Stellar parameters as key to exoplanet parameter accuracy

The main focus of the asteroseismology programme of PLATO will be to support exoplanet science by providing:

  • stellar masses with a precision better than 10%,
  • stellar radii to 1‒2% precision,
  • and stellar ages to 10% precision

for a reference star similar to the Sun at V= 10.

Starting with the Sun decades ago, it has long been demonstrated that seismology is the most efficient method to provide precise stellar parameters for solar-like oscillating stars (Chaplin & Miglio 2013; Lebreton et al. 2014a, b; Silva Aguirre et al. 2015). The precision and accuracy on the stellar mass, radius and age determination depend on the type, precision and number of seismic and non-seismic observational constraints. The quality of the seismic determinations also depends on the choice and on the efficiency of the procedure used to derive the parameters; the choice of a procedure in turn depends on the quality of the available data. In addition, the age determination relies on the use of stellar models. Significant improvements will come with the results of the Gaia mission and PLATO.

Seismic radius

Gaia provides the distances to the stars via direct, geometric measurements, allowing the absolute luminosity of the star to be derived with high precision. Combining the luminosity with the effective surface temperature of the star obtained from (ground-based) high-resolution spectroscopy, we will obtain the radii of target stars at a precision of 1‒2%. Note that Gaia will be complete down to V~20, while PLATO will observe stars at 4<V<16, so all PLATO targets will also be observed by Gaia. The first and second Gaia data releases were successfully presented in September 2016, and April 2018, respectively.  It should be noted that measurements of Teff to within 1% will be achievable through dedicated high-resolution, high signal-to-noise spectroscopic observations obtained as part of PLATO’s ground-based follow-up program. Determination of chemical abundances and Teff will be based on state-of-the-art spectral analysis techniques and model atmospheres, taking 3D and non-LTE effects into account (Bergemann et al. 2012; Magic et al. 2013, 2015a,b; Martins 2014). Taken together with the luminosities expected from Gaia, the effective temperature will lead to stellar radii with a relative precision within 1‒2% for un-reddened stars.

For the brightest stars, combining Gaia distances and interferometric measurement of angular radius will provide model-independent measurement of the stellar radius. This has already been achieved for a small set of host stars (Huber 2016). This provides an additional and independent constraint on stellar interiors. Such independent information can be used together with seismic constraints to provide further constraints on stellar interiors.

Seismic mass from scaling relations or inversion technique

Asteroseismology can provide the bulk density of the star via scaling relations even with data of relatively low signal-to-noise ratio (SNR), or via inversion techniques from data of high SNR.

Scaling relations based on solar values have already been tested and validated using Kepler targets by comparing the asteroseismic radii and distances with interferometric observations and Hipparcos parallaxes (Huber et al. 2012; Silva Aguirre et al. 2012; White et al. 2015). Fits to global averages of seismic quantities (assuming the effective temperature is known) provide a determination of stellar radius and mass. The fitting can be done in a model-independent way (e.g. Chaplin et al. 2011a) but typically is further constrained by the inclusion of a grid of models. An extensive analysis of this nature for 500 Kepler dwarfs and subgiants observed in short cadence for one month was carried out by Chaplin et al. (2014). Due to their importance, scaling relations are the subject of many studies to investigate their precision and possible biases (e.g. Huber et al. 2013; Coelho et al. 2015; Pinheiro et al. 2014; Mosser et al. 2013). Such studies require precise photometric data and will largely benefit from PLATO. A full understanding of the domains of validity and biases to be corrected will in turn make the seismic scaling relations efficient tools for deriving precise and accurate stellar mass and radius, even when the data are not at their optimum precision or for step-and-stare observations. Subsequently, with the help of stellar models and calibrated scaling relations, PLATO will allow to perform ensemble seismology on a very large number of stars by providing information far beyond the only bulk density.

Inversions of mode frequencies have been used extensively in helioseismology to infer solar internal properties, and are presently being adapted to asteroseismology (Reese et al. 2012; Buldgen et al. 2015a, b). One advantage of  inversion techniques is that the solutions are not restricted to those of a space of standard stellar models. In particular, they allow for the determination of a model-independent mass, if the stellar radius is known independently from the seismic data. These techniques require the highest quality data and are therefore applicable for the brightest stars and longest observing runs.

Hence, by combining the very precise bulk density values of PLATO’s asteroseismic analysis with stellar radii from Gaia, we will obtain precise stellar masses for a large set of stars.


Seismic mass, radius and age inferences from stellar modelling

Stellar age determination by asteroseismology is more complex, and requires invoking models of stellar evolution. Age will be estimated by comparing grids of stellar models computed for different initial parameters (mass, metallicities, helium abundances, convection parameters) to the combined non-asteroseismic and asteroseismic observational constraints. Given data with high SNR, it is indeed possible to determine individual oscillation frequencies.


Top – Solar power spectrum from 2 years of SOHO/VIRGO photometric data. Bottom – Power spectrum of HD52265 from 117 days of observation with CoRoT (Gizon et al. 2013). The insert figures zoom in on the sections of the power spectra highlighted in grey.

As illustrated in the diagrams, the parameters of the stars and their internal structure are inferred by fitting model oscillation frequencies to the observed oscillation frequencies. This procedure is now routinely applied by the scientific community to retrieve stellar radii, masses, and ages from space-based asteroseismic data. The fit may either use grids of stellar models and oscillation frequencies, or more sophisticated search techniques requiring repeated stellar model calculations.


Kepler has carried out asteroseismic observations for several transiting planets. Examples of such planetary systems containing rocky/icy planets are Kepler-36b (Carter et al. 2012), Kepler-68 (Gilliland et al. 2013), and the smallest planet detected so far, Kepler-37b (Barclay et al. 2013). Characterisation of the terrestrial and/or rocky planets via the seismic characterisation of their host stars has been successfully performed (Huber 2015), for instance, for Kepler-93b (Ballard et al. 2014; Dressing et al. 2015) and Kepler-10b (Fogtmann-Schulz et al. 2014).


Top – power spectrum of 16 Cyg A from 35 months of Kepler photometric data. Bottom – zoom over three groups of modes clearly showing individual l=0, l=1 and l=2 mode frequencies (Christensen-Dalsgaard 2016).

Based on the asteroseismic analysis of 66 Keplerplanet host stars, Huber et al. (2013) claim typical uncertainties of 3% and 7% in radius and mass, respectively, from the analysis of global asteroseismic parameters.

Metcalfe et al. (2014) provided a uniform seismic modelling of 42 solar-type Keplertargets observed for 9 months and analysed by Appourchaux et al. (2012). The authors found that using the individual frequencies in the search for an optimal stellar model typically doubles the precision of the resulting seismic radius, mass, and age compared to grid-based modelling of the global oscillation properties, and improves the precision of the radius and mass by about a factor of three over empirical scaling relation. Lebreton & Goupil (2014) derived the stellar parameters for the star HD 52265 observed by CoRoT for four months. This star has an effective temperature around 6100 K and was inferred to have a mass and radius around 1.2 Mand 1.3 R, respectively, with the central hydrogen abundance reduced to roughly half the original value. Lebreton & Goupil (2014) and Lebreton et al. (2014a, b) studied the impact of different choices of seismic diagnostics and of different assumptions for the input physics.


Ranges of ages derived from stellar model optimisation for HD 52265. In abscissae are listed the case numbers. For all cases (in abcissae), the non -seismic constraints Teff, L, [Fe/H] are included. Case a: no seismic constraints. Case b: only < Δνn,l>. Case c: < Δνn,l> and the averaged small separation. Case d: individual ratios of frequency combinations. Case e: individual frequencies corrected for surface effect. Symbols and colours refer to different assumptions about the input physics and the chemical composition. For instance, in case a, square and diamond are for α conv values changed by ±0.20 dex with respect to solar. In case e, the red cross is a model without corrections from surface effects. From Lebreton et al. (2014b).

The diagram on the side shows the results of the analyses for the age, for different choices of seismic diagnostics, observational (seismic and non-seismic) constraints, assumptions on the chemical composition, and physics input. A drastic decrease of the dispersion of the results occurs for the cases c, d, e, i.e. when some diagnostic(s) appropriate for the age is/are included. The final errors are found around 1.5, 7 and 13% in radius, mass and age, respectively. The analysis demonstrated the striking improvement in precision that results from including seismic data.


Silva Aguirre et al. (2015) analysed data on individual frequencies for 33 stars observed by Kepler to be (potential) planet hosts. The baseline of the observations is long enough that the uncertainties of individual frequencies are about 0.3 µHz for frequencies close to νmax (and down to 0.1 µHz in the best cases). The stars broadly sample the region of the HR diagram where asteroseismic inferences can be expected. The main analysis was based on ratios of frequency separations, which are insensitive to surface effects, but additional analyses were carried out with different techniques to test for the consistency of the results and the sensitivity to differences in model physics and numerical techniques.


Fractional uncertainties for the mass, radius and ages obtained with seismic modelling of 33 Kepler stars. From Silva Aguirre et al. (2015).

The diagram on the side shows the distribution of errors in the sample, in radius, mass, mean density, and age. The median errors in radius, mass, and age were 1.1, 3.3 and 14%, respectively. These uncertainties account for systematic effects of the use of different modelling or fitting techniques. As a specific example, let us consider the star KIC 4141376 observed by Kepler. It has a mass and evolutionary state very similar to the Sun, although a somewhat lower metallicity. Frequencies were obtained with typical errors around 0.5 µHz, while the error in Teff was assumed to be 91 K and the error in metallicity [Fe/H] was 0.1 dex. For this star, Silva Aguirre et al. (2015) determined mass and radius with precisions of 2.5% and 1%, while the precision of the age was around 20%.



Seismic rotation

The oscillation frequencies can also provide information about stellar interior rotation (e.g. Beck et al. 2012a, b; Deheuvels et al. 2012; Gizon et al. 2013). This requires a high SNR and frequency resolution because one must resolve and extract signatures of rotation (rotational frequency splittings) in the power spectra of non-radial modes of oscillation (e.g. Gizon & Solanki 2003; Ballot et al. 2006, 2008; Benomar et al. 2009; Campante et al. 2011). Seismic rotation periods were recently determined for 22 main-sequence CoRoT and Keplertargets with masses in the range 1.0‒1.6M(Benomar et al 2015). Nielsen et al. (2014, 2015) showed that seismic constraints can be placed on the internal differential rotation of Sun-like stars. For the bright star 16 Cygni A (V~ 6), the precise seismic measurements of the rotation period (Davies et al. 2015) and age (Metcalfe et al. 2015) after 928 days of Kepler short cadence observations can be used to calibrate gyrochronology relations (Davies et al. 2015). Such seismically validated calibrations will serve to derive ages for stars without seismic characterisation. This will be very valuable to the research fields that rely on accurate stellar ages, such as galactic evolution studies. Of course, several benchmark stars like 16 Cygni A must be used to establish and calibrate gyrochronology relations with a broad range of validity. PLATO will do so.


Spin-orbit angle

The angle between the stellar spin and the planetary orbital axes of the exoplanetary systems (also called spin-orbit angle or obliquity) is a unique observational indicator of the origin and evolution of planetary systems. For example, the existence of Jupiter-size planets orbiting in less than a week indicate that inward migration could be required to explain their formation and evolution. While a planet-disk interaction predicts orbital planes parallel to the stellar spin axis, a planet-planet scattering scenario or the Kozai mechanism predicts oblique orbits. Thus, a precise determination of the spin-orbit angle and its statistical distribution would give tight constraints on the contribution of different migration scenarios (Queloz et al. 2000; Winn et al. 2005). The measure of the spin-orbit angle is however not easy because its determination usually requires the knowledge of three angles:

  • The orbital inclination, which is determined with a precision of only a few per cents from the transit light curve.
  • The projected spin-orbit angle onto the sky plane. This is commonly measured using the Rossiter-McLaughlin effect (Rossiter 1924; McLaughlin 1924; Queloz et al. 2000; Winn 2011) and is determined with a precision of 1‒10%.
  • The stellar inclination, which can be directly determined by asteroseismology (e.g. Gizon et al. 2013; Benomar al. 2014, 2015) or indirectly by combining the spectroscopic vsinand the surface rotation measured using stellar activity signatures (surface spots) on the light curve (Hirano et al. 2014).

To date, most statistical studies rely on the projected spin-­orbit angle onto the plane of the sky (Albrecht et al. 2013; Xue et al. 2014). While this is an indicator of the planetary systems obliquity, the projection effect makes its interpretation difficult. In order to obtain tight and reliable constraints on the formation and evolution of exoplanetary systems, it is therefore important to measure the true obliquity, free of projection effects. Asteroseismology can contribute in achieving this goal, as shown by Chaplin et al. (2013), Benomar et al. (2014), and Campante et al. (2016).


Seismic measurement of the inclination angle or the Kepler star, Kepler 25. The plot shows the probability distribution function for the stellar inclination with the median (green). Blue represents region of lowest probability. Red areas are of highest probability.From Benomar et al. 2014.

In addition to the rotation, measurements of the relative amplitudes of the split components of the modes also reveal the stellar inclination, i.e., inclination of the rotation axis relative to the line of sight (Gizon & Solanki 2003; Gizon et al. 2013; Chaplin et al. 2013). This is because geometric cancellation effects, which depend on the stellar inclination, affect these visibilities. The necessary high SNR and the frequency resolution implies that one must observe bright enough targets for a long enough time. In that respect, the initial phase of the Kepler mission was remarkable for its four continuous years of observations of a specific region of the sky, which allows us to measure the stellar inclination of Sun-like stars with a precision of ~10%. However, the stellar inclination is only available for ~30 stars with known planets of the Kepler field (Campante et al. 2016) and one for the CoRoT field (HD52265). This represents approximately 10% of main-sequence stars with detected Sun-like pulsations, but corresponds to less than 1% of the total number of planet candidates. The example of the planet host Kepler25 is shown in the figure aboveAnother example is HAT 7 (Campante et al. 2016; Lund et al 2014; Benomar et al 2014).


The main reason for this low number is that most of the claimed detections of exoplanets are made around stars with apparent magnitude higher than 12, while high precision seismology often requires brighter stars. This is also due to the fact that the initial Kepler field corresponds to a very limited region of the sky.

By performing a high precision photometry of the bright stars (V<12) for long observation durations, and a global sky survey, PLATO circumvents these issues. Thus, one might expect the space mission to provide the first large-­scale statistical study for the orbit configurations of exoplanets around Sun-­like stars. This will enable us to discriminate clearly the scenario of formation and evolution of planets.

One must stress that the K2 mission observes several regions of the sky. However, its degraded observation mode (compared to the Kepler initial phase) limits the seismic observation to very bright stars during only ~75 days. This prevents us from measuring accurately the stellar inclination for a large ensemble of stars. In addition, none of the fields observed by K2 contains planetary systems with a known Rossiter-­McLaughlin effect, so that the true obliquity cannot be measured. Furthermore, for most of the targets it will observe, TESS is not expected to surpass K2 in terms of what will be possible from asteroseismology, because of 1) its small aperture, which will limit asteroseismology of solar-type stars to targets brighter than about V=7 (e.g. see Campante et al. 2016); and 2) the fact that most stars will be observed for only 27 days.




PLATO – Revealing habitable worlds around solar-like stars
Definition Study Report, ESA-SCI(2017)1, April 2017