In this chapter I implemented a global fit of stellar oscillation power spectra. The global fit was tested using 4 month blocks of Sun-as-a-star observations from SoHO/VIRGO which cover a total time span of about 14 years. The parameterization of the expectation value of the power spectrum takes into account the regular pattern of the global p-mode spectrum in terms of smooth functions of the radial ordernand the frequencyν. With the global fit I am able to measure the oscillation parameters of the solar p modes as accurate as expected for a 4 month observation. The large and small frequency variations are measured without bias over the radial orders 15 ≤ n ≤ 28. Variations of the p-mode parameters over the course of the solar cycle are recovered. Since the Sun is a slowly rotating star, only Ωsinicould be measured accurately. The independent estimates onΩ andi are biased.
The simplified parameterization of the expectation value of the global oscillation power spectrum reduces the number of free parameters for the global fit significantly. In this analysis, the global fit includes 14 radial orders and modes with` ≤2 and contains 20 free parameters. A fit with independent parameters for frequencies, linewidths, and amplitudes contains 128 free parameters. The reduction of the number of free parameters reduces the computation time significantly and minimize the risk of premature convergence of the fit. However, due to the simplified model some information is lost, e.g. the details of the variation of the large separation with frequency.
The global fit can be applied to time series of distant stars. However, the time series and the star itself have to fulfill several requirements. Time series of distant stars have a significantly higher noise level than observations of the Sun. The observed star has to be bright enough so that the oscillation signal is above the noise background. Furthermore, the data have to be continuous and long enough so that modes with particular radial order nand angular degree`are resolved in the power spectrum. To adopt a simple parameter-ization of the expectation value of the oscillation power spectrum, the mode parameters of stellar oscillations have to vary smoothly with radial order. Thus, stars that are very Sun-like seem to be particularly suited. The stars must provide an oscillation spectrum that allows one an unambiguous identification of the angular degree` of the modes. For instance, HD 49933 (Appourchaux et al. 2008) is an F5 dwarf which shows a clear oscil-lation power spectrum of solar-like p modes but the mode linewidth is comparable to the small separation such that an unambiguous mode identification is not possible.
Furthermore, the star must not be too evolved. When a solar-like star evolves from the main sequence towards its subgiant phase it may exhibit so-called mixed modes which have the characteristics of p modes near the surface and g modes near the stellar core.
These modes show a clear deviation from the regular frequency spacing of modes with subsequent radial orders. In such a case, the mode frequencies cannot be described with a simple smooth function. HD 49385 (Deheuvels et al. 2010) is an evolved solar-like star which is in the transition from the main sequence to the post-main sequence phase. It exhibits mixed modes which show a clear deviation from the regular frequency spacing.
In the next chapter I will present the analysis of the time series of the Sun-like star HD 52265 which was observed with CoRoT for 4 months. HD 52265 delivers an excellent solar-like power spectrum that fulfills all requirements discussed above and allows one to adopt the global fit.
Figure 2.22: Map of the joint PDF in theΩ/2π-siniplane (top panel). The colors rep-resent the value of the joint PDF for a pair of parameters, (Ω/2π, sini), determined with global fits of 35 blocks of 120 day VIRGO time series. The symbols show the results forΩ/2πandiof the global fits of the 35 blocks of VIRGO data. The solid and dashed black linesrepresent the fit result ofΩ/2π sini= 0.424+−0.0360.038µHz. Thehorizontal dotted lines mark the position of the two peaks in the low frequency power spectrum in Fig-ure 2.23 and are attributed to solar surface rotation. Thecontour line shows the 3σerror ofΩ/2π siniaround the minimum of the likelihood function.Bottom panel:Cut through the log-likelihood function at constantΩ/2π sini = 0.424µHz, i.e. a cut along the solid black line in the upper panel.
Figure 2.23: Solar surface rotation determined from the low-frequency power spectrum.
Thetop panelshows a 120 day VIRGO time series. The bottom paneldisplays the cor-responding power spectrum at low frequencies. Thedashed line shows the power spec-trum oversampled by a factor of 10. The vertical green lines mark two peaks which are attributed to the surface rotation of the Sun. The peaks are at ν = 0.40 µHz and at ν = 0.59 µHz corresponding to a rotation period of T = 29 days and T = 20 days respectively.
In this chapter, I analyze the solar-like star HD 52265 which was observed continuously for about four months with the CNES-ESA space telescope CoRoT. HD 52265 was pro-posed as a possible CoRoT target by L. Gizon in 2005 and selected as a primary target for a CoRoT long-run thanks to the efforts of the CoRoT Scientific Committee (LRa2, November 2008 to March 2009). The analysis of the data presented in this thesis is origi-nal. In parallel, the HD 52265 data are being analyzed independently by several members of the CoRoT Data Analysis Team (DAT), led by J. Ballot. The CoRoT observations are supplemented by spectroscopic follow-ups with the Narval1 spectropolarimeter installed on the Bernard Lyot Telescope at the Pic du Midi Observatory (France), led by C. Catala and H. Bruntt. My results were presented as a poster at the 2010 HELAS IV International Conference (Gizon et al. 2010a) and will be submitted for publication shortly. They were compared to the other results from the DAT in the poster by Ballot et al. (2010), which has resulted in another publication since then (Ballot et al. 2011).
The star HD 52265 is particularly interesting as it is very similar to the Sun with respect to its fundamental parameters and it is the only solar-like star which was known to host a planet before the launch of CoRoT. The known properties of both the central star and its companion are summarized in Section 3.1. The photometric observations with CoRoT are presented in Section 3.2. In Section 3.3, I will present the HD 52265 power spectrum. The determination of the stellar background is presented in Section 3.4. In Section 3.5 and 3.6, I describe the parameterization of the power spectrum model, the estimation of errors using Monte Carlo simulations, and the global fit itself. The results of the global fit are presented in Section 3.7.