Time delay analysis of the δ Scuti pulsations in the exoplanet host star β Pictoris based on space and ground-based photometry
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Master Thesis
Time delay analysis
of the δ Scuti pulsations
in the exoplanet host star β Pictoris
based on space and ground-based
photometry
Sebastian Zieba
supervised by
Konstanze Zwintz
co-supervised by
Matthew Kenworthy
A thesis submitted in fulfillment of the requirements
for the degree of Master of Science in Astrophysics
at the
University of Innsbruck
Faculty of Mathematics, Computer Science and Physics
Institute for Astro- and Particle Physics
December 2019
Abstract
Time delay analysis
of the δ Scuti pulsations
in the exoplanet host star β Pictoris
based on space and ground-based photometry
by Sebastian Zieba
Abstract. This thesis analyses the phase variations of the δ Scuti pulsations of β
Pictoris in order to search for time delays that might be caused by companions in the
system. The photometric data of the star was collected over approximately four years
by the BRITE-Constellation, bRing, ASTEP and TESS.
The uncertainty in the phases and frequencies, however, are too high to see the
influence of one of the two exoplanets β Pictoris b or c on the timing of the pulsations
of their host star. We compare the calculated time delays with a simulated data set and
find that both are in agreement with each other, showing that the current observations
are not able to give meaningful results.
Additional photometry of β Pictoris, which will be collected by TESS between 2021
and 2022, will lead to a better frequency determination and therefore to provide another
chance to search for the signature of both currently known planets in the system.
We also use the TESS observations to derive a frequency list of the strongest 37
significant δ Scuti pulsations. The residuals of the TESS photometry are investigated
for periodic transits using a Box Least Squares (BLS) algorithm showing no significant
detection.
iii
Danksagung
I would like to start my thesis by thanking the people who made this work possible
and helped me along the way.
First of all, my supervisor Konstanze Zwintz who was always there to give me feed-
back on the current status of my work. My conversions with Matthew Kenworthy were
also really helpful and I look back in great memory to the two weeks when I visited
Leiden Observatory. I also want to thank him, Grant Kennedy and Konstanze for
identifying the dimming events I noticed in the TESS data of β Pictoris as exocomets.
After that, I was able to share this discovery at various conferences and workshops. I
met many new colleagues and friends along the way which I also want to mention here.
I thank my great research group and office colleagues - Thomas, Marco and Laura - who
always assisted me with helpful discussion on my work. I also acknowledge Thomas
and Sebastian for giving me this template for my thesis.
And of course my parents who made everything possible.
Sebastian
v
Contents
Abstract iii
1 Introduction 1
2 The β Pictoris System 5
2.1 The star: β Pictoris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The planet: β Pictoris b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 The planet candidate: β Pictoris c . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Non-detections in the system . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Observations 13
3.1 BRITE-Constellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 bRing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2.1 Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 ASTEP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3.1 Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 TESS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4.1 Data reduction and frequency analysis . . . . . . . . . . . . . . . . . . 16
4 Theory and Methodology 23
4.1 δ Scuti stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 The ephemeris equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 O-C diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.4 Frequency modulation and Phase modulation: state-of-the-art . . . . . . . . . 25
4.5 Time Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.6 Phase Modulation Method: Methodology . . . . . . . . . . . . . . . . . . . . 28
4.7 Intrinsic Amplitude and Phase variations . . . . . . . . . . . . . . . . . . . . 30
4.8 Comparison to other methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.9 Barycentric correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.10 Light curve reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5 Results 35
5.1 Frequency stability between the different observations . . . . . . . . . . . . . 35
5.1.1 β Pictoris as seen by Kepler . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Expectations, Observations and Simulations . . . . . . . . . . . . . . . . . . . 40
5.3 Possibility with TESS’ extended mission . . . . . . . . . . . . . . . . . . . . . 44
6 Conclusions 45
7 Acknowledgements 47
viiCONTENTS CONTENTS
A TESS frequency analysis 49
A.1 Frequency List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A.2 Gaussian highpass filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
A.2.1 Residuals of every TESS Sector . . . . . . . . . . . . . . . . . . . . . . 60
A.3 γ Doradus pulsations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
B BLS search 63
C Transit time 63
D Nomenclature 65
References 67
viii1 Introduction
The study of companions around stars has a long history. Ptolemy called the stars ν 1 and
ν 2 Sagittarii being διπλoυ̃ς (double) (Toomer 1984). Those two stars do indeed form an
optical double star system. In contrast to other languages, English does strictly differentiate
between two stars being only an apparent close pair of stars due to the viewing angle from
the earth (double star) and two - or more - stars being gravitationally bound (binary star).
A review on double and binary stars including their history of discovery can be found in
Heintz (1978) and Aitken (1964).
The first binary system was then discovered in 1783 by John Goodricke when he tried to
explain the observed light variations of the star β Persei, better known under its Arabic
name Algol (Goodricke 1783). At that time two different scenarios were put forward to
explain these sudden brightness variations: either the star is being regularly eclipsed by
a big planet or the surface of Algol has a dark spot. The light variations were however
so severe that a transiting planet could be ruled out. The idea of a “transiting star” was
finally proven over one century later in 1889 when Hermann Carl Vogel observed that the
spectral lines of the star were moving, hinting to a orbital motion and thus to a binary star
system (Vogel 1890). Such a system with observable periodic shifts in the spectrum of the
star caused by a changing radial velocity relative to the observer are called SB1 stars. If
two sets of spectral lines are observed moving in anti-phase directions, we call this system
a double-lined spectroscopic binary (SB2). Antonia Maury observed a periodic spitting and
shift for Mizar A (ζ 1 UMa) as reported by Edward Pickering in 1889 (Pickering 1890). She
therefore found the first binary star system. Two independent reports by Ludendorff (1908)
and Frost (1908) reported a few years later also a radial velocity variation for the fainter
component Mizar B (ζ 2 UMa).
The same principle was used by Michel Mayor and Didier Queloz to discover the first
exoplanet orbiting an solar like star (Mayor & Queloz 1995). This discovery was recently
awarded with the Nobel Prize in physics1 and marks the discovery of an up to that
time unknown class of planets: Hot-Jupiters - gas-giants with a short orbital period and
therefore being highly irradiated leading to hot temperatures for the planet. This so-called
“radial-velocity method” for discovering exoplanets by the induced orbital motion on their
host star and modulation of the star’s spectral lines was the most successful method for the
longest time. This changed after the launch of the Kepler space telescope (Koch et al. 2010)
and the following discoveries of transiting exoplanets (Borucki et al. 2010; Borucki et al.
2011; Batalha et al. 2013). Today, as of October 2019, around 80% of all exoplanets have
been discovered using the transit method2 .
1
Press release: The Nobel Prize in Physics 2019. NobelPrize.org. Nobel Media AB 2019. Tue. 12 Nov
2019. https://www.nobelprize.org/prizes/physics/2019/press-release/
2
Statistics taken from the NASA Exoplanet Archive: https://exoplanetarchive.ipac.caltech.edu/
11 INTRODUCTION
There are, however, stars for which the transit method struggles. This is especially the
case for an early spectral type host star - which has a greater luminosity and radius - or
for intrinsically variable stars. δ Scuti stars tick off both boxes. They can be found at the
intersection region between the main-sequence and the instability strip on the Hertzsprung-
Ruessel diagram. The first Hot-Jupiter (with an orbital period of around one day) to be
discovered around such a δ Scuti star was WASP-33b (Collier Cameron et al. 2010; Herrero
et al. 2011). The proximity to its hot, rapidly-rotating host star opens great possibilities to
study the effects of star-planet interaction (SPI). The star’s temperature of around 3400 K
(von Essen et al. 2015) makes it the second hottest planet after KELT-9b (Gaudi et al. 2017).
WASP-33b was therefore reobserved several times resulting in multiple discoveries: The
possibility of directly observable SPIs - like the amplification of the δ Scuti pulsations of the
star by the exoplanet (Collier Cameron et al. 2010) - was analysed by several ground-based
observatories. This led to the discovery of multiple pulsational modes on the order of 1
millimagnitude (Kovács et al. 2013; von Essen et al. 2014), transit anomalies (Kovács et al.
2013) and a possible commensurability between a stellar pulsation frequency and the orbital
period of the planet (Collier Cameron et al. 2010). WASP-33b was also the first exoplanet
with observed Titanium Oxide (TiO) in its stratosphere (Haynes et al. 2015; Nugroho et al.
2017). A recent study by von Essen et al. (2019) report the first significant aluminium
oxide (AlO) in the atmosphere of the planet. The planet’s frequent transits were also
pre-discovered in the Hipparcos data from the early 1990s (McDonald & Kerins 2018).
These discoveries show the great possibilities having an exoplanet orbiting a bright (mV ≈
8 mag) and relatively young (Age ≈ 100 Myr; Moya et al. 2011) δ Scuti star. Additional
photometry of WASP-33 collected by the TESS mission will approximately be released at
the end of December 2019.
WASP-33b is, however, often wrongly mistaken as being the only planet orbiting a δ Scuti
star. β Pictoris is a close, southern hemisphere star visible by naked eye for which δ Scuti -like
pulsations were reported by Koen (2003). Its planet, β Pictoris b, was discovered using the
VLT/NaCo instrument with Direct Imaging (Lagrange et al. 2009a; Lagrange et al. 2010).
Evidence of a second planet in the β Pictoris system was recently published by Lagrange
et al. (2019a) using the radial-velocity method. If confirmed, this would mark the first sys-
tem containing a directly imaged planet and a second one discovered with a different method.
The lifetime and frequency stability of δ Scuti pulsations make them astronomical “stellar
clocks” and therefore great targets for applying timing techniques. The common orbital
motion of the star together with the companion around the barycenter of the system results
in a periodic early or late arrival of the signals. This principle led to the first detection of
planets outside our solar system orbiting a pulsar (Wolszczan & Frail 1992; Wolszczan 1994).
This periodic variation of the arrival times can be either seen as a frequency modulation
21 INTRODUCTION
(FM; Shibahashi & Kurtz 2012; Shibahashi et al. 2015) or phase modulation (PM; Murphy
et al. 2014; Murphy & Shibahashi 2015; Murphy et al. 2016b). The latter method works
better for companions in wider orbits. As a massive companion close to β Pictoris is
ruled out by other observations, we will use the PM method in this work. Applying the
PM method on Kepler data, Murphy et al. (2016a) was able to discover a massive planet
(m sin i ≈ 12MJ ) with an orbital period of about 840 days around a δ Scuti star.
The PM method led - besides to the discovery of this exoplanet - to the detection of about
one thousand binary star systems. It furthermore provides us with the full orbital solution
of the system just like the radial velocity (RV) method does. The PM method, however,
uses photometric data and can therefore be done on bigger scales as the RV method as was
shown with Kepler data.
Applying the same method on pulsating stars observed by the TESS mission, will lead
to many more binary systems with full orbital solutions. The majority of them will be
around the ecliptic poles where TESS observes approximately one year each in its primary
mission. TESS was recently approved for an extended mission by NASA3 . The extended
mission will start in July 2020 and observe β Pictoris again at the end of 2020.4 One of the
changes will be the higher cadence of 10 minutes for the full frame images (FFI) compared
to the primary mission cadence of 30 minutes. This increases the Nyquist frequency from
24 d−1 to 72 d−1 , meaning that the majority of δ Scuti -pulsations can be unambiguously
identified, as they usually pulsate below 80 d−1 (Aerts et al. 2010). This opens the way to
do a complete photometric analysis for δ Scuti stars over (nearly) the full sky with a high
precision instrument like TESS.
In this work, we will use the data collected by the TESS satellite in its primary mission
and data collected by the Hill sphere transit campaign, which was an international effort of
space (e.g. BRITE-constellation) and ground-based (e.g. bRing, ASTEP) observations in
order to search for signatures of material around the giant planet β Pictoris b (Kalas et al.
2019). This photometric data will then be analysed by searching for phase variations caused
by orbital motion and therefore time delays in the pulsational signals.
In Chapter 2 we describe the history and properties of the different components in the β
Pictoris system. Chapter 3 has a summary of all observational instruments and a frequency
analysis for the photometry collected by TESS. The theory and equations for this thesis can
be found in Chapter 4. Finally, we present the results (see Chapter 5) and follow it with a
discussion (see Chapter 6).
3
https://nspires.nasaprs.com/external/viewrepositorydocument/cmdocumentid=665982/
solicitationId=%7B21B53A82-2F78-9C26-3CA6-B9FEFE2AF929%7D/viewSolicitationDocument=1/D.10%
20TESS%20Cycle%203%20Final%20text%20101619%20clean.pdf
4
https://heasarc.gsfc.nasa.gov/cgi-bin/tess/webtess/wtv.py?Entry=beta+Pictoris
32 The β Pictoris System
β Pictoris is definitely one of the most studied and intriguing star - planet systems. The
Infrared Astronomical Satellite (IRAS) discovered an infrared excess (Aumann et al. 1984) for
this bright and close southern star, which was attributed to the the presence of a circumstellar
disk. This was first imaged by B. A. Smith & Terrile (1984) and clearly showed the edge-on
geometry of this system (see Figure 1). Such a dust disk should however not be confused with
an optically thick protoplanetary disk. The planets in a “Vega / β Pictoris - like” system are
already formed. The gas and dust is mostly “second generation”, i.e. constantly replenished
by collisions of comets and asteroids (Lagrange et al. 2000). A warp in this disk (Augereau
et al. 2001; Mouillet et al. 1997; Nesvold & Kuchner 2015) and signatures of evaporating
exocomets (also called Falling Evaporating Bodies - FEBs) in spectroscopy (Ferlet et al.
1987; Beust & Morbidelli 2000) were attributed to an exoplanet orbiting the star interacting
dynamically with its environment. The planet β Pictoris b was found using Direct Imaging
with a separation to its star already predicted by dynamical simulations (Lagrange et al.
2009a; Lagrange et al. 2010). Evidence of a second planet detected in the radial velocity of
the star was published very recently by Lagrange et al. (2019a).
The following Subsections will focus on the major components of that system: The star
(Section 2.1), the confirmed planet β Pictoris b (Section 2.2), the candidate planet β Pictoris
c (Section 2.3) and various non-detections (Section 2.4).
2.1 The star: β Pictoris
The star β Pictoris was the first one discovered to exhibit quickly varying absorptions in
the Calcium H&K lines which are nowadays attributed to so-called “FEB activity”, i.e. gas
evaporating from comets that changes the star’s spectrum (Ferlet et al. 1987). Many other
stars have been discovered to show these FEB-like features (Rebollido et al. 2018). Like β
Pictoris, all of these stars are of spectral type A. The young age of β Pictoris with about 23
Myrs was confirmed after it was found to be a part of a moving group, which now carries the
name of the star (Mamajek & Bell (2014) and references therein). Photometric detections of
exocomets were enabled with the start of the era of high precision photometry obtained by
the Kepler and TESS satellites. The host stars are of spectral type F (Rappaport et al. 2018;
Kennedy et al. 2019; Boyajian et al. 2016) except for β Pictoris which is the only A-type star
(Zieba et al. 2019).
Koen (2003) discovered δ Scuti type pulsations at the millimagnitude level originating
from β Pictoris. Further analysis by Mékarnia et al. (2017), Zwintz et al. (2019) and Zieba
et al. (2019) showed dozens of additional frequencies in a range between 20 and 80 cycles per
day. Those pulsations however also induce intrinsic variations in the radial velocity at . 1
km s−1 peak-to-peak (Lagrange et al. 2009b; Lagrange et al. 2012; Galland et al. 2006) which
hampers the search for planets with the radial velocity method in this system.
A selection of fundamental properties of β Pictoris is listed in Table 1. This thesis will also
reanalyse the photometric data collected by the TESS satellite for β Pictoris and provide a
52.2 The planet: β Pictoris b 2 THE β PICTORIS SYSTEM Figure 1: The circumstellar disk around β Pictoris was the first one to be imaged by B. A. Smith & Terrile (1984). frequency table for the pulsational modes in Section 3.4.1. 2.2 The planet: β Pictoris b The warp of the inner disk of β Pictoris observed in 1998 by the Hubble Space telescope (see Figure 2) was one of the indirect hints for a massive companion orbiting the star. Signatures of infalling exocomets in the spectra of the star also needed a “perturber”. The planet, β Pictoris b, was then discovered using the VLT/NaCo instrument data in 2003 (Lagrange et al. 2009a) and was later confirmed by Lagrange et al. (2010). A transit-like event has also been observed in 1981 and attributed to a planet (Lecavelier Des Etangs et al. 1995) however, a better orbit determination with the VLT/SPHERE instrument ruled out β Pictoris b as the cause for it (Lagrange et al. 2019b). Furthermore, using data from the Gemini Planet Imager, Wang et al. (2016) were able to rule out a transit of the planet β Pictoris b during 6
2 THE β PICTORIS SYSTEM 2.2 The planet: β Pictoris b
Table 1: Various stellar parameters of the star β Pictoris.
Parameter Value Source
catalogue information
05h 47m 17.09s
RA (J2000.0) Stassun et al. (2019)
86.8212165826392◦
-51h 03m 59.41s
DEC (J2000.0) Stassun et al. (2019)
-51.0665035714537◦
82:32:37.05
Ecl. lon. (J2000.0) Stassun et al. (2019)
82.5436240336648◦
-74:25:24.63
Ecl. lat. (J2000.0) Stassun et al. (2019)
-74.4235079666848◦
HD ID 39060
HR ID 2020
HIP ID 27321
TIC 270577175 Stassun et al. (2019)
Gaia DR2 ID 4792774797545105664 Gaia Collaboration et al. (2018)
photometric properties
V (mag) 3.86 Ducati (2002)
T (mag) 3.696 Stassun et al. (2019)a
derived properties
age (Myr) 23 ± 3 Myr Mamajek & Bell (2014)
parallax (mas) 50.62(33) Gaia Collaboration et al. (2018)
distance (pc) 19.75(13) Gaia Collaboration et al. (2018)
spectral class A6V Gray et al. (2006)
Radius (R ) 1.497(25) Zwintz et al. (2019)
1.80+0.03
−0.04 Wang et al. (2016)
Mass (M ) 1.797 ± 0.035 Zwintz et al. (2019)
1.84 ± 0.05 Dupuy et al. (2019)
a
Note that T is the TESS magnitude as described in Stassun et al. (2019).
the conjunction in 2017 at a 10σ level. However, a Hill sphere transit5 was predicted for the
time between late 2017 and early 2018 (Lecavelier des Etangs & Vidal-Madjar 2016; Wang
et al. 2016). Various observational campaigns were initiated by e.g. bRing (Kenworthy 2017)
and the BRITE constellation (Weiss et al. 2014) in order to photometrically observe possible
material around the planet; however without any significant detection. The mass and the
orbital solution of β Pictoris b from various sources are listed in Table 2.
5
The Hill sphere is the region around a planet where masses - like moons and planetary rings - are
gravitationally bound to it.
72.3 The planet candidate: β Pictoris c 2 THE β PICTORIS SYSTEM
Figure 2: The warp in the dust disk around the star β Pictoris as seen by the Hubble Space
Telescope. Credit: A. Schultz (Computer Sciences Corp.), S. Heap (NASA/ESA Goddard
Space FlightCenter) and NASA/ESA
2.3 The planet candidate: β Pictoris c
Evidence of an additional planet in the β Pictoris system was recently published by Lagrange
et al. (2019a). Over 6000 spectra of the star taken between 2003 and 2018 by the HARPS
instrument at the ESO La Silla 3.6 m telescope have been analyzed. After removing the
stellar pulsations, the periodogram of the radial velocity (RV) curve shows a peak which
is hard to explain if one does not assume a second planet in the system. If confirmed, this
would mark the first RV detected planet orbiting an A-type star. The difficulty of discovering
planets around such a star is further discussed in Section 4.8. The planet is predicted to be
closer to the star than β Pictoris b. A list of the mass and orbital parameters is given in
Table 3.
Using the semi-major axis given in Table 3 and assuming a coplanar configuration for the
two planets in the system (i = 88.81◦ ), it is evident that β Pictoris c is a non-transiting
planet just like β Pictoris b. Wang et al. (2016) found that for β Pictoris b to transit an
inclination of |i − 90◦ | < 0.05◦ would be needed. The value for β Pictoris c - by applying
simple geometry - is around |i − 90◦ | < 0.2◦ .
82 THE β PICTORIS SYSTEM 2.3 The planet candidate: β Pictoris c
Table 2: Various parameters of the planet β Pictoris b.
Parameter Value Source
Properties
12.7 ± 0.3 Morzinski et al. (2015)
12.9 ± 0.2 Chilcote et al. (2017)
Mass (MJ ) 11 ± 2 Snellen & Brown (2018)a
13.1+2.8
−3.2 Dupuy et al. (2019)
9.9 Lagrange et al. (2019a)
Orbital Parameters
>22.2 (2σ) Snellen & Brown (2018)
Period (years) 20.29+0.86
−1.35 Lagrange et al. (2019b)
29.9+2.9
−3.2 Dupuy et al. (2019)
9.66+1.12
−0.64 Wang et al. (2016)
a (AU) 8.90+0.25
−0.41 Lagrange et al. (2019b)
11.8+0.8
−0.9 Dupuy et al. (2019)
0.08+0.091
−0.053 Wang et al. (2016)
e 0.01+0.029
−0.01 Lagrange et al. (2019b)
0.24 ± 0.06 Dupuy et al. (2019)
88.81+0.12
−0.11 Wang et al. (2016)
i 89.08+0.16
−0.19 Lagrange et al. (2019b)
88.87 ± 0.08 Dupuy et al. (2019)
205.8+52.6
−13 Wang et al. (2016)
$ (◦ ) -170 Lagrange et al. (2019b)
189.3+3.0
−2.9 Dupuy et al. (2019)
τ 0.73+0.14
−0.41 Wang et al. (2016)b
31.76+0.08
−0.09 Wang et al. (2016)
Ω (◦ ) -148 Lagrange et al. (2019b)
31.65 ± 0.09 Dupuy et al. (2019)
JD 2,355,992 Wang et al. (2016)
tp year 2003 Lagrange et al. (2019b)
JD 2,456,380+80−60 Dupuy et al. (2019)
a
Note that the results given by Snellen & Brown (2018) assume only one
planet in the system.
b
The reference epoch for Wang et al. (2016) is MJD 55000 (priv. comm.),
which can be then used to derive the time of periastron tp .
a ... major semi axis
e ... eccentricity
i ... inclination
$ ... argument of periastron
τ ... epoch of periastron
Ω ... positional angle of the ascending node
tp ... time of periastron
Using the full orbital solution for β Pictoris c given in Lagrange et al. (2019a) it is possible
92.4 Non-detections in the system 2 THE β PICTORIS SYSTEM
Table 3: Various parameters
of the candidate planet β Pic-
toris c following Lagrange et
al. (2019a).
Parameter Value
Properties
Mass (MJ ) 8.9
Orbital Parameters
Period (days) 1220
a (AU) 2.7
e 0.24
$ (◦ ) -95
tp (JD-2,450,000) 4117
a ... major semi axis
e ... eccentricity
i ... inclination
$ ... argument of periastron
tp ... time of periastron
to predict an approximate time of inferior conjunction. If the inclination is favourable - as
described above - this yields the transit time. Due to the uncertainties in the parameters a
precise value cannot be given as noted in Lagrange et al. (2019a). The equations needed to
calculate the transit time from the orbital parameters is given in Appendix C. Figure 3 shows
the radial velocity curve of β Pictoris c using the values in Table 3 and Equation 10. The
best solution given in Lagrange et al. (2019a) leads to a transit time that falls approximately
four days after the start of observation of β Pictoris by TESS in its primary mission (see
Figure 3). Assuming an approximate radius of 1 RJ for β Pictoris c and a radius of around
1.5 R for the star (Zwintz et al. 2019) the approximate expected transit depth is 2.5 %.
Such a dip cannot be observed in the beginning of the TESS observations. But due to the
uncertainties in the orbital parameters this does not conclusively rule out the possibility of
β Pictoris c being a transiting planet.
2.4 Non-detections in the system
Due to the frequent spectroscopic and photometric observations of the star β Pictoris, upper
limits to not-yet discovered planets in the system can be given.
Using photometric observations collected by the BRITE constellation, Mol Lous et al.
(2018) were able to rule out an inner planet larger than 0.6, 0.75, 1.05 RJ for a major semi
axis less than 0.07, 0.11, 0.18 AU, respectively.
Combining the HARPS data with high resolution imaging from the VLT/NaCo instrument,
102 THE β PICTORIS SYSTEM 2.4 Non-detections in the system
Time [yr]
2018 2019 2020 2021 2022
200
100
RV [m/s]
0
RV Pic c
100 TESS (pm)
TESS (em)
pred. transit
200
4200 4400 4600 4800 5000 5200 5400
Time [JD - 2 454 000]
Figure 3: The radial velocity curve and the predicted transit time of β Pictoris c using the
orbital parameters given in Table 3. The TESS observations of β Pictoris are marked in
blue (for the primary mission) and in orange (for the expected observations in the extended
mission).
planets heavier than 4 MJ closer than 1 AU and further away than 10 AU were ruled out
(Lagrange et al. 2018; Lagrange et al. 2019a). This is consistent with earlier constraints given
by Galland et al. (2006). A further analysis by Kervella et al. (2019) of the astrometric data
collected by the Hipparcos and Gaia missions show the allowed mass and period regime for
companions other than β Pictoris b around the star.
113 Observations
Due to anticipated 2017 - 2018 Hill Sphere Transit of β Pictoris b an international campaign
of space and ground-based observations was launched in order to search for signatures of
material around the giant planet (Kalas et al. 2019). Table 4 summarizes various properties
of the different provided light curves. Changes to those light curves other than the Gaussian
highpass procedure - which will be explained in Section A.2 - are noted in the corresponding
Subsections (3.2.1, 3.3.1, 3.4.1). The data provided by the BRITE-constellation was left
unchanged. A detailed analysis of the photometry of β Pictoris collected by the BRITE-
constellation and bRing was published by Zwintz et al. (2019).
Table 4: A summery of the properties of the various instruments and corresponding light curves:
T denotes the timebase of the observations, the reciprocal value 1/T corresponds to the Rayleigh
criterion. fNy. is the Nyquist frequency and DC the duty cycle.
Wavelength Observation Observation T 1/T fNy. cadence DC
Observation
(nm) start end (days) (10−3 d−1 ) ( d−1 ) (s) (%)
BHr 550 - 700 16 March 2015 2 June 2015 78.32 12.77 4167 10.37 6.78
BTr + BHr 550 - 700 4 Nov 2016 17 June 2017 224.6 4.453 2128 20.30 7.07
BHr 550 - 700 9 Nov 2017 25 April 2018 167.3 5.976 2128 20.30 7.48
bRing 463 - 639 2 Feb 2017 1 Sept. 2018 575.5 1.738 135.4 319.1 27.0
ASTEP17 695 - 844 28 March 2017 14 Sept. 2017 170.0 5.881 495.8 87.13 18.9
ASTEP18 695 - 844 28 March 2018 15 July 2018 109.3 9.150 502.8 85.92 29.2
TESS 600 - 1000 19 Oct 2018 1 Feb. 2019 105.2 9.507 360.0 120.0 85.3
Note that BRITE Lem (BLb) - equipped with a blue filter - also observed β Pictoris from December 2016
until June 2017 but due to significantly higher noise in the time series, the data was regarded from the
analysis. See Zwintz et al. (2019) for an analysis of the BLb observations.
3.1 BRITE-Constellation
The BRITE-Constellation (Weiss et al. 2014) consists of five nanosatellites6 collecting pho-
tometry for the brightest stars on the sky. Being in a low-earth orbit the orbital periods of
the satellites are all around 100 minutes. At minimum of 15 minutes per orbit are dedicated
to observations.
Three different runs where conducted in the constellations around Pictor and Vela which
also included the star β Pictoris. A summary of the durations and various properties of those
observations can be found in Table 4. The pipeline for the photometry reduction is described
in Popowicz et al. (2017).
An analysis of all BRITE observations was conducted in Zwintz et al. (2019). For the three
runs by BHr, BTr+BHr and BHr which all used the red BRITE filter, 6, 13 and 8 significant
frequencies were extracted, respectively. The only run with a blue filter by BLb suffered
from higher noise compared to the other BRITE-observations. Zwintz et al. (2019) reports
4 frequencies for the collected photometry. The blue observations were therefore discarded
from this analysis.
6
For the abbreviated designations of the in this work relevant satellites see Appendix D.
133.2 bRing 3 OBSERVATIONS Figure 4: The full light curve of all available observations of the star β Pictoris used in this work. 3.2 bRing bRing, standing for “the β Pictoris b Ring project”, was initiated in order to collect pho- tometry of β Pictoris during the Hill sphere transit of β Pictoris b at the end of 2017 (Stuik et al. 2017). For that, two stations in South Africa and Australia were built each consisting of two wide field cameras. Their design is based on the Multi-Site All-Sky CAmeRA (MAS- CARA) (Snellen et al. 2012; Talens et al. 2017). The capability of bRing to monitor bright stars and to find previously unknown variables was recently shown by Mellon et al. (2019). More information on the observing strategy and design of bRing can be found in Stuik et al. (2017). The reduction pipeline for the MASCARA and bRing instruments is described in Talens et al. (2018). With a passband of 463 - 639 nm, bRing is the bluest of all in this work considered observatories so we expect to see the highest pulsational amplitudes in these data. 3.2.1 Data Reduction Due to some evident outliers in the data, one 5-σ clip with respect to the median of the dataset was applied. This significantly weakens the one day aliases in the spectral window. An iterative sigma clipping procedure was not conduced due to noticeable changes in the amplitudes of the pulsations in this case. A discussion of sigma clipping in order to remove outliers can be found in Hogg et al. (2010). The observations by bRing were separated into two equally sized segments to gain more time delay measurements and keeping the precision in frequency and phase comparable to the ASTEP observations. 14
3 OBSERVATIONS 3.3 ASTEP
Zwintz et al. (2019) found 6 significant frequencies in the photometry collected by bRing.
All of them can be also found in the data collected by BRITE, ASTEP and TESS.
3.3 ASTEP
ASTEP, standing for the Antartica Search for Transiting Extrasolar Planets, is an autom-
atized telescope with an aperture of 40 cm located at the Concordia station at Dome C in
Antarctica (Abe et al. 2013; Guillot et al. 2015; Mékarnia et al. 2017). It uses a Sloan i’ filter.
3.3.1 Data Reduction
Only measurements with a sun elevation lower than -18◦ were used. Furthermore, datapoints
where the centroid of the star did not not fall on the central pixel suffer from strong outliers.
The removal of those and a 5-σ clip with respect the median weakens aliases significantly
without noticeable changes in the amplitude of the strongest pulsational frequencies.
Mékarnia et al. (2017) conduced a frequency analysis of the β Pictoris photometry collected
by the ASTEP observatory. The 31 significant frequencies identified in that work was the
longest list of pulsational frequencies of β Pictoris prior to the TESS observations.
3.4 TESS
The Transiting Exoplanet Survey Satellite (TESS; Ricker et al. 2015) was launched in April
2018 in order to find transiting exoplanets around nearby, bright stars. It has completed its
survey of the southern ecliptic hemisphere within its main mission and is currently (i.e., as of
October 2019) observing the northern hemisphere. The data of β Pictoris (TIC 270577175,
T = 3.696 mag) was collected from 19 October 2018 to 1 February 2019 in the sectors 4
through 7. Those four sectors make up the total observation of β Pictoris during TESS’ main
mission. The star will however be reobserved in the extended mission starting in mid 2020
(see Section 5.3 for a further discussion of the extended mission). β Pictoris is one of the
preselected targets for which short cadence (2 minute) data are provided. This Candidate
Target List (CTL) is a subset of the TESS Input Catalogue (TIC; Stassun et al. 2019) with
about 200,000 targets for TESS 2 min cadence observations, which were primarily chosen in
order to maximise the yield of transiting exoplanets. Due to this high cadence data, the high
photometric precision of TESS, its high duty cycle and the long baseline, δ Scuti pulsations
can by resolved and identified to a high precision. A frequency analysis of the pulsations
was conducted using the pipeline light curve (PDCSAP) available on the MAST archive7 in
Section 3.4.1. A second analysis (see Section A.2) using a Gaussian highpass filtered light
curve in order to remove long term trends and produce a light curve better suited for the
search of transiting exoplanets (see Section B) can be found in the appendix of this work.
7
https://archive.stsci.edu/tess/
153.4 TESS 3 OBSERVATIONS
3.4.1 Data reduction and frequency analysis
The photometric data of β Pictoris as observed by TESS was accessed and modified with
the Python package lightkurve (Lightkurve Collaboration et al. 2018), which retrieves the
data from the MAST archive7 . For stars on the CTL additional data products are provided.
Those are the Target Pixel Files (TPF), Simple Aperture Photometry (SAP) light curves
and Presearch Data Conditioning Simple Aperture Photometry (PDCSAP; J. C. Smith et al.
2012; Stumpe et al. 2012) light curves. The latter are produced from the Science Processing
Operations Center (SPOC) pipeline (Jenkins et al. 2016; Jenkins 2017), which was originally
developed for the Kepler mission (Jenkins et al. 2010). These PDCSAP light curves were
corrected for systematics by the SPOC pipeline.
The full SAP and PDCSAP light curves of β Pictoris can be seen in Figure 5. The red
triangles mark momentum dumps, i.e. thruster firings that reduce the speed of the reaction
wheels and decrease the pointing jitter. During this time TESS loses the fine attitude control
mode for about 15 min (see the TESS Instrument Handbook8 ). Consequently, the pointing
is less stable during these particular times, resulting in potential changes in the photometric
fluxes of the target stars.
The beginnings of each sector are indicated with a vertical dashed red line. The visible
gaps in the light curve are related to the data downlink at the perigee of the orbit of TESS,
which occurs every 13.7 days and during which TESS halts its observations for about one
day. The PDCSAP are designed to remove - especially long term - systematic trends in the
data. This can be seen in three days of sector 4 between BTJD 1421 and BTJD 1424 owing to
higher than normal rates of spacecraft jitter. This occurred just after an instrument anomaly
(see the Data Release Note of sector 4 for more information9 ).
However, PDCSAP does not always outperform the SAP light curve. This can specifically
be seen in the end of sector 4 where the flux of the PDCSAP light curve increases where the
SAP light curve stays flat. Such shortcomings of the pipeline will be hopefully be mitigated
in the future after we learn more about the systematics of TESS and new pipelines will be
developed and tested. This shows the importance of comparing all available light curves in
order to rule out the possibility of a flaw in the pipeline.
8
https://archive.stsci.edu/missions/tess/doc/TESS_Instrument_Handbook_v0.1.pdf
9
https://archive.stsci.edu/tess/tess_drn.html
163 OBSERVATIONS 3.4 TESS
Figure 5: The full TESS SAP light curve (blue in the background) and the PDCSAP light
curve (black). Upper panel: The full flux range is shown in order to show the extend of the
systematic event in the middle of sector 4 which is clearly visible in the SAP light curve.
Lower panel: A zoom of the light curves showing the full PDCSAP light curve. The blue
triangles mark the identified exocomets by Zieba et al. (2019), red triangles the times of
momentum dumps and the green triangle marks a change in the attitude of the satellite due
to an updated startable.
The TPFs were visually inspected in order to rule out various instrumental and astrophys-
ical effects like solar system asteroids or comets crossing the field of view, “CCD crosstalk”
or “rolling CCD bands”.
A comparison of the Lomb Scargle periodogram (Lomb 1976; Scargle 1982) of the SAP
and PDCSAP light curve can be seen in Figure 6. The upper two panels include sector 4
and the lower two exclude sector 4. One can clearly see a significant change in the noise at
173.4 TESS 3 OBSERVATIONS
low frequencies. This is due to the systematic effects present in this sector. Figure 7 shows
a zoom into the region between 0 and 6 cycles per day. The lowest noise in this frequency
range can be found for the PDCSAP light curve with a completely removed fourth sector.
This light curve was then used for the main frequency analysis.
The individual sectors were normalized by dividing each of the sectors by their respective
median flux and combined into one light curve. Furthermore, every measurement with a non-
zero quality flag (see Sect. 9 in the TESS Science Data Products Description Document)10
was removed. They mark anomalies like cosmic ray events or instrumental issues.
1000
SAP PDCSAP 1.09
800 0.87
Amplitude [mmag]
Amplitude [ppm]
600 0.65
400 0.43
200 0.22
0 0.00
1000 1.09
800 0.87
Amplitude [mmag]
Amplitude [ppm]
600 0.65
400 0.43
200 0.22
0 0.00
0 20 40 60 80 100 0 20 40 60 80 100
Frequency [d 1] Frequency [d 1]
Figure 6: Amplitude spectra of the SAP (left column) and PDCSAP (right column) light
curve. Upper panel: The amplitude spectrum of all four available sectors. Lower panel: The
amplitude spectrum of the TESS light curve without sector four which suffers from systematic
effects as visible in Figure 5.
10
https://archive.stsci.edu/missions/tess/doc/EXP-TESS-ARC-ICD-TM-0014.pdf
183 OBSERVATIONS 3.4 TESS
400
SAP PDCSAP 434
Amplitude [ mag] Amplitude [ mag] Amplitude [ mag]
Amplitude [ppm] Amplitude [ppm] Amplitude [ppm]
300 326
200 217
100 109
600 065.1
40 43.4
20 21.7
600 Frequency [1d ] Frequency [1d ]
0.0
65.1
40 43.4
20 21.7
0 0.0
0 1 2 3 4 5 0 1 2 3 4 5
Frequency [d 1] Frequency [d 1]
Figure 7: Same as Figure 6 with a zoom towards lower frequencies. Upper panel: Amplitude
spectrum of all four sectors. Middle panel: Amplitude spectrum after removing the instru-
mental event in sector four between BTJD 1421 and 1424. Lower panel: Amplitude spectrum
after removing the whole fourth sector. It clearly shows the lowest noise at low frequencies
and was therefore used for the main frequency analysis.
193.4 TESS 3 OBSERVATIONS
1.0
0.8
0.6
0.4
0.2
0.0
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Frequency [d ]
1
Figure 8: The spectral window of the TESS light curve of β Pictoris.
The frequency analysis was conducted using the Python package SMURFS11 and checked
with the software package Period04 (Lenz & Breger 2005). All pulsation frequencies down
to a signal to noise ratio of 4 following Breger et al. (1993) were extracted. The frequency
range analysed is between 0 and the Nyquist frequency of 360 cycles per day. From this, 37
significant p-modes in the frequency range from 34 to 76 d−1 were identified. The extracted
frequencies are marked in red in Figure 9 and the corresponding residual amplitude spectrum
can be seen in Figure 10 which clearly illustrates that additional pulsational signal is likely
still buried in the noise. A one-day zoom into the light curve illustrating the pulsational
behaviour is shown in the upper panel of Figure 11. A full list of the extracted frequencies
can be found in Table 6.
11
https://github.com/MarcoMuellner/SMURFS
203 OBSERVATIONS 3.4 TESS
1.0
0.8
Amplitude [mmag]
0.6
0.4
0.2
0.0
0 10 20 30 40 50 60 70 80 90 100
Frequency [d 1]
Figure 9: Pulsation frequency analysis of β Pictoris. The original amplitude spectrum is
shown in black and the 37 identified δ Scuti pulsations are shown in red.
30 32.6
25 27.1
20 21.7
Amplitude [ mag]
Amplitude [ppm]
15 16.3
10 10.9
5 5.4
0 0.0
0 10 20 30 40 50 60 70 80 90 100
Frequency [d 1]
Figure 10: The residual amplitude spectrum after prewhitening the 37 identified frequencies.
213.4 TESS 3 OBSERVATIONS
6
4
inst. mag [mmag]
2
0
2
4
6
2
residuals
0
2
1472.0 1472.2 1472.4 1472.6 1472.8 1473.0
Time - 2457000 [BTJD days]
Figure 11: One-day zoom of the β Pictoris light curve. Upper panel: TESS photometric
time series (red points) and multi-sine fit using the 37 identified δ Scuti frequencies. Lower
panel: residual time series after subtracting the multi-sine fit using all 40 identified pulsation
frequencies. One still clearly sees high frequency variations in the residuals indicating that
more δ Scuti pulsations are present.
Figure 12: The residuals of the PDCSAP light curve after removing the 37 identified δ Scuti
frequencies. The systematic effects in Sector 4 - which was excluded from the frequency
analysis - are clearly visible. The three exocomets identified by Zieba et al. (2019) are
marked with blue triangles. Plots showing the residuals for every individual sector are shown
in Figure 30 (for the SAP light curve) and Figure 31 (for the PDCSAP light curve).
224 Theory and Methodology
In this chapter we want to discuss the theory behind time delays and the methods used in
order to finally arrive at the time delay plot which can be used to search for companions
around pulsating stars.
4.1 δ Scuti stars
δ Scuti stars can be found at the intersection region between the main-sequence and the
instability strip on the Hertzsprung-Russel diagram (see Figure 13). Thanks to the nearly
uninterrupted, high-precision photometry of Kepler’s main 4-years mission our understanding
of pulsating stars has been revolutionized. It was for example shown that no more than
50% of the stars in this region of the Hertzsprung Ruessel diagram are pulsating (Balona
& Dziembowski 2011). At the middle of the instability strip the peak with 70% is reached
(Murphy et al. 2019). Those δ Scuti stars have masses between 1.5 and 2.5 M . They
pulsate in radial and non-radial, low-degree, low-order, pressure (p) modes and are excited
by an opacity mechanism (also called κ-machanism) in their HeII zone. Those oscillations
have periods between 18 minutes and 8 hours respectively 3 and 80 cycles per day (Aerts
et al. 2010). Linear combinations of those oscillations can however create peaks at lower
frequencies (Breger & Montgomery 2014). Such linear combinations can also be seen in the
case of δ Scuti stars (see Table 7 for possible combinations).
Murphy et al. (2019) were recently able to show a discrepancy between the theoretical
and observed δ Scuti instability strip and argue that this may be caused by contribution of
turbulent pressure to pulsational driving. According to this study, the δ Scuti population can
be found around A3-F0 type stars. Besides main-sequence and more evolved stars, δ Scuti
pulsations were observed in pre-main-sequence stars, thus giving us the possibility to learn
about early stellar evolution (Zwintz et al. 2014).
4.2 The ephemeris equation
The search for time delays in certain astrophysical signals requires a (quasi-)periodic process
in space. A review on this and the related equations can be found in Hermes (2018). There
are different processes which are “clock-like” under the assumption of a closed system: the
exceptional stable signals of pulsars, the eclipse time of binary stars or certain pulsating
stars as in our case. Those processes can be described by a simple linear equation called the
ephemeris equation:
TE = T0 + P ∗ E, (1)
where T0 is the midpoint time for epoch E = 0 for a regular process with period P . TE then
gives us the midpoint time for a given epoch E. However, no process in space perfectly follows
Equation 1. In the case of pulsars the deviation from this equation is due to a rotational spin
234.2 The ephemeris equation 4 THEORY AND METHODOLOGY
Spectral type
O B A F G K M
6
PVSG
5
20 M
RSG
12 M β Cep δ Cep
4
DOV 7 M
3 SPB Mira
4 M
Semi-
2 BLAP 3 M Regular
log (L / L )
RR Lyr
2.1 M Red Giant
1 sdBV δ Sct
γ Dor
Solar-like
0 1 M
DBV
-1
-2
Z=0.02 DAV
5.0 4.8 4.6 4.4 4.2 4.0 3.8 3.6 3.4
log (Teff / K)
Figure 13: The asteroseismic Hertzsprung Ruessel diagram. Backslash (\) filled areas are
excited by pressure (p) modes and slash (/) filled areas by gravity (g) modes. The dashed
curve represents the zero age main-sequence (ZAMS). A 2.1 M MESA model from the ZAMS
to a white dwarf is shown by the solid purple line. δ Scuti stars can be found at the
intersection region between the main-sequence and the instability strip. Figure from Paxton
et al. (2019) and design by Papics (2013).
244 THEORY AND METHODOLOGY 4.3 O-C diagrams
down, mainly caused by its rotating magnetic field. The ephemeris equation considering the
change of the period with time is then described by:
1
TE = T0 + P ∗ E + Ṗ P̄ E 2 , (2)
2
where P̄ is the average period over the observed time interval. Finally, we can consider
the case that the periodic process is convolved with a cyclic or quasi-sinusoidal motion with
period Porb :
1 2πE
TE = T0 + P ∗ E + Ṗ P̄ E 2 + A sin +φ , (3)
2 Porb
where A describes the amplitude and φ the phase of the sinusoidal variation.
4.3 O-C diagrams
Equation 3 is well established nowadays and can be used to analyse the spin down of
pulsars or to discover companions around pulsars (Wolszczan & Frail 1992; Wolszczan
1994), eclipsing binaries (Barnes & Moffett 1975) or pulsating stars (Silvotti et al. 2007)
(for a review of pulsating stars in binary systems see Murphy 2018). This is done by
creating O-C (observed minus calculated) diagrams (Sterken 2005) in order to search for
deviations from the predicted ephemeris in the observations. This led to the discovery of a
binary pulsar system (Hulse & Taylor 1975). This discovery, which also gave the first indi-
rect hint for gravitational waves, was awarded with the nobel price in physics in the year 1993.
O-C diagrams were used in order to find indications of a planetary companion around the
sub dwarf B-type (sdB) star V391 Pegasi (Silvotti et al. 2007). However, the analysis of 13
years of photometric data by Silvotti et al. (2018) seem to refute this detection and the status
of this planet was downranked to “putative”. This shows that finding low mass companions
around pulsating stars is not an easy task. O-C diagrams work the best if the star is pulsating
in only a single mode and if the maxima are sharp. The pulsation maxima are easy to track
in that case. They struggle however especially with multi-mode pulsators.
4.4 Frequency modulation and Phase modulation: state-of-the-art
Building on those established methods of O-C diagrams two new and complementary tech-
niques arose in order to find companions around pulsating stars. The frequency modulation
method (FM) first described in Shibahashi & Kurtz (2012) searches and analyses the varia-
tions in the frequency of the pulsating star induced by a companion. The periodic frequency
modulation creates multiples around every pulsation peak in the frequency spectrum. Their
frequency and relative height and phase can be used to get the full orbital solution as de-
scribed in Shibahashi et al. (2015). Its effectiveness was validated by comparison with a
eclipsing binary system (Kurtz et al. 2015). The FM method is best suited for data sets with
a baseline which exceeds the orbital period of the companion.
254.5 Time Delays 4 THEORY AND METHODOLOGY
The phase modulation (PM) method developed by Murphy et al. (2014) and improved in
Murphy & Shibahashi (2015) and Murphy et al. (2016b) works better for companions in wider
orbits. Compton et al. (2016) showed that δ Scuti stars and white dwarfs are best suited
for this PM method. Its effectiveness was demonstrated by Schmid et al. (2015). They were
able to show the binary nature of KIC10080943 using the PM method and attribute certain
pulsations to the corresponding star in the binary due to the antiphase modulation in the
time delays. Such a system with observable time delays in both components is called PB2.
This is in analogy to spectroscopy, where binary star systems are called SB2s if both stars
show observable radial velocities. Other proof of the functionality of the PM method was
shown by Derekas et al. (2019) by comparing the orbital parameters derived from RV with
those from PM. They were in good agreement and a combined analysis of the two methods
was conducted in order to determine more accurate parameters.
Finally, Murphy et al. (2016a) discovered a massive planet (m sin i ≈ 12MJ ) with an
orbital period of about 840 days in the Kepler 4-year main field data around a δ Scuti star.
However, due to the lack of knowledge of the orbital inclination it might even be a brown
dwarf. Nevertheless, this is the only convincing discovery of a low mass companion with a
pulsation timing variation method.
An additional advantage of the PM method is its easier automatization for many stars.
Applying this method on 2224 main-sequence A/F stars in the 4-year main Kepler data,
Murphy et al. (2018) was able to find 317 PB1 systems, where only one component is pulsating
and showing time delays, and 24 PB2 systems, where two stars are pulsating. It is worth
noting here that archiving orbital solutions using spectra and generating radial velocity curves
for the same amount of stars would be much more time intensive.
Other methods developed by Koen (2014) and Balona (2014) were also developed to search
for binary systems by tracing the δ Scuti pulsations of stars. They are not able however, in
contrast to the FM and PM method, to provide a full orbital solution which is usually gained
by analysing radial velocity (RV) curves of spectroscopic binaries.
4.5 Time Delays
Time delays arise when a signal (in our case always an electromagnetic wave with the prop-
agation velocity defined by the speed of light) has to travel different distances at different
times. Following Smart (1977) and Balona (2014), the distance r between the pulsating star
and the center of gravity of its system can be described by
a1 1 − e 2
r= (4)
1 + e cos f
where a1 denotes the major-semi axis of the star, e its eccentricity and f the true anomaly.
The distance to the star varies relative to the earth by
z = r sin(f + $) sin i (5)
264 THEORY AND METHODOLOGY 4.5 Time Delays
with $ being the the argument of periapsis, i.e. the angle between the nodal point and the
periapsis12 and i the inclination of the system.
We can now insert Equation 4 in Equation 5. The time delay τ = −z/c is then completely
described by the following equation:
a1 sin i sin f cos $ + cos f sin $
τ (t, x) = − (1 − e2 ) . (6)
c 1 + e cos f
The set x = (Ω = 2π/P, a1 sin i/c, e, $, tp ) in Equation 6 includes all system specific
parameters which are needed to describe the time delay for a given time t. P is the orbital
period of the system, or equivalently 1/P = νorb the orbital frequency and thus Ω the
angular orbital frequency. The projected semi-major axis of the pulsating star is described
by a1 sin i. Dividing this quantity by the speed of light c gives us the size of the orbit for the
pulsating star in light seconds. The argument of periapsis is described by $ and the time of
periapsis passage by tp . For a graphical visualisation of the orbital parameters see Murphy
& Shibahashi (2015).
The two trigonometric functions of the true anomaly, sin f and cos f , can be expressed in
terms of series expansions and Bessel functions:
∞
2 1 − e2 X
cos f = −e + Jn (ne) cos nΩ (t − tp ) (7)
e
n=1
p ∞
X
sin f = 2 1 − e 2 Jn0 (ne) sin nΩ (t − tp ) (8)
n=1
with Jn0 (x) = dJn (x)/dx (the derivation of Equation 7 and 8 can be found in appendix A of
Shibahashi et al. 2015) As it turns out, the changing distances between us and the clock in
space are deeply connected with varying radial velocities:
dτ
vrad = c . (9)
dt
Inserting Equation 6 in Equation 9 gives us:
Ωa1 sin i
v rad = − √ [cos(f + $) + e cos $] (10)
1 − e2
where vrad is the radial velocity, c the speed of light and τ the time delay (following the sign-
convention introduced in Murphy & Shibahashi (2015) as seen in Table 5). Given Equation
9 and the convention that a positive radial velocity corresponds with an receding object and
a negative with an approaching one, we can deduce the following: a negative time delay is
due to an early arrival of the signal, i.e. the star is closer to us and visa versa13 .
12
The argument of periapsis is usually denoted with ω. The latter symbol is however used in asteroseismol-
ogy to denote the angular oscillation frequency. One should also not confuse $ with the longitude of periapsis
which is the sum of the longitude of the ascending node Ω and the argument of periapsis.
13
This convention for the time delays was established with Murphy & Shibahashi (2015). As (Murphy et al.
2014) uses reversed signs, their plots there are basically upside down.
274.6 Phase Modulation Method: Methodology 4 THEORY AND METHODOLOGY
Table 5: Sign convention for the radial velocity vrad and the time delays τ .
+ -
vrad moving away approaching
τ further away / late arrival closer / early arrival
We see in Equation 6 and 10 that the time delay as well as the radial velocity of a system
can be completely described by the orbital parameters. If we obtain those parameters by
one method we can predict what we should observe with the other one. Furthermore, if we
generate the time delay plot from our observations, we can apply a chi-squared minimization
technique in order to get the parameters in set x. This idea was introduced with Murphy
& Shibahashi (2015) and is a major improvement to Murphy et al. (2014) where the time
delay measurements were numerically differentiated in order to derive the parameters from
the obtained radial velocity curve.
Finally, by using two of the derived orbital parameters, a1 sin i/c and Porb , we can calculate
the mass function f (m1 , m2 , sin i) for the binary system:
3
(m2 sin i)3 4π 2 c3 2
a1 sin i
f (m1 , m2 , sin i) := = v orb (11)
(m1 + m2 )2 G c
with m2 being the mass of the (usually non - pulsating) companion and G the gravitational
constant.
4.6 Phase Modulation Method: Methodology
Before we can create the time delay plot, we have to analyse the change in phase of the
various pulsation modes with time. The basic equations for that can be found in Murphy
et al. (2014) and are reviewed in the following. We start by dividing the light curve in n
equally sized segments. Then, we calculate the phase in every bin for each frequency. This
leaves us with a series of phases Φj for every bin (1, 2, . . . , n) for a fixed frequency νj :
Φj = [φ1j , φ2j , . . . , φij , . . . , φnj ] (12)
Numerically, the phase in a segment is derived by calculating the argument of the Fourier
Transformation in the respective bin:
−1 Im(F(t; ν, δt))
Φ(t; ν) = tan , (13)
Real(F(t; ν, δt))
where F(t; ν, δt) is the value of the Fourier Transformation of the time series for frequency
ν in segment δt. As phases are frequency dependent they will have different amplitudes for
different frequencies. To get rid of this effect we first calculate the relative phase shifts:
∆φij = φij − φj , (14)
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