Seismic diagnostics of solarlike oscillating stars
1 Introduction
Recent space missions, such as CoRoT and Kepler, have led us to a golden epoch when large scale asteroseismic analysis of stars can be carried out. Thanks to the high precision and longlasting observations provided by these space missions, new previously unavailable areas of the frequency domain have been opened (e.g. Appourchaux et al. 2008 ; Borucki et al. 2007 ; Gilliland et al. 2010 ; Gruberbauer et al. 2013 ). With detected oscillations, the following asteroseismic studies are able to provide us with a unique approach to constrain a star’s fundamental properties, and even to test the theory of stellar structure and evolution. They enrich our knowledge not only on stars, but also on clusters and the Galaxy, or even broader, the whole universe (e.g. Soriano et al. 2007 ; Doǧan et al. 2013 ; Campante et al. 2015 ; Casanellas 2015 ; Sharma et al. 2016 ).
Solarlike oscillations refer to stars oscillating with the same mechanism as the Sun, where they are stochastically excited and damped by convection motion in the nearsurface convection zone (e.g. ChristensenDalsgaard 1982 ; ChristensenDalsgaard & Frandsen 1983 ; Houdek et al. 1999 ). Study of oscillations could yield worthful conceptions on stellar structures and evolutionary stages. Mainsequence stars behave as p modes (pressure dominated) in the envelope. Subgiant stars behave as mixed modes, which are characterized by g modes (gravity dominated) in the core and p modes in the envelope (Tassoul 1980 ), when “avoided crossing” commences (Aizenman et al. 1977 ; Benomar et al. 2014 ; Lagarde et al. 2015 ). Therefore, oscillations are capable of distinguishing different types of stars with their identical signatures. Mixed modes have further shown potential on constraining stellar models in a powerful way (e.g. Deheuvels et al. 2012 ; Montalbán & Noels 2013 ; Silva Aguirre et al. 2013 ; Mosser et al. 2014 ; Mosser 2015 ), since some stellar parameters are particularly sensitive to them, e.g. stellar age (Metcalfe et al. 2010 ) and mass (Benomar et al. 2012 ). It is even possible to determine the presence and size of the convective core with the help of asteroseismology (Liu et al. 2014 ; Yang et al. 2015 ).
Accurate data analysis of oscillations is one crucial prerequisite for detailed stellar diagnostics (e.g. Ozel et al.
2010
; Deheuvels et al.
2012
; Silva Aguirre et al.
2013
,
2014
; Chaplin et al.
2014
). Two global asteroseismic parameters, mean large separation
The article is organized as follows. In Section
2 List of targets
We revisited the topic explored by Chaplin et al. (
2014
), who derived the values of mean large separation
The Kepler mission provides photometric time series of the targets with long cadence (LC; 29.43min sampling) and short cadence (SC; 58.84 s sampling). The pulsation frequency range is estimated to be above the Nyquist frequency of LC data. Here, we obtained SC time series over one year, which were collected from the Kepler Asteroseismic Science Consortium website
Notes: “Q” represents a threemonth long observation quarter. Atmospheric parameters were derived by LSP3 (Xiang et al. 2015b ). 

Atmospheric parameters of the stars are crucial since they serve as constraints on stellar models. We noticed that the six targets were covered by the LAMOSTKepler project, and were observed by LAMOST low resolution (∼1800) optical spectra in the waveband of 3800∼9000 Å by September 2014. Three atmospheric parameters,
Table
3 Data analysis
3.1 Preprocessing of Data
First, for the six targets, we concatenated all the time series of the six targets and preprocessed them using the method described by García et al. (
2011
), correcting outliers, jumps and drifts on the flux, and then passed the light curve through a highpass filter with width of one day. The highpass filter was built based on a movingaverage smoothing function with Gaussian weights. It only affects frequencies lower than
Second, we obtained the power spectra of the six targets by applying the LombScargle Periodogram (Lomb 1976 ; Scargle 1982 ) method, which is especially suitable for irregularly spaced discrete data with gaps.
Figure
3.2 Global Asteroseismic Parameters
Mean large separation
Figure
Frequency of maximum power
Figure
After we checked our results with different approaches for quality assurance, we compared our results, both
Table
Notes: (a) results in this work; (b) results from Chaplin et al. ( 2014 ). 

3.3 Oscillation Frequencies
In order to excavate deeper seismic information, we extracted oscillation frequencies of the six targets. We started by reviewing characteristics of solarlike oscillation. The signature of p modes can be well described using asymptotic theory controlled by radial order n and angular degree l. The approximate expression can be written as (Tassoul 1980 )
This indicates that g mode frequencies are equally spaced in period (i.e.
Several methods to extract oscillation frequencies in a global way have been put forward, for example, Bayesian Markov Chain Monte Carlo (Handberg & Campante 2011 ; Benomar 2008 ) and Maximum Likelihood Estimation (Appourchaux et al. 1998 ); however, when the power spectrum reveals p and g mixed modes with low S/N, global analysis is not advantageous, because there exist several frequencies that are hard to determine, and it is easy for an automatic program to wrongly determine these frequencies. Therefore, here we derived them separately based on asymptotic theory and visual inspection. The identified modes are fitted with Lorentzian profiles using the leastsqaures minimization.
The Lorentzian model is
Values for frequency centroid
We present individual mode frequencies


4 Grid modeling
Some stellar fundamental parameters, e.g. M and R, can be directly deduced by seismic parameters (discussed below). However, to further investigate and analyze the six targets comprehensively, we constructed stellar grid models. The main theme of grid modeling is to construct models in a large range and select models which meet the constraints, including seismic constraints (e.g.
4.1 Modeling Parameters and Input Physics
We computed stellar models with the Yale Rotating Stellar Evolution Code (YREC, Demarque et al.
2008
; Pinsonneault et al.
1990
,
1992
). The input parameter, mass M, was estimated with scaling relations. Mean large separation
This requires the grid should at least cover


The input physics is set as follows. We adopted NACRE nuclear reaction rates in Bahcall et al. (
1995
), equation of state tables in Rogers & Nayfonov (
2002
), OPAL hightemperature opacities in Iglesias & Rogers (
1996
) and lowtemperature opacities in Ferguson et al. (
2005
). Atomic diffusion was considered only under initial masses
4.2 Constraining Models
We selected qualified models of the six targets, which match the requirements imposed by observational constraints:
We assigned an overall probability for each model
The prior probability is set to a uniform value
By canceling out the constant prior probabilities, Bayes’ theorem simplifies to
The above equation is used to derive posterior probability for each model. By constructing the marginal probability distribution of each parameter, we estimated their values and assigned a 1σ deviation from median values as the uncertainties.
The oscillation patterns revealed from the corresponding échelle diagram, i.e. Figure
Figure
Table
Notes: (a) results in this work; (b) results from Chaplin et al. (
2014
) with input 

Also, we compared the log g derived from LSP3 and this work. Figure
5 Conclusions
We performed data processing and asteroseismic analysis on six solarlike stars observed by both the Kepler mission and LAMOST, and combined stellar grid models to determine stellar fundamental parameters.
We derived asteroseismic parameters
According to numerical solutions of the stellar models, we looked into the evolutionary stages of six solarlike targets and categorized them into one mainsequence star (KIC 10079226) and five subgiant stars (the others), four of which show strong characteristics of mixed p and g modes. Grid modeling indicates that the five subgiant stars are in the range of
In this work, we do not use individual oscillation frequencies to constrain stellar models.
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