Determining the Best Method of Calculating the Large Frequency Separation For Stellar Models
Lucas S. Viani, Sarbani Basu, Enrico Corsaro, Warrick H. Ball, William J. Chaplin
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DETERMINING THE BEST METHOD OF CALCULATING THE LARGE FREQUENCY SEPARATION FORSTELLAR MODELS
Lucas S. Viani , Sarbani Basu , Enrico Corsaro , Warrick H. Ball , and William J. Chaplin Department of Astronomy, Yale University, New Haven, CT, 06520, USA INAF − Osservatorio Astrofisico di Catania, via S. Sofia 78, 95123 Catania, Italy School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK Stellar Astrophysics Centre (SAC), Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C,Denmark
ABSTRACTAsteroseismology of solar-like oscillators often relies on the comparisons between stellar models andstellar observations in order to determine the properties of stars. The values of the global seismicparameters, ν max (the frequency where the smoothed amplitude of the oscillations peak) and ∆ ν (the large frequency separation), are frequently used in grid-based modeling searches. However, themethods by which ∆ ν is calculated from observed data and how ∆ ν is calculated from stellar modelsare not the same. Typically for observed stars, especially for those with low signal-to-noise data, ∆ ν is calculated by taking the power spectrum of a power spectrum, or with autocorrelation techniques.However, for stellar models, the actual individual mode frequencies are calculated and the averagespacing between them directly determined. In this work we try to determine the best way to combinemodel frequencies in order to obtain ∆ ν that can be compared with observations. For this we usestars with high signal-to-noise observations from Kepler as well as simulated TESS data of Ball et al.(2018). We find that when determining ∆ ν from individual mode frequencies the best method is touse the (cid:96) = 0 modes with either no weighting or with a Gaussian weighting around ν max . Keywords: stars: fundamental parameters — stars: interiors — stars: oscillations INTRODUCTIONIn the field of asteroseismology, stellar models play akey role in determining the properties of observed stars.Just by knowing the basic seismic parameters, ν max (thefrequency where the smoothed amplitude of the oscilla-tions peak) and ∆ ν (the large frequency separation), aswell as T eff for a star, models can be used to place con-straints on stellar age, radius, and mass. Since thereis such frequent reliance on matching the seismic pa-rameters determined from observations to the seismicparameters extracted from stellar models, we need tobe sure that the methods by which we calculate ∆ ν and ν max from models produce accurate representations ofthe observed global values. Determining the correct wayto extract the value of ∆ ν and ν max from a stellar modelis therefore of great importance.The large frequency separation, ∆ ν , is the averagefrequency spacing between modes of adjacent radial or-der ( n ), of a given degree ( (cid:96) ). The radial order n is thenumber of nodes in the radial direction and (cid:96) is the num- [email protected] ber of node lines on the star’s surface. This quantity∆ ν arises from the asymptotic relation (Tassoul 1980;Gough 1986), which is applicable for modes of low (cid:96) andhigh n . The relation is not exact and the spacings be-tween the modes have some variability. Therefore, thevalue of ∆ ν will depend on the method by which thisaverage spacing is calculated.The value of ∆ ν can be approximately related to thedensity of the star, as ∆ ν ∝ √ ¯ ρ (see, e.g. Tassoul 1980;Ulrich 1986; Christensen-Dalsgaard 1988, 1993). Thisleads to the ∆ ν scaling relation,∆ ν ∆ ν (cid:12) (cid:39) (cid:115) M/M (cid:12) ( R/R (cid:12) ) . (1)For an observed star, the value of ∆ ν can be deter-mined from the excited p-modes in the star’s powerspectrum. This observed ∆ ν is not usually calculateddirectly from individual mode spacings, as this wouldrequire high signal-to-noise, but instead through othermethods. For example, ∆ ν is often determined by tak-ing a power spectrum of a power spectrum (Mathur et al.2010; Hekker et al. 2010) or autocorrelation techniques(Roxburgh & Vorontsov 2006; Roxburgh 2009; Mosser a r X i v : . [ a s t r o - ph . S R ] M a y & Appourchaux 2009; Huber et al. 2009; Kiefer 2013;Verner & Roxburgh 2011).It is important to note that this is not how the valueof ∆ ν is determined for stellar models. Since for a stel-lar model the radius and mass are known quantities, asimple approach to determine ∆ ν would be to make useof the ∆ ν scaling relation in Eq. 1. However, studieshave shown that the ∆ ν scaling relation has deviations,is a function of T eff and [Fe / H], and only holds to a fewpercent (White et al. 2011; Mosser et al. 2013; Miglioet al. 2013; Guggenberger et al. 2016; Sharma et al. 2016;Yıldız et al. 2016; Rodrigues et al. 2017; Ong & Basu2019). Therefore, ∆ ν for stellar models is not usuallydetermined using the scaling relation, but by calculat-ing the model’s individual mode frequencies. The valueof ∆ ν can then be calculated by finding the averagespacing between these frequencies (of a certain (cid:96) ). Thisaverage is usually determined as the slope of a linear fitto the ν – n relationship for modes of a given (cid:96) . Since thespacing of modes is not exactly the same throughoutthe excited mode envelope, then the manner in whichthe averaging is performed is important. This leavessome ambiguity as to what the best method of calculat-ing ∆ ν from these individual mode frequencies is. Forexample, which (cid:96) modes to include in the averaging orwhether to weight the modes around ν max more heavily,are decisions which can produce important differencesin the value of ∆ ν . Rodrigues et al. (2017) shows thatthere is a difference between Gaussian weighted and er-ror weighted values of ∆ ν up to about 1%. Roxburgh(2014) also discusses that depending on the method ofcalculating ∆ ν , the results can differ by about 1%. Ifrelying on the scaling relation to determine mass, a 1%deviation in ∆ ν can have a meaningful impact.The frequency where the smoothed amplitude of theoscillations peak, ν max , can be shown to be proportionalto the acoustic cutoff frequency, ν ac (Belkacem et al.2011), and goes as ν max ∝ ν ac ∝ gT − / (Brown et al.1991; Kjeldsen & Bedding 1995; Bedding & Kjeldsen2003; Belkacem et al. 2011). This leads to the ν max scaling relation, given by ν max ν max , (cid:12) (cid:39) (cid:18) MM (cid:12) (cid:19) (cid:18) RR (cid:12) (cid:19) − (cid:18) T eff T eff , (cid:12) (cid:19) − / . (2)While one could determine the approximate value of ν max for a stellar model using Eq. 2, the ν max scalingrelation must be used with caution, as many studieshave shown that deviations do exist (Bedding & Kjeld-sen 2003; Stello et al. 2009; Bruntt et al. 2010; Miglio2012; Bedding 2014; Coelho et al. 2015; Silva Aguirreet al. 2015; Yıldız et al. 2016; Viani et al. 2017). Forstellar models, a more accurate way to calculate ν max isto avoid the ν max scaling relation and instead use theacoustic cutoff frequency, as described in Viani et al. (2017).In this paper we investigate various methods of cal-culating the large frequency separation from individ-ual mode frequencies (referred to as ∆ ν freq from nowon) to determine which method gives values of ∆ ν thatbest represent the global observational ∆ ν values. Thiswill be done using observations of high signal-to-noisestars, with individual mode frequencies already deter-mined from previous studies in the literature. For thesestars, the individual mode frequencies will serve as aproxy for the frequency values that one would have ifthey were modeling the star. For each star in our sam-ple, the value of ∆ ν freq will be calculated using a varietyof methods and compared to the global value of ∆ ν , cal-culated using standard observational methods. This willallow us to determine the optimal way to calculate ∆ ν from individual mode frequencies and develop a betterunderstanding of how ∆ ν freq of stellar models should bedetermined. We then verify these results using the Ballet al. (2018) simulations of lightcurves for NASA’s Tran-siting Exoplanet Survey Satellite , TESS (Ricker et al.2015). This issue is especially important as a multitudeof new observations from TESS become available.The paper is organized as follows, Sec. 2 gives anoverview on the stars used in the study, explains themethods used to determine the seismic parameters fromthe observed data, and discusses the various methodsused to calculate ∆ ν from the individual mode frequen-cies. Sec. 3 presents the results and compares the dif-ferent values of ∆ ν . Sec. 4 discusses the findings andSec. 5 provides concluding remarks. DATA AND ANALYSIS2.1.
Sample of Stars in the Study
The stars used in this study consist of the 66main sequence stars from the
Kepler
AsteroseismicLEGACY Sample from Lund et al. (2017), 34 solar-type planet-hosting stars from Davies et al. (2016), the23 main sequence and subgiant stars from Appourchauxet al. (2012) that were not already included from theLEGACY Sample, and 17 red giant stars from NGC6791 that were in Corsaro et al. (2017b) and McKeeveret al. (2019). It should be noted that while Davies et al.(2016) examined 35 stars, we excluded KIC 8684730 asit did not have a readily available multi-quarter powerspectrum. For each of the 140 stars in our sample, the
Kepler power spectrum was obtained from the KASOCwebsite . For the main sequence and subgiant stars theshort cadence KASOC weighted version of the powerspectra were used while for the RGB stars the “Work- kasoc.phys.au.dk Figure 1 . The power spectrum for KIC 6116048. The blueline shows the smoothed background estimate. ing Group 8” long cadence data were used. The seismicparameters were extracted from the power spectrum us-ing several different methods as described in Sec 2.2.2.2.
Determining Seismic Parameters from PowerSpectra
2D Autocorrelation Method
One method to determine the value of ν max and ∆ ν from a power spectrum is the 2D autocorrelation func-tion (ACF) method, as in Huber et al. (2009), Verner& Roxburgh (2011), and Kiefer (2013). The premiseof this technique is to perform a series of autocorrela-tions on segments of the power spectrum to determinethe frequency range of the envelope of excited modes.First the power spectrum is smoothed, to estimate thebackground, using a median filter with a window size of100 µ Hz for the main sequence and subgiant stars anda window of 10 µ Hz for the giants. An example of thissmoothing for star KIC 6116048 can be seen in Fig. 1.Then the power relative to the smoothed backgroundspectrum (PBS) is determined using Eq. 3 (Verner &Roxburgh 2011; Kiefer 2013)
P BS ( ν ) = P ( ν ) − Bg ( ν ) Bg ( ν ) , (3)where P ( ν ) is the power at a given frequency and Bg ( ν )is the smoothed background.A series of autocorrelations are then performed on dif-ferent segments of the PBS. Starting with the lowest fre-quency in the power spectrum, an autocorrelation is cal-culated for a 250 µ Hz wide window (25 µ Hz window forthe giant stars). The window size for the main sequencestars was chosen to match that of Kiefer (2013). Thecentral frequency of the window is then shifted by 1 µ Hz(as in Verner & Roxburgh (2011) and Kiefer (2013)), to-wards higher frequencies, and another autocorrelation is performed. This is continued until the window reachesthe end of the power spectrum. The results of this pro-cess can be seen in the top pannel of Fig. 2, where theautocorrelation power for each frequency lag can be plot-ted as a function of the central window frequency.As can be seen in the top pannel of Fig. 2, when in thefrequency range of the excited p-modes, the autocorre-lation power spikes with a regular spacing which corre-sponds to ∆ ν/
2. This clear pattern in the autocorrela-tion power is not present outside of the frequency rangeof the p-mode envelope. Thus, by examining where theautocorrelation shows this spacing, we can determinethe frequency range of the envelope of excited modes aswell as the value of ν max .To make this more clear, we can collapse the top panelin Fig. 2 to examine just the total average autocor-relation power at each central frequency. This quan-tity, called the mean collapsed correlation (MCC: Kiefer(2013)), is calculated for each central frequency as,MCC = ( (cid:80) n lags i =1 | ACF i | ) − n lags (4)where n lags is the number of lags in the autocorrelationfunction. In the numerator 1 is subtracted because atlag 0 the autocorrelation is 1 since the spectrum hasnot been shifted. The presence of the absolute valuein the equation is because a negative correlation poweralso holds valuable information. We plot the MCC asa function of central window frequency in the middlepanel of Fig. 2, again for KIC 6116048.From the collapsed 2D autocorrelation, the frequencyrange in which the excited p-modes reside can clearlybe seen. A Guassian is then fit to the MCC peak, withthe Gaussian’s center being ν max , as done for exampleby Huber et al. (2009), Verner & Roxburgh (2011), andKiefer (2013). The envelope of excited p-modes is thendefined, following Kiefer (2013), to be the frequencyrange around ν max where the MCC value is at least 10%of the Gaussian peak height.Once the frequency range of the excited p-mode en-velope is determined, the value of ∆ ν can be calcu-lated by taking a power spectrum of the power spectrum(PS ⊗ PS) for this frequency range (see, e.g., Mathuret al. 2010; Hekker et al. 2010). A Lomb-Scargle peri-odogram (Lomb 1976; Scargle 1982) is computed on thePBS spectrum for the frequency range of the excited en-velope. For the example star, KIC 6116048, the PS ⊗ PScan be seen in the bottom panel of Fig. 2. A Gaussian isthen fit to the periodogram, with the Gaussian’s centercorresponding to ∆ ν/
2. The (cid:96) = 1 peak falls betweenthe (cid:96) = 0 peaks and so the power maximizes at ∆ ν/ ν . To help fit the Gaussian, and determinethe correct peak in the PS ⊗ PS, we estimate ∆ ν expected using our calculated value of ν max . Many studies have Figure 2 . Top:
The 2D autocorrelation results for KIC6116048. The abscissa shows the central window frequency,the ordinate shows the autocorrelation lag, and the colorsindicate the autocorrelation power.
Middle:
The mean col-lapsed correlation (MCC). The abscissa shows the centralwindow frequency and the ordinate shows the average abso-lute value of the autocorrelation power for each window (seeEq. 4).
Bottom:
The power as a function of ∆ ν/ ⊗ PS of the p-mode envelope for the example star KIC6116048. shown a relationship where ∆ ν ∝ ν β max where β is be-tween about 0.7 and 0.8 (see, e.g., Hekker et al. 2009;Stello et al. 2009; Huber et al. 2011; Yu et al. 2018),which allows us to determine which of the peaks in thePS ⊗ PS corresponds to ∆ ν/
2. The uncertainty in thelocation of the peak is determined using the standarddeviation of grouped data as in Hekker et al. (2010), s = (cid:118)(cid:117)(cid:117)(cid:116) (cid:80) f x − ( (cid:80) fx ) (cid:80) f (cid:80) f − ≥
1% of the peak’s height. In the remainder of thepaper we will refer to the seismic parameters determinedusing the ACF method as ν max , ACF and ∆ ν ACF .2.2.2.
Determining Seismic Parameters using theCoefficient of Variation Method
While the 2D autocorrelation method has been shownto provide reliable measurements of seismic parametersit can be computationally time consuming. To obtainanother set of ν max and ∆ ν measurements for our datawe also determined the seismic parameters using a moreefficient technique. The recent work of Bell et al. (2019)has shown that ν max can be quickly determined usingwhat is called the coefficient of variation, or CV. Thecoefficient of variation is the ratio of the standard de-viation to the mean of the power spectrum. The basicpremise is that in a power spectrum of pure noise, thisratio should be about 1. Thus, examining where in thepower spectrum this ratio is greater than 1 can be usedto determine the location of solar-like oscillations.Our implementation of the CV method to determine ν max and the frequency range of the excited p-mode en-velope is as follows. First, the power spectrum is brokenup into a series of segments and the CV value is cal-culated for each segment. Starting with a window cen-tered at 1 µ Hz, the window size is set to be the sameas the estimated value of ∆ ν if the central frequencywere assumed to be the value of ν max . This is done toensure that the window size is large enough so that ifthere were oscillations present, some would fall withinthe window. The ∆ ν estimate value is calculated assum-ing that ∆ ν estimate = 0 . ν . as in Yu et al. (2018).With the window size for this central frequency defined,the CV ratio is calculated for this window. The centralfrequency then shifts to higher frequencies by 1/6 of theprevious window size. A new window size is calculatedbased on the new central frequency and the CV valueis found again. The process is repeated until the endof the power spectrum is reached. The CV value foreach window can be seen as the blue diamond points inFig. 3.Next, the CV values from the different overlappingwindows are smoothed. For each central frequency (eachblue point in Fig. 3) the width of the smoothing win-dow is given by 0 . ν . (referred to as W Mosser inthe remainder of the paper), based on the FWHM ofthe excited mode envelope from Mosser et al. (2012).The CV values within this window are then averagedtogether. The resulting smoothed CV trend can be seenas the yellow points in Fig. 3. The location of the high-est smoothed value (the highest yellow point in Fig. 3)is used as the initial estimate of ν max . From this initial ν max , estimate , a weighted mean is performed, using thepoints that are within a window of 0 . ν . , estimate , todetermine the true value of ν max . The weighted meanof the peak is calculated by,Peak Centroid = (cid:80) ji =1 ν i × CV ( ν i ) (cid:80) ji =1 CV ( ν i ) (6)where ν is the frequency, CV ( ν ) is the smoothed CV Frequency (μHz) C V Figure 3 . The coefficient of variation for KIC 6116048. Theblue diamonds show the CV value for each window and theyellow points show the smoothed trend. value at that frequency, and j is the number of fre-quency bins that are within the window defined by0 . ν . , estimate . The uncertainty in ν max was again cal-culated using the standard deviation of grouped data,Eq. 5.The value of ∆ ν was then calculated in a manner sim-ilar to the method used in the 2D autocorrelation ap-proach. A Gaussian was fit to the smoothed CV ν max peak and the envelope of excited p-modes was deter-mined to be the frequency range where the value wasat least 10% of the peak height. A Lomb-Scargle peri-odogram was then computed on the power spectrum forthis frequency range, following the traditional PS ⊗ PSmethod of determining ∆ ν (see, e.g., Mathur et al.2010; Hekker et al. 2010). As previously described inSec. 2.2.1, a Gaussian was then fit to the peak in theperiodogram, were the Gaussian’s center correspondsto ∆ ν/
2. As before, the uncertainty in the location ofthe peak is determined using the standard deviation ofgrouped data as in Hekker et al. (2010). The seismicparameters determined using the CV method will be re-ferred to as ν max , CV and ∆ ν CV for the remainder of thepaper.It should be noted that our implementation of the CVmethod is not identical to that of Bell et al. (2019). Thisis necessary because the Bell et al. (2019) CV methodwas designed for red giant stars, using long-cadence lightcurves, while our sample also contains main sequenceand subgiant stars that have short-cadence data. Forexample, Bell et al. (2019) use 2000 overlapping binsspaced evenly in log-frequency (for their “oversampled”spectrum) while we implement the moving window over-lapping by an amount based on the previous windowsize. This allows our windows to behave in the samemanner regardless of whether we are using long or short- cadence data. Since our implementation is different, theparameter choices which we used were tested (see Ap-pendix A) to ensure that we were determining the lo-cation of ν max correctly. For example, we examine theimpact of changing the window size and smoothing size.As can be seen in Appendix A, the CV method describedin this section provided the best ν max values.2.3. Comparing ∆ ν Results
Since the ACF and CV methods define the frequencyrange of the excited p-mode envelope slightly differently,the resulting ∆ ν value from the PS ⊗ PS will be affected.Figure 4 shows the difference between ∆ ν CV and ∆ ν ACF .As can be seen in Fig. 4, the value of ∆ ν CV tends to beslightly lower than the value of ∆ ν ACF , however, the val-ues of ∆ ν from the two methods are in excellent agree-ment, with the difference being less than 0.5% for thevast majority of stars, and less than 1.5% in all cases.The spread in the difference between the values of ∆ ν CV and ∆ ν ACF is less than the spread when comparing our∆ ν values to those in the literature (as seen in Fig. 5).It should also be noted that the CV method offers asignificant speed advantage over the ACF method.Additionally, we can compare the value of ∆ ν deter-mined using the CV method to the value of ∆ ν from theliterature for our set of stars. For the comparison, valuesof ∆ ν for our sample of stars were obtained from Lundet al. (2017), Davies et al. (2016), Appourchaux et al.(2012), Huber et al. (2013), Bellamy & Stello (2015), andBellamy (2015). A histogram of the fractional differencebetween our calculated value of ∆ ν CV and the corre-sponding literature value of ∆ ν can be seen in Fig. 5.As can be seen in Fig. 5, our calculated values of ∆ ν agree very well with the literature values of ∆ ν . Over94% of the stars have calculated values of ∆ ν within 1%of their corresponding ∆ ν value in the literature and thedistribution is centered around zero. Even in the mostextreme case, the difference in the value of ∆ ν is around3%.2.4. Testing the Seismic Parameter Extraction onSolar Data
Both the 2D autocorrelation method and the coeffi-cient of variation method to extract seismic parame-ters were then tested using solar data. This exerciseserved as a check to ensure our methods of seismic pa-rameter extraction were providing reasonable values for∆ ν and ν max and as a way to compare the two meth-ods. The Solar data was obtained from the 1 minutecadence photometric observations from the VIRGO in-strument (Fr¨ohlich et al. 1995; Frohlich et al. 1997) onthe ESA/NASA spacecraft SOHO . A segment of VIRGOdata was used that was the same length as our
Kepler data. Using the CV method, the value of ν max and ∆ ν −0.4−0.20.00.20.40.6 ( ∆ ν C V − ∆ ν A C F ) / σ d i ff e r e n ce Main SequenceSubgiantsRGB − . − . − . . . . . N (∆ν CV −∆ν ACF )/∆ν CV ν max (µHz) Figure 4 . The difference between ∆ ν CV and ∆ ν ACF normalized by the uncertainty, σ difference (left). σ difference is the propagateduncertainty for the value of ∆ ν CV − ∆ ν ACF . The black points show the RGB stars, the orange triangles are the subgiants, andthe blue circles are the main sequence stars. The right panel shows a histogram of the fractional difference between ∆ ν CV and∆ ν ACF for all the stars in the sample. The red dashed line marks 0 difference. −0.01 0.00 0.01 0.02 0.03 (∆ν CV −∆ν Lit )/∆ν CV N Figure 5 . The fractional difference between ∆ ν calculatedusing the CV method and ∆ ν from the literature for oursample of stars. The red dashed line marks 0 difference. for the Sun were 3091.4 ± µ Hz and 135.0 ± µ Hzwhile for the 2D autocorrelation method the value of ν max and ∆ ν were 3417.6 ± ± µ Hz. Thetypical accepted values of ν max , (cid:12) and ∆ ν (cid:12) are 3090 and135.1 µ Hz (e.g., see Huber et al. 2011). The value of∆ ν (cid:12) for both the 2D ACF method and the CV methodare in good agreement with the accepted value of ∆ ν (cid:12) ,with the CV method’s value being slightly closer to theaccepted ∆ ν (cid:12) . Looking at the ν max , (cid:12) value, while theCV ν max , (cid:12) value is in good agreement with the acceptedvalue, the 2D ACF ν max , (cid:12) value is too large. While the focus of the paper is on measurements of ∆ ν , obtainingthe correct value of ν max is important as the value of ν max can in some cases affect the calculation of ∆ ν freq (see Sec. 2.5). Therefore, due to our inability to repro-duce ν max , (cid:12) using our ACF prescription, along with thefact that it is time consuming, we use the seismic param-eters determined using the CV method for the remainderof this work.2.5. Determining ∆ ν from Individual ModeFrequencies Since the goal of this work is to compare the value of∆ ν CV to ∆ ν freq , we also must calculate the large sepa-ration using the individual mode frequencies for thesestars. Observed individual mode frequencies for theLund et al. (2017), Davies et al. (2016), and Appour-chaux et al. (2012) stars were obtained from the cor-responding publications. For the NGC 6791 red giantstars the individual mode frequencies were determinedby peak-bagging using the Diamonds code (Corsaro& De Ridder 2014) and the methodology for red giants(Corsaro et al. 2015), for a sample of cluster red giantsfrom Corsaro et al. (2017b). The Diamonds code deter-mines parameters using a nested sampling Monte Carloalgorithm. Software and
Diamonds code description are available athttps://github.com/EnricoCorsaro/DIAMONDS
From the individual mode frequencies ∆ ν freq can becalculated by determining the slope of the line of best fitfor a plot of frequency versus n for modes of the same (cid:96) .The issue however, is that performing this fit in differentways will alter the values of ∆ ν freq . For example, theslope of the line will be different depending on which (cid:96) modes are being used. Additionally, there is the ques-tion if there should be any weighting on the frequencies.Should modes closer to ν max be more heavily weighted?Should the modes be weighted by their uncertainties?Should only modes closest to ν max be used? All of theseoptions will result in a different value of ∆ ν freq .In the literature there are many different methodsused to determine the value of ∆ ν from individual modefrequencies. For example, Handberg et al. (2017) calcu-lated the average ∆ ν by weighting the frequencies bytheir observational errors. Other studies have imple-mented some type of Gaussian weighting around ν max (White et al. 2011; Rodrigues et al. 2017). Hekker et al.(2013) used both a Gaussian weighted linear fit, an un-weighted fit, as well as the median of the pairwise dif-ferences of the modes. It is also possible to just usethe modes closest to ν max , for example Corsaro et al.(2017a) determined ∆ ν using a Bayesian linear regres-sion on the asymptotic relation using the central 3 (cid:96) = 0mode frequencies.For each star in our sample, we calculate ∆ ν freq in 7different ways for each (cid:96) . So, for stars with (cid:96) = 0 , ν freq was calculated 21 differenttimes. The different methods of determining ∆ ν aresummarized in Table 1 and explained in more detail asfollows:I. No Weighting : The best-fit slope of the frequencyvs. n plot is simply calculated without taking anyweighting or uncertainties into account. The fullset of observed modes for a given (cid:96) are used.II. Error Weighting : The best-fit slope takes intoaccount the uncertainties in the observed frequencyvalues. The full set of observed modes for a given (cid:96) are used.III.
Gaussian Weighting : High signal-to-noise datashows that the power envelope of the excitedmodes is a Gaussian with a full-width-half-max of0 . ν . (Mosser et al. 2012). Therefore, it maybe reasonable to weight those modes closer to ν max more heavily in the best-fit slope. Here each fre-quency is given a weight, where the weighting func-tion is a Gaussian centered on ν max with a FWHMgiven by 0 . ν . . The weight, W , for each pointis then given by W = e − ( ν − ν max ) / (2 σ ) (7) where σ = FWHM / (2 (cid:112) σ used. Rodrigueset al. (2017) use σ = 0 . ν . , while we use thatas our FWHM, thus making our values of σ differby a factor of 2 (cid:112) k (cid:88) i =1 W i [ ν i − (slope × n i + intercept)] , (8)where k is the number of modes. The full set ofobserved modes for a given (cid:96) are used.IV. No Weighting, 4 Points : Only 4 frequencies areused in the fit, 2 on each side of ν max . No errors orweighting equation is used.V. Error Weighting, 4 Points : Only 4 frequenciesare used in the fit, 2 on each side of ν max . The un-certainties in the observed frequencies are includedin the best-fit slope calculation.VI. No Weighting, 10 Points : Only 10 frequenciesare used in the fit, 5 on each side of ν max . No errorsor weighting equation is used.VII. Error Weighting, 10 Points : Only 10 frequen-cies are used in the fit, 5 on each side of ν max .The uncertainties in the observed frequencies areincluded in the best-fit slope calculation.For each star, and for each (cid:96) , these 7 methods wereused to compute ∆ ν freq . For the remainder of the paperthe methods will be referred to by their correspondingRoman numeral. Note that when determining the in-dividual mode frequencies in actual stellar models thenmethods II, V, and VII cannot be used since there is noassociated observational uncertainty on the frequencies. RESULTS3.1.
Comparing ∆ ν CV and ∆ ν freq The large frequency spacing calculated using the CVmethod, ∆ ν CV , can be compared to the different valuesof ∆ ν freq calculated in Sec. 2.5. Figure 6 shows the dif-ference between the values of ∆ ν freq and ∆ ν CV dividedby the uncertainty in the difference, as a function of ν max , for each different method of calculating ∆ ν freq forthe (cid:96) = 0 , , and 2 modes. As can be seen in Figure 6,the difference between the values of ∆ ν are mostly allwithin 1 σ . The exception to this is ∆ ν calculated withthe (cid:96) = 1 modes in the subgiant stars, which have a largescatter. This is due to the fact that the subgiant starshave mixed-modes. For the sake of making the ordinatescale in Fig. 6 small enough to easily view the data, some Table 1 . A summary of the various different methods used to calculate ∆ ν freq . Method Weighting Used In Slope Determination Number of Frequencies UsedI
None Full Set of Observed Modes Used II Error Weighted Full Set of Observed Modes Used
III
Gaussian with a FWHM of 0.66 ν . (Mosser et al. 2012) Full Set of Observed Modes Used IV None 4 total, 2 on each side of ν max V Error Weighted 4 total, 2 on each side of ν max VI None 10 total, 5 on each side of ν max VII
Error Weighted 10 total, 5 on each side of ν max of the values of ∆ ν calculated using the (cid:96) = 1 frequen-cies for the subgiant stars fall outside the range of thefigure and are not visible. Additionally, one should notethat for the RGB stars there were not enough modes oneither side of ν max for the value of ∆ ν freq to be calculatedusing methods VI and VII.It can also be seen in Figure 6 that there is a largerscatter and disagreement for the method of determining∆ ν freq using only 2 frequencies on either side of ν max (methods IV and V). This suggests that using only 4frequencies in the ∆ ν determination is not ideal whenattempting to match the value of ∆ ν one would calculatefrom the observed power spectrum. This larger scattermake sense due to the fact that the value of ∆ ν CV usesnearly the entire region of oscillations from the powerspectrum and therefore takes many more than 4 frequen-cies into account. So, one might expect that the value of∆ ν CV and ∆ ν freq would match more poorly when ∆ ν freq only uses a few modes.Additionally, for some of the methods it does appearthat the value of ∆ ν freq tends to be smaller than thevalue of ∆ ν CV . This can be seen in Fig. 6 for the (cid:96) = 0modes in methods II, III, VI, and VII as well as the (cid:96) = 1 and (cid:96) = 2 modes for method VII. Despite thedifferences among the methods, with exception to someof the subgiant stars, the value of ∆ ν CV and ∆ ν freq agreeto within a few percent.3.2. Simulated TESS Data
So far our comparisons have used observed data withthe determined mode frequencies acting as a proxy forthe frequencies one would have from a stellar model.Since the properties of the observations, for example theS/N or time-series length, will determine which modeswere observed, we need to make sure that our resultsalso hold when using stellar models. Additionally, sincethe goal of this project is to determine the best methodof calculating ∆ ν from stellar models, it is critical thatwe repeat the experiment using frequencies from actualstellar models. To accomplish this, we make use of thesimulated TESS data from Ball et al. (2018). Ball et al.(2018) created a mock catalog of lightcurves to simulate data from NASA’s Transiting Exoplanet Survey Satel-lite , TESS (Ricker et al. 2015). From this mock cat-alog we selected 34 main sequence stars, 37 subgiants,and 47 red giants to analyze. The selected stars canbe seen in the Kiel diagram in Figure 7. Note that theBall et al. (2018) simulated TESS data was restricted tostars which would be observed with the short cadence(2 minutes) mode of TESS and as a result the red gi-ant stars available in this catalog are those which arenot evolved too far along the red giant branch. All ofthe selected stars and lightcurves used are from “Sector1” in Ball et al. (2018). Additionally, it should be notedthat the lightcurves used did not have white noise addedto them. While Ball et al. (2018) does provide the ex-pected value of the white noise for each lightcurve, thescope of this project is not concerned with the actualobserving capabilities of TESS, rather the comparisonbetween the value of ∆ ν from the power spectrum and∆ ν from frequencies. Hence, the clean lightcurves fromBall et al. (2018) were used without added noise.From the simulated TESS data, the “observed” val-ues were calculated from the lightcurves. The valuesof ν max and ∆ ν CV were calculated using the coefficientof variation method as discussed in Sec. 2.2.2. For thesimulated stars Ball et al. (2018) also models each starand provides individual mode frequencies. It should benoted that the frequencies from the models were used inthe creation of the simulated spectra. So, unlike typi-cal model frequencies which may disagree with the truefrequency values, these modeled frequencies are actu-ally those found in the simulated spectra. Using thesemodel frequencies, the value of ∆ ν freq was calculated asdiscussed in Sec. 2.5. The frequencies provided from theBall et al. (2018) models are for values between 0.15and 0.95 ν ac . Thus, for the model stars, when calculat-ing ∆ ν freq using methods I and III, which utilize the fullset of modes, this corresponds to all modes between 0.15and 0.95 ν ac .As in Sec. 3.1, we then compare the values of ∆ ν CV and ∆ ν freq . Figure 6 can be remade, but for the sampleof stars from the simulated TESS data. This can be −202 I ℓ = 0 ℓ =1 ℓ =2 −202 II −202 III −202 I V −202 V −202 V I V II ν max (µHz) ( ∆ ν f r e q − ∆ ν C V ) / σ D i ff e r e n ce Figure 6 . The difference, (∆ ν freq − ∆ ν CV ) /σ Difference , as a function of ν max for each of the different methods of determining∆ ν freq and for each (cid:96) . Note that some of the subgiant stars are not in the ordinate range of this plot. Colors are the same asFig. 4. For reference, the dashed line is at 0 and the dotted lines are at ± T eff (K) l o g ( g ) Figure 7 . Kiel diagram of the stars selected from Ball et al.(2018). The black points are the red giant stars, the orangetriangles are the subgiant stars, and the blue circles are themain sequence stars. The background gray lines show tracksof mass 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, and 2.0 M (cid:12) , created usingYREC (Demarque et al. 2008). seen in Figure 8. Note that methods II, V, and VIIare not used in this case. This is due to the fact thatthese methods use the observational uncertainties whendetermining the value of ∆ ν freq and the model values donot have observational uncertainties. As seen in Fig. 8,again the values of ∆ ν CV and ∆ ν freq agree well and forthe most part are within 1 or 2 σ . Additionally, it canbe seen that for method I the value of ∆ ν freq tends tobe smaller than the value of ∆ ν CV . DISCUSSIONWe can put the information in Figures 6 and 8 on amore quantitative footing to determine which method ofcalculating ∆ ν freq is most in agreement with the valueof ∆ ν CV . The root mean square (rms) value of the per-cent difference, 100 × (∆ ν freq − ∆ ν CV ) / ∆ ν CV , can becompared for each method of calculating ∆ ν freq . Ta-ble 2 shows the rms value for each method of calculat-ing ∆ ν freq for each set of stars for the observed sample.As can be seen in Table 2, the best method to calculate∆ ν freq depends on the star’s evolutionary stage and the (cid:96) of interest.For the main sequence stars using the (cid:96) = 0 modes,methods I, III, and VI perform equally well and producevalues of ∆ ν freq closer to the values of ∆ ν CV than theother methods. Using the (cid:96) = 1 modes instead, methodsI, III, and VI again outperform the other methods, withmethod VII being equally good as well. When using onlythe (cid:96) = 2 modes methods VI and VII perform equallywell and better than the other methods. However, wemust be careful with comparing all the methods at once,since not every star in the sample can be included in ev-ery method of calculating ∆ ν freq . For example, not allthe stars had enough observed modes to use methods VI and VII. Since all the stars are included in methodsI, II, and III, then it is safest to compare these threeagainst each other. Doing this we see that for the mainsequence stars using the (cid:96) = 0 or (cid:96) = 1 modes are muchbetter than using the (cid:96) = 2 modes. Also, we see thatusing methods I and III are nearly equivalent and bet-ter than method II. So, this means that for the mainsequence stars using either no weighting or the Mosseret al. (2012) Gaussian weighting is better than weightingby the observational uncertainties.For the subgiant stars, using the (cid:96) = 0 modes, meth-ods I, III, and VI perform nearly equivalently and arebetter than the other methods. The (cid:96) = 1 modes do notprovide good values, as expected due to the presence ofmixed-modes. For the (cid:96) = 2 modes, all the methodsperform about the same except for methods I and VIIbeing worse.For the red giant stars, using the (cid:96) = 0 modes, methodIII is the best, however all methods except II do equallywell. Note that methods VI and VII are not included forthe RGB stars because there were not enough modes tohave 5 frequencies on each side of ν max . Using the (cid:96) = 2modes, methods I and III perform equally well and bet-ter than the other methods. For every method, using the (cid:96) = 0 modes provided the best results for the RGB stars.Similar to the main sequence stars, we see that using ei-ther no weighting or the Mosser et al. (2012) Gaussianweighting is better than using the observational uncer-tainties.We can perform the same investigation for the sim-ulated TESS data, again looking at the rms value ofthe percent difference, 100 × (∆ ν freq − ∆ ν CV ) / ∆ ν CV ,for each method. Table 3 shows the rms value for eachmethod of calculating ∆ ν freq for each set of stars andeach (cid:96) . There are a few important differences to notewith using the simulated TESS data compared to theobservational data. First of all, for the simulated TESSdata, we only selected the (cid:96) = 1 modes which were notmixed-modes. So, the rms value of the subgiant (cid:96) = 1modes are unrealistically good. Additionally, since themode frequencies are from models, we have many moremodes than we would actually have from observationsfor these stars. This also means that all of the stars inour sample of simulated TESS data had enough modesthat all stars could be put through each method of calcu-lating ∆ ν freq for every (cid:96) . However, since the modes arefrom stellar models, methods II, V, and VII could not becalculated since there was no observational uncertaintyon the modes.As can be seen in Table 3, for the main sequence stars,when using the (cid:96) = 0 modes, methods III and VI per-form nearly equally well and better than the other meth-ods. Methods III and VI again outperform the otherswhen using the (cid:96) = 1 modes or the (cid:96) = 2 modes as well.1 −4−2024 I ℓ =0 ℓ =1 ℓ =2 −4−2024 III −4−2024 I V
500 1500 2500−4−2024 V I
500 1500 2500 ν max (µHz)
500 1500 2500 ( ∆ ν f r e q − ∆ ν C V ) / σ D i ff e r e n ce Figure 8 . The difference, (∆ ν freq − ∆ ν CV ) /σ Difference , as a function of ν max for each of the different methods of determining∆ ν freq and for each (cid:96) for stars from the TESS simulated data. Colors are the same as Fig. 4. For reference, the dashed line isat 0 and the dotted lines are at ± Table 2 . The rms value for the percent difference, 100 × (∆ ν freq − ∆ ν CV ) / ∆ ν CV , for each method and each (cid:96) for our set of Kepler stars. (cid:96) = 0 (cid:96) = 1 (cid:96) = 2 Average of (cid:96) = 0 , (cid:96) = 0 , Table 3 . The rms value for the percent difference, 100 × (∆ ν freq − ∆ ν CV ) / ∆ ν CV , for each method for the simulated TESS stars. (cid:96) = 0 (cid:96) = 1 (cid:96) = 2 Average of (cid:96) = 0 , (cid:96) = 0 , (cid:96) = 0 and (cid:96) = 2 modes forthe main sequence stars we see that for methods III, IV,and VI the resulting rms values are very similar. Formethod I using the (cid:96) = 0 modes is better than using the (cid:96) = 2 modes.For the subgiant stars methods III and VI are slightlybetter for the (cid:96) = 0 modes. For the (cid:96) = 1 and (cid:96) = 2modes method III performs the best. For every method,using the (cid:96) = 0 modes for the subgiants outperformsusing the (cid:96) = 2 modes.For the red giant branch stars, regardless of whichmethod is being used, using the (cid:96) = 0 modes give alower rms value than using the (cid:96) = 1 or (cid:96) = 2 modes.Additionally, regardless of which (cid:96) modes are being used,method I is the best in all cases. However, since for thesesimulated stars we have a lot more modes than wouldget in observations, then perhaps using all available fre-quencies as in method I is unrealistic. If instead for theRGB stars we are restricted to only using 5 points oneither side of ν max , as in method VI, then the resultingrms values are either better or nearly the same as usingthe Gaussian weighting of method III.Finally, for our observed RGB stars we can compareour ν max , CV and ∆ ν CV values to those from Corsaroet al. (2017a). Corsaro et al. (2017a) determined ∆ ν using a Bayesian linear regression on the asymptotic re-lation using the central 3 (cid:96) = 0 mode frequencies. Wesee that the values of ∆ ν agree to within 1.5% and val-ues of ν max agree to within 10%. Comparing the valuesof ∆ ν freq to the ∆ ν values from Corsaro et al. (2017a)we see that for the (cid:96) = 0 modes, method V is in the mostagreement. This is as expected since method V uses 2points on each side of ν max with error weights and is themost similar method compared to the technique of us-ing the 3 central frequencies performed in Corsaro et al.(2017a). SUMMARY AND CONCLUSIONWhen comparing observed values of ∆ ν to values of∆ ν calculated from stellar models, we must be awarethat the manner in which these two values of ∆ ν arebeing determined are different. The observed value of∆ ν from photometric time series data is typically deter-mined using autocorrelation techniques or PS ⊗ PS meth-ods, while this is not the case when calculating ∆ ν fromstellar models. In stellar models the actual individual mode frequencies are calculated and then the value of∆ ν freq can be determined by fitting a line to the fre-quency versus n data. There are many different meth-ods by which to perform this linear fit, for example toweight the frequencies closer to ν max more heavily, toonly use a few points around ν max , and deciding which (cid:96) modes to include. It is critical that when we determinethe value of ∆ ν for stellar models that we are doing so ina way that will provide consistent results with the valuesof ∆ ν calculated through observations. Otherwise thecomparison between the two values of ∆ ν looses accu-racy.In this work we took high signal-to-noise Kepler ob-servations and determined the seismic parameters usingstandard methods. Each of these stars also had individ-ual mode frequencies determined in the literature. Us-ing these individual mode frequencies as a proxy for thefrequencies one would have from modeling a star, we de-termined ∆ ν freq in several different ways to compare tothe value of ∆ ν CV . We also made use of simulated TESSlightcurves and stellar models to compare the methodsof calculating ∆ ν values. From the results of compar-ing ∆ ν CV and ∆ ν freq both from the observed stars andthe simulated TESS data, we show that using the (cid:96) = 0modes with either no weighting or a Gaussian weightingas in Mosser et al. (2012) provides the best agreement.Additionally, we see that using the coefficient of varia-tion method as in Bell et al. (2019) provides a quick andaccurate way to identify the frequency range of excitedmodes.This work was partially supported by NSF grant AST-1514676 and NASA grant NNX16AI09G to SB. EC isfunded by the European Union’s Horizon 2020 researchand innovation program under the Marie Sklodowska-Curie grant agreement No. 664931. WHB and WJC ac-knowledge support from the UK Science and TechnologyFacilities Council (STFC). WHB acknowledges supportfrom the UK Space Agency (UKSA). Funding for theStellar Astrophysics Centre is provided by The DanishNational Research Foundation (grant DNRF106). Theauthors thank the anonymous referee for the helpfulcomments and suggestions. Software:
Diamonds (Corsaro & De Ridder 2014)and YREC (Demarque et al. 2008)REFERENCES
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APPENDIX A. TESTING THE VARIOUS CV METHODSAs mentioned in Sec 2.2.2, many variations of the CV method were tested. These different implementations of theCV method will be described here, where we refer to the method described in Sec 2.2.2 as the “base” method. Inthe “base” method, each step we shift the window towards higher frequencies by 1 / /
4, 1 /
2, 1 /
10, and 1 /
20 of the previous window size. This effectively changes thedensity of the blue points in Fig. 3. Also, we tested using different window sizes when calculating the CV value alongthe spectrum. In the base implementation the window size is determined by estimating the value of ∆ ν at the centralfrequency. We also tested window sizes of 2∆ ν , 4.2∆ ν , and W Mosser . Additionally, when smoothing the CV valuesthe size of the smoothing window was tested. In the base method we smoothed with a window of width W Mosser andhere we test using a window of width 2 W Mosser , W Mosser /
2, 2∆ ν , 4 . ν , 6∆ ν , and 8∆ ν . The final parameter that wastested was the window sizes used when determining the weighted centroid of the peak. In the base implementation weuse a window of W Mosser and here we also test a window of 2 W Mosser .To test which method works best a set of 63 randomly selected red giant stars from the Second APOKASC Catalog4(Pinsonneault et al. 2018) were used. Using the power spectrum from KASOC, the ν max values of the 63 stars werecalculated using the base CV method as described in Sec. 2.2.2. Then the various different implementations of theCV method, described in the previous paragraph, were used to again calculate ν max . The resulting ν max values werecompared to the values of ν max from the Second APOKASC paper (Pinsonneault et al. 2018), as well as the differentpipelines in the Second APOKASC Catalog: A2Z (Mathur et al. 2010; Garc´ıa et al. 2014), CAN (Kallinger et al.2010), COR (Mosser & Appourchaux 2009), OCT (Hekker et al. 2010), and SYD (Huber et al. 2009). The rms valuefor the percent difference, 100 × ( ν max , pipeline − ν max , CV ) /ν max , CV , was calculated for each CV implementation and canbe seen in Table A1. While the “best” method depends on which pipeline the ν max value is being compared to, thebase CV method was in good agreement across all sets. Additionally, if the rms values for the pipelines are averagedtogether, as seen in the last column of Table A1, then the base CV method performs the best. Therefore, this basemethod was the one selected as the best implementation of the CV method and is the one used throughout the paper. Table A1 . The rms value for the percent difference, 100 × ( ν max , pipeline − ν max , CV ) /ν max , CV for each CV implementation forour sample of RGB stars from the Second APOKASC Catalog (Pinsonneault et al. 2018). The columns represent the various ν max pipeline values from the catalog. The last column is an average of the rms value for all the pipelines. PipelineCV Method APOKASC A2Z CAN COR OCT SYD Average
Base Method 2.83 3.20 3.10 3.20 2.97 3.02 3.05Window Shift: 1 / / /
10 Previous 3.16 3.67 3.36 3.57 2.94 3.44 3.35Window Shift: 1 /