Detecting sub-stellar companions using the stellar pulsation timing method

PLATO Mission Conference 2021

Felix Mackebrandt, Sonja Schuh
Max Planck Institute for Solar System Research
Unpublished,

Abstract

We investigate variations in the arrival time of coherent stellar pulsations due to the light travel time effect to test for the presence of sub-stellar companions. This timing method is particularly sensitive to planets at large distances and complementary to other exoplanet detection methods which are not efficient for systems with small transit probabilities or high intrinsic radial velocity noise and thus enhances the diversity of potential exoplanet host stars that can be probed with PLATO.

We apply this method to subdwarf B stars within the EXOTIME project and Kepler observations of \(\delta\) Scuti stars.

Classical exoplanet detection methods...

...and why we need alternatives

Radial velocity

Pulsating stars often pose a challenge to detect radial velocity changes due to companions, because the stellar pulsations modulate the radial velocity of the stellar surface and can mask additional signals. Additionally, a large number of stellar lines helps to improve on statistical uncertainties, but especially sdB stars show very few and broad absorption lines, mostly by hydrogen.

Transits

The probability for a transiting exoplanet depends on the orbital separation and the size of the host star, which makes a transit unlikely for small stars (like sdB stars) or far out planets, as they are expected for \(\delta\) Sct stars (Johnson et al. 2011; Lloyd 2011).

Pulsation timing

The motion of a pulsating star with mass M around the common centre of mass implies a variation in the projected line of sight and thus a change in light travel time. The periodic pulsation of the star allow for a measurement of this change to conclude on the minimum mass of the companion m with an orbital period P.

Fig.: Schematic picture of the orbital configuration.

From a sufficiently long observational time span, we compare the average phase of a light curve model to the phase of models from shorter sub-sets (epochs) and generate an \(O-C\) diagram.

Fig.: Schematic picture of the \(O - C\) diagram construction. Top: A model to the full light curve yields the reference phase \(C\). Middle: Light curve models to each epoch yield individual phase information \(O\). Bottom: Construction of the \(O - C\) diagram from above measurements.

\(\kappa\) mechanism for sdB and \(\delta\) Scuti stars

Fig.: Visualization of a l=2 m=0 pulsation.

Changes in the opacity \(\kappa\) of a layer in the stellar atmosphere are driving non-adiabatic pulsations. This occurs where layers of hydrogen (\(\delta\) Scuti stars) or metals (sdB stars) is partially ionized. These stellar pulsations feature following properties:

Especially a coherent pulsation phase is of most importance in order to make use of the pulsation timing method.

EXOTIME project - sdB stars

Companions to subdwarf B stars (sdB) are the key to a possible formation scenario of apparently single sdB stars. During the RGB the star would develop a common envelope with a giant planet that leads to the loss of the envelope and leaves a helium burning sdB star.

A Sun-like star with a planetary evolves into a red giant, the planets are engulfed in the expanding atmoshpere and the planets exchange angular momentum to expell the envelope
Fig.: Proposed formation scenario for the planetary system orbiting KIC 05807616 (Kempton, 2011; Soker, 1998).

The EXOTIME project conducted long term photometric observations of five sdB stars to answer this question. This e-poster shows exemplary the results for DW Lyn (Mackebrandt et al., 2020). For results on the other targets please refer to Mackebrandt et al., 2020, Silvotti et al. 2007 and Silvotti et al. 2018.

EXOTIME project - DW Lyn observations

DW Lyn is a p- and g-mode pulsator (Dreizler et al., 2002; Schuh et al., 2006), observed from 2007 to 2012 within the EXOTIME project for almost 1000 hours by 12 different observatories.

Amplitude spectrum of DW Lyn
Fig.: Amplitude spectrum of DW Lyn showing the two main pulsation frequencies \(f_1 \approx\ 237 \text{d}^{-1}\) and \(f_2 \approx\ 225 \text{d}^{-1}\).

EXOTIME project - DW Lyn results

We derive an evolutionary time scale from the change in period \(\dot{P}\). By comparing to models from Charpinet et al. (2002), we conclude that the star is likely still in the first evolutionary stage of central He burning.

O-C diagram for DW Lyn
Fig.: Results for DW Lyn. Top panel: Amplitudes. Middle panel: Fits of the \(O - C\) data with second order polynomials in time. Lower panel: Residuals.

After subtracting the long term trends, a significant periodic signal in the arrival times of of \(f_1\) is not visible in the measurements of \(f_2\), but likely an artefact due to mode beating. We can not confirm a tentative periodic signal at 80 days as described by Lutz et al. (2011). We compare our measurements with simulated \(O-C\) orbits and put constrains on companion masses within the detection limits.

Minimum companion mass for DW Lyn
Fig.: Minimum companion mass as a function of orbital period (\(f_1\) left and \(f_2\) right panel). Greyscale shows the difference between the \(O - C\) measurements and artificial \(O - C\) data generated for a given combination of companion mass and orbit in units of \(O - C\) uncertainties.

\(\delta\) Scuti stars

Planet occurrence rates predict a maximum for host stars with masses of around \(2 \text{M}_\odot\) (Reffert et al., 2015), close to the mass of main sequence A stars in the instability strip where \(\delta\) Sct pulsators are common. But actual planet detections are affected by observational selection effects. This motivates a search for planets orbiting \(\delta\) Sct stars using the pulsation timing method.

Fig.: Planet occurrence rate as a function of metallicity and stellar mass in the Lick RV sample (Reffert et al., 2015).

\(\delta\) Scuti - KIC 7917485 observations

KIC 7917485 was observed by Kepler and Murphy et al. (2016) detected a planet in an 840 day orbit using the timing method. We confirm these measurements with our pipeline.

Amplitude spectrum of KIC 7917485
Fig.: Amplitude spectrum of KIC 7917485, high pass filtered data to remove instrumental effects and respective residuals after subtracting the two main pulsation frequencies \(f_1 \approx\ 15 \text{d}^{-1}\) and \(f_2 \approx\ 20 \text{d}^{-1}\) and other signals with amplitudes above 0.005%.

\(\delta\) Scuti - KIC 7917485 analysis

We measure the pulsation arrival times and compute the radial velocity of the star for further analysis, via \(v_{rad} = -c \frac{\mathrm{d}\tau}{\mathrm{d}t}\).

O-C diagram for KIC 7917485
Fig.: Analysis for KIC 7917485. Top panel: Amplitudes. Lower panel: \(O - C\) measurements.

\(\delta\) Scuti - KIC 7917485 results

The amplitude spectra of the radial velocity data show a significant periodicity around 800 days in both analysed pulsation frequencies. We model the Keplerian orbit using the RadVel package (Fulton et al., 2018).

Radial velocities for KIC 7917485
Fig.: Radial velocity measurements for KIC 7917485 and model in black (top). Residuals (bottom).
Amplitude spectra of the RV data for KIC 7917485
Fig.: Amplitude spectra of the RV data for KIC 7917485 (top) and residuals (bottom).

The results of our MCMC analysis are in agreement to the orbital parameters derived by Murphy et al. (2016).

this work Murphy et al. (2016)
\(P/\text{d}\) \(884^{+31}_{-35}\) \(840^{+22}_{-20}\)
\(e\) \(0.183^{+0.051}_{-0.06}\) \(0.15^{+0.13}_{-0.10}\)
\(K \sin i/\text{ms}^{-1}\) \(169^{+12}_{-13}\) \(186^{+17}_{-13}\)
\(M_b \sin i/\text{M}_{\text{J}}\) \(10.8\pm 1.1\) \(11.8^{+0.8}_{-0.6}\)
\(a_b \sin i/\text{AU}\) \(2.127^{+0.083}_{-0.087}\) \(2.03^{+0.27}_{-0.22}\)

Outlook

The pulsation timing method proves it strength with long term photometric observations, especially from space. Although the TESS mission covers most of the sky, targets need to be revisited multiple times to be of use for this method. The PLATO mission on the other hand, will observe targets for a much longer period of time. This makes the timing method very valuable to investigate sub-stellar companions to evolved stars.

TESS mission
Plato mission
TESS mission
Credit: NASA/Goddard Space Flight Center
Plato mission
Credit: ESA/PLATO mission consortium
Acknowledgements: Funded trough the Volkswagen Foundation (grant number VWZN3020) and conducted in the International Max Planck Research School for Solar System Science at the University of Göttingen.

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