Determining stellar parameters of asteroseismic targets: going beyond the use of scaling relations

2.4 MB·

by  Thaíse S. Rodrigues

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed18January2017 (MNLATEXstylefilev2.2)
Determining stellar parameters of asteroseismic targets:
going beyond the use of scaling relations
7
1 Tha´ıse S. Rodrigues1,2, Diego Bossini3, Andrea Miglio3,4, L´eo Girardi1,
0
Josefina Montalb´an2, Arlette Noels5, Michele Trabucchi2,
2
n Hugo Rodrigues Coelho3,4, Paola Marigo2
a 1 Osservatorio Astronomico di Padova – INAF, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy
J
2 Dipartimento di Fisica e Astronomia, Universita` di Padova, Vicolo dell’Osservatorio 2, I-35122 Padova, Italy

7 3School of Physics and Astronomy, Universityof Birmingham, Edgbaston, Birmingham B15 2TT, UnitedKingdom
1 4Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University,DK-8000 Aarhus C, Denmark
5 Institut d’Astrophysique et de G´eophysique, All´ee du 6 aout, 17 – Bat. B5c, B-4000 Li`ege 1 (Sart-Tilman), Belgium

]
R
S
Accepted 2017January13forpublicationinMNRAS
.
h
p
- ABSTRACT
o
Asteroseismic parameters allow us to measure the basic stellar properties of field gi-
r
t ants observed far across the Galaxy. Most of such determinations are, up to now,
s
based on simple scaling relations involving the large frequency separation, ∆ν, and
a
[ the frequency of maximum power, νmax. In this work, we implement ∆ν and the pe-
riod spacing, ∆P, computed along detailed grids of stellar evolutionary tracks, into
1
stellarisochronesandhenceinaBayesianmethodofparameterestimation.Testswith
v
synthetic data reveal that masses and ages can be determined with typical precision
1
of 5 and 19 per cent, respectively, provided precise seismic parameters are available.
9
Adding independent information on the stellar luminosity, these values can decrease
7
4 downto3and10percentrespectively.TheapplicationofthesemethodstoNGC6819
0 giants produces a mean age in agreement with those derived from isochrone fitting,
. and no evidence of systematic differences between RGB and RC stars. The age dis-
1
persionof NGC 6819 stars,however,is largerthan expected, with at least partof the
0
spread ascribable to stars that underwent mass-transfer events.
7
1
Key words: Hertzsprung–Russell and colour–magnitude diagrams – stars: funda-
:
v mental parameters
i
X
r
a
1 INTRODUCTION to theso-called scaling relations:
∆ν ρ1/2 M1/2/R3/2
∝ ∝
ν gT−1/2 (M/R2)T−1/2 . (1)
max ∝ eff ∝ eff
With the detection of solar-like oscillation in thousands of It is straightforward to invert these relations and derive
red giant stars, Kepler and CoRoT missions have opened masses M and radii R as a function of ν , ∆ν, and T .
max eff
the way to the derivation of basic stellar properties such The latter has to be estimated in an independent way, for
as mass and age even for single stars located at distances instance via the analysis of high-resolution spectroscopy.
of several kiloparsecs (e.g. Chaplin & Miglio 2013, and ref- M and R can then be determined either (1) in a model-
erences therein). In most cases this derivation is based on independentway by the“direct method”, which consists in
the two more easily-measured asteroseismic properties: the simplyapplyingthescalingrelationswithrespecttotheso-
large frequency separation, ∆ν, and thefrequency of maxi- larvalues,or(2)viasomestatisticalmethodthattakesinto
mum oscillation power, ν . ∆ν is the separation between account stellar theory predictions and other kinds of prior
max
oscillation modeswiththesameangulardegreeandconsec- information. In the latter case, the methods are usually re-
utiveradialorders,andscalestoaverygoodapproximation ferred to as either “grid-based” or “Bayesian” methods.
with the square root of the mean density (ρ), while ν Determining the radii and masses of giant stars brings
max
is related with thecut-off frequency for acoustic waves in a consequences of great astrophysical interest: The radius
isothermal atmosphere, which scales with surface gravity g added to a set of apparent magnitudes can be used to esti-
andeffectivetemperatureT .Thesedependenciesgiverise matethestellardistanceandtheforegroundextinction.The
eff
2 Rodrigues et al.
massofagiantisgenerallyveryclosetotheturn-offmassof
Table 1. Initial masses and chemical composition of the com-
itsparentpopulation,andhencecloselyrelatedwithitsage; putedtracks.
the latter is otherwise very difficult to estimate for isolated
fieldstars.Inaddition,thesurfacegravitiesofasteroseismic Mass(M⊙)
targetscanbedeterminedwithanaccuracygenerallymuch
0.60,0.80,1.00,1.10,1.20,1.30,1.40,1.50,1.55,1.60,
betterthan allowed byspectroscopy. 1.65,1.70,1.75,1.80,2.00,2.15,2.30,2.35,2.40,2.45,2.50
Although these ideas are now widely-recognized and
largely used in the analyses of CoRoT and Kepler sam- [Fe/H] Z Y
ples, there are also several indications that asteroseis-
−1.00 0.00176 0.25027
mology can provide even better estimates of masses and −0.75 0.00312 0.25164
ages of red giants, than allowed by the scaling relations −0.50 0.00555 0.25409
above. First, there are significant evidences of correc- −0.25 0.00987 0.25844
0.00 0.01756 0.26618
tions of a few percent being necessary (see White et al. 0.25 0.03123 0.27994
2011; Miglio 2012; Miglio et al. 2013; Brogaard et al. 2016; 0.50 0.05553 0.30441
Miglio et al. 2016; Guggenberger et al. 2016; Sharma et al.
2016; Handberget al. 2016) in the ∆ν scaling relation. Al-
though such corrections are expected to have little impact ferentmetallicitiesrangingfrom[Fe/H]= 1.00to0.50(Ta-
−
on the stellar radii (and hence on the distances), they are ble1)1. Thefollowing points summarize therelevant physi-
expected to reduce the errors in the derived stellar masses, cal inputsused:
hence on the derived ages for giants. Second, there are
The tracks were computed starting from the pre-main
other asteroseismic parameters as well – like for instance •
sequence(PMS) uptothefirstthermalpulseoftheasymp-
the period spacing of mixed modes, ∆P (Beck et al. 2011;
totic giant branch (TP-AGB).
Mosser et al.2014)–thatcanbeusedtoestimatestellarpa-
WeadoptGrevesse & Noels(1993)heavyelementspar-
rameters, although not via so easy-to-use scaling relations •
tition.
as those above-mentioned.
The OPAL equation of state (Rogers & Nayfonov
In this paper, we go beyond the use of simple scaling •
2002) and OPAL opacities (Iglesias & Rogers 1996) were
relationsintheestimationofstellarpropertiesviaBayesian
used, augmented by low-temperature opacities from
methods, first by replacing the ∆ν scaling relation by us-
Ferguson et al. (2005). C-O enhanced opacity tables were
ing frequencies actually computed along the evolutionary
considered duringthehelium-core burning(HeCB) phase.
tracks, and second by including the period spacing ∆P in
A custom table of nuclear reaction rates was used
the method. We study how the precision and accuracy of •
(NACRE,Anguloet al. 1999).
theinferredstellarpropertiesimprovewithrespecttothose
The atmosphere is taken according to KrishnaSwamy
derivedfrom scaling relations, and howthey dependon the •
(1966) model.
set of available constraints. The set of additional parame-
Convectionwastreatedaccordingtomixing-lengththe-
ters to be explored includes also the intrinsic stellar lumi- •
ory,using thesolar-calibrated parameter (α =1.9657).
nosity, which will be soon determined for a huge number MLT
Overshooting was applied during the core-convective
of stars in the Milky Way thanks to the upcoming Gaia •
burningphasesinaccordancewithMaeder(1975)stepfunc-
parallaxes (Lindegren et al. 2016, and references therein).
tion scheme. We use overshooting with a parameter of
The results are tested both on synthetic data and on the
α =0.2H during the main sequence, while we consider
starclusterNGC6819,forwhichKepler hasprovidedhigh- ovH p
α = 0.5H penetrative convection in the HeCB phase
quality oscillation spectra for about 50 giants (Basu et al. ovHe p
(following the definitions in Zahn 1991 and the result in
2011;Stello et al.2011;Corsaro et al.2012;Handberg et al.
Bossini et al. 2015).
2016).
Element diffusion, mass loss, and effects of rotational
The structure of this paper is as follows. Section 2 •
mixing were not taken in account.
presents the grids of stellar models used in this work, de-
Metallicities [Fe/H] were converted in mass fractions
scribes how the ∆ν and ∆P are computed along the evo- •
of heavy elements Z by the approximate formula Z =
lutionary tracks, and how the same are accurately interpo-
latedinordertogenerateisochrones. Section3employsthe Z⊙ ·10[Fe/H], where Z⊙ = 0.01756, coming from the solar
calibration. The initial helium mass fraction Y depends on
isochrone sets incorporating the new asteroseismic proper-
Z and was set using a linear helium enrichment expression
ties to evaluate stellar parameters by means of a Bayesian
approach.Themethodistestedbothonsyntheticdataand ∆Y
Y =Y + Z (2)
on real data for the NGC 6819 cluster. Section 4 draws the p ∆Z
final conclusions.
with theprimordial helium abundanceY =0.2485 and the
p
slope ∆Y/∆Z =(Y⊙ Yp)/Z⊙ =1.007. Table 1 shows the
−
relationship between [Fe/H], Z, and Y for the tracks com-
puted.
2 MODELS
2.1 Physical inputs
The grid of models was computed using the MESA code 1 AccordingtothesimulationsbyGirardietal.(2015),lessthan
(Paxton et al. 2011, 2013). We computed 21 masses in a onepercentofthegiantsintheKeplerfieldsareexpectedtohave
rangebetweenM =0.6 2.5M⊙,incombinationwith7dif- masseslargerthan2.5M⊙.
−
Beyond scaling relations 3
2.2 Structure of the grid 0.03
TobuildthetracksactuallyusedinourBayesian-estimation
code, we select from the original tracks computed with 0.02
MESAabouttwohundredstructureswell-distributedinthe
HthResedimagordaemlsawnederxeptrraecstengtlionbgaallqlueavnotluittiieosn,asruychstaagsetsh.eFarogme, ∆νgauss 0.01
/
tthheeppheroitoodspsphaerciicngluomfignroasviittyy,tmhoedeeffse(c∆tivPe,tseemepSeecratitounre2.(4T)e.ffI)n, ∆ν)fit 0
addition, each structure is also used to compute individual −
radial mode frequencies with GYRE (Townsend & Teitler νgauss−0.01
2013) in order to calculate large separations (∆ν), as de- ∆
(
scribed in Section 2.3.
−0.02
2.3 Average large frequency separation
−0.03
25 30 35 40 45 50 55 120 140 160
2.3.1 Determination of the large frequency separation νmax(µHz)
Inafirstapproximation,thelargeseparation∆νcanbeesti- Figure 1. Comparison between the average large separation
matedinthemodelsbytheequation1.However,thisestima- h∆νfitiofthestarinNGC6819,estimatedbylinearfittingwith
tioncanbeinaccurate,sinceisaffectedbysystematiceffects theactual error,andtheoutputofthemethoddescribedinSec-
tion2.3.1,forwhichtheactualerrorsweresubstitutedbyaGaus-
whichdepende.g.ontheevolutionaryphaseand,moregen-
erally,onhowthesoundspeedbehavesinthestellarinterior. sianfunctioncentredinνmax.
Togobeyondtheseismicscalingrelations,wecalculateindi-
vidualradial-modefrequenciesforeachofthemodelsinthe function in equation 3. Figure 1 shows the comparison be-
grid. Based of the frequencies we compute an average large tween ∆ν , determined from the method above, with
gauss
frequency separation ∆ν . We adopt a definition of ∆ν ∆ν hestimateid in the paper using the actual errors. The
as close as possible toh thie observational counterpart.hThei hmethfitoid estimate ∆ν with relative differences within
gauss
average∆ν asmeasuredintheobservationsdependsonthe theerrorbarsforthhemajoriityofthestars.Althoughthedef-
number of frequencies identified around νmax and on their inition of ∆ν may seem a minor technical issue, it plays
uncertainties. Therefore, with the aim of a self-consistent an importahnt riole in avoiding systematic effects on e.g. the
comparison between data and models, any ∆ν calculated mass and age estimates.
h i
from stellar oscillation codes must take in account the re-
strictions given bytheobservations. Handberg et al. (2016)
estimated the quantity ∆ν for the stars in the Kepler’s 2.3.2 Surface effects
fit
cluster NGC 6819. In that paper, ∆ν is estimated by a
fit It is well known that current stellar models suffer from an
simple linear fit of the individual frequencies (weighted on
inaccurate description of near-surface layers leading to a
theirerrors)asfunctionoftheradialorder.Thevalueofthe
mismatch between theoretically predicted and observed os-
slope resultingfrom thefittingline givestheestimated ∆ν.
cillation frequencies. These so-called surface effects have a
However,thesamemethodcannotbeappliedtotheoretical
sizable impact also on the large frequency separation, and
modelssincetheirfrequencieshavenoerrorbars.Therefore
on its average value. When using model-predicted ∆ν it is
weneedtotakeintoaccounttheuncertaintiesassociated to
therefore necessary to correct for such effects. As usually
each frequency in order to give them a consistent weight.
done,afirstattemptatcorrectingistousetheSunasaref-
Observationalerrorsdependprimarilyonthefrequencydis-
erence, hence by normalising the ∆ν of a solar-calibrated
tance between a given oscillation mode and νmax, with a model with theobserved one. h i
trend that follows approximately the inverse of a Gaussian
envelope (smaller errors near ν , larger errors far away In our solar model, αMLT and X⊙ are calibrated to
max reproduce, at the solar age t⊙ = 4.57 Gyr, the observed
from νmax; Handberget al. 2016).Forthis reason weadopt luminosity L⊙ = 3.8418 1033 erg s−1, the photospheric
a Gaussian function, as described in Mosser et al. (2012a), ·
radius R⊙ = 6.9598 1010 cm (Bahcall et al. 2005), and
to calculate theindividual weights: ·
the present-day ratio of heavy elements to hydrogen in the
(ν ν )2 photosphere (Z/X = 0.02452, Grevesse & Noels 1993). We
w=exp − max , (3)
(cid:20)− 2 σ2 (cid:21) used the same input physics as described in Section 2. A
· comparison between the large frequency separation of our
wherewistheweightassociatedtotheoscillationfrequency calibrated solar model and that from solar oscillation fre-
ν, and quencies(Broomhall et al.2014)isshowninFig.2.Wefind
σ=0.66·νm0.a8x8 . (4) t1h3a6t.1thµeHzpr(eddeicfitneeddacvfe.rSageectiloanrge2.3se)p,aisra0ti.o8np,ehr∆cνe⊙n,tmoladrige=r
The ∆ν is then calculated by a linear fittingof theradial than the observed one ( ∆ν⊙,obs = 135.0 µHz). We then
h i h i
frequencies ν as function of the radial order n, with the follow the approach by White et al. (2011) and adopt as
n,0
weightstakenateachν frequency.Inordertotestoures- a solar reference value that of our calibrated solar model
n,0
timationsweusetheobservedfrequenciesinHandberg et al. ( ∆ν ⊙ = ∆ν mod,⊙ =136.1µHz).
h i h i
(2016) simulating their errors using the Gaussian weight Thisisanapproximationwhichshouldbekeptinmind,
4 Rodrigues et al.
142 degree as in the Sun (see e.g. Belkacem et al. 2013, for a
Model
BiSON data more detailed explanation).
140 Thecombinationofthesetwofactorsiswhateventually
determinesadeviationfromthescalingrelationitself.Cases
whereasmallcorrectionisexpectedarelikelytheresultofa
138
fortuitous cancellation of thetwo effects (e.g. in RCstars).
Hz] We would like to stress that beyond global trends
∆νµ [136 e.g. with global properties, such corrections are also ex-
pected to be evolutionary-state and mass dependent, as
134 discussed e.g. in Miglio (2012), Miglio et al. (2013), and
Christensen-Dalsgaard et al.(2014).Aspointedoutinthese
papers, the mass distribution is very different inside stars
132
with same mass and radius but in RGB or RC phases. A
RGB model has a central density 10 times higher than a
13105 00 2000 2500 3000 3500 4000 4500 RC one; the former has a radiativ∼e degenerate core of He,
ν [µHz]
while the latter has a very small convective core inside a
Figure 2. Large frequency separation (∆ν) of radial modes as
He-core. The mass coordinate of the He-core is roughly a
function of frequency, as observed in the Sun (Broomhalletal.
factor2largerfortheRCmodel,whilethefractionalradius
2014, dots connected by a blue line) and in our calibrated solar
of this core is very small ( 2.5 6 10−3) in both cases.
model(redline).Thegraygaussianprofilerepresentstheweights ∼ − ×
Thefrequencies of radial modes are dominated byenvelope
givenbyeachpointof∆ν whenestimatingh∆νi(accordinglyto
themethoddescribedinSection2.3.1). properties,whichhavenotverydifferenttemperatures.How
thedifferenceinthedeepinteriorofthestaraffectsthenthe
relation between mean density and the seismic parameter?
and an increased accuracy when using ∆ν can only be As suggested in the above mentioned papers, different dis-
h i
achieved by both improving our theoretical understand- tribution of mass implies a lower density of the envelope of
ing of surface effects in stars other than the Sun (e.g. see theRCwith respect to theRGB one, and hence a different
Sonoi et al. 2015; Ball et al. 2016), and by trying to mit- sound speed in the regions effectively probed by radial os-
igate surface effects when comparing models and observa- cillations. As shown by Ledoux & Walraven (1958, and ref-
tions. In this respect a way forward would be to determine erences therein), the oscillation frequencies of radial modes
the star’s mean density by using the full set of observed dependnotonlyonthemeandensityofthestar,butalsoon
acoustic modes, not just their average frequency spacing. the mass concentration, with mode frequencies (and hence
This approach was carried out in at least two RGB stars separations) increasing with mass concentration. Although
(Huberet al. 2013; Lillo-Box et al. 2014), and led to deter- intheRGBmodelthecenterdensityis10timeslargerthan
mination of the stellar mean density which is 5 6 per the same in the RC one, the latter is a more concentrated
∼ −
cent higher than derived from assuming scaling relations, model since for 1 M⊙, for instance, half of the stellar mass
andwith amuchimprovedprecision of 1.4percent.Fur- is inside some thousandths of its radius. Moreover, as mass
∼
thermore,theimpactofsurfaceeffectsontheinferredmean concentrationincreases, theoscillation modestendtoprop-
densityismitigatedwhendeterminingthemeandensityus- agate in more external layers. Hence,not only theenvelope
ingindividualmode frequencies ratherthan usingtheaver- of the RC model has a lower density,in addition the eigen-
agelargeseparation (e.g.,seeChaplin & Miglio 2013).This functions propagate in more external regions with respect
approachishowevernotyetfeasibleforpopulationsstudies, totheirbehaviourinRGBstars.Theadiabaticsoundspeed
mostlybecauseindividualmodefrequenciesarenotavailable of the regions probed by these oscillation modes is smaller
yetforsuchlargeensembles,butitisapathworthpursuing in the RC than in the RGB, leading to differences in large
to improve both precision and accuracy of estimates of the frequency separations, and corrections with respect to the
stellar mean density. scaling relation.
Figure 3 shows the ratio between the large separation
obtainedfromthescalingrelation,∆ν ,and ∆ν (calcu-
scal.
2.3.3 ∆ν: deviations from simple scaling lated as described in Sec. 2.3.1) as a function ohf νi for a
max
large number of tracks in our computed grid. These panels
Small-scale deviations from the ∆ν scaling relation have
h i illustrate thedependenceof ∆ν corrections on mass, evolu-
been investigated in several papers. This is usually done
tionarystateandalsochemicalcomposition thataffectsthe
by comparing how well model predicted ∆ν scales with
ρ1/2, taking the Sun as a reference point (hseeiWhite et al. mass distribution inside thestar.
2011; Miglio 2012; Miglio et al. 2013; Brogaard et al. 2016; As shown in Fig. 3 the deviation of ∆ν with respect
Miglio et al. 2016; Guggenberger et al. 2016; Sharma et al. to the scaling relation tends to low values for stars in the
2016;Handberg et al. 2016). secondary clump. We must keep in mind however, that the
Suchdeviationsmaybeexpectedprimarilyfortworea- masses of stars populating the secondary clump depend on
sons. First, stars in general are not homologous to theSun, the mixing processes occurred during the previous main-
hence the sound speed in their interior (hence the total sequencephase,andalso on thechemical composition, that
acoustic travel time) does not simply scale with mass and is metallicity and initial mass fraction of He. Therefore, a
radius only. Second, the oscillation modes detected in stars straightforward parametrization ofcorrection asfunctionof
donotadheretotheasymptoticapproximationtothesame mass and metallicity is not possible.
Beyond scaling relations 5
1.05
1.04 [Fe/H] = -1.00
1.03
M=0.80M⊙
M=1.00M⊙ 1.02
MM==11..1200MM⊙⊙ ∆νscal1.01
M=1.30M⊙ / 1
MM==11..4500MM⊙⊙ ∆νfreq0.99
M=1.60M⊙ 0.98
M=1.70M⊙
M=1.80M⊙ 0.97
M=2.00M⊙ 0.96
M=2.50M⊙
0.95
104 103 102 101 100 10−1
νmax (µHz)
0 0
1.05 1.05
1.04 [Fe/H] = -0.75 1.04 [Fe/H] = -0.50
1.03 1.03
1.02 1.02
∆νscal1.01 ∆νscal1.01
∆ν/freq0.919 ∆ν/freq0.919
0.98 0.98
0.97 0.97
0.96 0.96
0.91504 103 102 101 100 10−1 0.91504 103 102 101 100 10−1
νmax (µHz) νmax (µHz)
0 0
1.05 1.05
1.04 [Fe/H] = -0.25 1.04 [Fe/H] = 0.00
1.03 1.03
1.02 1.02
∆∆νν/freqscal01..90191 ∆∆νν/freqscal01..90191
0.98 0.98
0.97 0.97
0.96 0.96
0.91504 103 102 101 100 10−1 0.91504 103 102 101 100 10−1
νmax (µHz) νmax (µHz)
0 0
1.05 1.05
1.04 [Fe/H] = 0.25 1.04 [Fe/H] = 0.50
1.03 1.03
1.02 1.02
∆νscal1.01 ∆νscal1.01
∆ν/freq0.919 ∆ν/freq0.919
0.98 0.98
0.97 0.97
0.96 0.96
0.91504 103 102 101 100 10−1 0.91504 103 102 101 100 10−1
νmax (µHz) νmax (µHz)
0 0
Figure 3.Correctionofscalingrelation∆ν infunctionofνmax,forasubsectionofthegridoftrackspresentedinSection2.2.
6 Rodrigues et al.
2.4 Period spacing where yi′ and σyi′ are the mean and standard deviation, for
each of thei quantities considered in the data set.
It has been shown by Mosser et al. (2012b) that is possible
In orderto obtain thestellar quantityx , theposterior
to infer the asymptotic period spacing of a star, by fitting i
PDF is then integrated over all parameters, except x , re-
a simple pattern on their oscillation spectra. This is par- i
sulting a PDF for this parameter. For each PDF, a central
ticularly relevant for those stars that present a rich forest
tendency (mean, mode, or median) is calculated with their
of dipole modes (l = 1), like, for instance, the red giants.
credible intervals. Therefore this method requires not only
Theasymptotictheoryofstellaroscillation tellsusthatthe
trustingthestellarevolutionarymodelsbutalsoadoptinga
g-modes are related by an asymptotic relation where their
minimumsetofreasonablepriors(instellarage,mass,etc.).
periodsareequallyspacedby∆P .Therelation statesthat
l In addition to avoid the scaling relations, the method re-
theasymptoticperiodspacingisproportionaltotheinverse
quiresthat theasteroseismic quantitiesare tabulated along
of the integral of the Brunt-V¨ais¨al¨a frequency N inside the
asetofstellarmodels,coveringthecompleterelevantinter-
trapping cavity:
valof masses, ages, and metallicities.
2π2 r2 N −1
∆P = dr , (5)
l l(l+1)(cid:18)Zr1 r (cid:19) 2.6 Interpolating the ∆ν deviations to make
p isochrones
where r and r are the coordinates in radius of turning
1 2
points that limit the cavity. It is easy to see that its value The ∆ν computed along the tracks appropriately sample
depends, among other things, on the size and the position stars in themost relevant evolutionary stages, and over the
of the internal cavity, fact that will become particularly interval of mass and metallicity to be considered in this
relevant in the helium-core-burning phase, giving the un- work. However, in order to be useful in Bayesian codes, a
certainties on the core convection (Montalba´n et al. 2013; furtherstepisnecessary:suchcalculationsneedtobeinter-
Bossini et al.2015).OntheRGBtheperiodspacingisanex- polated for any intermediate value of evolutionary stage,
cellent tooltoset constrainsonotherstellar quantities,like mass, and metallicity. This would allow us to derive de-
radius, and luminosity (see for instance Lagarde et al. 2016 tailed isochrones, that can enter easily in any estimation
and Davies & Miglio 2016). Moreover the period spacing code which involves age as a parameter. Needless to say,
gives an easy and immediate discrimination between stars such isochrones may find many other applications.
inhelium-core-burningandinRGBphases,sincetheformer Thecomputationalframeworktoperformsuchinterpo-
havea∆P systematicallylargerofabout 200 300sthan lations is already present in our isochrone-making routines,
∼ −
the latter, while after the early-AGB phase it decreases to which are described elsewhere (see Marigo et al. 2016). In
similar or smaller values. short,thefollowing stepsareperformed:ourcodereadsthe
evolutionarytracksofallavailableinitialmassesandmetal-
licities; these tracks contain age (τ), luminosity (L), T ,
eff
2.5 A quick introduction to grid-based and
∆ν, and ∆P from the ZAMS until TP-AGB. These quan-
Bayesian methods
tities are interpolated between the tracks, for any interme-
Havingintroducedtheway∆ν (hereafter,tosimplify∆ν = diate value of initial mass and metallicity, by performing
∆ν ) and ∆P are computed in the grids of tracks, let us linearinterpolationsbetweenpairsof“equivalentevolution-
hfirstiremind how they enter in the grid-based and Bayesian ary points”, i.e., points in neighbouring tracks which share
methods. similar evolutionary properties. An isochrone is then built
Intheso-calleddirectmethods,theasteroseismicquan- bysimplyselectingasetofinterpolatedpointsforthesame
titiesareusedtoprovideestimatesofstellarparametersand age and metallicity. In the case of ∆ν, the interpolation is
their errors, by directly entering them either in formulas doneinthequantity∆ν/∆νSR,where∆νSR isthevaluede-
(like the scaling relations of equation 1) or in 2D diagrams finedbythescalingrelationinequation1.Infact,∆ν/∆νSR
built from grids of stellar models. In grid-based methods varies along the tracks in a much smoother way, and has a
with Bayesian inference, this procedure is improved by the muchmorelimitedrangeofvaluesthanthe∆ν itself;there-
weighting of all possible models and by updating the prob- forethemultipleinterpolationsofitsvalueamongthetracks
ability with additional information about the data set, de- alsoproducewell-behavedresults.Ofcourse,intheendthe
scribed approximately as: interpolated values of ∆ν/∆νSR are converted into ∆ν, for
every point in thegenerated isochrones.
p(xy) p(yx)p(x), (6) Figure4showsasetofevolutionarytracksuntiltheTP-
| ∼ |
where p(xy) is the posterior probability density function AGB phase in the range [0.60,1.75] M⊙ for [Fe/H] = 0.25
| (Z =0.03123) and interpolated isochrones of 2 and 10 Gyr
(PDF), p(yx) is the likelihood function, which makes the
| both in the Hertzsprung–Russell (HR), the ratio ∆ν/∆ν
connectionbetweenthemeasureddatayandthemodelsde- SR
versus ν , and the ∆P versus ∆ν diagrams. The middle
scribedasafunctionofparameterstobederivedx,andp(x) max
panelshowsthedeviation of thescaling ∆ν offew percents
is the prior probability function that describes the knowl-
mainly overtheRGB and early-AGBphases. Deviations at
edgeaboutthederivedparametersobtainedbeforethemea-
the stages of main sequence and core helium burning are
sureddata.Theuncertaintiesofthemeasureddataareusu-
generally smaller than oneper cent.
ally described as a normal distribution, therefore the likeli-
The Fig. 4 also shows that our interpolation scheme
hood function is written as
works very well, with the derived isochrones reproducing
p(y′x)= 1 exp −(yi′−yi)2 , (7) thebehaviourexpected from theevolutionary tracks.
| Yi √2πσyi × (cid:18) 2σy2i (cid:19) Nosimilarprocedurewasnecessaryfortheinterpolation
Beyond scaling relations 7
in ∆P, since it does not follow any simple scaling relation,
and it varies much more smoothly and covering a smaller
total range than ∆ν. The interpolations of ∆P are simply
linear ones using parameters such as mass, age (along the
tracks), and initial metallicity as the independent parame-
ters.
3 APPLICATIONS
We derived the stellar properties using the Bayesian tool
PARAM(da Silvaet al. 2006;Rodrigues et al. 2014).From
the measured data – T , [M/H], ∆ν, and ν – the code
eff max
computes PDFs for the stellar parameters: M, R, logg,
meandensity,absolutemagnitudesinseveralpassbands,and
asasecondstep,it combinesapparent andabsolutemagni-
tudestoderiveextinctionsA intheV bandanddistances
V
d. The code uses a flat prior for metallicity and age, while
for the mass, the Chabrier (2001) initial mass function was
adopted with a correction for small amount of mass lost
nearthetipoftheRGBcomputedfromReimers(1975)law
with efficiency parameter η = 0.2 (cf. Miglio et al. 2012).
The code also has a prior on evolutionary stage that, when
applied, separates the isochrones into 3 groups: core-He
burners (RC), non-core He burners (RGB/AGB), and only
RGB (till the tip of the RGB). The statistical method and
someapplicationsaredescribedindetailsinRodrigues et al.
(2014).
We expanded the code to read the additional seismic
informationoftheMESAmodelsdescribedinSection2.We
implemented new variables to be taken into account in the
likelihoodfunction(equation7),suchas∆ν fromthemodel
frequencies, ∆P,logg, and luminosity.Hencetheentire set
of measured data is
y=([M/H],T ,∆ν,ν ,∆P,logg,L),
eff max
where ∆ν can still be computed using the standard scal-
ing relation (hereafter ∆ν(SR)). Therefore PARAM is now
abletocomputestellar propertiesusingseveraldifferentin-
put configurations, i.e., the code can be set to use different
combinations of measured data. Someinteresting cases are,
together with T and [M/H],
eff
∆ν and ν from scaling relation (equation 1);
max
•
∆ν from modelfrequenciesandν from scaling rela-
max
•
tion;
∆ν (either from model frequencies or scaling relation),
•
together with some other asteroseismic parameter, such as
∆P;
logg;
•
any of the previous options together with the addition
•
of a constraint on thestellar luminosity.
Figure 4. MESA evolutionary tracks color coded according to
massintheHR(toppanel),∆ν/∆νSRversusνmax(middle),and The first two cases constitute the main improvement we
∆P versus∆ν(bottom)diagrams.Thesolidandthedashedblack consider in this paper, which is already subject of signifi-
lines are examples of interpolated isochrones of 2 and 10 Gyr, cantattentionintheliterature(seee.g.Sharma et al.2016;
respectively. Guggenberger et al.2016).Thethirdcaseisparticularlyim-
portant given the fact that the ν scaling relation is ba-
max
sically empirical and may still reveal small offsets in the
future. Finally, the fourth and fifth cases are aimed at ex-
ploring the effect of lacking of seismic information, when
only spectroscopic data is available for a given star; and
addingindependentinformation inthemethod,likee.g.the
8 Rodrigues et al.
knowndistanceofacluster,orofupcomingGaiaparallaxes, represents the resulting PDF when prior on evolutionary
respectively. stageisapplied,whileintherightside(whitecolor)theprior
is not being used. The black dots and error bars represent
the mode and its 68 per cent credible intervals of the PDF
3.1 Tests with artificial data with prior on evolutionary stage (cyan distributions).
In most cases, we recover the stellar masses and ages
Totesttheprecisionthatwecouldreachwithatypicalsetof
observationalconstraintsavailableforKeplerstars,wehave within the 68 percent credible intervals. Using only ∆ν re-
sults in wider and more skewed PDFs (case i in the plots),
chosen6modelsfromourgridofmodelsandconsideredvar-
while adding ν confines the solution in a much smaller
iouscombinationsofseismic,astrometric,andspectroscopic max
region (cases ii and iii). When combining with ∆P, the so-
constraints (see Table 2).
lutionistiedbetter(casev).Inmostcases,thecombination
The seismic constraints taken from the artificial data
of ∆ν and ν provides narrower PDFsthan ∆ν and ∆P,
are ∆ν, ν , and ∆P. The latter is used by taking its max
max
which indicates that ∆P does not constrain the solution as
asymptoticvalueasadditionalconstraintinequation7,and
tightly as ν (cases ii and iv) even for RC stars. As ex-
notasonlyadiscriminantfortheevolutionaryphaseasdone max
pected,addingmoreinformation asluminosity,narrowsthe
inpreviousworks(e.g.Rodrigues et al.2014).Uncertainties
searching“area” intheparameterspacewhichprovidesthe
on∆νandν weretakenfromHandberget al.(2016)and
max
narrowest PDFs when all asteroseismic parameters and lu-
on ∆P from Vrard,Mosser & Samadi (2016). We adopted
minosity are combined (case vii). The usage of only ν
0.2 dex as uncertainties on logg based on average values max
andluminosity(caseviii)isveryinteresting,becauseitpro-
coming from spectroscopy. For luminosity, we adopted un-
vides PDFs slightly narrower than using the typical combi-
certainties of the order of 3 per cent based on Gaia paral-
nation of ∆ν and ν or ∆ν and luminosity (case xi), and
laxes, where a significant fraction of the uncertainty comes max
it is similar to the v and vi cases. The lack of asteroseis-
from bolometric corrections (Reeseet al. 2016).
mic information (cases ix) worsens the situation, providing
We derived stellar properties using 11 different com-
a significant larger error bar than most other cases, simply
binations as input to PARAM, in all cases using T and
eff
becauseof large uncertaintieson gravity coming from spec-
[Fe/H], explained as following:
troscopic analysis. The case x results in PDFs very similar
i ∆ν – only ∆ν from model frequencies; with case i. This is to be expected: including ∆ν as a con-
ii ∆ν and νmax – to compare with the previous item in straint (case i) leads to a typical σ(logg) 0.02 dex (see
orderto test if we can eliminate theusage of νmax; alsothediscussioninMorel et al.2014,pag≃e4),i.e.,adding
iii ∆ν(SR) and νmax – traditional scaling relations, to the spectroscopic logg (σ(logg) 0.2 dex) as a constraint
≃
compare with the previous item and correct the offset (case x) has a negligible impact on thePDFs.
introduced byusing ∆ν scaling; Finally,theprioronevolutionarystagedoesnotchange
iv ∆ν and ∆P – in order to test if we can eliminate the the shape of the PDFs in almost all cases, except for the
usage of νmax and improve precision using the period RGB star S4. Regarding this case, it is interesting to note
spacing not only as prior, but as a measured data; that S4 and S5 have similar ∆ν and ν , but different
max
v ∆ν, νmax, and ∆P – using all the asteroseismic data ∆P, that is, they are in a region of the ∆ν versus νmax
available; diagram that is crossed by both RC and RGB evolution-
vi ∆ν, ∆P, and L – in order to test if we can eliminate ary paths. In similar cases, not knowing the evolutionary
theusageofνmax,whenluminosityisavailable(fromthe stage causes the Bayesian code to cover all sections of the
photometry plusparallaxes); evolutionary paths, meaning that there is a large parame-
vii ∆ν,νmax,∆P,andL–usingalltheasteroseismicdata ter space to cover, which often causes the PDFs to become
available and luminosity, simulating future data avail- multi-peakedorspreadforallpossiblesolutionsasthecases
able for stars with seismic data observed by Gaia; ixandx.Furtherexamplesofthiseffectaregiveninfigure5
viii νmaxandL–inthecasewhenitmaynotalwaysbepos- of Rodrigues et al. (2014). Knowingtheevolutionary stage,
sible to derive ∆ν from lightcurves, simulating possible instead, limits the Bayesian code to weight just a fraction
datafrom K2 and Gaia surveys; of the available evolutionary paths, hence limiting the pa-
ix logg and L–in thecase when only spectroscopic data rameter space to be explored and, occasionally, producing
are available (in addition to L); narrower PDFs. This is what happens for star S4, which,
x ∆ν andlogg–againinordertotestifwecaneliminate despitebeingaRGBstarof1.6M⊙,happenstohaveaster-
theusageofνmax,replacingitbythespectroscopiclogg; oseismicparameterstoosimilartothosethemorelong-lived
RCstars of masses 1.1 M⊙.
xi ∆ν and L – again in order to test if we can eliminate Table 3 present∼s the average relative mass and age un-
theusage of νmax, when luminosity is available. certainties for RGB and RC stars, which summarizes well
the qualitative description given above. Cases i (very sim-
Inallcases, theprioron evolutionarystagewasalso tested.
ilar to case x) and ix results the largest uncertainties: 17
TheresultingmassandagePDFsforeachartificialstarare
presentedusingviolinplots2 inFigure5and6,respectively. and 12 per cent for RGB, and 8 and 11 for RC masses; up
70 and 40 per cent for RGB, and 22 and 31 for RC ages,
Thexaxis indicateseach combination ofinputparameters,
respectively. From the traditional scaling relations (case ii)
as discussed before; the left side of the violin (cyan color)
to the addition of period spacing and luminosity (case vii,
theuncertaintiescandecreasefrom8to3percentforRGB
2 Violinplotsaresimilartoboxplots,butshowingthesmoothed and5to3forRCmasses;29to10percentforRGBand14
probabilitydensityfunction. to 8 for RC ages. It is remarkable that we can also achieve
Beyond scaling relations 9
Figure5.PDFsofmassforthe6artificialstarspresentedinTable2usingviolinplots.Eachpanelshowstheresultsofonestar,named
inthetoptogetherwithitsevolutionarystage.ThexaxisindicateseachcombinationofinputparametersforPARAMcodeasdescribed
inSection3.1.Theleftsideoftheviolin(cyancolor)representstheresultingPDFwhenprioronevolutionarystageisapplied,whilein
the right side (white color) the prior is not being used. The black dots and error bars represent the mode and its 68 per cent credible
intervalsofthePDFwithprioronevolutionarystage(cyandistributions).Thedashedlineindicatesthemassoftheartificialstars.
Table 2.SetofartificialdataconsideredinSection3.1.
Label M/M⊙ logAge/yr Teff (K) [Fe/H] logg L/L⊙ νmax (µHz) ∆ν (µHz) ∆P (s) Ev.State
S1 1.00 9.8379 4813±70 -0.75±0.1 2.38±0.20 54.77±1.64 30.26±0.58 3.76±0.05 61.40±0.61 RGB
S2 1.00 9.8445 5046±70 -0.75±0.1 2.39±0.20 64.52±1.94 30.31±0.58 4.04±0.05 304.20±3.04 RC
S3 1.60 9.3383 4830±70 0.0±0.1 2.92±0.20 25.57±0.77 105.01±1.83 8.66±0.05 70.80±0.71 RGB
S4 1.60 9.3461 4656±70 0.0±0.1 2.55±0.02 51.36±1.54 45.99±0.84 4.56±0.05 62.00±0.62 RGB
S5 1.60 9.3623 4769±70 0.0±0.1 2.54±0.20 58.40±1.75 43.97±0.81 4.60±0.05 268.30±2.68 RC
S6 2.35 8.9120 5003±70 0.0±0.1 2.85±0.20 51.41±1.54 86.79±1.54 6.86±0.05 251.20±2.51 RC
10 Rodrigues et al.
Figure 6.Thesameas figure5butforlogarithmof theages. Therighty-axisgives theage inGyr.Thedashed lineindicates theage
oftheartificialstars.
aprecisionaround10percentonagesusingν andlumi- the resulting mass and age PDFs for stars S2 and S5 with
max
nosity (case viii), and 15 per cent using ∆ν and luminosity the efficiency parameter η = 0.2 (cyan colors) and η = 0.4
(case xi). (white colors). For the cases v, vi, vii, viii, and xi, a mass
loss with efficiency η = 0.4 produces differences on masses
Averagerelativedifferencesbetweenmassesare61per
of 1 per cent, while on ages, may be greater than 47 per
centforcases v,vi,vii,andviii,around1percentforcases ∼
centforS2andthan18percentforS5.Thesmalldifference
ii and xi, 6 per cent for case iii, and greater than 6 per
∼ in masses results from the fact that, in these cases, mass
cent for cases i, ix, and x. Regarding ages, relative absolute
valuesfollowalmostdirectlyfromtheobservables–roughly
differencesarelesserthan5percentforcasesv,vi,vii,viii,
speaking, they represent the mass of the tracks that pass
and xi, around 10 per cent when using ∆ν and ν , 20
max
∼ closertotheobservedparameters.Aswellknown,redgiant
percentwhenusing∆ν(SR)andν ,andgreaterthan40
max
stars quickly lose memory of their initial masses and follow
per cent for cases i,ix, and x.
evolutionary tracks which are primarily just a function of
Wealsoappliedmasslossonthemodels.Figure7shows