CHAPTER: ABSTRACTABSTRACTABSTRACT

Stars spend most of their life on the main sequence (MS). But their most substantial changes occur off the MS stage, either before on the pre-MS or beyond at the post-MS phase.
Due to very complex and varied dynamical processes, the evolution of planetary systems orbiting non-MS stars significantly differs from those of MS planetary systems.

This work focusses on the search for sub-stellar companions in post-MS systems and determination of the evolutionary state of their host stars, especially subdwarf B stars (sdB stars). These are stripped Helium-burning cores of red giants with a thin hydrogen atmosphere. The canonical model involves binary evolution to explain the existence of sdB stars. Formation scenarios for single sdBs are more controversially discussed and can be hard to reconcile with observational properties. Besides the merger of two helium white dwarfs or other merger processes for apparently single sdB stars, an alternative formation channel involves planetary systems. During the red giant phase, the star would develop a common envelope with a giant planet that leads to the loss of the envelope. Thus, sdB stars are laboratories to test how planets survive and influence the late phases of stellar evolution.

The rapid pulsations of sdB stars can be used to detect sub-stellar companions from periodic variations in the expected arrival times of the pulsation maxima. This timing method is particularly sensitive to companions at large distances and complementary to other exoplanet detection methods because they are not efficient for stars with small radii and high gravities. Thus, the timing method opens up a new parameter range in terms of the host stars and helps to understand the formation process of single sdBs.

In this work I implemented, tested and applied the pulsation timing analysis to search for sub-stellar companions in late evolutionary stage stellar systems. The method is already established in the literature but not to an extent which is capable of automatically processing long-time series of high-cadence data, i.e., from space born observations.

Part <ref> provides an introduction to extrasolar planets, and to the formation and properties of sdB stars.

Part <ref> and <ref> describe the long-term ground-based observations of four rapidly pulsating sdB stars DW Lyn, V1636 Ori, QQ Vir and V541 Hya. The data are used to measure the secular drifts in pulsation periods. The results constrain the evolutionary state of these stars and are compared to theoretical predictions of stellar evolutionary models. Furthermore, the measurements set limits to masses and orbital periods of sub-stellar companions. In contrast to previous studies, tentative companion detections are not confirmed.

Part <ref> describes the application of the implemented timing analysis to other pulsating stars and data sets. Compared to ground-based observatories, satellite-based telescopes offer the advantage of uninterrupted observations. Observatories like Kepler, TESS or the upcoming PLATO mission provide a large sample of targets. Besides sdB stars,  Scuti ( Sct) pulsators are excellent stars to apply the timing method on.  Sct stars are evolved beyond the MS and of spectral type A. From Kepler observations, previous studies revealed a planetary companion orbiting the  Sct star KIC 791748. The implemented timing analysis of this work is applied to these data and can recover the planetary signature, validating the implementation at hand and independently confirming the planetary companion discovery.

Part <ref> discusses the results of this thesis and provides an outlook to further applications.

CHAPTER: ZUSAMMENFASSUNGZUSAMMENFASSUNGZUSAMMENFASSUNG

Sterne verbringen den Großteil ihres Lebens auf der Hauptreihe (HR). Die gravierendsten Veränderungen geschehen jedoch abseits der HR, entweder zuvor auf der pre-HR oder folgend auf der post-HR. Durch sehr komplexe und variierende dynamische Prozesse, unterscheidet sich die Entwicklung von Planeten in nicht-HR Sternsystemen signifikant von der Entwicklung von Planeten in HR Sternsystemen.

Diese Arbeit legt ihren Fokus auf die Suche nach sub-stellaren Begleitern in post-HR Systemen und auf die Bestimmung des evolutionären Stadiums derer Zentralsterne, insbesondere bei heißen Unterzwergen des Spektraltyps B (engl.: subdwarf B, kurz: sdB), auch Blaue-Unterzwerge genannt. Solche sdB-Sterne sind freigelegte, heliumbrennende Kerne, von vormals Roten-Riesen, mit einer dünnen Wasserstoffatmosphäre. Deren kanonische Entstehung beschreibt die Entwicklung in einem Doppelsternsystem. Entstehungsszenarien für einzelne sdB-Sterne werden jedoch kontrovers diskutiert und sind nur schwer mit Beobachtungen zu verifizieren. Neben der Verschmelzung von zwei heliumdominierten Weißen Zwergen, könnten auch Planetensysteme zur Bildung von sdB-Sternen beitragen. Während der Roten-Riesen-Phase, würde der Stern einen seiner Riesenplaneten in eine gemeinsame Hülle einschließen, was letztendlich zum Verlust der Hülle führt. Somit erweisen sich sdB-Sterne als Testobjekte, das Überleben von Planeten zu prüfen, sowie den Einfluss der Planeten auf die Spätphasen stellare Entwicklung zu beobachten.

Die schnellen Pulsationen von sdB Sternen können, mit Hilfe von periodischen Veränderungen in den zu erwarteten Ankunftszeiten der Pulsationsmaxima, zur Detektion von sub-stellaren Begleitern genutzt werden. Diese Methode der Lichtlaufzeitmessungen ist besonders sensitiv für Begleiter mit großer Distanz zu ihrem Zentralstern und damit komplementär zu anderen Detektionsmethoden. Denn andere Detektionsmethoden sind bei Sternen mit hoher Oberflächengravitation und kleinen Radien weniger aussagekräftig. Damit eröffnet die Methode der Lichtlaufzeitmessungen eine Möglichkeit für Entdeckungen von bisher nicht erforschbaren Planetensystemen und hilft somit den Entstehungsprozess von sdB-Sternen besser zu verstehen.

In dieser Arbeit habe ich die Methode der Pulsations-Lichtlaufzeitmessungen implementiert, getestet und zur Suche nach sub-stellaren Begleitern bei Systemen in den Spätphasen stellarer Entwicklung angewandt. Die Methode ist bereits etabliert, bisher jedoch nicht für eine automatische Anwendung auf lange Beobachtungsreihen in hoher zeitlicher Auflösung, wie z.B. von Weltraumteleskopen, optimiert.

Teil <ref> dieser Arbeit gibt eine Einleitung zu extrasolaren Planeten, sowie auch zur Entstehung und Eigenschaften von sdB-Sternen.

Teil <ref> und <ref> beschreiben die bodengebundenen Langzeitbeobachtungen der vier schnell pulsierenden sdB-Sterne DW Lyn, V1636 Ori, QQ Vir und V541 Hya. Mit Hilfe dieser Daten werden die Veränderungen der Pulsationsperioden gemessen. Diese Ergebnisse lassen auf das evolutionäre Stadium der Sterne schließen und werden mit theoretischen Sternentwicklungsmodellen verglichen. Weiterhin setzen die Ergebnisse der Beobachtungen Grenzwerte für die Masse und Umlaufdauer möglicher sub-stellarer Begleiter. Im Widerspruch zu vorhergehenden Studien, können potentielle Detektionen von Begleitern nicht bestätigt werden.

Teil <ref> beschreibt die Anwendung der implementierten Methode der Pulsations-Lichtlaufzeitmessungen auf andere Sterne und Datensätze. Im Vergleich zu bodengebundenen Observatorien bieten weltraumgestützte Teleskope den Vorteil der ununterbrochenen Beobachtung. Observatorien wie das Kepler Weltraumobservatorium, TESS oder die geplante PLATO Mission, liefern eine große Anzahl von Beobachtungszielen. Neben sdB-Sternen ist die Klasse der pulsierenden   Scuti ( Sct) Sterne ein hervorragendes Ziel für Lichtlaufzeitmessungen.  Sct Sterne sind in einem Entwicklungsstadium jenseits der HR und in der Spektralklasse A zu finden. Vorhergehende Studien haben aus Beobachtungen des Kepler Weltraumteleskops einen planetaren Begleiter um den Stern KIC 791748 offenbart. Mit der in dieser Arbeit implementierten Methode der Pulsations-Lichtlaufzeitmessungen kann das Signal dieses Begleiters aus den Daten gewonnen und somit die Implementierung validiert, sowie eine unabhängige Bestätigung dieser Planetenentdeckung erbracht werden.

Teil <ref> diskutiert die vorhergehenden Ergebnisse und liefert einen Ausblick auf weitere Anwendungen.

CHAPTER: EXTRASOLAR PLANETS

§ HISTORICAL OVERVIEW

Speculations about the existence of other worlds or solar systems other than our own go as far back as the ancient times. Greek philosophers debated about the possibility of countless worlds:

And he maintained worlds to be infinite, and varying in bulk; and that in some there is neither sun nor moon, while in others that they are larger than with us, and with others more numerous. [...] And that some worlds are destitute of animals and plants, and every species of moisture.

Democritus <cit.>

He was supported by Epicurus (314270 BCE) who debated about an infinite number of worlds both like and unlike this world of ours [Bailey, 1926] in a letter to Democritus. But the atomists of this time believed in the geocentric universe, and thus the work of Aristotle (341270 BCE) outshone their ideas. This started to change with the introduction of the Heliocentric universe theory of Nicolaus Copernicus in 1543. However, the Italian philosopher Giordano Bruno was burned during the Inquisition because of his belief in an infinite number of stars which have an infinite number of terrestrial worlds orbiting them. Finally, Galileo Galilei discovered with his telescope, invented in 1606, the true nature of the Moon and planets and paved the way for the discovery of the remaining planets in our solar system.

Our solar system shaped the understanding of planetary evolution and system architecture, until Wolszczan and Frail, 1992 discovered the first exoplanetary system later confirmed by Wolszczan, 1994. The two Earth-mass like planets orbit the millisecond pulsar PSR 1257+12, a rather unusual host star. With the discovery of the first extrasolar planet, 51 Pegasi b orbiting a sun-like star [Mayor and Queloz, 1995], the architecture of exoplanetary systems remained very different than expected from the solar system. The planet has a mass comparable to the mass of Jupiter but a very close-in orbit of about 0.05, leading to a new category of planets called hot Jupiters.

So far, more than 4000 exoplanets have been discovered by various detection methods. They exhibit a wide range of planetary masses and radii as shown in Figure <ref>. Besides hot Jupiters, Neptune- and Saturn-sized planets as well as super-Earths were discovered.

The first transiting exoplanet HD 209458 b was discovered in 1999 [Charbonneau et al., 2000] which later became the first planet with a spectroscopically analysed atmosphere [Charbonneau et al., 2002]. HD 28185 b was discovered as the first exoplanet in the habitable zone, a region around a star in which a planet may retain liquid water on its surface. Using the phase-mapping technique Knutson et al., 2007 reconstructed a rough map of HD 189733 b showing the temperature of the cloud deck. An image of Formalhaut b in 2008 was the first direct image of an exoplanet [Kalas et al., 2008]. With Kepler-186 f, an Earth-sized planet orbiting within its star's habitable zone was discovered [Quintana et al., 2014]. With the discovery of Kepler-452 b and its 1.6 Earth-radii, another category not yet present in our solar system called super Earths was created. Another super Earth orbiting our closest neighbour star, Proxima Centauri, even in the habitable zone was discovered by Anglada-Escudé et al., 2016. Future missions aim to characterise exoplanetary systems and to understand the formation, evolution and chemical composition in more detail.

[Exoplanet mass and radius distribution.]Exoplanet mass and radius distribution as function of their orbital period and effective temperature of their host star (color coded). Out of the 4130 discovered exoplanets, there are 994 mass and 1631 radius measurements <cit.>.

§ DEFINITION OF AN EXOPLANET

The word planet was introduced by the ancient Greeks as the term
àstéres planh̃ta
(asteres planetai) which can be translated as wandering star. This was inspired by the apparent movement of the planets in the solar system as opposed to the fixed stars. In a more scientific way the International Astronomical Union (IAU) defined a planet in the solar system in its resolution 5A as follows:

A planet is a celestial body that

* is in orbit around the Sun,

* has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and

* has cleared the neighbourhood around its orbit.

IAU, 2006

Additionally the working group on extrasolar planets of the IAU published the following statement, not yet adopted as a resolution:

* Objects with true masses below the limiting mass for thermonuclear fusion of deuterium (currently calculated to be 13 Jupiter masses for objects of solar metallicity) that orbit stars or stellar remnants are planets (no matter how they formed). The minimum mass/size required for an extrasolar object to be considered a planet should be the same as that used in our Solar System.

* Substellar objects with true masses above the limiting mass for thermonuclear fusion of deuterium are brown dwarfs, no matter how they formed nor where they are located.

* Free-floating objects in young star clusters with masses below the limiting mass for thermonuclear fusion of deuterium are not planets, but are sub-brown dwarfs (or whatever name is most appropriate).

Boss et al., 2005, reproduced with permission.

The designation of exoplanets consists of the name of the host star followed by a lowercase letter. The first part can be an astronomical catalogue, the scientific instrument or project that discovered the exoplanet. The following letter indicates the order of the exoplanets discovery around its host star usually beginning with b. This is analogous to the designation of multiple star systems.

§ OBSERVATIONAL TECHNIQUES

The vast variety of exoplanets necessitates the need for different detection techniques. They can be separated into three broad categories as the Perryman tree, a visual overview on the detection methods, illustrates in Fig. <ref>.

Widely separated exoplanets can be imaged directly. Also, protoplanetary disks can be detected this way. In case the orbital plane of the exoplanet aligns with the line of sight of the observer, the planet transits in front of the stellar disk which can be detected as flux variations in primary and secondary eclipse.

Gravitational lensing is known for very massive objects like galaxy clusters but can also be detected using a single star as a lens for a background star. This induces a characteristic signal in the light curve. If the foreground star hosts an exoplanet, it will act as an additional lens.

Dynamical effects are based on the gravitational interaction of host star and planet. This can be measured directly as an astrometric change of position on the sky, or indirectly via Doppler shifts in the host star's spectrum or timing variations of transits, eclipses or stellar pulsations. The latter method is used in particular in this work and explained in detail in section <ref>.

[Exoplanet detection methods.]Exoplanet detection methods. The lower limits of the lines indicate masses within reach of present measurements	(solid lines), and those that might be expected within the next few years (dashed). The (logarithmic) mass scale is shown on the left.	Miscellaneous signatures to the upper right are less well quantified in mass terms. Solid arrows show relevant discoveries. Open arrows indicate measurements of previously-detected systems. Numbers are from the NASA Exoplanet Archive, 2018 January 1 <cit.>.

CHAPTER: SUBDWARF B STARS

The majority of known exoplanets orbit cool host stars, i.e., G and M type stars. So far, only a small fraction of planets orbiting evolved host stars have been discovered (compared to effective temperatures of host stars in Fig. <ref>). This raises questions on the evolution and fate of planetary systems. The following chapter revises the stellar evolution with focus on the post-main sequence phase of the host star, arriving at subdwarf B stars (sdB stars). These stars are laboratories to test how planets survive and influence the late phases of stellar evolution.

§ CLASSIFICATION

Humason and Zwicky, 1947 discovered sdB stars in a photometric survey of the North Galactic Pole region.
Their position in the Hertzsprung-Russel diagram (HRD) in Fig <ref>, in between the main sequence (MS) and the white-dwarf (WD) sequence, was later determined by Greenstein and Sargent, 1974. They show effective temperatures $ T_{\text{eff}} $ from ∼2000050000K with surface gravities $ \log (g/ \si{\centi\meter \per \second\squared}) $ of 5.06.5.
Heber et al., 1984, Heber, 1986 could connect the spectroscopic sdB class with the extreme horizontal branch (EHB) evolutionary stage.
Thus, sdB stars are stripped helium-burning cores ($ \SI{\sim 0.5}{M_{\odot}} $) of red giants with a thin hydrogen (H) atmosphere. Their spectra are therefore dominated by the broad Balmer lines of H, while helium (He) is usually depleted.
But their structure differs from horizontal-branch stars, as their H envelopes are too thin to sustain H burning.
They evolve directly on to the WD cooling sequence, as they lack the envelope mass to pass via the asymptotic giant branch (AGB).

[Sketch of a Hertzsprung-Russell diagram.]Sketch of a Hertzsprung-Russell diagram highlighting the position of hot subdwarf (sdB and sdO) stars and the extreme horizontal branch (EHB) located to the left and below the hot end of the main sequence but above the white dwarf cooling sequence. The EHB is separated from the blue horizontal branch (BHB). The location of stars having evolved from the postasymptotic giant branch is shown for comparison. The hot subdwarf stars have nothing in common with traditional cool subdwarfs found below the lower main sequence [Heber, 2009]. Reprinted by permission from Copyright Clearance Center: Annual Reviews, Annual Review of Astronomy and Astrophysics Hot Subdwarf Stars, Heber, ©2009.

§ CANONICAL STELLAR AND PLANETARY EVOLUTION

The progenitors of sdB stars are assumed to be solar-mass stars. They form in molecular clouds, consisting mostly of H, about 2030 per cent of He and a few percent of heavy elements. The gas remains in hydrostatic equilibrium as long as the gas pressure is in balance with gravitational force (virial theorem) but a gravitational collapse can be triggered by shock waves (e.g. nearby supernova). As the cloud contracts, it breaks into smaller fragments, in which the collapsing gas radiates the gravitational potential as heat. As temperature and pressure increase, the fragment condenses into a protostar which is in hydrostatic equilibrium. This way, stars with different masses form mostly in groups. Stellar winds of more massive stars ionize the remaining H in between the stars, creating H II regions, and ultimately disrupt the cloud which prevents further star formation.

The protostar continues to accrete material from its protoplanetary disk (Fig. <ref>) until finally H fusion starts in the core and the star begins its phase on the main-sequence of the HRD (Fig. <ref>).

[Protoplanetary discs.]Left: Protoplanetary disc imaged by the Hubble Space Telescope in the Orion nebula. The image is about 1800 across. Credit: NASA, C.R. O'Dell and S.K. Wong. Right: ALMA image of HL Tauri showing th protoplanetary disc at a wavelength of 1.3mm. The image is about 250 across. Credit: ALMA (ESO/NAOJ/NRAO).

Inside the protoplanetary disk, planetesimals form from electrostatic and gravitational interactions as building blocks for planets. During their formation, planets migrate within the disk and accrete material until the disk is evaporated by the central stars radiation. On the main-sequence, the star will evolve only slowly and remain there for about 10^10 years. As the hydrogen in the core eventually depletes, the fusion rate cannot be maintained and the core contracts. This continues to fuse H in a shell outside the He core. The core contracts and the outer layers begin to expand. The star becomes a red giant, moving up the red giant branch (RGB) in the HRD. The He core compacts into degenerate matter and once core pressure and temperature are high enough, the He starts to fuse with a He flash at the tip of the RGB in the HRD. The core mass during the helium flash is about half a solar-mass. The star decreases in radius, increases its surface temperature, and moves to the horizontal branch (HB) of the HRD. After the He is fused in the core, outer shells fuse further He and the star moves to the asymptotic giant branch (AGB) and becomes red giant again. The produced carbon (C) can fuse with He and oxygen (O) to build a CO core. The H and He shells are consecutively fusing in a cyclic process. The star ejects most of its envelope during these thermal pulses. The expelled material is ionized by the UV emission of the hot remnant core and can be observed as a planetary nebula. After this short phase, the remnant star cools and enters the white dwarf cooling sequence.

Planet occurrence rates, estimated from Kepler observations, suggest that 30 per cent of Sun-like stars host Kepler-like planetary systems[Kepler-like planets are planets that have radii $ R_P \gtrsim R_{\earth} $ and orbital periods $ P > \SI{400}{\day} $. Our Solar system is not detectable by Kepler. ] [Zhu et al., 2018]. Several studies investigated occurrence rates from radial velocity surveys. Jupiter-like planets[Planets with masses $ 0.15 R_{\jupiter} \lesssim M_P \lesssim 13 M_{\jupiter} $ and orbital periods of 37.] may occur at six per cent of Solar-like stars [Wittenmyer et al., 2016]. Systems with giant planets from the Kepler mission show outer gas giants within orbital periods of less than 400 days at a 5 per cent occurrence rate [Santerne et al., 2016]. Bryan et al., 2016 suggest the total occurrence rate of companions[For companions in the range of 120M_♃ and 520] from radial velocity measurements and direct imaging with about 50 per cent. Recently Grunblatt et al., 2019 estimated the occurrence rate of planets larger than Jupiter for low-luminosity red giant branch stars from the K2 mission to be 0.5 per cent.
Observations of polluted WDs, debris disks orbiting WDs like WD 1145+017 [Vanderburg et al., 2015] or even in-situ accretion of a giant planet to WD J0914+1914 [Gänsicke et al., 2019] show that these disks must form during the WD phase since they are located within the preceding giant stellar radii. Spectroscopic analysis of polluted WD atmospheres find metals, as to be expected from planetary composition. Since the sinking times for such heavy metals are orders of magnitude shorter than the WD cooling age, the expected detection rate of polluted WDs is at about 0.1 per cent. But the actual observed rate of metals is at 2550 per cent. Thus, planetary systems appear to survive the RGB phase of their host star and may get disrupted and accreted during the WD phase. Veras, 2016, Farihi, 2016 give a broad overview on post-main-sequence planetary system evolution and polluted WDs.

§ SUBDWARF B STARS FORMATION SCENARIOS

The canonical formation of Subdwarf B stars is not explained by the typical evolution described in the section above since some mechanism during the progenitors giant phase must have removed enough of the envelope to not sustain a H burning after the He flash.
The formation scenario in a binary system, first proposed by Mengel et al., 1976, gives rise to different formation channels [Han et al., 2002, Han et al., 2003, Podsiadlowski, 2008] depending on the initial mass ratios. The scenarios are illustrated in Fig. <ref> and <ref>. The most simple scenario is a stable Roche-lobe over flow (RLOF), as in the left panel of Fig. <ref>. The sdB progenitor fills its Roche lobe near the tip of the RGB. The mass transfer is dynamically stable and the companion accretes the matter. This will form a sdB in a long-period binary system with a main-sequence companion. Observations find periods from 7001300, leading to an improvement of the RLOF model [Chen et al., 2013]. If the mass transfer during  the RLOF is too high, the companion cannot accrete all the matter and a common envelope (CE) is formed (see Fig. <ref>). Friction in the CE let the stars spiral inwards and until the CE gets ejected. This will form a close binary system.

While about 50 per cent of sdB stars are found in close binary systems with periods of less than ten days, observations show about 30 per cent appear as apparent single sdB stars. There are several possible formation scenarios including atmosphere loss due to stellar winds, the merger of two He WD (right panel of Fig. <ref>), or undetected low-mass companions.

Hall and Jeffery, 2016 estimated the H mass in the remnants of He WD mergers. In the effective temperature-surface gravity plane, they found a region occupied by these stars during the core-He-burning phase and located the majority of apparently single sdB stars inside this region. Nevertheless, this conclusion depends on assumptions made to the model, regarding the initial H mass and loss of H during the merger. Subdwarf B stars located outside of this parameter region are found to be rapidly rotating low-gravity sdB stars.

Alternatively, the sdB progenitor could lose mass near the RGB tip due to strong stellar winds or internal rotation.
While the He flash occurs typically at the tip of the RGB, a sufficient mass loss can lead a star to depart from the RGB so that the He flash is delayed [Castellani and Castellani, 1993].
The remnants of these hot flashers are located close to the He main sequence, which coincides with the sdB location. Although an early hot flasher scenario may explain He and C abundances in sdB stars, observations show some inconsistencies [Moehler et al., 2011, Latour et al., 2014].

[Formation of sdB stars I.]Stable Roche-lobe channel (left) and single-star/merger channels (right) for the formation of sdB stars [Podsiadlowski, 2008]. By the kind permission of ASP Conference Series, vol. 401, P. Podsiadlowski, p. 63.

[Formation of sdB stars II.]Common-envelope (CE) channels for the production of sdB stars. The CE phase can be either the first (right panel) or the second (left panel) mass-transfer phase, producing a tight sdB binary with either a normal-star or white-dwarf companion, respectively [Podsiadlowski, 2008]. By the kind permission of ASP Conference Series, vol. 401, P. Podsiadlowski, p. 63.

Finally, substellar objects, e.g., brown dwarfs or even planets, can lead to enough mass loss in order to form sdB stars. As described in the previous section, the abundance of planets is high enough to be considered to survive and influence the late stellar evolution.	For a single red giant, planets with orbital separations smaller than about 5, will enhance the mass loss when the planet is engulfed by the red giant's atmosphere. Since planets have a comparable high angular momentum, parts of it can be transferred to the envelope. In consequence of this momentum transfer, the planet spirals inwards and stellar mass loss increases, which leads to the formation of a sdB star. In this scenario, the planet would either evaporate, merge with the core, or survive while most of the planet's envelope is lost [Soker, 1998]. With the merger scenario, an enhanced stellar rotation could mix more He to the envelope and thus increase the RGB tip luminosity and further enhance the total mass loss [Sweigart, 1997]. The hypothesis of surviving planetary systems is supported by planetary-mass companions like the candidates V391 Peg b [Silvotti et al., 2007], KIC 05807616 b,c [Charpinet et al., 2011], KIC 10001893 b,c,d [Silvotti et al., 2014] or brown dwarf companions like V2008-1753 B [Schaffenroth et al., 2015] or CS 1246 [Barlow et al., 2011]. Figure <ref> illustrates the process of forming a sdB planetary system.

From observational data it cannot be ruled out that these planets may have formed as second generation planets. After the merger of two He WDs, the remaining circumstellar disk could provide enough material to form planets, in analogy to the pulsar planets [Wolszczan and Frail, 1992, Thorsett et al., 1993].

[Possible formation mechanism of close-in planets.]Possible formation mechanism of close-in planets. a, The two planets identified by Charpinet et al., 2011 were probably massive gas-giant planets during the early part of their host star's life, when the star burned hydrogen into helium as our Sun does today. At this time, the planets resided farther away from their host star than they do today. b, When the star exhausted the supply of hydrogen in its core, it expanded to become a red-giant star. The outer layers of the star then engulfed the orbits of the two planets. The planets lost their outer gaseous layers as their orbits spiralled inwards. c, For unknown reasons, the red giant expelled its cool outer layers, leaving behind a small hot B subdwarf star. The small rocky cores of the initial planets were left behind in close-in orbits with periods of 5.8 and 8.2 hours. (Figure not drawn to scale.) [Kempton, 2011]. Reprinted by permission from  Copyright Clearance Center: Springer Nature, Nature, Planetary science: The ultimate fate of planets, Kempton, ©2011.

Finally, there is evidence that some sdB stars are not core He-burning objects. Heber et al., 2003 discovered a sdB with insufficient mass to ignite He-burning in the core.

§ ASTEROSEISMOLOGY

Asteroseismology describes the study of the interiors of stars by using their oscillations as seismic waves. The first pulsating star was discovered in 1596 by David Fabricius. He noticed that the star  Ceti (subsequently named Mira, the wonderful) vanished from the visible sky. Later on, it was realized that the star did so every eleven months and became the first known (semi) periodic variable star. Subsequently, more variable stars like  Cephei (related stars are summarized as Cepheids) were discovered and their variations linked to radial pulsations. By now, a large number of groups of pulsating star is known to exist. Figure <ref> shows their occurrences in a HRD. Historically, they have been classified on a phenomenological basis and mostly named after their prototype star. The physical reasons for the classification were uncovered later and are related to the different types of excited pulsation mode, mass and evolutionary state. Since this work relates mostly to sdB stars, the following describes their driving mechanism. Extensive overviews can be found, e.g., in Cunha et al., 2007, Aerts et al., 2010.

[Hertzsprung-Russell diagram showing different classes of pulsating stars.]Hertzsprung-Russell diagram showing different classes of pulsating stars. Some of these are named after a particular member of the class. Others are acronyms, respectively, for: rapidly oscillating	Ap (roAp); Slowly Pulsating B (SPB); subdwarf B variables (sdBV). The group labelled GW Vir includes what has formerly been known as the PNNV stars (for Planetary Nebulae Nuclei Variables), and the variable hot DO white dwarfs (DOV); the DBV and DAV stars are variable DB (helium-rich) and DA (hydrogen-rich) white dwarfs. The parallel long-dashed lines indicate the Cepheid instability strip [Cunha et al., 2007].
Reprinted by permission from  Copyright Clearance Center: Springer Nature, Astron Astrophys Rev., Asteroseismology and interferometry, Cunha, ©2007.

§.§  mechanism

Baker and Kippenhahn, 1962 proposed the  mechanism to explain  Cephei pulsations.
For stellar matter than can be described as an ideal gas, an increase in compression of the atmosphere causes an increase in temperature and density. This usually results in a decrease of the opacity , allowing energy to be transported more efficiently. This describes an equilibrium where temperature and pressure are balanced.
But in layers of the star where  increases with increasing temperature, the incoming flux from inner layers can be temporally stored. These layers are associated with regions where (partial) ionization of chemical elements occur.
When the layer expands in order to reach its equilibrium, the additionally accumulated energy is  released and the star can expand beyond its equilibrium radius.
When the material recedes, energy is again stored in the stellar interior, and the cycle can repeat as a periodic stellar oscillation.

Depending on the restoring force of the pulsations, modes are either of the nature of standing acoustic waves (referred to as pressure modes or p modes) or internal gravity waves (g modes). However, especially in evolved stars, both types of modes can be found.

Unlike solar oscillations, which are exited stochastically, pulsations excited by the  mechanism are coherent in phase; their lifetime is very long, compared to observations spanning over years and change slowly due to stellar evolution. These modes are suitable for a pulsation timing analysis.

In variable sdB stars (sdBV) the bump in opacity is related to elements of the iron group [Charpinet et al., 1997], which are accumulated in the driving layers by diffusion processes like gravitational settling and radiative levitation <cit.>. Charpinet et al., 1996 predicted the nonradial pulsations theoretically almost simultaneously to their discovery by Kilkenny et al., 1997. Effective temperatures $ T_{\text{eff}} $ typically range from 2800035000 and surface gravity $ \log g $ from 5.2 to 6.1. Observed pulsation amplitudes are usually smaller than 50. Hybrid pulsators, showing both p and g modes <cit.>, locate at the temperature boundary around 28000 between both classes of pulsators. About ten per cent of the sdB stars, located in the instability strip of the $ T_{\text{eff}} - \log g$ plane, exhibit pulsations above a few . This fraction might be biased due to detection limits of ground based observations. However, Kepler observations confirmed the deficiency of p-mode pulsators in the instability strip <cit.>.

Due to the He fusion into C  and O in the core of sdB stars, the internal structure varies slowly and thus modifies the condition of hydrostatic equilibrium. This results into slow, secular changes of the surface gravity and effective temperature, and slow changes in the pulsation period [Charpinet et al., 2002]. Using long-term observations, these period changes can be measured.

§.§ Asteroseismic observations

Stellar oscillations generate motions and temperature variations on the surface and result in observable variability. These variations cause changes in brightness and colour, radial velocity and line profiles. Thus, pulsating stars can be studied using photometric and spectroscopic time series measurements, which can be used to generate frequency spectra - one of the most important tools in asteroseismology. The following focuses on time series photometry.

Since the observations are usually discrete and contain gaps, a harmonic analysis is performed using the Discrete Fourier Transformation

\begin{align*}
F(f) = \sum_{k=1}^{N} x(t_k) \exp\left( i 2 \pi f t_k \right),
\end{align*}

of the measurements $ x(t_k) $, which leads to an amplitude spectrum. The observations oppose limitations to this spectrum. Due to the discrete nature, the highest useful frequency to search for is called the Nyquist frequency

\begin{align*}
f_{\text{Nyq}} = \frac{1}{2\Delta t},
\end{align*}

with the time between two data points $ \Delta t $. In the case of unevenly spaced data, the Nyquist frequency does not exist, higher frequencies might be detectable. With a finite length of the observations, the spectrum has a finite frequency resolution proportional to the inverse of the total time span. Moreover, observations are likely to show gaps (e.g. day/night or seasonal observation windows, technical problems, poor observing conditions), some of them with a regular nature, which produce additional peaks in the Fourier space. These effects combined are known as the window function in the time domain or as spectral window in the Fourier space. The spectral window can show multiple peaks even if the underlying signal is monochromatic and complicate the interpretation of the data. Additionally, long gaps produce unwanted signals in the low frequency regime. In order to overcome the complex response in frequencies and amplitudes of the signal, even without noise, different forms of periodograms have been defined.
A well-known and widely applied technique is the Lomb-Scargle periodogram [Scargle, 1982].
Horne and Baliunas, 1986, Schwarzenberg-Czerny, 1997 showed the equivalence of the variance reduction between a Lomb-Scargle periodogram and a least squares fit of sinusoids at test frequencies. Thus, fits of a harmonic series of sinusoids at test frequencies can be used to search for non-sinusoidal signals as well. The Lomb-Scargle periodogram does not require regular sampled data and is used for the analysis in this work (see Press et al., 2007, VanderPlas, 2017; c.f. section <ref>).

The EXOTIME project

CHAPTER: TARGETS

The EXOTIME program (EXOplanet search with the TIming MEthod) was born out of long-term monitoring of the object HS 2201+2610 in order to measure long-term drifts in the pulsation periods $ \dot{P} $. The large amount of data made it possible to not only constrain sdB star evolution from $ \dot{P} $ but resulted in the postulation of the substellar companion candidate V391 Peg b using the pulsation timing method [Silvotti et al., 2007]. With the puzzling formation history of sdB stars, the authors initiated the EXOTIME program in 2008. The project extends the long-term monitoring to the objects HS 0702+6043 (DW Lyn), HS 0444+0458 (V1636 Ori), PG 1325+101 (QQ Vir), EC 09582-1137 (V541 Hya) and HS 2201+2610 (V391 Peg). While the majority of exoplanet research is strongly focussed on planets (and moons) in the habitable zone and formation of planetary systems, the main goal of the EXOTIME project is to detect exoplanets orbiting evolved stars, using the stellar pulsations to conclude on timing variations (see section <ref>). Transit and radial velocity searches are not efficient for stars with small radii and high gravities like sdB stars. Additionally, the project wants to characterize the target stars using asteroseismic methods and in particular the measurement of $ \dot{P} $.

The following sections describe the targets and their observations in detail. Their selection followed some basic properties: The targets must be observable from the Northern hemisphere and bright enough for a sufficient signal-to-noise ratio in the Johnson B band with 13m class telescopes, since sdB stars emit the maximum energy in the near UV regime. A suitable target for the timing method should show two to four independent pulsation frequencies, coherent in phase and stable in amplitude ($ \SI{>1}{\milli\mag} \simeq \SI{0.1}{\percent} $). Each frequency leads to an independent timing measurement. But coherence and stability can only be verified with the long-term monitoring. To ensure the sdB star is not accompanied by a low-mass star in a binary system, there should be no strong IR excess compared to a black body spectrum. The observational parameters of the five EXOTIME targets are summarized in Table <ref>.
Section <ref> describes DW Lyn, V1636 Ori, QQ Vir, V541 Hya in detail and lists the atmospheric parameters in Table <ref>.

Observations and results on a substellar companion for V391 Peg are published by Silvotti et al., 2007, alternative scenarios are discussed by Silvotti, 2008 and additional observations are published by Silvotti et al., 2018. The latter publication placed the first interpretation of the $ O-C $ variations under discussion because of possible non-linear interactions between different pulsation modes that change arrival times. We proposed for continued observations at the Las Cumbres Observatory <cit.>, using 2m class telescopes in order to connect observations from Silvotti et al., 2018 with upcoming TESS observations in 2020. In 2017, we observed for about 52 hours, in 2019 for eight hours, each in the Johnson B band with exposure times of 1020. The observations are listed in Table <ref>.

CHAPTER: MULTI-EPOCH PHOTOMETRIC DATA

§ TIME SERIES PHOTOMETRY

The timing method applied to the data and described in the next section imposes requirements on the photometric time series. Each observation run needs a high temporal resolution of less than about 30 in order to sample the p-modes correctly and a high signal-to-noise ratio to detect their low amplitudes. A single phase measurement in the later analysis is required to reach an uncertainty in the order of one second. Therefore, the observations should cover about three consecutive nights, each with about three hours for one phase measurement.

§.§ Photometry

In order to make use of the raw CCD images from the observations, the data need to be calibrated properly. This includes detector specific corrections (bias and dark field subtraction), as well as instrumental and sensitivity corrections (flat field division). The following photometric measurements were performed using the IDL package TRIPP (Time Resolved Imaging Photometry Package). Some data have been reduced using different software packages but the concept does not differ. A circular aperture is placed around the target star and measures the flux. An annulus around this aperture or a further field measures the background flux which is subtracted from the target flux. Since variations in the Earth's atmosphere on short and long term time scales (sky transparency, air mass) affect this measurement, the target flux is compared to the flux of multiple comparison stars. The field of view is assumed to be small enough that all atmospheric variations influence the flux of all stars equally. This procedure is applied to all images of an observation run and yields the light curve. Atmospheric extinction is correction by a second order polynomial in time.

§.§ Timing accuracy

Accurate time stamps for each observation is crucial for the analysis of timing effects. Acquisition equipment is usually synced via network with a time server to the universal time (UTC). However, some of the first observations used for the project do not record the fraction of the second. Thus, the accuracy of individual observations is assumed to be better than ±0.5.

Additional to the recording of precise time stamps, the observer needs to be positioned in a rest frame compared to the target. Without the necessary corrections, the result would be superimposed by the Earth's motion around the barycentre of the Solar system and discontinuous time standards. Eastman et al., 2010 describes the necessary formalism in detail. The Barycentric Coordinate Time (TDB) time frame including barycentric corrections for the Solar system is accurate down to 3.4 and applied to all observational time stamps (hereafter refereed as BJD(TDB)).

CHAPTER: METHODS

§ $ O-C $ PIPELINE

Already in the 17th century, Ole Rœmer made use of the travel time of light in order to measure the speed of light <cit.>. An inversion of this scenario allows to measure changes of orbits using the finite speed of light. In astronomy, this idea is widely used for different applications, e.g., on pulsars, eclipsing binary systems, timing variations on planetary transits or stellar pulsations. In the context of stellar pulsations, this concept is referred to as the $ O-C $ method (Observed minus Calculated). The approach compares the phase of a periodic function to a reference phase and thus allow conclusions on variations in the observed phase. While the following describes the method illustratively, the specific implementation of the $ O-C $ method in this work is explained in section <ref>.

The schematic construction of an $ O-C $ diagram is pictured in Fig. <ref>. Requirement for a $ O-C $ analysis is a sufficiently long observational time span of the stellar pulsation. A light curve model fit to full data set provides the reference phase, referred to as $ C $. Similar fits to small sub-sets of the observation provides an independent light curve model. Thus, each sub-set, called epoch, provides a phase measurement, referred to as $ O $. Iterating through all epochs in time yield the differences between the average phase of the reference model and the phase measurements of the epochs, called $ O-C $ diagram.

[Schematic picture of the $ O-C $ diagram construction.]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.

An $ O-C $ diagram with no variations implies an agreement between observed phases and the reference phase. Linear changes in the diagram suggest a wrong frequency for the reference model and thus a constant error added per cycle. More complex deviations are described below.

§.§ Linear change in period

The following considerations can be found in Kepler et al., 1991. In order to make use of the timing method, a sufficiently stable timer in the observed system is required. Sufficiently refers to accuracy compared to the length of observations. In the case of stellar pulsations, this timer is a periodic brightness variation. The observed time of the $ E $th cycle can be expressed as

\begin{align*}
T_E = T_0 + PE,
\end{align*}

with the period $ P $ at $ T_0 $ and the epoch $ E $, as in the cycle number counted from $ T_0 $.

Due to physical processes, the stellar pulsation period will change slowly over time (see section <ref>). If this change is small, $ T_E $ can be expand in a Taylor series:

\begin{align*}
T_E = T_0 + \diff{T}{E} \left( E-E_0 \right) + \frac{1}{2} \diff{^2T}{E^2} \left( E-E_0 \right)^2 + \mathcal{O}\left( \left( E-E_0 \right) \right)^2.
\end{align*}

Terms of higher order are neglected. Expanding the quadratic term, using $ \dif{T}/\dif{E} = P $, leads to

\begin{align*}
\diff{^2 T}{E^2} = \diff{P}{E} = \diff{t}{E} \diff{P}{t} = P \diff{P}{t}.
\end{align*}

Assuming $ 2E_0 \ll E $ finally yields

\begin{align*}
T_E = T_0 + PE + \frac{1}{2}P\dot{P}E.
\end{align*}

In order to compare the model of a slowly changing period with the null hypotheses (the period is stable), both identify with

\begin{align*}
O &\coloneqq T_0 + PE + \frac{1}{2} P \dot{P}E \\
C &\coloneqq T_0' + P'E,
\end{align*}

where $ O $ stands for observed ephemerides with a constant period $ P' $ and $ C $ for the calculated or computed ephemerides. With $ \Delta T = T_0 - T_0' $ and $ \Delta P = P - P' $ and $ E = tP^{-1} $ this leads to

\begin{align*}
O-C = \Delta T_0 + \frac{\Delta P}{P} t + \frac{1}{2} \frac{\dot{P}}{P} t^2.
\end{align*}

In an $ O-C $ diagram, a fit with a parabolic function yields the evolutionary time scale $ \dot{P}/P $.

§.§ Light travel time effect

Assuming a two body system consisting of the pulsating star and a companion, both orbit their common centre of mass. The motion of pulsating star with mass $ M $ around the barycentre implies a variation in the projected line of sight, as show in Fig. <ref>, which translates into a change in light travel time. The periodic pulsation of the star allow for measuring this change and hence, concluding on the minimum mass of the companion $ m $. Assuming a circular orbit due to the proposed formation history of companions of subdwarf B stars, the change in light travel time varies periodically and imposes a sinusoidal signal in the $ O-C $ diagram. The following describes the derivation of the expected amplitude of this light travel time effect <cit.>.

(18,6)$ x $
(45,57)$ z $
(88,30)$ y $
(44,9)$ a $
(54,15)$ a_{comp} $
(91.5,40)$ i $
(37,13)$ M $
(61,54)$ m $
(49,38)$ +\Delta T $
(49,29)$ -\Delta T $

[Schematic picture of the orbital configuration.]Schematic picture of the orbital configuration.

Let the line of sight be aligned with the $ z $ direction. With the inclination of the orbit $ i $, the argument of periapsis $ \omega $ and the true anomaly $ \vartheta $ of the Kepler ellipse, the $ z $ component of the radius vector $ r $ reads as

\begin{align*}
r_z = r \sin\left( i \right) \sin\left( \omega + \vartheta \right).
\end{align*}

Kepler's laws of planetary motion yield

\begin{align*}
r = \frac{a \left( 1 - e^2 \right)}{1 + e \cos \left( \vartheta \right)}, \label{eq:rz1}
\end{align*}

with the semi mayor axis $ a $ and the eccentricity $ e $. Thus,

\begin{align*}
r_z = \frac{a \left( 1 - e^2 \right)}{1 + e \cos \left( \vartheta \right)} \sin\left( i \right) \sin\left( \omega + \vartheta \right).
\end{align*}

With Kepler's Third Law follows for the semi major axis

\begin{align*}
a^3 = G \frac{m^3 P^2}{4 \pi^4 \left( M+m \right)^2},
\end{align*}

where $ P $ is the orbital period. Assuming a circular orbit with $ e=0 $. The maximum displacement $ \Delta r_z $ in the orbit occurs at $ \sin\left( \omega + \vartheta \right) = \pm1 $:

\begin{align*}
\Delta r_z = 2 G^{1/3} m \left( \frac{P}{2\pi \left( M+m \right)} \right)^{2/3} \sin i .
\end{align*}

The light travel time follows from

\begin{align*}
\Delta T &= \frac{r_z}{c} \\
&= \frac{2 G^{1/3} m}{c} \left( \frac{P}{2\pi \left( M+m \right)} \right)^{2/3} \sin i,
\end{align*}

which is the semi-amplitude of the sinusoidal signal in the $ O-C $ diagram induced by a sub-stellar companion due to the light travel time effect.

§ ARTIFICIAL DATA TESTING

Once such a fast pipeline is deployed, it provides a convenient way to analyse artificial data with precisely known input parameters and test for the expected results. In order to avoid additional effects due to irregular observations and gaps in the data, the artificial data are mostly generated according to the PLATO scheme: cadence of 25 for three years (i.e. $ 1.86\times10^6 $ measurements), and a signal to noise ratio of about 6, as expected for a 14 star in the B band observed with all 24 telescopes [ESA/SRE(2011)13, 2011]. A different set of artificial data is created for a simplified ground-based observational scheme (three consecutive nights per month, each eight hours, for ten years, i.e. $ 1.24\times10^5 $). Since the results between these two sets are very similar, merely the PLATO-like scheme is discussed below.

In total, about 50 different scenarios of frequency spacings and amplitudes were investigated. Some interesting cases are explained below.

Resonant frequencies
Pulsations with frequency-spacing in integer multiples might cause an unstable fitting process. To test for such scenarios, a set of several light curves was created. They simulate pulsation frequencies in 1:2, 1:3 and 1:5 resonances with a constant phase and amplitude. The resulting fits to individual modes tend to show oscillating phases but the simultaneous fit of all frequencies is stable and recovers successfully phase- and amplitude-information. Fig. <ref> shows an example of resonant modes with a base frequency of $ \SI{240}{\per\day} $. Some modes tend to show an constant offset in the $ O-C $ diagram but are on average still close to the null hypothesis.

[Amplitude spectrum for four resonant pulsations with frequencies.]Left: Amplitude spectrum for four resonant pulsations with frequencies of 240;120;80;48 and amplitudes of 0.5. Right: Window function.

[Results for four resonant pulsations.]Results for four resonant pulsations with frequencies of 240;120;80;48. Each epoch contains data from one month. Top: Reconstructed Amplitudes. Bottom: $ O-C $ diagram.

Close frequencies
Unresolved frequencies are a problem in the analysis of data and interpretation of their results. The frequency resolution is intrinsic to the observation pattern. Unfortunately, it is very difficult to conclude from an observational $ O-C $ pattern on unresolved frequencies. The vast range of possible combinations in number of frequencies, their spacing and amplitude ratios makes it almost impossible to conclude on a specific configuration to explain the observations. However, if there is a resolved multiplet in the observations, but the signal-to-noise ratio of individual amplitudes is not good enough to provide useful phase measurements, a simulated multiplet on the observational time grid can provide insights on the $ O-C $ behaviour of the main frequency. Additionally the window function can help to interpret amplitude spectra.

Artificial test cases can raise awareness of possible $ O-C $ patterns and caution the interpretation of observational results. The following tests were conducted with two pulsation frequencies and a constant phase which should result in a $ O-C $ diagram not different from zero.

With a sufficient frequency separation and signal-to-noise ratio for both pulsations (see Fig. <ref>), the $ O-C $ diagram might show mode-beating effects like in Fig. <ref>. The recovered amplitude and phase of the main pulsation mode show periodic variations, although they should appear constant. Thus, the simultaneous occurrence of amplitude- and phase-modulations in an $ O-C $ diagram should caution the interpretation in favour of a light travel time effect.

[Amplitude spectrum for two close pulsations.]Left: Amplitude spectrum for two close pulsations with frequencies of 240.000;240.002 and amplitudes of 0.5;0.2 (top), after subtraction of the mode at 240.000 (middle) and residuals after subtraction of the mode at 240.002 (bottom). Right: Window function.

[Results of the simultaneous fit of two close pulsations]Results of the simultaneous fit of two close pulsations with frequencies of 240.0;240.002 and similar amplitudes. Each epoch contains data from one month. Top:Reconstructed  Amplitudes. Bottom: $ O-C $ diagram.

When the signal-to-noise ratio for one frequency is barely over the noise level, it might be considered as noise after the pre-whitening of the dominant mode and neglected.
Using the example above, but reducing the amplitude of the second pulsation from 0.5 to 0.001 (see Fig. <ref>), leads to the $ O-C $ diagram in Fig. <ref>. The second pulsation is not taken into consideration because the amplitude does not elevate above the noise level. The main pulsation shows periodic variations in the phase, as to be expected from a substellar companion. At the same time, the amplitude appears to be modulated as well, indicating a mode-beating effect.
These scenarios strengthen the need for at least one additional phase variation measurement on an independent pulsation mode in order to conclude on reflex motion effects due to a substellar companion.

[Amplitude spectrum for two close pulsations and very different amplitudes.]Left: Amplitude spectrum for two close pulsations with frequencies of 240.0;240.002 and amplitudes of 0.5;0.001 (top), residuals after subtraction of the mode at 240.0 (bottom). Note the different scale. The residual peaks are only slightly above the average noise level. Right: Window function.

[Results for two close pulsations and very different amplitudes.]Results for two close pulsations with frequencies of 240.0;240.002 and very different amplitudes. Each epoch contains data from one month. Top: Reconstructed Amplitude. Bottom: $ O-C $ diagram for the main pulsation.

Variable amplitude
Previous observational studies show that amplitude modulations are a common phenomenon for sdB stars Kilkenny, 2010. But due to the relatively short observational baseline of such investigations, it is not clear whether amplitude modulations can appear on their own or together with phase modulations (as result of mode beating effects). Thus, the $ O-C $ method needs to be able to distinguish between the two scenarios, which can be tested by using a simulated mode with an amplitude modulation but no phase modulation. The example data set shown in Fig. <ref> and <ref> exhibits an amplitude modulation of 50. The $ O-C $ analysis recovers successful this amplitude variation. Also, the phase shows no simultaneous variation which enables the distinction between a mode beating and pure amplitude modulated scenario.

[Amplitude spectrum for one pulsation of varying amplitude.]Left: Amplitude spectrum for one pulsation with a frequency of  240.0 and an amplitude of 0.5, varying by 50 with a 300 period (top), residuals after pre-whitening of the mode at 240.0 (bottom). Right: Window function.

[Results for one simulated pulsation with varying amplitude.]Results for one simulated pulsation with a frequency of  240.0 and an amplitude of 0.5. The amplitude is varying by 50 with a 300 period. Each epoch contains data from one month. Top: Reconstructed Amplitude. Bottom: $ O-C $ diagram.

§ ALTERNATIVE APPROACHES

§.§ Analytic signal

The analytic signal (AS) is a complex-valued function, where real and imaginary part are related to each other by the Hilbert transform. In polar coordinates the AS can be expressed in terms of its amplitude and phase as a function of time, called envelope and instantaneous phase. The instantaneous frequency can be obtained by differentiating the instantaneous phase in respect to time. From the instantaneous phase, an $ O-C $ diagram can be inferred, which makes the AS a potentially useful tool to investigate phase, amplitude and frequency variations in stellar pulsations. See III, 2007 for a comprehensive overview.

The AS $ \tilde{s}(t) $ of a signal $ s(t) $ is defined as

\begin{align*}
\tilde{s}(t) = s(t) + i \mathcal{H}\left( s(t) \right),
\end{align*}

where the Hilbert transformation $ \mathcal{H} $ is defined over the convolution

\begin{align*}
\mathcal{H} = \frac{1}{\pi t} * s(t).
\end{align*}

While the original signal generates one amplitude value for each point in time, the analytical representation generates a rotating vector in the complex plane. Thus, the analytical representation provides instantaneous values for amplitude, frequency and phase, with only a single sample of a discrete signal.

The AS is implemented in the  library [Jones et al., 2011]. Since it is based on Fourier transforms, it requires equidistant steps in time. Thus, a space born observatory with a regular and continuous observing scheme is preferred over ground based observations by this method. Due to barycentric motion of the spacecraft the time steps will be distorted after correcting for the barycentric motion. Therefore, an interpolation in time is necessary which makes the analysis computationally very expensive.

In order to test the implementation, the artificial light curves should represent this temporal behavior. The data were generated sampling one stellar pulsation mode according to the PLATO scheme (cadence of 25 for three years) and an inverse barycentric correction was applied in order to create the irregular temporal spacing.

For the analysis, the artificial data are interpolated on a grid with a spacing of the shortest time between two measurements, close to 25. From an amplitude spectrum, the frequency of the pulsation mode is selected, very similar to the classical $ O-C $ approach implemented for this work. In the presence of multiple pulsation frequencies, a frequency filtering is necessary. After the computation of the envelope, instantaneous phase and frequency, the expected linear cycle count is subtracted from the instantaneous phase to yield the $ O-C $ diagram (see Fig. <ref>). Since the AS is computed without a binning of data, the result appears to be scattering much more than the classical $ O-C $ result. Averaging the results over a sufficient amount of time smooths the measurements. Nevertheless, the process of interpolation before the actual analysis can start is computationally very expensive; about a factor 1000 more than the classical $ O-C $ calculation. This makes the Hilbert transform not practical for analysing irregular high cadence data. This approach has not been further pursued in this work.

[Results of the Hilbert transform for one pulsation.]Results of the Hilbert transform for one pulsation with a frequency of  240.0 and an amplitude of 0.5 Top: Flux measurements (black) sampled from the pulsation (grey) and amplitude envelope (red). Middle: Instantaneous frequency Bottom: Instantaneous phase minus reference phase.

§.§ Independent confirmation via direct imaging

Direct imaging with adaptive optics allow for observations near the refraction limit of the telescope. This might offer the possibility to confirm sub-stellar companion candidates where a clear detection using the pulsation timing is affected by non-linear effects of stellar pulsations.
However, possible companions are still not resolvable with current technology. V391 Peg and DW Lyn have a distance of $ d = \SI{1238.7 \pm 82.2422}{pc} $ and $ d = \SI{1495.4389 \pm 130.3785}{pc} $, respectively [Gaia Collaboration et al., 2018]. With semi major axis of about $ \SI{1.7}{\au} $ [Silvotti et al., 2007] and $ \SI{1.96}{\au} $, the hypothetical companions would be separated by $ \SI{1.37}{mas} $ and $ \SI{1.31}{mas} $ from the host star, respectively. This is about one order of magnitude smaller than the theoretical angular resolution of $ \SI{8}{\meter} $ class telescopes. The remaining EXOTIME targets are not significantly closer to Earth. Thus, the sub-stellar companion candidates in the EXOTIME sample can not be confirmed via direct imaging.

The EXOTIME project: Signals in the $ O-C $ diagrams of the rapidly pulsating subdwarfs DW Lyn, V1636 Ori, QQ Vir, and V541 Hya

This chapter is published as a journal article The EXOTIME project: Signals in the $ O-C $ diagrams of the rapidly pulsating subdwarfs DW Lyn, V1636 Ori, QQ Vir and V541 Hya in Astronomy and Astrophysics (A&A, 638 (2020) A108, DOI: <https://doi.org/10.1051/0004-6361/201937172>). © F. Mackebrandt et al. 2020.

Authors: F. Mackebrandt[Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany],[Institut für Astrophysik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany]
, S. Schuh[1]
, R. Silvotti[INAF – Osservatorio Astrofisico di Torino, strada dell'Osservatorio 20, 10025 Pino Torinese, Italy]
, S.-L. Kim[Korea Astronomy and Space Science Institute, Daejeon 34055, South Korea]
, D. Kilkenny[Department of Physics and Astronomy, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa]
, E. M. Green[Steward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA]
, R. Lutz[German Aerospace Center (DLR), Remote Sensing Technology Institute, Münchener Str. 20, 82234 Weßling, Germany]
, T. Nagel[Institute for Astronomy and Astrophysics, Kepler Center for Astro and Particle Physics, University of Tübingen, 72076 Tübingen, Germany]
, J. L. Provencal[University of Delaware, Department of Physics and Astronomy Newark, DE 19716, USA],[Delaware Asteroseismic Research Center, Mt. Cuba Observatory, Greenville, DE 19807, USA]
, T. Otani[Embry-Riddle Aeronautical University, Department of Physical Science and SARA, Daytona Beach, FL 32114, USA],[Florida Institute of Technology, Department of Physics & Space Sciences, Melbourne, FL 32901, USA]
, T. D. Oswalt[11],[12]
, S. Benatti[INAF – Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy]
, L. Lanteri[3]
, A. Bonanno[INAF – Osservatorio Astrofisico di Catania, Via S. Sofia 78, 95123 Catania, Italy]
, A. Frasca[14]
, R. Janulis[Institute of Theoretical Physics and Astronomy, Vilnius University, Gostauto 12, Vilnius 01108, Lithuania]
, M. Paparó[Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly-Thege M. út 15-17, 1121 Budapest, Hungary]
, L. Molnár[16],[MTA CSFK Lendület Near-Field Cosmology Research Group, Konkoly Observatory]
, R. Claudi[13]
, R. H. Østensen[Department of Physics, Astronomy, and Materials Science, Missouri State University, Springfield, MO 65897, USA]

Contributions: F.M. analysed the data and wrote the manuscript; S.S. is CoI of the EXOTIME project, coordinated observations at CAHA and supervised the findings of this work; R.S. is PI of the EXOTIME project, coordinated observations at TNG and contributed Fig. <ref>, and the according description in section <ref>. The following authors observed and reduced the data: S.-L.K. (LOAO); D.K. (SAAO); E.M.G. (MtB); R.L. (M/N); T.N. (Tue); J.L.P. (WET); T.O. (SARA-KP); T.D.O. (SARA-KP); S.B. (Asi); L.L. (Loi); A.B. (Serra la Nave); A.F. (Serra la Nave); R.J. (Mol); M.P. (Kon); L.M. (Kon); R.C. (Asi); R.H.Ø. (NOT)

The manuscript has been read, discussed and approved by all named authors.

Photometric data of Fig. <ref>, results in Fig. <ref>, <ref>, <ref> and <ref> and figures in the appendix are only available at the CDS via anonymous ftp to <cdsarc.u-strasbg.fr> (<130.79.128.5>) or via <http://cdsarc.

Based on observations obtained at the 0.9 m SARA-KP telescope,
which is operated by the Southeastern Association for Research in
Astronomy (<saraobservatory.org>).

We thank Wen-Shan Hsiao for observing at the Lulin Observatory and Mike D. Reed for observing at the Baker Observatory and Elia Leibowitz for observing at the WISE observatory.
F.M. conducted the work in this paper in the framework of the International Max-Planck Research School (IMPRS) for Solar System Science at the University of Göttingen (Volkswagen Foundation project grant number VWZN3020).
DK thanks the SAAO for generous allocations of telescope time and the National Research Foundation of South Africa and the University of the Western Cape for financial support.
TDO gratefully acknowledges support from the U.S. National Science Foundation grant AST-0807919.
L.M. was supported by the Premium Postdoctoral Research Program of the Hungarian Academy of Sciences. This project has been supported by the Lendület Program  of the Hungarian Academy of Sciences, project No. LP2018-7/2019.
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias.
Based on observations collected at the Centro Astronómico Hispánico en Andalucía (CAHA) at Calar Alto, operated jointly by the Andalusian Universities and the Instituto de Astrofísica de Andalucía (CSIC).
Based on observations collected at Copernico telescope (Asiago, Italy) of the INAF - Osservatorio Astronomico di Padova.
Based on observations made with the Nordic Optical Telescope, operated by the Nordic Optical Telescope Scientific Association at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias.
This paper includes data collected by the TESS mission. Funding for the TESS mission is provided by the NASA Explorer Program.

Application to larger target pool

CHAPTER: FURTHER ASTEROSEISMIC TARGETS

The search for sub-stellar companions using the pulsation timing method is not only limited to sdB stars but can also be applied to a greater variety of pulsating stars, as long as the pulsations are driven by a  mechanism, i.e., coherent in phase. However, modes of long periods decrease the precision of timing measurements. Additionally, large pulsation amplitudes tend to show non-linear effects, which excludes the application of this technique. A case study of  Cephei, RR Lyrae and  Scuti ( Sct) stars can be found in Silvotti et al., 2011. Among these,  Sct stars are the most promising pulsators to follow-up. The following describes  Sct stars and the application of the pulsation timing method in more detail.

§  SCUTI STARS

Planet occurrence rates predict a maximum for host stars with masses of 1.90.10.5$\SI{}{M_\odot}$ [Reffert et al., 2015], close to the mass of main sequence A stars in the instability strip where  Sct pulsators are common (see Fig. <ref>). However, actual planet detections are affected by observational selection effects. Planetary transits are difficult to observe because of the pulsational luminosity variations and due to wider orbits of planets around A stars [Johnson et al., 2011, Lloyd, 2011], resulting in lower transit probabilities. For the radial velocity method, A stars are not well suited because of their fast rotation, with the mean of the equatorial rotational velocity distribution exceeding 100 [Abt and Morrell, 1995, Royer et al., 2007]. Their high effective temperatures compared to solar-like stars leads to fewer and shallower absorption lines, which can additionally be distorted by pulsations. This leaves the pulsation timing method as a promising alternative to search for companions.  Sct stars show radial and non-radial p-modes with amplitudes up to 0.5 and with pulsation periods between about 20 and 8 <cit.>. The pulsations are driven by the  mechanism due to partial ionization of He II.

The mostly uninterrupted observations of space missions like Kepler and TESS provide a large sample of  Sct observations for this purpose. The two missions and some of their applications are described in the following sections.

CHAPTER: KEPLER

Sections of this chapter are part of a paper in preparation to be submitted as an article to Astronomy and Astrophysics.

§ KEPLER MISSION

The Kepler mission [Borucki et al., 2010] was launched in 2009 to observe a fixed patch of sky in the Cygnus constellation with more than 100^2 field of view to discover transiting Earth-like exoplanets in the habitable zone. The telescope's mirror measures 0.95m in diameter and the camera is sensitive for a wavelength range of 430890. Asteroseismic observations of pulsating stars were organized by the Kepler Asteroseismic Science Consortium (KASC). Some were observed in a high cadence mode of 60 (short cadence), the nominal cadence measures 30 (long cadence).

§.§ sdB stars

The Kepler field contained a small number of sdB stars <cit.>. None of them showed indications for a planetary transit. However, KIC 05807616 and KIC 10001893, two pulsating sdB stars in the Kepler field, show some evidence for the presence of planetary companions [Charpinet et al., 2011, Silvotti et al., 2014]. However, interpretation of these signals is questioned, as already discussed in chapter <ref>.

§.§  Scuti stars

Kepler observed around one thousand  Sct stars in its field of view. The primary goal of their observations are conclusions about the stellar structure and evolution via asteroseismology, as the transition from deep to shallow convective envelopes takes place in the region of the HRD where  Sct are located (see Fig. <ref>). This opens the opportunity to use the extensive Kepler data set and apply the pulsation timing method to the observations. The following describes one particular system as a benchmark for which a planetary companion was already discovered.

§.§.§ KIC 7917485

Murphy et al., 2014, Murphy et al., 2016 used the Kepler data to search for binary systems using the light travel time effect. Murphy et al., 2016 (hereafter M16) detected a planet orbiting its host star KIC 7917485 in a 840 orbit. The following validates this finding using the $ O-C $ pipeline implemented for this work.

KIC 7917485 was observed between May 2009 and May 2013 by Kepler. We use the light curves provided by the MAST archive[<https://archive.stsci.edu/>] which had common instrumental trends removed by the Pre-Search Data Conditioning Pipeline <cit.>. In order to remove remaining long term trends, we make use of the  routine of the PyKE package [Still and Barclay, 2012, Vinícius et al., 2018]. The light curve and the corresponding amplitude spectrum are shown in Fig. <ref>.

In order to reproduce the results of M16, we apply a high-pass frequency filter. This removes frequencies below $ \unit[5]{d^{-1}} $, which are most likely  artefacts due to instrumental effect. We apply this filter forward and backward in time to compensate for phase delays.

KIC 7917485 light curve.

KIC 7917485 amplitude spectrum.

[KIC 7917485 data.]KIC 7917485 data. Top: Light curve. Grey points are considered outliers and partially exceeding the plotting range. Bottom: Amplitude spectrum. The top panel shows the observations. The middle panel the amplitude spectrum after applying the high pass filter. Note the different scale on the y-axis in order to make small amplitudes visible. The bottom panel shows the residuals after modelling six pulsation modes.

For the $ O-C $ analysis, we applied our pipeline and fitted all pulsations with amplitudes above 0.005. Although the signal-to-noise ratio is sufficient for only two modes, the simultaneous fitting process of four more pulsations prevents any unaccounted interactions between the modes. The time dependent variation in pulsation amplitude and phase is shown in Fig. <ref>. Each epoch measures about ten days. The main pulsation frequency $ f_1 = \SI[separate-uncertainty = false]{15.3830074\pm0.0000004}{\per\day} $ exhibits large semi-periodic variations which correlate with the observational quarters of the Kepler mission. Thus, they are likely to be instrumental systematics. The amplitude of $ f_2 = \SI[separate-uncertainty = false]{20.2629053\pm0.0000008}{\per\day} $ shows a similar behaviour on top of a general trend of decreasing amplitude with time. The phase variations for both pulsations show clearly a sinusoidal modulation with an amplitude of about 10 and a period of about 800. We fit simultaneously a quadratic long term trend and a sinusoidal model for a circular companion orbit to the data. The results are presented in Table <ref>.

[Results for the two main pulsations of KIC 7917485.]Results for the two main pulsations of KIC 7917485. Top panel: Amplitudes. Middle panel: Fits of the $ O-C $ data with second order polynomials only (dashed lines) and full fit including the light travel time effect (solid lines). Lower panel: Residuals.

[Orbital parameters for the KIC 7917485 system.]Orbital parameters for the KIC 7917485 system. Values for M16 are taken from Table 2 in Murphy et al., 2016. M16 did not account for a change in pulsation period $ \dot{P} $. The model of this work uses a circular orbit model for the companion, i.e. $ e=0 $.
table-text-alignment = center,

M16     this work $ f_1 $      this work $ f_2 $

$ \Delta T /\si{\second} $     7.10.50.4     6.8 ±0.3     6.6 ±0.4

$ P_{\text{orbit}}/\si{\day} $     8402220     821.50 ±0.08     726.72 ±0.08

$ e $    0.150.130.10     0     0

$ \dot{P}/\si{\second\,Myr^{-1}}  $          -0.001 ±0.001     -0.035 ±0.002

While a change in period $ \dot{P} $ for $ f_1 $ is almost not detectable, $ f_2 $ shows a significant $ \dot{P} $, which coincides with the decrease in pulsation amplitude of 0.04 per cent (amplitude) or about 18 per cent (relative). M16 did not account for a change in period, thus we cannot compare our measurements.

For the orbital fit of the companion, M16 used the weighted mean of both phase measurements. Since we want to confirm the nature of the companion with two independent measurements, we do not follow this procedure. Our sinusoidal orbit fit for $ f_1 $ agrees within the uncertainties with the measurement of M16. However, the orbital period measured for $ f_2 $ is about 100 days significantly shorter than for $ f_1 $. This might be attributed to our additional fit for the change in period, combined with larger $ O-C $ uncertainties towards the end of the observations (approximately two times larger), due to the decreasing pulsation amplitude.

In order to compare the M16 results better with the measurements of this work, we will be implementing a full orbit fit on both measurements simultaneously. Additionally, the uncertainties appear rather small in comparison to M16. A different treatment of uncertainties might give more reliable error bars.

CHAPTER: TESS

§ TESS MISSION

The Transiting Exoplanet Survey Satellite (TESS) by NASA was launched in April 2018 as an all-sky survey for transiting exoplanets. The 24$ \times $96 field of view observes each sector for about 27 days and sweeps across the sky. This creates some observations lasting longer than 27 days due to the overlapping sectors. The four telescopes (each with a field of view of 24^2) measure 105mm in diameter and the cameras are sensitive for a wavelength range of 6001000. This is slightly redder than the Kepler bandpass due to the mission's differing priorities: detecting transiting planets orbiting Sun-like stars for Kepler and small, cool stars for TESS. The full frame cadence measures 30 minutes while the nominal cadence for transit detections two minutes. The TESS Asteroseismic Science Consortium (TASC) coordinates 20 seconds short cadence observations of bright asteroseismic target stars.

However, observations with such a short time span of 27 days (and up to one year but only around the ecliptic pole, e.g., 1.7 of the sky) do not allow an extensive search of sub-stellar companions using the pulsation timing method on their own. But they can be used as additional observations to existing data sets, especially the EXOTIME project data. Without additional ground-based observations, a joint $ O-C $ analysis of  Sct data from Kepler and TESS is likely to fail due to the large observational gap of about seven years.

§.§ sdB stars

TESS observations of V1636 Ori and V541 Hya are already discussed in chapter <ref>. The data are taken at two minutes cadence and thus do not allow the investigation of p-modes.

DW Lyn was observed by TESS with a cadence of $ \unit[120]{s} $ between December 25, 2019 and January 20, 2020. Fig. <ref> shows the light curve and amplitude spectrum. Besides the 120 alias at 720, the spectrum shows $ f_1 $ and $ f_2 $ at about 238 and 225 respectively, as well as their Nyquist alias above 360 <cit.>. For the g-mode regime, Lutz et al., 2008 reported pulsations at 271.7;318.1;206.3 (23.47;27.48;17.83). The TESS data show only the pulsation at 318.1 as a detection of about two times above the noise level. While the 271.7 pulsation is visible but not above the noise level, the 206.3 pulsation is not present in the TESS data set. Since we have no data between the last EXOTIME observations in 2013 and the TESS observations in 2020, the time span for a joint $ O-C $ analysis is too large.

DW Lyn light curve.

DW Lyn amplitude spectrum.

[TESS DW Lyn data.]TESS DW Lyn data. Top: Light curve. Grey points are considered outliers and partially exceeding the plotting range. Bottom: Amplitude spectrum. The inset shows a magnification of the g-mode regime. Modes reported by Lutz et al., 2008 are indicated as dots.

V391 Peg will be observed by TESS and the data can be used to extend the set of observation from Silvotti et al., 2018. In order to span the observational gaps between 2013 and 2020, we performed additional ground-bases observations, as described in chapter <ref>.

CHAPTER: DISCUSSION

Studying post main-sequence stars offers a window into the fate of planetary systems, including our own solar system. Can planets survive the final stages of stellar evolution and how do they in turn affect their host star's evolution? One discussed formation scenario of single sdB stars involves the influence of giant planets during the red giant phase of their progenitor. Observations can put constrains on the presence of sub-stellar companions orbiting sdB stars.

Chapter <ref> investigated a unique set of long term ground-based observations of the four rapidly pulsating sdB stars DW Lyn, V1636 Ori, QQ Vir and V541 Hya. The main goals of the EXOTIME project are the determination of the evolutionary state of these stars and the constraints on sub-stellar companions to draw conclusions on possible formation scenarios.

The individual secular period drifts for different modes per target, derived from the $ O-C $ measurements, do not always agree with each other. These results therefore need to be interpreted with caution with regard to the evolutionary state of the star and hold only for the assumption of a linear change in period. For DW Lyn, $ \dot{P}_{f1} $ is comparable to values derived from stellar models, and place the star in the evolutionary phase of core He burning. $ \dot{P}_{f2} $ on the other hand, can be explained by the apparent mode splitting, whose origin cannot be confirmed with certainty. A possible explanation is the slow decay of one pulsation mode and an excitation of a mode with close frequency but not resolvable by the observations. Additionally, amplitude and phase variations indicate a mode beating effect. As shown in section <ref>, two close - but not necessarily resolvable, pulsation modes can cause such effects. Measurements for the period changes of V1636 Ori need to be cautioned by the same arguments. For QQ Vir, the identified radial order and degree of modes do not allow a direct comparison with stellar models but the sign of the $ \dot{P} $ measurements places QQ Vir in the state of He burning. The $ O-C $ measurements of V541 Hya show no significant change of period. If these results are not affected by observational limitations but originate from stellar evolution, the star might be in the transition phase between core He burning and contraction.

Atmospheric measurements of all four stars can place the stars in the $ \log g - T_{\text{eff}} $ plane relatively close to stellar evolutionary models of Charpinet et al., 2002, as illustrated in Fig. <ref>. This independently confirms the tentative interpretations above. DW Lyn, V1636 Ori and QQ Vir overlap each with several models of different envelope mass in the phase of core He burning within their measured uncertainties. Although $ T_{\text{eff}} $ and $ \log g $ measurements of V541 Hya agree with two models in that same evolutionary phase, the measurements agree within $ \SI{3}{\sigma} $ with an evolutionary model in the transition between core He burning and contraction.

[EHB evolutionary tracks of sdB stars.]EHB evolutionary tracks covering the $ \log g - T_{\text{eff}}$ region where sdB stars are observed. Data points taken from Appendix B of Charpinet et al., 2002 (crosses) and Table <ref> (circles). The seven models correspond, respectively, from left (high $ T_{\text{eff}} $) to right (low $ T_{\text{eff}} $), to objects with a ZAEHB envelope mass $ M_{\text{env}} =$ 0.0001, 0.0002, 0.0007, 0.0012, 0.0022, 0.0032, and 0.0042 $ M_{\odot} $. The core mass is $M_{\text{c}} = 0.4758 M_{\odot} $ for all sequences except the first and third (from the left), which have $M_c = 0.4690 M_{\odot} $. For a positive (negative) slope in the models, the change in pulsation period is positive (negative). See Charpinet et al., 2002 for a detailed explanation.

In principle, proper motions of stars can give an alternative explanation for the apparent change in pulsation period, as described by Pajdosz, 1995, Pajdosz, 1995. They derive the effect of proper motion to be

\begin{align*}
\frac{\dot{P}_{\text{pm}}}{\si{\second\per\second}} = 2.430 \times 10^{-18} \frac{P}{\si{\second}} \frac{\mu}{ \si{''\,yr}^{-1}} \frac{d}{\si{pc}},
\end{align*}

where $ \mu $ is the proper motion in arcseconds per year and $ d $ the distance in parsec. With measurements available from the Gaia [Gaia Collaboration et al., 2018], the derived values are about three orders of magnitude smaller than the measured $ \dot{P} $ and thus neglectable.

After subtraction of the long term trends in the $ O-C $ measurements, the residuals show no clear evidence for signatures of sub-stellar companions. The comparison with synthetic $ O-C $ curves, indicates a companion-induced signal in the main pulsation mode of DW Lyn. But this is not confirmed by the independent measurement of the second pulsation mode, as would be expected. Furthermore, previously described periodic signals in the $ O-C $ measurements by Lutz et al., 2011 are ruled out to be caused by sub-stellar companions. Nevertheless, the data set is limited in detection sensitivity. Depending on the observational gaps, the data cannot set limitations for possible companion orbits shorter than approximately 30 days (DW Lyn and V1636 Ori), 4 days (QQ Vir) or 120 days (V541 Hya), or longer than the total length of observations, roughly 1500 days. Photometric precision additionally limits the detection threshold to companions signals larger than about 1. This corresponds to a companion with a mass of $ \SI{1}{M_{\jupiter}} $ ($ \SI{0.5}{M_{\jupiter}} $) and an orbital period of about 0.5years (1.5years).

Chapter <ref> enlarges the target selection and applies the $ O-C $ analysis to one example. Pulsations of  Sct  stars are driven by the  mechanism, allowing stable pulsations over long periods of time. Planet detection methods, like the transit or radial velocity method, cannot be easily applied to those stars, which favours the pulsation timing analysis and thus opens a new parameter space in the exoplanet detection. The  Sct pulsator KIC 7917485 was found to be orbited by a sub-stellar companion, whose signal can be recovered in the pulsation arrival times.

CHAPTER: OUTLOOK

As chapter <ref> applied the $ O-C $ analysis to one example observation beyond the EXOTIME project, there are more extensive long time series of high cadence data already available, or with upcoming space missions. The next sections highlight data sets of interest.

§ K2 MISSION

After the failure of two reaction wheels on the Kepler space craft, its mission could be extended from 2014 onward as K2. The space craft was balanced against the radiation pressure from the sun, which limits each campaign to a duration of about 80 days. Thus, K2 was not staring at only one field in the sky but sweeping along the ecliptic and could observe many more targets for signs of transiting exoplanets. Armstrong et al., 2015 created a catalogue of variable stars from the K2 observations and categorized the targets into different classes of pulsators. From this catalogue, about 500 variables are classified as  Sct stars. Additionally, 33 sdB stars were observed by K2 [Reed et al., 2018]. Although the observations span only 80 days, they can set constraints on close in sub-stellar companions.

§ TESS

The TESS mission aims to detect transiting exoplanets around bright stars. Asteroseismic targets are observed with a cadence of 20 seconds if they are bright enough. Most of the EXOTIME targets are already observed or are going to be observed. Due to the cadence of two minutes for these stars and the long duration in between the EXOTIME observations, the data cannot be used for a common $ O-C $ analysis. Nevertheless, they provide constraints on g-modes. For the upcoming V391 Peg observations, the gaps to ground based observations are not too big. With the planetary candidate claimed by Silvotti et al., 2007 but cautioned later by Silvotti et al., 2018, the new observations by ground based telescopes and TESS are expected to clarify the interpretations.

For other asteroseismic targets, ether sdB or  Sct stars, an $ O-C $ analysis is reasonable only for targets with overlapping sectors, i.e., more than 27 days of observing time.

§ PLATO

PLAnetary Transits and Oscillations of stars (PLATO) is an ESA mission scheduled for launch in 2026. Its primary goal is to discover and characterize terrestrial exoplanets orbiting bright solar-type stars, especially  planets in the habitable zone. Bright host stars allow the determination of the planets' masses using the radial velocity method from ground-based observations and thus a mean density. This can be used to constrain planetary interior models. Additionally, bright targets allow transmission spectroscopy observations in order to measure chemical abundances of the planetary atmospheres. A secondary goal of the PLATO mission is asteroseismic observations to determine stellar masses, radii, and ages. These parameters are crucial in order to correctly determine the planetary masses, radii and their evolution.

PLATO's payload module consists of 24 cameras, each with an aperture of 120 and a 1100^2 field of view (see Fig.<ref>). The cameras are arranged in four groups of six, where each group has the same field of view but their angle is slightly offset - this allows a total field view of about 2250^2 per pointing. The camera arrays overlap in some parts, resulting in different sensitivities over the field. They will observe with a cadence of 25. Additionally, two fast cameras will observe with a cadence of 2.5. PLATO's nominal life time sums up to four years but could be in orbit for double the time. For most of the time it will stare to one region in the sky. Additionally, an step and stare phase, comparable to K2 is possible.

[Artist's impression of ESA's PLATO spacecraft.]Artist's impression of ESA's PLATO spacecraft. Credit: ESA/ATG medialab, CC BY-SA IGO 3.0

With the explicit goal of asteroseismic observations, PLATO is expected to collect many data on evolved pulsating stars allowing for an investigation with the pulsation timing method.
The pipeline developed for this work can be easily adopted and used in an automated way in order to process many targets and search for sub-stellar companions.

[Abt and Morrell, 1995]
Abt, H. A., Morrell, N. I., 1995, The Relation between Rotational
Velocities and Spectral Peculiarities among A-Type Stars, The
Astrophysical Journal Supplement Series, 99, 135, ISSN 0067-0049

[Aerts et al., 2010]
Aerts, C., Christensen-Dalsgaard, J., Kurtz, D. W., 2010, Asteroseismology,
Astronomy and Astrophysics Library, Springer Netherlands, Dordrecht,
ISBN 978-1-4020-5178-4 978-1-4020-5803-5

[Anglada-Escudé et al., 2016]
Anglada-Escudé, G., Amado, P. J., Barnes, J., Berdiñas, Z. M.,
Butler, R. P., Coleman, G. A. L., de la Cueva, I., Dreizler, S., Endl, M.,
Giesers, B., Jeffers, S. V., Jenkins, J. S., Jones, H. R. A., Kiraga, M.,
Kürster, M., López-González, M. J., Marvin, C. J., Morales, N.,
Morin, J., Nelson, R. P., Ortiz, J. L., Ofir, A., Paardekooper, S.-J.,
Reiners, A., Rodríguez, E., Rodr$\acute{\iota}$guez-López, C.,
Sarmiento, L. F., Strachan, J. P., Tsapras, Y., Tuomi, M., Zechmeister, M.,
2016, A terrestrial planet candidate in a temperate orbit around Proxima
Centauri, Nature, 536, 437–440, ISSN 1476-4687

[Armstrong et al., 2015]
Armstrong, D. J., Kirk, J., Lam, K. W. F., McCormac, J., Walker, S. R., Brown,
D. J. A., Osborn, H. P., Pollacco, D. L., Spake, J., 2015, K2 Variable
Catalogue: Variable stars and eclipsing binaries in K2 campaigns 1
and 0, A&A, 579, A19, ISSN 0004-6361, 1432-0746

[Astropy Collaboration et al., 2013]
Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., Greenfield, P.,
Droettboom, M., Bray, E., Aldcroft, T., Davis, M., Ginsburg, A.,
Price-Whelan, A. M., Kerzendorf, W. E., Conley, A., Crighton, N., Barbary,
K., Muna, D., Ferguson, H., Grollier, F., Parikh, M. M., Nair, P. H., Unther,
H. M., Deil, C., Woillez, J., Conseil, S., Kramer, R., Turner, J. E. H.,
Singer, L., Fox, R., Weaver, B. A., Zabalza, V., Edwards, Z. I.,
Azalee Bostroem, K., Burke, D. J., Casey, A. R., Crawford, S. M., Dencheva,
N., Ely, J., Jenness, T., Labrie, K., Lim, P. L., Pierfederici, F., Pontzen,
A., Ptak, A., Refsdal, B., Servillat, M., Streicher, O., 2013, Astropy: A
community Python package for astronomy, Astronomy and Astrophysics, 558,
A33, ISSN 0004-6361

[Astropy Collaboration et al., 2018]
Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M.,
Günther, H. M., Lim, P. L., Crawford, S. M., Conseil, S., Shupe, D. L.,
Craig, M. W., Dencheva, N., Ginsburg, A., VanderPlas, J. T., Bradley, L. D.,
Pérez-Suárez, D., de Val-Borro, M., Aldcroft, T. L., Cruz, K. L.,
Robitaille, T. P., Tollerud, E. J., Ardelean, C., Babej, T., Bach, Y. P.,
Bachetti, M., Bakanov, A. V., Bamford, S. P., Barentsen, G., Barmby, P.,
Baumbach, A., Berry, K. L., Biscani, F., Boquien, M., Bostroem, K. A., Bouma,
L. G., Brammer, G. B., Bray, E. M., Breytenbach, H., Buddelmeijer, H., Burke,
D. J., Calderone, G., Cano Rodríguez, J. L., Cara, M., Cardoso, J. V. M.,
Cheedella, S., Copin, Y., Corrales, L., Crichton, D., D'Avella, D., Deil, C.,
Depagne, É., Dietrich, J. P., Donath, A., Droettboom, M., Earl, N.,
Erben, T., Fabbro, S., Ferreira, L. A., Finethy, T., Fox, R. T., Garrison,
L. H., Gibbons, S. L. J., Goldstein, D. A., Gommers, R., Greco, J. P.,
Greenfield, P., Groener, A. M., Grollier, F., Hagen, A., Hirst, P., Homeier,
D., Horton, A. J., Hosseinzadeh, G., Hu, L., Hunkeler, J. S., Ivezić, Ž., Jain, A., Jenness, T., Kanarek, G., Kendrew, S., Kern, N. S.,
Kerzendorf, W. E., Khvalko, A., King, J., Kirkby, D., Kulkarni, A. M., Kumar,
A., Lee, A., Lenz, D., Littlefair, S. P., Ma, Z., Macleod, D. M.,
Mastropietro, M., McCully, C., Montagnac, S., Morris, B. M., Mueller, M.,
Mumford, S. J., Muna, D., Murphy, N. A., Nelson, S., Nguyen, G. H., Ninan,
J. P., Nöthe, M., Ogaz, S., Oh, S., Parejko, J. K., Parley, N., Pascual,
S., Patil, R., Patil, A. A., Plunkett, A. L., Prochaska, J. X., Rastogi, T.,
Reddy Janga, V., Sabater, J., Sakurikar, P., Seifert, M., Sherbert, L. E.,
Sherwood-Taylor, H., Shih, A. Y., Sick, J., Silbiger, M. T., Singanamalla,
S., Singer, L. P., Sladen, P. H., Sooley, K. A., Sornarajah, S., Streicher,
O., Teuben, P., Thomas, S. W., Tremblay, G. R., Turner, J. E. H., Terrón,
V., van Kerkwijk, M. H., de la Vega, A., Watkins, L. L., Weaver, B. A.,
Whitmore, J. B., Woillez, J., Zabalza, V., Astropy Contributors, 2018, The
Astropy Project: Building an Open-science Project and
Status of the v2.0 Core Package, The Astronomical Journal, 156, 123,
ISSN 0004-6256

[Bailey, 1926]
Bailey, C., 1926, Epicurus The Extant Remains Bailey Oxford 1926
Optimized For Greek On Left

[Baker and Kippenhahn, 1962]
Baker, N., Kippenhahn, R., 1962, The Pulsations of Models of $\delta$
Cephei Stars. With 17 Figures in the Text, Zeitschrift fur
Astrophysik, 54, 114, ISSN 0372-8331

[Ballard et al., 2011]
Ballard, S., Fabrycky, D., Fressin, F., Charbonneau, D., Desert, J.-M., Torres,
G., Marcy, G., Burke, C. J., Isaacson, H., Henze, C., Steffen, J. H., Ciardi,
D. R., Howell, S. B., Cochran, W. D., Endl, M., Bryson, S. T., Rowe, J. F.,
Holman, M. J., Lissauer, J. J., Jenkins, J. M., Still, M., Ford, E. B.,
Christiansen, J. L., Middour, C. K., Haas, M. R., Li, J., Hall, J. R.,
McCauliff, S., Batalha, N. M., Koch, D. G., Borucki, W. J., 2011, The
Kepler-19 System: A Transiting 2.2 R $\oplus$ Planet
and a Second Planet Detected via Transit Timing Variations, The
Astrophysical Journal, 743, 200, ISSN 0004-637X

[Balona et al., 2012]
Balona, L. A., Lenz, P., Antoci, V., Bernabei, S., Catanzaro, G.,
Daszyńska-Daszkiewicz, J., Di Criscienzo, M., Grigahcène, A.,
Handler, G., Kurtz, D. W., Marconi, M., Molenda-Żakowicz, J., Moya, A.,
Nemec, J. M., Pigulski, A., Pricopi, D., Ripepi, V., Smalley, B., Suárez,
J. C., Suran, M., Hall, J. R., Kinemuchi, K., Klaus, T. C., 2012, Kepler
observations of the high-amplitude $\delta$ Scuti star V2367 Cyg:
Kepler observations of V2367 Cyg, Mon. Not. R. Astron. Soc., 419,
3028–3038, ISSN 00358711

[Baran et al., 2005]
Baran, A., Pigulski, A., Kozieł, D., Ogłoza, W., Silvotti, R., Zoła,
S., 2005, Multicolour photometry of Balloon 090100001: Linking the two
classes of pulsating hot subdwarfs, Monthly Notices of the Royal Astronomical
Society, 360, 737–747, ISSN 0035-8711

[Baran et al., 2018]
Baran, A. S., Østensen, R. H., Telting, J. H., Vos, J., Kilkenny, D., Vučković, M., Reed, M. D., Silvotti, R., Jeffery, C. S., Parsons, S. G.,
Dhillon, V. S., Marsh, T. R., 2018, Pulsations and eclipse-time analysis of
HW Vir, Monthly Notices of the Royal Astronomical Society, 481,
2721–2735, ISSN 0035-8711

[Barlow et al., 2011]
Barlow, B. N., Dunlap, B. H., Clemens, J. C., 2011a, Radial
Velocity Confirmation of a Binary Detected from Pulse Timings,
Astrophys. J., 737, L2

[Barlow et al., 2011]
Barlow, B. N., Dunlap, B. H., Clemens, J. C., Reichart, D. E., Ivarsen, K. M.,
Lacluyze, A. P., Haislip, J. B., Nysewander, M. C., 2011b,
Fortnightly fluctuations in the O-C diagram of CS 1246, Mon. Not.
R. Astron. Soc., 414, 3434

[Benatti et al., 2010]
Benatti, S., Silvotti, R., Claudi, R. U., Schuh, S., Lutz, R., Kim, S.-L.,
Janulis, R., Paparò, M., Baran, A., Østensen, R., 2010, The
EXOTIME project: A status report on PG 1325+101 (QQ Vir),
ArXiv10120747 Astro-Ph, 1012.0747

[Bischoff-Kim et al., 2019]
Bischoff-Kim, A., Provencal, J. L., Bradley, P. A., Montgomery, M. H.,
Shipman, H. L., Harrold, S. T., Howard, B., Strickland, W., Chandler, D.,
Campbell, D., Arredondo, A., Linn, R., Russell, D. P., Doyle, D., Brickhouse,
A., Peters, D., Kim, S.-L., Jiang, X. J., Mao, Y.-N., Kusakin, A. V.,
Sergeev, A. V., Andreev, M., Velichko, S., Janulis, R., Pakstiene, E., Aliçavuş, F., Horoz, N., Zola, S., Ogłoza, W., Koziel-Wierzbowska,
D., Kundera, T., Jableka, D., Debski, B., Baran, A., Meingast, S., Nagel, T.,
Loebling, L., Heinitz, C., Hoyer, D., Bognár, Z., Castanheira, B. G.,
Erdem, A., 2019, GD358: Three Decades of Observations for the
In-depth Asteroseismology of a DBV Star, The Astrophysical
Journal, 871, 13, ISSN 0004-637X

[Blokesz et al., 2019]
Blokesz, A., Krzesinski, J., Kedziora-Chudczer, L., 2019, Analysis of
putative exoplanetary signatures found in light curves of two sdBV stars
observed by Kepler, Astronomy and Astrophysics, 627, A86, ISSN 0004-6361

[Bonanno et al., 2003]
Bonanno, A., Catalano, S., Frasca, A., Mignemi, G., Paternò, L.,
2003a, PG 1613+426: A new sdB pulsator, Astronomy
and Astrophysics, 398, 283–285, ISSN 0004-6361

[Bonanno et al., 2003]
Bonanno, A., Frasca, A., Lanza, A. F., Ventura, R., Mignemi, B., Silvotti, R.,
2003b, Asteroseismology of Pulsating sdB Stars Observed at
Catania Astrophysical Observatory, Baltic Astronomy, 12, 287–294, ISSN

[Borucki et al., 2010]
Borucki, W. J., Koch, D., Basri, G., Batalha, N., Brown, T., Caldwell, D.,
Caldwell, J., Christensen-Dalsgaard, J., Cochran, W. D., DeVore, E.,
Dunham, E. W., Dupree, A. K., Gautier, T. N., Geary, J. C., Gilliland, R.,
Gould, A., Howell, S. B., Jenkins, J. M., Kondo, Y., Latham, D. W., Marcy,
G. W., Meibom, S., Kjeldsen, H., Lissauer, J. J., Monet, D. G., Morrison, D.,
Sasselov, D., Tarter, J., Boss, A., Brownlee, D., Owen, T., Buzasi, D.,
Charbonneau, D., Doyle, L., Fortney, J., Ford, E. B., Holman, M. J., Seager,
S., Steffen, J. H., Welsh, W. F., Rowe, J., Anderson, H., Buchhave, L.,
Ciardi, D., Walkowicz, L., Sherry, W., Horch, E., Isaacson, H., Everett,
M. E., Fischer, D., Torres, G., Johnson, J. A., Endl, M., MacQueen, P.,
Bryson, S. T., Dotson, J., Haas, M., Kolodziejczak, J., Van Cleve, J.,
Chandrasekaran, H., Twicken, J. D., Quintana, E. V., Clarke, B. D., Allen,
C., Li, J., Wu, H., Tenenbaum, P., Verner, E., Bruhweiler, F., Barnes, J.,
Prsa, A., 2010, Kepler Planet-Detection Mission: Introduction and
First Results, Science, 327, 977, ISSN 0036-8075

[Boss et al., 2005]
Boss, A. P., Butler, R. P., Hubbard, W. B., Ianna, P. A., Kürster, M.,
Lissauer, J. J., Mayor, M., Meech, K. J., Mignard, F., Penny, A. J.,
Quirrenbach, A., Tarter, J. C., Vidal-Madjar, A., 2005, Working Group
on Extrasolar Planets, Proc. Int. Astron. Union, 1, 183–186, ISSN
1743-9221, 1743-9213

[Brown et al., 2013]
Brown, T. M., Baliber, N., Bianco, F. B., Bowman, M., Burleson, B., Conway, P.,
Crellin, M., Depagne, É., Vera, J. D., Dilday, B., Dragomir, D.,
Dubberley, M., Eastman, J. D., Elphick, M., Falarski, M., Foale, S., Ford,
M., Fulton, B. J., Garza, J., Gomez, E. L., Graham, M., Greene, R., Haldeman,
B., Hawkins, E., Haworth, B., Haynes, R., Hidas, M., Hjelstrom, A. E.,
Howell, D. A., Hygelund, J., Lister, T. A., Lobdill, R., Martinez, J.,
Mullins, D. S., Norbury, M., Parrent, J., Paulson, R., Petry, D. L., Pickles,
A., Posner, V., Rosing, W. E., Ross, R., Sand, D. J., Saunders, E. S.,
Shobbrook, J., Shporer, A., Street, R. A., Thomas, D., Tsapras, Y., Tufts,
J. R., Valenti, S., Horst, K. V., Walker, Z., White, G., Willis, M., 2013,
Las Cumbres Observatory Global Telescope Network, PASP, 125, 1031–1055,
ISSN 1538-3873

[Bryan et al., 2016]
Bryan, M. L., Knutson, H. A., Howard, A. W., Ngo, H., Batygin, K., Crepp,
J. R., Fulton, B. J., Hinkley, S., Isaacson, H., Johnson, J. A., Marcy,
G. W., Wright, J. T., 2016, Statistics of Long Period Gas Giant Planets
in Known Planetary Systems, ApJ, 821, 89, ISSN 0004-637X

[Castellani and Castellani, 1993]
Castellani, M., Castellani, V., 1993, Mass Loss in Globular Cluster Red
Giants: An Evolutionary Investigation, Astrophys. J., 407, 649, ISSN

[Charbonneau et al., 2000]
Charbonneau, D., Brown, T. M., Latham, D. W., Mayor, M., 2000, Detection of
Planetary Transits Across a Sun-like Star, Astrophys. J., 529,
L45, ISSN 0004-637X

[Charbonneau et al., 2002]
Charbonneau, D., Brown, T. M., Noyes, R. W., Gilliland, R. L., 2002, Detection
of an Extrasolar Planet Atmosphere, The Astrophysical Journal, 568,
377–384, ISSN 0004-637X

[Charpinet et al., 1996]
Charpinet, S., Fontaine, G., Brassard, P., Dorman, B., 1996, The Potential
of Asteroseismology for Hot, Subdwarf B Stars: A New Class of
Pulsating Stars?, The Astrophysical Journal Letters, 471, L103, ISSN

[Charpinet et al., 1997]
Charpinet, S., Fontaine, G., Brassard, P., Chayer, P., Rogers, F. J., Iglesias,
C. A., Dorman, B., 1997, A Driving Mechanism for the Newly Discovered
Class of Pulsating Subdwarf B Stars, Astrophys. J., 483, L123

[Charpinet et al., 2002]
Charpinet, S., Fontaine, G., Brassard, P., Dorman, B., 2002, Adiabatic
Survey of Subdwarf B Star Oscillations. III. Effects of
Extreme Horizontal Branch Stellar Evolution on Pulsation Modes, The
Astrophysical Journal Supplement Series, 140, 469–561, ISSN 0067-0049

[Charpinet et al., 2006]
Charpinet, S., Silvotti, R., Bonanno, A., Fontaine, G., Brassard, P., Chayer,
P., Green, E. M., Bergeron, P., Bernabei, S., Leccia, S., Kjeldsen, H.,
Janulis, R., Frasca, A., Østensen, R., Kim, S.-L., Park, B.-G., Jiang, X.,
Reed, M. D., Patterson, R. S., Gietzen, K. M., Clark, P. J., Wolf, G. W.,
Lipkin, Y., Formiggini, L., Leibowitz, E., Oswalt, T. D., Rudkin, M.,
Johnston, K., 2006, The rapidly pulsating subdwarf B star PG
1325+101: II. Structural parameters from asteroseismology, Astron.
Astrophys., 459, 565–576, ISSN 0004-6361, 1432-0746

[Charpinet et al., 2011]
Charpinet, S., Fontaine, G., Brassard, P., Green, E. M., Van Grootel, V.,
Randall, S. K., Silvotti, R., Baran, A. S., Østensen, R. H., Kawaler,
S. D., Telting, J. H., 2011, A compact system of small planets around a
former red-giant star, Nature, 480, 496–499, ISSN 1476-4687

[Charpinet et al., 2018]
Charpinet, S., Giammichele, N., Zong, W., Van Grootel, V., Brassard, P.,
Fontaine, G., 2018, Rotation in sdB stars as revealed by stellar
oscillations, Open Astron., 27, 112–119, ISSN 2543-6376

[Chen et al., 2013]
Chen, X., Han, Z., Deca, J., Podsiadlowski, P., 2013, The orbital periods of
subdwarf B binaries produced by the first stable Roche Lobe overflow
channel, Monthly Notices of the Royal Astronomical Society, 434, 186–193,
ISSN 0035-8711

[Cunha et al., 2007]
Cunha, M. S., Aerts, C., Christensen-Dalsgaard, J., Baglin, A., Bigot, L.,
Brown, T. M., Catala, C., Creevey, O. L., de Souza, A. D., Eggenberger, P.,
Garcia, P. J. V., Grundahl, F., Kervella, P., Kurtz, D. W., Mathias, P.,
Miglio, A., Monteiro, M. J. P. F. G., Perrin, G., Pijpers, F. P., Pourbaix,
D., Quirrenbach, A., Rousselet-Perraut, K., Teixeira, T. C., Thévenin,
F., Thompson, M. J., 2007, Asteroseismology and interferometry, Astron
Astrophys Rev, 14, 217–360, ISSN 1432-0754

[Dreizler et al., 2002]
Dreizler, S., Schuh, S. L., Deetjen, J. L., Edelmann, H., Heber, U., 2002,
HS0702+6043 – A new large amplitude sdB variable at
the cool end of the instability region, Astron. Astrophys., 386, 249–255,
ISSN 0004-6361, 1432-0746

[Eastman et al., 2010]
Eastman, J., Siverd, R., Gaudi, B. S., 2010, Achieving Better Than 1
Minute Accuracy in the Heliocentric and Barycentric Julian Dates,
Publications of the Astronomical Society of the Pacific, 122, 935, ISSN

[ESA/SRE(2011)13, 2011]
ESA/SRE(2011)13, P., 2011, PLATO Definition Study Report (Red Book),
vol. Issue 1.0, ESA

[Farihi, 2016]
Farihi, J., 2016, Circumstellar debris and pollution at white dwarf stars, New
Astronomy Reviews, 71, 9–34, ISSN 1387-6473

[Fontaine et al., 2003]
Fontaine, G., Brassard, P., Charpinet, S., Green, E. M., Chayer, P.,
Billères, M., Randall, S. K., 2003, A Driving Mechanism for the
Newly Discovered Long-Period Pulsating Subdwarf B Stars, Astrophys.
J., 597, 518

[Gaia Collaboration et al., 2018]
Gaia Collaboration, Brown, A. G. A., Vallenari, A., Prusti, T., de Bruijne,
J. H. J., Babusiaux, C., Bailer-Jones, C. A. L., Biermann, M., Evans,
D. W., Eyer, L., Jansen, F., Jordi, C., Klioner, S. A., Lammers, U.,
Lindegren, L., Luri, X., Mignard, F., Panem, C., Pourbaix, D., Randich, S.,
Sartoretti, P., Siddiqui, H. I., Soubiran, C., van Leeuwen, F., Walton,
N. A., Arenou, F., Bastian, U., Cropper, M., Drimmel, R., Katz, D., Lattanzi,
M. G., Bakker, J., Cacciari, C., Castañeda, J., Chaoul, L., Cheek, N.,
De Angeli, F., Fabricius, C., Guerra, R., Holl, B., Masana, E., Messineo, R.,
Mowlavi, N., Nienartowicz, K., Panuzzo, P., Portell, J., Riello, M.,
Seabroke, G. M., Tanga, P., Thévenin, F., Gracia-Abril, G., Comoretto,
G., Garcia-Reinaldos, M., Teyssier, D., Altmann, M., Andrae, R., Audard,
M., Bellas-Velidis, I., Benson, K., Berthier, J., Blomme, R., Burgess, P.,
Busso, G., Carry, B., Cellino, A., Clementini, G., Clotet, M., Creevey, O.,
Davidson, M., De Ridder, J., Delchambre, L., Dell'Oro, A., Ducourant, C.,
Fernández-Hernández, J., Fouesneau, M., Frémat, Y., Galluccio,
L., García-Torres, M., González-Núñez, J.,
González-Vidal, J. J., Gosset, E., Guy, L. P., Halbwachs, J.-L.,
Hambly, N. C., Harrison, D. L., Hernández, J., Hestroffer, D., Hodgkin,
S. T., Hutton, A., Jasniewicz, G., Jean-Antoine-Piccolo, A., Jordan, S.,
Korn, A. J., Krone-Martins, A., Lanzafame, A. C., Lebzelter, T.,
Löffler, W., Manteiga, M., Marrese, P. M., Martín-Fleitas, J. M.,
Moitinho, A., Mora, A., Muinonen, K., Osinde, J., Pancino, E., Pauwels, T.,
Petit, J.-M., Recio-Blanco, A., Richards, P. J., Rimoldini, L., Robin,
A. C., Sarro, L. M., Siopis, C., Smith, M., Sozzetti, A., Süveges, M.,
Torra, J., van Reeven, W., Abbas, U., Abreu Aramburu, A., Accart, S.,
Aerts, C., Altavilla, G., Álvarez, M. A., Alvarez, R., Alves, J.,
Anderson, R. I., Andrei, A. H., Anglada Varela, E., Antiche, E., Antoja, T.,
Arcay, B., Astraatmadja, T. L., Bach, N., Baker, S. G.,
Balaguer-Núñez, L., Balm, P., Barache, C., Barata, C., Barbato, D.,
Barblan, F., Barklem, P. S., Barrado, D., Barros, M., Barstow, M. A.,
Bartholomé Muñoz, S., Bassilana, J.-L., Becciani, U., Bellazzini, M.,
Berihuete, A., Bertone, S., Bianchi, L., Bienaymé, O., Blanco-Cuaresma,
S., Boch, T., Boeche, C., Bombrun, A., Borrachero, R., Bossini, D.,
Bouquillon, S., Bourda, G., Bragaglia, A., Bramante, L., Breddels, M. A.,
Bressan, A., Brouillet, N., Brüsemeister, T., Brugaletta, E.,
Bucciarelli, B., Burlacu, A., Busonero, D., Butkevich, A. G., Buzzi, R.,
Caffau, E., Cancelliere, R., Cannizzaro, G., Cantat-Gaudin, T., Carballo,
R., Carlucci, T., Carrasco, J. M., Casamiquela, L., Castellani, M.,
Castro-Ginard, A., Charlot, P., Chemin, L., Chiavassa, A., Cocozza, G.,
Costigan, G., Cowell, S., Crifo, F., Crosta, M., Crowley, C., Cuypers, J.,
Dafonte, C., Damerdji, Y., Dapergolas, A., David, P., David, M., de
Laverny, P., De Luise, F., De March, R., de Martino, D., de Souza, R.,
de Torres, A., Debosscher, J., del Pozo, E., Delbo, M., Delgado, A.,
Delgado, H. E., Di Matteo, P., Diakite, S., Diener, C., Distefano, E.,
Dolding, C., Drazinos, P., Durán, J., Edvardsson, B., Enke, H., Eriksson,
K., Esquej, P., Eynard Bontemps, G., Fabre, C., Fabrizio, M., Faigler, S.,
Falcão, A. J., Farràs Casas, M., Federici, L., Fedorets, G.,
Fernique, P., Figueras, F., Filippi, F., Findeisen, K., Fonti, A., Fraile,
E., Fraser, M., Frézouls, B., Gai, M., Galleti, S., Garabato, D.,
García-Sedano, F., Garofalo, A., Garralda, N., Gavel, A., Gavras, P.,
Gerssen, J., Geyer, R., Giacobbe, P., Gilmore, G., Girona, S., Giuffrida, G.,
Glass, F., Gomes, M., Granvik, M., Gueguen, A., Guerrier, A., Guiraud, J.,
Gutiérrez-Sánchez, R., Haigron, R., Hatzidimitriou, D., Hauser, M.,
Haywood, M., Heiter, U., Helmi, A., Heu, J., Hilger, T., Hobbs, D., Hofmann,
W., Holland, G., Huckle, H. E., Hypki, A., Icardi, V., Janßen, K.,
Jevardat de Fombelle, G., Jonker, P. G., Juhász, Á. L., Julbe, F.,
Karampelas, A., Kewley, A., Klar, J., Kochoska, A., Kohley, R., Kolenberg,
K., Kontizas, M., Kontizas, E., Koposov, S. E., Kordopatis, G.,
Kostrzewa-Rutkowska, Z., Koubsky, P., Lambert, S., Lanza, A. F., Lasne, Y.,
Lavigne, J.-B., Le Fustec, Y., Le Poncin-Lafitte, C., Lebreton, Y., Leccia,
S., Leclerc, N., Lecoeur-Taibi, I., Lenhardt, H., Leroux, F., Liao, S.,
Licata, E., Lindstrøm, H. E. P., Lister, T. A., Livanou, E., Lobel, A.,
López, M., Managau, S., Mann, R. G., Mantelet, G., Marchal, O., Marchant,
J. M., Marconi, M., Marinoni, S., Marschalkó, G., Marshall, D. J.,
Martino, M., Marton, G., Mary, N., Massari, D., Matijevič, G., Mazeh,
T., McMillan, P. J., Messina, S., Michalik, D., Millar, N. R., Molina, D.,
Molinaro, R., Molnár, L., Montegriffo, P., Mor, R., Morbidelli, R.,
Morel, T., Morris, D., Mulone, A. F., Muraveva, T., Musella, I., Nelemans,
G., Nicastro, L., Noval, L., O'Mullane, W., Ordénovic, C.,
Ordóñez-Blanco, D., Osborne, P., Pagani, C., Pagano, I., Pailler,
F., Palacin, H., Palaversa, L., Panahi, A., Pawlak, M., Piersimoni, A. M.,
Pineau, F.-X., Plachy, E., Plum, G., Poggio, E., Poujoulet, E., Prša,
A., Pulone, L., Racero, E., Ragaini, S., Rambaux, N., Ramos-Lerate, M.,
Regibo, S., Reylé, C., Riclet, F., Ripepi, V., Riva, A., Rivard, A.,
Rixon, G., Roegiers, T., Roelens, M., Romero-Gómez, M., Rowell, N.,
Royer, F., Ruiz-Dern, L., Sadowski, G., Sagristà Sellés, T.,
Sahlmann, J., Salgado, J., Salguero, E., Sanna, N., Santana-Ros, T.,
Sarasso, M., Savietto, H., Schultheis, M., Sciacca, E., Segol, M., Segovia,
J. C., Ségransan, D., Shih, I.-C., Siltala, L., Silva, A. F., Smart,
R. L., Smith, K. W., Solano, E., Solitro, F., Sordo, R., Soria Nieto, S.,
Souchay, J., Spagna, A., Spoto, F., Stampa, U., Steele, I. A.,
Steidelmüller, H., Stephenson, C. A., Stoev, H., Suess, F. F., Surdej,
J., Szabados, L., Szegedi-Elek, E., Tapiador, D., Taris, F., Tauran, G.,
Taylor, M. B., Teixeira, R., Terrett, D., Teyssandier, P., Thuillot, W.,
Titarenko, A., Torra Clotet, F., Turon, C., Ulla, A., Utrilla, E., Uzzi, S.,
Vaillant, M., Valentini, G., Valette, V., van Elteren, A., Van Hemelryck,
E., van Leeuwen, M., Vaschetto, M., Vecchiato, A., Veljanoski, J., Viala,
Y., Vicente, D., Vogt, S., von Essen, C., Voss, H., Votruba, V., Voutsinas,
S., Walmsley, G., Weiler, M., Wertz, O., Wevers, T., Wyrzykowski, Ł.,
Yoldas, A., Žerjal, M., Ziaeepour, H., Zorec, J., Zschocke, S., Zucker,
S., Zurbach, C., Zwitter, T., 2018, Gaia Data Release 2. Summary of
the contents and survey properties, Astronomy and Astrophysics, 616, A1, ISSN

[Gänsicke et al., 2019]
Gänsicke, B. T., Schreiber, M. R., Toloza, O., Fusillo, N. P. G., Koester,
D., Manser, C. J., 2019, Accretion of a giant planet onto a white dwarf star,
Nature, 576, 61–64, ISSN 0028-0836, 1476-4687

[Green et al., 2003]
Green, E. M., Fontaine, G., Reed, M. D., Callerame, K., Seitenzahl, I. R.,
White, B. A., Hyde, E. A., Østensen, R., Cordes, O., Brassard, P., Falter,
S., Jeffery, E. J., Dreizler, S., Schuh, S. L., Giovanni, M., Edelmann, H.,
Rigby, J., Bronowska, A., 2003, Discovery of A New Class of Pulsating
Stars: Gravity-Mode Pulsators among Subdwarf B Stars,
Astrophys. J., 583, L31

[Greenstein and Sargent, 1974]
Greenstein, J. L., Sargent, A. I., 1974, The Nature of Faint Blue Stars
in the Halo. II, Astrophys. J. Suppl. Ser., 28, 157, ISSN 0067-0049

[Grunblatt et al., 2019]
Grunblatt, S. K., Huber, D., Gaidos, E., Hon, M., Zinn, J. C., Stello, D.,
2019, Giant planet occurrence within 0.2 AU of low-luminosity red giant
branch stars with K2, arXiv, p. arXiv:1910.05346

[Hall and Jeffery, 2016]
Hall, P. D., Jeffery, C. S., 2016, Hydrogen in hot subdwarfs formed by double
helium white dwarf mergers, Mon. Not. R. Astron. Soc., 463, 2756, ISSN

[Han et al., 2002]
Han, Z., Podsiadlowski, P., Maxted, P. F. L., Marsh, T. R., Ivanova, N., 2002,
The origin of subdwarf B stars - I. The formation channels, Mon.
Not. R. Astron. Soc., 336, 449

[Han et al., 2003]
Han, Z., Podsiadlowski, P., Maxted, P. F. L., Marsh, T. R., 2003, The origin of
subdwarf B stars - II, Mon. Not. R. Astron. Soc., 341, 669

[Heber, 1986]
Heber, U., 1986, The atmosphere of subluminous B stars. II -
Analysis of 10 helium poor subdwarfs and the birthrate of sdB stars,
Astronomy and Astrophysics, 155, 33–45, ISSN 0004-6361

[Heber, 2009]
Heber, U., 2009, Hot Subdwarf Stars, Annu. Rev. Astron. Astrophys., 47, 211

[Heber, 2016]
Heber, U., 2016, Hot Subluminous Stars, Publications of the Astronomical
Society of the Pacific, 128, 082 001, ISSN 0004-6280

[Heber et al., 1984]
Heber, U., Hunger, K., Jonas, G., Kudritzki, R. P., 1984, The atmosphere of
sublumious B stars., Astron. Astrophys., 130, 119, ISSN 0004-6361

[Heber et al., 2003]
Heber, U., Edelmann, H., Lisker, T., Napiwotzki, R., 2003, Discovery of a
helium-core white dwarf progenitor, Astronomy and Astrophysics, 411,
L477–L480, ISSN 0004-6361

[Hilditch, 2001]
Hilditch, R. W., 2001, An Introduction to Close Binary Stars,
Cambridge University Press, first edn., ISBN 978-0-521-79800-6
978-0-521-24106-9 978-1-139-16357-6

[Horne and Baliunas, 1986]
Horne, J. H., Baliunas, S. L., 1986, A prescription for period analysis of
unevenly sampled time series, The Astrophysical Journal, 302, 757–763, ISSN

[Humason and Zwicky, 1947]
Humason, M. L., Zwicky, F., 1947, A Search for Faint Blue Stars.,
Astrophys. J., 105, 85, ISSN 0004-637X

[Hutchens et al., 2017]
Hutchens, Z. L., Barlow, B. N., Soto, A. V., Reichart, D. E., Haislip, J. B.,
Kouprianov, V. V., Linder, T. R., Moore, J. P., 2017, New Pulse Timing
Measurements of the sdBV Star CS 1246, ArXiv171202392 Astro-Ph,

[IAU, 2006]
IAU, 2006, RESOLUTION B5 - Definition of a Planet in the Solar
System, https://www.iau.org/static/resolutions/Resolution_GA26-5-6.pdf

[III, 2007]
III, J. O. S., 2007, Mathematics of the Discrete Fourier Transform
(DFT): With Audio Applications —- Second Edition, W3K
Publishing, North Charleston, 2 edition edn., ISBN 978-0-9745607-4-8

[Johnson et al., 2011]
Johnson, J. A., Clanton, C., Howard, A. W., Bowler, B. P., Henry, G. W., Marcy,
G. W., Crepp, J. R., Endl, M., Cochran, W. D., MacQueen, P. J., Wright,
J. T., Isaacson, H., 2011, Retired A Stars and Their Companions.
VII. 18 New Jovian Planets, The Astrophysical Journal Supplement
Series, 197, 26, ISSN 0067-0049

[Jones et al., 2011]
Jones, E., Oliphant, T., Peterson, P., 2011, SciPy: Open source
scientific tools for Python, https://www.scipy.org/

[Kalas et al., 2008]
Kalas, P., Graham, J. R., Chiang, E., Fitzgerald, M. P., Clampin, M., Kite,
E. S., Stapelfeldt, K., Marois, C., Krist, J., 2008, Optical Images of an
Exosolar Planet 25 Light-Years from Earth, Science, 322,
1345–1348, ISSN 0036-8075, 1095-9203

[Kempton, 2011]
Kempton, E. M. R., 2011, Planetary science: The ultimate fate of planets,
Nature, 480, 460–461, ISSN 1476-4687

[Kepler et al., 1991]
Kepler, S. O., Winget, D. E., Nather, R. E., Bradley, P. A., Grauer, A. D.,
Fontaine, G., Bergeron, P., Vauclair, G., Claver, C. F., Marar, T. M. K.,
Seetha, S., Ashoka, B. N., Mazeh, T., Leibowitz, E., Dolez, N., Chevreton,
M., Barstow, M. A., Clemens, J. C., Kleinman, S. J., Sansom, A. E., Tweedy,
R. W., Kanaan, A., Hine, B. P., Provencal, J. L., Wesemael, F., Wood, M. A.,
Brassard, P., Solheim, J.-E., Emanuelsen, P.-I., 1991, A detection of the
evolutionary time scale of the DA white dwarf G117 - B15A with
the Whole Earth Telescope, ApJ, 378, L45, ISSN 0004-637X, 1538-4357

[Kilkenny, 2010]
Kilkenny, D., 2010, Amplitude variations in pulsating sdB stars, Astrophys
Space Sci, 329, 175–181, ISSN 0004-640X, 1572-946X

[Kilkenny, 2014]
Kilkenny, D., 2014, The orbital periods of three sdB eclipsing binary
systems, Monthly Notices of the Royal Astronomical Society, 445, 4247–4251,
ISSN 0035-8711

[Kilkenny et al., 1997]
Kilkenny, D., Koen, C., Stobie, R. S., O'Donoghue, D., Lynas-Gray, A. E.,
Kawaler, S. D., 1997, A New Class of Pulsating Star Discovered in the
EC Survey, Third Conf. Faint Blue Stars, p. 77

[Kilkenny et al., 2006]
Kilkenny, D., Stobie, R. S., O'Donoghue, D., Koen, C., Hambly, N.,
MacGillivray, H., Lynas-Gray, A. E., 2006, Three new pulsating sdB
stars from the Edinburgh-Cape survey, Mon. Not. R. Astron. Soc., 367,

[Knutson et al., 2007]
Knutson, H. A., Charbonneau, D., Allen, L. E., Fortney, J. J., Agol, E., Cowan,
N. B., Showman, A. P., Cooper, C. S., Megeath, S. T., 2007, A map of the
day–night contrast of the extrasolar planet HD 189733b,
Nature, 447, 183–186, ISSN 1476-4687

[Krzesinski, 2015]
Krzesinski, J., 2015, Planetary candidates around the pulsating sdB star
KIC 5807616 considered doubtful, Astron. Astrophys., 581, A7, ISSN
0004-6361, 1432-0746

[Latour et al., 2014]
Latour, M., Randall, S. K., Fontaine, G., Bono, G., Calamida, A., Brassard, P.,
2014, A Helium-Carbon Correlation on the Extreme Horizontal
Branch in $\omega$ Centauri, Astrophys. J., 795, 106, ISSN 0004-637X

[Lee et al., 2014]
Lee, J. W., Hinse, T. C., Youn, J.-H., Han, W., 2014, The pulsating
sdB+M eclipsing system NY Virginis and its circumbinary planets,
Mon. Not. R. Astron. Soc., 445, 2331–2339, ISSN 0035-8711, 1365-2966

[Litwa, 2016]
Litwa, M. D., 2016, Refutation of All Heresies, Society of Biblical
Literature, ISBN 978-0-88414-087-0

[Lloyd, 2011]
Lloyd, J. P., 2011, "Retired" Planet Hosts: Not So Massive, Maybe
Just Portly After Lunch, The Astrophysical Journal Letters, 739, L49, ISSN

[Lutz, 2011]
Lutz, R., 2011, The search for substellar companions to subdwarf B stars in
connection with evolutionary aspects, PhD Thesis

[Lutz et al., 2008]
Lutz, R., Schuh, S., Silvotti, R., Dreizler, S., Green, E. M., Fontaine, G.,
Stahn, T., Hügelmeyer, S. D., Husser, T.-O., 2008a, Light
Curve Analysis of the Hybrid SdB PulsatorsHS 0702+6043 and HS
2201+2610, Hot Subdwarf Stars Relat. Objects, 392, 339

[Lutz et al., 2008]
Lutz, R., Schuh, S., Silvotti, R., Kruspe, R., Dreizler, S.,
2008b, Long-term photometric monitoring of the hybrid subdwarf
B pulsator HS 0702+6043, Commun. Asteroseismol., 157, 185

[Lutz et al., 2011]
Lutz, R., Schuh, S., Silvotti, R., 2011, The EXOTIME Targets HS 0702+6043
and HS 0444+0458, Am. Inst. Phys. Conf. Ser., 1331, 155

[Maxted et al., 2001]
Maxted, P. F. L., Heber, U., Marsh, T. R., North, R. C., 2001, The binary
fraction of extreme horizontal branch stars, Mon. Not. R. Astron. Soc., 326,

[Mayor and Queloz, 1995]
Mayor, M., Queloz, D., 1995, A Jupiter-mass companion to a solar-type star,
Nature, 378, 355–359, ISSN 0028-0836, 1476-4687

[McNamara et al., 2012]
McNamara, B. J., Jackiewicz, J., McKeever, J., 2012, THE CLASSIFICATION OF
KEPLER B-STAR VARIABLES, The Astronomical Journal, 143,
101, ISSN 0004-6256, 1538-3881

[Mengel et al., 1976]
Mengel, J. G., Norris, J., Gross, P. G., 1976, Binary Hypothesis for the
Subdwarf B Stars, Astrophys. J., 204, 488, ISSN 0004-637X

[Moehler et al., 2011]
Moehler, S., Dreizler, S., Lanz, T., Bono, G., Sweigart, A. V., Calamida, A.,
Nonino, M., 2011, The hot horizontal-branch stars in $\omega$ Centauri,
Astron. Astrophys., 526, A136, ISSN 0004-6361

[Murphy et al., 2013]
Murphy, S. J., Shibahashi, H., Kurtz, D. W., 2013, Super-Nyquist
asteroseismology with the Kepler Space Telescope, Mon. Not. R. Astron.
Soc., 430, 2986–2998, ISSN 0035-8711, 1365-2966

[Murphy et al., 2014]
Murphy, S. J., Bedding, T. R., Shibahashi, H., Kurtz, D. W., Kjeldsen, H.,
2014, Finding binaries among Kepler pulsating stars from phase modulation
of their pulsations, Mon. Not. R. Astron. Soc., 441, 2515–2527, ISSN
0035-8711, 1365-2966

[Murphy et al., 2016]
Murphy, S. J., Bedding, T. R., Shibahashi, H., 2016a, A Planet
in an 840 Day Orbit around a Kepler Main-sequence A Star Found
from Phase Modulation of Its Pulsations, Astrophys. J. Lett., 827,

[Murphy et al., 2016]
Murphy, S. J., Shibahashi, H., Bedding, T. R., 2016b, Finding
binaries from phase modulation of pulsating stars with Kepler. IV.
Detection limits and radial velocity verification, Mon. Not. R. Astron.
Soc., 461, 4215–4226, ISSN 0035-8711, 1365-2966, 1607.07879

[Østensen et al., 2001]
Østensen, R., Heber, U., Silvotti, R., Solheim, J.-E., Dreizler, S.,
Edelmann, H., 2001, Four new subdwarf B pulsators, Astron. Astrophys.,
378, 466

[Østensen et al., 2011]
Østensen, R. H., Bloemen, S., Vučković, M., Aerts, C., Oreiro, R.,
Kinemuchi, K., Still, M., Koester, D., 2011, At Last—A V777
Her Pulsator in the Kepler Field, The Astrophysical Journal Letters,
736, L39, ISSN 0004-637X

[Østensen et al., 2014]
Østensen, R. H., Reed, M. D., Baran, A. S., Telting, J. H., 2014, Stochastic
pulsations in the subdwarf-B star KIC 2991276, Astronomy and
Astrophysics, 564, L14, ISSN 0004-6361

[Otani et al., 2018]
Otani, T., Oswalt, T. D., Lynas-Gray, A. E., Kilkenny, D., Koen, C., Amaral,
M., Jordan, R., 2018, Orbital Characteristics of the Subdwarf-B
and F V Star Binary EC 20117-4014 (=V4640 Sgr), ApJ, 859, 145, ISSN

[Pajdosz, 1995]
Pajdosz, G., 1995a, The Effect of Proper Motion on P
in Pulsating Stars, 83, 439

[Pajdosz, 1995]
Pajdosz, G., 1995b, Non-evolutionary secular period increase in
pulsating DA white dwarfs, Astronomy and Astrophysics, 295, L17–L19,
ISSN 0004-6361

[Perryman, 2018]
Perryman, M., 2018, The Exoplanet Handbook:, Cambridge University Press,
second edn., ISBN 978-1-108-41977-2 978-1-108-30416-0

[Podsiadlowski, 2008]
Podsiadlowski, P., 2008, The Evolution of Close Binaries, in RS
Ophiuchi (2006) and the Recurrent Nova Phenomenon, vol. 401, p. 63

[Postnov and Yungelson, 2014]
Postnov, K. A., Yungelson, L. R., 2014, The Evolution of Compact Binary
Star Systems, Living Rev. Relativ., 17, ISSN 2367-3613, 1433-8351

[Press et al., 2007]
Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P., 2007,
Numerical Recipes 3rd Edition: The Art of Scientific
Computing, Cambridge University Press, ISBN 978-0-521-88068-8

[Provencal et al., 2009]
Provencal, J. L., Montgomery, M. H., Kanaan, A., Shipman, H. L., Childers, D.,
Baran, A., Kepler, S. O., Reed, M., Zhou, A., Eggen, J., Watson, T. K.,
Winget, D. E., Thompson, S. E., Riaz, B., Nitta, A., Kleinman, S. J., Crowe,
R., Slivkoff, J., Sherard, P., Purves, N., Binder, P., Knight, R., Kim,
S. L., Chen, W.-P., Yang, M., Lin, H. C., Lin, C. C., Chen, C. W., Jiang,
X. J., Sergeev, A. V., Mkrtichian, D., Andreev, M., Janulis, R., Siwak, M.,
Zola, S., Koziel, D., Stachowski, G., Paparo, M., Bognar, Z., Handler, G.,
Lorenz, D., Steininger, B., Beck, P., Nagel, T., Kusterer, D., Hoffman, A.,
Reiff, E., Kowalski, R., Vauclair, G., Charpinet, S., Chevreton, M., Solheim,
J. E., Pakstiene, E., Fraga, L., Dalessio, J., 2009, 2006 Whole Earth
Telescope Observations of GD358: A New Look at the Prototype
DBV, The Astrophysical Journal, 693, 564–585, ISSN 0004-637X

[Qian et al., 2015]
Qian, S. B., Han, Z. T., Laj?s, E. F., Zhu, L. Y., Li, L. J., Liao, W. P.,
Zhao, E. G., 2015, LONG-TERM DECREASE AND CYCLIC VARIATION IN THE
ORBITAL PERIOD OF THE ECLIPSING DWARF NOVA V2051 OPH, Astrophys. J. Suppl.
Ser., 221, 17, ISSN 1538-4365

[Qian et al., 2017]
Qian, S. B., Li, L. J., He, J. J., Zhang, J., Zhu, L. Y., Han, Z. T., 2017,
LAMOST Views $\delta$ Scuti Pulsating Stars,
ArXiv170704006 Astro-Ph, 1707.04006

[Quintana et al., 2014]
Quintana, E. V., Barclay, T., Raymond, S. N., Rowe, J. F., Bolmont, E.,
Caldwell, D. A., Howell, S. B., Kane, S. R., Huber, D., Crepp, J. R.,
Lissauer, J. J., Ciardi, D. R., Coughlin, J. L., Everett, M. E., Henze,
C. E., Horch, E., Isaacson, H., Ford, E. B., Adams, F. C., Still, M., Hunter,
R. C., Quarles, B., Selsis, F., 2014, An Earth-Sized Planet in the
Habitable Zone of a Cool Star, Science, 344, 277–280, ISSN
0036-8075, 1095-9203

[Randall et al., 2009]
Randall, S. K., Van Grootel, V., Fontaine, G., Charpinet, S., Brassard, P.,
2009, Observations and asteroseismological analysis of the rapid subdwarf
B pulsator EC 09582-1137, Astron. Astrophys., 507, 911

[Reed et al., 2007]
Reed, M. D., O'Toole, S. J., Terndrup, D. M., Eggen, J. R., Zhou, A.-Y., An,
D., Chen, C.-W., Chen, W. P., Lin, H.-C., Akan, C., Cakirli, O., Worters, H.,
Kilkenny, D., Siwak, M., Zola, S., Kim, S.-L., Gelven, G. A., Harms, S. L.,
Wolf, G. W., 2007a, Follow-Up Observations of Pulsating
Subdwarf B Stars: Multisite Campaigns on PG 1618+563B and
PG 0048+091, ApJ, 664, 518–535, ISSN 0004-637X, 1538-4357

[Reed et al., 2007]
Reed, M. D., Terndrup, D. M., Zhou, A.-Y., Unterborn, C. T., An, D., Eggen,
J. R., 2007b, Resolving the pulsations of subdwarf B stars:
HS 0039+4302, HS 0444+0458 and an examination of the group properties
of resolved pulsators, Mon. Not. R. Astron. Soc., 378, 1049

[Reed et al., 2018]
Reed, M. D., Baran, A. S., Telting, J. H., Østensen, R. H., Jeffery, C. S.,
Kern, J. W., Ketzer, L., Crooke, J., Slayton, A., 2018, A review of seismic
observations of Kepler and K2-Observed sdBV stars, Open Astron.,
27, 157–166, ISSN 2543-6376

[Reffert et al., 2015]
Reffert, S., Bergmann, C., Quirrenbach, A., Trifonov, T., Künstler, A.,
2015, Precise radial velocities of giant stars. VII. Occurrence rate
of giant extrasolar planets as a function of mass and metallicity, Astronomy
and Astrophysics, 574, A116, ISSN 0004-6361

[Ricker et al., 2015]
Ricker, G. R., Winn, J. N., Vanderspek, R., Latham, D. W., Bakos, G. Á.,
Bean, J. L., Berta-Thompson, Z. K., Brown, T. M., Buchhave, L., Butler,
N. R., Butler, R. P., Chaplin, W. J., Charbonneau, D.,
Christensen-Dalsgaard, J., Clampin, M., Deming, D., Doty, J., De Lee, N.,
Dressing, C., Dunham, E. W., Endl, M., Fressin, F., Ge, J., Henning, T.,
Holman, M. J., Howard, A. W., Ida, S., Jenkins, J. M., Jernigan, G., Johnson,
J. A., Kaltenegger, L., Kawai, N., Kjeldsen, H., Laughlin, G., Levine, A. M.,
Lin, D., Lissauer, J. J., MacQueen, P., Marcy, G., McCullough, P. R., Morton,
T. D., Narita, N., Paegert, M., Palle, E., Pepe, F., Pepper, J., Quirrenbach,
A., Rinehart, S. A., Sasselov, D., Sato, B., Seager, S., Sozzetti, A.,
Stassun, K. G., Sullivan, P., Szentgyorgyi, A., Torres, G., Udry, S.,
Villasenor, J., 2015, Transiting Exoplanet Survey Satellite (TESS),
Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014 003

[Royer et al., 2007]
Royer, F., Zorec, J., Gómez, A. E., 2007, Rotational velocities of
A-type stars. III. Velocity distributions, Astronomy and
Astrophysics, 463, 671–682, ISSN 0004-6361

[Santerne et al., 2016]
Santerne, A., Moutou, C., Tsantaki, M., Bouchy, F., Hébrard, G., Adibekyan,
V., Almenara, J.-M., Amard, L., Barros, S. C. C., Boisse, I., Bonomo, A. S.,
Bruno, G., Courcol, B., Deleuil, M., Demangeon, O., Díaz, R. F., Guillot,
T., Havel, M., Montagnier, G., Rajpurohit, A. S., Rey, J., Santos, N. C.,
2016, SOPHIE velocimetry of Kepler transit candidates. XVII.
The physical properties of giant exoplanets within 400 days of period,
A&A, 587, A64, ISSN 0004-6361

[Scargle, 1982]
Scargle, J. D., 1982, Studies in astronomical time series analysis. II -
Statistical aspects of spectral analysis of unevenly spaced data, The
Astrophysical Journal, 263, 835–853, ISSN 0004-637X

[Schaffenroth et al., 2015]
Schaffenroth, V., Barlow, B. N., Drechsel, H., Dunlap, B. H., 2015, An
eclipsing post common-envelope system consisting of a pulsating hot subdwarf
B star and a brown dwarf companion, Astron. Astrophys., 576, A123, ISSN
0004-6361, 1432-0746