together with the spectral response function S 𝜆 for Kepler which is taken from Thompson et al. (2016). From Fig. 3 we cansee that for both stars, the predicted spectral flux distributions agreereasonably well with observations, both in magnitude and in overalltrend.The time-averaged (average over all selected snapshots) spec-tral flux 𝐹 ( 𝜆 ) is used to calculate 𝑐 𝑃 − bol through Eq. (40). Thederivative term 𝜕𝐹 ( 𝜆 )/ 𝜕𝑇 eff can be evaluated from the same sim-ulation snapshots. More specifically, by synthesising the flux spec-trum for each individual simulation snapshot, we can obtain 𝐹 ( 𝜆 ) asa function of 𝑇 eff because different simulation snapshots correspondto different bolometric flux, hence effective temperature. This thenpermits the numerical evaluation of 𝜕𝐹 ( 𝜆 )/ 𝜕𝑇 eff at the referenceeffective temperature of the star. The 𝑐 𝑃 − bol values computed basedon the aforementioned simulation configuration are presented in Ta-ble 2. In the case of 𝜖 Tau, the conversion factor calculated fromthe 3D model is 0.68, more than 10% less than the correspondingATLAS9 result given in Lund (2019). We have verified for boththe Sun and 𝜖 Tau that our 𝑐 𝑃 − bol value is robust, as (1) increas-ing the number of polar angles in the radiative transfer calculation from five to ten has negligible effect on the final 𝑐 𝑃 − bol value; and(2) neither increasing the wavelength resolution from 1 nm to 1Å nor doubling the simulation time sequence in the flux spectrumcalculation changes the outcome. Different model atmospheres is alikely cause of the discrepancy in 𝑐 𝑃 − bol , as 𝑐 𝑃 − bol computed fromthe black body spectrum also differs clearly from the one computedfrom ATLAS9, as seen in Table 2.It is worth noting that in principle the conversion factor 𝑐 𝑃 − bol is not strictly a constant because the flux emitted by a star fluctuateswith time. The fluctuation is caused by stellar granulation and os-cillation for solar-type stars with the contribution from granulationgenerally being much larger (Kallinger et al. 2014). According toKallinger et al. (2014), the measured solar bolometric granulationamplitude is 𝐴 gran =
41 ppm, corresponding to a fluctuation of ap-proximately 0.06 K in effective temperature. Assuming a black bodyspectrum, the 0.06 K fluctuation in 𝑇 eff will result in a relative changeof less than 10 − in 𝑐 𝑃 − bol . For 𝜖 Tau, the bolometric granulationamplitude, estimated from the scaling relation 𝐴 gran ∝ ( 𝑔 𝑀 ) − / (Kallinger et al. 2014 Eq. 5), is roughly 250 ppm, corresponding toa ∼ . 𝑐 𝑃 − bol of less than 10 − . Therefore, regarding the conversion factor as aconstant is a suitable approximation. In the spectroscopic method of measuring stellar oscillations, vari-ations in the radial (i.e. line-of-sight) velocity near the stellar pho-tosphere are quantified by analysing the Doppler shift of certainspectral lines. Two representative efforts based on this method arethe Birmingham Solar Oscillations Network (BiSON, Chaplin et al.1996) and Stellar Oscillations Network Group (SONG, Grundahlet al. 2006). As implied by its name, BiSON focus solely on he-lioseismology: it detects solar oscillations using the Doppler shiftof the disk-integrated solar potassium (K i) 7698 Å line. The spec-trograph operates by imposing a magnetic field on a sample ofpotassium gas; the anomalous Zeeman effect produces line splittingwhere the 𝜎 − and 𝜎 + line components located in the blue and redwings of the solar K i line exhibit circular polarisation with oppo-site orientation. Incident sunlight fed through the instrument passesthrough a linear polarizer and a quarter-wave plate, which inducescircular polarization that can be rapidly switched between left- andright-handedness in order to produce resonant scattering with ei-ther line component. This allows the measurement of the relativeintensity between the blue and red wing, which reflects the Dopplershift of solar K i line that is caused by the velocity field of the solarsurface. A thorough explanation of the BiSON instrumentation andobservation technique can be found in Chaplin et al. (1996).In contrast, SONG measures the radial velocity signal simul-taneously from a large number of spectral lines through a tra-ditional echelle spectrograph that covers the wide spectral range4400–6900 Å (Grundahl et al. 2017). The incident starlight passesthrough an iodine cell, which superimposes on the stellar spectruma large number of weak absorption lines; these act as a highly ac-curate simultaneously recorded wavelength reference. The overallDoppler shift is inferred by cross-correlating the observed spectrumwith a reference spectrum that was recorded without the iodine cell.Because radial velocity quantified in this manner takes many spec-tral lines into account, the result reflects the mean velocity of thephotosphere rather than the velocities at the specific heights wherecertain lines are formed. A more detailed description of the iodinecell method is given by Butler et al. (1996), and its application to MNRAS , 1–16 (2021) Y. Zhou et al. N o r m a li z e d f l u x Observation
Figure 4.
Spatially averaged K i line profiles predicted from our 3D non-LTEline formation calculations. The line formation calculation is done for everysimulation snapshot, but only one in every 100 snapshots are shown here toavoid over-crowded figure. The observed solar K i line profile is plotted inblack line for comparison (Neckel 1999). The effects of gravitational redshift(633 m s − for the Sun, Dravins 2008) are included in both the theoretical andthe observed line profiles. The blue and red Gaussian profiles schematicallyrepresent the two laboratory potassium lines used in BiSON, whose centralwavelengths are marked by vertical dotted lines. P S [][ m / s ] Figure 5.
The power spectrum of radial velocity amplitude, with three peakscorrespond to three simulation modes. The power spectrum here should notbe confused with the grey line in Fig. 1(a), the plot here reflects radialvelocity variation in the K i line forming regions while the latter is thefluctuation of vertical component of fluid velocity near the photosphere.
SONG is described in detail by Antoci et al. (2013). To make the-oretical predictions as consistent as possible with observations, weextract the radial velocity from our 3D atmosphere models throughline formation calculations with a method resembling the BiSONand SONG observational setups in the following subsections.
Although 3D hydrodynamic simulations are able to realisticallypredict the convective velocities throughout the simulation domain(e.g. Asplund et al. 2000; Nordlund et al. 2009; Pereira et al. 2013),what BiSON measures is a radial velocity as imprinted on a par-ticular spectral line, which is connected but not equivalent to thefluid velocity given by 3D simulations. In practice, spectral linesform over a range of atmospheric heights, and velocity fields arethus imprinted to varying extent on the core and wings (see e.g. Asplund et al. 2000; Chiavassa et al. 2018). We therefore opt tocarry out a forward-modelling approach by performing line for-mation calculations for the solar K i line. Due to the pronounceddepartures from LTE for the K i 7698 Å resonance line (Bruls et al.1992; Reggiani et al. 2019), it is crucial to carry out full 3D non-LTE radiative transfer computations to obtain realistic atmosphericvelocity information.The line formation calculations are performed using balder(Amarsi et al. 2016a,b, 2018), a 3D non-LTE radiative transfer codebased on the multi3d code (Botnen & Carlsson 1999; Leenaarts& Carlsson 2009). Our model atom contains 29 levels of K i plusthe K ii ground state, and resolves the fine structure in all doubletsof K i. Atomic energy levels and oscillator strengths originate fromNIST (Sansonetti 2008), and collisional line broadening is com-puted following the method of Barklem et al. (1998). We implementtransition rates due to collisions with electrons and hydrogen atomsfollowing Reggiani et al. (2019). For radiative transitions our simpli-fied atom considers only the 7664–7698 Å resonance line doublet,as their departure from LTE is almost entirely explained throughphoton losses in the resonance lines themselves which producesa characteristic sub-thermal source function that deepens the line(Reggiani et al. 2019). This simplification was a necessary trade-offto obtain line profiles across the entire simulation time series. Inany case, test calculations indicated that this two-line atom differsfrom a comprehensive K i model atom with 134 levels and 250bound-bound radiative transitions (Reggiani et al. 2019) by about1 % in the depth of the 7698 Å line and with negligible differencesin inferred radial velocities. Moreover, we rescaled the hydrody-namic simulations from a resolution of 240 to 120 × − .We adopted a solar abundance 𝐴 ( K ) = .
10 as this producedgood agreement with observations of the K i 7698 Å line, but did notfine-tune this value. Figure 4 demonstrates the spatially averaged,disk integrated K i line profiles computed with balder, which are inexcellent agreement with the observed line profile. The calculationswere carried out for every snapshot in the 3D solar simulation (2880in total) to obtain the temporal evolution of the solar K i line. Thelocation and shape of the line varies from snapshot to snapshot asa consequence of varying line-of-sight velocity fields in the lineforming region.Radial velocities are extracted from our theoretical K i lines ina way that is fully consistent with the BiSON observational setup.Under a magnetic field of 0.2 T, which is approximately the mag-netic field strength imposed in the BiSON spectrometer (Brookeset al. 1978), the K i 7698 Å line is split into two components centredat 𝜆 − = . 𝜆 + = . − for the two laboratory lines. TheirGaussian profiles are schematically shown in Fig. 4. We computethe convolution between the spatially and temporally averaged the-oretical solar K i line profile and each of the laboratory lines: 𝐹 B = 𝑓 𝜆 ∗ 𝐺 𝜆 − 𝐹 R = 𝑓 𝜆 ∗ 𝐺 𝜆 + , (41) MNRAS000
10 as this producedgood agreement with observations of the K i 7698 Å line, but did notfine-tune this value. Figure 4 demonstrates the spatially averaged,disk integrated K i line profiles computed with balder, which are inexcellent agreement with the observed line profile. The calculationswere carried out for every snapshot in the 3D solar simulation (2880in total) to obtain the temporal evolution of the solar K i line. Thelocation and shape of the line varies from snapshot to snapshot asa consequence of varying line-of-sight velocity fields in the lineforming region.Radial velocities are extracted from our theoretical K i lines ina way that is fully consistent with the BiSON observational setup.Under a magnetic field of 0.2 T, which is approximately the mag-netic field strength imposed in the BiSON spectrometer (Brookeset al. 1978), the K i 7698 Å line is split into two components centredat 𝜆 − = . 𝜆 + = . − for the two laboratory lines. TheirGaussian profiles are schematically shown in Fig. 4. We computethe convolution between the spatially and temporally averaged the-oretical solar K i line profile and each of the laboratory lines: 𝐹 B = 𝑓 𝜆 ∗ 𝐺 𝜆 − 𝐹 R = 𝑓 𝜆 ∗ 𝐺 𝜆 + , (41) MNRAS000 , 1–16 (2021) uminosity and velocity oscillation amplitudes Table 3.
Selected parameters of fictitious Fe i lines. Here log ( gf ) values aredistributed in steps of 0.5. There are totally 49 lines covering typical Fe ilines in the Sun and 𝜖 Tau within the SONG wavelength range.Wavelength (Å) 𝐸 low (eV) log ( gf ) − − −
4, 0]4.5 [ − − − − −
3, 0]6500 2.5 [ − − −
3, 0] and subsequently the normalised flux difference R = 𝐹 B − 𝐹 R 𝐹 B + 𝐹 R , (42)where “*” is the convolution operator, 𝑓 𝜆 is the solar flux spectrumand 𝐺 𝜆 −(+) is the blue (red) component of laboratory potassiumline after Zeeman splitting . The normalised flux difference R isproportional to the radial velocity (Chaplin et al. 1996). In orderto quantify the proportionality constant, we translate the averagedK i line back and forth in velocity space in steps of 3 m s − . Theaforementioned calculation (Eqs. (41) and (42)) is repeated eachtime to obtain the velocity shift as a function of R , which is welldescribed by a linear function 𝔳 = . R − .
903 m s − , (43)over the interval [− , ] m s − . The slope of Eq. (43) is ingood agreement with the proportionality constant used in BiSON ,which is typically 3000 m s − (Chaplin et al. 1996), implying thatour synthesised line profile describes the solar K i line in a realisticway. The above procedure is identical to the means by which BiSONextracts radial velocity from the solar K i line. We therefore applyEqs. (41)-(43) to every snapshot in the 3D solar simulation. The thusevaluated radial velocity 𝔳 as a function of time has the same physicalmeaning as what BiSON measures. The fluctuation of radial velocityis mainly caused by radial oscillations in the simulation box. Wenote that temporal variations of granulation also contribute to avelocity fluctuation. However, by performing a horizontal averageover more than ten granules, the influence of granulation on velocityfluctuation largely cancels out (Asplund et al. 2000, Sect. 4.2).Finally, we apply a Fourier transform to the temporal evolution of 𝔳 into the frequency domain to obtain the radial velocity powerspectrum presented in Fig. 5. The radial velocity amplitude of thedominant simulation mode is 81.6 m s − at frequency 3.299 mHz. SONG is a high-resolution spectrograph that covers a broad wave-length range, 4400–6900 Å, containing thousands of spectral ab-sorption lines. For computational reasons, it is not feasible to per-form high-resolution 3D radiative transfer over such a broad wave-length range for our very long hydrodynamic time series. Instead, We note that the magnitude of 𝐺 𝜆 −(+) has no effect on our results, as itcancels out in Eq. (42) In practice, the diurnal change of the measured normalized flux difference R due to the rotation of the Earth is used to calibrate the proportionalityconstant. A third-order polynomial relation, calibrated on a daily basis, isused. we develop a simplified method that relies on computing syntheticspectra for representative lines covering a range of atomic line prop-erties.First, in order to have a macroscopic perception of absorptionlines in the wavelength region covered by SONG, we computed thestrength of every atomic absorption line between 4400 and 6900Å with the TurboSpectrum code (v15.1; Plez 2012) for the stellarparameters of the Sun and 𝜖 Tau, using an atomic linelist from theVienna Atomic Line Database (VALD, Ryabchikova et al. 2015).Selecting only lines with a line strength 𝑊 𝜆 / 𝜆 ≥ − (correspond-ing to an equivalent width 𝑊 𝜆 = 𝜆 = 𝜖 Tau. Of these,40 % are due to Fe i, and another 40 % are due to neutral species ofother Fe-peak elements. These numbers are in good agreement withthe list of lines identified over the same wavelength range in thespectrum of Arcturus by Hinkle et al. (2000). As the vast majorityof spectral lines in the optical region are due to the neutral speciesof iron or elements with similar electron structure to iron, we useFe i as a representative species.The problem now is to determine a set of Fe i lines that canreasonably simulate SONG observations. The strength and shapeof an absorption line is mainly governed by three parameters: thewavelength 𝜆 , which controls the background opacity due largely toH − , the excitation potential of the lower ionization state 𝐸 low , whichdetermines the population of the level in LTE and thus the numberof absorbers, and the oscillator strength log ( gf ) , which representsroughly the likelihood that a photon is absorbed by the line. Itis therefore necessary to select representative values of 𝜆 , 𝐸 low and log ( gf ) combinations, ensuring the corresponding artificial Fe ilines cover the properties of most observed Fe i lines. Firstly, wechoose three wavelength ranges: 4400–4600 Å, 5400–5600 Å, and6400–6600 Å. In a given wavelength region, we pick all Fe i lineswith equivalent width greater than 10 mÅ from the aforementionedsolar and 𝜖 Tau line lists. All selected lines from the two theoreticalline lists are then classified into different groups based on their 𝐸 low . For example, the thus determined typical 𝐸 low values for Fe ilines between 4400 and 4600 Å are 2.5, 3.5 and 4.5 eV. Next, for agiven 𝐸 low , we further select a set of log ( gf ) values that span theentire log ( gf ) range seen in the theoretical line list. The selectionprocedure gives 49 line parameter combinations listed in Table 3,which are reasonable representation of Fe i lines from the Sun and 𝜖 Tau between 4400 and 6900 Å.We carry out 3D LTE line formation calculations for these 49fictitious Fe i lines using the scate code (Hayek et al. 2011). In thesehigh-resolution calculations, we tabulate continuum opacities andphoton destruction probabilities for a set of temperatures and den-sitites and use these to compute the effects of continuum scatteringat run time. Line opacities are likewise evaluated at run time, takinginto account local velocity fields that produce Doppler shifts in theline profiles. Specific intensities are computed for 20 rays, along fivedifferent polar and four azimuthal angles; the former are distributedon a Gauss-Legendre quadrature and the latter are equidistant. Wecompute the spectral lines at high resolution, with a velocity step ofjust 40 m s − , over a range ±
10 km s − that samples the entire line.We perform these calculations on each snapshot from the simula-tions of the Sun and 𝜖 Tau, and note that in the case of 𝜖 Tau, the Feabundance used in the line formation calculation is adopted fromthe reference metallicity [ Fe / H ] = .
15. Example theoretical lineprofiles, i.e. the normalized emergent flux as a function of wave-length, computed from one simulation snapshot of the Sun and 𝜖 Tau are shown in Fig. 6. The theoretical lines in 𝜖 Tau are broader
MNRAS , 1–16 (2021) Y. Zhou et al. = 4500 Å, E low = 2.5 eV = 6500 Å, E low = 2.5 eV0.10.20.30.40.50.60.70.80.91.0 = 4500 Å, E low = 3.5 eV = 5500 Å, E low = 3.5 eV0.12 0.06 0.00 0.06 0.120.10.20.30.40.50.60.70.80.91.0 = 4500 Å, E low = 4.5 eV 0.12 0.06 0.00 0.06 0.12 = 5500 Å, E low = 4.5 eV 0.12 0.06 0.00 0.06 0.12 = 6500 Å, E low = 4.5 eV [Å] N o r m a li z e d f l u x log(gf) = -5.0log(gf) = -4.5 log(gf) = -4.0log(gf) = -3.5 log(gf) = -3.0log(gf) = -2.5 log(gf) = -2.0log(gf) = -1.5 log(gf) = -1.0log(gf) = -0.5 log(gf) = 0.0 Figure 6.
Spatially averaged Fe i line profiles of all line parameters tabulated in Table 3 computed from scate. Results from one example simulation snapshotare shown here for both the Sun (solid lines) and 𝜖 Tau (dashed lines). Reference wavelength 𝜆 and 𝐸 low are marked in the figures, and log ( gf ) values arecolour-coded as indicated in the legend. The top middle and middle right panels are left blank, because the corresponding { 𝜆, 𝐸 low } combinations are notselected as representative line parameters (see Table 3). than the corresponding solar lines for all line parameters, due to thelarger velocity field in red giant stars.The line formation calculation introduced above gives the timeevolution of all 49 fictitious Fe i lines, from which we can then ex-tract the corresponding radial velocity variation. As reference, weuse template spectra computed as the temporal averages from thetwo simulation sequences. For each { 𝜆, 𝐸 low , log ( gf )} combination,a certain theoretical Fe i line from a given snapshot is fitted to thetemplate line profile using a 𝜒 technique to obtain the wavelength shift Δ 𝜆 of this line. This method closely resembles the cross corre-lation technique often used in observational work (e.g. Butler et al.1996).The thus obtained radial velocity as a function of time is trans-lated to the frequency domain through a Fourier transform, whichgives radial velocity amplitude at the frequency of the dominantsimulation mode. Radial velocity amplitudes for the 49 lines rangefrom roughly 60 to 80 m s − in the solar case and 80–110 m s − inthe case of 𝜖 Tau, as shown in Fig. 7. The next question is how
MNRAS000
MNRAS000 , 1–16 (2021) uminosity and velocity oscillation amplitudes R a d i a l v e l o c i t y a m p li t u d e [ m / s ] Sun Tau
Figure 7.
Radial velocity amplitude at the frequency of the dominant sim-ulation mode evaluated from 49 fictitious Fe i lines are plotted against theequivalent width of these lines. Results from the solar and red giant simu-lations are shown in black dots and red asterisks, respectively. Linear fits tothese data points are presented in dashed lines. then to reliably determine a final radial velocity amplitude, giventhese 49 different values. Recall that strong absorption lines tendto form higher up in the stellar photosphere than weak lines (e.g.Rutten 2003), where the velocity fields are typically larger due tothe substantially smaller densities. It is therefore anticipated thatthe magnitude of radial velocity amplitude is correlated with thestrength of the line, and Fig. 7 indeed follows an approximatelylinear relationship between the radial velocity amplitude and equiv-alent width for the Fe i lines considered for both stars.We select from our theoretical line list every Fe i line inthe SONG spectral range with equivalent width between 10 and200 mÅ, and use our fitted linear relations to estimate the radial ve-locity variation amplitude for each line. We exclude weaker lines asthese are unlikely to significantly influence the radial velocity deter-mination in a real stellar spectrum. Very strong lines with 𝑊 𝜆 > 𝜖 Tau, respectively. Theensemble of estimated radial velocity amplitudes are averaged to afinal value, weighted by the equivalent width. The weighted averageis performed with the understanding that the signal to noise ratioof weak lines is typically smaller than stronger lines, meaning arelatively larger error and thus a smaller influence on the final result(see Fig. 2 of Antoci et al. 2013).Although the method developed here is not identical to themeans by which SONG determines radial velocity from the ob-served spectra, it simulates the SONG observations sufficiently well.First, the set of fictitious Fe i lines carefully chosen in this work isable to represent the properties of most Fe i lines seen between4400 and 6900 Å, which constitute a large part of all lines in thiswavelength interval. Second, the procedure to extract radial velocityfrom theoretical spectral lines is similar to how radial velocities aretypically obtained from observed spectra. Third, the evaluation ofour final radial velocity amplitude includes the information of manyspectral lines that span the whole range in observation. The majoruncertainty in our method is associated with the linear relationshipbetween radial velocity amplitude and equivalent width. Due to thecomplicated physical processes involved in spectral line formationin a 3D atmosphere (e.g. Asplund et al. 2000), it is difficult to quan-tify higher order effects beyond the linear relation between 𝔳 and Table 4.
Summary of predicted and observed oscillation amplitudes andamplitude ratios for the Sun and 𝜖 Tau. Here we emphasize again thatindividual oscillation amplitudes from 3D atmosphere simulations are notcomparable to the corresponding observations (cf. Sect. 3 and 4).Sun Modelling ObservationBolometric (ppm) 1714 . ± . . ± .
16 (a)BiSON (m s − ) 81.6 0 . ± .
007 (b)SONG (m s − ) 72 . ± . . ± .
004 (c)Bolometric/BiSON (ppm/[m s − ]) 21 . ± .
04 19 . ± . − ]) 23 . ± . . ± . 𝜖 Tau Modelling ObservationBolometric (ppm) 3070 . ± . K2 (ppm) 4515 . ± . . ± . − ) 93 . ± . . ± .
04 (d) K2 /SONG (ppm/[m s − ]) 48 . ± . . ± . 𝑊 𝜆 ; that is, the systematic uncertainty of the linear fitting. Never-theless, it is still illuminating to provide the statistical uncertainty.The statistical uncertainty is quantified using the bootstrap method.The data set considered here is the radial velocity amplitude andequivalent width of 49 fictitious Fe i lines. We conduct 10000 boot-strap samplings, that is, generating 10000 data sets each containing49 randomly sampled 𝔳 and 𝑊 𝜆 pairs. A linear regression betweenequivalent width and radial velocity is then performed for each re-sampled data set. For each fitting, we compute the equivalent widthweighted mean radial velocity amplitude for all selected Fe i lines.The bootstrap method therefore results in 10000 weighted meanradial velocity amplitudes, their mean and variance is the desiredfinal radial velocity amplitude and its statistical uncertainty, whichis 72 . ± . − for the 3D solar model and 93 . ± . − forthe 𝜖 Tau model.
Our results, together with the corresponding observations, are sum-marised in Table 4.The predicted ratio between luminosity and BiSON radialvelocity amplitude at approximately 3.3 mHz is 1714.0 ppm ÷ − ≈ − ]. Observationally, the measuredmaximum bolometric amplitude per radial mode for the Sun is3 . ± .
16 ppm according to Michel et al. (2009); the value pre-sented in their paper is multiplied by √ . ± . − (Kjeldsen et al. 2008). Therefore, the ampli-tude ratio determined from observation is approximately 19 . ± . − ], in good agreement with the result predicted by oursimulations. However, we caution that the above observed solaramplitude ratio is evaluated at the frequency of maximum power 𝜈 max , that is, 3.1 mHz for the Sun (Kjeldsen et al. 2008), whereasour theoretical result is obtained at the frequency of the dominantsimulation mode (3.3 mHz). The two values are hence not strictlycomparable because amplitude ratio depends on frequency in prin-ciple. Nevertheless, the frequency dependence is weak, especiallyfor frequencies near 𝜈 max , as shown in detailed asteroseismic obser-vations (for example, Fig. 13 of Arentoft et al. 2019). Therefore, asan initial effort to this topic, a single amplitude ratio value is likely MNRAS , 1–16 (2021) Y. Zhou et al. to be sufficient to describe the relationship between luminosity andradial velocity variation for a given star.The predicted ratio between luminosity and SONG radialvelocity amplitudes (at approximately 3.3 mHz) is 23 . ± . − ] for the Sun. We emphasise that the presented un-certainty reflects the combined statistical error of the granulationbackground fitting (Sect. 5.1) and the linear fit to radial velocities(Sect. 6.2). On the other hand, the measured solar maximum bolo-metric amplitude is 3 . ± .
16 ppm, whereas the maximum radialvelocity amplitude obtained from SONG observations of the Sunis 16 . ± . − (Fredslund Andersen et al. 2019). Together,the observed solar amplitude ratio is 21 . ± . − ], beingconsistent with our theoretical result. For 𝜖 Tau, the amplitude ra-tio measured by Arentoft et al. (2019) is 42 . ± . − ],which is the ratio between K2 luminosity amplitude and the SONGradial velocity amplitude. The intrinsic luminosity amplitude com-puted from our simulations is 3070 . ± . Kepler luminosity amplitude is 0.68 for 𝜖 Tau as quan-tified in Sect. 5.2. According to Eq. (39), the theoretical luminosityamplitude in the
Kepler band-pass turns out to be 4515 . ± . Kepler ( K2 )and SONG for 𝜖 Tau, which is 48 . ± . − ]. Again, wefind reasonable agreement between our theoretical calculations andthe observations.We note that the amplitude ratio estimated from the widelyused empirical relationship of Kjeldsen & Bedding (1995, theirEq. 5) is 23.2 ppm/[m s − ] for 𝜖 Tau, being significantly lower thanthe observed value by almost a factor of two. The good agreementbetween observation and our theoretical result based on detailedmodelling therefore shows great potential to accurately quantifythe relationship between luminosity and radial velocity amplitude,especially for red giant stars where the empirical amplitude ratiorelation may fail. Nonetheless, we are aware that our predictedratios are systematically larger than corresponding observations byabout 10%. The underlying reason for this small discrepancy is notentirely clear and will be investigated in future work.
In this work, we investigated the relationship between photometricand spectroscopic measurements of solar-like oscillations using 3Dradiative-hydrodynamical stellar atmosphere simulations with theStagger code. We used as test cases the Sun and the Hyades redgiant 𝜖 Tau. Our simulations provide realistic descriptions of fluidmotions from first principles, hence naturally yield compressibleeffects such as sound waves. Although sound waves emerging inthe simulation domain are analogous to 𝑝 -modes in solar-type os-cillating stars, the simulation modes have much larger oscillationamplitudes than observed stellar 𝑝 -modes due to the limited extentof simulation box. Therefore, we first analytically demonstrated that3D simulations are still able to reliably predict the ratio between lu-minosity and velocity amplitudes, despite the individual amplitudevalues not being comparable with observations.Having established the basis of our analysis, we computed thespectrum of luminosity variation based on bolometric fluxes pre-dicted by the state-of-the-art radiative transfer module in our sim-ulation. Contribution from the granulation background was mod-elled in a way similar to what is applied to real observations. Themodelled granulation background was then subtracted from the lu- minosity spectrum to obtain the intrinsic luminosity amplitude ofthe dominant simulation mode. To enable comparison with ampli-tudes measured with a given spacecraft, it was necessary to quantifythe conversion factor (also called bolometric correction) betweenthe intrinsic and measured luminosity amplitudes. We adopted thetheoretical formulation of Michel et al. (2009), Ballot et al. (2011)and Lund (2019) for the evaluation of the conversion factor, whosecomponents are consistently computed via 3D spectrum synthesisusing the code scate. As an initial step, we evaluated the conver-sion factor between intrinsic and Kepler luminosity amplitude forthe two stars studied in this work. For 𝜖 Tau, our result differs fromLund (2019) by roughly 10%, implying that the conversion factorsof red giant stars are likely to be sensitive to the choice of modelatmosphere.In turn, we have developed novel methods to simulate thespectroscopic measurement of stellar oscillations from numericalsimulations for the first time. Theoretical radial velocities are ob-tained from realistic spectral line formation calculations with 3Dtime-dependent atmosphere models as input. In order to simulateBiSON, which measures solar oscillations through the K i line,we performed detailed 3D non-LTE K i line formation calculationswith balder for our solar model atmosphere. The computed K iline profile is in excellent agreement with observation, its temporalevolution (that is, Doppler shift) gives radial velocities whose phys-ical meaning is identical to what measured by BiSON. In addition,we carried out 3D LTE line formation calculations for a large setof fictitious Fe i lines using scate to simulate SONG observationswhich determines radial velocities from a forest of absorption linesbetween 4400 and 6900 Å. The parameters of the chosen Fe i lineswere carefully selected such that their properties cover most linestypically seen in the Sun and a warm giant within the SONG wave-length range. For each selected line, radial velocities were extractedaccording to the Doppler shift of line profiles with method that re-semble observations. This procedure was repeated for all selectedlines, thereby giving rise to a set of independent radial velocityamplitudes. With the insight that the radial velocity amplitude com-puted from a certain line is correlated with its strength, we fit alinear function between radial velocity amplitude and equivalentwidth based on results from all selected fictitious lines. The linearrelation was further used to estimate radial velocity amplitudes forall visible lines within the SONG wavelength range in our theo-retical line list, which were subsequently reduced to a final radialvelocity amplitude value via weighted average.In concert, the calculations gave us the ratio between luminos-ity and radial velocity amplitude, which characterize the relationshipbetween photometric and spectroscopic measurement of stellar os-cillations. Given the 3D atmosphere simulations and line formationcalculations presented in this work, our approach to quantify theamplitude ratio is free from any empirical parameters that have tobe assumed or calibrated from observations. The ab initio natureof our numerical modelling therefore not only reveals the underly-ing physics behind asteroseismic observations but also enables anindependent comparison between theoretical results and observedamplitude ratio. For the Sun, our theoretical bolometric and Bi-SON ratio as well as bolometric and SONG ratio are comparedwith helioseismic observations with good agreements, thus validateour numerical approach. In the case of 𝜖 Tau, the predicted ratiobetween K2 and SONG amplitude matches corresponding obser-vations as well, which is particularly encouraging as the observedamplitude ratio of this star cannot be explained by the widely usedempirical amplitude ratio scaling relation. The good theoretical–observational consistency achieved for both the Sun and a red giant MNRAS000
In this work, we investigated the relationship between photometricand spectroscopic measurements of solar-like oscillations using 3Dradiative-hydrodynamical stellar atmosphere simulations with theStagger code. We used as test cases the Sun and the Hyades redgiant 𝜖 Tau. Our simulations provide realistic descriptions of fluidmotions from first principles, hence naturally yield compressibleeffects such as sound waves. Although sound waves emerging inthe simulation domain are analogous to 𝑝 -modes in solar-type os-cillating stars, the simulation modes have much larger oscillationamplitudes than observed stellar 𝑝 -modes due to the limited extentof simulation box. Therefore, we first analytically demonstrated that3D simulations are still able to reliably predict the ratio between lu-minosity and velocity amplitudes, despite the individual amplitudevalues not being comparable with observations.Having established the basis of our analysis, we computed thespectrum of luminosity variation based on bolometric fluxes pre-dicted by the state-of-the-art radiative transfer module in our sim-ulation. Contribution from the granulation background was mod-elled in a way similar to what is applied to real observations. Themodelled granulation background was then subtracted from the lu- minosity spectrum to obtain the intrinsic luminosity amplitude ofthe dominant simulation mode. To enable comparison with ampli-tudes measured with a given spacecraft, it was necessary to quantifythe conversion factor (also called bolometric correction) betweenthe intrinsic and measured luminosity amplitudes. We adopted thetheoretical formulation of Michel et al. (2009), Ballot et al. (2011)and Lund (2019) for the evaluation of the conversion factor, whosecomponents are consistently computed via 3D spectrum synthesisusing the code scate. As an initial step, we evaluated the conver-sion factor between intrinsic and Kepler luminosity amplitude forthe two stars studied in this work. For 𝜖 Tau, our result differs fromLund (2019) by roughly 10%, implying that the conversion factorsof red giant stars are likely to be sensitive to the choice of modelatmosphere.In turn, we have developed novel methods to simulate thespectroscopic measurement of stellar oscillations from numericalsimulations for the first time. Theoretical radial velocities are ob-tained from realistic spectral line formation calculations with 3Dtime-dependent atmosphere models as input. In order to simulateBiSON, which measures solar oscillations through the K i line,we performed detailed 3D non-LTE K i line formation calculationswith balder for our solar model atmosphere. The computed K iline profile is in excellent agreement with observation, its temporalevolution (that is, Doppler shift) gives radial velocities whose phys-ical meaning is identical to what measured by BiSON. In addition,we carried out 3D LTE line formation calculations for a large setof fictitious Fe i lines using scate to simulate SONG observationswhich determines radial velocities from a forest of absorption linesbetween 4400 and 6900 Å. The parameters of the chosen Fe i lineswere carefully selected such that their properties cover most linestypically seen in the Sun and a warm giant within the SONG wave-length range. For each selected line, radial velocities were extractedaccording to the Doppler shift of line profiles with method that re-semble observations. This procedure was repeated for all selectedlines, thereby giving rise to a set of independent radial velocityamplitudes. With the insight that the radial velocity amplitude com-puted from a certain line is correlated with its strength, we fit alinear function between radial velocity amplitude and equivalentwidth based on results from all selected fictitious lines. The linearrelation was further used to estimate radial velocity amplitudes forall visible lines within the SONG wavelength range in our theo-retical line list, which were subsequently reduced to a final radialvelocity amplitude value via weighted average.In concert, the calculations gave us the ratio between luminos-ity and radial velocity amplitude, which characterize the relationshipbetween photometric and spectroscopic measurement of stellar os-cillations. Given the 3D atmosphere simulations and line formationcalculations presented in this work, our approach to quantify theamplitude ratio is free from any empirical parameters that have tobe assumed or calibrated from observations. The ab initio natureof our numerical modelling therefore not only reveals the underly-ing physics behind asteroseismic observations but also enables anindependent comparison between theoretical results and observedamplitude ratio. For the Sun, our theoretical bolometric and Bi-SON ratio as well as bolometric and SONG ratio are comparedwith helioseismic observations with good agreements, thus validateour numerical approach. In the case of 𝜖 Tau, the predicted ratiobetween K2 and SONG amplitude matches corresponding obser-vations as well, which is particularly encouraging as the observedamplitude ratio of this star cannot be explained by the widely usedempirical amplitude ratio scaling relation. The good theoretical–observational consistency achieved for both the Sun and a red giant MNRAS000 , 1–16 (2021) uminosity and velocity oscillation amplitudes star suggested that our method of connecting the luminosity andradial velocity measurements of solar-like oscillations is robust, ef-fective, and likely applicable to a wide range of stellar parameters.This demonstrates great potential in the era of simultaneous obser-vations of stellar oscillation with both space-photometry (such asTESS) and ground-based spectroscopy (such as SONG).In the future we plan to extend our analysis to cover the pa-rameter space of solar-like oscillating stars across the HR diagram(i.e. dwarfs, subgiants and red giants), which will give amplituderatios as a function of basic stellar parameters. These theoreticalamplitude ratios can provide valuable insight to asteroseismic ob-servations by helping to determine whether a star is better observedin photometry or spectroscopy. Conversely, it is possible to deter-mine radial velocity oscillation amplitude from Kepler or TESS datathrough the theoretical amplitude ratio relation, thereby quantifyingthe oscillation part of the so-called “radial velocity jitter” (see Yuet al. 2018 for a pioneering study in this direction) which is of greatimportance in exoplanet science.
ACKNOWLEDGEMENTS
The authors are grateful to Remo Collet, Tim Bedding and SaskiaHekker for valuable comments and fruitful discussions. This projecthas been supported by the Australian Research Council (projectDP150100250 awarded to MA and LC). LC is the recipient ofan ARC Future Fellowship (project FT160100402). MJ was alsosupported by the Research School of Astronomy and Astrophysicsat the Australian National University and funding from AustralianResearch Council grant No. DP150100250. AMA acknowledgessupport from the Swedish Research Council (VR 2016-03765),and the project grant ‘The New Milky Way’ (KAW 2013.0052)from the Knut and Alice Wallenberg Foundation. MA gratefullyacknowledges additional funding through an ARC Laureate Fellow-ship (project FL110100012). This work was supported by compu-tational resources provided by the Australian Government throughthe National Computational Infrastructure (NCI) facility under theANU Merit Allocation Scheme and the National ComputationalMerit Allocation Scheme. Parts of this research were conductedby the Australian Research Council Centre of Excellence for AllSky Astrophysics in 3 Dimensions (ASTRO 3D), through projectnumber CE170100013.
DATA AVAILABILITY
Data available on request. The data underlying this article will beshared on reasonable request to the corresponding author.
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