arXiv:2001.10949v2 [astro-ph.SR] 31 Jan 2020 MNRAS g,a i cln eain (e.g. relations scaling via as ogy, ahlHowe, Rachel asteroseismology for implications ⋆ that damped explicitly and suggested excited first are oute modes the the of where characteristics ( proper- the star by stellar the set of fundamental is layers constrain it though help even to ties, used be can 1995 oa yl aito of variation cycle Solar u .Davies, R. Guy etdasaigwt ufc gravity, surface sug- with turn scaling in a which gested frequency, cut-off acoustic photospheric © perature, ee facrc fti cln eg,see (e.g., scaling this of accuracy of level with atnB Nielsen, Bo Martin cetdXX eevdYY noiia omZZZ form original in YYY; Received XXX. Accepted twihteevlp fteplainsetu a its has spectrum as pulsation frequency known the is the of power oscillators, in envelope solar-like maximum the other which and at Sun the In INTRODUCTION 1 ν ν 4 yMneae10 K80 ahsC Denmark C, Aarhus DK-8000 3 120, Munkegade Ny 2 1 ekcme l 2011 al. et Belkacem max max tla srpyisCnr SC,Dprmn fPyisa of Department Birmingham, (SAC), of Centre University , Stellar and Physics of School -al .oeba.cu (RH) [email protected] E-mail: etrfrSaeSine YA nttt,NwYr Univers York New Institute, NYUAD Science, Space for Center eateto srnm,Yl nvriy ..Bx208101, Box P.O. University, Yale Astronomy, of Department 09TeAuthors The 2019 .Teqatt ly oei nebeasteroseismol- ensemble in role a plays quantity The ). ∝ ocntanselrpoete et o nyo the on only not rests properties stellar constrain to M g T h asand mass the 000 eff − T 1 eff / , 2 fteform the of , 1 ∝ – 6 MR 21)Pern eray22 oplduigMRSL MNRAS using Compiled 2020 February 3 Preprint (2019) − 1 2 , T , 2013 2 eff R − 1 ⋆ 1 , h aiso h tr h s of use The star. the of radius the 2 / 2 ila .Chaplin, J. William , .I was It ). vneElsworth, Yvonne 1 , ν 2 max , OGadGL,adpooer aafo IG/P nteESA/ the velo on Doppler VIRGO/SPM significant Sun-as-a-Star nevertheless from to but solar data the weak fit of a photometry discover the We and apply spacecraft. SOHO also GOLF, binned We and to data. GONG model wit the simple observations of a velocity fitting tra by Doppler (BiSON), Sun-as-a-Star Network of Solar-Oscillations years 25 using as ae ndrc s fatrsimcsaigrltoscalibrat relations scaling asteroseismic of use mean direct on based mass, of h edfrcuinwe using when caution for need the frequency, The ABSTRACT iltr sa motn uniyi seoesooy emeasure We asteroseismology. in quantity important an is cillators e words: Key 4 ν hpi ilo2013 Miglio & Chaplin max n ua .Viani S. Lucas and rw tal. et Brown ≃ ih erltdt the to related be might 25 ( g jlsn&Bedding & Kjeldsen ν n ffcietem- effective and , max oloe l 2015 al. et Coelho µ Hz ntedffrn aaesi loceryosti rqec.Orresu Our frequency. in offset clearly also is datasets different the in ν rnltst nucranyo . e eti aisad24prce per 2.4 and radius in cent per 0.8 of uncertainty an to translates , max ( u:hloesooy–Sn ciiy–asteroseismology – activity Sun: – Sun: 1991 ν ν ihslratvt.Teucvrdsitbtenlwadhg activity high and low between shift uncovered The activity. solar with max max twihteevlp fplainpwrpasfrslrlk os- solar-like for peaks power pulsation of envelope the which at , who ) 1 it ) , (1) 2 1 dAtooy ahsUniversity, Aarhus Astronomy, nd , r ; nhloesi aaadits and data helioseismic in 2 tvnJ Hale, J. Steven imnhm 1 T,Uie Kingdom United 2TT, B15 Birmingham, e ae,C 62-11 USA 06520-8101, CT Haven, New abn Basu, Sarbani t b hb,P o 218 b hb,Uie rbEmirates Arab United Dhabi, Abu 129188, Box PO Dhabi, Abu ity 3 t xeddK iso hs ( phase mission K2 extended its .. rmteGoa silto ewr ru (GONG) Group Network Oscillation ( Global telescopes the from e.g., ntems o ie tr ae ndrc s fscaling of use cent direct per 8 on and based star, radius given the a (e.g. in relations for cent mass the per 2.5 in about of ference in ewr ru SN)setorp e ihscat- with fed ( spectrograph sunlight tered (SONG) Group Network tions estimated instead fromhave NASA sunlight the reflected by used captured estimate Neptune photometric other 2014 ue ta.2011 al. et Huber 2017 al. et Viani n htmti u-sasa aa( data Sun-as-a-star photometric ing h eeec.Rcn siae fi nteltrtr cove literature the in to 3080 it range of the estimates Recent reference. the S/AASH pccat( spacecraft SOHO ESA/NASA airt h eain h solar The relation. the calibrate nterpre aus ieecsi ehdlg a play may methodology in Differences values. reported the in ν max ayetmtso h solar the of estimates Many hr r eea atr htcnrbt otespread the to contribute that factors several are There olce yteVROSMisrmn nthe on instrument VIRGO/SPM the by collected ) nasteroseismology. in ifre l 2018 al. et Kiefer hpi ilo2013 Miglio & Chaplin rdln nesne l 2019 al. et Andersen Fredslund ,btas nhvn outrfrneto reference robust a having on also but ), 1 3 , 3160 2 ; arc .Ball, H. Warrick nraMiglio, Andrea osre l 2013 al. et Mosser µ ν Hz max hssra rnltst dif- a to translates spread This . sn aai ope velocity, Doppler in data using ;o rmteSelrObserva- Stellar the from or ); alee l 2016 al. et Gaulme ν Fr max Kepler hihe l 1995 al. et ¨ ohlich airtdt h Sun. the to calibrated ) ν alne ta.2010 al. et Kallinger suulyaotdas adopted usually is max oiiecorrelation positive h Birmingham the h dt h u.The Sun. the to ed ; A ν T max eecp during telescope oefo us- from come iydt from data city E 1 alne tal. et Kallinger tl l v3.0 file style X 1 , , 2 ). oe spec- power 2 o h Sun the for .Others ). t flag lts NASA .An- ). tin nt r , ; 2 R. Howe et al.

2 a role, e.g., the parametric model used to fit the data. For ex- factor of 4 sin (πν/2νNyq), where νNyq is the Nyquist fre- ample, fits to stars and solar data typically assume a Gaus- quency – 8333 µHz for the 60-second cadence of GONG. For sian envelope for the pulsation spectrum (e.g. Lund et al. VIRGO/SPM, we used up-to-date level-1 data2 from the 2017); however, the solar envelope is clearly asymmetric Red channel. The 22-year GOLF dataset comes from a new (Kiefer et al. 2018). calibration and is an average of signals from the PM1 and Differences related to the data are undoubtedly im- PM2 detectors (Appourchaux et al. 2018). portant, beginning with how the oscillations are observed. The procedure for measuring νmax over time was as fol- Doppler results tend to give a higher νmax than do photomet- lows. The time series of observations were divided into over- ric observations. There are also differences between different lapping segments – each of length 1 year, with start-times Doppler datasets. Doppler observations are sensitive to per- offset by 3 months – and a Fourier power spectrum was com- turbations at different heights in the stellar atmosphere, de- puted for each segment. Each spectrum was then averaged pending on the Fraunhofer line (or lines) used. The locations over a number of frequency bins of width 135 µHz, corre- of the outer boundaries of the mode cavities lie beneath sponding to the separation between modes of the same de- the photosphere, and depend on mode frequency. Pertur- gree and adjacent radial order, the so-called large frequency bations due to the modes are therefore evanescent in the separation ∆ν. Binning gives data that have an underlying photosphere, with the observed signal suffering frequency smooth trend in frequency and statistics that tend to Gaus- dependent attenuation with increasing height. This has the sian. potential to affect the observed νmax. Also of note is that We used the lmfit python package (Newville et al. the ratio of the amplitudes of signals due to oscillations and 2018) to perform a non-linear least-squares fit of a custom granulation is much lower in photometry than in Doppler model to each binned power spectrum. Taking inspiration velocity. This means that for photometric data, the back- from Kiefer et al. (2018), the model consists of an asymmet- ground power spectral density is dominated in frequency by ric pseudo-Voigt profile (Stancik & Brauns 2008). There is the granulation, and the oscillations are usually observed at also a frequency-dependent background term, and a constant a significantly lower signal-to-noise ratio than in Doppler ve- background offset. The oscillation envelope profile takes the locity. Most results come from Sun-as-a-star data, which are form sensitive to modes of low angular degree, l; but one recent P(ν) = f × PL(ν) + [1 − f ]× PG(ν), (2) estimate has been made using a much wider set of modes (2 ≤ l ≤ 150; see Kiefer et al. 2018). Finally, even though where ν is frequency, PL(ν) and PG(ν) are respectively the data from the same instrument may have been used, the se- Lorentzian and Gaussian parts of the profile, and f is the lections are not usually contemporaneous. It is on this last factor governing the balance between the Lorentzian and point that we focus here. Gaussian contributions. Specifically, we use While most stellar observations cover relatively short X = [(ν − ν )/Γ(ν)]2, (3) time intervals, for the Sun we have more than twenty years max of high-quality observations from multiple instruments, in where multiple observables. It is therefore of interest to see how [a(ν−νmax)] Γ(ν) = 2Γ0/[1 + exp ] (4) precisely we can in fact measure the solar νmax over long periods of time using different instruments, as this has im- describes the frequency dependence of the width, with Γ = Γ0 plications for the use of the quantity in scaling relations, and at ν = νmax and a being an asymmetry term. The Lorentzian to see whether there is any intrinsic variability related to the term is . That is the objective of this paper. 2H PL(ν) = , (5) (1 + 4X)πΓ(ν) and the Gaussian term is 2 DATA AND METHODS Hp4 ln(2) PG(ν) = , (6) Our primary dataset consists of Sun-as-a-star Doppler ve- πΓ(ν) exp[4X ln(2)] locity observations from the Birmingham Solar-Oscillations where H governs the height of the oscillation power envelope. Network (BiSON; Chaplin et al. 1996; Hale et al. 2016). A weakness of this parametrisation is that if both νmax The data we used cover the period 1995 to early 2018, and a are allowed to vary independently the fit results for or nearly two full solar cycles. For comparison, we also these two parameters are highly correlated. For the final fits considered velocity data from the Global Oscillation Net- we therefore selected a fixed value of a, by repeating the fits work Group (GONG; Harvey et al. 1996) and from the for a range of a and selecting the one that gave the lowest Global Oscillations at Low Frequency (GOLF) instru- overall χ2 for the whole dataset. For the background we ment (Gabriel et al. 1995) on SOHO; and photometry data tried both a Lorentzian model centred on zero frequency, from the Red channel of the VIRGO/SPM instrument and a non-parametric background given by smoothing with (Frohlich et al. 1997), also on SOHO. For GONG we used a median filter. l = 0 the spherical harmonic time series provided by Fig. 1 shows the mean of the bin-averaged power spec- the GONG project1. The GONG data have been treated tra from each full dataset, all normalized to show the same with a first-difference filter; to correct for the effects of power spectral density at the peak of the five-minute en- this on the spectral power we divide the spectrum by a velope (arbitrarily scaled to unity on the plot). The figure

1 Available from gong.nso.edu 2 SOHO.nascom.nasa.gov/data/data.html

MNRAS 000, 1–6 (2019) Solar cycle variation of νmax 3

Table 1. Results of fits of the best-fitting νmax from Fig. 2 to the 10.7-cm radio flux, using the linear model described by Equa- VIRGO tion 7. 1.00 GONG BiSON GOLF red Dataset c0 c1 GOLF blue (µHz) (µHz RF−1)

VIRGO 3085 ± 4 0.18 ± 0.06 0.10 GONG 3138 ± 2 0.14 ± 0.03 BiSON 3169 ± 2 0.11 ± 0.03 GOLF blue 3261 ± 4 0.15 ± 0.07 Normalized smoothed PSD

0.01 2000 3000 4000 5000 6000 3 RESULTS Frequency (µHz) We comment first on the variability of νmax over time, which Figure 1. Mean of the bin-averaged power spectra from each full is the focus of the paper. Fig. 2 shows the best-fitting νmax dataset, normalized to show the same power spectral density at from each dataset as a function of time (coloured points the peak of the five-minute envelope (arbitrarily scaled to unity with associated error bars). The results for BiSON (black on the ordinate). points), GONG (green points) and VIRGO (violet points) show a similar pattern of variation, and a significant posi- tive correlation with solar activity. This correlation is shown clearly in Fig. 3, which plots the best-fitting νmax as a func- tion of the 10.7 cm radio flux, F10.7 (Tapping 2013), which we adopt as a proxy of global solar activity. The dotted lines in Fig. 3 are from fits of a simple linear model of the form: shows clearly differences in the shape of the observed power ν (t) = c + c ×(F (t)− 110) . (7) spectra. The final fits adopted fixed values of the asymme- max 0 1 10.7 try parameter of a = −0.5 (BiSON), −0.3 (GONG) and −0.7 The offset of 110 radio flux units corresponds to the average (VIRGO). Note a negative asymmetry implies more power 10.7-cm flux over the full period of the data; its introduction on the high-frequency side of the oscillation envelope. Two to the model means c0 corresponds to νmax at average activ- mean spectra are shown for GOLF: one made from data ity. The solid lines in Fig. 2 show the scaled 10.7 cm radio collected after 1998 September and before 2002 November, flux from this linear model. Table 1 reports the best-fitting when in our final fits a was fixed at −1.2; and another made coefficients of each fit. from data collected at earlier and later dates, when it was We also performed the analysis using two independent fixed at a = −1.9. This reflected a change in the instrument’s pipelines to verify the results. One pipeline (Nielsen et al. observing mode, from detecting Doppler shifts before 1998 2017) used a Markov chain Monte Carlo sampler to fit the September in the blue wing of the Sodium doublet at 589 nm; raw (unaveraged) power spectra to a model comprising three to then detecting them in the red wing up to 2002 Novem- background terms (two Lorentzians and a flat offset) and a ber; and finally to detecting them back in the blue wing Gaussian for the mode envelope. The other pipeline adopted thereafter. We found that the free parameter f converged a very different approach, following Huber et al. (2009). In on values typically around 20 % or lower for VIRGO, in the brief, after dividing out the background – as estimated using range 30 % to 40 % for BiSON and GONG, around 60 % for a moving-median filter – we computed the autocorrelation of GOLF red wing, and even higher for GOLF blue wing. 675 µHz (≃ 5∆ν) wide ranges of the power spectrum, sliding a The statistics of the bin-averaged spectrum mean that frequency window through the full spectrum. We then fitted the uncertainties in each power average are highest near a Gaussian to the sum over all lags to estimate νmax, with the νmax, where the modes are most prominent. That is because correlation between overlapping segments included in the fit. individual bins in each ∆ν-wide segment that contribute to Both pipelines uncovered the same temporal variations as the re-binned averages will span a considerable range in our main pipeline, showing our results on the time variation power, from high-power spectral densities across individ- of νmax are robust against the choice of fitting model and ual modes down to lower-power spectral densities between fitting procedure. modes. An error-weighted fit would then be dominated by We see a positive shift in νmax of ≃ 25 µHz between low the averaged bins in the wings of the power envelope and and high activity. The results for GOLF are however more the background, where the dispersion from individual bins complicated, and are dominated by the change in observ- contributing to each average is lower. We therefore choose ing mode. νmax is around 80 µHz lower in the red-wing data to fit the logarithmic spectrum, because this gives similar than it is in the blue-wing data. If we separate out the blue- relative uncertainties in every bin. We used a Monte Carlo wing data, which cover a much longer period, it too shows method to estimate the uncertainties, in which the fit was a positive correlation with activity. repeated for 1000 realizations of each spectrum taken from a Barban et al. (2013) studied the solar νmax in data from normal distribution of width σP(ν) centered on the observed GOLF and VIRGO collected between 1996 and 2004 and value P(ν) at each frequency bin, and the uncertainty was found that the GOLF νmax appeared to be anti-correlated taken to be the standard deviation of the resulting estimated with the sunspot number but that from VIRGO was not. parameters. Based on our results above, it now seems clear that the ap-

MNRAS 000, 1–6 (2019) 4 R. Howe et al.

3300

3250 GOLF blue

3200 GOLF red BiSON Hz) µ (

max 3150 ν

GONG 3100

3050 VIRGO

1995 2000 2005 2010 2015 2020 Date (year)

Figure 2. Best-fitting νmax from each dataset as a function of time (coloured points with associated error bars). The results are for BiSON (black points), GONG (green points), VIRGO (violet points), GOLF blue wing (blue points) and GOLF red wing (red points). The solid lines show the scaled 10.7 cm radio flux from fits of the linear model defined by Equation 7.

30 BiSON 30 GONG 20 20 10 10 Hz) Hz) µ µ

( 0 ( 0 max −10 max −10 δν δν −20 −20 −30 −30 50 100 150 200 50 100 150 200 10.7−cm flux (radio flux units) 10.7−cm flux (radio flux units)

30 VIRGO 30 GOLF blue 20 20 10 10 Hz) Hz) µ µ

( 0 ( 0 max −10 max −10 δν δν −20 −20 −30 −30 50 100 150 200 50 100 150 200 10.7−cm flux (radio flux units) 10.7−cm flux (radio flux units)

Figure 3. Best-fitting νmax as a function of the 10.7 cm radio flux, for BiSON (top left-hand panel), GONG (top right-hand panel), VIRGO (bottom left-hand panel) and GOLF blue wing (bottom right-hand panel. The dotted lines show fits of the linear model defined by Equation 7. parent anti-correlation with activity they reported is actu- low and high activity (centered on epochs around 2001 and ally due to the change in operation from one wing to another 2017, respectively). The cycle-related shift is clearly visible. (and back again). That Barban et al. (2013) found no appar- ent change in the VIRGO νmax is likely due to them having had less data than are available to us now, i.e., the changes we uncover, whilst significant, are nonetheless quite weak. 4 DISCUSSION AND CONCLUSIONS We have discovered a weak but nevertheless significant pos- Our results also show significant differences in the aver- itive correlation of the solar νmax with solar activity. The age νmax for the different datasets (e.g. see the c0 estimates uncovered shift, of ≃ 25 µHz between low and high activity, in Table 1). The left-hand panel of Fig. 4 shows a zoom of translates to an uncertainty of up to 0.8 per cent in radius the mean spectra from Fig. 1 after a mean best-fitting back- and 2.4 per cent in mass, based on using the Sun as a cali- ground model was removed from each. The residual spectra brator in direct use of the scaling relations. Our result also have then been offset on the ordinate to show more clearly suggests that we might expect to find variations of νmax in the differences in νmax. The absolute spread is noticeably other stars, which would add additional uncertainty to any larger than the activity-related variations. The photometric inferences made using this global asteroseismic parameter (a VIRGO results show the lowest average νmax. The right-hand point we come back to below). panel shows a zoom of mean BiSON spectra from epochs of The underlying causes of the variations we have uncov-

MNRAS 000, 1–6 (2019) Solar cycle variation of νmax 5

VIRGO low activity high activity BiSON BiSON GONG BiSON GOLF red GOLF blue Residual smoothed and offset PSD Residual smoothed and offset PSD

2800 3000 3200 3400 3600 2800 3000 3200 3400 Frequency (µHz) Frequency (µHz)

Figure 4. Left-hand panel: Zoom of the mean spectra from Fig. 1 after a best-fitting background model was removed from each spectrum. The residual spectra have then been offset on the ordinate. Right-hand panel: zoom of mean BiSON spectra from epochs of low and high activity (centered on epochs around 2001 and 2017, respectively).

ered in the solar νmax must be intimately tied to variations ACKNOWLEDGEMENTS in the power and damping of the modes. There is an exten- We would like to thank all those who are, or have been, sive literature (see Howe et al. 2015, and references therein) associated with BiSON, in particular, P. Pall´eand T. Roca- showing that p-mode powers decrease whilst damping rates Cort´es in Tenerife and E. Rhodes Jr and S. Pinkerton at increase as levels of solar activity rise. The interplay be- Mt. Wilson. BiSON is funded by the Science and Tech- tween the relative sizes of these changes with frequency will nology Facilities Council (STFC). This work utilizes data determine the variation of νmax. obtained by the GONG program, managed by the Na- As noted earlier, the scaling relation for νmax arose tional Solar Observatory, which is operated by AURA, by assuming a proportionate scaling with the photospheric Inc. under a cooperative agreement with the National Sci- acoustic cut-off frequency. It is therefore intriguing to note ence Foundation. The GOLF and VIRGO instruments on that Jim´enez et al. (2011) found a clear positive correlation board SOHO are a cooperative effort of many individu- of the solar cut-off frequency with solar activity. The frac- als to whom we are indebted. SOHO is a project of in- tional change they found is about four-times larger than the ternational collaboration between ESA and NASA. Fund- fractional change we have uncovered in νmax. ing for the Stellar Astrophysics Centre is provided by the Danish National Research Foundation (grant agree- We also find significant differences in the average νmax ment No.: DNRF106). This work made use of the Python given by the various datasets. The ordering of νmax does not packages SciPy (Jones et al. 2001), NumPy (Oliphant respect a simple correlation (or anti-correlation) with the 2006), and Astropy,3 a community-developed core Python average height above the base of the photosphere at which package for Astronomy (Astropy Collaboration et al. 2013; the observations are effectively made. A correlation with in- Price-Whelan et al. 2018). creasing atmospheric height would, moving outwards, sug- gest (e.g., see Jim´enez-Reyes et al. 2007) an ordering in νmax of VIRGO/SPM, GONG, BiSON, GOLF blue, and finally REFERENCES GOLF red (and a reversed order for an anti-correlation); this is not quite what we see. Nevertheless, that there is Appourchaux T., Boumier P., Leibacher J. W., Corbard T., 2018, such a dramatic change between GOLF red and GOLF blue A&A, 617, A108 shows that the way the observations are made matters (see Astropy Collaboration et al., 2013, A&A, 558, A33 also Garc´ıa et al. 2005). Understanding these absolute dif- Barban C., Beuret M., Baudin F., Belkacem K., Goupil M. J., ferences more clearly will be the focus of future work. Samadi R., 2013, in Journal of Physics Conference Series. p. 012031, doi:10.1088/1742-6596/440/1/012031 Our results clearly flag the need for caution when using Belkacem K., Goupil M. J., Dupret M. A., Samadi R., Baudin F., Noels A., Mosser B., 2011, A&A, 530, A142 νmax on other stars. They sound a warning over making sure Belkacem K., Samadi R., Mosser B., Goupil M. J., Ludwig H. G., one has a solar reference extracted from data with the same 2013, in Shibahashi H., Lynas-Gray A. E., eds, Astronomical instrumental response as the stellar data, using the same Society of the Pacific Conference Series Vol. 479, Progress analysis methodology, and for a clearly defined level of solar in Physics of the Sun and Stars: A New Era in Helio- and activity. Since changing levels of activity perturb the solar Asteroseismology. p. 61 (arXiv:1307.3132) νmax, a desirable reference would be one commensurate with Brown T. M., Gilliland R. L., Noyes R. W., Ramsey L. W., 1991, minimal levels of activity (corresponding to minimal impact ApJ, 368, 599 on νmax). But the stellar νmax must then also be compati- ble with regard to activity levels, which may not always be possible to achieve. 3 http://www.astropy.org

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This paper has been typeset from a TEX/LATEX file prepared by the author.

MNRAS 000, 1–6 (2019)