Impact of magnetic activity on inferred stellar properties of main sequence Sun-like stars
Alexandra E. L. Thomas, William J. Chaplin, Sarbani Basu, Ben Rendle, Guy Davies, Andrea Miglio
MMNRAS , 1–13 (2021) Preprint 5 February 2021 Compiled using MNRAS L A TEX style file v3.0
Impact of magnetic activity on inferred stellar properties of main sequenceSun-like stars
Alexandra E. L. Thomas , ★ , William J. Chaplin , , Sarbani Basu ,Ben Rendle , , Guy Davies , , Andrea Miglio , School of Physics & Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Stellar Astrophysics Centre, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark Department of Astronomy, Yale University, PO Box 208101, New Haven, CT 06520-8101, USA
Accepted 2021 February 4. Received 2021 January 11; in original form 2020 October 30
ABSTRACT
The oscillation frequencies observed in Sun-like stars are susceptible to being shifted by magnetic activity effects. The measuredshifts depend on a complex relationship involving the mode type, the field strength and spatial distribution of activity, as wellas the inclination angle of the star. Evidence of these shifts is also present in frequency separation ratios which are often usedwhen inferring global properties of stars in order to avoid surface effects. However, one assumption when using frequency ratiosfor this purpose is that there are no near-surface perturbations that are non-spherically symmetric. In this work, we studied theimpact on inferred stellar properties when using frequency ratios that are influenced by non-homogeneous activity distributions.We generate several sets of artificial oscillation frequencies with various amounts of shift and determine stellar properties usingtwo separate pipelines. We find that for asteroseismic observations of Sun-like targets we can expect magnetic activity to affectmode frequencies which will bias the results from stellar modelling analysis. Although for most stellar properties this offsetshould be small, typically less than 0.5% in mass, estimates of age and central hydrogen content can have an error of up to 5%and 3% respectively. We expect a larger frequency shift and therefore larger bias for more active stars. We also warn that for starswith very high or low inclination angles, the response of modes to activity is more easily observable in the separation ratios andhence will incur a larger bias.
Key words: asteroseismology – stars: activity – stars: fundamental parameters
Sun-like stars with outer convective zones pulsate due to turbu-lent motion within these layers. These perturbations excite acousticwaves within the star producing a rich spectrum of modes of os-cillation which can be studied via asteroseismology to investigateinternal stellar physics and determine global properties. Thanks tothe high-resolution photometric observations from CoRoT (Baglinet al. 2006),
Kepler (Borucki et al. 2010; Howell et al. 2014), andmore recently
TESS (Ricker et al. 2015), we have measured acousticoscillations for thousands of stars.Surface magnetic activity is known to affect acoustic modes,changing their frequencies, enlarging damping rates and suppressingamplitudes (e.g. Christensen-Dalsgaard 2002; Chaplin et al. 2007;Metcalfe et al. 2007; Fuller et al. 2015; Kiefer et al. 2019). Sev-eral studies have found evidence of acoustic modes being shifted infrequency, an effect that varies with the 11-year solar activity cycle(e.g. Woodard & Noyes 1985; Palle et al. 1989; Elsworth et al. 1990;Howe et al. 2002; Chaplin et al. 2007; Broomhall et al. 2014) as wellas with a quasi-biennial period (e.g. Fletcher et al. 2010; Broomhallet al. 2012; Simoniello et al. 2012). Similar behaviour has also beenfound in other solar-type stars (e.g. García et al. 2010; Salabert et al. ★ E-mail: [email protected] (AELT) © a r X i v : . [ a s t r o - ph . S R ] F e b A. E. L. Thomas et al. near-surface field strengths and spatial distributions of surface activ-ity. We studied the difference between estimations of stellar modelsfitted to shifted and non-shifted sets of frequency separation ratios.Comparisons made between these results would reveal the impact ofactivity. The justification for using separation ratios is explained inSection 2. In Sections 3 and 4 we outline the model used to calculateactivity-induced frequency shifts and our process for generating setsof frequencies. Section 5 contains a description of the two stellarmodelling pipelines used to infer global properties. Our results areshown in Section 6 followed by discussion and conclusions.
Asteroseismology is a powerful tool to infer fundamental proper-ties of solar-type stars. With long-timebase photometry from, forexample,
Kepler and
TESS it is possible to resolve individual modesof oscillation in stellar spectra giving us a window into the innerworkings of stars. Asteroseismic modelling pipelines can be usedto obtain precise properties of stars using inputs of mode frequen-cies along with complimentary non-seismic data, typically, but notlimited to, effective temperature, metallicity, and luminosity derivedusing parallaxes. More robust and higher accuracy constraints onstellar properties, including mass, radius and age, are possible withthe inclusion of individual mode frequencies, or combinations of fre-quencies, as opposed to solely using global seismic quantities (e.g.Gai et al. 2011; Lebreton & Goupil 2014; Metcalfe et al. 2014; Reeseet al. 2016; Silva Aguirre et al. 2017). Using a set of input physicsand evolutionary codes stellar models are computed, either on the flyor to build a predefined grid, where each stellar model corresponds toa combination of properties for a model star. For each model pulsa-tion codes are then used to predict theoretical oscillation frequencies.Theoretical observables, including frequencies and additional non-seismic data, are fit to actual observations to obtain the best matchingmodel and the corresponding stellar properties.The oscillations are acoustic modes where pressure perturbationsdrive standing waves within a main sequence star. Spherical harmon-ics are used to describe the appearance of these modes on a spherewith oscillations usually described by three numbers:the radial or-der, 𝑛 , the angular degree, 𝑙 , and the azimuthal order, 𝑚 . Solar-likeoscillators produce a spectrum of modes, the frequencies of whichdepend on the star’s properties and internal structure. However, forsolar-type stars observed by Kepler and
TESS it is only possible tomeasure modes with 𝑙 ≤ 𝑟 separation ratio defined as 𝑟 ( 𝑛 ) = 𝑑 ( 𝑛 ) Δ 𝜈 ( 𝑛 ) , where 𝑑 ( 𝑛 ) = 𝜈 𝑛, − 𝜈 𝑛 − , , Δ 𝜈 ( 𝑛 ) = 𝜈 𝑛, − 𝜈 𝑛 − , . (1)Here 𝜈 𝑛,𝑙 is the frequency of a mode with radial order 𝑛 and angulardegree 𝑙 , 𝑑 ( 𝑛 ) is the small frequency separation between 𝑙 = 𝑙 = Δ 𝜈 ( 𝑛 ) is the large frequency separation for 𝑙 = 𝑙 modes to near-surface condi-tions is independent of the degree. Therefore the small separationis already fairly insensitive to surface layers since it calculates thedifference between two modes of very similar frequency which bothpropagate in the near-surface regions. This sensitivity is reduced evenfurther when taking the ratio of small to large separations. Roxburgh& Vorontsov (2003) compared stellar models with the same inte-rior structure but different surface layers to illustrate that frequencyratios are much less sensitive to the poorly-modelled outer layers’conditions. As a result, despite some loss of information when tak-ing ratios, separation ratios can be used to isolate the effects of thedeep stellar interior, a key focus for those determining ages and evo-lutionary states of stars. Due to this sensitivity to central conditionsand being almost unaffected by surface regions Silva Aguirre et al.(2013) argue that more reliable stellar properties can be obtain by us-ing separation ratios rather than oscillation frequencies themselves.Basu & Kinnane (2018) showed that, as long as the surface effectis somehow compensated for when inferring stellar properties frommodels, then the obtained results are robust.Nevertheless, one underlying assumption when using frequencyratios is that there are no non-spherically symmetric near-surfaceperturbations (Otí Floranes et al. 2005) which would induce fre-quency shifts that depend on the degree of the mode. One examplesource would be a surface magnetic activity distribution such as thatseen on the Sun. The Δ 𝜈 𝑙 are calculated from modes with the samecombination of 𝑚 and 𝑙 therefore changes in Δ 𝜈 𝑙 due to activity willbe negligible. However, 𝑑 uses frequencies with different combi-nations of 𝑚 and 𝑙 which occupy different spatial regions on a starand hence experience different size shifts in the presence of non-spherically symmetric magnetic activity. For this reason 𝑑 , andhence 𝑟 , will carry a signature of magnetic field changes; for ex-ample the solar cycle. Chaplin et al. (2005) studied this effect usingSun-as-star observations from BiSON (Chaplin et al. 1996) and mea-sured variations in the ratios with solar activity level. They attributethis change to acoustic asphericity from surface activity and advisedthat care must be taken to account for biases when using ratios fromlong data sets. For this reason in this study we use separation ratioswhen fitting stellar models, avoiding the need for a surface term andwe know that any remaining effect will be caused by non-sphericallysymmetric activity. The shift in frequencies due to surface activity will depend not onlyon the star’s magnetic field strength but also the spatial distribution ofthe activity on the stellar surface. What’s more, our ability to observethese shifts will depend on the inclination angle of the star since thisdictates the mode visibility and hence our ability to observe modecomponents of different 𝑙 and 𝑚 . For this work we build on the modelfrom Thomas et al. (2019) but summarise the main principles here. MNRAS , 1–13 (2021) agnetic activity asteroseismic impact The response of each mode depends on the field strength in theregion where the corresponding acoustic wave propagates so theimpact of a non-homogeneous distribution of activity on a modewill depend on the mode’s spatial distribution. Assuming that thefrequency shifts are caused by a source in the near-surface regions ofthe star, the shift experienced by modes of different 𝑚 and 𝑙 is givenby (Moreno-Insertis & Solanki 2000): 𝛿𝜈 𝑙𝑚 ∝ (cid:16) 𝑙 + (cid:17) ( 𝑙 − | 𝑚 |) ! ( 𝑙 + | 𝑚 |) ! 𝜃 max ∫ 𝜃 min | 𝑃 | 𝑚 | 𝑙 ( cos 𝜃 )| 𝐵 ( 𝜃 ) sin 𝜃 𝑑𝜃. (2)From this we can see that a mode’s sensitivity to activity hasa term that depends on the mode 𝑚 and 𝑙 whose spatial responseis described by the associated Legendre polynomials, 𝑃 | 𝑚 | 𝑙 ( cos 𝜃 ) .This is combined with the magnetic field strength, 𝐵 ( 𝜃 ) , which isa function of the distribution of activity. The above describes thisdistribution using 𝜃 , the co-latitude on the sphere, however, for therest of this paper we will use the latitude, 𝜆 , where 𝜃 = ( 𝜋 − 𝜆 ) . Thearrangement of activity on the stellar surface therefore determinesthe relative magnitudes of shifts for modes of different 𝑙 and 𝑚 . Tocalculate these shifts we assume the same top-hat model as Thomaset al. (2019), defined such that the magnetic activity is distributed ineach hemisphere as a band of uniform field strength, 𝐵 , lying betweenlatitudes 𝜆 min and 𝜆 max , i.e. 𝐵 ( 𝜃 = 𝜋 − 𝜆 ) = (cid:40) 𝐵, if 𝜆 min ≤ 𝜆 ≤ 𝜆 max , , otherwise . (3)The southern hemisphere is assumed to be a reflection of the northernhemisphere, since the globally coherent modes have no sensitivity todifferences between hemispheres.Although theoretically we can describe the response of individual 𝑚 modes to activity, it is not always possible to isolate them in a fre-quency spectrum. Typically, stellar modelling pipelines use only onefrequency per 𝑛 and 𝑙 mode as input so the frequencies of the 𝑚 com-ponents must be combined. The relative visibility of each azimuthalmode depends on the inclination angle of the star, 𝑖 . By introducinga weighting factor, 𝛼 ( 𝑖 ) , which is a function of the inclination angle,Thomas et al. (2019) defined how the contributions combine to givethe central frequency of an 𝑙 = 𝑙 = 𝑖 . Therefore the mea-sured frequency shift of the combined 𝑙 peak will depend not only onthe magnitudes of the individual 𝑚 shifts but also on the inclinationangle.It is well known that activity-induced frequency shifts have a de-pendence on the radial order of the mode (Libbrecht & Woodard1990; Chaplin et al. 1998). Higher frequency modes have shallowerupper reflection points than their lower-frequency counterparts andare therefore more sensitive to the perturbations in the layers closerto the stellar surface. That they therefore show larger frequency shiftshas been observed for the Sun and also other solar-like oscillators(Salabert et al. 2011, 2016; Kiefer et al. 2017; Salabert et al. 2018).To account for this in our artificial data we adjusted the shifts to havea Sun-like frequency dependence as represented by a polynomialrelation in frequency (Chaplin et al. 1998; Howe et al. 2017).Using the same frequency shift model, Thomas et al. (2019) foundthat the active latitudes required to produce the observed solar shifts(for solar cycle 23) extend between 𝜆 min = . ° and 𝜆 max = . ° .This describes the spread of significant, large-scale field on the solarsurface. We therefore re-parameterise Equation 3 to replace 𝐵 by a Inclination angle, i ( ◦ ) . . . . . . F r e q u e n c y s h i f t( µ H z ) [ λ min , λ max ]:[0,20] ◦ [ λ min , λ max ]:[30,50] ◦ [ λ min , λ max ]:[60,80] ◦ [ λ min , λ max ]:[10,50] ◦ [ λ min , λ max ]:[70,90] ◦ Figure 1.
Example frequency shifts of 𝑙 = relative field strength 𝐵 rel , normalised to unity for the Sun: 𝐵 ( 𝜃 = 𝜋 − 𝜆 ) = (cid:40) 𝐵 rel , if 𝜆 min ≤ 𝜆 ≤ 𝜆 max , , otherwise . (4)To enable this we introduced a multiplicative calibration constant 𝐶 𝛿𝜈 into the frequency shift calculation, turning the proportionalityin Equation 2 to an equality giving 𝛿𝜈 𝑙𝑚 = 𝐶 𝛿𝜈 𝐵 rel (cid:16) 𝑙 + (cid:17) ( 𝑙 − | 𝑚 |) ! ( 𝑙 + | 𝑚 |) ! 𝜃 max ∫ 𝜃 min | 𝑃 | 𝑚 | 𝑙 ( cos 𝜃 )| sin 𝜃 𝑑𝜃. (5)The value of this calibration constant is fixed to 𝐶 𝛿𝜈 = . 𝐵 rel = 𝑖 = ° , and the above mentioned latitude parameterspertaining to the Sun.Our model uses Equation 5 to calculate the frequency shift of each 𝑚 mode. As described above we used the weighted contributions ofthe azimuthal components to calculate the central frequencies of theshifted 𝑙 = 𝑙 = 𝑙 , inclination angle 𝑖 , relativemagnetic field strength 𝐵 rel , and distribution of activity 𝜆 min and 𝜆 max . Increasing the magnetic field strength will induce a largermagnitude frequency shift. At certain inclination angles it will beeasier to observe particular 𝑚 components due to the relative modevisibility. The response of modes and how we measure them is com-plex. Figure 1 shows an example of how the measured frequency ofa 𝑙 = 𝜆 min and 𝜆 max combinations. We can see that the response is notnecessarily straightforward. The 𝑙 = 𝑚 components to combine which each depend on theactivity distribution and whose relative contribution to the measured 𝑙 = MNRAS000
Example frequency shifts of 𝑙 = relative field strength 𝐵 rel , normalised to unity for the Sun: 𝐵 ( 𝜃 = 𝜋 − 𝜆 ) = (cid:40) 𝐵 rel , if 𝜆 min ≤ 𝜆 ≤ 𝜆 max , , otherwise . (4)To enable this we introduced a multiplicative calibration constant 𝐶 𝛿𝜈 into the frequency shift calculation, turning the proportionalityin Equation 2 to an equality giving 𝛿𝜈 𝑙𝑚 = 𝐶 𝛿𝜈 𝐵 rel (cid:16) 𝑙 + (cid:17) ( 𝑙 − | 𝑚 |) ! ( 𝑙 + | 𝑚 |) ! 𝜃 max ∫ 𝜃 min | 𝑃 | 𝑚 | 𝑙 ( cos 𝜃 )| sin 𝜃 𝑑𝜃. (5)The value of this calibration constant is fixed to 𝐶 𝛿𝜈 = . 𝐵 rel = 𝑖 = ° , and the above mentioned latitude parameterspertaining to the Sun.Our model uses Equation 5 to calculate the frequency shift of each 𝑚 mode. As described above we used the weighted contributions ofthe azimuthal components to calculate the central frequencies of theshifted 𝑙 = 𝑙 = 𝑙 , inclination angle 𝑖 , relativemagnetic field strength 𝐵 rel , and distribution of activity 𝜆 min and 𝜆 max . Increasing the magnetic field strength will induce a largermagnitude frequency shift. At certain inclination angles it will beeasier to observe particular 𝑚 components due to the relative modevisibility. The response of modes and how we measure them is com-plex. Figure 1 shows an example of how the measured frequency ofa 𝑙 = 𝜆 min and 𝜆 max combinations. We can see that the response is notnecessarily straightforward. The 𝑙 = 𝑚 components to combine which each depend on theactivity distribution and whose relative contribution to the measured 𝑙 = MNRAS000 , 1–13 (2021)
A. E. L. Thomas et al.
We use two stellar modelling pipelines to infer properties for artificialstars from their ‘observed’ frequencies: Asteroseismic Inference ona Massive Scale (AIMS) and another grid-based approach whichwe call Yale Grid-based Modelling (YGM) (see Section 5 for moredetails). Both methods can fit to separation ratios thereby avoidingthe need for a surface term.To build our artificial data sets we started with a set of ‘pristine’frequencies, free from any simulated magnetic activity effects. At anactivity minimum we expect a more uniform distribution of activityacross the stellar surface so all modes experience the same size ofshift. This cancels out when taking the ratio of frequencies. Thereforeby using ratios when fitting stellar models, our pristine separationratios are equivalent to what we would observe on a field-free star,and also what we would expect to observe at minimum levels ofstellar activity. Chaplin et al. (2019) showed that minimum-epochsolar p modes should have frequencies very close to field-free case.For the AIMS analysis presented in Section 6 the pristine frequen-cies were taken from the model in the grid which was most similar tothe Sun in terms of mass and age (4.61 Gyr). The pristine frequenciesfor the YGM analysis were taken from a calibrated Standard SolarModel (SSM). This was created with the same input physics as thegrid that was used for fitting, except for the atmospheric model, whichwas that of Krishna Swamy (1966) (see Section 5 for details of thephysics of the grid). As is usual in constructing SSMs, we iteratedover the mixing length parameter and the initial helium abundance inorder to get a 1M (cid:12) model that has the correct radius and luminosityat the solar age (4.57 Gyr). The converged model has a mixing-lengthparameter of 2.1566 and an initial helium abundance of 0.2734. Themodel has a convection-zone helium abundance of 0.2447, and 𝑍 / 𝑋 of 0.02299. The base of the convection zone is at 0.71317R (cid:12) . Forcompleteness we repeated the analysis with the pristine frequencysets swapped; i.e. the AIMS pipeline was also run with data setsbased on the SSM frequencies, and the YGM analysis using fre-quency sets based on those from the most solar-like AIMS model.The results were in agreement whichever set of pristine frequencieswere used as a base.Data sets were comprised of the 10 overtones of degrees 𝑙 = 𝜈 max (to match the procedure of Ball & Gizon 2014).Frequency uncertainties were taken from BiSON 1-year data and arecomparable to uncertainties given by Kepler data of duration a yearor more from high-quality SNR targets.Using the pristine data as the base, activity-affected frequencysets were generated by shifting the pristine modes according to ourmodel and the chosen combination of 𝐵 rel , 𝑖 , 𝜆 min and 𝜆 max . Theartificial data were created to represent solar-like oscillators at variousinclination angles and with a variety of magnetic activity strengthsand distributions.To choose interesting combinations we first determined thosewhich would produce a set of shifted separation ratios that were,on average, discernibly different from those of the pristine set, i.ebeyond the uncertainty of the pristine ratios (Figure 2 shows an ex-ample set of shifted 𝑟 compared to the pristine 𝑟 using 𝐵 rel = . 𝑖 = ° , 𝜆 min = ° and 𝜆 max = ° ). This was motivated by our goalto find the combinations that would incur a significant bias in stellarproperty estimates from modelling pipelines. We constructed a gridof 𝑖 , 𝜆 min and 𝜆 max , each in the range 0 − ° with increments of 1 ° ,and calculated the minimum field strength, 𝐵 min , needed to producethe desired shifted separation ratios. 𝐵 min was in fact taken to bethe weighted average minimum field strength over all of the ratios.Figure 3 shows the latitudinal positions of activity bands (shaded re-
16 17 18 19 20 21 22 23 24 n . . . . . . r ShiftedPristine
Figure 2.
Example of artificially shifted 𝑟 with respect to the pristine ratios,calculated using 𝐵 rel = . 𝑖 = ° , 𝜆 min = ° and 𝜆 max = ° . An effect ofthis size produces an average shifted 𝑟 which lies just outside of uncertaintyof the average pristine 𝑟 . gions) used to produce the required separation ratios across the rangeof inclination angles. For each element in the grid (correspondingto a particular 𝑖 , 𝜆 min and 𝜆 max ) we shaded the region in latitudeand inclination space with a colour intensity that was proportionalto 1 / 𝐵 min for that element. This was repeated for the entire grid tobuild up Figure 3. Therefore the darker regions indicate where a lower 𝐵 min was necessary to sufficiently shift separation ratios beyond thepristine frequencies, and lighter areas where a much greater 𝐵 min was needed.For each inclination angle we took the [ 𝜆 min , 𝜆 max ] pair corre-sponding to the smallest 𝐵 min thereby compressing a 3D grid to 1D.We chose combinations to study both when the 𝑙 = 𝛿𝜈 ,were larger than the 𝑙 = 𝛿𝜈 , and vice versa. The relativemagnitudes of these shifts affect the direction of the bias we get inthe stellar property estimations from models. Figure 3a shows theresults for 𝛿𝜈 < 𝛿𝜈 shifts and Figure 3b for 𝛿𝜈 > 𝛿𝜈 shifts. We cansee that for each case the 𝐵 min is produced by activity bands at verydifferent latitudes; for example to achieve 𝐵 min at high inclinations alower latitude activity band would produce 𝛿𝜈 < 𝛿𝜈 whereas a bandsituated at higher latitudes would cause 𝛿𝜈 > 𝛿𝜈 . The solid bluecurve in each plot shows the smallest 𝐵 min value at each inclinationangle and the dashed blue curve shows the same but for the oppositesign of shifts. We can see that at intermediate inclination angles themagnetic field strength would need to be larger than at low or highangles to induce the same size 𝑟 shift. Smaller field strengths arenecessary for the lowest and highest inclinations.A variety of combinations were chosen to generate several setsof shifted frequencies. In addition, we also created artificial data toproduce shifts that we would expect from the Sun using 𝑖 = ° and[ 𝜆 min , 𝜆 max ] = [ . , . ] ° as found by Thomas et al. (2019) for theSun’s activity distribution. For one set of frequencies we chose asolar-like magnetic field strength (i.e. 𝐵 rel = 𝐵 min value required to produce shiftedratios discernibly different from the pristine frequencies (which ishigher than we see in the Sun). The red bars in Figure 3 correspondto the chosen latitudes and inclination angles. The collections ofparameters we used to produce each data set are summarised inTable 1.We fitted stellar models to our sets of artificial frequencies andcompared deviations between their estimations of stellar proper-ties from application to the pristine frequencies and those from the MNRAS , 1–13 (2021) agnetic activity asteroseismic impact Table 1.
Combinations of parameters used to calculate frequency shifts forartificial data sets. The method for choosing parameters is explained in themain text. The pristine data set is representative of a field-free star. The Sunmodel uses the parameters necessary to produce solar-like frequency shiftsfrom our model. The Sun 2 model is the same but for a Sun with strongermagnetic field strength in order to make significantly shifted separation ratios(i.e. 𝐵 rel = 𝐵 min ).Relative magneticfield strength, 𝐵 rel Inclinationangle, 𝑖 ( ° ) Minimumlatitude, 𝜆 min ( ° ) Maximumlatitude, 𝜆 max ( ° ) pristine 0.0 - - -i 1.2 0 11 53ii 2.2 30 0 46iii 9.7 53 0 20iv 10.7 54 58 90v 2.2 90 26 85vi 1.7 0 53 90vii 2.2 30 46 90viii 6.1 58 11 53ix 2.2 90 0 26Sun 1.0 90 3.3 40.6Sun 2 4.0 90 3.3 40.6 activity-shifted sets. Any differences must be due to the simulatedmagnetic effects. If activity does not affect stellar model predictionsthen all results will be similar to those obtained from pristine fre-quencies. We used two different pipelines to fit stellar models to oscillation fre-quencies. Both used a predefined grid of models. There are many dif-ferent choices for how to carry out the analysis and the constraints touse when fitting which will impact the uncertainty on estimated stel-lar properties. In particular, anchoring the lowest frequency modescan reduce error bars as we will show later. Below we detail the inputphysics and briefly cover the methods of each pipeline, one of whichimplemented anchoring.
The AIMS pipeline (Reese 2016) uses individual oscillation frequen-cies, or in this case frequency ratios, along with classical constraintsto determine global stellar properties. The grid of models we usedwas the same as the MS grid from Rendle et al. (2019). Modelswere computed using the CLÉS (Code Liégeois d’Évolution Stel-laire, Scuflaire et al. (2008a)) stellar evolution code and the grid wasparameterised by mass in the range 0.75M (cid:12) to 2.25M (cid:12) with an inter-val of 0.02M (cid:12) , initial metallicity ( 𝑍 init ) from 0 . − . 𝑋 init ) in the range 0 . − . 𝑍 init and 𝑋 init values used can be found in Table 1 of Rendle et al. (2019).Microscopic diffusion with a fixed solar-calibrated mixing length of1 .
81 was included since Rendle et al. (2019) found it to producemore closely matching values for the Sun (Thoul et al. 1994). Theconvective overshoot was 0 .
05 times the local pressure scale height,Grevesse & Noels (1993) abundances were used to convert [ Fe / H ] to 𝑍 / 𝑋 , and nuclear reaction rates from taken Adelberger et al. (2011).The models were computed using opacities from Iglesias & Rogers Inclination angle, i ( ◦ ) L a t i t ud e , λ ( ◦ ) B m i n . . . (a) 𝛿𝜈 < 𝛿𝜈 Inclination angle, i ( ◦ ) L a t i t ud e , λ ( ◦ ) B m i n (b) 𝛿𝜈 > 𝛿𝜈 Figure 3.
Determining the combination of parameters to produce separationratios discernible from the pristine set, where 𝐵 min is the minimum fieldstrength needed to do this. The shaded areas show the latitudinal distributionof activity with darker regions indicating where a lower 𝐵 min was neces-sary. The colorbar shows how the shading is inversely proportional to thefield strength with the maximum value of 1.0 corresponding to 𝐵 min forthat inclination angle. Top: the case for ( 𝛿𝜈 )<( 𝛿𝜈 ). Bottom: the case for( 𝛿𝜈 )>( 𝛿𝜈 ). The solid blue lines are the smallest 𝐵 min needed for a particu-lar inclination angle. The dashed blue line is the same but for the opposite signof shift. The red vertical bars indicate the latitudes occupied by the active bandfor each set of parameters we chose to focus on (see Table 1). Results wereconstructed from a grid covering 0 < 𝑖, 𝜆 min , 𝜆 max < ° , hence containing91 models. (1996) and the equation of state from FreeEOS (Irwin 2012). Fre-quencies were calculated using the LOSC (Liège Oscillation Code,Scuflaire et al. (2008b)) pulsation code.AIMS combines approximating a set of best fitting models using aMarkov Chain Monte Carlo (MCMC) algorithm ( emcee , Foreman-Mackey et al. (2013)) with interpolation implemented within the gridof models in order to refine constraints on properties. Interpolationis conducted using multidimensional Delaunay tessellation (see e.g.Field (1991)) both linearly along an evolutionary track and betweentracks.Further details may be found in Rendle et al. (2019). MNRAS000
Determining the combination of parameters to produce separationratios discernible from the pristine set, where 𝐵 min is the minimum fieldstrength needed to do this. The shaded areas show the latitudinal distributionof activity with darker regions indicating where a lower 𝐵 min was neces-sary. The colorbar shows how the shading is inversely proportional to thefield strength with the maximum value of 1.0 corresponding to 𝐵 min forthat inclination angle. Top: the case for ( 𝛿𝜈 )<( 𝛿𝜈 ). Bottom: the case for( 𝛿𝜈 )>( 𝛿𝜈 ). The solid blue lines are the smallest 𝐵 min needed for a particu-lar inclination angle. The dashed blue line is the same but for the opposite signof shift. The red vertical bars indicate the latitudes occupied by the active bandfor each set of parameters we chose to focus on (see Table 1). Results wereconstructed from a grid covering 0 < 𝑖, 𝜆 min , 𝜆 max < ° , hence containing91 models. (1996) and the equation of state from FreeEOS (Irwin 2012). Fre-quencies were calculated using the LOSC (Liège Oscillation Code,Scuflaire et al. (2008b)) pulsation code.AIMS combines approximating a set of best fitting models using aMarkov Chain Monte Carlo (MCMC) algorithm ( emcee , Foreman-Mackey et al. (2013)) with interpolation implemented within the gridof models in order to refine constraints on properties. Interpolationis conducted using multidimensional Delaunay tessellation (see e.g.Field (1991)) both linearly along an evolutionary track and betweentracks.Further details may be found in Rendle et al. (2019). MNRAS000 , 1–13 (2021)
A. E. L. Thomas et al.
For YGM analysis we constructed a uniform grid of models formasses in the range 0.95M (cid:12) to 1.05M (cid:12) with a spacing of 0.01M (cid:12) .For each mass, models were created with fifteen values of the mix-ing length parameter spanning 𝛼 MLT = . 𝑍 / 𝑋 . The stars were mod-elled using the Yale Stellar Evolution Code, YREC (Demarque et al.2008). For each of the parameters, the models were evolved from thezero-age main sequence to an age of 8 Gyr. Models were output atintermediate ages.The models were constructed using the Opacity Project (OP) opac-ities (Badnell et al. 2005) supplemented with low temperature opaci-ties from Ferguson et al. (2005). The OPAL equation of state (Rogers& Nayfonov 2002) was used. All nuclear reaction rates are obtainedfrom Adelberger et al. (1998), except for that of the 𝑁 ( 𝑝, 𝛾 ) 𝑂 reaction, for which we use the rate of Formicola et al. (2004). Allmodels included gravitational settling of helium and heavy elementsusing the formulation of Thoul et al. (1994). The frequencies of themodels were calculated with the code of Antia & Basu (1994).To determine stellar properties, we defined a goodness of fit foreach model in the grid as follows. For each of the spectroscopicobservables, [Fe/H], 𝑇 eff and luminosity 𝐿 , we define a likelihood.For instance, the likelihood for effective temperature was define as L( 𝑇 eff ) = 𝐶 exp (− 𝜒 ( 𝑇 eff )) , (6)with 𝜒 ( 𝑇 eff ) = ( 𝑇 obseff − 𝑇 modeleff ) 𝜎 𝑇 , (7)where 𝜎 𝑇 is the uncertainty on the effective temperature, and 𝐶 theconstant of normalisation. We define the likelihoods for [Fe/H] and 𝐿 in a similar manner.We considered the seismic data using the separation ratio 𝑟 . Forthis we need to take error correlations into account and thus 𝜒 ( 𝑟 ) = ( 𝑟 − 𝑟 ) 𝑇 C − ( 𝑟 − 𝑟 ) , (8)where 𝑟 is the vector defining the observe 𝑟 , 𝑟 is thevector defining the 𝑟 for the model at the observed frequency, and C is the error-covariance matrix. Thus L( 𝑟 ) = 𝐷 exp (− 𝜒 ( 𝑟 )) , (9) 𝐷 being the normalisation constant.The total likelihood is then L total = L( 𝑟 )L( 𝑇 eff )L([ Fe / H ])L( 𝐿 ) . (10)The likelihood was normalised by the prior distributions of eachproperty in order to convert to a probability density. The mediansof the marginalised likelihoods of the ensemble of models was thenused to determine the parameters of the star.However, the total likelihood defined in Equation 10 can result inerroneously high likelihood for some models. The surface term issmaller at low frequencies than at high frequencies, but the seismiclikelihood function defined above does not take this into account.Presently, it could be possible to have a model with low 𝜒 ( 𝑟 ) but where the low frequency modes are badly fit. In order to downweight models for which frequency differences are large, we multiply Equation 10 with the term L reg = 𝐸 exp (− 𝜒 ( 𝜈 low )) , (11)where 𝜒 ( 𝜈 low ) is the 𝜒 for the two lowest frequency modes of eachdegree and 𝐸 is another normalisation constant. Note that Equation11 is not a true likelihood function; the division of the 𝜒 by 100rather than 2 ensures that this term does not dominate the finalselection process. One can set this anchoring of the low frequencymodes in AIMS, however in order to show the range of results toexpect using different analysis approaches here we ran AIMS withoutthis constraint. We determined stellar properties for the pristine and activity-affectedstars using the pipelines described in the previous section. Bothmethods were supplied with the artificial sets of frequencies and fittedusing the separation ratios 𝑟 . Observational constraints of effectivetemperature 𝑇 eff = ±
80K and metallicity [ Fe / H ] = ± . 𝐿 = . ± . (cid:12) . The assumed use of a luminosityuncertainty of 3% was based on Gaia (Gaia Collaboration 2018)parallaxes for Sun-like stars. Figures 4 and 5 show results fromfitting using just frequencies, 𝑇 eff and [ Fe / H ] ; Figures 6 and 7 showresults from fitting including a luminosity constraint.Presented are the median results taken from the posteriors of eachproperty, either from the AIMS (Panels (a)) or the YGM pipeline(Panels (b)). The black circles indicate the median value of stellarproperties obtained by fitting to a ‘pristine’, i.e. field-free, set of fre-quencies which act as a reference. The results obtained from otherdata sets have been spread along the x-axis for clarity. The greyband shows the uncertainties from 68% confidence intervals on theestimates from stellar models applied to pristine frequencies. Thehorizontal grey dashed lines illustrate the underlying properties fromthe corresponding model used to generate pristine frequencies. Thereis a systematic offset between the underlying properties used to com-pute the pristine data set and the results from fitting to the pristinefrequencies, however they are generally well within error. The focusof this work is on how the results from fitting to shifted frequencyratios differ to those from fitting to pristine frequency ratios sincethis will be due to magnetic activity effects.For the majority of properties the median estimates for the activity-affected data lie within the uncertainties of the equivalent pristinevalues. However, for all runs we can see that the largest differencesbetween estimates from the pristine frequencies and those for dif-ferent data sets are in age and central hydrogen abundance. For thecase of the YGM pipeline we can see there is also a considerablespread in the 𝑇 eff values. The spread is more significant for the fitsthat included a luminosity constraint since the uncertainty bars aresmaller.It is clear that for stars experiencing this amount of activity-inducedfrequency shift, some of the stellar properties we infer will have anotable bias. Focusing on mass and age we find that the bias can beup to 5% in age, but only up to 0.5% in mass. This offset is thereforenot a concern since for the analysis carried out in this work, the ageparameter typically has a 4.5% uncertainty and we see a 2.5% erroron mass.As described in Section 4, there are four combinations of re-sults: AIMS fitted to AIMS model frequencies, AIMS applied to MNRAS , 1–13 (2021) agnetic activity asteroseismic impact P r e d i c t i o n . . . . ( M / M (cid:12) ) Mass ( M y r s ) Age . . . . . ( R / R (cid:12) ) Radius . . . . ( c m . s − ) log g . . . . ( g . c m − ) Density ( K ) T eff . . . . () CentralH . . . ( L / L (cid:12) ) Luminositypristinei iiiii ivv vivii viiiix Underlying (a) AIMS fitting. P r e d i c t i o n . . . . ( M / M (cid:12) ) Mass ( M y r s ) Age . . . . . ( R / R (cid:12) ) Radius . . . . ( c m . s − ) log g . . . . ( g . c m − ) Density ( K ) T eff . . . . () CentralH . . . ( L / L (cid:12) ) Luminositypristinei iiiii ivv vivii viiiix Underlying (b) YGM fitting.
Figure 4.
Predictions from fitting without a luminosity constraint. The pristine data set was based on the most solar-like model from the AIMS grid. Blackcircles indicate median results from the pristine data set with the grey band showing the uncertainty on the stellar model estimates from the pristine frequencies.The coloured circles correspond to the results from frequency sets shown in Table 1, and are spread along the x-direction for clarity; their position along thex-axis has no meaning. The same y-axis ranges have been used in Figures 4-8 to more easily allow a like-for-like comparison between plots.MNRAS000
Predictions from fitting without a luminosity constraint. The pristine data set was based on the most solar-like model from the AIMS grid. Blackcircles indicate median results from the pristine data set with the grey band showing the uncertainty on the stellar model estimates from the pristine frequencies.The coloured circles correspond to the results from frequency sets shown in Table 1, and are spread along the x-direction for clarity; their position along thex-axis has no meaning. The same y-axis ranges have been used in Figures 4-8 to more easily allow a like-for-like comparison between plots.MNRAS000 , 1–13 (2021)
A. E. L. Thomas et al. P r e d i c t i o n . . . . ( M / M (cid:12) ) Mass ( M y r s ) Age . . . . . ( R / R (cid:12) ) Radius . . . . ( c m . s − ) log g . . . . ( g . c m − ) Density ( K ) T eff . . . . () CentralH . . . ( L / L (cid:12) ) Luminositypristinei iiiii ivv vivii viiiix Underlying (a) AIMS fitting. P r e d i c t i o n . . . . ( M / M (cid:12) ) Mass ( M y r s ) Age . . . . . ( R / R (cid:12) ) Radius . . . . ( c m . s − ) log g . . . . ( g . c m − ) Density ( K ) T eff . . . . () CentralH . . . ( L / L (cid:12) ) Luminositypristinei iiiii ivv vivii viiiix Underlying (b) YGM fitting.
Figure 5.
The same as in Figure 4 but with the pristine data set based on the SSM.MNRAS , 1–13 (2021) agnetic activity asteroseismic impact P r e d i c t i o n . . . . ( M / M (cid:12) ) Mass ( M y r s ) Age . . . . . ( R / R (cid:12) ) Radius . . . . ( c m . s − ) log g . . . . ( g . c m − ) Density ( K ) T eff . . . . () CentralH . . . ( L / L (cid:12) ) Luminositypristinei iiiii ivv vivii viiiix Underlying (a) AIMS fitting. P r e d i c t i o n . . . . ( M / M (cid:12) ) Mass ( M y r s ) Age . . . . . ( R / R (cid:12) ) Radius . . . . ( c m . s − ) log g . . . . ( g . c m − ) Density ( K ) T eff . . . . () CentralH . . . ( L / L (cid:12) ) Luminositypristinei iiiii ivv vivii viiiix Underlying (b) YGM fitting.
Figure 6.
The same as in Figure 4 with the pristine data set based on the most solar-like model from the AIMS grid but applying a luminosity constraint.MNRAS000
The same as in Figure 4 with the pristine data set based on the most solar-like model from the AIMS grid but applying a luminosity constraint.MNRAS000 , 1–13 (2021) A. E. L. Thomas et al. P r e d i c t i o n . . . . ( M / M (cid:12) ) Mass ( M y r s ) Age . . . . . ( R / R (cid:12) ) Radius . . . . ( c m . s − ) log g . . . . ( g . c m − ) Density ( K ) T eff . . . . () CentralH . . . ( L / L (cid:12) ) Luminositypristinei iiiii ivv vivii viiiix Underlying (a) AIMS fitting. P r e d i c t i o n . . . . ( M / M (cid:12) ) Mass ( M y r s ) Age . . . . . ( R / R (cid:12) ) Radius . . . . ( c m . s − ) log g . . . . ( g . c m − ) Density ( K ) T eff . . . . () CentralH . . . ( L / L (cid:12) ) Luminositypristinei iiiii ivv vivii viiiix Underlying (b) YGM fitting.
Figure 7.
The same as Figure 4) but with the pristine data set based on the SSM and applying a luminosity constraint.MNRAS , 1–13 (2021) agnetic activity asteroseismic impact frequencies built on the SSM data, YGM applied to the AIMS modelfrequencies, and finally YGM analysis of the SSM frequency sets. Ingeneral we see similar results for all combinations. The uncertaintiesfrom YGM fitting are consistently smaller than those from AIMS butthis can be attributed to the additional constraints placed on the lowfrequency modes (see Section 5). AIMS uncertainties match whatwe would expect from Rendle et al. (2019) (Table 3) when fittingusing separation ratios and a luminosity constraint.In the plots where the absolute spread of stellar property estimatesis discernible from the pristine values we can see that there are twodistinct groups of points above and below the pristine results. Thesecorrespond to where the underlying frequency shift at 𝑙 = 𝑙 = 𝑑 is greater or smaller thanfor the pristine set. For example, if the shifted 𝑑 is larger than thepristine 𝑑 , i.e. 𝛿𝜈 > 𝛿𝜈 , then the fitting will find a smaller age.To verify that models were equally well fitted to shifted data setsas to the pristine frequency ratios we calculated the ratios of log-likelihoods between fits and found them to be approximately unity.We also tested the analysis methods described above by replacingthe 𝑟 frequency constraint with 𝑟 data, and separately using both 𝑟 and 𝑟 data simultaneously. The resulting posterior estimatesof stellar properties showed very similar patterns to the 𝑟 resultspresented here. In addition to the various combinations tested above, we also studiedthe bias we would expect for frequency shifts from a Sun-like starcompared to the pristine frequencies. As described earlier, shiftswere calculated using 𝑖 = ° , 𝐵 rel = 𝜆 min , max = [ . , . ] ° .Another set of frequencies was computed for the same parametervalues except using 𝐵 rel = . 𝐵 rel = 𝜎 level for the estimated age and central hydrogen content. We have demonstrated that for some distributions and strengths ofsurface stellar activity the oscillation frequencies would experiencea shift that impacts the properties obtained from stellar modellingpipelines when applied to separation ratios. The shifts we measuredepend on a star’s magnetic field strength, the activity distributionon the stellar surface, and it’s inclination angle (the angle affectingwhich azimuthal mode components are detectable). Measured shiftstherefore show a complex relationship between these variables. Wegenerated several artificial sets of ‘measured’ frequencies using shiftsarising from various combinations of the above. By fitting to separa-tion ratios ( 𝑟 ) constructed from the frequencies, global propertiesfor these fake stars were estimated by two pipelines and compared toresults from a field-free star.Our results showed that estimates on stellar properties split intotwo groupings either side of the pristine result based on whether theshifted 𝑑 is greater or less than the pristine 𝑑 . The most noticeabledivide is in the age parameter which is lower for an increased 𝑑 .By extension, given the small range of metallicities here, a lower agewill automatically result in a higher central hydrogen abundance. In general we see a greater mass for those shifted data sets with larger 𝑑 which is as expected given we are taking a cut in 𝑇 eff .The division into two groups implies that by measuring the shift insmall frequency separation, it is possible to determine the directionof the biases, i.e. whether the property is an under- or over- estimate.Since the size of the bias depends on the frequency shifts experiencedby the modes, and is therefore a complex function of the inclinationangle, activity strength and distribution, it is more difficult to estimatethe size of the bias. By pairing this with the methods of Thomas et al.(2019) it is possible to constrain the active latitudes present on thestar using observations of frequency shifts over time along with thestellar inclination angle. If there is some way to estimate the star’smagnetic field strength relative to the Sun then it could be possibleto understand the expected size of the bias on properties, albeit withfairly large uncertainty.We found that, in general, to experience a bias in property es-timation larger magnetic field strengths are necessary. The devia-tions from the underlying properties would be larger for stars with astronger magnetic field since this simply increases the magnitude offrequency shifts experienced by the modes. For the case of the Sun( 𝐵 rel =
1) the frequency shifts due to activity would not produce asignificantly biased estimation of solar properties at the levels of pre-cision tested here. In order for these to be affected the field strengthwould need to be approximately four times stronger.The measured frequencies from stars with intermediate inclinationangles are least susceptible to magnetic activity effects. As discussedin Section 3, this is due to the relative visibilities of the individualazimuthal modes and how their contributions to a central mode fre-quency are balanced. At these inclinations the field strength wouldneed to be high for the observed separation ratios to be shifted farenough from the pristine 𝑟 to have an impact on the estimatedstellar properties. This can be seen in Figure 3 where the 𝐵 min valuepeaks at 𝑖 ∼ ° , where 𝐵 min is the field strength required to produceshifted ratios discernibly different to the pristine ratios. However,these inclinations only account for ∼
20% of stars (between 45 − ° ).Assuming an underlying isotropic distribution of inclination angles,the relative number of stars observed as a function of 𝑖 is propor-tional to sin ( 𝑖 ) , therefore observing a star with high inclination ismore likely. For the lowest ( < ° ) or highest ( > ° ) inclinations 𝐵 min is lower. This means that for stars at these inclination anglesmore care must be taken to consider the bias on stellar properties dueto activity.For this analysis we took frequency uncertainties commensuratewith Kepler data of a year or more. For shorter duration observationsthe frequency resolution will be reduced thereby minimising theseeffects since estimates of stellar properties will have larger uncer-tainties. The significance of the bias in properties will depend onthe quality of data provided to the modelling pipeline. This includesadditional spectroscopic measurements and whether low-frequencymodes are constrained separately to separation ratios.We find that for asteroseismic observations of Sun-like targets wecan expect magnetic activity to affect mode frequencies which willbias the results from stellar modelling analysis. For most stellar prop-erties we studied this offset should not be an issue since it is smallerthan the uncertainties, including those on mass. However, for ageand central hydrogen content the effect could be significant. Particu-lar care must be taken when analysing long duration observations ofstars with stronger magnetic field strengths than the Sun for whichwe expect higher magnitude frequency shifts. The same is true forstars with very high or low inclination angles where, for the samefield strength, the shift in measured frequency separation ratios iseasier to observe and therefore will produce a more significant bias.
MNRAS , 1–13 (2021) A. E. L. Thomas et al. P r e d i c t i o n . . . . ( M / M (cid:12) ) Mass ( M y r s ) Age . . . . . ( R / R (cid:12) ) Radius . . . . ( c m . s − ) log g . . . . ( g . c m − ) Density ( K ) T eff . . . . () CentralH . . . ( L / L (cid:12) ) Luminositypristine Sun Sun 2 Underlying
Figure 8.
Results from fitting to two additional artificial sets: one using frequency shifts that we would expect to see in the Sun, and the other with sameinclination angle and activity distribution as we would observe but with four times the magnetic field strength, i.e. 𝐵 =
4. Fitting was conducted by AIMSincluding a luminosity constraint and the pristine data set was based on the most solar-like AIMS model.
An obvious next step is to assess the fraction of asteroseismictargets in the
Kepler and
TESS samples that might be susceptible tothese effects based on results from asteroseismic signatures of stellaractivity cycles (e.g. Salabert et al. 2011; Régulo et al. 2016; Salabertet al. 2016; Kiefer et al. 2017; Santos et al. 2018; Kiefer et al. 2019)and proxies of magnetic activity (e.g. see Mathur et al. 2019 andreferences therein).
ACKNOWLEDGEMENTS
We would like to thank Josefina Montelbán for her useful discussionsregarding the use of AIMS. A.E.L.T., W.J.C. and G.R.D. acknowl-edge the support of the Science and Technology Facilities Council(STFC). Funding for the Stellar Astrophysics Centre is providedby The Danish National Research Foundation (Grant agreementno.:DNRF106). A.M. acknowledges support from the ERC Consol-idator Grant funding scheme (project ASTEROCHRONOMETRY, , G.A. n. 772293).
DATA AVAILABILITY
The data underlying this article were generated with publicly avail-able software: AIMS, https://gitlab.com/sasp/aims .The data underlying this article will be shared on reasonable re-quest to the corresponding author.
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