The asteroseismic surface effect from a grid of 3D convection simulations. I. Frequency shifts from convective expansion of stellar atmospheres
Regner Trampedach, Magnus J. Aarslev, Günter Houdek, Remo Collet, Jørgen Christensen-Dalsgaard, Robert F. Stein, Martin Asplund
aa r X i v : . [ a s t r o - ph . S R ] N ov MNRAS , 000–000 (0000) Preprint 27 August 2018 Compiled using MNRAS L A TEX style file v3.0
The asteroseismic surface effect from a grid of3D convection simulations. I. Frequency shifts fromconvective expansion of stellar atmospheres
Regner Trampedach , ⋆ , Magnus J. Aarslev , G¨unter Houdek , Remo Collet ,Jørgen Christensen-Dalsgaard , Robert F. Stein and Martin Asplund Space Science Institute, 4750 Walnut Street, Suite 205, Boulder, CO 80301 USA Stellar Astrophysics Centre, Dept. of Physics and Astronomy, Ny Munkegade 120, Aarhus University, DK–8000 Aarhus C, Denmark Department of Physics and Astronomy, Michigan State University East Lansing, MI 48824, USA Research School of Astronomy and Astrophysics, Mt. Stromlo Observatory, Cotter Road, Weston ACT 2611, Australia
Received 27 August 2018/ Accepted 27 August 2018
ABSTRACT
We analyse the effect on adiabatic stellar oscillation frequencies of replacing the near-surface layers in 1D stellar structure models with averaged 3D stellar surface con-vection simulations. The main difference is an expansion of the atmosphere by 3Dconvection, expected to explain a major part of the asteroseismic surface effect; asystematic overestimation of p-mode frequencies due to inadequate surface physics.We employ pairs of 1D stellar envelope models and 3D simulations from a pre-vious calibration of the mixing-length parameter, α . That calibration constitutes thehitherto most consistent matching of 1D models to 3D simulations, ensuring that theirdifferences are not spurious, but entirely due to the 3D nature of convection. The re-sulting frequency shift is identified as the structural part of the surface effect. Theimportant, typically non-adiabatic, modal components of the surface effect are notincluded in the present analysis, but relegated to future papers.Evaluating the structural surface effect at the frequency of maximum mode am-plitude, ν max , we find shifts from δν = − . µ Hz for giants at log g = 2 . − µ Hzfor a ( T eff = 6901 K, log g = 4 .
29) dwarf. The fractional effect δν ( ν max ) /ν max , rangesfrom − .
1% for a cool dwarf (4185 K, 4.74) to −
6% for a warm giant (4962 K, 2.20).
Key words:
Stars: atmospheres – stars: evolution – convection
The asteroseismic surface effect is a systematic differencebetween measured stellar p-mode frequencies and theo-retical, adiabatic frequencies of stellar models, known toarise from differences in the surface layers (Brown 1984;Christensen-Dalsgaard, D¨appen & Lebreton 1988).Rosenthal et al. (1999) first analysed the helioseismicsurface effect in terms of frequency differences between 1Dmodels and averaged 3D surface simulations. They con-cluded that most of the effect is due to a convective ex-pansion of the atmosphere, compared to 1D convective at-mospheres. Similar analyses have now been carried out byPiau et al. (2014) and Magic & Weiss (2016) for the Sun,Sonoi et al. (2015) for a grid of 10 simulations, and byBall et al. (2016) for four simulations on the main sequence,all for solar metallicity. ⋆ E-mail: [email protected]
Houdek et al. (2017) recently presented an analysis ofvarious components of the surface effect for the solar case.We use the same 3D solar simulation, but extend ouranalysis to the whole grid of solar metallicity simulations(Trampedach et al. 2013), to explore the behaviour with at-mospheric parameters. On the other hand, we limit our anal-ysis to only include the stratification contributions to theseismic surface effect (see Section 3), and defer the evalua-tion of modal components to a future paper.
We use the grid of 37 fully compressible 3D radiation-coupled hydrodynamic simulations of stellar surface convec-tion by Trampedach et al. (2013). The grid covers effectivetemperatures, T eff = 4 200–6 900 K on the main sequence,and T eff = 4 300–5 000 K and log g = 2 . . c (cid:13) R. Trampedach, et al. for solar metallicity. The simulations evolve the conserva-tion equations of mass, momentum and energy in a smallbox straddling the stellar photosphere extending up to alogarithmic, Rosseland optical depth of log τ = − . The above simulations were previously used to calibrate themain parameter, α , of the mixing-length formulation (MLT)of 1D convection (Trampedach et al. 2014b). This was car-ried out as a matching of temperature and density at thecommon total pressure of the matching point. The match-ing point is chosen as deep in the simulation as possible(less than a pressure scale-height from the bottom), whilestill avoiding boundary effects (See Fig. 1). The 1D modelsinclude a turbulent pressure (See Sect. 3.1), p = β̺v ,with form-factor β . This pressure is smoothly suppressed to-wards the surface, as the MLT version would give an unphys-ically sharply peaked p . The approach is therefore to in-clude it at the fitting point and below, to ensure a consistentmatch to the 3D simulations, as shown in Fig. 1, but sup-press it before it becomes important for the hydrostatic equi-librium (see Trampedach et al. 2014b, for details). Formu-lating a realistic turbulent pressure for 1D models is a sep-arate project. We iterated for α and β until both convergedto within 10 − resulting in deviations of log T and log ̺ atthe matching point of less than 10 − and 10 − , respectively.The 1D models employ the exact same EOS and abundancesas the simulations, and in the atmosphere use the opacitiesof the simulations, and the temperature stratification, T ( τ ),extracted from the simulations (Trampedach et al. 2014a).This ensures that the 1D envelope model and the averaged3D simulation can be patched together for a continuous andsmooth model across the matching point. This is illustratedin Fig. 1 for the warmest dwarf simulation, which deviatesmost strongly from its calibrated 1D model.The 3D simulations are slightly more extended thanthe (un-patched) 1D models (giving rise to the structuralsurface effect), and hence have slightly lower T eff and log g ,corresponding to their common mass and luminosity. Thesimulations are carried out in the plane-parallel approxima-tion (constant surface gravity) and the averages are there-fore corrected for sphericity consistent with the radius of the1D model, to avoid glitches at the matching point. Of thesesteps, only the consistent EOS and abundances have beenimplemented in previous work (Piau et al. 2014; Sonoi et al.2015; Ball et al. 2016; Magic & Weiss 2016).The 1D models are computed with the stellar envelopecode by Christensen-Dalsgaard & Frandsen (1983), which isclosely related to the ASTEC stellar structure and evolution Figure 1.
Top panel: the pressures (black) of the 3D simulation(also shown in Fig. 2) and the calibrated 1D model (magenta).The insert panel is a zoom around the matching point (verti-cal, black dashed, shared with main panel). The surface (where h T i = T eff ) is shown for the 3D simulation (vertical, black dot-ted) and the 1D model (magenta dotted), The difference causesthe structural surface effect. Bottom panel: Same as top panelbut for the logarithmic temperature. code (Christensen-Dalsgaard 2008a). Being envelope mod-els, they ignore the innermost 5% of the star, as well asnuclear reactions and any other composition-altering pro-cesses. The limited extent constrains the modes availablefor our analysis to those with turning points well inside theenvelope model. Our results will, however, not be affected,as the surface effect is indeed confined close to the surfaceand fully contained even in the 3D, deep atmosphere simu-lations. We compare adiabatic mode frequencies from two cases:
UPM: pure 1D models calibrated against the 3D simula-tions as detailed in Section 2.1 (un-patched models), and
PM: those same 1D models, but with the surface layers sub-stituted by the averaged 3D simulations (patched models).The two models are by construction identical interior to the3D simulations. The frequency differences between PM andUPM will be the asteroseismic surface effect due to con-vective effects on the average stratification of the atmo-spheres. This is in contrast to effects from the mode dy-namics through direct interactions between 3D convectionand modes. We refer to these two classes of seismic surfaceeffects as structure effects and modal effects , respectively.The modal effects include the response to the pulsationsof turbulent pressure and non-adiabatic energetics, includingthe convective flux. Houdek et al. (2017) computed modalcomponents for the solar case, based on a non-local, time-dependent mixing-length formulation of convection (Gough1977), and found them to be of the opposite sign and about30% of the structure effects, bringing the total into remark-
MNRAS , 000–000 (0000) urface effect of convective expansion of atmosphere L3 Figure 2.
Ratios of turbulent to total pressures (solid lines, left-hand scale) for four simulations, spanning the range of behavioursin the grid of simulations, with T eff and log g indicated in paren-thesis. The right-hand axis shows the atmospheric expansion dueto the turbulent pressure only, Λ t (dashed lines), as fraction ofstellar radius. The total Λ t is indicated for each curve. able agreement with observations. Using just the structuraleffects, as presented here, will therefore give frequency shiftsthat are larger than the total seismic surface effect (assumingthat modal effects are always positive). Computing modalcomponents directly from the 3D simulations is a significantproject and will be the subject of future papers. The struc-ture effect itself has two components, as detailed below. The horizontally and temporally averaged (denoted by h . . . i )turbulent pressure p t = h ̺u z i , with p = h p g i + p t , (1)contributes about half of the total convective expansion,where ̺ is the density, u z is the vertical velocity, p g is thegas pressure and p the total pressure. This expansion, Λ t ,by p t can be directly quantified by integrating hydrostaticequilibrium over just that component of the pressured p d z = g h ̺ i ⇔ Λ t ≡ ∆ z t = Z d p t g h ̺ i , (2)where z is the depth in the atmosphere. This Λ t is exact inthe sense that ̺ , T and p g do not change with p t , only thelocation where those values occur are shifted. The turbulentcontribution to the total pressure, p t /p , peaks at between 4%for the coolest dwarf in our grid, and 30% for our warmestgiant (see Fig. 2), just below the top of the convection zone.The upturn in p t /p above the photosphere, is not con-vective but rather the effect of travelling waves escaping theacoustic cavity above the acoustic cut-off frequency. Noticethat in local MLT formulations of convection, the convectivevelocities would drop to zero from the peak of the p t /p -ratioin a small fraction of a pressure scale-height, missing abouthalf of the atmospheric expansion from turbulent pressure. Another, less straightforward, contribution to the convectiveexpansion of the atmosphere is caused by convective fluctu-ations in the opacity. Since the top of convective envelopesoccurs at temperatures and densities where the opacity, κ , is extremely sensitive to temperature (about κ ∝ T for the Sun) the convective temperature fluctuations willcause much larger fluctuations in opacity. The warm upflowswill be shielded from cooling until (geometrically) close tothe photosphere, as the high opacity constitutes a geomet-ric compression of the optical depth scale. This effectivelycauses a warming below the photosphere, compared to amodel based on the opacity of the average stratification.The high power in T means the opposing cooling effect inthe downdrafts will be smaller. The upflows also occupy alarger fractional area. Coupled with the non-linear natureof radiative transfer, the cooler downdrafts do not cancelthe effect in the upflows, resulting in a net warming belowthe photosphere. This in turn gives a larger pressure scale-height and hence an expansion of the atmosphere, denotedΛ κ . The effect has a similar magnitude as that from theturbulent pressure. The two effects are also correlated, asthe amplitude of convective velocities and temperature fluc-tuations are correlated. The total convective expansion bythe two mechanisms is denoted Λ and is shown in Fig. 5of Trampedach et al. (2013). We compute this as the radialoff-set of pressure stratifications between the PM and UPMmodels, high in the atmosphere. p t Rosenthal et al. (1999) considered the effect of p t on modesusing two simple cases as illustrative examples: a): p t reacts exactly as p g , i.e., is in phase with the densityfluctuations and proportional to them by γ . b): p t has a completely incoherent response to modal den-sity fluctuations, and over time has no net effect on mode fre-quencies or eigen-functions, i.e., exhibits no modal response.Case b) result in Lagrangian pressure fluctuations δ ln p =(0 · p t /p + γ p g /p ) δ ln ̺ , where γ is the adiabatic expo-nent of the gas, and the parenthesis is referred to as the reduced γ . This should not be viewed as a reduction ofthe thermodynamic quantity, but rather a statement aboutthe turbulent pressure response to modes. Previous calcu-lations (e.g., Piau et al. 2014; Sonoi et al. 2015; Ball et al.2016; Magic & Weiss 2016) have all used case a), which isboth an unjustified choice of the modal response to p t , aswell as an incomplete accounting of modal components.To isolate the structural surface effect, we shall here usecase b), as did Houdek et al. (2017), to assume that there isno modal response to p t , This results in a structural part ofthe surface effect which, for the solar case, is about 1.4 timeslarger than the total surface effect at the acoustic cut-off fre-quency, and about 3 times larger at ν max (see Houdek et al.2017). We analyse the differences between adiabatic frequencies(computed with ADIPLS; Christensen-Dalsgaard 2008b) for
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Figure 3.
Scaled frequency differences in the sense: patched mi-nus un-patched models (Sect. 3) shown with ⋄ , for T eff = 6 569 Kand log g = 4 .
45. We show both the two-term, BG14-fit (solidline), and the residual of that fit (+). the patched and unpatched models, which we identify as thesurface effect due to differences in the average atmosphericstructure of the two cases. We do this for modes with degree l = 20–23 and all orders, n , that have frequencies, ν nl , belowthe acoustic cut-off frequency, ν ac . This l -range ensures themodes are confined well within the envelope models.As first suggested by Ball & Gizon (2014, BG14), we fitthe frequency differences, δν nl , to expressions of the form I nl δν nl = c − ( ν/ν ac ) − + c ( ν/ν ac ) , (3)where we evaluate the acoustic cut-off frequency as ν ac = 5 100 µ Hz ( g/g ⊙ ) p T eff ⊙ /T eff , (4)scaled by the solar value (Jim´enez 2006). Equation (3) wasmotivated by Gough (1990), exploring the origins of the so-lar-cycle modulation of frequencies. He found that a changeof scale-height in the superadiabatic layer would give rise tothe first term, while the ν -term arises from a change to thesound speed that keeps the density unchanged. In terms ofthe convective expansion, these would arise from the convec-tive backwarming and the turbulent pressure, respectively.The frequency shifts in Eq. (3) are scaled by the modeinertia, I nl (e.g., Aerts, Christensen-Dalsgaard & Kurtz2010). This scaling renders the frequency shifts independentof l , and likewise for the fit (to within 0.25%). This con-firms that the restrictions on l , from using envelope mod-els, do not limit the validity of our results. Rather, it is animprovement, since a particular atmosphere simulation cancorrespond to several interior models, in different stages ofevolution, potentially affecting the mode inertia. Our proce-dure effectively separates the surface part, c − and c , fromthe interior part, I nl (supplied by the user), of δν nl .In Figures 3–6, the frequency shifts are reduced to l = 20 by scaling with Q ∗ nl = I nl /I ( ν nl ), where I is I n interpolated to ν nl . An example of our fit to Eq. (3)is shown in Fig. 3 for a warm dwarf. A power-law fit(Kjeldsen, Bedding & Christensen-Dalsgaard 2008) is obvi-ously unable to fit the frequency differences in Fig. 3 overthe full frequency range, as discussed by Sonoi et al. (2015).How the two terms and the mode inertia contribute to Figure 4.
The various contributions to the fit to Eq. (3), for thecase shown in Figure 3. The ν − -term (solid) and the ν -term(dashed), and the total fit (dotted) is compared to the actualfrequency shift ( ⋄ ). The inverse mode inertia (+) is also shown. Figure 5.
The amplitude of the frequency shift (patched minusunpatched) at ν max , as function of T eff and log g . The colour-scale is logarithmic. The location of the solar ( ⊙ ) and stellar ( ∗ )simulations are indicated in white. MESA (Paxton et al. 2011)stellar evolution tracks are over-plotted for masses as indicated.The dashed part shows the pre-main-sequence contraction. the BG14-fit, is shown in Figure 4. It is apparent that thevarious bumps in the frequency shift are due to the modeinertia, and the main reason the BG14 fit is so successful.The amplitude of the surface effect at the frequency ofmaximum power, estimated as ν max ≃ . ν ac , is shown inFigure 5. This qualitatively agrees with analysis of Kepler observations (Metcalfe et al. 2014) of 42 F–G dwarfs andsub-giants. We performed linear regression of their surfaceeffects at ν max , and of ours interpolated to their targets, giv-ing similar increases with log T eff and log g . Our amplitudesare 2–8 times larger, however, partly due to our omission of MNRAS , 000–000 (0000) urface effect of convective expansion of atmosphere L5 Figure 6.
As Fig. 5, except the amplitude is normalised by ν max ,to highlight the relative significance of the surface effect. modal effects, expected to result in a surface effect largerthan the total. Another important factor is how stellar fitsto seismic observations often exhibit coupled parameters. Inparticular the surface effect, mixing length and helium con-tent can be strongly correlated, stressing the importance ofconstraining these quantities independently.Fig. 5 shows that the magnitude of the surface effectincreases roughly as g . To take out this variation and high-light the relative importance of the effect we show in Figure6 the fractional surface effect, in units of ν max . This is seento be predominantly, but not exclusively, a function of theatmospheric expansion, based on the near proportionalitybetween Figure 6 and Figure 5 of Trampedach et al. (2013).The ratio of the two terms in Eq. (3) at ν max , is shownin Figure 7, and illustrates a general change of shape withatmospheric parameters. The c − -term dominates along aridge running parallel with the warm edge of our grid.We have evaluated the stuctural part of the asteroseis-mic surface effect, as the effect on frequencies of the atmo-spheric expansion by realistic 3D convection, relative to 1DMLT stellar models. Contrary to recent studies, we isolatethe structural part from the modal part of the surface ef-fect by ignoring the turbulent pressure response to modes,through the use of the so-called reduced γ . For the solarcase, this gives a frequency shift that is larger than the to-tal, as the modal part turns out to have the opposite sign.Our results are well fit by BG14’s expression, which alsoeliminates first-order dependencies on l , so we can benefitfrom using envelope models instead of full evolution models. ACKNOWLEDGEMENTS
We thank the referee for helpful comments. RT acknowl-edges funding from NASA grant NNX15AB24G. Fundingfor the Stellar Astrophysics Centre is provided by The Dan-ish National Research Foundation (Grant DNRF106).
Figure 7.
As Fig. 5, but showing the ratio of the two terms ofEq. [3]) at ν max , ( c − /c ) · ( ν max /ν ac ) − . REFERENCES
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