Modelling Β Pictoris' Pulsations to Weigh Its Giant Planet

Modelling β Pictoris’ pulsations to weigh its giant planet

M.Sc. Research in Astronomy - Cosmology: Minor Research Project

Author: Martijn S. S. L. Oei B.Sc.

Supervisor: Prof. dr. Ignas A. G. Snellen

Last revision: July 31, 2016 Title page background: Visualisation of asteroseismological pulsations in a star orbited by a hot, rocky planet. Credits: Gabriel P´erez D´ıaz,Instituto de Astrof´ısica de Canarias (Servicio MultiMedia)

Author contact details:

Martijn Simon Soen Liong Oei B.Sc. Leiden Observatory, Leiden University, The Netherlands

E-mail: [email protected] [email protected]

Website: home.strw.leidenuniv.nl/~oei

1 Contents

1 Introduction 11 1.1 The exoplanet revolution ...... 11 1.2 Planets around massive stars ...... 11 1.3 Asteroseismology ...... 12 1.4 Aims of this work ...... 12 1.5 Case study: Beta Pictoris ...... 13

2 Pulsational noise on stellar radial velocities 14 2.1 Data set ...... 14 2.2 From spectra to radial velocities ...... 15 2.3 Reducing the measured spectra ...... 15 2.4 Reducing the reference spectrum ...... 16 2.5 The spectral cross correlation function ...... 17 2.5.1 Deﬁning cross correlation ...... 17 2.5.2 Deﬁning autocorrelation ...... 17 2.5.3 Types of spectral CCFs ...... 18 2.6 Creating measured spectral CCFs ...... 18 2.7 Determining radial velocities ...... 20 2.7.1 Approximating the no-pulsation MCCF ...... 21 2.7.2 Fitting the no-pulsation MCCF to other MCCFs ...... 21 2.8 High frequency variations ...... 21

3 Building a stellar model 23 3.1 Model idealisations ...... 23 3.2 Coordinate system conventions ...... 24 3.3 From observer plane to stellar surface ...... 25 3.4 The stellar coordinate axes ...... 26

3.5 From cartesian ΣW coordinates to spherical ΣS coordinates ...... 28 3.6 Pulsational velocity ...... 29 3.7 Rotational velocity ...... 30 3.8 Radial velocity ...... 31

2 3.9 Limb darkening ...... 31 3.10 Rest frame spectrum ...... 32 3.11 Observable spectrum ...... 35 3.12 Spectral cross correlation function ...... 36 3.13 Spectral cross correlation function residuals ...... 37 3.14 Physical parameter overview ...... 38 3.14.1 Basic physical parameters ...... 38 3.14.2 Pulsational and starspot-related physical parameters ...... 38

4 Pulsational eﬀects on the spectral CCF 40 4.1 Gaussian blobs ...... 41 4.2 An equatorial blob moving along ˆr ...... 42 4.3 An equatorial blob moving along φˆ ...... 44 4.4 An equatorial blob moving along θˆ ...... 46 4.5 Sine waves ...... 46 4.6 A sine wave moving along ˆr ...... 48 4.7 A sine wave moving along φˆ ...... 51

5 Attenuating pulsational noise 53 5.1 Determining stellar parameters ...... 53 5.1.1 Apparent equatorial rotational velocity ...... 53 5.1.2 Fitting the no-pulsation MCCF ...... 55 5.2 Creating SCCRs ...... 57 5.2.1 Superposition of 2 waves ...... 57 5.2.2 Superposition of 3 or more waves ...... 58 5.2.3 Linearity of SCCRs ...... 60 5.3 Fitting SCCRs to MCCRs ...... 61 5.3.1 Parameter estimation ...... 62 5.3.2 Fitting procedure ...... 63 5.3.3 Fitting results ...... 63 5.4 Improving on RV measurements ...... 65 5.4.1 Creating CMCCF ...... 65 5.4.2 Recalculating RVs ...... 65

6 Discussion 69 6.1 Stellar model inconsistencies ...... 69 6.2 High frequency variations ...... 70

7 Summary and conclusion 71

8 Acknowledgements 73

9 Bibliography 75

3 A Additional ﬁgures 79 A.1 Chapter 4 ...... 79 A.2 Chapter 5 ...... 81

4 List of Figures

2.1 Upper graph: More than 1000 HARPS β Pictoris spectral cross correlation functions (MCCFs) plotted on top of eachother show RV-dependent variability. Lower graph: This variability between the MCCFs is quantified at each radial velocity via the MCCF standard deviation...... 19 2.2 Upper graph: Hundreds of HARPS β Pictoris radial velocities spread over several years determined by fitting a no-pulsation spectral CCF to each of the MCCFs. Lower graph: Radial velocities seem to fluctuate during a single night (MJD 54542) with a frequency ∼ 101 min. Similar patterns are observed in other nights. . . . 22

3.1 Stellar latitude maps for various inclination angles made with our model. Each band covers a 10° range of latitudes. As far as we know, the fourth image from left represents the way that β Pictoris is viewed from the Solar System...... 25 3.2 Longitude maps for a rotating star at an inclination angle of 45° for various times as made with our model. The sharp red-blue boundary indicates the direction of

xˆS(t), which of course rotates a full 2π rad during a stellar day. For β Pictoris,

td =16h...... 28 3.3 Radial velocity maps for a star rotating at a constant angular velocity seen under the same inclination angles as used in Figure 3.1, again made with our model. No pulsations are introduced. Note that the only diﬀerence between the maps is the direction from which the star is observed. The colours indicate the Dopper shift of a green emission line - red signals redshift while blue means blueshift. . . 31 3.4 Relative intensity maps for stars with various limb darkening coeﬃcients, again made with our model. No starspots are introduced. The inclination angle is irrelevant, as are the pulsation and rotation parameters. The limb darkening co-

efficients a0, a1, a2, ... are given directly underneath each map’s description. From left to right: Lambertian radiator, exempli gratia a blackbody; star with constant emissivity throughout and vanishing absorption coefficient (for some wavelength); our estimate of β Pictoris limb darkening profile from fitting the σ-clipped mean MCCF (see Chapter 5); Solar limb darkening profile at 550 nm (Pierce, 2002); Solar limb darkening profile at 2 µm (Pierce, 2002)...... 33

5 3.5 Comparison between normalised spectra with a single absorption line. Upper graph: a Gaussian profile is indicated with a thick green line, with various Voigt profiles overplotted. Lower graph: differences between the Gaussian profile and each Voigt profile. The lower γ is, the closer the Voigt profile resembles the Gaussian. 35 3.6 Two simultaneously plotted simulated single-lined stellar spectra. In green: spectrum with single Gaussian absorption line that would be measured when looking at stellar surface patches at rest with respect to the observer. In blue: the rotationally broadened spectrum, which an observer infinitely far from the rotating star would observe. No pulsations were added to the star...... 36

4.1 Basic lay-out of our pulsation visualiser, which provides a graphical overview of the effect of pulsations and star spots on stellar spectra and spectral CCF-derived quantities. Upper row, from left to right: the first three maps indicate the pulsational velocity of each point of the stellar surface in the local radial, azimuthal and polar directions, respectively. The fourth map graphically represents the radial velocity due to pulsations and, most importantly for stars like β Pic, due to rotation. In order to raise the association with redshift and blueshift, positive radial velocity is denoted by red and negative radial velocity is denoted by blue. The fifth map shows the limb darkening profile or relative intensity map of the stellar disk. Lower row, from left to right: the rest frame and observable spectrum overplotted. The big central graph shows the raw spectral CCF. The above of the rightmost graphs gives the time-evolving part that is due to pulsations, and the other shows the partial derivative with respect to time thereof...... 41 4.2 Pulsation velocity maps for a star with 3 blobs of moving material at various times at an inclination angle of 90°, which is considered suitable for β Pictoris. Over the course of half a stellar rotation period, the static pulsation pattern moves in and out of view, causing a time-dependent perturbation to the observable spectrum. Exactly what changes are made depends on whether the pulsation velocity displayed is pointing in the local radial, azimuthal or polar directions. . 42 4.3 Three pulsation visualiser snapshots of the process in which a single equatorial Gaussian blob turning in and out of view by stellar rotation distorts the observable spectrum and CCF. In this case, the blob moves in the radial direction. For clarity, −1 vA = 50 km s was used...... 43 4.4 Three pulsation visualiser snapshots of the process in which a single equatorial Gaussian blob turning in and out of view by stellar rotation distorts the observable spectrum and CCF. In this case, the blob moves in the azimuthal direction. Just −1 like for Figure 4.3, vA = 50 km s was used...... 45

6 4.5 Sinusoidal pulsation velocity maps for various wave numbers N for a star seen under an inclination angle of 60°. Upper row: pulsation patterns on the visible part of the stellar surface. Lower row: plate caréeprojection world maps of the pulsation patterns that were ‘wrapped around’ the stellar spheres in the upper row. These maps allow one to see the whole pulsation pattern imprinted on the star at once - the azimuthal angle coordinate goes from 0 to 2π rad from left to right in these maps, the polar angle from 0 to π rad...... 47 4.6 A pulsation visualiser snapshot of the process in which a single, static, sinusoidal wave pattern seemingly changing due to stellar rotation distorts the observable spectrum and CCF. In this case, the material in the wave moves in the radial direction...... 48 4.7 Simulated spectral cross correlation function residuals for a rotating star to which sinusoidal pulsations in the radial direction are added, shown simultaneously for various times. The positive and negative SCCR amplitudes are also shown for each radial velocity via the dashed lines. In general, these amplitudes do not lie equally far away from 0, although superficial visual inspection might suggest so. The amplitudes are computed via hundreds of time slices, but for the sake of clarity just 5 are shown. Note that the vertical scale is different from the one of Figure 4.12...... 49

4.8 Simulated relation between the number of local maxima in the SCCR NC and the wavenumber of a sinusoidal pulsation in radial direction of wavenumber N. The naive expectation motivated in the text has been drawn as well, alongside two polynomial ﬁts of degree 1 and 2...... 49

4.9 Discrete approximations to probability density functions of / vrad . for various wavenumbers N, corresponding to a single sinusoidal pulsation in radial direction. 50 4.10 Simulated relation between the ‘distance’ in radial velocity between two subse-

quent local maxima / vrad . in an SCCR time slice and the wavenumber N of the responsible single sinusoidal pulsation in radial direction...... 51 4.11 A pulsation visualiser snapshot of the process in which a single, static, sinusoidal wave pattern seemingly changing due to stellar rotation distorts the observable spectrum and CCF. In this case, the material in the wave moves in the azimuthal direction...... 52 4.12 Simulated spectral cross correlation function residuals for a rotating star to which sinusoidal pulsations in the azimuthal direction are added, shown simultaneously for various times. The positive and negative SCCR amplitudes are also shown for each radial velocity via the dashed lines. In general, these amplitudes do not lie equally far away from 0, although superﬁcial visual inspection might suggest so. The amplitudes are computed via hundreds of time slices, but for the sake of clarity just 5 are shown. Note that the vertical scale is diﬀerent from the one of Figure 4.7...... 52

7 5.1 Simulated spectral cross correlation functions for stars with various equatorial rotational velocities. As an example, Solar limb darkening coefficients at 550 nm are used, together with a Gaussian line shape of width σ = 0.1 A.˚ ...... 54 5.2 Apparent equatorial rotational velocity determination from the spectral CCF FWHM for a star with a Gaussian-shaped mean absorption line and neither pulsations nor starspots. The various limb darkening choices have been visualised previously in Figure 3.4. Red curves: star with constant emission and no absorption. Yellow curves: Solar limb darkening coefficients at 550 nm. Green curves: fitted β Pictoris limb darkening coefficients in the optical. Blue curves: Lambertian radiators...... 54 5.3 2970 out of a set of 5940 SCCFs for a rotating star and various parameter sets

{||v~rot||max sin i, (a0, a1, a2), σ, γ}, to be fitted to the sigma-clipped mean MCCF. −1 Probed parameter space: ||v~rot||max sin i ∈ [122.3, 122.7] km s ,(a0, a1, a2) ∈ (0.5, [1, 0.6], [−0.5, −0.1]), σ ∈ [0.025, 0.035] A˚ and γ ∈ [0.03, 0.06] A.˚ ...... 55 5.4 Best fit of a spectral CCF generated by our model and the σ-clipped mean MCCF. −1 The parameters of the best fitting SCCF were ||v~rot||max sin i = 122.4 km s ,

(a0, a1, a2) = (0.5, 0.82, −0.32), σ = 0.035 A˚ and γ = 0.042 A.˚ Notice that the vertical scales between the CCFs and the residuals diﬀer by a factor of 10. . . . . 56 5.5 Simulated spectral cross correlation function residuals for a rotating star to which a superposition of ‘radial’ and ‘azimuthal’ sinusoidal pulsations is added, shown simultaneously for various times. The positive and negative SCCR amplitudes are also shown for each radial velocity via the dashed lines. Upper plot: the wave inducing azimuthal motion has a phase shift of 0 with respect to the wave inducing π radial motion. Central plot: phase shift of 2 rad. Lower plot: phase shift of π rad. 58 5.6 Spectral CCF residuals time evolution for a superposition of sinusoidal pulsations −1 −1 in the radial (vA = 1 km s , N = 15, φ0 = 0) and azimuthal (vA = 4 km s ,

N = 18, φ0 = 0) directions. The basic stellar parameters used to generate the SCCRs are listed...... 59 5.7 Simulated spectral cross correlation function residuals for a rotating star to which superpositions of ‘radial’ and ‘azimuthal’ sinusoidal pulsations are added. Upper plot: wave inducing radial motion with N = 27 and wave inducing azimuthal motion with N = 34. Lower plot: wave inducing radial motion with N = 34 and wave inducing azimuthal motion with N = 27. Note that the vertical scales are equal...... 59 5.8 HARPS β Pictoris spectral CCF residuals time evolution for a single night: MJD 54542. This approach to plotting spectral CCF residuals was adopted from (Koen et al., 2003)...... 61 5.9 HARPS β Pictoris spectral CCF residuals amplitude spectra for a single night:

MJD 54542. Left: stacked normalised amplitude spectra, one for each value of vrad. Right: normalised cumulative or mean amplitude spectrum. Both normalisations are such that the peak value is 1...... 62

8 5.10 HARPS β Pictoris spectral CCF residuals of the night MJD 54542 ﬁtted with various sets of pulsations, of which 1 induces radial and 1 induces azimuthal motion.

The best fit occurs for (Nrad,Nazi) = (9, 17)...... 64 5.11 HARPS β Pictoris spectral cross correlation function residuals fitting for the night 54542 MJD. The MCCRs are fitted with SCCRs generated from applying the linearity principle to SCCRs of a single sinusoidal pulsation in radial direction and SCCRs of a sinusoidal pulsation in azimuthal direction. The overview shows the best fit...... 64 5.12 Upper graph: More than 1000 pulsation-corrected HARPS β Pictoris spectral cross correlation functions (CMCCFs) plotted on top of eachother show RV-dependent variability. Lower graph: This variability between the CMCCFs is quantified at each radial velocity via the CMCCF standard deviation. Note that the same vertical scale was used as in Figure 2.1...... 66 5.13 Map with fitting errors for a range of potential radial velocity measurements, calculated for the HARPS β Pictoris spectra taken during MJD 54542. The image is overlaid with a curve connecting the points of lowest fitting error, and results in an oscillation reminiscent to the one in the lower plot of Figure 2.2...... 67 5.14 Comparison between the RV-dependent variability between MCCFs and CMC- CFs. For most radial velocities, the pulsation subtraction indeed decreases the spectral cross correlation variability. For some radial velocities near the edges though, the variability increases...... 68

A.1 Simulated relation between the number of local maxima in the SCCRs NC and the wavenumber of a sinusoidal pulsation in azimuthal direction of wavenumber N. The naive expectation motivated in the text has been drawn as well, alongside two polynomial ﬁts of degree 1 and 2...... 79

A.2 Discrete approximations to probability density functions of / vrad . for various wavenumbers N, corresponding to a single sinusoidal pulsation in azimuthal direction...... 80 A.3 Simulated relation between the ‘distance’ in radial velocity between two subse-

quent local maxima / vrad . in an SCCR time slice and the wavenumber N of the responsible single sinusoidal pulsation in azimuthal direction...... 80 A.4 Inductive demonstration of approximate linearity of SCCRs. Upper graph: additivity is shown to hold to high degree in case of a mixed radial-azimuthal pulsation pattern. Lower graph: homogeneity is demonstrated to hold to high degree for a radial pulsation pattern. Per graph, the color scales are the same...... 81 A.5 Simulated spectral CCF residuals amplitude spectra for a single pulsation in radial direction (N = 20). Left: stacked normalised amplitude spectra, one for each

value of vrad. Right: normalised cumulative or mean amplitude spectrum. Both

normalisations are such that the peak value is 1. td = 17.2 h was used...... 82

9 A.6 Simulated spectral CCF residuals amplitude spectra for a single pulsation in azimuthal direction (N = 20). Left: stacked normalised amplitude spectra, one for

each value of vrad. Right: normalised cumulative or mean amplitude spectrum.

Both normalisations are such that the peak value is 1. td = 17.2 h was used. . . . 82 A.7 HARPS β Pictoris spectral cross correlation function residuals fitting for the night 54542 MJD. The MCCRs are fitted with SCCRs of a single sinusoidal pulsation in azimuthal direction. Upper image: the overview shows the best fit. Lower image: MCCR time slice most helped by SCCR fitting...... 83 A.8 HARPS β Pictoris spectral cross correlation function residuals fitting for the night 54799 MJD. The MCCRs are fitted with SCCRs of a single sinusoidal pulsation in radial direction. Upper image: the overview shows the best fit. Lower image: MCCR time slice most helped by SCCR fitting...... 84 A.9 HARPS β Pictoris spectral cross correlation function residuals fitting for the night 55597 MJD. The MCCRs are fitted with SCCRs of a single sinusoidal pulsation in radial direction. Upper image: the overview shows the best fit. Lower image: MCCR time slice most helped by SCCR fitting...... 85

10 Chapter 1

Introduction

1.1 The exoplanet revolution

The last decade of the twentieth century has seen a historic leap forward in questioning one of humanity’s most stinging philosophical questions: ‘Are we alone?’ After the ﬁrst unambiguous exoplanet discovery (Wolszczan and Frail, 1992), thousands of worlds in other planetary systems have been found using both ground-based telescopes and spacecraft such as CoRoT, Spitzer and - most notably - Kepler. The exoplanet revolution has yielded statistical evidence that the probability for stars to harbour planets is close to unity, at least for Sun-like stars (Winn and Fabrycky, 2015). Planets, and possibly moons, constitute the stages where the play of life unfolds, and the study of their characteristics therefore has attracted major scientiﬁc and popular interest. Exoplanetary properties deduced so far are inter alia orbital period, radius, mass, density, surface gravity, atmospheric composition, wind speeds and temperature distributions (Knutson et al., 2007) and, recently, even day length (Snellen et al., 2014) and the structural parameters of a satellite ring system such as Saturn’s (Kenworthy and Mamajek, 2015).1

1.2 Planets around massive stars

As the first human generation capable of studying worlds around remote stars, addressing the issue of alien life is profoundly tempting but likely premature in the first decades of the twenty- first century. Currently, progress in assessing the likelihood of our uniqueness can more feasibly be made by probing the prevalence of exoplanets in general. Most searches have up till now focused on F, G and K star systems for various reasons. The heavier O, B and A stars are not only less likely to have continuously habitable companions,2 they also feature time-varying pulsations in both brightness (bolometric luminosity) and color (normalized spectral flux). Consequently,

1The studies listed describe the ﬁrst measurements of the relevant property. 2Consider a model in which the stellar bolometric luminosity is given by

2 4 L∗ = 4πR∗σT∗ . (1.1)

11 it is harder to detect the probably desolate planets orbiting these stars with both the transit and the radial velocity (RV) method. We should, however, not ignore heavy MS stars in our exoplanet searches, as detections could prove beneﬁcial to our understanding of the formation and prevalence of giant planets in general. Accurate methods to detect planets around A stars (and heavier) are thus valuable.

1.3 Asteroseismology

As stated quite brieﬂy in the previous section, the pulsations that perturb the stellar structure as set by hydrostatic equilibrium form a major obstacle to the detection of planets around massive stars. At the same time though, these pulsations provide an extraordinary opportunity to garner insight into the processes that take place in the deep interiors of stellar bodies. Asteroseismology, a ﬁeld that has been developed since the second half of the twentieth century, aims to establish in an increasingly precise manner the relation between observable pulsation features and the thermodynamics and chemical makeup of stars. A key principle used is that the speed by which pressure waves propagate depends on the density of the material through which is travelled. Also, some oscillations reach deeper into the star than others. The discipline is recognised as important, as its techniques are one of the very few3 capable of shedding light on nuclear fusion processes that drive stellar evolution. These take place in the otherwise unobservable hearts of stars during the lion’s share of their lifetime.

1.4 Aims of this work

In this work, we endeavour to enhance the accuracy by which the mass of planets around pulsating stars can be deduced via the RV method. We do so by attenuating the eﬀect of these pulsations on observed stellar spectra. In concreto, we will show how to construct a simulation that can produce stellar spectra for

Calling the bolometric ﬂux at distance r from the star F∗(r), the power a planet of radius R• and bolometric albedo A receives at such locus is 2 L•(r) = F∗(r)πR•(1 − A). (1.2) Equating this to the planetary blackbody power output, we obtain for the planetary temperature gradient

1 1 3 dT• 1 1 − A 2 − = − ( ) 4 T∗R r 2 < 0. (1.3) dr 2 4 ∗ As expected, the planetary equilibrium temperature decreases as the planet is located further away from its star. The larger the absolute value of this gradient, the narrower the circumstellar band called habitable zone (HZ) in which liquid water can exist under some given pressure. For hotter and larger stars, we readily see that the HZ is more tight.

Moreover, hot stars have brief lives (O stars typically reside on the main sequence (MS) for ∼ 10 million years), so that life forms would likely not have enough time to emerge and develop. Also, hotter stars have HZs that move away radially at a higher rate over the course of MS evolution. 3Id est, along with the observation of Solar neutrinos that can escape from our star without interaction after their creation in nuclear fusion processes in the core.

12 a rotating star with a given set of surface pulsations. Subsequently, these synthetic spectra are compared with actual spectra, and the simulated pulsations altered until a good ﬁt to the actual spectra is found. The pulsation signature of the measurements is then removed, and consequently better RV signals acquired. In doing so, we hope to demonstrate that asteroseismological modelling can help in constraining the mass of exoplanets via the RV method. Also, via the simulations, we aim to establish a tool for a more insightful understanding of how various pulsation patterns generate spectral line shape variations.

1.5 Case study: Beta Pictoris

As a case study, we will demonstrate our method on High Accuracy Radial Velocity Planet Searcher (HARPS) spectra of the young,4 dust-girdled A6V star Beta Pictoris (β Pic). This relatively well-studied star is orbited by the giant gas planet Beta Pictoris b (β Pic b), which is hitherto one of the few exoplanets that is both directly imaged (Lagrange et al., 2009) and suitable for potential detection by means of RV measurements. This puts it in a unique position, where accurate and independent luminosity and mass determinations could lead to tests of evolutionary models of giant planet formation (like the ones in (Baraﬀe et al., 2003)). Currently, the mass +4 +3 estimate for β Pic b is constrained to M = 7−3 MJ (Currie et al., 2013) and M = 10−2 MJ (Bonnefoy et al., 2013) by ﬁtting giant planet models to high-contrast images, and to be lower than 12 MJ for a semi-major axis of 9 AU by RV measurements (Lagrange et al., 2012). More precise RV mass estimates are needed to falsify the aforementioned models. Another motivation to select β Pic as target of choice is that there is a possibility of ∼ 4% (Macintosh et al., 2014) or ∼ 0.06% (Millar-Blanchaer et al., 2015) that the planet will transit the star in late 2017, which could constrain the planetary radius. With an accurate mass estimate of β Pic b available, we could then quickly gain more insight in the planet’s composition via the mean density.

4(Zuckerman et al., 2001) suggests that Beta Pictoris is ∼ 12 million years old.

13 Chapter 2

Pulsational noise on stellar radial velocities

To date, inferring the existence of planets by measuring periodic stellar motion along the line of sight via the Doppler shift of spectral lines - id est, the RV method - has proven to be very effective.1 Unfortunately, such radial velocity measurements intended to find planets suffer from noise caused by stellar oscillations (for β Pic, see (Koen et al., 2003), (Galland et al., 2006) and (Lagrange et al., 2012)). This chapter aims to elucidate how this noise source manifests itself in RV data, and how these pulsation effects could potentially be attenuated in an effort to increase the fidelity of detection of exoplanets orbiting seismologically active stars. To illustrate the effect of pulsations on RV data, we will use high-quality spectroscopic observations of β Pic.

2.1 Data set

In this work, we have utilised a publicly available data set of roughly 1,800 optical spectra of β Pic (∼ 380 − 690 nm), whose abscissae have units of wavelength and ordinates units of spectral ﬂux density.2 They were measured by the High Accuracy Radial Velocity Planet Searcher (HARPS) spectrograph, which is mounted on ESO’s 3.6-metre telescope at La Silla Observatory, located at the outskirts of Chile’s barren Atacama Desert. The measurements were obtained from 1 to 2 minutes of integration, resulting in a signal-to-noise ratio of 150 - 250 at 400 nm, and were taken over the course of more than a decade, starting from October 2003 and continuing into 2014. 1098 of the spectra were also included in an earlier analysis (Lagrange et al., 2012) that led to a semi-major axis dependent upper bound to the β Pic b mass. No one has yet achieved an actual RV β Pic b mass determination with this (or any other) data set.

1According to the exoplanet database of the University of California, Santa Cruz in July 2016 (Han et al., 2014), 475 exoplanets have been detected with the RV method (and possibly via other methods, too) out of a conﬁrmed total of 2933. This amounts to ∼ 16% of the currently recognised population. 2Exempli gratia Wm−2m−1 in SI units.

14 2.2 From spectra to radial velocities

Each spectrum that meets a certain quality criterium (speciﬁed later) eventually leads to an RV data point. The next sections systematically elaborate on the steps involved. These are, respectively:

1. The reduction of the measured spectra;

2. The reduction of a reference spectrum, to which the measured spectra with their Doppler shifted spectral lines are compared;

3. The selection of the cross correlation function (CCF) as a suitable similarity measure;

4. The computation of measured spectral cross correlation functions (MCCFs), that quantify the similarity between each of the measured spectra and the reference spectrum, artiﬁcially shifted for various Doppler shifts;

5. The determination of the RVs from each of the MCCFs.

2.3 Reducing the measured spectra

The velocity of the HARPS instrument with respect to β Pic changes during the night and during the year due to Earth’s axial rotation and orbital motion. As this introduces a variability in RV measurements that is often not of interest, the HARPS collaboration modiﬁes the spectra it generates to appear measured in the Solar System barycentric reference frame.3 We download these β Pic HARPS ‘pipeline-processed’ data from the European Southern Observatory (ESO) archive. We subsequently reduce the spectra in three steps.

1. The overall shape of the spectra is hard to interpret as it is determined by the stellar spectral ﬂux, telescope and instrument sensitivity and transmission by Earth’s atmosphere, which all vary with wavelength. Luckily, we are only interested in the shape and position of the stellar spectral lines detected by HARPS. To allow these to be clearly examined, we normalise the initial spectra in the following way. First, the overarching shape is found for each spectrum using median ﬁltering. A dimensionless normalised spectrum is then found by dividing the original spectrum by its smoothed version. This leaves sharp features like spectral lines intact, while broad trends disappear.

2. The spectra lack measurements from ∼ 5302 to 5339 A,˚ due to a ﬁnite spacing between HARPS’ detector chips. We deal with the data gap by setting the normalised spectral ﬂux between the mentioned wavelengths to 1.

3. As a simplistic way to combat the adverse eﬀects of telluric absorption lines, we trim the spectra, discarding all data below 410 nm and above 670 nm.

3Of course, the spectra are only Doppler shifted, not corrected for e.g. gravitational blueshift and other relativistic eﬀects that arise if one would actually take observations in the curved spacetime of the Solar System barycentre (which lies frighteningly close to the Solar surface anyways).

15 Unfortunately, the stellar spectra are - after these steps - still contaminated with telluric absorption lines.4 For our purposes, though, we do not need to take further measures. The reference spectrum described in the next section does not contain telluric signatures, and thus the cross correlation signal induced by such lines will be small. The ﬁnal RV noise due to the presence of optical telluric lines is of the order of ∼ m s−1 (Wang et al., 2015), which is negligible compared to the ∼ km s−1 RV noise due to stellar pulsations that we will demonstrate to be present in the HARPS β Pic data shortly.

2.4 Reducing the reference spectrum

To assess the motion of β Pic with respect to the observer,5 we must compare the position of the stellar lines in the HARPS spectra with their rest frame position. We thus need a reference spectrum, that describes the amount of light that escapes from a small surface element of stellar atmosphere quantified in units of energy per unit of time per unit of surface area per unit of wavelength, as measured by an observer that is macroscopically at rest with this surface element.6 We use a detailed reference spectrum apt to represent β Pic containing spectral lines of many particle species, as generated by sophisticated computer models (Buser and Kurucz, 1992). This template does not represent a spectrum of the star as a whole, but only of a single surface patch in its rest frame. Therefore, it does not contain the effects of stellar rotation, stellar pulsations, stellar bulk motion due to planetary gravity, bulk motion between the β Pic planetary system and observers in the Solar System, telluric atmospheric effects or telescope and instrumental noise and artefacts.

We have modiﬁed this reference spectrum - which we will use to cross-correlate the reduced HARPS spectra with - as well.

1. Just like the HARPS spectra, the reference spectrum was normalised by division by a median ﬁltered version of itself.

2. The original reference spectrum contains big and broad Balmer lines. As Doppler shifts of spectral lines are more easily detected for narrow lines, we increase the sensitivity of our RV measurements by removing these Balmer lines from the reference spectrum. In

concreto, we erased the NB first Balmer lines from the normalised reference spectrum. To do this, we first calculated the Balmer line centres from Bohr’s simplistic hydrogen atom 7 model. We then set the normalised flux density to 1 for a region of width ∆λB around

4 For our HARPS spectra, molecular oxygen (O2) and water (H2O) are the contaminating species. 5This is, technically, an observer in the Solar System barycentre. 6Again, the ordinates of the spectra have units of spectral ﬂux density. 7In this model, the electronic energy levels within the hydrogen atom are determined by one quantum number only, and are therefore suitably denoted by En. Here, n ∈ N \{0}. Assume the electron falls back from a state characterised by nold to a lower energy state characterised by nnew. The photon consequently emitted will have energy ∆E(n , n ) = E1 − E1 , where the zero-point energy E ∼ −13.6 eV. Thus, for the wavelength λ old new 2 n2 1 nold new of the photon, hc λ(nold, nnew) = . (2.1) ∆E(nold, nnew)

16 the line centre.8

2.5 The spectral cross correlation function

In principle, an RV measurement can already be made by measuring the shift of a single stellar spectral line from its rest frame wavelength. Of course, the precision of RV measurements increases as the shifts of more (non-telluric) lines are included in the analysis. A convenient solution to take into account all such lines in the HARPS spectra is the employment of the spectral cross correlation function (CCF). The spectral CCF is closely related to the mean spectral line shape, and can be used to infer the mean line shift (with respect to the rest frame, that is).

2.5.1 Deﬁning cross correlation

The CCF for two general one-dimensional complex continuous functions with real arguments f(x) and g(x) is deﬁned by

∞ Z (f ? g)(x) := f ∗(ξ)g(ξ + x)dξ, (2.2)

−∞ apart from a possible normalisation factor in front of the integral. From this deﬁnition, it can be understood why the CCF is also sometimes called the sliding inner product. First, one of the curves is shifted a horizontal amount x w.r.t. the other, and then the complex conjugated values of the ﬁrst function are multiplied with the values of the second, after which all multiplication results are added up. This is also what happens in the computation of inner products. In practice, CCFs are calculated in computers via discrete functions. The one form employed in this research project is

N−1 P ¯ (fn − f)(gn − g¯) n=0 (f ? g)(0) := s s , (2.3) N−1 N−1 P ¯ 2 P 2 (fn − f) (gn − g¯) n=0 n=0 where f and g are both functions deﬁned at N points.9 The normalisation is built-in, and it is clear that [(f ? g)(0)] = 1 irrespective of the dimensions of f and g.

2.5.2 Deﬁning autocorrelation

Consider now an autocorrelation: a cross correlation of a function with itself. (Understanding this particular type of cross correlation is of importance to our work.) Assume that the function

The Balmer line wavelengths in the Bohr model are given by varying nold > 2 while ﬁxing nnew to 2. 8 We have used NB = 10 and ∆λB = 10 A˚ in the ﬁnal run of our code. 9This expression is equal to IDL’s C CORRELATE function (as retrieved in July 2016) in the absence of a shift.

17 under consideration is real and has positive values only. An autocorrelation always peaks at zero displacement, because then peaks match up with peaks, so that maximal contributions to the cross correlation are realised. Likewise, the minima match up nicely with minima in this case, so that each minimum has the lowest growth-stopping inﬂu- ence on the value of the sum of products. In other words, high function values aren’t ‘wasted’ by being multiplied by low function values.

2.5.3 Types of spectral CCFs

Having appreciated this line of thought, it is not hard to see that the CCF between two similar spectra peaks when the spectral lines are neatly overlapping. Therefore, we cross-correlate the real spectra with the template spectrum, and call the results measured spectral cross correlation functions (MCCFs) in this work.10 Any such MCCF that peaks at zero displacement of the template spectrum with respect to the real spectrum corresponds to a time when the star was at rest relative to the observer.11

For the sake of completeness, we also brieﬂy mention simulated spectral cross correlation functions (SCCFs) here, whose generation is discussed in Chapter 3. These are computed by cross correlating a simulated single-lined rest frame spectrum with a corresponding observable spectrum given certain stellar rotation and oscillation parameters. Although this observable spectrum is equally single-lined and so hardly comparable to the HARPS spectra, measured and simulated spectral CCFs can be compared directly. We will use this fact later on.

2.6 Creating measured spectral CCFs

As a next step, we calculated the measured spectral cross correlation function for each HARPS spectrum. This was done by selecting ﬁrst a range of radial velocities, for which the cross correlation was to be calculated. Then, for each radial velocity, the reference spectrum was ﬁrst Doppler shifted and then utilised in a zero-shift cross correlation computation.

More formally, we ﬁnd the CCF C(vrad, t) for some spectrum F at time t by

1. generating a set of radial velocities {vrad,min, vrad,min+∆vrad, ..., vrad,max−∆vrad, vrad,max} for which the value of the CCF is to be calculated;12

2. ﬁnding the shifted template Dvrad Ftem(λ) for some RV value vrad in this domain according to Equation 3.48;

10As all CCFs in this work are spectral CCFs, specifying this by including an ‘S’ in our abbreviations is considered superﬂuous. 11Actually, in this case, the observer would not necessarily be at rest with the star at the time of observation, but with the star as it was many years ago. This is, of course, due to the vast size of interstellar space and the +0.1 ﬁnite nature of the speed of light. β Pictoris, for instance, is 63.4−0.1 light years away (Crifo et al., 1997). 12 A suitable value for ∆vrad is obtained by considering that the spectral resolution is related to the discernible c −1 radial velocity shift δvrad by R = . For HARPS, this gives δvrad ≈ 3 km s . Keeping the Nyquist-Shannon δvrad −1 −1 sampling theorem in mind, we should settle for ∆vrad ≈ 1.5 km s . As a safe choice, we set ∆vrad = 1 km s .

18 Figure 2.1: Upper graph: More than 1000 HARPS β Pictoris spectral cross correlation functions (MC- CFs) plotted on top of eachother show RV-dependent variability. Lower graph: This variability between the MCCFs is quantiﬁed at each radial velocity via the MCCF standard deviation.

3. estimating the values of the shifted template at the wavelengths of the measured spectrum F using interpolation;

4. computing the zero-shift cross correlation of discrete functions as formulated in Equa- tion 2.3;

5. proceeding to the next RV in the deﬁned domain.

Figure 2.1 shows many of the MCCFs thus produced at once. It can be seen by eye that the symmetry centres of the curves clearly have positive RVs, indicative for bulk radial motion of the β Pic planetary system away from the Solar System barycentre with a velocity ∼ 101 km s−1. Notice the signiﬁcant variability increase near the peaks of the MCCFs (see the lower graph of Figure 2.1), which is caused by pulsations. Crucially, it has been demonstrated before that infalling comets or starspots cannot be responsible in the case of β Pic (Galland et al., 2006). The contrast in variability is especially clear when a comparison is drawn between regions near the MCCF symmetry centres and regions further away.

Some spectra are too noisy to produce reliable MCCFs and - consequently - reliable RVs. We identify noisy spectra by means of their corresponding unreliable MCCFs. These will be less peaked than the reliable ones, because high noise levels in a HARPS spectrum eﬀect that no Doppler shift of the reference spectrum works particularly great to matching it up with the rel-

19 evant measured spectrum. A rough and simple rule we therefore implement to discard noisy data is that all MCCFs are rejected whose peak lies below a certain threshold above the MCCF baseline of ∼ −0.05. This threshold was set at 90% of the diﬀerence in spectral cross correlation between the highest and lowest point of the median MCCF. This median MCCF was obtained by taking, for each RV, the median cross correlation value among all those provided by the MCCFs.

2.7 Determining radial velocities

Naively, one would expect that the radial velocities of β Pictoris with respect to the observer are now simply obtained by picking, for each MCCF, the RV of maximal cross correlation. In practice though, this approach does not work, as the MCCFs feature waves with crests and troughs appearing at negative RV and disappearing at positive RV sometime later. Picking the RV of the highest peak of the MCCF will cause a drift in the RV measurements on the order of ∼ 102 km s−1 on a ∼ 1 h time scale, before discontinuously jumping to a lower value as a newly appearing wave crest topples the decaying, formerly dominant one. Such behaviour is clearly unacceptable, as was also pointed out by (Koen et al., 2003).

Although the MCCFs (mathematically denoted by C(vrad, t)) exhibit rapidly moving ripples of signiﬁcant amplitude (∼ 10−2) and various morphologies, they also clearly feature a common static shape, whose position varies slowly with time and with low amplitude.13 Appreciating the dichotomy between the component of the spectral CCF that varies strongly on short time scales on the one hand, and the component that varies weakly on long time scales on the other, we split up C(vrad, t) into two terms. In concreto,

C(vrad, t) = C0(vrad, t) + C (vrad, t), (2.4) where the subdivision of C into the weakly and slowly varying C0(vrad, t) and the strongly and rapidly varying C (vrad, t) is apparent. 14 The shape of C0(vrad, t) is not determined by pulsations, but rather by stellar rotation and the mean stellar absorption line shape. We will thus describe it as the no-pulsation MCCF. It is the 15 symmetry centre RV of C0(vrad, t) that we’d like to measure as RV datum for time t, as its time evolution is solely due to the gravity of the planet(s) orbiting β Pictoris. In contrast, for investigating the asteroseismology of β Pictoris (which we will do later), we only need C (vrad, t), the other component.

We thus proceed by

13 To be precise, the gravitational tug of β Pictoris b will make this base curve shift along the vrad-axis with a on the order of ∼ 101 m s−1 over the course of a ∼ 101 yr period. 14In concreto, the relevant parameters in our model are the equatorial rotational velocity and the inclination angle of the rotation axis. We tacitly assume here that the star is a rigid, spherical rotator. See Chapter 3. 15Without good reason, the singular form of the plural ‘data’ is seldom used in Modern English.

20 1. ﬁnding an approximation to C0 at some arbitrary time tref ;

2. computing an RV datum for each MCCF by establishing to what extent C0(vrad, tref ) should

be shifted along the vrad-axis in order to best ﬁt the MCCF in question.

2.7.1 Approximating the no-pulsation MCCF

A straightforward way to calculate an approximation to C0(vrad, tref ) is to compute the mean or median MCCF from a set of many not-too-noisy MCCFs, where we hope that the pulsations more or less cancel out. Preferably, the time stamps of the corresponding spectra lie close to each other so that tref is well-determined. The drawback of using the mean is that it is sensitive to outlying values, while the median can be unnecessarily coarse. As a more optimal choice, we use the sigma-clipped mean. First, in an iterative process, outliers are removed. Then, the mean of the remaining MCCFs values at each RV is computed, leaving us a σ-clipped mean MCCF.

2.7.2 Fitting the no-pulsation MCCF to other MCCFs

After an approximation to C0(vrad, tref ) has been found, the RV datum of time t follows readily.

We use the least squares method to ﬁnd the RV shift ∆Svrad that makes C0(vrad − ∆Svrad, tref ) the best approximation of C(vrad, t), the MCCF of time t.

The RVs are measured relative to the symmetry centre of C0(vrad, tref ), which is the radial velocity of β Pictoris with respect to a Solar System barycentre observer at time tref . Therefore, if the eﬀect of planetary gravity on the motion of the star can be neglected, the radial velocities presented are given - to good approximation - with respect to the β Pictoris system barycentre.

2.8 High frequency variations

Radial velocity measurements obtained in the way specified in this chapter are shown in Fig- ure 2.2. From the upper graph, it is clear that the RVs are too noisy (the variability is on the order of ∼ 102 m s−1 within a night) to identify a definite trend by eye, even though the time span is long. Upon closer inspection of the radial velocities measured during a single night, we see that the RV noise is not random, but follows a sinusoidal pattern with a frequency ∼ 101 min. These high frequency variations (HF variations) were noted before in this data set by (Galland et al., 2006) and (Lagrange et al., 2012).16 In the last-mentioned research, the HF variations were mitigated by changing the strategy of observation. Instead of taking just two spectra every night, observation runs were extended to last hours. In this way, if a HF pattern would arise in a well-sampled night’s data, a sinusoidal fit could be made in order to remove the stellar pulsation effect and

16We will avoid using the terminology RV jitter, which is also commonly used, on the basis that ‘jitter’ suggests that the RV variability is highly random. This is misleading, as the RV variability mainly consists of regular oscillatory patterns.

21 Figure 2.2: Upper graph: Hundreds of HARPS β Pictoris radial velocities spread over several years determined by ﬁtting a no-pulsation spectral CCF to each of the MCCFs. Lower graph: Radial velocities seem to ﬂuctuate during a single night (MJD 54542) with a frequency ∼ 101 min. Similar patterns are observed in other nights. attain more accurate RV measurements. For the sparsely sampled nights though, this was not an option, and the RVs remained uncorrected for HF variations.

It will be a goal of this research to see whether we can also remove the HF variations from the measurements of sparsely sampled nights by means of effective asteroseismological modelling. The stellar model via which this is done, is discussed in Chapter 3. If successful in suppressing most HF variations, we hope to improve on the mass constraints currently known for β Pictoris b. Ideally, sufficient HF variation mitigation could lead to the first RV determination of the β Pictoris b mass.

22 Chapter 3

Building a stellar model

Axial rotation, asteroseismological pulsations and starspots, inter alia, can all signiﬁcantly aﬀect stellar spectra as measured from the ground. To understand precisely how these phenomena alter observations, we have constructed a basic model of a rotating star to which arbitrary position- and time-dependent patterns of pulsations and starspots can be added. After the stellar surface and its orientation relative to the observer have been characterised, the model then simulates, for a given time interval, the observed mean stellar atmospheric absorption line measured by a far-away observer. From this simulated observable spectrum and a simulated rest frame spectrum, spectral cross correlation functions are computed which can be directly compared with observations.

3.1 Model idealisations

The model outlined in this chapter relies on several unrealistic assumptions that are expected to only mildly impair the model’s predictive power. Crucially, these idealisations greatly simplify the mathematics involved. The most notable include:

• A spherically symmetric shape. As stars are not rigid bodies, every rotating star is oblated in reality. This causes inter alia gravity darkening, which leaves a signature on stellar spectra. The effect of oblateness has long been neglected as a second order effect in stellar models (Royer, 2009), and will also be disregarded as such in this work. Stellar pulsations will cause the stellar surface to deviate from maximal sphericity as well. Also this surface deformation effect is neglected in our treatment. To justify this, we note that our work indicates that pulsational velocities with respect to the stellar ‘surface’ are ∼ 1 km s−1, and that pulsation periods as measured by a clock at rest with the star are ∼ 1 h. This suggests that any surface material in oscillatory motion will not ascend ∼ 1 km s−1 · 103 s = 103 km above the mean stellar surface level. As stellar radii of A stars are typically ∼ 106 km, pulsations do not significantly affect the shape of the star.1

1The deviation from sphericity is thus analogous to the case of a sphere with a radius of 1 metre, whose surface

23 • Uniform angular rotation. Stars are known to exhibit diﬀerential rotation, again as a consequence of non-rigidity. Exempli gratia the Solar equator completes a full rotation in ∼ 24.5 Earth days, while the Solar poles do so in ∼ 35 Earth days (Brajˇsaet al., 2004).

• Uniform chemical make-up. We assume for simplicity that each patch of stellar surface has the same ion abundances in its plasma. In this way, one can roughly assume that the normalised spectrum emitted by each surface patch observed in the local rest frame is equal. Note that in reality, the plasma temperature also inﬂuences the shape of spectral lines.

• An inﬁnitely far away observer. This assumption is formally needed as the model is such that exactly half of the stellar surface emits light seen by the observer. This requires the solid angle covered by the star to vanish. Thus, given that the stellar radius is on the order of the Solar radius, the observer must be far away. This assumption is justiﬁed by a mock calculation. The solid angle covered by a star of Solar radius at a distance of 10 light years is ∼ 10−17 1.

• Wavelength-independent limb darkening. Although it is not true that the stellar opacity is the same for each wavelength, we assume that the limb darkening parameters are constant over the visible wavelength range.

3.2 Coordinate system conventions

For clarity, we first explicitly state definitions for the line of sight, for the stellar rotation axis and for the inclination angle of some axis. The line of sight in this work starts at the observer and points towards the source under consideration, implying that astronomical photons detected at Terra were moving antiparallel to the line of sight. Moreover, the stellar rotation axis is understood to be represented by a vector pointing from the stellar south pole to the stellar north pole. The stellar north pole is defined as the pole around which the star seems to revolve anticlockwise (in positive mathematical direction) when viewed from above. Finally, the inclination angle of an axis is defined as the angle between the axis and a vector antiparallel to the line of sight.

The alignment between the Solar System and β Pictoris is such that our line of sight is almost perpendicular to the stellar rotation axis. This is inferred from the fact that the inclination +0.26 angle of the system’s debris disk is 85.27° −0.19 (Millar-Blanchaer et al., 2015), which has been obtained by fitting disk models to polarised light images.2 An even stronger motivation for assuming a stellar inclination angle of 90° is the near edge-on orbit measured for β Pic b, where exhibits height variations ∼ 1 millimetre. 2Analogous to the definition of the stellar rotation axis, the rotation axis of the debris disk is defined such that the constituent particles rotate in anticlockwise direction when looking antiparallel to the axis. For a more insightful view of the 3-dimensional orientation of the β Pictoris system with respect to the Solar System, see Figure 10 of (Millar-Blanchaer et al., 2015).

24 Figure 3.1: Stellar latitude maps for various inclination angles made with our model. Each band covers a 10° range of latitudes. As far as we know, the fourth image from left represents the way that β Pictoris is viewed from the Solar System.

(Macintosh et al., 2014) calculated i = 90.7° ± 0.7 and (Millar-Blanchaer et al., 2015) found i = 89.01° ± 0.36. Although it would therefore in the case of β Pic be tempting to create a model in which the stellar rotation axis lies exactly in the plane of the sky, it is desirable to construct a more general model in which the stellar rotation axis can have any orientation with respect to the line of sight - this with future application of our approach to other (A) stars in mind. See Figure 3.1.

Let us call the orthogonal coordinate system of some observer - at rest relative to the star of interest - the world coordinate system ΣW , as is customary. The z-axis of ΣW points from the star to the observer, coalescing with but antiparallel to the line of sight. An observer at the origin of ΣW then sees the 3D scene projected onto the Oxy-plane.

It is also convenient to deﬁne an orthogonal, stellar coordinate system ΣS that serves to describe the pulsations of the star’s surface in a convenient way. From the vantage point of ΣW ,ΣS is seen rotating at a constant angular velocity around its z-axis, which is represented by the unit vector zˆS. This vector therefore remains constant in both coordinate systems. Likewise, the x- and y-axes of ΣS are represented by xˆS and yˆS, which do rotate in ΣW and span the equatorial plane of the star. The pulsations that are ultimately added to the stellar surface are expressed in spherical coordinates of ΣS.

In the analysis that follows, we will work from the vantage point of the world coordinate system only.

3.3 From observer plane to stellar surface

Our initial task will be to see what point on the stellar surface corresponds to a point (x, y) in the plane of an observer’s eyes.

The centre of the star is given by

r~C = [xC , yC , zC ], (3.1)

25 so that the stellar surface is the set of all points (x, y, z) satisfying

2 2 2 2 (x − xC ) + (y − yC ) + (z − zC ) = R , (3.2) where R is the stellar radius. The visible part of the surface is

p 2 2 2 zvis(x, y) = R − ((x − xC ) + (y − yC ) ) + zC , (3.3) where the correct root is taken from the previous equation by recalling that the positive z-axis is pointing towards us as observers. In this way, any part of the visible stellar surface is always at z-values equal to zC or greater.

Now imagine we let all points on the visible hemisphere of the star ‘collapse’ unto the observer’s plane. The point on the stellar surface projected to (x, y, 0) is - by construction - simply represented by the vector

~r(x, y) = [x, y, zvis(x, y)]. (3.4)

p 2 2 Note that all observer plane points (x, y, 0) for which (x − xC ) + (y − yC ) > R lack a corresponding point on the surface of the star. The vector pointing from r~C to ~r(x, y) is r~rel(x, y):

r~rel(x, y) := ~r(x, y) − r~C, (3.5) so that p 2 2 2 r~rel(x, y) = [x − xC , y − yC , R − ((x − xC ) + (y − yC ) )]. (3.6) Note that p ||r~rel|| = r~rel · r~rel = |R| = R (3.7) as it should be.

3.4 The stellar coordinate axes

Next, we consider the rotation of the star around its axis. This rotation axis is for convenience made to coincide with the z-axis of ΣS, as indicated before. It can be unambiguously characterised by a static, real, 3-component unit vector zˆS = [ax, ay, az], that points to the stellar north pole. Perpendicular to zˆS are all vectors [x, y, z] satisfying

[x, y, z] · [ax, ay, az] = 0. (3.8)

The points (x, y, z) corresponding to the vectors [x, y, z] form a plane V through the origin:

axx + ayy + azz = 0. (3.9)

As the stellar coordinate system is rotating in ΣW , its x- and y-axes lie in V , but rotate continuously as the star spins around its axis zˆS. We can concretely describe the time evolution of the stellar x- and y-axes only given some initial conditions: a speciﬁcation of the x- and y-axes at

26 some arbitrary moment in time that we use as zero-point. Let’s call the time coordinate t, and construct two vectors xˆS(t = 0) and yˆS(t = 0). Then, we can alternatively deﬁne V as

V := span (xˆS(t = 0), yˆS(t = 0)), (3.10) or - more generally3 - as

V := span (xˆS(t), yˆS(t)). (3.11)

We will now rigidly define xˆS(t = 0) = xˆS(0). xˆS(0) should lie in the plane described by Equation 3.9. Quite arbitrarily, we might want this vector at t = 0 to have no component in the z-direction of the world coordinate system, so that z = 0. Using Equation 3.9, we then find a y = − x x, (3.12) ay still leaving freedom. Now choosing x = 1, and dividing the resulting vector by its own length, we find 1 ax xˆS(0) = q [1, − , 0]. (3.13) 1 + ( ax )2 ay ay

It is evident that this computation fails if ay = 0, or, equivalently, when the rotation axis vector lies in span (xˆ, zˆ). Then yˆ is obviously a vector perpendicular to aˆ, so that we can use it as a deﬁnition for xˆS(0) in case ay = 0:

xˆS(0) = yˆ. (3.14)

Having deﬁned xˆS(0) for an arbitrary zˆS, we can proceed to calculate yˆS(0), for which there is now no freedom left. From the right hand rule, we see that we end up with a right-handed set of axes xˆS(0), yˆS(0), zˆS if

yˆS(0) := −xˆS(0) × zˆS. (3.15) Now, as the stellar coordinate system rotates, we can describe the time evolution of the x-axis of the stellar coordinate system in the world coordinate system in terms of the angular velocity ω as

xˆS(t) = cos(ωt) xˆS(0) + sin(ωt) yˆS(0), (3.16) 2π rad or, in terms of stellar day length td := ω , as t t xˆS(t) = cos(2π rad ) xˆS(0) + sin(2π rad ) yˆS(0). (3.17) td td The y-axis of the stellar coordinate system in world coordinates can be found painlessly at any time via

yˆS(t) := −xˆS(t) × zˆS, (3.18) which is a generalisation of Equation 3.15. See Figure 3.2.

3It is not redundant to put t here, as without t, it would be ambiguous what vectors we mean. There would be no problem if we would always pick the vectors at the same time: then the vectors span V . But if the vectors are taken at different times, say, at t and at t + ∆t, then there exist infinitely many time differences 3 1 1 3 (∆t = {..., − 4 P, − 4 P, 4 P, 4 P, ...}, in concreto) for which xˆS(t) = ± yˆS(t + ∆t). If the vectors xˆS and yˆS were to be evaluated with such time difference, the vectors would coincide or be antiparallel and the span of the vectors would not be a plane but a line instead.

27 Figure 3.2: Longitude maps for a rotating star at an inclination angle of 45° for various times as made with our model. The sharp red-blue boundary indicates the direction of xˆS(t), which of course rotates a full 2π rad during a stellar day. For β Pictoris, td = 16 h.

3.5 From cartesian ΣW coordinates to spherical ΣS coordinates

Having found the time evolution of the ΣS x- and y- coordinate axes, we can proceed to calculating the spherical coordinates φ(x, y, t) and θ(x, y) belonging to r~rel(x, y). Recall that r~rel(x, y) is the vector pointing from the stellar centre to the surface point whose projection on the ΣW observer plane is (x, y, 0).

Let PV r~rel(x, y) represent the projection of r~rel(x, y) onto V , the stellar equatorial plane. PV is thus a projection matrix. φ(x, y, t) is deﬁned as the rotation that a vector starting at xˆS(t) should undergo (in anticlockwise direction, as seen from the stellar north pole) in order to end up parallel to PV r~rel(x, y).

To ﬁnd φ, let us decompose r~rel(x, y) into two vectors, one perpendicular to the equatorial plane and one lying in it - the latter is the projection we’d like to ﬁnd. After subtracting the perpendicular component from r~rel(x, y), the projection should remain. Appreciating this thought, we thus write

PV r~rel(x, y) = r~rel(x, y) − (r~rel(x, y) · zˆS) zˆS, (3.19) where the second of the terms at the right-hand side is the projection of r~rel(x, y) on the rotation axis. The deﬁnition of φ above means inter alia that

cos φ(x, y, t) = cos ∠(xˆS(t),PV r~rel(x, y)), (3.20) and so xˆS(t) · PV r~rel(x, y) ˆ cos φ(x, y, t) = = xˆS(t) · PV r~rel(x, y). (3.21) ||PV r~rel(x, y)|| The inverse cosine function must be invoked to obtain φ. This function can be deﬁned in multiple ways. One could e.g. choose the version that sends each number in the domain [−1, 1] to some number in the range [0, π rad]. Notice that this function would not yield angles in the range hπ rad, 2π radi, even though φ is in this range for half of the surface of the star. It is evident that we should alter the output of the inverse cosine function for some PV r~rel(x, y) at any time

28 t.

When the 3-tuple of vectors (xˆS(t),PV r~rel(x, y), zˆS) is a right-handed triple, a rotation in the plane V of less than π rad (starting from xˆS(t)) is enough to let a dummy vector overlap with

PV r~rel(x, y). In this case, the output of the chosen inverse cosine function does not have to be altered. Alternatively, when the mentioned 3-tuple is a left-handed triple, the inverse cosine function output needs to be subtracted from 2π rad to yield the correct φ. The criterion thus is  ˆ arccos(xˆS(t) · PV r~rel(x, y)) if (xˆS(t) × PV r~rel(x, y)) · zˆS > 0 φ(x, y, t) = (3.22) ˆ 2π rad − arccos(xˆS(t) · PV r~rel(x, y)) otherwise

The second spherical coordinate θ(x, y), representing the angle between rrel(x, y) and the rotation axis, is easier to ﬁnd. Using the same convention for the meaning of the inverse cosine function, we have

θ(x, y) = arccos(zˆS · rrelˆ (x, y)). (3.23) Notice the lack of time-dependence.

3.6 Pulsational velocity

Let us recall what we have done so far. Given some point (x, y, 0) in the observer plane at time t, we have found the corresponding azimuthal and polar spherical coordinates φ(x, y, t) and θ(x, y).

Imagine now that we have some vector function vpuls~ (φ, θ, t), that assigns a velocity vector indicating the local direction and magnitude of movement of the stellar surface plasma to each spot (φ, θ) at time t. Such vectors could of course be decomposed into a sum of the form

ˆ ˆ vpuls~ (φ, θ, t) = vpuls,r(φ, θ, t) ˆr + vpuls,φ(φ, θ, t) φ + vpuls,θ(φ, θ, t) θ, (3.24)

ˆ ˆ where the orthonormal vectors ˆr, φ and θ depend on φ and θ in ΣS (they all three vary from spot to spot); and on φ, θ and t in ΣW , where the rotation of the star has to be taken into account. The functions vpuls,r, vpuls,φ and vpuls,θ represent the pulsational velocities in the local radial, azimuthal and polar directions, respectively. After working out φ(x, y, t) and θ(x, y), we can thus compute:

• vpuls,r(φ(x, y, t), θ(x, y), t),

• vpuls,φ(φ(x, y, t), θ(x, y), t),

• vpuls,θ(φ(x, y, t), θ(x, y), t). ˆ The last step remaining in order to obtain vpuls~ explicitly is, then, to ﬁnd expressions for ˆr, φ ˆ and θ in ΣW . The ﬁrst of these is just the unit vector pointing out of the star:

ˆr(x, y) = rrelˆ (x, y). (3.25)

29 The local azimuthal unit vector φˆ is orthogonal to the local ˆr by deﬁnition, but also to the rotation axis. It can moreover be chosen to point parallel to the local direction of rotation (instead of antiparallel). Thus, φˆ must be equal to the normalised cross product of the aforementioned vectors. In concreto, zˆ × r~ (x, y) φˆ(x, y) = S rel . (3.26) ||zˆS × r~rel(x, y)|| The orthonormality condition ﬁxes θˆ:

θˆ(x, y) = φˆ(x, y) × ˆr(x, y). (3.27)

Note that for a given (x, y), the set {ˆr, φˆ, θˆ} is time-independent.

The exact functional forms of vpuls,r, vpuls,φ and vpuls,θ, which represent the magnitudes of the pulsational velocity in each of the local directions, can be chosen freely and determine partly the type of pulsating star one simulates with the model. The functional forms necessary to mimic the actual pulsations on β Pictoris will be the subject of extensive investigation, and will be discussed in detail in Chapter 4. Lastly, it is worthwhile to emphasise that due to the general nature of our treatment, the net pulsational velocity vector does not necessarily have to point radially outward from the stellar surface.

3.7 Rotational velocity

The total velocity at some point of the surface is

~v(x, y, t) := v~rot(x, y) + vpuls~ (x, y, t), (3.28) where we acknowledged that vpuls~ = vpuls~ (φ, θ, t) = vpuls~ (φ(x, y, t), θ(x, y), t) = vpuls~ (x, y, t).

The magnitude of the rotational velocity vector v~rot at some point is directly proportional to the distance of that point to the axis of rotation. When we consider a point on the stellar surface parametrised by (x, y), its distance to the rotation axis is given by PV r~rel(x, y). Therefore,

||v~rot(x, y)|| = ω||PV r~rel(x, y)||, (3.29) so that

||v~rot||max = ω||PV r~rel||max = ωR. (3.30) Combining these two equations yields ||P r~ (x, y)|| ||v~ (x, y)|| = ||v~ || V rel . (3.31) rot rot max R Because we deﬁned φˆ to always point in the local direction of rotation, we have

ˆ v~rot(x, y) = ||v~rot(x, y)||φ(x, y). (3.32)

30 Figure 3.3: Radial velocity maps for a star rotating at a constant angular velocity seen under the same inclination angles as used in Figure 3.1, again made with our model. No pulsations are introduced. Note that the only diﬀerence between the maps is the direction from which the star is observed. The colours indicate the Dopper shift of a green emission line - red signals redshift while blue means blueshift.

Combining the expressions for the magnitude and direction of the rotational velocity vector and performing algebra, we can write ﬁnally that

||v~ || v~ (x, y) = rot max zˆ × P r~ (x, y) = ω zˆ × P r~ (x, y). (3.33) rot R S V rel S V rel Giving this expression a quick thought exposes this indeed had to be the correct formula.

3.8 Radial velocity

Having calculated both the pulsational and rotational velocity vectors, we can obtain the total velocity vector from Equation 3.28. The radial velocity vrad is now deﬁned by

vrad(x, y, t) := −vz(x, y, t), (3.34) where vz is the third component of the total velocity vector ~v. We use a minus sign here, as radial velocity is measured along the line of sight, in the direction that points away from the observer. The line of sight is, in our construction, antiparallel to the z-axis of the world coordinate system. Under this deﬁnition, the radial velocity is positive when a body moves away from the origin of

ΣW , and negative when a body moves towards it. See Figure 3.3. Note that the radial velocity map of a rotating star is just a linear gradient. We moreover observe that a swiftly rotating star seen under a low inclination angle has the same radial velocity pattern as a slowly rotating star seen under a high inclination angle; id est, stars with equal ||v~rot||max sin i have indistinguishable RV patterns.

3.9 Limb darkening

A limb darkening profile quantifies how the apparent brightness varies across the stellar disk. In practice, it is wavelength-dependent, but we assume the profile to be the same for each wavelength considered in the model.

31 By the assumption of spherical symmetry, the stellar limb darkening proﬁle can only depend on ψ, the angle between the position vector drawn from the stellar core to a point on the surface and a vector antiparallel to the line of sight. In concreto,

r~rel(x, y) zvis(x, y) cos(ψ) = · zˆ = := zrel(x, y), (3.35) ||r~rel(x, y)|| R where we have introduced zrel(x, y) as the line-of-sight elevation of the stellar surface relative to the stellar radius. One intensity function I(ψ) that is used to ﬁt actual stellar limb darkening proﬁles, is

∞ X k I(ψ) = I(0) ak cos (ψ), (3.36) k=0

π where the intensity at the disk edge (ψ = 2 rad) vanishes as it should. Note that ψ = 0 fixes P∞ the sum of coefficients: k=0 ak = 1. Now define I(ψ) I (ψ) := (3.37) rel I(0) as the intensity ‘at’ ψ relative to the central intensity, so that

∞ X k Irel(x, y) = akzrel(x, y). (3.38) k=0

Numerically, we can only use a finite number of limb darkening coefficients, forcing ak to vanish from some value of k onwards. E.g. for the Sun, truncating the sum after considering 3 terms gives a good approximation to the 550 nm limb darkening profile, and a0 = 0.3, a1 = 0.93 and a2 = −0.23 (Pierce, 2002). These terms indeed sum up to unity, as required for self-consistency.

See Figure 3.4. Note that the homogeneity, or optical thickness of the stars increase as a0 increases. The idealised Lambertian radiator has inﬁnite optical depth.

3.10 Rest frame spectrum

Now imagine that an observer looks at our model star. As we assumed the star to be infinitely far away, the object cannot be resolved, and thus the observer cannot distinguish what photon came from what patch of stellar surface. She or he could rather only hope to construct a normalised spectrum of the star as a whole. In other words, such a spectrum quantifies how common photons of various energies are with respect to each other in an observed global sample, and does not reflect the relative abundances of the various photon types reaching the observer from any single patch of stellar surface.

In order to calculate the average line shape that an observer looking at the rotating, pulsating model star would measure, we ﬁrst need the average line shape as emitted by a body of stellar plasma that is macroscopically at rest with respect to this observer. Such body is not

32 Figure 3.4: Relative intensity maps for stars with various limb darkening coefficients, again made with our model. No starspots are introduced. The inclination angle is irrelevant, as are the pulsation and rotation parameters. The limb darkening coefficients a0, a1, a2, ... are given directly underneath each map’s description. From left to right: Lambertian radiator, exempli gratia a blackbody; star with constant emissivity throughout and vanishing absorption coefficient (for some wavelength); our estimate of β Pictoris limb darkening profile from fitting the σ-clipped mean MCCF (see Chapter 5); Solar limb darkening profile at 550 nm (Pierce, 2002); Solar limb darkening profile at 2 µm (Pierce, 2002). quite at rest on the level of atoms and ions, which run amok with typical particle speeds of km 4 ∼ 10 s due to the considerable temperatures in stellar atmospheres. The effective temperature +30 of β Pic (denoted Teff,βPic) is, for instance, 8052−30 K (Gray et al., 2006). The internal motion of a batch of microscopic particles whose bulk motion is zero with respect to an observer does of course influence the spectrum seen by such observer. As most particles have a non-zero velocity component along the line of sight, the light they emit is Doppler shifted. The Doppler shift is different for each particle. For this reason, the spectral line broadening caused by internal thermal motion of a gas or plasma is called Doppler broadening.

The Doppler broadened line shape has the form of a simple (inverted) Gaussian function.5 If we would denote the absorption spectrum of a body of plasma macroscopically at rest w.r.t. an

4We can obtain a simple order of magnitude estimate for the average speed v of a non-relativistic particle of mass m in thermal equilibrium at temperature T by equating the classical kinetic with the mean thermal energy:

1 2 3 mv = kB T. (3.39) 2 2 Solving for v and using the proton mass for the estimate, we ﬁnd s 3k T v = B , (3.40) mp

km resulting in v ≈ 14 s for β Pic. We see that indeed v c, as was required for self-consistency. 5Consider a cloud of non-relativistic particles of mass m in thermal equilibrium at temperature T . The cloud is macroscopically at rest w.r.t. the observer. According to the Boltzmann distribution, the fraction of particles with energy between E and E + dE, which we call f(E)dE, is equal to the probability to be in a state of energy − E E (id est Z(T )e kB T , where Z(T ) is the partition function), times the number of available states between E

33 observer by F (λ), we would have

2 − (λ−λc) 2σ2 F (λ) = F0,λ(1 − Ae G ), (3.44) where F0,λ represents the spectral ﬂux density ‘far away’ from the line centre λc, A the maximal relative depth of the spectral line and σG the Gaussian line width. In SI units, [F (λ)] = [F0,λ] = Wm−2m−1. The Doppler line width is given by r k T σ = λ B , (3.45) G c mc2 where m is the mass of the particle emitting light at wavelength λc in its rest frame. By taking −1 the proton mass mp for m and Teﬀ,βPic for T , we estimate σG ∼ 10 A˚ for a spectral line in the visible part of the electromagnetic continuum (λc ∼ 500 nm).

The other main line broadening eﬀects in stars are collisional or impact pressure broadening, in which collisions between atoms or ions distort the electronic energy levels of both partners, causing a variability in the photon energies that the particles can absorb; and natural broadening, arising from the combination of short-lived quantum state occupations and the uncertainty principle. Both processes give rise to a so-called Lorentzian line proﬁle.

The net macroscopic rest frame line profile is therefore probably better represented by a middle- ground between a Gaussian and a Lorentzian profile. Such a line profile is called a Voigt profile,6 defined as the convolution between a Gaussian and Lorentzian profile. In this work, the Voigt profile is calculated via Re(w(z(λ))) V (λ) = √ , (3.46) σ 2π where Re(w(z)) is the real part of the Faddeeva function7 evaluated at z defined as z(λ) := λ−λ√c+iγ . σ 2 and E + dE (denoted by g(E)dE, where g(E) is the density of states):

− E f(E)dE = Z(T )e kB T g(E)dE. (3.41) The energy we consider here is only the part of a particle’s kinetic energy due to motion along the line of sight, 1 2 so that we set E = 2 mvrad. Recognizing that f(E)dE = f(vrad)dvrad and g(E)dE = g(vrad)dvrad, we ﬁnd that

1 mv2 2 rad − k T f(vrad) = Z(T )e B g(vrad). (3.42)

Now we note that g(vrad) is constant, so that the only vrad-dependence resides in the exponent. Assume that the particles can decay from one quantum level to another, thereby emitting a photon of wavelength λc in their rest λ frames. From considering the non-relativistic Doppler eﬀect, we have that vrad = c( − 1). This ﬁnally leads to λc

2 (λ(v )−λ )2 − mc rad c 2kB T λ2 f(vrad) ∝ e c . (3.43) As each particle within the cloud has the same emission rate of this photon type, the emission spectrum is proportional to f(vrad), and thus also Gaussian. 6Named after the German physicist Woldemar Voigt (b 1850 - d 1919). 7Named after the Russian mathematician Vera Faddeeva (b 1906 - d 1983), the Faddeeva function (also called 2 Kramp function) is deﬁned as w(z) := e−z erfc(−iz).

34 Figure 3.5: Comparison between normalised spectra with a single absorption line. Upper graph: a Gaussian profile is indicated with a thick green line, with various Voigt profiles overplotted. Lower graph: differences between the Gaussian profile and each Voigt profile. The lower γ is, the closer the Voigt profile resembles the Gaussian.

A comparison between a Gaussian proﬁle and several similar Voigt proﬁles with σ = σG and increasing γ is seen in Figure 3.5. For larger γ, the broad Lorentzian line wings are more pronounced. The shape of the centre of the spectral line is dominated by the Gaussian component.

3.11 Observable spectrum

To find the observable spectrum as seen by an observer infinitely far away from the star, we have to sum up the spectral contributions from each patch of visible stellar surface. Not each surface element contributes equally dominant to the overall spectrum, as some are brighter than others. The ‘weight’ of a patch’s contribution can be quantified as the intensity of the patch relative to the disk centre. Also, we should keep in mind that stellar rotation causes the local spectrum to Doppler shift. In concreto, to find the net flux density of the star at some wavelength λ, we should

1. Doppler shift the rest frame spectrum of a tiny surface patch according to the local radial velocity;

2. ﬁnd the ﬂux density at λ;

35 Figure 3.6: Two simultaneously plotted simulated single-lined stellar spectra. In green: spectrum with single Gaussian absorption line that would be measured when looking at stellar surface patches at rest with respect to the observer. In blue: the rotationally broadened spectrum, which an observer inﬁnitely far from the rotating star would observe. No pulsations were added to the star.

3. multiply this number by the relative intensity of the surface patch;

4. proceed to the next surface patch and add up all contributions.

To obtain a spectrum, the procedure above is repeated for the desired range of wavelengths. To normalize the resulting spectrum, one divides all spectral ﬂux values by the highest occuring value.

Mathematically, this is expressed as Z

Ftot(λ, t) = dxdy Irel(x, y, t)(Dvrad(x,y,t)F (λ))(λ, x, y, t) (3.47) stellar disk

where Dvrad(x,y,t) is the operator that Doppler shifts spectra according to vrad at (x, y) at time v t. With β(v) := c , it can be deﬁned as s ! 1 − β(vrad) Dvrad F (λ) := F λ . (3.48) 1 + β(vrad)

−1 In SI units, [Ftot(λ, t)] = Wm .

The resulting observable spectrum is symmetric when the radial velocity function is determined by stellar rotation only.8 An example of such a spectrum can be seen in Figure 3.6.

3.12 Spectral cross correlation function

The previous sections have explained the principles underlying our model of a pulsating, rotating star for which artiﬁcial spectra can be generated at any moment in a simulated time interval. It

8Some pulsation patterns will, momentarily, also give rise to symmetric observed spectra.

36 would be desirable to compare these artiﬁcial spectra to the real, measured spectra.

The artificial spectra describe the mean spectral line shape for a star with certain pulsations on it, and thus feature only one line. The real spectra on the other hand contain many lines, some of which even (partially) overlap due to broadening effects - leading to dubious hybrid lines. It is therefore not at all straightforward to find the mean spectral line shape for real spectra.

As a solution, we employ both measured and simulated spectral cross correlation functions (MC- CFs and SCCFs) as described in Chapter 2, which allow for direct quantitative comparison. It is also explained there how MCCFs are computed. The SCCFs are computed analogously; in this case though by cross correlating the artificial rest frame spectrum with the artificial observable spectrum. Both of these quantities have been discussed in the preceding sections of this chapter. Therefore, after our model obtains an artificial observable spectrum, it also computes the corresponding SCCF to allow for comparison with MCCFs.

3.13 Spectral cross correlation function residuals

The stellar model as described hitherto has been developed to understand the origin of ripples on MCCFs in terms of no more than a few pulsational modes. The idea is to change the pulsations applied to our computer star continuously until we ﬁnd SCCFs that display the same ripples. When this happens, we might have found the actual pulsational modes of the real star, or at least an eﬀective description of the actual pulsations involved.

Although the MCCFs show ripples, and the SCCFs of pulsating model stars do so to, there is a clear static shape present in both CCF types. Recall how we discussed in Chapter 2 that such static shape weakly and slowly shifted along the vrad-axis due to the gravitational pull of planets in the β Pictoris system. As we simulate a star in isolation, this no-pulsation CCF is in fact really static. Again denoting a spectral CCF by C(vrad, t), we could thus write

C(vrad, t) = C0(vrad) + C (vrad, t), (3.49) where the subdivision of C into a time-dependent and a time-independent part is clear. The time-independent part is not determined by pulsations, but rather by the rest frame spectrum and stellar rotation parameters. To investigate the eﬀect of stellar pulsations of the CCF, we therefore only need C (vrad, t).

Thus C (vrad, t) is our final and preferred measurable quantity and simulation product, for which we will extensively study the behaviour as a result of applied stellar pulsations. In simulations, it is found by subtracting C0 from C. As C0 is determined by stellar rotation only, we find it by running the simulation of the star under consideration, but with all pulsations turned off. We will call the values of C spectral cross correlation function residuals (CCRs).

37 Parameter description Parameter dimensions Parameter symbol

Day length T td −1 Equatorial rotational velocity LT ||v~rot||max

Rotation axis orientation 1 zˆS

Limb darkening coeﬃcients 1 (a0, a1, a2, ...)

Voigt spectral line centre L λc Voigt spectral line central relative depth 1 A Voigt spectral line Gaussian width L σ Voigt spectral line Lorentzian width L γ

Table 3.1: Overview of all non-pulsational physical stellar input parameters used in the simulation.

3.14 Physical parameter overview

In this concluding section, we brieﬂy discuss all physical parameters used in our model to generate artiﬁcial spectra, SCCFs and SCCRs. The author can be contacted for details on the technical implementation.

3.14.1 Basic physical parameters

Table 3.1 lists all physical input parameters unrelated to pulsations and starspots, as used in the simulation. From these input parameters, several others are readily inferred:

• the stellar angular velocity ω can be obtained from the day length td;

• the stellar radius R is ﬁxed by td and the equatorial rotational velocity ||v~rot||max via ||v~ || t R = rot max d ; (3.50) 2π rad

• the inclination angle i can be deduced from the rotation axis orientation zˆS (in the world

coordinate system). zˆS is our preferred input parameter as it contains more information, namely also the orientation of the stellar disk in the plane of the sky. Note that this latter angle does not influence the spectra (so that the SCCFs and SCCRs are insensitive to it as well). It should be noted that although both the centre and the central relative depth of the Voigt spectral absorption line determine the artificial rest frame and observable spectrum, they do not influence the SCCF - except in the unphysical case λc = 0 and the trivial case A = 0. In the absence of pulsations and starspots, the SCCF is also independent of td. The rotation-only

SCCF is then determined only by ||v~rot||max sin i,(a0, a1, a2, ...), σ and γ.

3.14.2 Pulsational and starspot-related physical parameters

The stellar model can be enriched with pulsations in the radial, azimuthal and polar directions. Moreover, starspots can be added via excitations and depressions in the relative intensity

38 map. The types of pulsations and starspots used in this research and their parametrisations are discussed in the next chapter.

39 Chapter 4

Pulsational eﬀects on the spectral CCF

In Chapter 2, it was argued that waves going over measured spectral cross correlation functions hampered the acquisition of accurate RV measurements and were caused by pulsations. We did not, however, constrain the nature of these pulsations. In Chapter 3, we introduced the functions vpuls,r, vpuls,φ and vpuls,θ, which determine the (possibly time-dependent) pulsation pattern imprinted on the stellar surface. Consequently, now our stellar model is built, we must familiarise ourselves with the eﬀects of pulsations on C , the part of the CCF containing the pulsation signature. In this chapter, we explore various pulsation patterns and discuss their parametrisations.

To help attain intuition, a pulsation visualiser was built. A still is shown in e.g. Figure 4.1.1 The visualiser is run with the same stellar parameters throughout the chapter. The simulated star will be reminiscent of β Pictoris, which is the star for which we’ll demonstrate our CCF pulsation-attenuation method in Chapter 5, the ﬁnal part of this work. Exempli gratia, we will choose the rotation axis to be perpendicular to the line of sight2 and adopt an equatorial rotational velocity of 122.4 km s−1.3

Ignoring for the moment the possible time-dependencies of the pulsations, we consider the case of a pulsation pattern ﬁxated to the stellar surface. Due to the rotation of the star, the visible part of such a pulsation pattern will change over time. This causes the observable spectrum to be time-dependent even though vpuls,r(φ, θ), vpuls,φ(φ, θ) and vpuls,θ(φ, θ) are static. See Figure 4.2.

1Animated visualisations can be viewed and downloaded from the author’s website. 2The choice of i = 90° does not only facilitate the interpretation of the spectral eﬀects of the pulsations, but also constitutes a highly evidence-based guess (see Chapter 3). 3How this latter number has been obtained from the data is explained in Chapter 5 as well.

40 Figure 4.1: Basic lay-out of our pulsation visualiser, which provides a graphical overview of the eﬀect of pulsations and star spots on stellar spectra and spectral CCF-derived quantities.

Upper row, from left to right: the first three maps indicate the pulsational velocity of each point of the stellar surface in the local radial, azimuthal and polar directions, respectively. The fourth map graphically represents the radial velocity due to pulsations and, most importantly for stars like β Pic, due to rotation. In order to raise the association with redshift and blueshift, positive radial velocity is denoted by red and negative radial velocity is denoted by blue. The fifth map shows the limb darkening profile or relative intensity map of the stellar disk.

Lower row, from left to right: the rest frame and observable spectrum overplotted. The big central graph shows the raw spectral CCF. The above of the rightmost graphs gives the time-evolving part that is due to pulsations, and the other shows the partial derivative with respect to time thereof.

4.1 Gaussian blobs

The first sensible question to ask is how the direction of movement influences C for some elementary pulsation pattern. To get a feeling for this, we simulate a single blob of plasma moving either purely in the radial, azimuthal or polar direction and which is fixated to a certain point on the stellar equator. As discussed, this point turns in and out of the observer’s view due to the rotation of the star alone. The blob’s velocity profile was chosen to be two-dimensional Gaussian. If we denote the direction of the movement with x (so that the expression covers vpuls,r, vpuls,φ and vpuls,θ all at once), then this profile is determined by 4 parameters according

41 Figure 4.2: Pulsation velocity maps for a star with 3 blobs of moving material at various times at an inclination angle of 90°, which is considered suitable for β Pictoris. Over the course of half a stellar rotation period, the static pulsation pattern moves in and out of view, causing a time-dependent perturbation to the observable spectrum. Exactly what changes are made depends on whether the pulsation velocity displayed is pointing in the local radial, azimuthal or polar directions. to 2 2 − (φ−φc) +(θ−θc) 2σ2 vpuls,x(φ, θ, t) = vpuls,x(φ, θ) = vAe ang , (4.1) where vA is the pulsational velocity amplitude, (φc, θc) the angular location of the blob centre, and σang the angular width of the blob. Note that any pulsation pattern can be understood as a collection of these blobs simultaneously occupying the surface.

4.2 An equatorial blob moving along ˆr

We ﬁrst consider the corotating Gaussian plasma blob to move radially outwards, away from the stellar core. The process is visualised in Figure 4.3, and described hereafter in a stepwise fashion:

1. Initially, just as the blob becomes part of the visible surface hemisphere, no measurable

signature on C occurs. The reason for this is threefold:

• on the one hand, the projected area of the blob unto the observer’s plane vanishes. This holds irrespective of the blob’s direction of movement; • on the other hand, as in this case the blob moves radially outwards, the direction of movement is perpendicular to the line of sight, and therefore the pulsation does not inﬂuence the radial velocity map; • the intensity at the stellar disk edge with respect to the disk centre vanishes.

2. Then, as the blob continues to rotate into view, its projected size becomes larger, spanning a larger fraction of the perceived stellar disk. Also, the pulsational velocity vector points increasingly towards the observer, causing a blueshift. Some parts of the stellar disk that would otherwise have only a slightly blueshifted line as their spectrum, now emit a more

42 Figure 4.3: Three pulsation visualiser snapshots of the process in which a single equatorial Gaussian blob turning in and out of view by stellar rotation distorts the observable spectrum and CCF. In this −1 case, the blob moves in the radial direction. For clarity, vA = 50 km s was used.

43 extremely blueshifted line. Consider the simulated observable spectrum as a sum of a ﬁnite population of diﬀerently Doppler shifted lines. The more extremely blueshifted lines become excessively numerous at the expense of the only slightly blueshifted lines. The

result is that C (vrad, t) features a maximum adjacent to a minimum, where the maximum occurs at a lower radial velocity than the minimum.

3. The blob appears largest when it is in the centre of the perceived stellar disk. The pulsation velocity vector is now also oriented completely antiparallel to the line of sight, so that the inﬂuence of the pulsation on the radial velocity map is at its peak. This is the reason that pulsations in the radial direction have maximum amplitude when they cross the centre of the stellar disk.

4. When the blob has crossed the disk centre and is on its way to egress, the blueshifting

phenomenon still occurs, so that the shape of C (vrad, t) remains unaltered. Its amplitude decreases though, as the blob’s apparent area diminishes, the inclination angle of the pulsational velocity vector increases, and the blob moves towards the darker disk edge. These eﬀects are exactly opposite to those occurring during ingress.

Note that the inclination angle, hitherto assumed to be i = 90°, is also important. Not only do we lose a meaningful lower bound to the stellar rotational velocity near i = 0°, also equatorial blobs moving in radial direction become invisible in the spectrum. Instead, radially moving blobs near the visible pole become important. Some might even be circumpolar - visible throughout the stellar rotation cycle due to their extreme latitude and a small inclination angle.

4.3 An equatorial blob moving along φˆ

We then consider the corotating Gaussian plasma blob to move azimuthally, along the stellar surface and in the local direction of rotation. The process is visualised in Figure 4.4, and described hereafter in a stepwise fashion:

1. Although the CCF is not distorted by the pulsation right at ingress due to the vanishing apparent blob size and relative intensity,4 the pulsation signature grows more rapidly as the blob turns into view in the case of azimuthal movement than in the case of radial movement. This is due to the fact that at ingress the azimuthal pulsational velocity vector is antiparallel to the line of sight, maximising the magnitude of Doppler shifts, whereas the radial pulsational velocity vector is perpendicular to it.

2. As the blob turns into view, its projected area increases, as well as the relative intensity of the disk region it occupies. Meanwhile, though, the component of the azimuthally-oriented pulsational velocity vector pointing antiparallel to the line of sight diminishes in absolute

value. The result is that the amplitude of C peaks somewhere between ingress and the disk centre. 4Both of these reasons were also valid in our discussion of the spectrally unobservable ingress of a radially moving blob.

44 Figure 4.4: Three pulsation visualiser snapshots of the process in which a single equatorial Gaussian blob turning in and out of view by stellar rotation distorts the observable spectrum and CCF. In this −1 case, the blob moves in the azimuthal direction. Just like for Figure 4.3, vA = 50 km s was used.

45 3. When the blob passes the perceived stellar disk centre, C = 0 for all radial velocities, as the component of the pulsational velocity vector along the line of sight vanishes. Note that this behaviour is markedly diﬀerent from the maximal-amplitude behaviour of a radially moving pulsation blob at this point.

4. As the blob heads for egress, the component of the pulsational velocity vector along the line of sight is now positive, so that the redshift of the already receding blob constituents is boosted. Again by considering the simulated observable spectrum as the sum of a finite number of differently Doppler shifted lines, we can think of this redshifting effect as causing an excess of heavily redshifted lines by ‘stealing’ lines from the lower redshift

population. This results in a C maximum at a radial velocity higher than that of the adjacent minimum, which is opposite to the ingress case. Note that, again, this behaviour

is in stark contrast to the radially moving blob behaviour, where the shape of C remains the same throughout.

The detectability of azimuthal pulsations diminishes as the inclination angle decreases. After all, the signature of the stellar rotational velocity fades at low inclination angles, and the direction of azimuthal pulsations is the same as the direction of rotation.

4.4 An equatorial blob moving along θˆ

We ﬁnally consider the corotating Gaussian plasma blob to move in polar direction, along the stellar surface and from north to south pole. When in this case the blob is placed at the equator, the pulsational velocity vector is always perpendicular to the line of sight, irrespective of the moment between ingress and egress under consideration. For this reason, such pulsations would not leave a measurable trace on the stellar spectrum. Note that blobs moving in polar direction at other than equatorial latitude for i = 90° do leave visible traces. Conversely, equatorial blobs are detectable under lower inclination angles.

4.5 Sine waves

A second, slightly more advanced pulsation pattern we will consider is one consisting of sine waves. Each sine wave is determined only by three parameters - wavelength, amplitude and phase (oﬀset). We will consider only sinusoidal waves that vary along the azimuthal coordinate φ. Waves varying along the polar coordinate do not give rise to time-dependent spectral signatures as rotation leaves the apparent pulsation pattern invariant. See Figure 4.5. The deﬁning equation, again without specifying the direction of pulsation, is

vpuls,x(φ, θ, t) = vpuls,x(φ) = vA sin (Nφ + φ0), (4.2) where vA is the sine wave amplitude, N the number of complete sine cycles along the stellar surface and φ0 the phase of the wave at φ = 0. Instead of wavelength, we will actually use the

46 Figure 4.5: Sinusoidal pulsation velocity maps for various wave numbers N for a star seen under an inclination angle of 60°. Upper row: pulsation patterns on the visible part of the stellar surface. Lower row: plate caréeprojection world maps of the pulsation patterns that were ‘wrapped around’ the stellar spheres in the upper row. These maps allow one to see the whole pulsation pattern imprinted on the star at once - the azimuthal angle coordinate goes from 0 to 2π rad from left to right in these maps, the polar angle from 0 to π rad. wave number N as a defining parameter. Straightforward boundary conditions restrict the possible values of N, as if it were a quantum number. For arbitrary, non-integer and non-half-integer N, adopted exempli gratia in the first and third column (from left) of Figure 4.5, the pulsation pattern features a discontinuity - the velocity of stellar surface material would in such case exhibit a sudden and sharp change between two adjacent locations. This clearly is unphysical. Now consider the pulsation pattern corresponding to a sine wave with half-integer N, as shown in the second column of Figure 4.5. Although the discontinuity has disappeared, the pattern is not differentiable along the complete prime meridian. This is also unakin to realistic solutions to virtually all physical differential equations. We are therefore bound to adopt integer N only, for which an example pulsation pattern is shown in the fourth column of Figure 4.5.

Note that the aforementoined sine wave pulsation pattern is discontinuous at the poles. We therefore don’t expect it to be a realistic approximation of a possible pulsation pattern in these regions. For simulations on β Pictoris however, the inclination angle is likely so close to 90° that its poles are barely visible from the Solar System. The problem could be dealt with more generally by applying an amplitude modulation that depends on polar angle, exempli gratia

k vpuls,x(φ, θ, t) = vpuls,x(φ, θ) = vA sin (θ) sin (Nφ + φ0), (4.3) for some arbitrary positive value of k.

47 Figure 4.6: A pulsation visualiser snapshot of the process in which a single, static, sinusoidal wave pattern seemingly changing due to stellar rotation distorts the observable spectrum and CCF. In this case, the material in the wave moves in the radial direction.

4.6 A sine wave moving along ˆr

We now simulate the eﬀect of a single sinusoidal pulsation in radial direction on C (vrad, t). An appropriate value for N is chosen, ﬁxing the simulated stretch of time to a length t t = d . (4.4) s N Simulating a longer time span is not sensible, as the pulsation pattern and thus the SCCRs will start to repeat after this period. See Figure 4.6 for an overview of the discussed simulation.

Just like for the equatorial blob, the perturbation of the observable spectrum is most signiﬁ- cant at the absorption line centre, so that the CCR amplitude peaks at vrad = 0. Figure 4.7 shows the amplitude for each radial velocity.

It is, on ﬁrst sight, tempting to relate each local maximum of C to a crest in the pulsation pattern. If true, we could infer the wavenumber of a pulsation by counting the number of local maxima in its corresponding C , opening a way to characterising a star’s eﬀective pulsational modes from the shape of its MCCRs.5

5 In practice, it can be hard to identify the correct number of SCCR peaks for each pulsation when C is determined by a superposition of pulsations.

48 Figure 4.7: Simulated spectral cross correlation function residuals for a rotating star to which sinusoidal pulsations in the radial direction are added, shown simultaneously for various times. The positive and negative SCCR amplitudes are also shown for each radial velocity via the dashed lines. In general, these amplitudes do not lie equally far away from 0, although superﬁcial visual inspection might suggest so. The amplitudes are computed via hundreds of time slices, but for the sake of clarity just 5 are shown. Note that the vertical scale is diﬀerent from the one of Figure 4.12.

Figure 4.8: Simulated relation between the number of local maxima in the SCCR NC and the wavenumber of a sinusoidal pulsation in radial direction of wavenumber N. The naive expectation motivated in the text has been drawn as well, alongside two polynomial ﬁts of degree 1 and 2.

As the total number of waves running over the surface is N, the time-averaged number of crests N on the visible hemisphere will be 2 . The actual number of crests visible varies slightly with time, as it is possible that a crest moves out of view due to stellar rotation before another has moved N into view. If each crest leads to a local maximum in the SCCR, we expect that also NC = 2 of these local maxima should be present in C on average. This hypothesis has been tested and found approximately true only for N . 20. For larger N, the number of local maxima in the SCCR still increases, although at a lower pace. A local maximum was considered only if its cross correlation value exceeded a threshold value of 10−4. Figure 4.8 shows the described behaviour. The number of local maxima in the SCCR is the same as the number of local minima in the wave perturbing the mean absorption line spectrum. Note also that the wavenumber of a single

49 Figure 4.9: Discrete approximations to probability density functions of / vrad . for various wavenumbers N, corresponding to a single sinusoidal pulsation in radial direction. sinusoidal pulsation (in radial direction) can still be unambiguously determined from the number of local maxima in the corresponding SCCR, despite the fact that the relation established is not exactly linear.

As N increases, the number of local maxima in an SCCR time slice increases, so that it is natural to expect that the slice becomes ‘more crowded’ with peaks. This in turn leads to the straightforward idea that the distance between two peaks along the radial velocity axis / vrad . must decrease as N increases. In light of the potential of effective pulsation characterisation based on the peak morphology of the MCCRs, we first inquire how / vrad . is distributed for various wavenumbers. See Figure 4.9. We find discretised approximations to the probability 6 density functions (PDFs), which were inaccessible as C has been finitely sampled. The PDFs suggest bimodal distributions. The low-/ vrad . peaks lie at high |vrad| in the SCCF and SCCR, and are caused by pulsations occurring at the edges of the visible stellar hemisphere. The fact that the bimodality seems to decrease as N increases, could be caused by the fact that local maxima increasingly fall below the threshold cross correlation value of 10−4 imposed.

On average, / vrad . clearly decreases as N increases. The trend is shown in Figure 4.10. We conclude that the distribution of radial velocity diﬀerences between adjacent CCR peaks could be used as an indication for the wavenumber of the radial sinusoidal pulsations that eﬀectively describe the actual pulsations on a star’s surface.

6 PDFs are deﬁned only for continuous random variables. Theoretically, / vrad . is continuous, and although / vrad . is not a variable related to chance, we regard it as such in the spirit of the frequentist interpretation of probability.

50 Figure 4.10: Simulated relation between the ‘distance’ in radial velocity between two subsequent local maxima / vrad . in an SCCR time slice and the wavenumber N of the responsible single sinusoidal pulsation in radial direction.

4.7 A sine wave moving along φˆ

Analogously, the effect on C (vrad, t) of a single sinusoidal pulsation of wavenumber N in az- td imuthal direction is explored via a simulation probing a time span of length ts = N . Figure 4.11 provides a simulation overview, while Figure 4.12 shows that the line variations and therefore the CCRs vanish at vrad = 0. The CCRs have their peak amplitudes at a value for |vrad| that represents surface loci sufficiently far away from the disk centre that the velocity perturbation points significantly along the line of sight (either parallel or antiparallel), while on the other hand not being so far away that limb darkening effects or the projected area of pulsation crests and troughs become too problematic. −1 Note that the although the velocity amplitude used is the same as in Figure 4.7 (vA = 1 km s ), the radial SCCR peak amplitude is ∼ 3 times higher than the azimuthal SCCR peak amplitude.

Just as for the sine wave generating motion in radial direction, we have inquired the relation between the number of local CCR peaks and sinusoidal wavenumber in the azimuthal case. See

Figure A.1. Distributions of / vrad . for various wavenumbers are shown in Figure A.2, while Figure A.3 shows this data in graph-like fashion. The results are similar to those of sinusoidal pulsations in radial direction. This means that using the number of and radial velocity diﬀer- ences between peaks in CCR time slices might provide indications for the wavenumbers of both ‘radial’ and ‘azimuthal’ sinusoidal pulsations.

51 Figure 4.11: A pulsation visualiser snapshot of the process in which a single, static, sinusoidal wave pattern seemingly changing due to stellar rotation distorts the observable spectrum and CCF. In this case, the material in the wave moves in the azimuthal direction.

Figure 4.12: Simulated spectral cross correlation function residuals for a rotating star to which sinusoidal pulsations in the azimuthal direction are added, shown simultaneously for various times. The positive and negative SCCR amplitudes are also shown for each radial velocity via the dashed lines. In general, these amplitudes do not lie equally far away from 0, although superﬁcial visual inspection might suggest so. The amplitudes are computed via hundreds of time slices, but for the sake of clarity just 5 are shown. Note that the vertical scale is diﬀerent from the one of Figure 4.7.

52 Chapter 5

Attenuating pulsational noise

It is our objective to mitigate the pulsation-induced MCCF ripples shown in Chapter 2 by subtraction of suitable SCCRs, generated via Chapter 3’s stellar model and shown in Chapter 4. This chapter aims to describe the last step: ﬁtting the MCCF ripples with our model, leading to less noisy RV measurements. The implications of this procedure for the detection of β Pictoris b via the RV method are discussed as well.

5.1 Determining stellar parameters

It has been made clear previously that the functional forms of the SCCRs depend on several physical stellar parameters. These are - apart from the pulsation parameters - the equatorial rotational velocity ||v~rot||max, the inclination angle of the rotation axis i, the limb darkening coefficients (a0, a1, a2) and the Gaussian and Lorentzian mean absorption line widths σ and γ. In order to generate SCCRs apt to fit β Pictoris MCCRs, we should therefore determine reasonable values for these parameters first.

5.1.1 Apparent equatorial rotational velocity

The product of the equatorial rotational velocity of the star and the sine of the angle between the line of sight and the rotation axis, the apparent equatorial rotational velocity ||v~rot||max sin i, can be determined from Doppler broadened spectral lines (Slettebak et al., 1975). As spectral cross correlation functions are just different representations of such Doppler broadened spectral lines, they too can be used to measure ||v~rot||max sin i. The principle is shown in Figure 5.1, which shows that the CCF broadens and features a lower peak as ||v~rot||max increases. Although the peak height fixes the apparent equatorial rotational velocity,1 we cannot directly use the peak height in practice. This is due to noise in the spectra, which flattens the cross correlation function, thereby making the peak height a function of noise magnitude as well. Alternatively, we could seek to measure the apparent equatorial rotational

1Id est, given a set of line parameters and limb darkening coeﬃcients.

53 Figure 5.1: Simulated spectral cross correlation functions for stars with various equatorial rotational velocities. As an example, Solar limb darkening coeﬃcients at 550 nm are used, together with a Gaussian line shape of width σ = 0.1 A.˚

Figure 5.2: Apparent equatorial rotational velocity determination from the spectral CCF FWHM for a star with a Gaussian-shaped mean absorption line and neither pulsations nor starspots. The various limb darkening choices have been visualised previously in Figure 3.4. Red curves: star with constant emission and no absorption. Yellow curves: Solar limb darkening coefficients at 550 nm. Green curves: fitted β Pictoris limb darkening coefficients in the optical. Blue curves: Lambertian radiators. velocity from the full width at half maximum (FWHM) of the spectral CCF, which might be less sensitive to spectral noise. The approach is shown in Figure 5.2. Note that the dependency on the limb darkening coefficient tuple introduces an error ∼ 10 km s−1, while the dependency on the line width introduces an error ∼ 10−1 km s−1 for the no-absorption star and ∼ 1 km s−1 for the Lambertian radiator. The no-absorption star and the Lambertian radiator are the extreme cases; the curves for all real stars are expected to lie in between. It is evident that this method can be used to determine the apparent equatorial rotational velocity with a precision ∼ 1 km s−1

54 Figure 5.3: 2970 out of a set of 5940 SCCFs for a rotating star and various parameter sets

{||v~rot||max sin i, (a0, a1, a2), σ, γ}, to be fitted to the sigma-clipped mean MCCF. Probed param- −1 eter space: ||v~rot||max sin i ∈ [122.3, 122.7] km s ,(a0, a1, a2) ∈ (0.5, [1, 0.6], [−0.5, −0.1]), σ ∈ [0.025, 0.035] A˚ and γ ∈ [0.03, 0.06] A.˚ for a star with known limb darkening coefficients. The problem is that these parameters are not well-determined for most stars, including β Pictoris. Measuring the FWHM of C0(vrad, tref ), the no-pulsation or σ-clipped mean MCCF introduced in Chapter 2, results in FWHM = 195 km s−1 −1 −1 for β Pic, suggesting 120 km s . ||v~rot||max sin i . 130 km s for reasonable limb darkening coefficients.

5.1.2 Fitting the no-pulsation MCCF

Yet another approach is to ﬁt a large sample of simulated spectral cross correlation functions of a pulsation-free star to C0(vrad, tref ). We ﬁt not only ||v~rot||max sin i in this way, but also

(a0, a1, a2), σ and γ. The limb darkening tuple does not introduce 3 free parameters, as the coefficients cannot be varied at will - their definition requires their sum to be equal to unity. Thousands of candidate SCCFs used for this enterprise, each with its own set of parameter values, are shown in Figure 5.3. −1 The best fit is shown in Figure 5.4. It suggests ||v~rot||max sin i = 122.4 km s , consistent with the 120 km s−1 lower limit set by (Abt, 2000), but lower than the 130 km s−1 suggested in literature (Royer et al., 2007). The bulk radial velocity of the β Pictoris system with respect to the Solar System is denoted by vrad,bulk and was calculated by shifting the no-pulsation MCCF horizontally along the radial velocity axis as part of the fitting process. Our result suggests that −1 the β Pictoris system is receding from us at a velocity of vrad,bulk = 19.9 km s , consistent with −1 the earlier finding of vrad,bulk = 20.0 ± 0.7 km s (Gontcharov, 2006). Note that the σ-clipped mean MCCF is slightly asymmetric near the edges. This can be con- cluded from the fact that the SCCF is symmetric, and that on the negative RV edge, the SCCF underestimates the MCCF, while at the positive RV edge, the SCCF overestimates it.

55 Figure 5.4: Best fit of a spectral CCF generated by our model and the σ-clipped mean MCCF. The pa- −1 rameters of the best fitting SCCF were ||v~rot||max sin i = 122.4 km s ,(a0, a1, a2) = (0.5, 0.82, −0.32), σ = 0.035 A˚ and γ = 0.042 A.˚ Notice that the vertical scales between the CCFs and the residuals differ by a factor of 10.

Parameter β Pic estimate Source −1 Apparent equatorial rot. vel. ||v~rot||max sin i 122.4 km s this work (ﬁtting)

Limb darkening coefficients (a0, a1, a2, ...) (0.5, 0.8, −0.3) this work (fitting) Voigt spectral line Gaussian width σ 0.035 A˚ this work (fitting) Voigt spectral line Lorentzian width γ 0.042 A˚ this work (fitting) Rotation axis inclination angle i 90° this work (educated guess)

Day length td 17.2 h Equation 3.50 −1 β Pic system bulk radial motion vrad,bulk 19.9 km s this work (ﬁtting)

Photospheric radius R 1.732 ± 0.123 R (Kervella et al., 2004)

Table 5.1: Overview of used physical parameters for β Pictoris and their source, in three sections. Uppermost section: parameters relevant to the creation of the spectral CCF of a star without pulsations and starspots. Central section: parameters that become relevant when pulsations and starspots are introduced. Lowest section: parameters required in the analysis, although not relevant for the creation of SCCFs and SCCRs directly.

The final parameter values used in this work for β Pictoris, together with their origin, are conveniently listed in Table 5.1. The possibility that another SCCF with an appreciably different parameter set fits the no-pulsation MCCF roughly equally well, id est, whether the best fit model is unique or not, has not been explored in depth.

56 5.2 Creating SCCRs

It is evident from watching the HARPS β Pictoris MCCF evolve over the course of a single night that if sinusoidal pulsations can serve as an eﬀective description of the actual pulsations raging over the stellar surface, we need to deploy at least 2 sine waves in superposition.

5.2.1 Superposition of 2 waves

Consider therefore the case of a superposition of 2 pulsational sine waves, where we do not specify yet in which directions they induce surface motion. Let these waves be characterised by wavenumbers N1 and N2 and phase offsets φ0,1 and φ0,2. If suffices to simulate the star during a time span of td if only the first sine wave is considered N1 and we forget about the second for a moment; after that time the star has rotated just enough so that the initial pulsation pattern reemerges. If we now introduce the second sine wave, and

td td N2 = N1, then = still is enough time to simulate all possible visible patterns of the N1 N2 superposition for a fixed phase difference ∆φ = φ0,2 − φ0,1. To really obtain all patterns that two pulsations with equal frequency can produce, we should repeat the simulation for a multi- 2 tude of phase differences in [0, 2π radi. Figure 5.5 demonstrates that for N2 = N1 = 30, the

SCCRs vary considerably depending on the chosen phase oﬀset φ0,azi of the ‘azimuthal’ wave

(φ0,rad is kept at 0). For φ0,azi = 0, the SCCR amplitudes are damped at positive radial velocities, while for φ0,azi = π rad, the SCCR amplitudes are damped for negative radial velocities.

Let us now relax the condition that N2 = N1. In this case, the time we should simulate is

td ts = , (5.1) gcd(N1,N2) where gcd(N1,N2) represents the greatest common divisor of N1 and N2. In the most computationally intensive case, when the greatest common divisor is 1, the complete stellar day needs to be simulated to obtain all pulsation patterns for a fixed phase difference. Again, all pulsation patterns are obtained by repeating the simulation for many phase differences.

An example of SCCRs for a mixed radial-azimuthal pulsation pattern where N2 6= N1 at a single phase diﬀerence (∆φ = 0) is shown in Figure 5.6. Let us denote the wavenumbers of the ‘radial’ and ‘azimuthal’ pulsations by Nrad and Nazi, respectively. In this case, Nrad = 15 and Nazi = 18, whose greatest common divisor is 3. Therefore, Equation 5.1 dictates that a third of the stellar day must be simulated, which amounts to ts = 5.33 h for td = 16 h. Figure 5.7 shows two other examples of mixed radial-azimuthal pulsation patterns for N2 6= N1 and single phase diﬀerences. The alternative visualisation shows that for the SCCR waves, not all displacements are equally common for a given value of radial velocity. For both cases, relatively small displacements are the rule at high positive and negative radial velocities, while they are more rare near vrad = 0.

2Formally, to obtain ‘all’ patterns, an inﬁnite amount of phase diﬀerences should be probed.

57 Figure 5.5: Simulated spectral cross correlation function residuals for a rotating star to which a superposition of ‘radial’ and ‘azimuthal’ sinusoidal pulsations is added, shown simultaneously for various times. The positive and negative SCCR amplitudes are also shown for each radial velocity via the dashed lines. Upper plot: the wave inducing azimuthal motion has a phase shift of 0 with respect to the wave inducing π radial motion. Central plot: phase shift of 2 rad. Lower plot: phase shift of π rad.

5.2.2 Superposition of 3 or more waves

When k waves are superimposed instead of 2, Equation 5.1 must be generalised into

td ts = , (5.2) gcd(N1,N2, ..., Nk) generally making it more likely that the whole stellar day must be simulated in order to map the pattern for a ﬁxed set of k−1 phase diﬀerence parameters {φ0,2 −φ0,1, φ0,3 −φ0,1, ..., φ0,k −φ0,1}.

If we would allow for Nφ phase shifts per parameter, we would have to repeat the simulation k−1 3 Nφ times to generate a complete set of all possible SCCRs for the set of k pulsations. It has 3No ﬁnite set of SCCRs is ever complete, of course; we mean complete enough to adequately ﬁt the MCCRs.

58 Figure 5.6: Spectral CCF residuals time evolution for a superposition of sinusoidal pulsations in the −1 −1 radial (vA = 1 km s , N = 15, φ0 = 0) and azimuthal (vA = 4 km s , N = 18, φ0 = 0) directions. The basic stellar parameters used to generate the SCCRs are listed.

Figure 5.7: Simulated spectral cross correlation function residuals for a rotating star to which superpositions of ‘radial’ and ‘azimuthal’ sinusoidal pulsations are added. Upper plot: wave inducing radial motion with N = 27 and wave inducing azimuthal motion with N = 34. Lower plot: wave inducing radial motion with N = 34 and wave inducing azimuthal motion with N = 27. Note that the vertical scales are equal. turned out that this is a computationally demanding feat, even for modest numbers (exempli 4 gratia k = 3 and Nφ = 10), considering that we would like to generate the SCCRs for diﬀerent

4Id est, for a single average consumer computer anno 2016.

59 5 values of the k pulsational velocity amplitude parameters {vA,1, vA,2, ..., vA,k} as well. It is clear that generating all possible SCCRs for a superposition of even a few waves takes a lot of time and storage space. As we only approximately know beforehand which parameter values should be taken to ﬁt the MCCRs, the number of simulations we must run is considerable. It would be beneﬁcial if we could build up the SCCRs for a superposition of waves by adding up (and rescaling) the SCCRs generated from each of the pulsations in isolation. We will explore this possibility in the next section.

5.2.3 Linearity of SCCRs

To reduce the computational complexity of calculating the SCCRs of a superposition of pulsational waves, we inquire whether C has the property of linearity. Having many diﬀerent meanings, linearity is understood here in a mathematical fashion to encompass both additivity and homogeneity of degree 1, deﬁned by

~ ~ ~ ~ C (vpuls,1 + vpuls,2) = C (vpuls,1) + C (vpuls,2) (5.4) and ~ ~ C (α · vpuls) = α ·C (vpuls), (5.5) ~ respectively. In these expressions, C (vpuls) represents C (vrad, t) as induced by the 3-component vector function vpuls~ deﬁned in Equation 3.24, whose dependencies on φ, θ and t are dropped momentarily for notational simplicity. The homogeneity condition should hold for arbitrary α.

See Figure A.4 for a graphical inquiry into the linearity of C . Unfortunately, and contrary to conclusions drawn upon superficial visual inspection, the SCCRs only approximately obey the linearity conditions. For the precision required in this work though, the deviations from linearity, which are ∼ 10−4 − 10−3, can be neglected: the SCCRs have amplitudes ∼ 10−2 − 10−1. At least for finding an initial, approximate fit to the MCCRs, we can therefore invoke the linearity relations in order to bypass the tedious generation of a library of SCCRs. Having found the best

5 The number of complete cycles that wave j (j ∈ [1, 2, ..., k]) goes through during ts is

Nj Nts,j := . (5.3) gcd(N1,N2, ..., Nk) When adding more waves to the stellar surface without too carefully selecting the wavenumbers involved, the greatest common divisor will usually decline readily, so that the simulation time ts increases. This means that

Nts,j often increases as more waves are superimposed - id est, when k grows. Of course, Nts,j also increases for larger Nj . Now, without loss of generality, we define N1 to be the smallest wavenumber of all, so that Nts,1 is the lowest number of the set {Nts,1,Nts,2, ..., Nts,k}. Whenever wave 1 completes a cycle during the simulated time N −N interval, wave j has finished 1 + ts,j ts,1 cycles. (Using this expression, it is trivial to verify that after wave Nts,1 1 finishes Nts,1 cycles, wave j indeed finishes Nts,j cycles.) After n cycles of wave 1, wave j has gone through N −N n + n ts,j ts,1 cycles, meaning that each time wave 1 starts anew with a cycle, wave j is in a different stage Nts,1 of its oscillation. The result is that we effectively simulate various phase differences between the waves within a k−1 single simulation. Practically, this means that the added value of repeating simulations Nφ times diminishes as Nts,1 increases - id est, as the smallest number of simulated cycles of the set of waves goes up - established exempli gratia when pulsations with higher wavenumbers are considered.

60 Figure 5.8: HARPS β Pictoris spectral CCF residuals time evolution for a single night: MJD 54542. This approach to plotting spectral CCF residuals was adopted from (Koen et al., 2003).

fitting set of pulsational waves and their approximate parameter values, we could subsequently go back and run simulations for a superposition of these waves in the exact way, as described in the previous section. This could refine the provisional best fit.

The employment of linearity also brings about intuitive lucidity. To good approximation, exempli gratia, we can now reason that the superposition of a ‘radial’ wave (whose SCCRs are shown in Figure 4.7) and an ‘azimuthal’ wave (whose SCCRs are shown in Figure 4.12) will feature

SCCRs that have higher amplitudes near vrad = 0 than the azimuthal wave SCCRs do, while the amplitudes at large |vrad| are higher than they are for the radial wave SCCRs. That this is indeed the case can be glanced from Figure 5.7.

5.3 Fitting SCCRs to MCCRs

Having discussed the generation of SCCRs due to an arbitrary superposition of sinusoidal pulsations, we now turn our attention to the deployment of SCCRs to ﬁtting the MCCRs. As an illustration of the approach, we discuss the ﬁtting procedure for a single night of β Pic HARPS observations. In this case, we will focus on observations from MJD 54542 - March 17, 2008 - for which RV measurements are shown in the lower graph of Figure 2.2. The MCCRs for this night, obtained from subtracting the σ-clipped mean MCCF (C0(vrad, tref )) from each MCCF, −1 are shown in Figure 5.8. Note that the amplitudes appear to be larger for vrad . 20 km s −1 −1 than for vrad & 20 km s , and that a positive amplitude at vrad . 20 km s generally exceeds the negative amplitude at the same value of vrad.

61 Figure 5.9: HARPS β Pictoris spectral CCF residuals amplitude spectra for a single night: MJD 54542.

Left: stacked normalised amplitude spectra, one for each value of vrad. Right: normalised cumulative or mean amplitude spectrum. Both normalisations are such that the peak value is 1.

5.3.1 Parameter estimation

In order to determine initial guesses for the wavenumbers of the sinusoidal pulsations we will fit with, we first calculate amplitude spectra for the selected night’s MCCRs. For each value of vrad ∈ {vrad,min, vrad,min + ∆vrad, ..., vrad,max − ∆vrad, vrad,max}, the MCCR time evolution of Figure 5.8 is selected and Fourier transformed in order to obtain an amplitude spectrum. Stacking these amplitude spectra together, one obtains Figure 5.9. From the mean amplitude spectrum, we find two distinct peaks, the highest at a frequency of 69 d−1 and one of ∼ 75% that height at 43 d−1.6 The pulsation corresponding to the first frequency might be best represented by a sinusoid inducing motion in azimuthal direction, due to its prolonged importance even for radial velocities far away from ∼ 20 km s−1. As the MCCRs of Figure 5.8 are non-zero at ∼ 20 km s−1, we need a pulsation in radial direction as well. The second peak frequency is prominent in a more confined radial velocity domain, suggesting it might reasonable well be represented by a wave in radial direction. See Figure A.5 and Figure A.6 to compare Figure 5.9 to amplitude spectra of simulated CCF residuals. If the pulsation pattern is imprinted on the stellar surface, so that its appearance only changes due to stellar rotation, then the frequency f and the wavenumber N are related by 2N (N) f = C , (5.6) td where the functional form of NC(N) depends slightly on whether pulsations in radial or azimuthal direction are considered, as can be glimpsed from comparing Figure 4.8 to Figure A.1. To obtain N, this inverse function must be found. If a pulsation is not imprinted on the stellar surface but has angular velocity ωpuls in ΣS, the stellar corotating frame (see Chapter 3), then the generalisation is ω + ωpuls 2NC(N) ω + ωpuls f = · = NC(N), (5.7) ω td π rad 6 Just to be clear: d represents a telluric day. The length of the stellar day is denoted by td.

62 strictly only applicable to the non-relativistic regime. The relation neatly shows that the oscillation frequency f at any vrad is proportional to both the angular velocity of the pulsations

ω + ωpuls and the number of peaks in the residual spectrum and CCF residuals NC(N). From observing the time-evolution of the MCCRs during the night of MJD 54542, it is clear that the peaks move much faster than would be expected if they remained static with respect to the surface. It appears that ωpuls ∼ 3 ω. Solving Equation 5.7 for N, where we approximate for N 1 simplicity NC ≈ 2 , gives as an order of magnitude estimation that N ∼ 10 .

By comparing the amplitudes of the MCCRs in Figure 5.8 to the amplitudes of the SCCRs in Figure 4.7 and Figure 4.12, we infer that the pulsational velocity amplitudes on the surface of β Pictoris are ∼ 1 km s−1.

5.3.2 Fitting procedure

Guided by the rough estimates of the previous subsection, we ﬁt the SCCRs of a superposition of two sinusoidal waves to the MCCRs of MJD 54542. One of these is an wave inducing motion in the radial direction (parametrised by Nrad, vA,rad and φ0,rad), while the other induces motion in the azimuthal direction (parametrised by Nazi, vA,azi and φ0,azi). We generate SCCRs for the −1 ‘radial’ wave in isolation, for many values of Nrad (Nrad ∈ [5, 6, ..., 29, 30]) and vA,rad = 1 km s .

Similarly, we generate SCCRs for the ‘azimuthal’ wave in isolation, for many values of Nazi −1 (Nazi ∈ [5, 6, ..., 29, 30]) and vA,azi = 1 km s .

Then, exploiting the superposition principle, we build for each combination (Nrad,Nazi) the SCCRs of a mixed ‘radial-azimuthal’ pulsation. We ﬁt these to each of the MCCR time slices, using vA,rad, vA,azi, φ0,azi and t as the 4 parameters to ﬁt for each slice. By sticking with

φ0,rad = 0, the phase diﬀerence between the two waves is equal to the value of φ0,azi. We pretend that the pattern is imprinted on the stellar surface, but allow for better ﬁts by setting t free. This allows the mock star to be rotated around its axis at will for each slice.

5.3.3 Fitting results

Figure 5.10 shows the mean standard deviation reduction hOσi obtained by fitting SCCRs to the pulsations of MJD 54542, for 100 wavenumber sets (Nrad,Nazi). The best of the fits obtained in this way is shown in Figure 5.11. The corresponding SCCR set is generated by pulsations with (Nrad,Nazi) = (9, 17). We call this set SCCRbest. Let us now define the corrected measured spectral cross correlation function residuals, or CMC-

CRs for short, where CMCCR(t) := MCCR(t) − SCCRbest(t). The CMCCRs thus are MCCRs, corrected for unwanted pulsation signals by means of subtraction of the best ﬁtting SCCRs. As can be seen from Figure 5.11, the standard deviation of the CMCCRs is roughly 26% lower than the standard deviation of the MCCRs for night MJD 54542. Time slices for which our model is most successful feature a standard deviation reduction in excess of Oσ = 40%.

Logically, ﬁtting a night with a combined ‘radial-azimuthal’ pulsation always works better than

63 Figure 5.10: HARPS β Pictoris spectral CCF residuals of the night MJD 54542 ﬁtted with various sets of pulsations, of which 1 induces radial and 1 induces azimuthal motion. The best ﬁt occurs for

(Nrad,Nazi) = (9, 17).

Figure 5.11: HARPS β Pictoris spectral cross correlation function residuals fitting for the night 54542 MJD. The MCCRs are fitted with SCCRs generated from applying the linearity principle to SCCRs of a single sinusoidal pulsation in radial direction and SCCRs of a sinusoidal pulsation in azimuthal direction. The overview shows the best fit.

fitting a sole ‘radial’ or a sole ‘azimuthal’ pulsation: for parameter value vA,rad = 0, the pulsation reduces to an ‘azimuthal’ pulsation, while for vA,azi = 0 the pulsation is reduced to a ‘radial’ pulsation. Nonetheless, it is instructive to inquire how well single sinusoidal pulsations fit MC- CRs of MJD 54542 and other nights. Being able to use them effectively would be powerful, as a

64 single sinusoidal pulsation only has 2 free parameters - vA and t - to be ﬁtted for each MCCR time slice for a given N.

• Trying out a range of single pulsations in radial direction (Nrad ∈ [5, 6, ..., 29, 30]) results in a best fit with Nrad = 15 and a mean standard deviation reduction hOσi = 12%. When doing the same for single pulsations in azimuthal direction, we find a best fit characterised by Nazi = 17 and hOσi = 13%, shown in Figure A.7. The sawtooth-shaped time plot shows that for each MCCR time slice, the best fitting SCCR time slice is obtained from slightly rotating the mock star with respect to the optimal stellar orientation for the previous MCCR time slice. Considering we allow t to vary arbitrarily, this behaviour is consistent with our earlier idea of replicating MCCRs by imprinting a fixed pulsation pattern on the stellar surface, whose appearance to an observer changes due to stellar rotation only.

• Figure A.8 shows the best ﬁtting single pulsation in radial direction for MJD 54799 (Nrad = 14, hOσi = 25%). The best ‘azimuthal pulsation’ ﬁt has Nazi = 17 and hOσi = 13%.

• Figure A.9 shows the best ﬁtting single pulsation in radial direction for MJD 55597 (Nrad = 20, hOσi = 18%). The best ‘azimuthal pulsation’ ﬁt has Nazi = 25 and hOσi = 8%. The data were taken 3 years after those of MJD 54542.

The fact that the wavenumbers of the best ﬁtting sinusoidal pulsations vary only ∼ 1 over the course of several years, suggests a moderately stable pulsation morphology for β Pictoris.

Based on the analyses of MJD 54542, 54799, 55597 and other nights, we expect the existence of several wavenumber set (Nrad,Nazi) whose corresponding pulsation superposition leads to SC- CRs that could ﬁt all nights reasonably well. Performing the calculation, we ﬁnd that pulsations characterised by (Nrad,Nazi) = (14, 12) generate SCCRbest, for hOσi = 22%.

5.4 Improving on RV measurements

5.4.1 Creating CMCCF

After the pulsation correction set SCCRbest(t) has been found for all HARPS β Pictoris data, we create corrected measured spectral cross correlation functions, or CMCCFs for short. They are deﬁned by CMCCF(t) := MCCF(t) − SCCRbest(t), analogous to CMCCRs. See Figure 5.12, and compare with Figure 2.1.

5.4.2 Recalculating RVs

After we have obtained CMCCFs - MCCFs with pulsations partly attenuated - we recalculate the radial velocities found in Chapter 2. There, we adopted a rather simplistic approach to the determination of the radial velocities. As an RV data point, we chose the RV shift ∆Svrad that minimised the error between MCCF and ﬁtting curve C0(vrad, tref ). Of course, we could adopt the same approach now. Considering that the main aim of ﬁtting the pulsational signal in the

65 Figure 5.12: Upper graph: More than 1000 pulsation-corrected HARPS β Pictoris spectral cross correlation functions (CMCCFs) plotted on top of eachother show RV-dependent variability. Lower graph: This variability between the CMCCFs is quantiﬁed at each radial velocity via the CMCCF standard deviation. Note that the same vertical scale was used as in Figure 2.1.

MCCFs is a potential accuracy increase in the determination of the actual radial velocities, we proceed now with a more careful approach.

First, like before, a set of possible RV shifts ∆Svrad is generated. For each CMCCF, we

1. calculate the standard deviation of the diﬀerence between the CMCCF and C0(vrad −

∆Svrad, tref ), for all values of ∆Svrad generated;

2. retain all RV shifts with their corresponding ﬁt error, instead of remembering only the best ﬁtting RV shift. For MJD 54542, this results in data shown in Figure 5.13. By retaining all data, we possess information not only about which RV value is the most likely of all considered, but also about the likelihood that other, adjacent RV values in fact equal the actual radial velocity. After all, one RV shift might be the best, but there might be several almost equally good. This information would otherwise be discarded.

From Figure 5.13, one can glimpse that radial velocity data points with high outliers do not necessarily have a higher ﬁt error. Naively, one might expect this.

As a next step, we fit a linear path through the fitting error map, adopting as fitting criterion that good fits minimise the sum of all fitting errors encountered along the way. Any trend

66 Figure 5.13: Map with fitting errors for a range of potential radial velocity measurements, calculated for the HARPS β Pictoris spectra taken during MJD 54542. The image is overlaid with a curve connecting the points of lowest fitting error, and results in an oscillation reminiscent to the one in the lower plot of Figure 2.2. of RV increase or decrease over the course of years, indicative of the presence of β Pic b, is to +5.3 be found in this way. The linear fit is justified as the orbital period of β Pic b is P = 22.4−1.5 years (Millar-Blanchaer et al., 2015), so that only half of a sinusoidal cycle would be discernible with data taken over the course of ∼ 10 years. Moreover, the HARPS spectra were sampled at nights lying around MJD ∼ 54000, the estimated epoch of the secondary transit (the moment at which β Pictoris blocks our view of the planet). At transits, the radial velocity induced by the planet is 0, but the change in stellar radial velocity is at its maximum over time. The RV trend in our data should therefore correspond mostly to the linear part of the sinusoidal cycle.

The resulting ﬁt through the corrected radial velocity (CRV) data is undistinguishable from a direct linear ﬁt through the CRV data points. Moreover, the scatter in the radial velocities within nights is not reduced when compared to the original radial velocities of Chapter 2, before pulsation subtraction.

It is remarkable that the scatter in the radial velocities does not decrease, even though we remove 25% of the stellar pulsations. A possible explanation is that mostly the pulsational eﬀects near the centre of the MCCFs are removed, while more eccentric oscillations remain uncorrected. Figure 5.14 demonstrates this.

67 Figure 5.14: Comparison between the RV-dependent variability between MCCFs and CMCCFs. For most radial velocities, the pulsation subtraction indeed decreases the spectral cross correlation variability. For some radial velocities near the edges though, the variability increases.

68 Chapter 6

Discussion

6.1 Stellar model inconsistencies

Pulsational and rotational velocities in stars vary with depth, and therefore also along an indi- vidual sight line. Our stellar model does not recognise this - for instance, it produces maps of ‘the’ rotational velocity of the visible stellar surface. Such a notion is only well-defined assuming that the stellar plasma has infinite opacity, so that only an infinitesimally thin surface layer contributes to the observable spectrum. This is clearly not the case in reality. Moreover, our motivation to consider the effect of limb darkening in the model was indeed the recognition that stars have a considerable translucency, so that observers see light coming from a whole range of loci along a line of sight, and not just from an infinitesimally thin surface layer. As a result, we have uncovered within our model an internal conceptual inconsistency.

A proper way to resolve the conflict described is, of course, to change the model so that the varying rotational velocity of stellar plasma along each sight line is taken into account. Instead of using a single rotational velocity map, one could generate a set of such maps, each for another depth, thereby effectively dissecting the star into various ‘onion layers’ or shells. In addition, the limb darkening map could be replaced by an equally numerous collection of intensity maps. Each such map would express the perceived intensity of the corresponding layer, which is determined by two counteracting effects. On the one hand, the perceived intensity increases with depth as temperatures rise and emission is strongly temperature-dependent. On the other hand, the perceived intensity diminishes with depth due to absorption according to the radiative transfer equation solution1

R 0 0 0 − aλ(x,y,z )dz −τλ(x,y,z) Iλ,perceived(x, y, z) = Iλ,actual(x, y, z) e z = Iλ,actual(x, y, z) e , (6.1) where aλ is the absorption coeﬃcient and τλ is the optical depth. Moreover, the strong temperature gradient would necessitate making the broadening of the mean 1Inspired by the absorption-only solution, which can be found in exempli gratia (Rybicki and Lightman, 2004).

69 spectral line dependent on the layer depth. The ﬁnal observable spectrum would again be constructed by integrating over the apparent stellar disk, where the spectrum of each surface element is itself composite - namely the sum of spectra with diﬀerent line broadening, Doppler shifts, and perceived intensities.

In practice, resolving the aforementioned inconsistency may not signiﬁcantly improve the model’s eﬃcacy. To assess the necessity of introducing the complications set forth, we estimate the thickness of the photosphere, the layer from which we expect to see emission directly. For Sol, it is ∼ 500 km thick,2 which is only ∼ 10−1% of its total radius of ∼ 700, 000 km. As the rotational velocity of a plasma blob is proportional to its distance from the spin axis, material on the inner edge of the photosphere also rotates ∼ 10−1% slower. In light of the considerable increase in complexity described above and the marginal gain motivated here, we favour the simple model for the purposes of this work.

6.2 High frequency variations

The persistence of high frequency variations in the radial velocity data after pulsation subtraction is surprising, considering that no substantial decrease in the magnitude of the variations seems to occur. The reason for this should definitely be inquired. One might be tempted to say that HF variations emerge due to ripples near the centres of the spectral CCFs, where their variability is larger. This though, would suggest that pulsation attenuation as performed in this work should decrease these variations. If, on the other hand, pulsations near the CCF edges cause the HF variations, then we could decide to change our criterion of what SCCRs constitute good fits of a given MCCR time slice, exempli gratia by giving more priority to fitting (and therefore counteracting) pulsational signatures near the CCF edges than near the CCF centre.

2Exempli gratia, (Fraknoi et al., 2000) points out that just 4% of the light emerging from a depth of 400 km within the photosphere directly escapes Sol.

70 Chapter 7

Summary and conclusion

In the last two decades, curious and daring torchbearers of Homo sapiens have discovered thousands of planets scattered across the Galaxy. However, detecting planets orbiting stars with asteroseismological activity remains challenging. Oscillations and starspots change the shape of the mean stellar absorption line, leading to noise in the determination of the line-of-sight velocity of these stars with respect to the Solar System barycentre. This noise eﬀect can be so severe that it heavily limits our ability to detect exoplanets around these stars with high-resolution spectrographs and, subsequently, to measure their masses. In this work, we have a developed a new method to counteract the eﬀect of stellar pulsations on radial velocity measurements.

The method proposed amounts to fitting the time-dependent oscillations of the mean stellar line with synthetic line shape variations stemming from a model of a pulsating, rotating star. On this mock star, we imprint a simplistic pattern of effective pulsations, not meant to constitute an exact physical representation of the - oftentimes complex - actual oscillatory stellar surface motion, but merely to reproduce as many features of the line shape variations observed as possible with a minimal set of fit parameters. In particular, we explored the efficacy of sinusoidal pulsations varying over azimuthal angle.

To test our method, we applied the developed machinery to archival HARPS data of β Pic- toris, known to be orbited by a giant gas planet that has hitherto escaped unambiguous radial velocity detection due to asteroseismological activity. We show how ∼ 25% of the pulsational noise can removed with a stellar model featuring a superposition of two sinusoidal modes, one inducing motion in the radial direction, whilst the other induces motion along the stellar surface in the direction of rotation. To explain the observed line variations, the amplitudes of these velocity waves must be ∼ 1 km s−1. The pulsational velocity reaches this amplitude at ∼ 10 crests that move over the surface with an angular velocity ∼ 3 times the stellar rotational angular velocity.

The removal of the ﬁtted pulsational signatures from the data does not reduce the extent of

71 the high frequency radial velocity variations observed over the course of hours in well-sampled nights. Follow-up research should make clear why. Notwithstanding this unsatisfactory result, we have built a great tool to study the eﬀects of arbitrary patterns of stellar pulsations and starspots on the spectra of rotating stars, and have shown how stellar parameters such as the equatorial rotational velocity, limb darkening coeﬃcients and mean absorption line parameters can be measured using such a model.

72 Chapter 8

Acknowledgements

As my concluding remarks, I’d like to thank several people that have contributed with their substantive remarks, advice and eﬀort to making this research project a success!

• First of all, I would like to thank Professor Ignas Snellen, my supervisor during this project, for introducing me to the novel and exciting ﬁeld of exoplanets. In light of substantial public interest and the construction of promising new instrumentation, I’m sure there will be many impactful planetary discoveries in the years to come. While working in the research group, I obtained a much better sense of how science in general, and science on exoplanets in particular, is done. The group is friendly and inclusive, and the meetings were a good way to get suggestions for possible ways forward by employing the power of crowd intelligence. It was a great experience working together.

• From the start, Jens Hoeijmakers has been enthusiastically supporting me during the project. He was always so kind to free up time to discuss speciﬁc technicalities with me that better suited a personal conversation than a discussion during the group meeting. His attitude is both attentive and forward-thinking: as an example, Jens oﬀered to talk with attendants of the Nederlandse Astronomen Conferentie (NAC) 2016 to consult them about my project. I’d like to be so nice to M.Sc. students during my Ph.D. as well!

• Both Saskia Hekker and Professor Malcolm Fridlund, seasoned experts in the ﬁeld of asteroseismology, were so nice to spend some time discussing the feasibility of my approach to the project goals and possible alternative routes.

• I’d like to say thanks to Emanuelle di Gloria, who helped me by explaining a good way to compute spectral cross correlation functions in the Python programming language.

• My dear friend Jacob Bakermans has been quite helpful in various discussions outside of the walls of the university. It is great to see his ability to absorb a lot of information in a short time, and then giving valuable advice.

73 • Also, I am obliged to Arjaan Kooijman for his spiﬃng and sweet introduction to BIBTEX, of which I have become an aﬁcionado ever since.

• Finally, I’d like to thank my mum, Antoinette Kootte, for supporting me in so many ways along the project timeline, most notably during the last weeks.

This research has beneﬁtted from the Exoplanet Orbit Database and the Exoplanet Data Ex- plorer at exoplanets.org. Great free software used include typesetting system LATEX, Mac OS X code editor TextWrangler, Microsoft Windows code editor Spyder and programming language Python 3.

74 Chapter 9

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78 Appendix A

Additional ﬁgures

This appendix contains several ﬁgures that provide insight and understanding of the subject matter, but whose inclusion in the main text would make it more cluttered.

A.1 Chapter 4

Figure A.1: Simulated relation between the number of local maxima in the SCCRs NC and the wavenumber of a sinusoidal pulsation in azimuthal direction of wavenumber N. The naive expectation motivated in the text has been drawn as well, alongside two polynomial ﬁts of degree 1 and 2.

79 Figure A.2: Discrete approximations to probability density functions of / vrad . for various wavenumbers N, corresponding to a single sinusoidal pulsation in azimuthal direction.

Figure A.3: Simulated relation between the ‘distance’ in radial velocity between two subsequent local maxima / vrad . in an SCCR time slice and the wavenumber N of the responsible single sinusoidal pulsation in azimuthal direction.

80 A.2 Chapter 5

Figure A.4: Inductive demonstration of approximate linearity of SCCRs. Upper graph: additivity is shown to hold to high degree in case of a mixed radial-azimuthal pulsation pattern. Lower graph: homogeneity is demonstrated to hold to high degree for a radial pulsation pattern. Per graph, the color scales are the same.

81 Figure A.5: Simulated spectral CCF residuals amplitude spectra for a single pulsation in radial direction

(N = 20). Left: stacked normalised amplitude spectra, one for each value of vrad. Right: normalised cumulative or mean amplitude spectrum. Both normalisations are such that the peak value is 1. td = 17.2 h was used.

Figure A.6: Simulated spectral CCF residuals amplitude spectra for a single pulsation in azimuthal direction (N = 20). Left: stacked normalised amplitude spectra, one for each value of vrad. Right: normalised cumulative or mean amplitude spectrum. Both normalisations are such that the peak value is 1. td = 17.2 h was used.

82 Figure A.7: HARPS β Pictoris spectral cross correlation function residuals fitting for the night 54542 MJD. The MCCRs are fitted with SCCRs of a single sinusoidal pulsation in azimuthal direction. Upper image: the overview shows the best fit. Lower image: MCCR time slice most helped by SCCR fitting.

83 Figure A.8: HARPS β Pictoris spectral cross correlation function residuals fitting for the night 54799 MJD. The MCCRs are fitted with SCCRs of a single sinusoidal pulsation in radial direction. Upper image: the overview shows the best fit. Lower image: MCCR time slice most helped by SCCR fitting.

84 Figure A.9: HARPS β Pictoris spectral cross correlation function residuals fitting for the night 55597 MJD. The MCCRs are fitted with SCCRs of a single sinusoidal pulsation in radial direction. Upper image: the overview shows the best fit. Lower image: MCCR time slice most helped by SCCR fitting.