Resolving Power of Asteroseismic Inversion of the Kepler Legacy Sample
aa r X i v : . [ a s t r o - ph . S R ] F e b Solar and Stellar Magnetic Fields: Origins and ManifestationsProceedings IAU Symposium No. 354, 2020A.G. Kosovichev, K. Strassmeier & M. Jardine, eds. c (cid:13) Resolving Power of Asteroseismic Inversionof the Kepler Legacy Sample
Alexander G. Kosovichev and Irina N. Kitiashvili New Jersey Institute of Technology, Newark, NJ 07102, U.S.A.email: [email protected] NASA Ames Research CenterMoffett Field, CA 94035, U.S.A.email: [email protected]
Abstract.
The Kepler Asteroseismic Legacy Project provided frequencies, separation ratios,error estimates, and covariance matrices for 66 Kepler main sequence targets. Most of the pre-vious analysis of these data was focused on fitting standard stellar models. We present results ofdirect asteroseismic inversions using the method of optimally localized averages (OLA), whicheffectively eliminates the surface effects and attempts to resolve the stellar core structure. Theinversions are presented for various structure properties, including the density stratification andsound speed. The results show that the mixed modes observed in post-main sequence F-typestars allow us to resolve the stellar core structure and reveal significant deviations from theevolutionary models obtained by the grid-fitting procedure to match the observed oscillationfrequencies.
Keywords. stars: interiors, stars: late-type, stars: oscillations, methods: data analysis, stars:individual (KIC 10162436, KIC 5773345)
1. Introduction
The Kepler mission (Borucki et al. 2010) provided a wealth of stellar oscillation dataenabling asteroseismic investigation of the internal structure and rotation of many starsacross the HR diagram. A primary tool of asteroseismology employed for interpretation ofobserved oscillation frequencies used a method of grids of stellar models. In this method,the asteroseismic calibration of stellar models is performed by matching the observedoscillation frequencies to theoretical mode frequencies calculated for a grid of standardevolutionary models. In combination with spectroscopic constraints, this approach pro-vided estimates of stellar radius, composition, and age with unprecedented precision (seeChaplin et al. 2010; Metcalfe et al. 2010, and referencies therein). In addition, the highaccuracy measurements of oscillation frequencies opened a new opportunity for asteroseis-mic inversions which allows us to reconstruct the internal structure and test evolutionarystellar models. A similar approach has been used in helioseismology for more than twodecades.A specific feature of asteroseismology data is that only low-degree oscillations can beobserved, typically for modes of spherical harmonic degrees, ℓ = 0 , ,
2, and sometimes, ℓ = 3. In this situation, only the structure of stellar cores can be resolved by inversiontechniques. An additional difficulty is caused by uncertainties in the mass and radius ofdistant stars. Gough and Kosovichev (1993a) showed that this difficulty can be overcomeby an additional condition in the inversion procedure, which constraints the frequencyscaling factor, q = M/R , where M and R are the stellar mass and radius. They showed1 A.G. Kosovichev & I.N. Kitiashvili Table 1.
Characteristics of the stellar models that are used for inversion of the observedoscillation frequencies.KIC M/M ⊙ lg(R/R ⊙ ) Age(Gyr) lg(L/L ⊙ ) Ysurf Zsurf Teff alpha7206837 1.298 0.1980 2.90 0.5523 0.2800 0.0220 6320 1.7911435467 1.382 0.2436 2.56 0.6694 0.2637 0.0197 6414 1.69210162436 1.461 0.3082 2.51 0.8110 0.2460 0.0173 6494 1.6849353712 1.516 0.3261 2.03 0.9008 0.2603 0.0180 6665 1.7135773345 1.579 0.3074 2.07 0.7931 0.2593 0.0306 6400 1.71512069127 1.588 0.3403 1.76 0.9385 0.2669 0.0216 6700 1.672 that if the low- ℓ mode frequencies are measured with precision of 0 . µ Hz then the struc-ture of the stellar core can reconstructed even when the stellar mass and radius are notknown. When the measurement precision is relatively low (e.g. 1 µ Hz), the localization ofthe inversion accuracy is degraded. For such cases, Gough and Kosovichev (1993b) sug-gested a procedure of calibration of the averaging kernels to estimate averaged propertiesof the stellar core. The effectiveness of this procedure was demonstrated by applying it tothe low-degree solar oscillation frequencies observed by the IPHIR instrument that mea-sured the total solar irradiance onboard the PHOBOS spacecraft(Toutain and Froehlich1992). For analysis of the Kepler asteroseismology data, the inversion technique has beenrecently used by Bellinger et al. (2019), and the kernel calibration method was appliedby Buldgen et al. (2017) to estimate integrated properties of stellar structure.In this paper, we show that the detection of mixed modes in the oscillation spectra of F-type stars allows us to resolve the structure of the inner stellar cores. The mixed modeshave properties of internal gravity waves (g-modes) in the convectively stable heliumcore and properties of acoustic modes outside the core. The oscillation frequencies ofthese modes may be quite sensitive to the properties of the core. Including them in theinversion procedure allows us to localize the averaging kernels in the core region. Thisopens a unique opportunity for testing the stellar evolution theory for subgiant starswhere the energy release is in a hydrogen shell surrounding a helium core.
2. Target selection. Evolutionary models.
From the Kepler Asteroseismic Legacy Sample (Silva Aguirre et al. 2017) we selected6 stars in the mass range from about 1.3 to 1.6 solar masses, and, using the MESA stellarevolutionary code (Paxton et al. 2011), we calculated models of the internal structure,matching as close as possible the stellar parameters determined by grid fitting methods.Specifically, we chose the stellar mass, chemical composition, and the mixing parameterfrom the grid pipeline YMCM (Silva Aguirre et al. 2015) and evolved starting from a pre-main sequence phase to the age estimated from the grid fitted models. Basic parametersof the calculated models are shown in Table 1.In this paper, we present results for two models: KIC 10162436 and KIC 5773345, theradial profiles of the density, sound speed, and Brunt-V¨ais¨ail¨a frequency are shown inFigure 1. The Brunt-V¨ais¨ail¨a frequency displays a sharp peak at the helium core outerboundary, located at about 0 . R ⊙ in KIC 10162436, and at 0 . R ⊙ in KIC 577334.Figure 2 shows the difference between the observed and modeled frequencies plottedas a function of the radius of the acoustic inner turning points, r t , calculated from theasymptotic relation: r t /c S ( r t ) = ( l + 1 / /ω nl , where c S ( r ) is the sound-speed profile, ℓ esolving Power of Asteroseismic Inversion a) b) Figure 1.
Distributions of the density, the sound speed and the Brunt-V¨ais¨al¨a frequency inthe stellar models of: a) KIC 10162436, and b) KIC 5773345. l =0 l =1 l =2 l =3 a) b) l =0 l =1 l =2 l =3 Figure 2.
The differences between the observed and modeled frequencies as a function of theacoustic inner turning points for: a) KIC 10162436 and b)KIC 5773345. is the angular degree, and ω nl is the mode frequency. The acoustic turning points formthree branches corresponding to the modes of angular degree l = 0 , , and 2. Behavior ofthe frequency deviations for both models is similar, indicating a systematic difference ofthe evolutionary models from the real stellar structure. A.G. Kosovichev & I.N. Kitiashvili a) b)c) d) Figure 3.
Sensitivity kernels of oscillation frequencies for two mixed modes of ℓ = 1 (panels a − b ) and ℓ = 2 ( c − d ) to relative variations in stellar density ( a − c ), and variations in theabundance of helium ( b − d ), for KIC 10162436 )
3. Sensitivity kernels
The acoustic turning points of some ℓ = 0 modes are located in the stellar core.Nevertheless, the sensitivity of these modes to the core structure is low. Some of theobserved low-frequency non-radial modes of ℓ = 1 and 2 represent mixed acoustic-gravitymodes which have properties of internal gravity (g) modes in the stellar cores and acoustic(p) modes outside the core. The oscillation frequencies of these modes are very sensitiveto the core properties.Using explicit formulations for the variational principle, frequency perturbations canbe reduced to a system of integral equations for a chosen pair of independent variables;e.g. for ( ρ, γ ): δω ( n,l ) ω ( n,l ) = Z R K ( n,l ) ρ,γ δρρ dr + Z R K ( n,l ) γ,ρ δγγ dr, where K ( n,l ) ρ,γ ( r ) and K ( n,l ) γ,ρ ( r ) are sensitivity (or ‘seismic’) kernels. These are calculatedusing the initial solar model parameters, ρ , P , γ , and the oscillation eigenfunctions forthese model, ~ξ (for an explicit formulation, see e.g. Kosovichev 1999). The sensitivityfor various pairs of solar parameters, such the sound speed, Brunt-V¨ais¨al¨a frequency,temperature, and chemical abundances, can be obtained by using the relations amongthese parameters, which follow from the equations of solar structure (‘stellar evolutiontheory’). These ‘secondary’ kernels are then used for direct inversion of the various param-eters (Gough and Kosovichev 1988). A general procedure for calculating the sensitivitykernels can be illustrated in operator form (Kosovichev 2011). Consider two pairs of solarvariables, ~X and ~Y , e.g. ~X = (cid:18) δρρ , δγγ (cid:19) ; ~X = (cid:18) δρρ , δYY (cid:19) , where Y is the helium abundance. The linearized structure equations (the hydrostatic esolving Power of Asteroseismic Inversion a) b)c) d) Figure 4.
The same as Fig. 3 for KIC 5773345. equilibrium equation and the equation of state) that relate these variables can be writtensymbolically as
A ~X = ~Y . Let ~K X and ~K Y be the sensitivity kernels for X and Y ; then the frequency perturbationis: δωω = Z R ~K X · ~Xdr ≡ D ~K X · ~X E , where < · > denotes the inner product. Similarly, δωω = D ~K Y · ~Y E . Then from the stellar structure equation
A ~X = ~Y : D ~K Y · ~Y E = D ~K Y · A ~X E = D A ∗ ~K Y · ~X E , where A ∗ is an adjoint operator. Thus: D A ∗ ~K Y · ~X E = D ~K X · ~X E . This is valid for any ~X if A ∗ ~K Y = ~K X . This means that the equation for the sensitivity kernels is adjoint tothe stellar structure equations. An explicit formulation in terms of the stellar structureparameters and mode eigenfunctions was given by Kosovichev (1999).Examples of the sensitivity kernels to density and helium-abundance deviations for themixed acoustic-gravity modes of ℓ = 1 and ℓ = 2 for KIC 10162436, and KIC 5773345are shown in Figures 3 and 4, respectively. In the case of KIC 10162436, the sensitivityof the mode frequencies to the density stratification is high in the central core. In thecase of KIC 5773345, the sensitivity of the observed mixed modes in the core is not thathigh but still quite significant. In such cases the mixed oscillation modes open a uniqueopportunity for probing the stellar cores by direct structure inversion. The sensitivitykernels for helium abundance are concentrated in the helium and hydrogen ionizationzones. A.G. Kosovichev & I.N. Kitiashvili
4. Inversion procedure
In the inversion procedure it is important to take into account potential systematicuncertainties in the stellar mass and radius. Because the oscillation frequencies are scaledlinearly with the factor q = M/R , then, following (Gough and Kosovichev 1993a), themode frequencies ω i can be expressed in terms of their relative small difference δω i /ω i from those of a standard reference model of similar mass and radius according to thelinearized expression δω i /ω i = Z (cid:18) K if,Y δff + K iY,f δY (cid:19) dx − I iq δq, where x = r/R , q = M/R , Y is the helium abundance, and f can be any function of p and ρ (D¨appen et al. 1991); M and R are stellar mass and radius, in solar units, K f,Y and K Y,f are appropriate kernels, and I q is an integral over the reference model. In thispaper we consider two cases: f = ρ , and f = u ≡ P/ρ (Dziembowski et al. 1991).These constraints can provide localized averages of δ ln f and estimates of δY and δq of the kind δ ln f ≡ Z X i a i ( x ) K if,Y δ ln f dx ≡ Z A f,Y ( x, x ) δ ln f dx = X i a i ( x ) δω i ω i by minimizing over the coefficients a i ( x ) the functional (Backus and Gilbert 1968): , Z A f,Y ( x, x ) J f dx + λ Z X i a i K iY,f ! J Y dx + λ X i a i I iq ! + α X i a i ǫ i for tradeoff parameters λ , λ and α , where J f = ( x − x ) , and ǫ i are standard rela-tive errors in the data. The tradeoff parameters are chosen empirically, by selecting asufficiently smooth solution and using the L-curve criterion (Hansen 1992).The non-adiabatic frequency shift (aka ‘surface effect’) can be approximated by asmooth function of frequency, F ( ω ) scaled with the factor, Q ≡ I ( ω ) /I ( ω ), where I ( ω )is the mode inertia, and I ( ω ) is the mode inertia of radial modes ( l = 0), calculated atfrequency ω ; that is: δω onad,i ω i = F ( ω i ) /Q ( ω i ) . Function F ( ω ) can be approximated by a polynomial function of degree K (Dziembowski et al.1990): F ( ω i ) = K X k =0 c k P k ( ω i ) , where P k are the Legendre polynomials of degree k . Then the influence of nonadiabaticeffects is reduced by applying K + 1 additional constraints for a i : N X i =1 a i P k ( ω i ) Q ( ω i ) = 0 , k = 0 , ..., K.
5. Inversion results
The inversion results obtained by the Optimally Localized Averaging (OLA) proceduredescribed in the previous section are shown in Figure 5. They show the relative deviationsin stellar structure from the evolutionary models of density and parameter u = P/ρ as a esolving Power of Asteroseismic Inversion a) b)c) d) Figure 5.
The relative deviations of the stellar structure from the evolutionary models of density( a − b )and parameter u = P/ρ ( c − d ) as a function of radius for KIC 10162436 and KIC 5773345,obtained by inversion of the observed mode frequencies. Crosses show the center locations thelocalized averaging kernels, the horizontal bars show the spread of the averaging kernels, and thevertical bars show the uncertainties calculated using error estimates of the observed frequencies. function of radius for KIC 10162436 and KIC 5773345. Crosses show the center locationsthe localized averaging kernels, the horizontal bars show the spread of the averagingkernels, and the vertical bars show the uncertainties calculated using the observationalerror estimates (for the definitions of these properties see, Kosovichev 1999). It is re-markable that the mixed modes observed in the oscillation spectra of these stars allowus to obtain constraints on the density stratification in the central helium core and thehydrogen-burning shell. The averaging kernels for the parameter u (or, equivalently, thesound speed) are localized only outside the core. Therefore, it is very important to choosethe pairs of inversion variables that provide the best resolution. These variables can bedifferent for different regions of a star. In our cases only the density inversions providethe optimally localized averaging kernels centered in the stellar cores. The sound-speedinversions are incapable of resolving the core.For both stars, the inversion results show that the density of the core and the surround-ing shell are about 5% higher than in the stellar models, but lower outside the energy-release shell. The boundary of the helium core is located at 0 . R ⊙ in KIC 10162436,and at 0 . R ⊙ in KIC 5773345. Outside the helium cores, the nuclear energy produc-tion shells extend 0 . R ⊙ , with the peak rate at ∼ . R ⊙ in both models. Perhapsthese stellar regions involve physical processes that are not described by the evolution-ary models. For understanding these deviations it will be beneficial to perform moredetailed structure inversion studies for a large sample of post-main sequence stars withhydrogen-burning shells.
6. Conclusion
High accuracy measurements of oscillation frequencies for a large number stars as wellas identification of the observed oscillations in terms of normal modes for stellar models A.G. Kosovichev & I.N. Kitiashviliby the grid-fitting techniques open possibilities for performing asteroseismic inversionsand for testing evolutionary models. The discovery of the oscillation modes with mixedg- and p-mode characteristics in post-main sequence stars allows us to reconstruct thedensity stratification in the stellar helium cores and hydrogen burning shell.We applied the previously developed asteroseismic inversion method (Gough and Kosovichev1993a) to two F-type stars from the Kepler Asteroseismic Legacy Project (Silva Aguirre et al.2017) and performed inversions for the density and squared isothermal sound-speed pa-rameter. The background models were calculated using the MESA code, and the mass,composition, and age were previously determined by the model-grid fitting method.The inversion technique takes potential discrepancies in the estimated mass and radiusfrom the actual properties, as well as the potential frequency shifts due non-adiabaticnear-surface effects, and constructs optimally localized averaging kernels following the(Backus and Gilbert 1968) method.The inversion results for both stars showed that the density in the helium core and theinner part of the hydrogen shell may be about 5% higher than in the evolutionary models,and, in the outer part of the shell, lower. This suggests that the differences may be dueto physical processes not described by the evolutionary models. However, more detailedinversion studies for a large sample of stars are needed for quantifying the deviationsmore precisely.Acknowledgments. The work was partially supported by the NASA Astrophysics The-ory Program and grants: NNX14AB7CG and NNX17AE76A.
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