Frequency dependence of the large frequency separation of solar-like oscillators: Influence of the Helium second-ionization zone
aa r X i v : . [ a s t r o - ph . S R ] S e p Mon. Not. R. Astron. Soc. , 1–6 (2011) Printed 4 September 2017 (MN LaTEX style file v2.2)
Frequency dependence of the large frequency separation ofsolar-like oscillators: Influence of the Heliumsecond-ionization zone
S. Hekker , , Sarbani Basu , Y. Elsworth , W.J. Chaplin Astronomical Institute ‘Anton Pannekoek’, University of Amsterdam, Science Park 904, 1098 HX Amsterdam, the Netherlands School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Department of Astronomy, Yale University, P.O. Box 208101, New Haven CT 06520-8101, USA
ABSTRACT
The large frequency separation (∆ ν ) between modes of the same degree and consec-utive orders in a star is approximately proportional to the square root of its meandensity. To determine ∆ ν as accurately as possible a mean large frequency separation( h ∆ ν i ) computed over several orders is often used. It is, however, known that ∆ ν varies with frequency in a second order effect. From observations it has been shownthat this frequency dependence is more important for main-sequence stars than it isfor red-giant stars. Here we use YREC models to verify and explain this observationalresult. We find that for stars with R & ⊙ the effect of the Helium second ionisationzone is relatively small. For these stars the deep location of the He II zone induces afrequency modulation covering only a few ∆ ν , while the amplitude of the modulationis low due to the relatively weak and extended He II layer, causing a shallow widedepression in Γ . For less evolved stars the He II zone is located closer to the surface,and it is more confined, i.e. a deep narrow depression in Γ . This causes frequencymodulations with relatively high amplitudes covering up to about 20∆ ν , inducing arelatively large frequency modulation. Additionally, we find that for less evolved starsthe He II zone is stronger and more localised for more massive stars and for stars withlow metallicities further increasing the amplitude of the frequency modulation. Key words: stars: oscillations – stars: late type – stars: interiors
Stellar oscillations can be used to determine the internalstructures of stars. Solar-type stars, subgiants and red-giantstars exhibit solar-like oscillations, i.e., oscillations stochas-tically excited in the turbulent outer layer. These oscillations generally appear in regular patterns in frequency followingto reasonable approximation the asymptotic relation derivedby Tassoul (1980): ν n,l ≈ ∆ ν ( n + ℓ/ ǫ ) − ℓ ( ℓ + 1) D , (1)with ν n,l the frequency of an oscillation mode with radial order n and degree ℓ and ∆ ν the large frequency separationbetween modes of the same degree and consecutive orders. D is most sensitive to deeper layers in the star and ǫ to thesurface layers.The large frequency separation is inversely propor- tional to the sound travel time through the star and itcan be shown that this is directly proportional to thesquare root of the mean density of the star (Ulrich 1986;Kjeldsen & Bedding 1995). To a first order approximation the large frequency separation is constant over the observed frequency range and often a mean large frequency separa-tion ( h ∆ ν i ) is computed. However, for the Sun and othermain-sequence stars it has been shown that the frequencyrange over which h ∆ ν i is computed has a significant in-fluence on the resulting value. This frequency dependence can be attributed to acoustic glitches, i.e., sudden inter-nal property changes at the base of the convection zoneand at the Helium second-ionization zone. Also other moreslowly varying underlying variations, such as changing con-ditions close to the core, can cause departures from uni- form ∆ ν . Theory and methods considering acoustic glitcheswere first explored in the context of studies of solar oscil-lations, as initially proposed in Gough (1990), and devel-oped further by e.g. Basu & Antia (1994), Monteiro et al.(1994), Perez Hernandez & Christensen-Dalsgaard (1998), Basu & Mandel (2004). Different methods of analysis forextracting data on the glitches have been developed(e.g. see Mazumdar & Antia 2001a,b; Ballot et al. 2004;Verner et al. 2006; Houdek & Gough 2007; Roxburgh 2010), c (cid:13) S. Hekker et al.
Figure 1.
YREC evolutionary tracks in a Hertzsprung-Russell di-agram. Tracks with approximately solar metallicity are indicatedwith diamonds, while the solid lines show the tracks with [Fe/H]= +0.3 and the dashed lines indicate the tracks with [Fe/H] = − M = 1 . ⊙ (black) and 2.4 M ⊙ (red).Models with M = 1 . ⊙ , [Fe/H] = 0.0 including either dif-fusion, Krishna Swamy T - τ relation or α = 1 . H p (see textfor more details) are indicated with the black, mint and purpledashed-dotted lines, respectively. These lines nearly coincide withthe solar metallicity M = 1 . ⊙ models indicated with the blackdiamonds. The black cross indicates the model used in Fig. 2. Figure 2. ∆ ν computed using individual frequencies of ℓ = 0modes as a function of radial order (bottom axis) or frequency(top axis) for the 1.2 M ⊙ star indicated with the black cross inFig. 1. The intervals used to compute h ∆ ν i are indicated with thered and green vertical lines. in some cases for application to other stars. Recently, the HeII ionization zone glitch signal has been measured in thered giant HR7349 (Carrier et al. 2010; Miglio et al. 2010).Additionally, Mazumdar & Michel (2010) have claimed de-tection of acoustic glitches, from the HeII ionization zoneand the base of the convective envelope, in solar-like oscil- lations of the F-type star HD49933.In contrast to the frequency dependence of h ∆ ν i for theSun and main-sequence stars, first results from the Kepler mission (Borucki et al. 2010) showed that for red giants thevalue of h ∆ ν i is much less sensitive to the frequency range over which it was computed (Hekker et al. 2011). Here weinvestigate the reason for the difference in sensitivity of h ∆ ν i to the frequency range for main-sequence stars and red gi- ants using models constructed using YREC, the Yale stellarevolution code (Demarque et al. 2008). A full overview of the YREC code can be foundin Demarque et al. (2008). We use OPAL opacities(Iglesias & Rogers 1996) supplemented with low tempera-ture (log
T < .
1) opacities of Ferguson et al. (2005) and the OPAL equation of state (Rogers & Nayfonov 2002). Allnuclear reaction rates are obtained from Adelberger et al.(1998), except for that of the N ( p, γ ) O reaction, forwhich we use the rate of Formicola et al. (2004).In this work we use a sequence of models of solar metal- licity with masses between 1.0 and 2.4 M ⊙ . The sequencesrange from just after ZAMS to near the tip of the giantbranch for stars of masses up to 2.0 M ⊙ and up to the he-lium burning stage for the higher mass models. Additionallywe also use 1.0 and 2.4 M ⊙ models with [Fe/H]=+0.3 and − . Kepler . Themodels were constructed without diffusion and gravitationalsettling of helium and heavy elements. However, we do ex- amine the role of diffusion and gravitational settling usingthe Thoul et al. (1994) prescription for a sequence of 1.0 M ⊙ models. All models described here were constructed to havethe Eddington T - τ relation in their atmosphere. To studythe effect of atmospheric structure we use a sequence of 1.0 M ⊙ models constructed with the Krishna Swamy T - τ re-lation (Krishna Swamy 1966). Additionally, to examine theeffect of the mixing-length parameter, α , we have made asequence of 1.0 M ⊙ models with α = 1 . H p instead ofthe solar value of α = 1 . H p used in the other models. Evolutionary tracks of the models are shown in Fig. 1.Note that we consider solar-like oscillations in modelsof main-sequence stars up to masses of 2.4 M ⊙ , althoughsolar-like oscillations are generally not observed in main-sequence stars with masses above roughly 1.5 M ⊙ . Further- more, we investigate models with radii less than 20 R ⊙ andwith ∆ ν > µ Hz compatible with the observational resultsby Hekker et al. (2011). h ∆ ν i ON FREQUENCYRANGE To investigate the frequency dependence of h ∆ ν i from themodels, we derived h ∆ ν i from linear fits of the frequen-cies of radial modes in the intervals ν max ± ν ) scaling and ν max ± ν ) scaling as a function of their radial orders(see Fig. 2 and Eq. 1). ν max is the frequency of maximum oscillation power estimated from the mass, radius and ef-fective temperature of the models (Brown et al. 1991) and(∆ ν ) scaling is the mean large separation obtained from themass and radius of the star, using scaling relations (Ulrich1986; Kjeldsen & Bedding 1995). We also investigated the median ∆ ν values in the respective frequency intervals toinvestigate the symmetry of ∆ ν in frequency. This showedthat the behaviour of ∆ ν with frequency is antisymmetricabout ν max and that a linear fit can be used in a first order c (cid:13) , 1–6 requency dependence of ∆ ν : Influence of the He II zone Figure 3.
Left: Ratio of h ∆ ν i values computed from a linear fit to the frequencies versus the radial order of the modes in the intervals ν max ± ν and ν max ± ν as a function of stellar radius for different masses (top) and different metallicities or diffusion (bottom).Right: Same as right panels but now as a as a function of h ∆ ν i . The dashed black line indicates unity. approximation. We note that in those cases where it is not possible to extract individual radial-mode frequencies fromreal data, techniques are usually applied to the full oscil-lation power spectrum to estimate h ∆ ν i , in which case theestimated value is also affected by non-radial modes.The ratio of the h ∆ ν i values computed over each fre- quency range gives an indication of the sensitivity of h ∆ ν i tothe frequency range. Fig. 3 shows this ratio as a function ofstellar radius. The scatter in the ratio of h ∆ ν i computed overdifferent intervals symmetric around ν max decreases with in-creasing stellar radii and that the ratios converge to 1.0. From the ratios of h ∆ ν i we find that the scatter at lowradii ( . ⊙ ) reduces for masses below roughly 1.5 M ⊙ .On the other hand no significant change in the scatter ispresent due to different metallicities or due to the inclusionof diffusion, different atmospheres or mixing-length param- eters in the models (see bottom panels of Fig. 3). Note thatthe relative change in the values of h ∆ ν i for stars with radii . ⊙ as a function of the frequency range is of the sameorder as the precision as with which h ∆ ν i can be determinedfrom current state-of-the-art data. Hekker et al. (2011) suggested two possible reasons forthe decreased influence of the frequency range on h ∆ ν i forred giants: 1) the trend in ∆ ν over a typical frequency rangeis approximately linear, and/or 2) ∆ ν changes relativelyslowly with frequency. Other reasons might be that in red gi- ants fewer modes are observed, and these modes have lowerradial orders than in less evolved stars. We test these sug-gestions by fitting a linear polynomial through the values of ∆ ν over a range ν max ± ν ) scaling and investigate the slopeof the fit and the standard deviation of the values around the fit. These results are shown in Fig. 4 and show thatthe slope or linear trend of the variation in ∆ ν as a func-tion of frequency for stars with lower ∆ ν , i.e., stars withlarger radii, is larger than for stars with higher values of∆ ν , with little sensitivity to the mass, metallicity, diffusion, atmosphere and mixing-length parameter. For higher massmodels there is some variation visible in the trend, whichcoincides with the ‘hook’ in the H-R diagram (Fig. 1), i.e.,the contraction during the turn-off of the main sequence.The scatter around the fit is however much lower for stars with ∆ ν below ∼ µ Hz than for stars with larger valuesof ∆ ν . Thus we indeed conclude that stars with lower ∆ ν ( . µ Hz), i.e., larger radii ( & ⊙ ), have a predominantlylinear dependence on frequency, while for stars with larger∆ ν ( & µ Hz), i.e., smaller radii ( . ⊙ ), the trend is shallower and the scatter in ∆ ν increases. The amount ofscatter does not seem to depend critically on the inclusionof diffusion, different atmosphere or mixing-length param-eter in the models, while it increases with mass and withdecreasing metallicity. Acoustic glitches, i.e., regions of sharp-structure variationin the stellar interior, are known to cause a modulation of∆ ν with frequency. For the Sun and other main-sequence c (cid:13) , 1–6 S. Hekker et al.
Figure 4.
Left: Linear trend in ∆ ν over the range ν max ± ν for models with different masses (top) and different metallicity, diffusion,atmosphere or mixing-length parameter (bottom) as a function of ∆ ν derived from scaling relations. Right: Standard deviation of ∆ ν after subtracting the linear fit over the range ν max ± ν . stars both the Helium second-ionization zone (He II zone) and the base of the convection zone have been shownto contribute to the modulation of the frequencies (e.g.Monteiro & Thompson 2005; Verner et al. 2006, and refer-ences in the introduction). Generally, the signal of the baseof the convection zone is weaker than the signal from the He II zone, due to its location deeper in the star. Additionally,Miglio et al. (2010) already showed that for a red giant onlythe He II zone caused a measurable modulation.The He II zone causes a local depression in the firstadiabatic exponent γ = ( ∂ ln p/∂ ln ρ ) s , with p pressure, ρ density and s specific entropy. This glitch modulates thefrequencies in a sinusoidal manner with a so-called ‘period’inversely proportional to the acoustic depth ( τ ), with τ = Z Rr drc , (2)in which r indicates the radius of the glitch, R the ra- dius of the star and c sound speed. The ‘period’ due tothe bottom of the convection zone, located at the posi-tion where ∇ T becomes larger than ( ∇ T ) ad , has been com-puted similarly with respect to the closest acoustic boundary(Mazumdar & Antia 2001b). The period due to the He II zone depends on the loca-tion of the depression in γ , i.e., a depression close to thesurface will cause a lower value for τ than a depression lo-cated deeper in the star. We computed the expected period( τ − ) of the frequency modulation from the location of the He II zone for all our models and divided it by (∆ ν ) scaling to investigate how many orders one modulation period covers.This is shown in Fig. 5. For models with ∆ ν < µ Hz themodulation period is about 5∆ ν and not significantly depen-dent on mass, metallicity, diffusion, atmosphere or mixing- length parameter, while the period and the spread in periodsincrease for models with ∆ ν > µ Hz.To determine the dependence of h ∆ ν i on the frequencyrange we added/removed 2∆ ν on either side of the frequencyinterval (Section 3). This is about one full period for models with ∆ ν < µ Hz and no significant net effect in the meanvalue of ∆ ν has been observed. For stars with ∆ ν > µ Hzthe period depends on mass and is in some cases signifi-cantly longer than 5∆ ν implying that only part of a periodis added/removed for the different frequency ranges. This does have an effect on the resulting mean values of ∆ ν . Fromthis we can conclude that for larger stars ( R > ⊙ ) thesmooth stellar structure changes reflected in the linear trendof ∆ ν are dominant over the effect from the He II acousticglitch, while for smaller stars ( R < ⊙ ) the effect from the He II acoustic glitch is dominant.In addition to the period, we also investigated the am-plitude of the variations in ∆ ν , parametrized by the stan-dard deviation (see right hand side of Fig. 4). The amplitudeof the modulation is larger for stars with ∆ ν & µ Hz than for stars with ∆ ν . µ Hz and depends on mass and metal-licity (see Fig. 4). We expect the amplitude to depend on theshape of the glitch. We parametrize the shape of the glitchby the ratio of the depth of the He II depression in γ , i.e.its strength, to the width (in units of acoustic radius) of the c (cid:13) , 1–6 requency dependence of ∆ ν : Influence of the He II zone Figure 5.
Period of the frequency modulation expressed in unitsof ∆ ν as a function of ∆ ν . Figure 6.
The depth of the He II depression in γ divided by thewidth of the depression expressed in acoustic radius as a functionof ∆ ν . depression, i.e., the narrower the width for a given depththe bigger the effect on the frequencies. The ratios for allmodels are shown in Fig. 6 as a function of ∆ ν .The changes in the shape of the He II depression in γ at a given ∆ ν are clearly reflected in the changes in the stan- dard deviation of the variation in ∆ ν at that ∆ ν (see Fig. 7).The general trends in the curves for ∆ ν and the depth towidth ratio of the He II zone are the same, although thereare differences in the amplitude of the variations. Thereforewe conclude that the shape of the He II depression in γ , i.e. the ratio of the strength over the width of the He II zone,is a significant cause of the difference in the amplitudes ofthe variation in ∆ ν which vary with mass and metallicity.However, there must be additional effects that play a role.This could be the base of the convection zone, which induces a small modulation with a period that is of the same orderas the period of the He II zone.We note that the increasing value of ∆ ν at which theamplitude of the variation in ∆ ν increases is a function ofmass and metallicity (Fig. 4) and that this effect is consis- tently present in both the period and shape of the He II zone(Figs. 5 & 6). Figure 7.
For M = 1 . ⊙ models the standard deviation of thevariation in ∆ ν (asterisks) and the depth of the He II depressionin γ divided by the width of the depression expressed in acousticradius (diamonds) as a function of ∆ ν . In this work we confirm the statement by Hekker et al.(2011) that the value of h ∆ ν i computed for red giants is not as sensitive to the frequency range over which it is com-puted as is the case for the Sun or other main-sequencestars. The analysis of frequencies from YREC models ofstars with different masses and metallicities and with theinclusion of diffusion, a different atmosphere and different mixing-length parameter clearly show that for stars withlower ∆ ν ( . µ Hz), i.e., larger radii, the dependence of∆ ν follows a linear trend with relatively low scatter. Forstars with higher ∆ ν ( & µ Hz), i.e., smaller radii, thelinear trend diminishes, while the scatter increases, with a peak in the scatter at about 100 µ Hz. The magnitude of thispeak in the scatter increases with mass and with decreasingmetallicity, but does not seem to be sensitive to the inclu-sion of diffusion, a different atmosphere or mixing-lengthparameter in the models.
The lower sensitivity of h ∆ ν i on frequency is due todeeper location of a weaker and wider He II zone for ex-tended evolved stars. For less evolved stars the shallowerlocation of a stronger and thinner He II zone causes an in-creased sensitivity of h ∆ ν i on frequency. The strength and width of the He II zone in less evolved stars changes as afunction of mass and metallicity. ACKNOWLEDGEMENTS
SH acknowledges financial support from the Netherlands Or-ganisation for Scientific Research (NWO). YE and WJC ac- knowledge financial support of UK STFC. SH thanks M.Mooij for useful discussions.
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