Doctor of Philosophy

RICE UNIVERSITY

Alison O. Farrish

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE

APPROVED, THESIS COMMITTEE

David Alexander

Frank Toffoletto

Kirsten Siebach

HOUSTON, TEXAS April 2021 ABSTRACT

In the past two and a half decades, advances in the field of exoplanet detection have confirmed more than 4,000 known planets outside of our Solar System [Brennan, 2019]. With this wealth of data, the field is now poised to transition from a phase of detection to one of more in-depth characterization of planetary processes and evolution. Exoplanet systems are of interest not only for the potential for habitability, but also in the opportunity they provide for the study of comparative heliophysics - the similarities and differences in physical interactions between the central host star and any associated planets. In applying solar- and heliophysics- based knowledge and tools to the study of exoplanet systems, we can expand our understanding of the breadth of possible star-planet interactions and the influence of stellar behavior on planetary environments and processes such as atmospheric loss, planetary magnetosphere dynamics, ionospheric emission, and more. We present here a series of studies of solar-stellar connections and the heliophysics of exoplanet systems, employing a surface flux transport treatment of photospheric flux emergence, migration, and dispersal, and the application of this solar-based modeling framework to exoplanet host stars. In Chapter 1, we describe the state of the field of exoplanet characterization, relevant solar physics concepts, and the context for making comparisons between the Sun and other stars. Chapter 2 comprises the methodology employed in modeling stellar photospheres, coronae, and asterospheres with relevance to exoplanet space weather environments. Chapter 3 expands upon the solar-stellar connection, demonstrating the application of magnetic flux transport modeling to the simulation of stellar activity across a broad population of cool stars. Chapter 4 details our modeling of magnetic and energetic environments driving star-planet interaction. In Chapter 5, we present examples of applications of this work to planetary response modeling, detailed investigations of stellar extreme ultraviolet (EUV) emission, and young Sun analogues. Future applications of our integrated modeling approach, particularly in comparison with solar dynamo models and with upcoming observing campaigns from Parker Solar Probe and James Webb Space Telescope, are discussed in Chapter 6. Acknowledgments

I would like to thank my advisor Dr. David Alexander for his teachings, support, and encouragement over the course of my graduate studies. Thank you also to my committee members, Dr. Kirsten Siebach and Dr. Frank Toffoletto, for their participation and support in preparing this thesis. Additional support and helpful discussions from Dr. Chris Johns-Krull and Dr. Anthony Chan contributed greatly to the work presented in this thesis. Mei Maruo, Minjing Li, and J.J. Odell also supported this work as undergraduate researchers, and I wish them all well in their future careers. Thank you to Dr. Will Barnes for his patience and guidance in helping me to learn coronal loop physics and modeling. Thank you as well to Dr. Anthony Sciola for collaboration and support over the years, and I congratulate him wholeheartedly on his recent doctoral defense. Many thanks to Dr. Marc DeRosa for his support and advice in implementing the Surface Flux Transport modeling and for helpful scientific discussions. I am also grateful to Dr. Sofia Moschou, Dr. Jeremy Drake, and Dr. Ofer Cohen for fruitful conversations and collaboration. I owe my scientific career to the encouragement of mentors like Mr. John Powell of Tolland High School and Dr. Martha Haynes of Cornell University. Additionally, I would not be the person and scientist I am today without the emotional support, study help, and good times I have shared with Faye Elgart and Roberto Baquerizo. Thanks to my graduate school classmates like Anastasia Newheart, Laura Flagg, Maxwell Hummel, and Sarah Freed for their friendship and collaboration in coursework. Thank you also to Jason Ling and James Webster for being excellent partners in teaching, and to Dr. Lam Yu for his mentorship in teaching. Many thanks to my wonderful friends, including but not limited to Alix Macklin, iv

Amanda Woods, Andy Bhateja, Asa Stahl, Dr. Asante Hatcher, Brian Menegaz, Ian Kintslinger, Megan Chang, Dr. Meike van der Heijden, and Nia Christian. All my love and appreciation goes to my parents, Thomas and Katherine Farrish, for supporting me and my education. Thank you also to my best friends, Amanda Farrish and Ryan Hughes. Thank you as well to Joe Whalen, for everything you do. This thesis is dedicated in loving memory to my grandfather Dr. Raymond Farrish and to my dear friend Jadielle Ray. Contents

Abstract ii Acknowledgments iii List of Acronyms viii List of Illustrations x List of Tables xix

1 Introduction 1 1.1 Characterization of Exoplanet Systems ...... 2 1.1.1 Theories of Habitability ...... 3 1.1.2 Detection Methods and Observational Biases ...... 6 1.2 Solar Activity and Impacts on Solar System Planets ...... 9 1.2.1 The Solar Cycle and Dynamo Action ...... 10 1.2.2 Solar Coronal Emission ...... 12 1.2.3 The Solar Wind and Heliosphere ...... 15 1.3 Stellar Observations Context ...... 17

2 Solar and Stellar Modeling Capabilities 20 2.1 Surface Flux Transport ...... 20 2.2 Coronal Magnetic Structure ...... 26 2.3 Coronal XUV Emission ...... 30 2.4 Solar and Stellar Wind Propagation ...... 33 2.5 Planetary Magnetospheric Response ...... 34 2.6 Conclusion ...... 35 vi

3 Modeling the Activity of Stellar Populations 37 3.1 Emission and Rotation of Cool Stars ...... 38 3.2 Calibration of Solar-Based Modeling ...... 44 3.2.1 Magnetic Flux Transport Modeling ...... 44 3.2.2 Relationship between X-ray Luminosity and Magnetic Flux . . 46 3.2.3 Calibration of Stellar Bolometric Luminosity ...... 52 3.3 Modeling the Stellar Activity-Rotation Relation ...... 56 3.3.1 Incorporating Stellar Bolometric Luminosity Evolution . . . . 56 3.3.2 Comparison with Observed Stellar Populations ...... 59 3.4 Conclusions ...... 62

4 Impact of Stellar Activity on Exoplanet Environments 65 4.1 Simulating the Asterospheric Magnetic Field over Time ...... 66 4.2 Asterospheric Fields of Exoplanet Host Stars ...... 69 4.2.1 Stellar Magnetic Field Topology ...... 70 4.2.2 Stellar Alfv´enSurface ...... 73 4.2.3 Asterospheric Current Sheet ...... 78 4.3 Discussion of Results ...... 83

5 Applications to Star-Planet Interaction 88 5.1 Planetary Magnetospheric and Ionospheric Processes ...... 88 5.2 Solar and Stellar Coronal Emission ...... 90 5.3 Young Solar Analogs ...... 93

6 Future Work 97 6.1 Solar Variability and Anomalous Active Regions ...... 97 6.2 Sub-Alfv´enicRegime Characterization ...... 99 6.3 Potentially Habitable Exoplanet Systems of Interest ...... 102 vii

Bibliography 104 List of Acronyms

AAR: Anomalous active region ACS: Asterospheric current sheet AU: Astronomical Unit AWSoM: AlfvénWave Solar Model [van der Holst et al., 2014] CHZ: Circumstellar habitable zone CME: Coronal mass ejection DKIST: Daniel K. Inouye Solar Telescope EBTEL: Enthalpy-Based Thermal Evolution of Loops [Bradshaw and Cargill, 2013] EUV: Extreme ultraviolet FC: Fully-convective GAMERA: Grid-Agnostic MHD for Extended Research Applications [Zhang et al., 2019] IMF: Interplanetary magnetic field JWST: James Webb Space Telescope MAVEN: Mars Atmosphere and Volatile EvolutioN MHD: Magnetohydrodynamics MMS: Magnetospheric Multiscale NSO: National Solar Observatory PC: Partially-convective PFSS: Potential Field Source Surface [DeRosa et al., 2011] PSP: Parker Solar Probe RCM: Rice Convection Model [Toffoletto et al., 2003] RTV: Rosner, Tucker, Vaiana [Rosner et al., 1978] RV: Radial velocity ix

SBC: Sector boundary crossing SDO: Solar Dynamics Observatory AIA: Atmospheric Imaging Assembly HMI: Helioseismic and Magnetic Imager SFT: Surface Flux Transport SoHO: Solar and Heliophysics Observatory STEREO: Solar Terrestrial Relations Observatory TESS: Transiting Exoplanet Survey Satellite THEMIS: Time History of Events and Macroscale Interactions during Substorms WSA: Wang-Sheeley-Arge solar wind model [Arge and Pizzo, 2000] XUV: Collective term for X-ray, far UV and EUV emission (5-2000 A)˚ YaPSI: Yale-Potsdam Stellar Isochrone [Spada et al., 2013, Spada et al., 2017] ZDI: Zeeman Doppler Imaging Illustrations

1.1 From [Mendez, 2013], adapted from [Kopparapu et al., 2013], the range of circumstellar distances defining the habitable zone for host stars of different stellar temperatures and spectral types, based on two different planetary atmosphere models. The Earth’s location at 1

AU around a 1 MSun star is within the limits of the habitable zone by definition...... 4 1.2 A comparison of confirmed exoplanets as of April 2014, color-coded by detection method, from [Batalha, 2014]. NASA’s Kepler satellite revolutionized the field of exoplanet detection, confirming the existence of 2,414 extrasolar planets over the lifetime of the mission. Planets detected by the transit method with Kepler (yellow dots) are most commonly found in short-period orbits, less than about 100 days. Planets detected by the radial velocity method with other instruments are strongly biased towards giant planets of Neptune-size and greater...... 6 1.3 A magnetic ‘butterfly’ diagram showing cyclic patterns in the bipolar active regions on the solar photosphere over three solar cycle periods of about 11 years each. Data is aggregated from magnetometers at the National Solar Observatory (NSO) and onboard the SoHO spacecraft [Hathaway, 2010]. Yellow denotes positive polarity (outward from the photosphere) while blue denotes negative polarity (into the photosphere)...... 11 xi

1.4 Taken with the AIA instrument onboard SDO, an image of the Sun in the EUV (193 A),˚ with a prominent coronal hole. Coronal loop structures can also be seen, particularly in bright regions [NASA/SDO, 2010]...... 14 1.5 The solar wind distribution in speed and IMF orientation observed with the Ulysses spacecraft. In the polar regions, magnetic field lines are open to interplanetary space and solar wind particles stream quickly (up to ∼800 km/s) along open field; at low latitudes, more magnetic field lines are confined to the corona and escaping plasma moves more slowly (∼300-400 km/s) [Hathaway, 2014]...... 16 1.6 The Hertzsprung-Russell diagram of stellar luminosity vs. temperature or color. Top axis shows the stellar spectral classes [O,B,A,F,G,K,M]. The main sequence, labeled in the figure, is the prominent track of stars in the center of the diagram. Points on the plot are observed luminosities and temperatures from the Hipparcos Catalogue and the Gliese Catalogue of Nearby Stars [Powell, 2006]. . 18

2.1 At left, a magnetograph of the Sun taken on March 31, 2021 with SDO’s Helioseismic and Magnetic Imager [Zell, 2014], the most recent measurement from HMI at the time of writing, near solar minimum. At right, an HMI magnetograph from July 1, 2014, when the Sun was in its maximum phase. White concentrations are of positive magnetic polarity, oriented toward the viewer, while black concentrations are of negative polarity oriented away from the viewer. The mixed-polarity network of small ephemeral regions is apparent throughout the solar disk at both phases. At solar minimum only a few signiﬁcant bipolar concentrations are apparent; at maximum, many large active regions are present...... 21 xii

2.2 Comparison between the longitudinally averaged magnetic ﬁeld observed on the Sun (top) and modeled with the SFT (bottom) [Schrijver and Title, 2001] for a single magnetic polarity cycle of

period Pcyc ≈ 21.9 years. Patterns of observed solar behavior such as Spörer’s,Hale’s, and Joy’s laws, as well as the field reversals over a magnetic polarity cycle, are replicated with very good agreement in the solar flux transport model...... 25 2.3 At top, the modeled photospheric flux distributions for an SFT simulation with solar parameters (from left to right, at declining, minimum, rising, and maximum cycle phases). Positive magnetic polarity is shown in white and negative magnetic polarity is shown in black, as in HMI photospheric observations (see Figures 2.1 and 2.2). At bottom, the corresponding PFSS coronal field extrapolations for each of the cycle phases. Near solar minimum, the coronal magnetic field is largely dipolar; at solar maximum, the coronal structure is much more disordered and open-field coronal holes appear at lower latitudes...... 29 2.4 A coronal loop described by the one-dimensional spatial coordinate s; the background image is real imaging of the solar corona taken with the Transition Region and Coronal Explorer (TRACE) instrument in November 1999 [Reale, 2014]...... 32 2.5 At left, the modeled solar wind density and current sheet structure for a stellar minimum case using the AlfvénWave Solar Model (AWSoM); at right, the same modeled star at stellar maximum, with a highly disordered asterospheric current sheet and enhanced stellar wind pressure (the same color bar describes the density in both panels) [Alvarado-Gómezet al., 2016]...... 34 xiii

3.1 The activity-rotation distribution of a population of 824 observed stars of partially-convective (gray points) and fully-convective (pink and red points) interiors, from [Wright et al., 2018]. The presence of

FC stars in the unsaturated branch at Ro & 0.1 implies possible similarities in the production of magnetic ﬂux and its conversion to energetic coronal emission in both PC and FC populations of cool stars. 39 3.2 The solid line shows the saturation threshold Ro < 0.13 used in [Wright et al., 2011], as a function of stellar mass and age. A one solar mass star crosses the saturation threshold at ∼ 110Myr, while a

star of 0.3 MSun crosses the threshold and joins the unsaturated regime at about 1 Gyr...... 40 3.3 The X-ray luminosity vs. magnetic ﬂux for a variety of magnetic solar and stellar sources, from [Pevtsov et al., 2003]. Dots represent quiet Sun regions; squares denote X-ray bright points; diamonds represent solar active regions; pluses show full solar disk averages. Crosses denote the spatially-unresolved disk observations of active G, K, and M stars, and the open circles represent similar observations of young, accreting T Tauri stars. The power-law ﬁt to all data types has slope p=1.15 and is shown as the solid line...... 43 xiv

3.4 The variation of X-ray spectral radiance, in units of erg · s−1, with

scaled Rossby number (Ro/RoSun) for SFT simulations with

R∗ = RSun. The vertical spread in LX is a result of cyclic modulations in magnetic activity for each stellar model, producing variation in the associated X-ray emission. The linear ﬁt to the data

in the range 0.1 . Ro/RoSun . 1.2 displays the correlation between X-ray emission and Rossby number derived from the application of the [Pevtsov et al., 2003] scaling relationship. The dashed line

extends the range to 0.05 . Ro/RoSun . 2.5, where the linear scaling still appears valid, and allows for the variation in the assumed values

of RoSun in the literature (see discussion in main text)...... 50 3.5 Fractional X-ray luminosity as a function of scaled Rossby number,

for SFT simulated stars with R∗ = RSun. Diﬀerent values of Lbol for each Rossby number are determined based on a solar spin-down

model [Bahcall et al., 2001]. Lbol varies with Prot as the Sun-like star ages and spins down over time. This approach limits the data to a

narrower set of Rossby numbers corresponding to the range of Prot values achievable over the modeled lifetime of the Sun...... 53 3.6 Fractional X-ray luminosity as a function of scaled Rossby number, for SFT simulated stars of late-F, G, K, partially-convective (PC) M,

and fully-convective (FC) M types. Lbol is calculated using evolutionary tracks provided by YaPSI

[Spada et al., 2013, Spada et al., 2017]. The Lbol calibrations include the eﬀects of both stellar spindown of rotation period and the

evolution of τc with age. The slope of the linear ﬁt, p, is included for each stellar type...... 55 xv

3.7 One iteration of a random sampling of 175 partially-convective and 15 fully-convective stars from the full dataset in Figure 3.6. An ensemble of 10,000 random samplings was taken, and the mean slope of this ensemble of ﬁts is represented by the solid red line. The standard deviation of the slopes of all 10,000 iterations is represented by the grey shading. Modeled stars normalized by bolometric luminosities corresponding to partially-convective (late-F, G, K, and early-M) stellar types are displayed as blue dots; stars normalized by bolometric luminosities appropriate for fully-convective (FC) late-M stars are denoted by red crosses. For comparison, the observed stellar populations of [Wright et al., 2018] are included as grey dots...... 60

4.1 Three example stellar ﬂux transport simulations with associated coronal ﬁeld extrapolations. (a), (b), and (c) display the stellar

photosphere ﬂux distributions for Ro = 0.5 RoSun, Ro = RoSun, and

Ro = 4 RoSun, respectively. These flux distributions are represented in the form of line-of-sight magnetograms with white corresponding to positive magnetic polarity and black corresponding to negative magnetic polarity, as in Figure 2.3. (d), (e), and (f) display the associated coronal magnetic field line distributions for each stellar simulation, extrapolated using the PFSS method. Field lines in magenta (outward) and green (inward) represent fields that cross the source surface and extend into interplanetary space, forming the magnetic field of the inner asterosphere. Field lines depicted in black represent closed magnetic field lines with each end rooted in the stellar photosphere, forming the loop structures of the stellar corona. 68 xvi

4.2 (a) Ratio of open-field flux to total surface magnetic flux as a function of stellar activity at stellar maximum and stellar minimum. Error bars are associated with a ±10% variation in the simulation source strength, corresponding to variation in the open-to-total flux ratio for a given Rossby number, Ro. (b) Ratio of open-field flux to total stellar flux in three latitude bands: ± 0o − 30o (solid lines), ± 30o − 60o (dashed lines), and ± 60o − 90o (dotted lines). Each data point represents an average of the quantity over the northern and southern hemispheres...... 72 4.3 The Alfvénsurface of a Proxima Centauri-like star, shown in blue, is modeled for a maximum stellar magnetic field strength of 600G (left) and a mean field strength of 600G (right), using an MHD model of the stellar wind driven by Alfvénwave turbulence [Garraffo et al., 2016]. The plane denotes the location of the asterospheric current sheet; a theoretical orbit of the planet Proxima Centauri b, with a semi-major axis of 0.049 AU, is show in black. In the 600G-mean case at right, the orbit of Proxima Centauri b passes inside the Alfvénsurface once per orbit...... 74 xvii

4.4 (a) Alfv´ensurface radius as a function of stellar activity, at both stellar minimum and stellar maximum. Alfv´enradius is scaled to the

value for the Sun at solar maximum, equal to ∼20 RSun. Shaded error regions are associated with the uncertainty in the exoponent of Equation 4.1. For comparison, we show the relationship derived by [Schrijver et al., 2003]. (b) The orbital locations of several known close-in exoplanets are shown in relation to observational estimates of their host stars’ Rossby numbers. Many known exoplanets with orbits on the order of 1 AU or farther lie far above the upper limit of the plot and are expected not to interact measurably with the host star’s Alfv´ensurface...... 76

4.5 Radial magnetic ﬁeld component Br at the source surface for the

same simulations shown in Figure 4.1 (Ro/RoSun = [0.5, 1.0, 4.0]), shown at both stellar maximum (top) and minimum (bottom). The projection of two theoretical planetary orbits with 0o (dashed) and 30o (dotted) inclinations are shown...... 78 4.6 (a) Shown in red and blue, the magnitude of the average radial magnetic ﬁeld gradient at locations where the 0o inclination orbit (see Fig. 4.5) traverses a polarity inversion crossing for stellar maximum and minimum, respectively. The values are normalized to the largest

value in the set, the gradient at stellar maximum for Ro = 0.1 RoSun. (b) In red and blue, the magnitude of the average radial magnetic ﬁeld gradient at locations where the 30o inclination orbit crosses a polarity inversion, for stellar maximum and minimum, respectively. The values are normalized to the largest value in the set, the gradient

at stellar maximum for Ro = 0.1 RoSun...... 83 xviii

5.1 At left, a large active region from a surface flux transport model of the Sun, with a potential-field extrapolation of the coronal magnetic field above the solar surface. At right, the broadband EUV emission (100-1000A)˚ produced along the potential magnetic field lines in the left-hand panel...... 92 5.2 At left, a snapshot of solar maximum for a solar control case produced with the SFT. In the center, a magnetogram at stellar maximum of the 4G field strength representation of ι-Hor. At right, the stellar maximum phase of the 50G field strength representation of ι-Hor. For both ι-Hor test cases, the short activity cycle period leads to strong poleward flows (Eqn. 3.3) and therefore many active regions appear at higher latitudes than in the present-day solar case at left. . 96

6.1 Yearly averages of observed sunspot numbers are plotted vs. time; the prolonged period of low cycle amplitude known as the Maunder Minimum is readily apparent [Hathaway, 2010]...... 98 6.2 The insertion of a large anomalous active region in a solar dynamo simulation ‘killed’ the dynamo and triggered a grand minimum akin to the Maunder Minimum observed in archival solar data. This result could explain the causes of solar cycle amplitude variations [Nagy et al., 2017]...... 99 Tables

3.1 SFT model input parameters for a selection of the simulations presented in this study (see Figure 3.5). Each row represents a stellar magnetic simulation with the stated solar-scaled Rossby number and the corresponding input parameter values. SFT model parameters

include the ﬂux emergence strength, A0, the stellar equatorial

rotation speed, veq, the stellar magnetic cycle period, Pcyc (denoted as T in Eqn. 2.2), and the meridional flow multiplier, m. See Section 2.1 above for description of SFT input parameters. The input parameter values were calculated using Equations 3.1 - 3.3. The solar meridional flow profile v is given by Equation 3 of [Schrijver, 2001]. The me differential flow multiplier d (Equation 2.4) is not modified from a value of 1 in any of the simulations described here or in Chapter 4. . 47 3.2 X-ray emission and magnetic flux sources displayed in Figure 3.3,

p with the value p of the power-law ﬁt LX ∝ Φ for each type of source, from [Pevtsov et al., 2003]...... 48

5.1 SFT model input parameters for two test cases of an ι-Hor type star, for two possible observed magnetic ﬁeld strength (B) values. The

source strength A0, equatorial rotation speed veq, cycle period Pcyc, and meridional ﬂow multiplier m are deﬁned as before in Section 3.2.2. 94 xx

5.2 Alfv´enradius estimates for stellar minimum and maximum, calculated with Equation 4.1, for each ι-Hor test case in units of

RA,solar max ≈ 20RSun ≈ 0.1 AU. The ratios of open ﬂux to total surface magnetic ﬂux at stellar minimum and maximum are also included...... 95 1

Chapter 1

Introduction

Since the 1990’s, advances in astronomical observation techniques have led to the discovery of more than 4,000 exoplanets orbiting stars other than the Sun (at the time of writing, there are 4,375 confirmed detections with another 5,900 candidate exoplanets awaiting further study [Brennan, 2019]). Statistical arguments lead to the conclusion that most stars in the sky host at least one planet [Schrijver and Sojka, 2016]. As exoplanet detection becomes more commonplace, the study of these planets is evolving from a phase of basic discovery to one of more in-depth characterization. An important aspect of this characterization is the influence of stellar activity on planetary magnetospheric and atmospheric properties: namely, the heliophysics of exoplanetary systems. Placing constraints on star-planet interactions and the impact of host star magnetic behavior on associated exoplanets serves as motivation for the studies detailed in this thesis. We study the integrated system of the stellar surface and its impacts on coronal emission, stellar wind propagation, and related planetary responses in order to build a more detailed understanding of the stellar properties which influence exoplanet environments. Additionally, the combination of solar and stellar physics observations and modeling employed in this work can inform our understanding of solar dynamo activity and photospheric flux dynamics for a variety of stellar types. The interdisciplinary approach employed in these studies has yielded important advances in understanding of star-planet interactions and has opened several avenues for future work. 2

1.1 Characterization of Exoplanet Systems

The Sun-Earth system is the most well-studied and constrained example of star-planet interaction, with a wealth of solar and magnetospheric observations spanning a range of timescales and energetic conditions. A decades-long legacy of space-based solar missions such as the Solar and Heliophysics Observatory (SoHO), the twin spacecraft of the Solar Terrestrial Relations Observatory (STEREO), and the Solar Dynamics

Observatory (SDO), in tandem with ground-based solar telescopes such as the Na- tional Solar Observatory (NSO) and the recently activated Daniel K. Inouye Solar

Telescope (DKIST), have enabled constant high-resolution monitoring of the solar surface, atmosphere, and transient events like X-ray ﬂaring and coronal mass ejections (CMEs). In complement, geospace and planetary missions such as the Van

Allen Probes, Time History of Events and Macroscale Interactions during Substorms

(THEMIS) mission, and the more recent Magnetospheric Multiscale (MMS) constel- lation of four spacecraft; the Mars Atmosphere and Volatile EvolutioN (MAVEN) mission; and the Juno mission to observe Jupiter’s polar regions, have monitored the inﬂuence of solar variability on the magnetic ﬁelds and neutral and ionized atmospheres of the Solar System planets. This breadth of observations of space plasma dynamics and magnetism has also led to the development of detailed solar and magnetospheric modeling tools to describe the physical mechanisms producing observed

Sun-planet interactions.

Comparative heliophysics - the application of solar and magnetospheric physics to questions of stellar impacts on exoplanet environments - is increasingly important in order to understand the physical conditions of newly-discovered exoplanet systems [Schrijver and Sojka, 2016]. Current observational capabilities are not yet suﬃcient to detect observable signatures of star-planet interaction in other stellar 3 systems, such as auroral emission from exoplanet ionospheres reacting to changes in stellar wind conditions (though a possible preliminary detection of such emission was published recently [Turner et al., 2021]). Observational characterization of exoplanet atmospheres is also not yet a wide-spread reality, though the James Webb

Space Telescope (JWST), expected to launch in late 2021, will introduce the unprecedented capability to observe the presence and composition of exoplanet atmospheres in a variety of systems [Bean et al., 2018, Fauchez et al., 2019, Komacek et al., 2020,

Guzm´an-Mesaet al., 2020]. Given the current dearth of observed space weather dynamics in exoplanet systems, the application of solar and heliophysics-based modeling is thus an important tool for describing the possible dynamics of host star-exoplanet systems, and for predicting observational signatures for the next generation of exoplanet characterization missions such as JWST.

1.1.1 Theories of Habitability

The potential for a planet outside our Solar System to host extraterrestrial life is of obvious interest to the scientiﬁc community and general public. Questions of the habitability of exoplanets comprise a growing and increasingly complex ﬁeld of study.

By analogy with Earth, the ability for an exoplanet to retain liquid water on its surface over geologic timescales is considered fundamental to the planet’s potential for habitability [Kopparapu et al., 2013]. Loss of a planetary atmosphere over time due to stellar wind-driven escape mechanisms can cause a planet to lose its supply of surface liquid water, so star-planet interaction is a vital component of habitability that requires further study [Bolmont et al., 2017]. Conventional wisdom in geophysics states that Earth’s magnetosphere protects the planet from incident solar wind and prevents stripping of the Earth’s neutral atmosphere, though the relationship between 4

Figure 1.1 : From [Mendez, 2013], adapted from [Kopparapu et al., 2013], the range of circumstellar distances defining the habitable zone for host stars of different stellar temperatures and spectral types, based on two different planetary atmosphere models. The Earth’s location at 1 AU around a 1 MSun star is within the limits of the habitable zone by definition. 5 the presence of a planetary magnetosphere and atmospheric retention is the subject of much active research (e.g., [Gronoff et al., 2020]). Geomorphic observations indicate that Mars supported liquid water in its past and therefore must have lost roughly

1 bar of atmosphere in its lifetime [Jakosky et al., 2018]. Atmospheric escape and its dependence on magnetospheric behavior in response to stellar driving is therefore an important factor in the search for habitable conditions. An open problem in exoplanet characterization is the extent to which stellar winds and stellar magnetic variability contribute to atmospheric outﬂow processes. We will discuss the current understanding of the inﬂuence of stellar magnetism and winds on exoplanet space weather environments in more detail in Chapters 4 and 6.

Terrestrial exoplanets, of roughly Earth’s size and known to have Earth-like densities indicating rocky bodies, are of signiﬁcant interest to habitability studies since

Earth is the only planet with conditions known to support life. Additionally, astronomers employ the concept of a circumstellar habitable zone (CHZ) to constrain potentially habitable systems. The conventionally-defined CHZ comprises the range of orbital distances around a central host star at which a planet may host liquid water on its surface [Ramirez, 2018]. The location of the CHZ is strongly dependent on the temperature of the central host star, as shown in Figure 1.1. For main sequence stars, mass and temperature are directly related, so the largest, hottest stars have habitable zones in the range of two to three times the average Earth-Sun distance (known as an astronomical unit, AU); conversely, the smallest stars are much cooler and therefore their CHZs are found at much smaller orbital distances. It is crucial to note that this traditional definition of the CHZ, displayed in Figure 1.1, relies on the significant assumption of an Earth-like atmosphere shielding the planetary surface, and different models for planetary atmospheres result in different estimates of habitable zone lo- 6

Figure 1.2 : A comparison of confirmed exoplanets as of April 2014, color-coded by detection method, from [Batalha, 2014]. NASA’s Kepler satellite revolutionized the field of exoplanet detection, confirming the existence of 2,414 extrasolar planets over the lifetime of the mission. Planets detected by the transit method with Kepler (yellow dots) are most commonly found in short-period orbits, less than about 100 days. Planets detected by the radial velocity method with other instruments are strongly biased towards giant planets of Neptune-size and greater.

cations [Kopparapu et al., 2013, Kaltenegger and Traub, 2009, Kasting et al., 1993].

The traditionally-deﬁned CHZ also does not consider the potentially signiﬁcant effects of the magnetic structure and variability of the central host star, particularly in the case of the small, cool M-type stars where the CHZ may extend less than a tenth of an AU from the host star (indicated by the small red star in Figure 1.1).

We explore the relevance of this CHZ deﬁnition to extreme-proximity exoplanets in further detail in Chapter 5.

1.1.2 Detection Methods and Observational Biases

Exoplanet detection is a rapidly growing ﬁeld in astronomy which relies on several distinct methods for observing or inferring the presence of planets orbiting other 7

stars. One example is the transit method, wherein a planet passes between the

observer and the host star, causing periodic dimming in the brightness of the host

star (e.g., [Charbonneau et al., 2000]). The planet’s transit across the star’s surface

causes a dip in the star’s light curve; the depth of the transit gives the radius of the √ planet relative to the host star (Rp = R∗ Depth). NASA’s Kepler mission and its follow-up, K2, detected more than 2,000 extrasolar planets with the transit method

[Brennan, 2019]. As shown in Figure 1.2, exoplanets detected with Kepler are biased towards short orbital periods of less than about 100 days. Another approach is the radial velocity (RV) method, which measures the periodic red- and blue-shift of a star’s emission caused by the gravitational inﬂuence of the planet on the host star as they orbit their mutual center of mass; the minimum mass of the planet can be estimated from the strength of the radial velocity variations, though the unknown inclination angle i of the planetary system relative to the observer introduces an unknown factor (Mmin = Mpsin(i)). Both the transit and radial velocity methods have led to the detection of thousands of exoplanets in recent decades. Other detection methods, such as direct imaging or examining minute variations in the periodic emission of pulsars, exist but have contributed somewhat less to exoplanet conﬁrmations (see [Brennan, 2019] or Figure

1.2 reproduced from [Batalha, 2014]). However, both the transit and RV methods are biased towards planets that are large enough and/or close enough to impact the star’s brightness or motion along the line of sight at detectable levels. The earliest commonly observable class of exoplanets was the ‘hot Jupiters’, large gas giants that orbit close enough to their host stars that their transits or RV variations produce large signals (e.g., [Mayor and Queloz, 1995]). Current observational constraints limit our detection of habitable-zone terrestrial exoplanets to those found in close-in orbits 8 around small, cool M stars [Santos and Faria, 2018]. Since M dwarfs are the smallest of the main sequence stellar types, they are the host stars most readily observed to be inﬂuenced by their small, terrestrial planets. M stars are also the most numerous of the main sequence stellar types. There is therefore signiﬁcant interest in comparisons of the space weather environments established by M dwarfs to those of solar-type stars, and the resultant impacts on their associated planets [Cohen et al., 2014].

These observational restrictions are changing with the recently launched Transit- ing Exoplanet Survey Satellite (TESS) mission, which has added to the number of terrestrial planets detected and is expected to ﬁnd Earth-size planets around larger stars such as K and G dwarfs, e.g. [Huang et al., 2018]. As shown in Figure 1.1, the habitable zones of K stars are not in such extreme proximity to the central host star as those of M dwarfs, pointing to habitable-zone terrestrial exoplanets around K stars as a possible happy medium, with more hospitable space weather environments, while remaining detectable with current instrumentation. The detection of terrestrial exoplanets in habitable-zone orbits around Sun-like G stars is of obvious interest for direct comparisons to the Sun-Earth system, but these observational capabilities are still some years oﬀ.

Exoplanet characterization capabilities will be expanded by future observations;

JWST in particular is expected to resolve the atmospheres of a wide variety of transiting exoplanets. A planet with a suﬃciently thick atmosphere will show in its transit slighter dimming due to the atmosphere before and after the strong dimming due to the planetary body. As energetic light from the background star passes through the comparatively cool planetary atmosphere, spectral lines corresponding to the chem- ical makeup of the planetary atmosphere are detectable and the composition of the atmosphere can be inferred. There has already been some success in resolving atmo- 9 spheres of transiting hot Jupiters (e.g., [de Kok et al., 2013]), and JWST is expected to detect and characterize many more exoplanet atmospheres in the coming years.

The possible detection of biosignatures in exoplanetary atmospheres is an exciting avenue of further study in exoplanet characterization. Future studies of stellar systems of interest characterized by JWST are discussed more in Chapter 6.

Thus far we have discussed how terrestrial planets are particularly compelling in exoplanet studies due to analogies with Earth and the presumed favorable conditions for habitability. Other types of exoplanets are also of interest to help more fully understand planetary system dynamics and planetary characterization, including hot

Jupiters which may produce detectable auroral radio emission due to interaction with their nearby host stars; we discuss this concept further in Chapter 5. In addition, many exoplanets of roughly Neptune-size have been found in recent years. The true breadth of possible exoplanetary system architectures, with diﬀerent numbers and sizes of planets, is open for debate [Zhu, 2020, Weiss and Petigura, 2020]. However, many detailed models of planetary magnetospheric dynamics are restricted to Earth- like planets with a global magnetosphere and a neutral and ionized atmosphere. We discuss models of Earth-like planetary magnetospheres brieﬂy in Chapter 2 and in

Chapter 5.

1.2 Solar Activity and Impacts on Solar System Planets

The Sun is a G-type main sequence star: its photosphere has an eﬀective temperature of about 5,800K and its interior is composed of a radiative core, surrounded by an envelope where convective motion dominates the energy transport. Like all main sequence stars, the Sun was more active in its past; over stellar lifetimes, stars spin down and become less magnetically active as they lose angular momentum to 10 their stellar winds [Bahcall et al., 2001]. The enhanced activity of the young Sun has been linked to early critical Earth processes such as magnetospheric compression and atmospheric chemistry [Airapetian et al., 2016]. In addition to spindown over geologic time, the Sun’s magnetic ﬁeld also varies on a 22-year polarity cycle, man- ifesting observationally as an ∼11-year cyclical variation in sunspot numbers on the photosphere [Hathaway, 2010].

1.2.1 The Solar Cycle and Dynamo Action

The solar cycle was first observed and constrained in the 1800’s (e.g, [Schwabe, 1844]); early records of the solar cycle relied on astronomers manually counting and recording the number of sunspots on the Sun’s photosphere or observable surface. By the early 1900’s, solar observers had discerned predictable patterns in the placement and orientation of sunspots. Spörer’s Law states that at the minimum phase of a solar cycle, active regions appear at mid-latitudes (20°- 35°), and progressively emerge at lower latitudes as the cycle progresses through maximum and into a declining phase [Alexander, 2009]. Hale’s Polarity Law describes the general rule that solar active regions form as magnetic bipoles, where the leading and following edges of the bipole will have opposite polarity, and the orientation of the bipoles is flipped from the northern to southern hemisphere; the orientation of the bipoles also reverses on the order of every 11 years, so a full magnetic polarity cycle is equivalent to two periods of the sunspot cycle, or about 22 years [Hale et al., 1919]. Joy’s Law describes the observed pattern that the following pole of a sunspot appears at higher solar latitude than the leading pole and that this angle of inclination is steeper when the sunspot emerges at higher latitude [Hale et al., 1919]. The classic picture of sunspot distribution over many cycles is captured in a ‘butterfly diagram’ as shown in Figure 11

Figure 1.3 : A magnetic ‘butterﬂy’ diagram showing cyclic patterns in the bipolar active regions on the solar photosphere over three solar cycle periods of about 11 years each. Data is aggregated from magnetometers at the National Solar Observatory (NSO) and onboard the SoHO spacecraft [Hathaway, 2010]. Yellow denotes positive polarity (outward from the photosphere) while blue denotes negative polarity (into the photosphere).

1.3, where Sp¨orer’s law is readily apparent as a narrowing of the active latitudes from cycle minimum to maximum and into the declining phase, and Hale’s polarity law is evident in the reversal of the polarities of the bipolar active regions over time

[Babcock, 1959].

The cyclic nature of sunspot emergence and distribution on the Sun’s photosphere is a consequence of the underlying solar dynamo. The solar dynamo is the large- scale motion of plasma within the Sun’s interior which sustains the solar magnetic

field against dissipative forces (see, for example, [Charbonneau, 2020]). The solar magnetic field is generated by the interplay of poloidal- and toroidal-oriented large- scale magnetic fields. At solar minimum, the Sun’s magnetic field is largely dipolar; this is shown in Figure 1.3 as the periods where the strongest fields are concentrated at ± 90°latitude and very few active regions are present. At solar maximum, the field 12

is disorded, quadrupolar and higher-order ﬁeld components are strongly present, and

sunspot coverage is maximal.

Helioseismic observations of motions within the solar interior indicate the presence

of shearing at the tachocline, the boundary of the rigid-body radiative core and the convection-dominated outer envelope of the Sun [Dikpati and Charbonneau, 1999].

The convective envelope rotates differentially, with plasma flowing more quickly at the equator than at the poles; the interface of the differentially-rotating convective envelope and the rigidly-rotating radiative core causes shearing along the direction of the equatorial rotation, converting the poloidally-oriented field of solar minimum into toroidal magnetic field over the course of the cycle [Charbonneau, 2020]. The dynamo completes its magnetic cycle and returns to solar minimum when toroidal field is converted back to poloidal field. The precise mechanism by which this occurs on the

Sun is still unknown, though one possibility is that the cancellation of flux between oppositely oriented bipolar regions emerging close to the equator builds up excess magnetic flux of differing polarities in the northern and southern hemispheres, which is advected to high latitudes by meridional (poleward) flows, thereby re-establishing a magnetic dipole [Cameron and Schüssler,2010].

1.2.2 Solar Coronal Emission

Above the solar photosphere, the solar plasma becomes optically thin and magnetic forces dominate; this diﬀuse outer atmosphere of the Sun is called the corona. The

corona is made up of coronal loops, magnetic ﬁeld lines with footpoints rooted in

the solar photosphere which form the building blocks of the magnetically-dominated

regime. The corona is also many times hotter than the photosphere, heated to 1-

2MK in most regions and reaching up to 20MK in areas of dynamic energy transfer 13

in the form of solar ﬂaring [Aschwanden et al., 2001]. The coronal heating problem

describes the still largely unknown mechanism by which the coronal plasma is heated

to such high temperatures. The dominance of magnetic forces in the corona points to

motions of the magnetic ﬁeld lines as the most likely culprit of energy transfer sub-

stantial enough to explain the plasma temperatures in excess of 1MK. Ongoing mod-

eling studies [Barnes et al., 2016a, Barnes et al., 2016b] and observation campaigns

[Ishikawa et al., 2017] are investigating whether this transfer of magnetic energy to

thermal energy is accomplished by magnetic Alfv´enwave phenomena or by small,

frequent magnetic reconnection events called nanoﬂares.

In addition to closed-ﬁeld loop structures, the corona also has areas of magnetic

ﬁeld lines ‘open’ to interplanetary space. While no magnetic ﬁeld line is truly open-

ended, due to the solenoidal condition of Maxwell’s equations ∇·B~ = 0, the ﬁeld lines

of the corona that extend far into interplanetary space and form part of the solar wind

outﬂow are said to be open since they do not connect back to the Sun’s surface for

distances RSun. At solar minimum, the solar magnetic field is largely dipolar and the majority of open field lines are found at the poles; at solar maximum, the field is highly disordered and low-latitude regions of open field lines can appear. Since closed

field lines serve to confine the superheated plasma, they glow brightly in energetic emission. By contrast, the open field lines allow plasma to flow away from the Sun and appear dark in the X-ray and extreme ultraviolet ranges; large regions of open

ﬁeld lines and little coronal emission are therefore called coronal holes. Figure 1.4 shows the bright corona in the extreme ultraviolet (EUV), with a prominent coronal hole, taken with SDO’s Atmospheric Imaging Assembly (AIA) [Lemen et al., 2012].

The hot plasma of the corona produces energetic emission in the soft X-ray, far ultraviolet, and EUV ranges, collectively referred to as XUV emission. Such ener- 14

Figure 1.4 : Taken with the AIA instrument onboard SDO, an image of the Sun in the EUV (193 A),˚ with a prominent coronal hole. Coronal loop structures can also be seen, particularly in bright regions [NASA/SDO, 2010].

getic emission is a large factor in atmospheric chemistry and ionosphere dynamics at the Earth [Garcia-Sage et al., 2015, Cohen et al., 2014]. The coronal emission of exoplanet host stars is likely an important factor in the atmospheric and ionospheric processes occurring at their associated planets. The solar corona also experiences explosive, transient phenomena such as rapid bursts of X-ray emission called solar

flares and the spontaneous release of solar plasma and magnetic field lines in the form of CMEs. More active stars such as our nearest neighbor, Proxima Centauri, can experience enhanced flaring and possible associated CME activity multiple times per Earth day [Vida et al., 2019], which could provide highly volatile and dynamic space weather conditions at the associated exoplanets. 15

1.2.3 The Solar Wind and Heliosphere

The evolution of magnetic flux on the solar photosphere has far-reaching consequences: the solar magnetic field extends far outside the orbit of Pluto, forming a vast region of space dominated by solar magnetism called the heliosphere. The outflow of magnetized plasma from the Sun into the interplanetary space of the heliosphere is the solar wind.

Early theories of the existence of a solar wind, based on the trajectory of comet tails ([Durham, 2021] and references therein) and on the thermal expansion of the superheated corona [Parker, 1958], were confirmed observationally by the Mariner 2 mission [Neugebauer and Snyder, 1966]. The solar wind is composed of two populations: a fast solar wind, which follows the open field lines of the solar poles and coronal holes, and a slow solar wind of more uncertain origins. The Ulysses spacecraft constrained the solar wind velocity distribution; Figure 1.5 shows fast solar wind speeds at the open-field polar regions, with lower wind speeds in the low-latitude closed-field regions.

Coronal mass ejections are large eruptions of plasma from the solar corona, which expel particles and magnetic ﬂux from the corona into interplanetary space. When a CME moves faster than the ambient solar wind, shocks can form ahead of the

CME. Interplanetary CMEs traveling through the solar wind can rotate and produce rapid reversals in the orientation of the interplanetary magnetic ﬁeld (IMF). The ambient solar wind and propagating disturbances interact with the magnetosphere and ionosphere of Earth and the other solar system planets. Understanding the stellar wind is a vital component of the coupled star-planet system. We discuss solar wind coupling further in Chapters 2 and 6. 16

Figure 1.5 : The solar wind distribution in speed and IMF orientation observed with the Ulysses spacecraft. In the polar regions, magnetic field lines are open to interplanetary space and solar wind particles stream quickly (up to ∼800 km/s) along open field; at low latitudes, more magnetic field lines are confined to the corona and escaping plasma moves more slowly (∼300-400 km/s) [Hathaway, 2014]. 17

1.3 Stellar Observations Context

The same physical processes occurring on the Sun are apparent in many other stars.

Figure 1.6 shows the famous Hertzsprung-Russell diagram of stellar temperature,

luminosity, and spectral type or color; the spectral classes O, B, A, F, G, K, and M are

denoted along with the appropriate ranges of Teff for the stellar surface temperatures. The main sequence (MS) is denoted in the center of the diagram; this is the phase of stellar evolution where a star undergoes hydrogen fusion in its core. The Sun is in the main sequence phase of its lifetime and falls within the G spectral class. Other main sequence stars of classes F through M are expected to have moderate to close-in

CHZs as shown in Figure 1.1.

Stars other than the Sun which have interior convective zones and photospheres dominated by convection (main sequence classes F through M)∗ exhibit similar physical processes such as stellar winds, magnetic variability, and high-energy coronal emission in XUV. The variability of stellar magnetism and emission for these types of stars, and the impact of these variations on associated exoplanets, is the main focus of this thesis; we describe our methodology for applying solar-based modeling tools to these questions of stellar behavior in Chapter 2.

While continuous, detailed observing of the solar magnetic field is enabled by space- and ground-based missions, observations of stellar magnetic fields are more difficult. The Zeeman-Doppler Imaging (ZDI) technique has been used to constrain the surface magnetic fields of other stars. The Zeeman effect describes the manner in which a magnetic field polarizes spectral line emission; the strength of the polariza- tion depends upon the strength of the magnetic field, so magnetic properties can be

∗Main sequence stars of O, B, A, and early-F classes have radiative outer envelopes and do not exhibit solar-like surface convection and other related processes [Padmanabhan, 2001], so they will be treated as outside the scope of this thesis. 18

Figure 1.6 : The Hertzsprung-Russell diagram of stellar luminosity vs. temperature or color. Top axis shows the stellar spectral classes [O,B,A,F,G,K,M]. The main sequence, labeled in the figure, is the prominent track of stars in the center of the diagram. Points on the plot are observed luminosities and temperatures from the Hipparcos Catalogue and the Gliese Catalogue of Nearby Stars [Powell, 2006]. 19 inferred from the observed emission. Subtracting out the Doppler effect in the emission due to the star’s rotation provides a spatially-resolved map of the magnetic field on the stellar surface [Rosén et al., 2015]. As ZDI techniques improve, increasingly detailed maps of stellar magnetic fields will provide important inputs to planetary response models and will be able to validate stellar magnetic field modeling such as the methods detailed in Chapters 2-4. 20

Chapter 2

Solar and Stellar Modeling Capabilities

The solar magnetic field is responsible for shaping the structure of of the solar atmosphere and heliosphere and for the electromagnetic and kinetic energy that drives planetary responses throughout the Solar System. The energetic emission and transient activity of the corona, together with the magnetic field and plasma flows of the stellar wind, are all rooted in the emergence, migration, and cancellation of magnetic

flux on the surface of the host star. In this work, we employ an empirical-based modeling approach to the dynamics of the magnetic field at the stellar surface in order to explore this underlying driver of star-planet interaction in the Solar System and beyond. In expanding solar-based modeling tools to other stellar systems, we must take account of the distinct behavior of the magnetic flux emergence and migration on stars of different stellar types. We describe a detailed test of this approach in

Chapter 3. In this chapter, we detail the solar magnetism modeling methodology used throughout this thesis.

2.1 Surface Flux Transport

Nearly all magnetic and energetic phenomena in the heliosphere are inﬂuenced and driven by the motions of magnetic concentrations in the plasma of the Sun’s photosphere. Continuous monitoring of the solar photosphere with magnetographs such as

SDO’s HMI have allowed solar physicists to constrain patterns of magnetic ﬂux emer- 21

Figure 2.1 : At left, a magnetograph of the Sun taken on March 31, 2021 with SDO’s Helioseismic and Magnetic Imager [Zell, 2014], the most recent measurement from HMI at the time of writing, near solar minimum. At right, an HMI magnetograph from July 1, 2014, when the Sun was in its maximum phase. White concentrations are of positive magnetic polarity, oriented toward the viewer, while black concentrations are of negative polarity oriented away from the viewer. The mixed-polarity network of small ephemeral regions is apparent throughout the solar disk at both phases. At solar minimum only a few signiﬁcant bipolar concentrations are apparent; at maximum, many large active regions are present.

gence, ﬂow, and cancellation on the solar photosphere over many years [Schou et al., 2012].

Figure 2.1 displays the latest magnetograph of the Sun taken with SDO/HMI at the time of writing (near solar minimum) contrasted with an HMI magnetograph from solar maximum in 2014.

The solar winds and eruptive events that influence the Solar System planets are ultimately rooted in the motions of magnetic flux on the solar surface. In order to understand fully the interaction of a coupled star-planet system, we must first study the emergence and evolution of magnetic flux at the stellar surface which drives energetic and magnetic variability in the environment around the star on a variety of timescales. To this end, we employ a detailed empirical-based model of magnetic

ﬂux emergence and migration on the solar photosphere, designed to reproduce the 22 observed behavior of magnetic ﬂux on the Sun throughout a full solar magnetic cycle

[Schrijver and Title, 2001, Schrijver et al., 2003, Schrijver, 2001]. This Surface Flux

Transport model (SFT) captures the emergence and dynamics of solar magnetic flux, incorporating the flows, flux distributions, and timescales of variation exhibited by the Sun.

The SFT samples from a statistical distribution of magnetic flux concentrations and tracks these concentrations as they emerge on the photosphere, migrate according to large-scale meridional (poleward) and differential (latitudinal) flows, collide and fragment, and ultimately disperse. The SFT treatment employed here improves upon a classical model of flux diffusion in a few key aspects. First, the SFT includes not only the large-scale solar magnetic field but also ephemeral regions which populate the magnetic background of the solar surface (see Figure 2.1) and have been found to contribute strongly the overall flux budget and, importantly, to chromospheric and coronal heating [Schrijver, 2001]. A solar flux treatment including only large active regions and omitting small-scale ephemeral bipoles would underestimate the total solar flux by as much as an order of magnitude [Schrijver et al., 1998]. Secondly, the

SFT includes a diffusion coefficient for flux cancellation that is dependent on the flux in the bipolar region itself; this flux-dependent cancellation was required in order to agree with observed solar behavior [Schrijver, 2001]. Large flux concentrations, particularly in mature active regions referred to as plage, are observed to disperse more slowly than smaller ones.

The SFT treats magnetic ﬂux emergence numerically by selecting randomly from an underlying distribution of both active and ephemeral region concentrations. The magnetic concentrations are parametrized by ﬂux strength, latitude and longitude, and bipolar region tilt angle. The background distribution of the active region and 23

ephemeral region populations is given by:

−p α −p−1 n(S, A ) = (a0A S + a1A S )dSdt (2.1) where α = 1/3 represents the relative components of the flux in active and ephemeral regions, as observed on the Sun; a0 = a1 = 8 and p = 1.9 represent the active region flux spectrum observed at solar maximum; S is the area of the emerging bipoles in deg2; and A is a flux injection parameter which varies with the magnetic polarity cycle according to Eqn. 2.2 below. The first term of the right hand side represents the distribution of active regions, while the second term describes the ephemeral region distribution, following from [Hagenaar et al., 1999].

Additional constraints on the system include the assumption of a solar-like magnetic polarity cycle in which more active regions emerge at solar maximum than at solar minimum. By contrast, the population of ephemeral regions varies only weakly with the solar cycle; this is represented by the power α = 1/3 in the ﬂux emergence function A in the ephemeral region distribution term of Equation 2.1 [Schrijver and Title, 2001]. The solar magnetic polarity cycle is incorporated into the

SFT by treating the ﬂux injection parameter as function of time:

t t t2 A (t) = A0β sin 2π 2 mod 1 × exp −5 2 mod 1 (2.2) T T T 2

where T is the period of the magnetic polarity cycle (21.9 years for the Sun), and β is

a normalization factor such that max[A (t)] ≡ A0 for the Sun at solar maximum. The

quantity A0 is a user-defined input parameter in the SFT, defining the flux injection

multiplier parameter (φ∗/φSun,max) that we vary to simulate stars of diﬀerent ﬂux 24

strength, as described in more detail in Chapter 3.

After emerging through the photosphere, the bulk of the solar ﬂux is observed to

be passively advected by the large-scale zonal and meridional ﬂows. Treatment of

the large-scale meridional ﬂow of photospheric magnetic ﬂux concentrations is based

on analysis of more than a decade of solar magnetograms observed with the NSO

Vaccum Telescope [Komm et al., 1993a]. The observed solar ﬂows are approximated

by:

M(θ) = m[12.9 sin(2θ) + 1.4 sin(4θ)] (2.3) where the meridonal ﬂow speed as a function of latitude M(θ) is in units of m · s−1, and the meridional multiplier m is a user-adjusted parameter whose utility we detail further in Chapters 3 and 4.

In the SFT, an empirical rotation rate is used (cf. [Snodgrass and Ulrich, 1990],

[Komm et al., 1993b]) that applies to magnetic features of all sizes. The modeled diﬀerential rotation rate Ω is a function of latitude θ as follows:

Ω(θ) = d[a + b sin2 θ + c sin4 θ] (2.4) where (a, b, c) = (7.5984 × 10−2, −1.95, −2.17) deg day−1. The a coeﬃcient is near

zero because the model operates in a frame rotating close to the mean stellar rotation

rate. The diﬀerential rotation multiplier d is a controllable parameter in the SFT

model but is not modiﬁed for the simulations presented in this work (see Table 3.1).

The SFT has been veriﬁed against solar observations with very good agreement.

The modeled ﬂux distributions (bottom panel of Figure 2.2) obey the Sp¨orer,Hale,

and Joy laws of active region behavior, and accurately capture solar cycle variations, 25

Figure 2.2 : Comparison between the longitudinally averaged magnetic field observed on the Sun (top) and modeled with the SFT (bottom) [Schrijver and Title, 2001] for a single magnetic polarity cycle of period Pcyc ≈ 21.9 years. Patterns of observed solar behavior such as Spörer’s,Hale’s, and Joy’s laws, as well as the field reversals over a magnetic polarity cycle, are replicated with very good agreement in the solar flux transport model. 26 advection of decaying flux to the poles, and field polarity reversals as in real solar observations (top panel of Figure 2.2) [Schrijver and Title, 2001]. Another advantage of the SFT approach is its versatility; stellar activity parameters such as flux emergence strength, stellar cycle period, equatorial rotation period, meridional flow speed, and

ﬂux-dependent dispersal strength are all independently tunable.

An accurate representation of stellar ﬂux transport provides a laboratory in which we can investigate the interrelationships between solar physics-based parameters and compare quantitatively with observations of cool stars. Preliminary work in compar- ing this solar modeling approach to observable stellar phenomena was investigated in [Schrijver and Title, 2001], in which the model was used to reproduce large polar starspots seen in active stars, a phenomenon not observed on the Sun. We have conducted novel work in extending this model to other cool stars by introducing observational and dynamo modeling-based interrelationships between stellar parameters to provide realistic inputs to the SFT for a variety of cool star populations. The stellar relationships incorporated into the choice of SFT input parameters and the resulting good agreement of simulations with stellar observations are detailed in Chapter 3.

2.2 Coronal Magnetic Structure

Investigating the influence of stellar activity on exoplanet environments requires a treatment of the magnetic topology of the stellar corona. The magnetic field of the corona is established by the movements of photospheric magnetic flux, detailed in Sec- tion 2.1, and produces energetic emission which in turn affects planetary ionospheres and atmospheres (see Sections 2.3-2.5). Thus, a reliable and detailed treatment of coronal magnetic field structure is integral to our understanding of star-planet interactions. 27

The solar corona is strongly dominated by magnetic forces such that other forces, for example due to pressure gradients and gravity, are negligible in most locations within the corona. The assumption that the corona is ‘force-free’, i.e. has no other forces besides those of the coronal magnetic ﬁeld, implies that all currents within the coronal plasma must be either oriented only along magnetic ﬁeld lines or must be of zero magnitude:

∇ × B~ || B~ (2.5)

The ‘potential field’ assumption is the stronger condition that no currents are present in the coronal magnetic field and the magnetic field can be described by a scalar potential Φ:

∇ × B~ = 0 ⇒ ∇2Φ = 0 (2.6)

In energized active regions, the coronal field is generally assumed to be force-free with currents running parallel to the magnetic field B~ ; these currents provide the excess magnetic energy required to explain active region heating and flares. However, for the large-scale field, a more detailed magnetohydrodynamic (MHD) approach is necessary to take into account the non-negligible plasma forces that give rise to the solar wind.

A computationally inexpensive alternative that has been shown to provide good agreement with the large-scale magnetic structure of the Sun is the current-free Potential

Field Source Surface (PFSS) approach, originally proposed by [Schatten et al., 1969].

The PFSS model uses an observed or simulated flux distribution on the stellar photosphere as an inner boundary and adopts an artificial spherical outer boundary known as the source surface at some distance from the star at which the magnetic field becomes strictly radial (B~ = Brˆ). This assumption mimics the observed magnetic structure of the solar wind without recourse to more complicated MHD treatment 28 of the solar wind flows. The term ‘source surface’ indicates that forcing the coronal

field lines to become radial at the surface can violate the condition ∇ · B~ = 0 in some locations, introducing small sources and sinks of magnetic field. The source surface is therefore unphysical, but has been shown to provide a good approximation of the observed magnetic structure of the heliosphere, particularly field-line connectivity. As such, PFSS allows us to extrapolate the simulated surface flux to the asterospheric magnetic field topology without recourse to computationally expensive and heavily parameterized MHD solar wind models, while still preserving the key structural ele- ments of the magnetic field. See, for example, Figure 1.5 which shows the magnetic structure traced out by solar wind plasma flowing along open field lines.

Photospheric magnetic field distributions from solar observations (e.g., from HMI), or from appropriate photospheric models such as the SFT, form the inner boundary condition for the PFSS model of the corona. Setting the source surface condition that ~ B = Brˆ at some source surface radius RSS forms the outer boundary condition of the simulation volume. The assumption of a current-free region between these boundaries allows the coronal magnetic field to be solved as a scalar potential satisfying Laplace’s equation ∇2Φ = 0 in the three-dimensional simulation volume. The PFSS model of [DeRosa et al., 2011] solves for the coefficients of the spherical harmonics that describe the structure of the coronal magnetic field. We show in Figure 2.3 a solar test case of the SFT with the accompanying extrapolation of its coronal field via PFSS.

This approach has the power to yield a good representation of the connectivity of the large-scale coronal ﬁeld, and is therefore well-suited for our purposes of constraining the overall asterospheric conditions of exoplanet environments.

A spherical source surface located at 2.5RSun is typically assumed for the solar case, although there is some suggestion that the location of this surface should be 29

Figure 2.3 : At top, the modeled photospheric flux distributions for an SFT simulation with solar parameters (from left to right, at declining, minimum, rising, and maximum cycle phases). Positive magnetic polarity is shown in white and negative magnetic polarity is shown in black, as in HMI photospheric observations (see Figures 2.1 and 2.2). At bottom, the corresponding PFSS coronal field extrapolations for each of the cycle phases. Near solar minimum, the coronal magnetic field is largely dipolar; at solar maximum, the coronal structure is much more disordered and open-field coronal holes appear at lower latitudes. 30 dependent on the strength of the surface flux and therefore vary with solar cycle

[Lee et al., 2011]. For stars more active than the Sun, [Schrijver et al., 2003] found that in order for the source surface to adequately represent the large-scale coronal

field, its location had to be assumed proportional to the average surface flux, spanning from 2.5R∗ for solar-like activity to 19R∗ for a star ten times as active as the Sun. To date there has been no study considering the optimal distance of the source surface for stars much less active than the Sun, and no observations of stellar wind that could constrain the assumed behavior. However, a source surface distance comparable to that of the Sun is required to provide sufficient simulation volume to adequately represent the coronal field as it transitions from field- to wind-dominated flow. In the work presented in Chapter 4, we therefore adopt a minimum source surface radius of 2.5R∗ for simulations less active than the Sun and use the above scaling that RSS is dependent on the average surface flux for high-activity stars. This approach of allowing the source surface to scale with stellar magnetic field strength mitigates the error possibly introduced when applying PFSS with the standard source surface radius 2.5R∗ to a range of stellar activity levels.

2.3 Coronal XUV Emission

The solar corona is an optically-thin plasma of temperature in excess of 1MK. The high temperature of the coronal plasma leads to strong emission in X-rays, extreme ultraviolet, and far ultraviolet (collectively denoted as the XUV regime) from about

5-2000 A.˚ This high-energy emission impacts the Earth and other planets by ionizing atmospheric particles, in turn affecting conductance and current flows in planetary magnetic fields. It is therefore of direct relevance to planetary impacts to constrain the mechanisms which produce XUV emission in solar and stellar coronae. 31

A number of approaches have been developed for modeling the heating and emission of coronal plasma. The Rosner, Tucker, and Vaiana (RTV) [Rosner et al., 1978] formulation relies on the assumptions that solar plasma is conﬁned along coronal loops which are isolated from one another, that the plasma is in hydrostatic equi- librium, and that coronal heating is largely balanced with thermal conduction and radiation. For coronal loops shorter than one pressure scale height (about 50Mm for plasma of 1MK), the plasma pressure can also be treated as uniform along the coronal loop. The resulting RTV scaling law relates the maximum loop temperature T as a function of plasma pressure p and loop semi-length L:

T = 1.4 × 103(pL)1/3 (2.7)

More sophisticated treatments of coronal heating and emission have been developed [Aschwanden et al., 2008, Klimchuk et al., 2008, Bradshaw and Cargill, 2013].

Typical hydrodynamic treatments of coronal loops allow for plasma to ﬂow along the axis of the loop (see, for example, the s coordinate shown in Figure 2.4). Hydro- dynamic simulations of coronal emission must have a time-dependent heating input, typically either from wave heating or nanoﬂaring as discussed in Section 1.2.2.

We make use of the synthesizAR pipeline for forward modeling of coronal emission [Barnes et al., 2019]. This suite of tools incorporates coronal loop structures from magnetic ﬁeld extrapolation models such as PFSS. Coronal loops are treated as one- dimensional structures to which coronal plasma is conﬁned, as shown in Figure 2.4.

Then a treatment of the temperature and density proﬁles of the magnetically-conﬁned plasma is needed. The synthesizAR package allows for any number of methods for describing the temperature and density of loop plasma; the simple RTV scaling law is 32

Figure 2.4 : A coronal loop described by the one-dimensional spatial coordinate s; the background image is real imaging of the solar corona taken with the Transition Region and Coronal Explorer (TRACE) instrument in November 1999 [Reale, 2014].

one example. Another method for populating the coronal loop structures with plasma of appropriate temperature and density is the Enthalpy-Based Thermal Evolution of Loops (EBTEL) model which allows for time-dependent heating of loop structures and calculates radiative cooling from energy balance [Klimchuk et al., 2008,

Bradshaw and Cargill, 2013]. The synthesizAR package then incorporates the CHI-

ANTI database of atomic line emission proﬁles [Young et al., 2016], based on the temperature and densities calculated by the chosen heating model, and produces maps of the emission in loop structures at the desired viewing angle and wavelength or range of wavelengths. Examples of extrapolated coronal magnetic ﬁeld lines and the associated emission calculated via synthesizAR are displayed and discussed further in Chapter 5.

Work in applying the forward modeling of coronal emission for distributions of coronal magnetic ﬁelds from exoplanet host stars simulated with SFT+PFSS is still 33 in its early stages in collaboration with synthesizAR author Dr. Will Barnes, but will allow us to provide both magnetic and directly associated XUV energy inputs to planetary response modeling. We describe current and future applications of this modeling framework to questions of exoplanet host star emission and planetary impacts in Chapter 5.

2.4 Solar and Stellar Wind Propagation

The Sun sheds magnetic ﬂux, angular momentum, and ionized particles in the form of a solar wind [Parker, 1958, Gringauz, 1960]. The solar wind is the chief intermediary between the Sun and the planets of the Solar System. The propagation of solar wind disturbances such as CMEs impacts planetary magnetospheres, compressing the magnetopause boundary, inﬂuencing the size of polar cap regions, and contributing to conductance changes in the ionosphere. Atmospheric loss processes at Earth have been linked directly to these changes in the magnetosphere in response to stellar wind driving [Welling et al., 2015]. Thus, the behavior of the winds of the Sun and other stars is a key component of investigations into exoplanet conditions.

Since solar and stellar winds are made up of ionized plasma with embedded magnetic field lines, they can be described with MHD treatments. Figure 2.5 shows the results of an MHD simulation of the inner asterosphere of a modeled star, where heating of the plasma in the corona and stellar wind has been accomplished by the addition of low-frequency Alfvénwave turbulence via the AlfvénWave Solar Model

(AWSoM) [van der Holst et al., 2014]. Open and closed magnetic field lines develop self-consistently and the asterospheric current sheet, the interface where the stellar magnetic field reverses polarity, is highly structured and dependent on the phase of the stellar surface field. Several authors have made use of this model for prelimi- 34

Figure 2.5 : At left, the modeled solar wind density and current sheet structure for a stellar minimum case using the Alfv´enWave Solar Model (AWSoM); at right, the same modeled star at stellar maximum, with a highly disordered asterospheric current sheet and enhanced stellar wind pressure (the same color bar describes the density in both panels) [Alvarado-G´omezet al., 2016].

nary studies into exoplanet space weather environments (see [Garraﬀo et al., 2016,

Garraﬀo et al., 2017] and Chapter 4 for more details).

Another high-heritage model for the stellar wind is the Wang-Sheeley-Arge (WSA) model, which diﬀers from the MHD treatment of AWSoM and instead employs a potential ﬁeld description of the coronal magnetic structure and current sheet, and allows for propagation of stellar wind into the asterosphere via a velocity-dependent expansion factor [Arge and Pizzo, 2000]. We discuss further applications of stellar wind modeling in Chapter 6.

2.5 Planetary Magnetospheric Response

The Earth’s magnetosphere responds dynamically to changes in solar wind magnetic and kinetic energy densities, plasma pressure, and magnetic ﬁeld orientation.

In response to solar wind disturbances like large CMEs, the Earth’s magnetosphere may experience strong reconnection events that can carry away ionized atmospheric 35 material. Magnetospheric responses to changes in stellar winds are an important consideration for exoplanet habitability and characterization. The production of auroral emission at exoplanets due to interaction between the stellar wind and the planetary magnetic ﬁeld may also be an observable signature of star-planet interaction in the coming years.

As discussed in Chapter 1, many exoplanets of interest for studies of habitability, such as terrestrial planets close-in around M dwarfs, may reside in environments of enhanced stellar activity. Investigations into detailed modeling of planetary and exoplanetary magnetospheric response to stellar wind driving have been carried out by Anthony Sciola, a graduate student colleague at Rice University. Our group has investigated possible exoplanet magnetospheric dynamics via the application of the new Grid-Agnostic MHD for Extended Research Applications (GAMERA) model for magnetospheric response to stellar wind driving [Zhang et al., 2019]. Additionally, the dynamics and plasma population of the inner magnetosphere have been incorporated via the Rice Convection Model (RCM) [Toﬀoletto et al., 2003]. We discuss further applications in integrating our detailed modeling of stellar magnetic ﬁeld generation with planetary responses in Chapter 5.

2.6 Conclusion

The methodologies described in this chapter have covered a wide swath of stellar and space weather processes, ranging from the emergence and migration of magnetic ﬂux on the stellar surface, to the magnetic structure and energetic emission of the corona, to the propagation of the stellar wind and the impact on planetary magnetospheres.

The modeling frameworks described here can be thought of as pieces of the puzzle in understanding the complex, interconnected star-planet system. 36

We present here a flowchart detailing the separate components of the behavior and variability of stellar magnetic fields and emission at left, and potential impacts on the planetary space weather environment at right. The SFT, PFSS, and coronal XUV emission models discussed in this chapter capture the behavior of the stellar surface and the structure and heating of the stellar atmosphere. We discuss in further detail the possible impacts on the exoplanet environment, such as magnetic flux open to the interplanetary environment and the magnetic structure of the inner asterosphere

(Chapter 4) and the inﬂuence of energetic emission on planetary ionospheric and atmospheric dynamics (Chapters 5 and 6). 37

Chapter 3

Modeling the Activity of Stellar Populations

Investigations of the physical mechanisms giving rise to coronal plasma emission, and of the scaling of this emission with magnetic ﬁeld generation, have practical applications to the potential habitability of exoplanet systems. Water on planetary surfaces may be photolysed by stellar radiation in far UV wavelengths, and soft X- rays and EUV emission from the host star can contribute to atmospheric escape

[Ribas et al., 2016]. As a consequence, a treatment of the energetic emission produced in a host star’s corona is vital for characterization of exoplanet environments.

Below, we employ the SFT approach discussed in Chapter 2 to compare solar activity with that of other cool stars across a range of spectral types. We find that extrapolation of solar-like behavior readily captures the observed activity relationship in late-F through M stars, including fully-convective (FC) M dwarfs. Through the successful reproduction of the slope and spread of the observed rotation-activity relationship, we further demonstrate the effectiveness of the SFT for studying the energetic and magnetic environment of a variety of star-planet systems. The research efforts described in this chapter are currently under review by the Astrophysical Jour- nal [Farrish et al., 2021]. 38

3.1 Emission and Rotation of Cool Stars

We investigate the variation of coronal X-ray luminosity as a tracer of stellar mag-

netic activity in the context of cool stars. In this work, the term “cool star” indicates

a main-sequence star of a) late-F to early-M type, containing a radiative core sur-

rounded by a convective envelope as the Sun does, or b) later-M type, exhibiting a

fully-convective interior (refer back to Figure 1.2 for the distribution of stellar types).

The similarities or diﬀerences between partially- and fully-convective cool stars are

the subject of much interest in the study of solar-stellar connections [Yadav et al., 2015].

The Sun is a G star with a rigidly-rotating radiative interior and diﬀerentially-rotating

convective shell comprising the outer 30% of its radius [Dikpati and Gilman, 2001].

Recent observations and modeling have pointed to similarities in the Sun’s magnetic

behavior and that of other partially- and fully-convective stars [Wright et al., 2018,

Wright and Drake, 2016, Yadav et al., 2016]. The investigations discussed in this

chapter are motivated by this evidence indicating that a direct relationship between

magnetic activity and coronal emission is common to all cool stars, regardless of in-

terior structure. Building on our knowledge of magnetic ﬂux dispersal on the Sun’s

photosphere, we can use this commonality to simulate the ﬂux behavior of an array

of cool stars over their magnetic cycle durations [Farrish et al., 2019], and to explore

the nature of the activity-rotation relationship for these stellar populations.

Observations of cool stars, including partially-convective [Wright et al., 2011] and

fully-convective [Wright and Drake, 2016, Wright et al., 2018] stars, yield a linear re-

lationship between the logarithmic fractional X-ray luminosity, log(LX /Lbol), and logarithmic variation in Rossby number, log(Ro), for Ro = Prot/τc & 0.1. The fractional X-ray luminosity, RX ≡ LX /Lbol (where LX is the X-ray luminosity and Lbol is the bolometric or total luminosity over all wavelengths), has been used in the 39

Figure 3.1 : The activity-rotation distribution of a population of 824 observed stars of partially-convective (gray points) and fully-convective (pink and red points) interiors, from [Wright et al., 2018]. The presence of FC stars in the unsaturated branch at Ro & 0.1 implies possible similarities in the production of magnetic ﬂux and its conversion to energetic coronal emission in both PC and FC populations of cool stars. literature as a dimensionless tracer of observable stellar activity. In solar observations, X-ray luminosity and strong X-ray ﬂaring are correlated with the solar cycle

[Hathaway, 2010], and therefore vary intrinsically on cycle period timescales. The

Rossby number, the ratio of rotational period, Prot, to convective turnover time, τc, is a dimensionless indicator of magnetic activity resulting from the stellar rotation and convection that make up the dynamo action. Rossby number varies inversely with magnetic activity such that lower Rossby number values correspond to more active stars. The range Ro & 0.1 represents the unsaturated regime of stellar activity, where the brightness of the X-ray corona scales proportionally to the Rossby number (e.g.,

[Reiners, 2012], [Reiners et al., 2014]). For stars with Rossby numbers Ro . 0.1 (typically very fast-rotating stars or stars with very deep convection zones), the fractional 40

Figure 3.2 : The solid line shows the saturation threshold Ro < 0.13 used in [Wright et al., 2011], as a function of stellar mass and age. A one solar mass star crosses the saturation threshold at ∼ 110Myr, while a star of 0.3 MSun crosses the threshold and joins the unsaturated regime at about 1 Gyr.

−3 X-ray luminosity saturates at a value of RX ∼ 10 , and the relative X-ray emission becomes independent of Rossby number. Figure 3 of [Wright et al., 2018], reproduced

here as Figure 3.1, illustrates this observed behavior. The alignment of both partially-

and fully-convective stars along the same unsaturated branch is strong motivation for

the present investigation into the ability of the solar-based SFT to simulate both PC

and FC populations.

The precise mechanism producing the observed saturation for fast-rotating stars is

an area of active research ([Vidotto et al., 2014], [Shulyak et al., 2017] and references

therein). Here, we restrict our study to the behavior of cool stars in the unsaturated

regime. [Wright et al., 2011] found that since stars spin down as they age, stars in

the unsaturated regime, with Ro & 0.1 and therefore generally longer Prot values 41 than their saturated counterparts, correspond to stars that have had sufficiently long lifetimes to decrease in rotation speed over time. For instance, a typical G star must be of age &110 Myr to be unsaturated; an M star must be &1 Gyr to fall in the unsaturated regime (see Figure 4 of Wright et al. 2011 for greater detail, reproduced here as Figure 3.2). Pre-main sequence stars (of age .10 Myr) were excluded from their sample because of the differences in internal structure from more mature cool stars. Young stars generally do not exhibit regular stellar cycles and differ from mature stars in other respects such as the rotation-flux relation [Folsom et al., 2016,

Folsom et al., 2017]. In alignment with [Wright et al., 2011] and [Wright et al., 2018], in this work we restrict our treatment to main sequence age cool stars that can be modeled with our solar-based modeling framework.

The Sun is located squarely in the unsaturated regime, with an estimated Rossby number between ∼ 0.5 − 2.0 [Reiners, 2012]. Detailed helioseismic analyses of the near-surface layer of the Sun show that the solar Rossby number has a strong dependence with depth and heliographic location, with values ranging from & 2 near the surface to ∼ 0.1 at a depth of 80Mm, the base of the near-surface sheer layer

[Greer et al., 2015, Greer et al., 2016]. Figure 2 of [Wright et al., 2011] places the

Sun at Rossby number RoSun = 2.0, while [Stepien, 1994] quotes a value of RoSun =

∗ 1.85. In this work, we adopt the Prot,Sun = 25.3d value of [Schrijver, 2001] and the global convective rollover time τc = 27.7d given for a one solar-mass star at

∗The Rossby number of a rotating plasma is technically a local quantity; helioseismic observations allow for some constraints on variations in τc (and therefore Ro) with depth and latitude but make it difficult to settle on a single bulk Rossby number for the Sun. Stellar observations, by contrast, usually do not include spatially resolved estimates of Prot (though some mapping of stellar differential rotation profiles has been accomplished, e.g. [Donati and Cameron, 1997]) and estimates of global stellar Rossby number are more easily accomplished. It is worth noting, however, that most estimates of stellar τc values are model-based and not observationally constrained, so error in stellar Rossby numbers may also be high. 42

4.5 Gyr in the Yale-Potsdam Stellar Isochrone (YaPSI) database [Spada et al., 2013,

† Spada et al., 2017] ; together, these values yield a solar Rossby number RoSun = 0.91, which we use throughout this chapter. Despite the uncertainty in the precise value

of the solar Rossby number, the Sun clearly resides in the unsaturated regime of

the rotation-activity relationship. This placement, together with the commonalities

among cool stars discussed above, provides a basis for our application of a solar-

based surface ﬂux transport (SFT) model, extended to simulations of magnetic ﬂux

transport for other stars in this unsaturated regime.

In contrast to the unknown mechanism producing the saturated behavior of low-

Ro stars, the scaling of coronal activity with rotation for Ro & 0.1 is much more well-deﬁned. [Pevtsov et al., 2003] demonstrated that the X-ray luminosity is di-

rectly related to magnetic ﬂux by a simple power law for a variety of solar and stellar

regimes. This relation was found to hold true over a vast range of magnetic features,

from the ephemeral regions of the quiet Sun at the lowest observed X-ray emission

levels, up to the full disk of the Sun and the total X-ray emission output of active cool

stars. Figure 3.3 displays the magnetic source regions studied and the overall scal-

ing relationship determined in [Pevtsov et al., 2003]. Thus, for stars with Ro & 0.1, which, through dynamo action, produce a range of more moderate magnetic ﬂux distributions than their saturated fast-rotating counterparts, there exists a tight relationship of increasing fractional X-ray brightness, RX , with increasing dimensionless stellar dynamo activity, parametrized by Rossby number Ro. This well-observed scaling in unsaturated cool stars is the feature which we investigate here through stellar magnetic ﬂux modeling.

†The YaPSI database models the evolution of stellar luminosity, radius, and mass over main sequence lifetimes; we employ YaPSI simulations further in Section 3.3.1. 43

Figure 3.3 : The X-ray luminosity vs. magnetic ﬂux for a variety of magnetic solar and stellar sources, from [Pevtsov et al., 2003]. Dots represent quiet Sun regions; squares denote X-ray bright points; diamonds represent solar active regions; pluses show full solar disk averages. Crosses denote the spatially-unresolved disk observations of active G, K, and M stars, and the open circles represent similar observations of young, accreting T Tauri stars. The power-law ﬁt to all data types has slope p=1.15 and is shown as the solid line. 44

Additionally, while populations of cool stars are observed to follow a tight relationship in RX vs. Ro, there does exist a scatter of about one order of magnitude above and below the general trend (see Figure 3.1 above from [Wright et al., 2018]).

We explore here the explanation that this scatter is due to an intrinsic variation in coronal X-ray brightness that results from stellar magnetic activity cycles, by analogy with the Sun’s activity variation over the solar cycle. Ensembles of stars observed in studies such as those by Wright and collaborators can be expected to be randomly phased in their magnetic activity cycles from stellar minimum to maximum, thus producing a distribution in coronal X-ray brightness. We investigate this hypothesis via the application of a surface ﬂux transport model which incorporates intrinsic stellar activity cycles.

3.2 Calibration of Solar-Based Modeling

3.2.1 Magnetic Flux Transport Modeling

The SFT incorporates stellar cycles wherein the total unsigned ﬂux of the photosphere is modulated according to a prescription derived from the Sun’s observed magnetic activity (see Eqn. 2.2). Thus, for a given set of input parameters such as cycle duration,

flux emergence rate, and meridional (poleward) and differential flow rates, the model produces a self-consistent representation of the distribution of magnetic flux across the stellar photosphere with high spatial and temporal resolution over many stellar cycles. In [Farrish et al., 2019] (see Chapter 4), we produced a series of stellar models over a range of dynamo activity, parameterized by the Rossby number, which has been found to be interrelated empirically with other stellar parameters, including surface flows and the frequency of flux emergence (Equations 1-3 of [Farrish et al., 2019], 45

reproduced below).

The SFT model input parameters displayed below in Table 3.1 have been cho-

sen via stellar activity relations determined from astronomical observations and stel-

lar dynamo models [Dikpati and Charbonneau, 1999, W. Noyes et al., 1984] (see also

[Brun and Browning, 2017] and references therein):

1±0.25 Pcyc ∝ Prot (3.1)

−1.5 2 φ ∝ Ro R∗ (3.2)

−10/9 vme ∝ Pcyc (3.3)

where Pcyc is the period of the modeled star’s activity cycle, Prot is the equatorial stellar rotation period, φ is the peak unsigned stellar magnetic flux, Ro is the Rossby number, R∗ is the stellar radius, and vme is the meridional flow speed. Thus a star of particular Rossby number, which has a given rotation period when a convective rollover time τc can be estimated based on age and stellar type, can be modeled with a corresponding stellar activity cycle period (Eqn. 3.1), peak flux (Eqn. 3.2), and meridional flow speed (Eqn. 3.3).

We note here that Eqn. 3.3 was derived from a simplistic mean-field theory dynamo model [Dikpati and Charbonneau, 1999]. More complete dynamo modeling can provide very different relationships between meridional circulation speeds and cycle period, e.g. [Strugarek et al., 2017]. For the purposes of this work, and to remain consistent with [Farrish et al., 2019], we use Eqn. 3.3 as a simple relationship derived from a mean-field description of the solar dynamo. [Schrijver and Title, 2001] found that variability in meridional speed affected histograms of surface-integrated

flux density only weakly; an SFT simulation with zero meridional circulation differed 46 negligibly from the solar test case, while a model with 10 times solar meridional flow resulted in only slightly fewer of the largest active regions, presumably due to mildly enhanced shearing.

A given iteration of the SFT model generates a time series of simulated distributions of magnetic ﬂux in the stellar photosphere, modulated by the evolution of the ﬂux over several stellar cycles, following the prescription of [Schrijver, 2001] for the photospheric dynamics of the Sun, and shown to apply to other stellar types by

[Farrish et al., 2019]. Thus, for a speciﬁc choice of Rossby number and stellar radius, we generate the time evolution of the total unsigned photospheric magnetic ﬂux, Φ, and can sample the simulated star at any phase of its magnetic activity cycle. We detail in the following subsection the method for converting these Φ values to the corresponding X-ray luminosities.

Table 3.1 displays the input parameters used to construct models of cool stars for a subset of the Rossby numbers and stellar types simulated in this work. Several of these models were used to study the relations between stellar magnetic activity and magnetic properties of exoplanet host star asterospheres in [Farrish et al., 2019]. In the present work, we extend these simulations to represent late-F, G, K, early-M, and late-M stellar types in order to explore the relationship between cycle-phased magnetic activity and X-ray emission. These can then be directly compared with the observed patterns of magnetic and energetic behavior for cool star populations (see

Section 3.3 below).

3.2.2 Relationship between X-ray Luminosity and Magnetic Flux

We relate the simulated magnetic ﬂux, Φ, of the SFT models to an X-ray luminosity by applying the power-law relation determined in [Pevtsov et al., 2003], which com- 47

Ro/RoSun A0=φ∗/φ veq (km/s) Pcyc (yrs) m = vme /vme max ∗ 0.1 31.6 20.0 2.19 12.9 0.11 27.4 18.2 2.41 11.6 0.12 24.1 16.7 2.63 10.6 0.13 21.3 15.4 2.85 9.65 0.14 19.1 14.3 3.07 8.89 0.15 17.2 13.3 3.29 8.23 0.16 15.6 12.5 3.50 7.66 0.18 13.1 11.1 3.94 6.72 0.2 11.2 10.0 4.38 5.98 0.25 8.00 8.00 5.48 4.67 0.3 6.09 6.67 6.57 3.81 0.35 4.83 5.71 7.67 3.21 0.4 3.95 5.00 8.76 2.77 0.5 2.83 4.00 11.0 2.16 0.625 2.03 3.20 13.7 1.69 0.75 1.54 2.67 16.4 1.38 0.875 1.22 2.29 19.2 1.16 1.0 1.0 2.00 21.9 1.00 1.1 0.867 1.82 24.1 0.900 1.2 0.761 1.67 26.3 0.817

Table 3.1 : SFT model input parameters for a selection of the simulations presented in this study (see Figure 3.5). Each row represents a stellar magnetic simulation with the stated solar-scaled Rossby number and the corresponding input parameter values. SFT model parameters include the flux emergence strength, A0, the stellar equatorial rotation speed, veq, the stellar magnetic cycle period, Pcyc (denoted as T in Eqn. 2.2), and the meridional flow multiplier, m. See Section 2.1 above for description of SFT input parameters. The input parameter values were calculated using Equations 3.1 - 3.3. The solar meridional flow profile v is given by Equation 3 of [Schrijver, 2001]. me The differential flow multiplier d (Equation 2.4) is not modified from a value of 1 in any of the simulations described here or in Chapter 4.

piled observations of magnetic ﬂux strength and X-ray emission from a variety of source regions, including quiet Sun, solar X-ray bright points, solar active regions, the full solar disk, active dwarf stars, and fast-rotating T Tauri stars (see Figure

3.3 above). These data comprise an approximately linear relationship between X-ray 48

spectral radiance, log(LX ), and unsigned magnetic ﬂux, log(Φ), spanning twelve orders of magnitude, suggesting a universal relationship between photospheric magnetic

flux and the power dissipated through coronal heating. In particular, the inclusion of active dwarf stars in this linear relation points to a similarity in the X-ray emission/magnetic flux relation for both the Sun and other cool stars. We investigate this connection further using the stellar magnetic flux transport model described above.

[Pevtsov et al., 2003] found that the slope, p, of the log(LX ) vs. log(Φ) relation varied depending on the types of source regions considered. These results are sum- marized in their Table 1, reproduced here as Table 3.2.

Object Power-law index p Quiet Sun 0.93 ± 0.06 Quiet Sun 27-day avg. 1.84 ± 0.26 X-ray Bright Points 0.89 ± 0.19 Solar active regions 1.19 ± 0.04 Sun 2.02 ± 0.07 Sun-CH ‡ 1.61 ± 0.05 G, K, M dwarfs 0.98 ± 0.19 T Tauri stars -0.08 ± 0.51 All data 1.15 ± 0.00

Table 3.2 : X-ray emission and magnetic ﬂux sources displayed in Figure 3.3, p with the value p of the power-law ﬁt LX ∝ Φ for each type of source, from [Pevtsov et al., 2003].

We note, in particular, that the active dwarf stars studied in [Pevtsov et al., 2003] yielded a slope p = 0.98 ± 0.19, notably shallower than the average slope over all sources, p = 1.15, and the slope for the average solar disk, p = 2.02 ± 0.07. The authors themselves note that the relative shallowness of the relation for the active dwarf stars could be a result of some of the sampled stars being in the X-ray saturation

‡Solar disk with estimated coronal hole magnetic ﬂux subtracted. 49

regime and, therefore, not likely to follow the general trend. The same is also true for

the fast-rotating, very active T Tauri stars in the sample, which exhibit an essentially

ﬂat relationship, p = −0.08 ± 0.51, resulting from the saturation of X-ray emission

for these objects [Johns-Krull et al., 1999].

In converting the total unsigned magnetic ﬂux values of our SFT simulations to

p the resulting X-ray emission, we apply the relation LX ∝ Φ with p = 2.02, the power-law behavior measured for full solar disk observations. We use this particular

value because the SFT model produces full stellar disk magnetic ﬁeld distributions

commensurate with the full solar disk observations of [Pevtsov et al., 2003]. We in-

1.15 tentionally do not make use of the average ﬁt for all data types, LX ∝ Φ because of the aforementioned caveats with respect to the active dwarf and T Tauri star pop-

ulations. We also do not use the solar disk data with the coronal hole contribution

subtracted (p = 1.61 ± 0.05) because the SFT model outputs alone do not diﬀerenti-

ate between photospheric magnetic ﬂux associated with open- or closed-ﬁeld coronal

structures. Figure 3.4 shows the resulting relationship between LX and Ro/RoSun for the set of SFT simulations with R∗ = RSun. The observations of [Wright et al., 2011] and [Wright et al., 2018] fall predomi-

nantly at Rossby numbers less than 2 with a few sporadic cases in the 2 ≤ Ro ≤ 3 range. Adopting a solar value of RoSun ' 0.91 (see above), the bulk of our observational comparison is delineated by the linear ﬁt shown by the solid line in

Figure 3.4. There is some evidence that this relation extends to both higher and lower Rossby numbers (dashed line in Figure 3.4). We include a simulation at

Ro/RoSun = 0.05 in order to capture the full range of the unsaturated regime iden- tiﬁed by [Wright et al., 2011], where they have assumed a solar value of RoSun = 2.

The modeled data in Figure 3.4 show that the LX −Ro relationship appears to extend 50

Figure 3.4 : The variation of X-ray spectral radiance, in units of erg · s−1, with scaled Rossby number (Ro/RoSun) for SFT simulations with R∗ = RSun. The vertical spread in LX is a result of cyclic modulations in magnetic activity for each stellar model, producing variation in the associated X-ray emission. The linear ﬁt to the data in the range 0.1 . Ro/RoSun . 1.2 displays the correlation between X-ray emission and Rossby number derived from the application of the [Pevtsov et al., 2003] scaling relationship. The dashed line extends the range to 0.05 . Ro/RoSun . 2.5, where the linear scaling still appears valid, and allows for the variation in the assumed values of RoSun in the literature (see discussion in main text). 51

to these fast rotators. However, given the variation in the literature of the precise

value of Ro at which the transition to saturation occurs, coupled with the uncertainty in the value of RoSun, we adopt in this work the bounds 0.1 ≤ Ro/RoSun ≤ 1.2 where we can be conﬁdent the stars are unsaturated throughout their cycles and the Pevtsov

ﬂux scaling holds.

We note that the results of [Pevtsov et al., 2003] provide only the power of the

LX - Φ relation, not the direct scaling. In order to convert magnetic ﬂux values determined from the SFT modeling to X-ray brightnesses, we ﬁt a constant of pro-

2.02 portionality, a, to the relation LX = aΦ . This constant is derived using an estimate of the unsigned magnetic ﬂux, Φ, of the Sun, ∼ 1023Mx, and a solar X-ray spectral

radiance ∼ 5 × 1026erg · s−1 (c.f. Figure 3.3). These estimates are approximate but

consistent, to within an order of magnitude, with the Φ values produced by the solar

test case of the SFT model, as well as the solar LX /Lbol value in [Wright et al., 2011].

Applying the solar values for Φ and LX from [Pevtsov et al., 2003] yields a constant of proportionality a ' 1.7 × 10−20erg · s−1 · Mx−1. We note that any uncertainties

associated with estimating a would serve only to shift the data in Figures 3.4, 3.5,

and 3.6 up or down, but would change neither the functional form of the emission-ﬂux

relation, the best-ﬁt slope of the data, nor the magnitude of the cycle-based scatter.

For each stellar simulation, parametrized by a given Rossby number, the SFT

produces a time series of the global stellar photospheric ﬁeld over several stellar cycles.

Choosing ten simulation frames roughly evenly spaced in time over a magnetic cycle

for each Rossby number yields the vertical scatter seen in Figures 3.4 and 3.5. Each

simulated star has a distinct magnetic activity cycle deﬁned by its Rossby number

[Brun and Browning, 2017], which results in a variable X-ray radiance output over

time. We investigate in Section 3.3 below whether this cycle-induced variation in 52

stellar X-ray output suﬃciently explains the spread about the unsaturated power-law

behavior seen in cool star observations (e.g., [Wright et al., 2018, Wright et al., 2011,

Vidotto et al., 2014]).

3.2.3 Calibration of Stellar Bolometric Luminosity

To compare the simulation results directly to observations of cool stars, we must

express the X-ray luminosity as a fraction of the star’s bolometric luminosity, Lbol. Previous authors (e.g. [Wright et al., 2018, Vidotto et al., 2014]) have used this ap-

proach because the coronal X-ray emission (LX ) and the star’s rotation period (Prot) are expected to be interlinked via complex and under-constrained dynamo and coro-

nal heating processes; it is therefore useful to compare these observable parameters

scaled by functions of stellar mass, namely RX vs. Ro [Pallavicini et al., 1981]. For

instance, [Wright et al., 2011] found that the saturation level of LX is dependent on

stellar type, while scaling to RX causes the saturation level to become independent of

sat −3±0.5 stellar mass or stellar type (RX ∼ 10 for all stars in the sample). This involves the subtle but powerful realization that cool stars exhibit consistent behavior over

several orders of magnitude in rotation and activity when the stellar mass dependence

is taken into account [Pizzolato et al., 2003].

A star’s bolometric luminosity is dependent on its mass and evolves as the star

ages; similarly, a star’s rotation period evolves with age as the star spins down via an-

gular momentum loss to the magnetized wind [Skumanich, 1972]. The dependence of

Lbol on stellar age and mass means that it is not a simple function of Ro, but rather de-

pends multivariably upon Prot and τc, due to their own dependence on stellar age and stellar mass, respectively [Mamajek and Hillenbrand, 2008, Garraﬀo et al., 2018a].

As a consequence, the dependence of bolometric luminosities on Rossby number for 53

Figure 3.5 : Fractional X-ray luminosity as a function of scaled Rossby number, for SFT simulated stars with R∗ = RSun. Diﬀerent values of Lbol for each Rossby number are determined based on a solar spin-down model [Bahcall et al., 2001]. Lbol varies with Prot as the Sun-like star ages and spins down over time. This approach limits the data to a narrower set of Rossby numbers corresponding to the range of Prot values achievable over the modeled lifetime of the Sun.

diﬀerent stellar types over time is diﬃcult to calculate analytically.

In Figure 3.5, we display the modeled SFT X-ray brightness for the R∗ = RSun case

varying with Ro, normalized by the evolution in solar Lbol over the Sun’s lifetime. The values of LX are produced by sampling ﬂux values from the SFT models corresponding to speciﬁc Ro values over the course of the intrinsic stellar activity cycle in each model, then applying the LX − Φ relationship described above and in Figure 3.4

−20 −1 −1 2.02 (LX = [1.7 × 10 erg · s · Mx ]Φ ).

The Lbol values used to normalize the data points in Figure 3.5 correspond to modeled solar bolometric luminosity values over the solar lifetime. [Bahcall et al., 2001]

used a standard solar model calibrated to match current-epoch solar luminosity, ra- 54

dius, and heavy element abundances to derive numerically past solar radius and bolo-

metric luminosity values. Using the Lbol variation with solar age given in Table 2 of [Bahcall et al., 2001], and making use of the Skumanich relation between stellar age

1/2 and rotation period (Prot ∝ t , [Skumanich, 1972]), we estimate values of Lbol vs. Ro to normalize the LX vs. Ro relationship displayed in Figure 3.5. For data points corresponding to solar ages between the limited set of values given in [Bahcall et al., 2001]’s

Table 2, we interpolate by averaging between the available values in the table. It is important to note that the conversion from Prot to Ro in this approach relies on the assumption that τc does not vary over the solar lifetime. Additionally, we introduce in this calculation the variation of the solar radius over time using the RSun values over time calculated in [Bahcall et al., 2001]. Section 3.3.1 below describes a more

complete approach that accounts for the evolution in τc and R∗ over stellar lifetimes for diﬀerent stellar types.

The normalization approach described for Figure 3.5 produces a slope of the RX vs. Ro relation, p = −1.74 ± 0.037, close to or consistent with that of the obser-

+0.14 +0.4 vational studies cited above, e.g. p = −2.03−0.17 [N´u˜nezet al., 2015], p = −2.3−0.6 [Wright et al., 2018]. However, this approach is limited in scope only to the Rossby

numbers relevant to the Sun’s evolution as it spins down from its initial angular mo-

mentum to the end of its main sequence lifetime, resulting in a narrow range of Rossby

numbers that can be analyzed, i.e. Ro = [0.25−1.2]RoSun. Section 3.3 continues with a description of our method to incorporate normalization by bolometric luminosity

for a wider range of stellar types, with a treatment of the evolution in τc in order to compare our simulated stars to real stellar populations. 55

Figure 3.6 : Fractional X-ray luminosity as a function of scaled Rossby number, for SFT simulated stars of late-F, G, K, partially-convective (PC) M, and fully- convective (FC) M types. Lbol is calculated using evolutionary tracks provided by YaPSI [Spada et al., 2013, Spada et al., 2017]. The Lbol calibrations include the effects of both stellar spindown of rotation period and the evolution of τc with age. The slope of the linear ﬁt, p, is included for each stellar type. 56

3.3 Modeling the Stellar Activity-Rotation Relation

3.3.1 Incorporating Stellar Bolometric Luminosity Evolution

Previous studies of the empirical relation between magnetic ﬂux features on the Sun’s

photosphere and the X-ray emission of the solar corona have found strong correlation

between magnetic activity and energy dissipated via coronal heating [Pevtsov et al., 2003,

Pizzolato et al., 2003]. Further, observational studies have found corresponding links

between the magnetic ﬂux and the X-ray coronal emission of other active stars, up to

an observed saturation limit [Wright et al., 2018, Wright et al., 2011, Vidotto et al., 2014,

N´u˜nezet al., 2015].

By modeling the emergence, surface ﬂows, and annihilation of magnetic ﬂux

concentrations on a stellar photosphere, guided by observation-based prescriptions

for magnetic cycles, we can produce faithful representations of the expected large-

scale and long-term behavior of the magnetic activity of the Sun and other stars

[Schrijver, 2001, Farrish et al., 2019]. We have shown in Section 3.2 above that these

modeled stellar distributions can be related in a consistent manner to the correspond-

ing X-ray luminosity via the observed power-law relation between magnetic ﬂux and

X-ray luminosity [Pevtsov et al., 2003].

Here we incorporate variations in Lbol with Rossby number due to both the evolution of rotation period over a stellar lifetime as well as the variation of τc with stellar type. We use the YaPSI evolutionary tracks [Spada et al., 2017] to ﬁnd the variation of Lbol with Rossby number for ﬁve stellar types (partially convective F star with M = 1.10MSun, G star with M = 1.00MSun, K star with M = 0.70MSun, partially-convective M star with M = 0.48MSun, and fully-convective M star with

M = 0.26MSun). We also incorporate a scaling in LX for each stellar type by the 57 square of the age-dependent stellar radii given in the YaPSI evolutionary tracks for

2 these ﬁve stellar masses (see Eqn 3.2 for the dependence of stellar ﬂux on R∗).

Figure 3.6 displays the LX behavior from the SFT simulations, normalized by Lbol values exhibiting the complete evolutionary behavior expected from stellar spin-down and variations in convective rollover time with stellar mass and age. The LX values are again obtained via the application of the [Pevtsov et al., 2003] LX − Φ relation to the cycle-phased ﬂux values resulting from SFT simulations for each Ro value. The

LX values for the G star case are no longer identical to those presented in Figure 3.5; here, we have incorporated the fact that RSun varies by about 25% over the range

0.1 ≤ (Ro/RoSun) ≤ 1.2, where RoSun is the solar Rossby number at the present age.

2 For the other stellar types, the LX values are also obtained by scaling by R∗, using the age-dependent stellar radii in their respective YaPSI evolutionary tracks.

The Lbol normalization values for each stellar types’ RX − Ro relationship are obtained from the YaPSI evolutionary tracks mentioned above. The YaPSI evolutionary tracks provide age-dependent values for the bolometric luminosity Lbol, convective turnover time τc, and stellar radius R∗. Coupling the evolution of τc with a functional form of the stellar spindown in Prot provides the age-dependent evolution of Ro for each stellar type. We adopt the functional form of [Barnes, 2003],

1/2 Prot = t × f(B − V ), where the rotation period (in days) increases as the square root of stellar age in Myr, modiﬁed by a B − V color-dependent multiplier f. The function f(B − V ) = p(B − V ) − 0.5 − 0.15((B − V ) − 0.5) incorporates the observed behavior that stellar spindown happens on longer timescales for larger-mass stars [Barnes, 2003]. [Wright et al., 2011] similarly drew the conclusion that stars in the unsaturated regime are described appropriately by Skumanich-like spindown with

1/2 Prot ∝ t . 58

We thus find the Lbol values corresponding to each Ro/RoSun value for each stellar type necessary for the normalization of the stellar X-ray luminosities. The key differences from the approach used to produce Figure 3.5 is that the normalization method applied here builds in the numerical evolution of τc and R∗ over the lifetimes of the stellar types considered here (from the YaPSI evolutionary tracks, [Spada et al., 2017]), introducing a more realistic evolution in the Lbol vs. Ro relation. For each stellar type, we find good agreement with observations, both in slope and spread (Figure 3.6), indicating that a) the SFT simulations capture well the observed activity-rotation distribution, and are applicable to other stellar types beyond simple application to solar-like ∼4.5 Gyr G dwarfs, b) the [Pevtsov et al., 2003] LX -Φ relation applies across a wide array of stellar types for unsaturated stars, and c) stellar cycle variations provide a natural explanation for the observed scatter about the mean in the data.

Thus, the full dataset displayed in Figure 3.6 can be thought of as a demonstration of the evolution of coronal X-ray emission on two timescales. First, the vertical spread in X-ray emission for a given stellar type at a given Rossby number displays the short- term variation of X-ray brightness with the intrinsic magnetic activity of the modeled star on the timescale of the stellar activity cycle (typically a few years to decades).

Secondly, the range of Rossby numbers can be thought of as representing the range of X-ray and magnetic activity over stellar lifetimes. A star of a given stellar type and initial rotation period would follow one of the colored tracks to higher Ro values as it aged. Therefore, the dataset in Figure 3.6 contains several diﬀerent components of stellar magnetic behavior, a powerful representation of how stars evolve on short and long timescales. 59

3.3.2 Comparison with Observed Stellar Populations

To compare directly with observations (e.g., [Wright et al., 2018]), we take a random

sample of stars from the full set of simulations (Figure 3.6) where each Ro/RoSun value is populated with a number of stars at diﬀerent phases of their magnetic activity cycle.

In Figure 3.7, we show the results of a random selection of 175 partially-convective and 15 fully-convective stars from the distribution shown in Figure 3.6. These are sampled over an ensemble of 10,000 iterations with the mean slope and standard deviation given in the ﬁgure.

The fully-convective M stars are sampled more sparsely because these stars have, in general, very long convective rollover times, τc, meaning they are biased to low Ro values. Thus, unsaturated M stars are much rarer than unsaturated stars of other stellar types, and saturated M stars are more common than unsaturated ones.

We ﬁnd a mean ﬁt to the sampled data of slope p = −1.79±0.128 with a scatter of approximately ±1dex around this mean. This slope is in good agreement within errors

+0.4 with the observations of [Wright et al., 2018] (p = −2.3−0.6) and [N´u˜nezet al., 2015]

+0.14 (p = −2.03−0.17). The diﬀerence, in quadrature, of the p = −1.79 ± 0.128 value found

+0.460 here and the [Wright et al., 2018] value yields pdiff = 0.51−0.641, which is consistent with zero; similarly, the diﬀerence in quadrature between the p = −1.79 ± 0.128

+0.266 value found here and the [N´u˜nezet al., 2015] value yields pdiff = 0.24−0.283, which is again consistent with zero. However, the slope found in this work is noticeably not

in agreement with [Wright et al., 2011] (p = −2.70 ± 0.13), which did not include

fully-convective M stars in the sample considered. Additionally, the scatter of ±1dex

about the mean slope is comparable to the scatter in the observed populations, and is

fully explained by the intrinsic variation in X-ray brightness due to magnetic activity

cycles of the diﬀerent stellar types modeled. 60

Figure 3.7 : One iteration of a random sampling of 175 partially-convective and 15 fully-convective stars from the full dataset in Figure 3.6. An ensemble of 10,000 random samplings was taken, and the mean slope of this ensemble of ﬁts is represented by the solid red line. The standard deviation of the slopes of all 10,000 iterations is represented by the grey shading. Modeled stars normalized by bolometric luminosities corresponding to partially-convective (late-F, G, K, and early-M) stellar types are displayed as blue dots; stars normalized by bolometric luminosities appropriate for fully-convective (FC) late-M stars are denoted by red crosses. For comparison, the observed stellar populations of [Wright et al., 2018] are included as grey dots. 61

We note that due to their relatively low Lbol, the dim, cool fully-convective M stars (denoted by red crosses in Figure 3.7) generally lie above the trend line for the

full stellar sample. This is consistent with the observations of [Wright et al., 2018],

where the fully-convective stars also generally fall above the line of best ﬁt to the

full stellar population. This subtlety in the distribution is perhaps indicative of the

appropriateness of our approach for fully-convective M stars.

[Vidotto et al., 2014] extended the results of [Pevtsov et al., 2003] by considering

73 observed cool dwarfs from late-F to late-M type, i.e. partially- to fully-convective

1.80 stars; for the unsaturated stars in that sample they found a relation of LX ∝ Φ ,

0.98 significantly steeper than the [Pevtsov et al., 2003] fit to dwarf stars of LX ∝ Φ ; this is consistent with our findings because the [Pevtsov et al., 2003] value is likely to be an underestimate as a result of a) the very small sample used for the dwarf star fitting and b) the fact that those stars were near or at saturation, yielding a shallower relation. In this chapter, by applying the p = 2.02 power-law exponent

(derived from solar disk observations in [Pevtsov et al., 2003]) to the simulated stars, we find excellent agreement with the observed cool star populations. We note that the good agreement between the [Vidotto et al., 2014] fitting based on an observed stellar population and our results, using [Pevtsov et al., 2003]’s solar-based relation, is further evidence of the similarity in magnetic and coronal activity of the Sun and other cool stars. Additionally, if we combine [Vidotto et al., 2014]’s relationship of RX with unsigned magnetic flux, Φ, and Φ vs. Ro we find that their results

−2.142 yield RX = LX /Lbol ∼ Ro , in good agreement with the observations that we match as well [Wright et al., 2018, N´u˜nezet al., 2015]. Thus, the 73-star sample of

[Vidotto et al., 2014] containing partially- and fully-convective dwarfs matches the behavior of other observed populations, and of the stars simulated here. 62

3.4 Conclusions

In this chapter, we modeled stellar activity behavior for a range of stellar Rossby

numbers, and found a trend in fractional X-ray luminosity consistent with observa-

tions of unsaturated cool dwarf stars. Additionally, we incorporate the stellar cyclic

behavior of the SFT [Schrijver, 2001] to provide information on the scatter in the

RX vs. Ro relation. We ﬁnd that the variation of a stellar population’s X-ray activity due to the magnetic activity cycles of its constituent stars is consistent with the spread about the RX vs. Ro relation, indicating that observed distributions of cool star populations are well-described as an ensemble of stars randomly phased in their stellar cycles.

In using a tunable model of stellar ﬂux transport, based upon trends observed in real solar data and dynamo models, we capture the observed behavior of simulated stars with Ro-dependent magnetic activity cycles [Dikpati and Charbonneau, 1999].

In following the [Pevtsov et al., 2003] relation between magnetic activity and the resultant X-ray activity, and scaling to the evolution of cool star bolometric luminosities with age, we arrive at a fractional X-ray luminosity-Rossby number relation that is remarkably similar to observations of cool star populations in both slope and scatter about that slope. This is worth emphasizing: a set of solar-based prescriptions for magnetic ﬂux evolution, coupled with a solar-based empirical relation between that magnetic ﬂux and the Sun’s full-disk X-ray spectral radiance, can accurately reproduce the activity-rotation behavior of a set of unsaturated partially- and fully- convective cool stars. We consider this to be strong evidence for a universal activity- rotation relation for all unsaturated partially- and fully-convective main sequence stars, regardless of interior structure.

The good agreement between solar-based modeling prescriptions and observa- 63 tions of cool star populations, including fully-convective M stars, strengthens arguments presented in previous work (e.g., [Wright et al., 2018, Farrish et al., 2019,

Vidotto et al., 2014]) that the photospheric magnetic fields of fully-convective dwarf stars behave similarly to those of Sun-like partially-convective stars and that there may exist a universal scaling of stellar activity with Rossby number in the unsaturated regime for all cool stars. Further, this strong agreement supports the assertion made in [Farrish et al., 2019] that the surface evolution of stellar magnetic fields, captured in the SFT, is broadly applicable across all cool stars and not confined solely to solar-age G stars. This evident similarity in surface magnetic flux behavior may also point to the assertion of [Wright and Drake, 2016] that there is a single stellar dynamo process spanning both partially- and fully-convective dwarf stars. The analysis presented here refines the evidence for this argument by demonstrating that all cool dwarf stars may share a common flux emergence process in addition to a similarity in the surface dynamics of this flux.

While the adoption of solar-based flux transport and cycle variability successfully reproduces the observed activity relation, the connection to a universal dynamo process acting in both partially- and fully-convective stars requires further testing and validation. In this chapter, we vary the stellar magnetic flux, cycle duration, rotation period, and meridional flow profiles commensurate with observed relationships with respect to the stellar Rossby number (see Table 3.1). However, we did not explore the effects of differential rotation rate and latitude dependence, ratio of ephemeral region flux to active region flux, the latitude range of the flux emergence, and the dispersion function for the magnetic field, all of which are contained in the SFT model

(see [Schrijver, 2001]). We would expect all of these parameters to be directly dependent on the speciﬁc properties of the dynamo [Yadav et al., 2015]. For example, 64

[Rüdigeret al., 2019] suggest, from modeling of cross-correlations in meridional flow, that an antisolar differential rotation, with equatorial rotation slower than polar rotation [Künstleret al., 2015], can result for slowly rotating cool stars. Additionally, the presence of polar spots [Strassmeier, 2002, Xiang et al., 2020] further suggests differences in the flux emergence process resulting from differences in the dynamo action between fast and slow rotators [Yadav et al., 2015], although it is worth noting that

[Schrijver and Title, 2001] were able to generate polar spots from enhanced emergence and migration of low latitude magnetic ﬂux using the SFT model. The impact of modifying these parameters on the stellar ﬂux behavior, and the interpretation of this behavior with respect to the stellar dynamo in comparison to observations, may be explored in future works. 65

Chapter 4

Impact of Stellar Activity on Exoplanet Environments

An important consideration in mapping out the magnetic environment of exoplanet systems is to understand the structure, extent, and variability of the magnetic ﬁeld.

Building on the confirmation that the SFT adequately captures the magnetic evolution of cool stars of various stellar types, we now consider the application of a flux transport model to characterization of stellar interplanetary fields on cycle timescales for a range of stellar activity defined by the Rossby number. This framework allows us to examine the asterospheric environments of exoplanetary systems, and yields references against which exoplanetary observations can be compared. We examine several quantitative measures of star-exoplanet interaction: the ratio of open to total stellar magnetic flux, the location of the stellar Alfvénsurface, and the strength of interplanetary magnetic field polarity inversions, all of which influence planetary magnetic environments. For simulations in the range of Rossby numbers considered

(0.1 − 5 RoSun), we find that 1) the fraction of open magnetic flux available to interplanetary space increases with Rossby number, with a maximum of around 40% at stellar minimum for low-activity stars, while the open flux for very active stars

(Ro ∼ 0.1 − 0.25 RoSun) is ∼ 1 − 5%; 2) the mean Alfv´ensurface radius, RA, varies

between 0.7 and 1.3 RA,Sun and is larger for lower stellar activity; 3) at high activity, the asterospheric current sheet becomes more complex with stronger inversions,

possibly resulting in more frequent reconnection events (e.g., magnetic storms) at the 66 planetary magnetosphere. The simulations presented here serve to bound a range of asterospheric magnetic environments within which we can characterize the conditions impacting any exoplanets present. We relate these results to several known exoplanets and discuss how they might be aﬀected by changes in asterospheric magnetic ﬁeld topologies.

As mentioned in Chapter 1, current-day instrumentation has observational limits which conﬁne the detectable population of habitable-zone Earth-size exoplanets to those found in close-in orbits around small, cool M stars [Santos and Faria, 2018].

Though certain of these Earth-size exoplanets, such as Proxima Centauri b, Ross 128 b, and several of the TRAPPIST-1 planets, are thought to reside in the conventionally- deﬁned circumstellar habitable zone (CHZ) [Ramirez, 2018], the low surface temperatures of these M dwarf stars result in CHZs many times closer in to the star than the Earth-Sun distance. For example, Proxima Centauri b and Ross 128 b both have orbital semi-major axes of only about 0.05 AU [Anglada-Escud´eet al., 2016,

Bonﬁls et al., 2018], and all seven known TRAPPIST-1 planets orbit within about

0.062 AU of the star [Grimm et al., 2018]. For comparison, Mercury’s orbit has a semi-major axis of about 0.39 AU, so these ‘M-Earths’ - terrestrial planets orbiting M dwarfs - are indeed extremely close in to their host stars. We here constrain the space physics implications of such systems via a treatment of the magnetic and energetic properties of inner asterospheres for a range of stellar magnetic activity.

4.1 Simulating the Asterospheric Magnetic Field over Time

In this work we apply the same stellar parameter relationships detailed in Chapter 3

(Equations 3.1-3.3). Here, we consider stars in the range from 0.1 to 5 times the solar

Rossby number, RoSun. (Again, we scale to RoSun since there is some intrinsic uncer- 67 tainty in our knowledge of this value [Reiners, 2012]). We use the inter-relationships of Equations 3.1-3.3 to generate self-consistent simulations of the emergence of stellar

flux and its evolution over time for a wide range of stellar activity, extending the application of previous solar flux transport modeling efforts (e.g., [Schrijver et al., 2003]).

As described in Chapter 2, an understanding of the space environment in planetary systems must be built on detailed knowledge of the behavior of the magnetic ﬁeld of the star. To this end, we employ the SFT to treat the emergence, evolution, and dissipation of stellar magnetic ﬂux self-consistently across a range of stellar activity.

In order to simulate the large-scale magnetic ﬁeld structure of the stellar asterosphere, we employ the PFSS model for coronal magnetic structure.

In combining the stellar surface ﬂux transport simulations with potential ﬁeld coronal extrapolations, we obtain a set of self-consistent three dimensional magnetic

field distributions extending from the stellar surface into the corona and inner asterosphere. These models have finely-gridded spatial magnetic fields as well as complete temporal coverage as the field distributions evolve over several stellar cycles. Thus, we have constructed a set of virtual asterospheres, differentiated by their surface flux and related magnetic activity, that can be examined for their potential impacts on associated exoplanets. Figure 4.1 displays the results of a solar test case simulation as well as those for a more active (Ro = 0.5 RoSun) and a less active (Ro = 4 RoSun) star. Both the line-of-sight magnetograms of the simulated photospheres and the coronal magnetic field line distributions are displayed at stellar cycle maximum for direct comparison.

Figures 4.1(b, e) show the solar simulation at cycle maximum. The more active stellar simulation, again at cycle maximum (Figures 4.1(a,d)), contains many more strong and complex magnetic ﬂux concentrations as well as enhanced magnetic ﬂux 68

Figure 4.1 : Three example stellar flux transport simulations with associated coronal field extrapolations. (a), (b), and (c) display the stellar photosphere flux distributions for Ro = 0.5 RoSun, Ro = RoSun, and Ro = 4 RoSun, respectively. These flux distributions are represented in the form of line-of-sight magnetograms with white corresponding to positive magnetic polarity and black corresponding to negative magnetic polarity, as in Figure 2.3. (d), (e), and (f) display the associated coronal magnetic field line distributions for each stellar simulation, extrapolated using the PFSS method. Field lines in magenta (outward) and green (inward) represent fields that cross the source surface and extend into interplanetary space, forming the magnetic field of the inner asterosphere. Field lines depicted in black represent closed magnetic field lines with each end rooted in the stellar photosphere, forming the loop structures of the stellar corona. 69 in the quiescent regions outside of the starspots. Figures 4.1(c, f), representing cycle maximum for a less active star, have fewer large active regions, but still contain a global distribution of small-scale ephemeral regions. The clear increase of magnetic complexity with decreasing Rossby number is also evident in the coronal field distributions. The strength and complexity of the stellar magnetic field has implications for exoplanet environment: the effect of a star on a planet is largely governed by the nature of the global stellar magnetic field, its related dynamical behavior, and its evolution over time.

We examine a set of Rossby numbers in the unsaturated regime of stellar activity.

As discussed in Chapter 3, observations of cool stars have shown that for Rossby numbers in the range from about 0.1-3, stellar activity - chieﬂy in the form of indicators such as X-ray luminosity, chromospheric emission, or average magnetic ﬁeld strength - declines steadily with increasing Rossby number, scaling approximately logarithmically (e.g., [Wright and Drake, 2016, Reiners et al., 2014]). For smaller

Rossby numbers, Ro < 0.1, observations show that stellar activity becomes eﬀec- tively independent of Rossby number and the activity “saturates”. Stellar observations and our modeling in Chapter 3 have shown the dichotomy between the saturated and unsaturated regimes to be robust for populations of partially-convective cool stars like the Sun as well as for fully-convective stars like Proxima Centauri

[Wright et al., 2011, Wright and Drake, 2016]. In this chapter, we again restrict the application of the SFT model to stars in the unsaturated regime.

4.2 Asterospheric Fields of Exoplanet Host Stars

Treatment of magnetic flux transport, coupled to the PFSS coronal field extrapolation, allows us to characterize the coronal and asterospheric magnetic fields of active 70 stars as a function of stellar activity, via the Rossby number and the relationships de-

ﬁned in Equations 3.1 - 3.3. The topology and strength of the asterospheric ﬁeld is key to understanding the magnetic and energetic environment in which exoplanets reside.

Here we introduce and investigate several quantitative measures of star-exoplanet interaction: the ratio of magnetic ﬂux open to the asterosphere, the location of the stellar Alfv´enradius, and the strength of magnetic polarity inversion crossings for planetary orbits.

4.2.1 Stellar Magnetic Field Topology

Figure 4.2(a) displays the variation of the fraction of magnetic ﬂux contained in open

field lines, relative to total stellar flux as a function of Rossby number. This quantity is analogous to the asterospheric flux defined in [Schrijver et al., 2003], except that a) we here scale the open flux to the star’s total flux for direct comparison across a wide range of stellar activity, and b) we have included the variation of surface flows and stellar cycle parameters in the stellar models in accordance with the observationally-motivated relations of Equations 3.1-3.3. Additionally, this quantity reflects the complexity of the stellar magnetic field topology, with more complex fields containing lower fractions of open magnetic field lines; this magnetic field complexity has been studied for others stars such as young, fast rotators in open clusters

[Garraffo et al., 2018b]. In particular, [Garraffo et al., 2018b] found that the evolution of rotation rate in young stars depends strongly on magnetic complexity, as some fast-rotating stars transport angular momentum less efficiently to the stellar wind and thus have persistent fast rotation rates. While we limit our focus to stellar cycle timescales, where the Rossby number is constant for a given star, rather than the Myr-Gyr timescales on which stellar rotation period evolves, an understanding of 71 the longer-term effects of magnetic complexity on stellar activity and its implications for the evolution of the magnetic environments of exoplanets is an important consideration in determining the current state of the planet. Exploration of the relation between stellar evolution with age and exoplanetary magnetic environment will form the basis of future work in this area.

The SFT simulation results presented in Figure 4.2 are consistent with the observational trend of the logarithmic variation with Rossby number: all stellar simulations show more open flux at stellar minimum than at maximum, as a result of the dominant dipolar nature of the minimum field. As the star approaches starspot maximum, quadrupolar and higher-order components of the magnetic field grow, thereby contributing comparatively less magnetic flux that is open to the interplanetary medium

[Wright and Drake, 2016]. Figure 4.2(a) also shows that stars more active than the

Sun exhibit less variation in the proportion of open flux over the course of an activity cycle, since these very active stars will have complex and predominantly closed magnetic fields even at stellar minimum. Conversely, lower activity stars have dominant dipolar magnetic fields with more open field, principally at the stellar poles (see also the coronal field extrapolations of Figure 4.1).

All of the Solar System planets reside near the ecliptic, the plane of the Earth’s orbit around the Sun; the ecliptic plane coincides closely with the solar equator and the plane of other planetary orbits. Theories of planet formation in other stellar systems and observations of protoplanetary disks of dust and gas around young stars imply that most exoplanets form in coplanar orbits, roughly coincident with the rotational equator of the host star [Xie et al., 2016]. It is therefore useful to consider the variation of open ﬁeld with latitude as a function of stellar activity, particularly in a narrow range of latitudes around the stellar equator. We can examine the open 72

Figure 4.2 : (a) Ratio of open-field flux to total surface magnetic flux as a function of stellar activity at stellar maximum and stellar minimum. Error bars are associated with a ±10% variation in the simulation source strength, corresponding to variation in the open-to-total flux ratio for a given Rossby number, Ro. (b) Ratio of open-field flux to total stellar flux in three latitude bands: ± 0o − 30o (solid lines), ± 30o − 60o (dashed lines), and ± 60o − 90o (dotted lines). Each data point represents an average of the quantity over the northern and southern hemispheres.

ﬂux in the bands within 30o north or south of the stellar equator as a measure of the available ﬂux in the planet-hosting low-latitude regions of the asterosphere. Figure

4.2(b) demonstrates that the fraction of open flux at low latitudes (solid lines) does not vary significantly with activity, although there is a clear separation between stellar maximum and stellar minimum. It is clear that the open flux is predominantly defined by the high latitude field (e.g. polar coronal holes) and that the fraction of open flux at high latitudes increases dramatically with decreasing activity (see Figure

4.1(f) and [Reiners and Basri, 2006]). The low latitudes (±0o − 30o, solid lines) show the least variability, which is expected as the equatorial activity belt is typically composed of closed-field active regions for all activity levels. The largest variation in the asterospheric field topology with activity is associated with the higher latitude bands (the dashed and dotted trend lines in Figure 4.2(b), respectively), particularly 73 for stellar maximum (in red). At low Rossby number, the star is very active and likely dominated by disordered, closed field, even at stellar minimum when the dipole component of the field is expected to be relatively strong. This effect is even more pronounced at stellar maximum (see, for example, Figure 4.1(d)), when the open flux ratio rises sharply for high Rossby number because stellar maximum is still dominated by the overall dipole component.

We consider the variation of open-field magnetic flux as an indicator of the avail- ability of magnetic field to the asterosphere and the stellar wind. The closed magnetic

ﬁeld structures that make up the stellar atmosphere also play a signiﬁcant role in stellar activity with substantial planetary impact; the production of energetic EUV and

X-ray emission in closed coronal field structures has implications for planetary magnetosphere energization, variations in ionospheric conductance, atmospheric chemistry, and atmospheric loss [Airapetian et al., 2017]. Hydrostatic and hydrodynamic models of stellar coronal heating provide a means to relate the simulated flux distributions, and their associated 3D closed-field structures, to the expected energetic plasma emission. Such simulations provide self-consistent stellar XUV emission output that subsequently serve as inputs to associated planetary magnetospheric and ionospheric modeling, as well as providing opportunities for direct comparison with stellar observations (see Chapter 5 for more discussion).

4.2.2 Stellar Alfv´enSurface

An important consideration in examining the magnetic interactions between a star and a planet is the location of the planetary orbit relative to the stellar Alfv´ensurface.

We define the Alfvénradius as the average radius around a star at which the stellar wind energy density equals the asterospheric magnetic field energy density. At the 74

Figure 4.3 : The Alfvénsurface of a Proxima Centauri-like star, shown in blue, is modeled for a maximum stellar magnetic field strength of 600G (left) and a mean field strength of 600G (right), using an MHD model of the stellar wind driven by Alfvénwave turbulence [Garraffo et al., 2016]. The plane denotes the location of the asterospheric current sheet; a theoretical orbit of the planet Proxima Centauri b, with a semi-major axis of 0.049 AU, is show in black. In the 600G-mean case at right, the orbit of Proxima Centauri b passes inside the Alfvénsurface once per orbit.

radial distance of the Alfvénsurface, RA, the stellar wind transitions from sub- to super-Alfvénic.A planet orbiting periodically or continually inside this surface may be directly magnetically connected to the stellar corona, with potentially disastrous effects on atmospheric retention [Garraffo et al., 2017]; a planet with an orbit placed well outside this boundary will be decoupled from the coronal magnetic field and will interact with the stellar wind in a manner similar to that of the Earth.

Depending upon the assumptions one makes about the nature of the stellar wind

(e.g. thermally- or turbulently-driven) the Alfvénboundary may not be purely spherical but rather have a convoluted shape that depends strongly on photospheric magnetic field distributions and the variation of the Alfvénspeed in the stellar corona

[Scherer et al., 2001]. Previous modeling efforts have demonstrated, for example, that the extent of the Alfvénsurface may be smaller at the equator and larger at the poles 75 for a star of a given activity level and rotation rate [Révilleet al., 2015]. Figure 4.3 displays a test case using an MHD model of stellar wind expansion to estimate the

Alfvénsurface structure of Proxima Centauri [Garraffo et al., 2016]. In this chapter, we restrict our considerations to a spherical Alfvénsurface as a representative measure of the transition to stellar wind dominated magnetic field, enabling us to consider large-scale changes as a function of stellar activity and stellar cycle phase without recourse to a highly parameterized and computationally expensive wind model. We extend previous Alfvénsurface analyses to consider a range of potential host star activity and the associated effect on the Alfvénsurface, based on the spatially averaged magnetic flux of the star. While the precise impact that a direct magnetic connection to the stellar corona would have on atmospheric erosion is not within the scope of this work, it is certain that a planetary orbit embedded continually or periodically within its host star’s Alfvénsurface would reside in a significantly more hostile space weather environment than that of the Earth.

Following [Schrijver et al., 2003] we employ a scaling law for the calculation of the

Alfv´enradius, RA, under the assumption of a thermally-driven stellar wind:

R φ −0.16±0.13 A = ∗ (4.1) RA,Solarmax φSolarmax

where φ∗ is the average magnetic flux at the stellar surface. This relationship is derived from a detailed consideration of angular momentum loss via the stellar wind, while also including the simplifying assumption that the Alfvénspeed vA is equal to the terminal stellar wind speed at large distances from the stellar surface, v∞. We scale these values to the analogous values for the Sun at solar maximum. It is estimated that the solar Alfvénsurface has an average radius of ∼ 20 RSun (≈0.1 76

Figure 4.4 : (a) Alfvénsurface radius as a function of stellar activity, at both stellar minimum and stellar maximum. Alfvénradius is scaled to the value for the Sun at solar maximum, equal to ∼20 RSun. Shaded error regions are associated with the uncertainty in the exoponent of Equation 4.1. For comparison, we show the relationship derived by [Schrijver et al., 2003]. (b) The orbital locations of several known close-in exoplanets are shown in relation to observational estimates of their host stars’ Rossby numbers. Many known exoplanets with orbits on the order of 1 AU or farther lie far above the upper limit of the plot and are expected not to interact measurably with the host star’s Alfvénsurface.

AU) at solar maximum [Chhiber et al., 2019].

Figure 4.4(a) presents the Alfvénradii we find at both stellar minimum and maximum phases for our chosen range of stellar activity. Also shown in Figure 4.4(b) are the locations (planetary orbit semi-major axis vs. host star Rossby number) of several known exoplanets where we assume a fiducial solar Alfvénradius of 20Rsun and a solar Rossby number of 1.85 ([Stepien, 1994], consistent with [Wright et al., 2011]). We see that the orbits of some widely studied habitable-zone planets, Proxima Centauri b, Ross 128b, TRAPPIST-1 c, e, g, and h, all reside well within the Alfvénsurfaces of their host stars at all times over the stellar cycle. Of the two known planets within the

GJ 3323 system [Astudillo-Defru et al., 2017], GJ 3323 b lies well within the Alfvén radius while GJ 3323 c skirts the Alfvénsurface at stellar minimum. Different models of the stellar wind, or the assumption of a larger solar Alfvénradius, could result in 77

GJ 3323 c also falling within this crucial surface, signiﬁcantly aﬀecting its physical

state. The points representing Ross 128 b and the recently discovered planets around

Teegarden’s star [Zechmeister et al., 2019] were generated by estimating the Rossby

number of their respective stars using expression (11) in [Wright et al., 2011] to de-

termine the convective rollover time from the stellar mass and an assumed rotation

period of ≈100 days, reflecting the expectation for low mass stars (< 0.1MSun) with no significant Hα emission (see [Bonfils et al., 2018] and [Newton et al., 2017]). The error in the Rossby number is likely to be high. We scale to the assumed value

RoSun = 1.85 [Stepien, 1994] in each case [Wright et al., 2011]. These results compare well with previous modeling efforts incorporating a more detailed MHD stellar wind that demonstrated, for example, that Proxima Centauri b might cross in and out of its star’s Alfvénsurface over the course of each 11-day planetary orbit (see Figure 4.3, reproduced from [Garraffo et al., 2016]); see also similar arguments for the TRAPPIST-1 planets [Garraffo et al., 2017]. In the simulations presented here, we have assumed a thermally-driven wind expressed by a constant vA = v∞; a turbulently-driven wind results in a radial and latitudinal variation in the local Alfvénspeed [Cohen et al., 2014, Garraffo et al., 2017], resulting in a more structured Alfvénsurface (refer back to Section 2.4).

Figure 4.4(b) implies that however the Alfvénsurface is defined, many of the known Earth-sized planets of interest for habitability studies reside inside or near the Alfvénsurface boundary of their small, cool host stars, and therefore experience vastly different magnetic environments than does Earth. Our results are important to long-term prospects for habitability in these and other systems of extreme star-planet proximity, since the location of these planets relative to their stars is almost certainly an unfavorable configuration for a hospitable space weather environment, or at the 78

Figure 4.5 : Radial magnetic ﬁeld component Br at the source surface for the same simulations shown in Figure 4.1 (Ro/RoSun = [0.5, 1.0, 4.0]), shown at both stellar maximum (top) and minimum (bottom). The projection of two theoretical planetary orbits with 0o (dashed) and 30o (dotted) inclinations are shown.

very least an extremely exotic system compared to that of Earth and the Sun.

4.2.3 Asterospheric Current Sheet

Finally, we consider the complex structure of the interplanetary magnetic field and its potential for creating magnetic activity at the planet. The asterospheric current sheet (ACS; [Wilcox et al., 1980]) defines the boundary between opposite polarities in the large-scale stellar magnetic field; a simple stellar dipole field (e.g., Figure

4.1(f)) leads to an unstructured ACS centered on the stellar equator (see Figure

4.5(f)). Naturally, a more active star, or a star at a more active phase of its cycle, has higher-order components in its global magnetic ﬁeld, particularly in the low- latitude activity belt, and the ACS becomes more complex with signiﬁcant excursions

[Lundstedt et al., 1981, Nayar and Revathy, 1982]. Figure 4.5 shows the radial magnetic ﬁeld at the stellar source surface for each of the activity levels considered in

Figure 4.1 (Ro/RoSun = [0.5, 1.0, 4.0]). The increase in complexity is obvious as the 79

activity level increases and Ro decreases from right to left. Solar maximum activity

(Fig. 4.5(b)) shows a highly structured ACS with major poleward excursions in both

hemispheres. High-activity, low-Rossby number stars have a far more complex and

disorganized structure (Fig. 4.5(a)) and even experience signiﬁcant deviations from

dipolar magnetic structure at the minimum activity phase (Fig. 4.5(d)).

Also shown in each panel of Figure 4.5 are projected orbits for two hypothetical

planets with inclination angles to the stellar equator of 0o and 30o. The potential

for signiﬁcant planetary magnetic activity resulting from increased stellar magnetic

complexity has been extensively studied for the Sun with a number of case and sta-

tistical studies emphasizing the importance of sector boundary crossings (SBC) for

driving geomagnetic activity; SBCs signify sharp transitions in the sign of the stel-

lar radial ﬁeld. It has been shown [Asenovski, 2017] for the solar case that sec-

tor crossings (radial magnetic ﬁeld directed away/toward the Sun or toward/away)

are frequently associated with moderate to strong geomagnetic storms. Further,

[Khabarova and Zastenker, 2011] demonstrated that the rapidity of the transition

between magnetic polarities is also an important feature in driving strong magnetic

activity at the planet. Additionally, a comprehensive statistical analysis of the impact

of solar magnetic structures on geomagnetic activity by [Echer and Gonzalez, 2004]

has shown that SBCs alone are moderately geoeﬀective (∼ 26% of SBCs leading to moderate to intense activity), but have an increased impact when they are associated with other interplanetary magnetic structures such as magnetic clouds and corotat- ing streams: 100% of magnetic clouds at or near SBCs were found to be geoeﬀective

(compared to 77% of magnetic clouds as a whole). The strength of any planetary magnetic activity resulting from stellar magnetic complexity is dependent on a number of factors - the presence of sector boundary crossings, high speed streams and 80

magnetic clouds, as well as the strength and orientation of the planetary magnetic

ﬁeld. To fully constrain all of these factors would require detailed modeling coupled,

ideally, with high-resolution observations of a particular star-planet pairing. In this

chapter, we use the variation in the SBC structure with stellar activity as a diagnostic

for the potential for enhanced planetary magnetic activity, and as a consideration for

future possible observational searches for associated planetary emissions.

In Figure 4.6, we show the behavior with stellar activity of the magnetic ﬁeld

gradient averaged over all sector boundary crossings experienced by the hypothetical

planetary orbit of 0o inclination shown in Figure 4.5. Here we assume the ecliptic plane coincides with the equatorial plane of the star. While the number of ACS sector boundary crossings is a good representation of the magnetic complexity of the inner asterosphere, the significant variability in the structure of the ACS over a stellar cycle limits the utility of this number as a definitive measure of strong planetary magnetic activity. However, it has been shown for the Sun that the steepness of the transition from one polarity to the other has a significant bearing on the intensity of the planetary magnetic response [Khabarova and Zastenker, 2011]. As such, we have adopted the average spatial gradient of the radial magnetic field across polarity inversions (either away/toward or toward/away) as a discriminator of potential planetary magnetic impact. Figure 4.6(a) shows that there is likely to be a substantial increase in the magnetospheric activity at the planet with increasing stellar activity/complexity. For stars significantly less active than the Sun (Ro/RoSun ≥ 2), there is very little variation in the structure and complexity of the ACS, and therefore the average strength

of polarity inversion crossings is comparatively low and exhibits little variation with

Rossby number. Conversely, the trend shown in Fig. 4.6(a) demonstrates that the

typical crossing strength across boundaries between different magnetic polarities rises 81 sharply with higher stellar activity. For all activity levels modeled there is a clear distinction between stellar minimum and stellar maximum with the latter showing significantly stronger gradients due to the stronger fields present at stellar maximum.

To place the possible impact of SBCs in the exoplanetary context, we must consider the relative timescales of the orbital period and stellar rotation. The evolutionary timescale of the stellar magnetic ﬁeld itself may also need to be taken into account, particularly when considering planets with long orbital periods or when the orbital and stellar rotation periods are comparable - in this case the magnetic structure

“seen” by the planet over time is more dependent on the changing surface ﬁeld than on the planet’s location in its orbit. Here, we are interested in planets that are in close proximity to the star, where the potential for magnetic interactions is stronger; thus, we disregard the stellar ﬁeld evolution timescale as being long compared to planetary orbit timescales (on the order of 5-10 days for most of the planets displayed in Fig.

4.4(b)). We consider two cases: a) a fast-rotating star with a planet of reasonably long orbit period and b) a slow-rotating star with a close-in planetary orbit. In the former case, the planet can be regarded as eﬀectively stationary as the asterospheric

field sweeps past; in the latter case, the magnetic configuration can be assumed to be effectively stationary as the planet traverses the interplanetary magnetic structure

(see hypothetical orbital paths shown in Figure 4.5). For the zero-inclination orbit of

Figure 4.6(a), these two scenarios are equivalent and the planet eﬀectively encounters the entire equatorial open ﬁeld structure, so that the planetary magnetic activity is driven, in part, by the SBCs encountered. For an inclined orbit, e.g. the 30o case considered, the situation is more complex. For case a) - fast rotator, long-period orbit

- the planet is eﬀectively stationary at a speciﬁc asterographic latitude, depending upon where it is in its orbit, and the planet would experience any SBCs associated 82 with that latitude. For case b) - slow rotator, short-period orbit - the planet would sweep out a range of asterographic latitudes as shown in Figure 4.5 leading to a complex interaction with the ACS. In this case, a possible dominant signature in the average gradient would be the swing from the northern to southern hemisphere. This strong signature would appear at all activity levels and as a result would display only weak variation with activity.

In Figure 4.6(b), we show the results for a hypothetical planetary orbit with inclination 30o to the star’s equatorial plane (assumed orthogonal to the star’s rotation axis), following the sinusoidal curves displayed in the panels of Figure 4.5. Consistent with the reasoning above, there is little variation in the low-activity regime, where stellar rotation period is long compared to planetary orbital period and the polarity inversions are dominated by a simple crossing of the magnetic equator. Unlike the

0o inclination orbit shown in Fig. 4.6(a), there is no clear trend in the strength of polarity crossings at high stellar activity for the assumed 30o orbital inclination. This is due to the highly complex and disordered nature of the stellar magnetic fields at low Ro, complicated by the comparable timescales of stellar rotation and planetary orbit (low Ro stars tend to be fast rotators). In the absence of real planetary orbits with independently-defined inclinations and phases relative to stellar rotation, we must choose an arbitrary phasing of the orbit with respect to the magnetic structure, shown in Fig. 4.5, and thus cannot draw universal conclusions from an arbitrary sampling of the stellar magnetic field polarity inversions for high stellar activity. For the phasing selected, in Fig. 4.5(a), the planet traverses a large inversion in the radial magnetic field distribution at asterospheric longitudes of approximately 250o and 340o.

We consider these sharp transitions in the polarity of the stellar magnetic ﬁeld 83

Figure 4.6 : (a) Shown in red and blue, the magnitude of the average radial magnetic ﬁeld gradient at locations where the 0o inclination orbit (see Fig. 4.5) traverses a polarity inversion crossing for stellar maximum and minimum, respectively. The values are normalized to the largest value in the set, the gradient at stellar maximum for Ro = 0.1 RoSun. (b) In red and blue, the magnitude of the average radial magnetic ﬁeld gradient at locations where the 30o inclination orbit crosses a polarity inversion, for stellar maximum and minimum, respectively. The values are normalized to the largest value in the set, the gradient at stellar maximum for Ro = 0.1 RoSun.

because of the implications for strong and/or frequent opportunities for reconnection events in any planetary magnetosphere that may be present. The stronger the planetary magnetic response the stronger its expected radio signature (see Chapter 5 and

[Zarka, 2010]).

A more complete picture of the potential asterospheric magnetic ﬁeld eﬀects on exoplanets will emerge as we broaden our investigation in future work to consider full magnetohydrodynamic (MHD) stellar wind calculations. This research will provide additional knowledge of the behavior of the stellar wind plasma and the kinetic (ρv2) energy density of exoplanet host stellar winds (see Chapters 5 and 6).

4.3 Discussion of Results

We have here presented a description of our methodology in applying solar physics modeling capabilities to questions of exoplanet host star behavior. We bridge the gap 84 between heliophysics understanding of the Earth-Sun system and the open questions of characterizing the impact of star-exoplanet interactions. Assuming solar-like magnetic flux transport modified to take account of observed stellar physics relations, and a potential field treatment of the large-scale field of the stellar corona, we are able to simulate the spatial and temporal variations of stellar magnetic activity for a range of possible exoplanet host stars. We have characterized the scaling using the Rossby number as a measure of stellar activity and magnetic field behavior for the unsaturated regime of the activity-rotation relation, which observational evidence suggests is robust for both solar-like partially-convective stars and slowly-rotating fully-convective stars. This observational support for the scaling of the solar-based models to other stars is an encouraging result for this still developing field of comparative heliophysics for exoplanet systems.

We note that these simulations do not represent the exact surface flux distributions of any given star, due to factors such as large uncertainties in measured stellar parameters, e.g., stellar cycles, rotation periods, and large- and small-scale magnetic field strengths. In addition, the current state of stellar observations provides very limited knowledge of the surface flows and internal dynamics of other stars, though advances in ZDI imaging and asteroseismic observations may improve our understanding of these processes in the future [Reiners and Basri, 2006]. Rather than attempting an exact replication of magnetic flux distribution and evolution on a particular star’s surface, we instead consider a set of simulations that are representative of the levels of activity and the large-scale connectivity of asterospheric magnetic fields in general.

We complement these simulated stellar photospheric magnetic field distributions by coupling to a potential-field source surface model for the stellar corona and inner asterosphere; the PFSS extrapolations represent an ideal current-free magnetic field, 85 a reliable descriptor of the large-scale connectivity and topology of the solar corona.

Future work may incorporate more complex representations of coronal ﬁeld such as

MHD or force-free extrapolation methods to discern contributions to star-planet interactions from non-potential coronal magnetic ﬁelds with currents.

We find that with increasing stellar activity, the fraction of magnetic flux topo- logically open to the stellar wind decreases as the coronal magnetic field increases in complexity, but does become less restricted to the stellar poles; in addition, the variation across cycle timescales of this ratio decreases with decreasing Rossby number. One avenue of further investigation will be to combine this topological approach to wind structure with models of stellar wind speeds and pressures, to provide a fuller picture of both the strength and spatial distributions of stellar winds affecting exoplanets [Scherer et al., 2001].

We also demonstrate via a flux-dependent scaling relation that the average radius of the stellar Alfvénsurface decreases for small Rossby number but stays roughly constant at large Rossby number. We find that several close-in exoplanets around active host stars, e.g. Proxima Centauri b and the TRAPPIST-1 planets, reside entirely within their stars’ Alfvénsurfaces, effectively embedded in the potentially quite extreme environment of their host stars’ coronae. With a more structured, turbulent- driven stellar wind, the interaction between the planets and their star’s Alfvénsurface may be more complicated [Garraffo et al., 2017]. Investigation of the consequences for habitability of these planets is outside the scope of this work, but it is certain that such an environment would be very different than the magnetic environment at

Earth. However, given the thermally-driven wind assumption adopted here, GJ 3323 c would marginally lie outside the stellar Alfvénsurface, having an orbital semi-major axis of 0.126 AU, and if it has a sufficiently strong magnetic field it should be well- 86 protected from the stellar activity of GJ 3323 and an interesting target for future radio observations. For completeness, it is worth noting that GJ3323 c also lies significantly outside the star’s CHZ (estimated to be between 0.026 and 0.054 AU) and outside Kopparapu’s outer boundary for an Early Mars-like atmosphere (estimated to be 0.077AU for an M-type host star [Kopparapu et al., 2013]).

An important factor in the characterization of the physical nature of an exoplanet is the determination of its magnetospheric field. This requires modeling the planetary response to stellar activity. In the present work, we note that the increasing magnetic complexity associated with decreasing Ro is reflected in the number and intensity of the sector boundary crossings in the asterospheric magnetic field. While these characteristics of the ACS are by no means ‘fixed’ for a given star at a given phase of its activity cycle, the modeling presented here does provide some insight into the exoplanet magnetic environments expected. This also provides useful contextual information for possible searches for exoplanetary radio signatures.

With the rapid growth in exoplanet discovery, there is a need for in-depth characterization of planetary environments, significant aspects of which are driven by interaction with their parent star. It is important to understand the physical conditions that underpin these interactions if we are to fully characterize planetary conditions, interpret observational signatures, help guide future searches, and refine our knowledge of the potential for habitability. Since present observational capabilities are not sufficient to detect key signatures that define the planetary environment, it is necessary to guide future observations by constraining the expected signatures via modeling and simulation. In this chapter, we have considered the energetic interactions between stars and planets, their temporal evolution over stellar cycle timescales, and their expected impact on the magnetic environment of exoplanets. In future work 87 we will extend this analysis to the coronal XUV signatures associated with the stellar magnetic activity discussed here. Energetic XUV emission is an important component of the planetary response as it drives ionospheric currents that serve to influence the planet’s magnetospheric state. 88

Chapter 5

Applications to Star-Planet Interaction

In this chapter we discuss several ongoing applications of the work described in Chap- ters 3 and 4 in the area of solar-stellar connections and modeling of exoplanet systems.

5.1 Planetary Magnetospheric and Ionospheric Processes

Thus far we have discussed the magnetic and energetic behavior of the Sun and other stars with the aim of characterizing their inﬂuence on associated planets. The study of planetary responses to solar and stellar drivers is itself a rich and active

field of research. The study of the Earth’s magnetosphere describes the variability of magnetic fields, electrical currents, and plasma flows in the region of space dominated by the Earth’s magnetic field. In addition to the Earth, other Solar System planets such as Mercury, Jupiter, and Saturn host interior magnetic dynamos which produce extended magnetospheres.

To date there are no confirmed examples of extrasolar planets with planetary magnetospheres due to the difficulty of detecting star-planet interactions described in Section 1.1.2. However, it is known that in the Solar System, many bodies have electrically conducting fluid interiors and sufficiently fast rotation to sustain global magnetic fields - in addition to the Earth, Mercury, Jupiter, and Saturn, the ice giants Uranus and Neptune and the Galilean moon Ganymede produce global magnetospheres [Schrijver and Siscoe, 2009]. Even unmagnetized planets like Venus exhibit 89 dynamic interaction between the solar wind and the planet’s ionized atmosphere, and the resulting currents can produce a tenuous induced magnetopause on Venus’ dayside [Luhmann et al., 1991]. Mars once had a molten core that has since cooled, and remnants in the present-day Martian crust of strong magnetism imply the presence of a large-scale magnetosphere in Mars’ past; the fossil magnetic field of Mars interacts with the solar wind and can even display reconnection events similar to magnetospheric and ionospheric processes seen at Earth [DiBraccio et al., 2018]. With so many examples in our own Solar System, magnetospheres could be quite commonplace among exoplanets and the study of magnetospheric responses to solar and stellar wind driving are important components for understanding space weather activity around exoplanets.

The studies of stellar magnetic ﬁeld variability and coronal structure detailed in

Chapters 3 and 4 were conducted in parallel with complementary doctoral work by

Anthony Sciola in the ﬁeld of exoplanet magnetospheric responses to stellar wind and coronal emission. As part of the magnetospheric physics modeling group at Rice

University, Sciola has investigated a number of scenarios in which planets respond to variability in stellar wind conditions by employing the MHD models described in

Section 2.5.

[Sciola et al., 2021] demonstrated that the inclusion of important ionospheric physics in numerical models of the couple magnetosphere-ionosphere system is integral in determining estimates of exoplanet radio emission and predicting the feasibility of observing star-planet interaction. As discussed in Section 1.1.2, there are currently no conﬁrmed direct observations of star-planet space weather interaction, for example in the form of auroral radio emission as seen at Earth, Jupiter, and Saturn

[Zarka, 1998]. Detailed modeling of the coupled magnetosphere-ionosphere system 90

with reasonable assumptions for stellar wind conditions can place constraints on the

anticipated strength of radio emission from exoplanet aurorae. The brightest source

of auroral radio emission is due to the electron cyclotron maser instability (ECMI),

in which the frequency of emission is directly related to the local gyrofrequency of

the ionospheric plasma [Sciola et al., 2021, Winglee et al., 1986]. The gyrofrequency

ω is related to the strength of the magnetic ﬁeld present (ω = |q|B/m where q and m are the particle charge and mass, and B is the magnetic ﬁeld strength). Therefore, observations of exoplanet auroral radio emission would not only indicate the presence of star-planet interaction, but would also provide constraints on exoplanet magnetic

ﬁeld strengths. Such work is of great relevance to future radio observation campaigns.

A key factor in the ionospheric response to stellar activity is the EUV and X- ray irradiance. Solar EUV emission directly influences the conductance of Earth’s ionosphere, impacting current flows and magnetospheric dynamics. Therefore, reliable information about the energetic coronal emission is essential to the modeling of planetary response and other treatments of planetary magnetospheric-ionospheric systems. Exoplanets are expected to be heavily influenced by the EUV emission of their host stars; for example, Proxima Centauri b may receive more than 30 times the EUV flux than the Earth does [Ribas et al., 2016]. A more detailed treatment of the production and variation of stellar EUV emission over stellar cycles, as discussed below, could provide more realistic inputs to planetary response modeling.

5.2 Solar and Stellar Coronal Emission

The work described in Chapter 3 detailed the well-constrained relationship between

X-ray emission and magnetic ﬂux on a variety of spatial scales in the Sun and other stars. This strong relationship between the magnetic ﬁeld and the heating of the 91

solar atmosphere to temperatures suitable for X-ray emission is expected because

the magnetic ﬁeld is the only source of energy substantial enough to produce the

observed heating (see Section 1.2.2). The production of EUV ﬂux in stellar coronae

is less well-constrained: EUV emission is produced in both quiet Sun regions and in

active, ﬂaring regions [Ribas et al., 2016], and can also result from the subsequent

cooling of plasma after X-ray production. As such, the simple scaling law provided

p by the [Pevtsov et al., 2003] relationship LX ∝ Φ (see Section 3.1) cannot be applied to solar EUV emission. The importance of stellar EUV for understanding planetary response, contrasted with the diﬃculty of estimating bulk EUV properties of stellar coronae, means that a detailed modeling treatment of coronal EUV emission is necessary to ﬁll the gap in our understanding of solar-stellar connections.

We have described in Section 2.3 the coronal XUV modeling framework developed by [Klimchuk et al., 2008, Barnes et al., 2019] that we employ here. Figure 5.1(a) displays a large active region from a solar test case of the SFT (red shows positive magnetic polarity directed radially outward; blue concentrations have negative polarity directed radially inward). A potential ﬁeld extrapolation of the coronal magnetic

field in the region above the active region on the solar photosphere is overlaid in black lines. Figure 5.1(b) displays the broadband EUV emission (100-1000A)˚ calculated from a hydrostatic model of plasma heating and emission along the extrapolated magnetic field lines. The emission is calculated by solving the hydrostatic equations along each field line (axis coordinate is s as in Figure 2.3):

dp R 2 dr = −ρg Sun (5.1) ds Sun r ds dF E = − + n2Λ(T ) (5.2) H ds 92

Figure 5.1 : At left, a large active region from a surface flux transport model of the Sun, with a potential-field extrapolation of the coronal magnetic field above the solar surface. At right, the broadband EUV emission (100-1000A)˚ produced along the potential magnetic field lines in the left-hand panel.

where p is the pressure, ρ is the density, gSun is the acceleration due to solar gravity,

EH is the heating, F is the conductive ﬂux, and n is the number density of the plasma particles.

The EBTEL model for coronal heating and emission described in Section 2.3 has been optimized for the spatial scales of a single active region, as shown in Figure

5.1. Solar observations and modeling are often spatially-resolved and detailed enough to model the dynamics of coronal magnetic ﬁelds and plasma in individual active regions. By contrast, stellar observations of coronal emission are point sources without spatially-resolved stellar disks, as described in Section 3.2.2. We have therefore begun work in collaboration with synthesizAR author Dr. Will Barnes to model the global

X-ray and EUV emission of exoplanet host stars with the aim of providing realistic inputs to the magnetospheric-ionospheric modeling of [Sciola et al., 2021] described in Section 5.1 above. 93

5.3 Young Solar Analogs

Another application of interest to planetary studies is the evolution of the Sun over

its lifetime, particularly its younger, more active state in the past that may have

inﬂuenced the evolution of the Solar System planets (see Section 1.2). While we

cannot observe the Sun millions of years ago, there are many available observations

of young G stars which may serve as analogues for the Sun’s past. A combination

of observations of young G stars and modeling of enhanced-activity solar magnetism

can contribute greatly to our understanding of past solar behavior and inﬂuences on

the early Solar System. In this section we demonstrate our application of the SFT

to a speciﬁc young solar-like star of interest. We discuss in Section 6.3 future work

in applying this modeling framework to a variety of exoplanet systems of interest as

more observations become available.

ι-Horologium is a young G star (∼625 Myr compared to the Sun’s age of ∼4.5

Gyr) observed to rotate rapidly and vary on an extremely short activity cycle of 1.6 years (or a full magnetic polarity cycle of Pcyc = 3.2 years). According to Equation 3.3, this rapid cycle variation could imply strong meridional ﬂows, though the depen-

dence of vme can only be modeled and not observed on other stars, and is therefore subject to debate [Strugarek et al., 2017]. We have conducted preliminary work in

modeling the surface magnetic ﬂux distributions of ι-Hor with the SFT treatment

described in previous chapters. Table 5.1 shows the input parameters selected for

two SFT test runs of an ι-Hor case, with the source strength A0 determined from two different observational estimates of the large-scale field: a 4G dipole field strength

[Alvarado-G´omezet al., 2017], and a later measurement that the large-scale dipole

ﬁeld may be as strong as 50G (private communication with Dr. Soﬁa Moschou, a

co-author of [Alvarado-Gómezet al., 2017]). The large uncertainty in magnetic field 94 strength is a consequence of the still-developing ZDI measurement technique (Sec- tion 1.3) and the difficulty of resolving higher-order multipoles of the magnetic field, where much of the surface flux is found [Schrijver et al., 1998]. The equatorial rotation speed is determined from the observed stellar rotation period Prot = 7.70 d and the stellar radius R∗ = 1.16RSun; the meridional flow multiplier m is determined from

Equation 3.3 and the Pcyc = 3.2yr value observed by [Alvarado-G´omezet al., 2017].

The resulting mean Alfv´ensurface radius, RA, at stellar minimum and maximum, and the open-to-total ﬂux ratios described in Section 4.2.3 are also included as Table

5.2.

B ﬁeld A0=φ∗/φSunmax veq (km/s) Pcyc (yrs) m = vme∗ /vmeSun 4G 0.297 7.6 3.2 8.5 50G 3.72 7.6 3.2 8.5

Table 5.1 : SFT model input parameters for two test cases of an ι-Hor type star, for two possible observed magnetic field strength (B) values. The source strength A0, equatorial rotation speed veq, cycle period Pcyc, and meridional flow multiplier m are defined as before in Section 3.2.2.

ι-Hor is known to host at least one planet, of roughly 2.5 times the mass of Jupiter at an orbital radius of around 1 AU [Alvarado-Gómezet al., 2017]. The Alfvénradii calculated for the two ι-Hor test cases are well within the orbit of the associated planet at all phases of the stellar cycle. Treating ι-Hor as an analogue of the young Sun, we see that the Alfvénradius does not exceed ∼1.27 RA,solar max or about 0.127 AU; even Mercury’s orbit at 0.387 AU is well outside this distance, implying that direct magnetic coupling with the solar corona has never been an influence on the Solar

System planets in the Sun’s history. However, many terrestrial exoplanets around M stars likely do interact directly with their host star’s corona within the Alfv´ensurface

(see Chapter 4 and [Farrish et al., 2019]). We discuss possible future studies of the 95

implications of this direct interaction in Section 6.2.

B ﬁeld RA (min) RA(max) Open/Total Flux (min) Open/Total Flux (max) 4G 1.27 1.11 0.0359 0.0185 50G 1.09 0.880 0.0991 0.0280

Table 5.2 : Alfvénradius estimates for stellar minimum and maximum, calculated with Equation 4.1, for each ι-Hor test case in units of RA,solar max ≈ 20RSun ≈ 0.1 AU. The ratios of open flux to total surface magnetic flux at stellar minimum and maximum are also included.

We display in Figure 5.2 a set of modeled magnetograms of a present-day solar case, and the two test representations of ι-Hor, all at stellar maximum. The short cycle period of ι-Hor leads to a strongly enhanced meridional ﬂow (see Equation

3.3), so the active regions occur at comparatively higher latitudes in the ι-Hor cases than in the Sun. Again, the simplistic mean-field dynamo modeling which led to the development of Equation 3.3 possibly does not hold true when more detailed dynamo flows are incorporated. As ZDI techniques with higher spatial resolution of stellar magnetic fields develop, observations of the latitudinal distribution of magnetic concentrations on ι-Hor could provide a powerful test of the variety of hypotheses about the relationship between vme and Pcyc. 96

Figure 5.2 : At left, a snapshot of solar maximum for a solar control case produced with the SFT. In the center, a magnetogram at stellar maximum of the 4G field strength representation of ι-Hor. At right, the stellar maximum phase of the 50G field strength representation of ι-Hor. For both ι-Hor test cases, the short activity cycle period leads to strong poleward flows (Eqn. 3.3) and therefore many active regions appear at higher latitudes than in the present-day solar case at left. 97

Chapter 6

Future Work

6.1 Solar Variability and Anomalous Active Regions

The Sun’s variability on cycle timescales and over multiple solar cycles can impact the climate of Earth and other Solar System planets. Periods of solar maximum contribute to increased substorms in Earth’s magnetosphere, and the ionosphere of

Mars reacts to changing solar wind conditions and EUV emission throughout the cycle [Hall et al., 2019].

The solar cycle is also known to exhibit variability in the amplitudes of successive maxima across many activity indicators such as sunspot number and coronal emission

[Hathaway, 2010]. One prominent feature of historical solar cycle data is a pronounced period of extremely low-amplitude solar cycles in the years 1645-1715 known as the

Maunder Minimum (see Figure 6.1). Understanding the causes of periods of very low activity over multiple solar cycles, known as grand minima, is an open problem in solar dynamo research. Though the precise relationship between variation in the

Sun’s activity on the timescale of decades to centuries and the impacts on planetary climates is not completely understood, constraining the root causes of cycle amplitude variability and, moreover, predicting the amplitude of future activity cycles, is of interest to the solar physics and geophysics communities alike.

Solar cycle variability can serve as a diagnostic for changes in the solar dynamo.

The behavior of the dynamo additionally drives the appearance and dissipation of 98

Figure 6.1 : Yearly averages of observed sunspot numbers are plotted vs. time; the prolonged period of low cycle amplitude known as the Maunder Minimum is readily apparent [Hathaway, 2010].

magnetic flux on the solar photosphere. A series of recent studies has investigated the role that magnetic flux features on the solar photosphere may play in feedback to the dynamo and, in particular, how ‘rogue’ bipolar concentrations may disrupt the dynamo in such a way that a grand minimum forms. [Nagy et al., 2017] examined a solar dynamo model in which a large anomalous active region (AAR) effectively

‘killed’ the solar dynamo and triggered a grand minimum. The simulated AAR which produced this dramatic effect had very high flux and an unfavorable tilt angle which served to cancel enough surrounding flux that the seed field for the next solar cycle was diminished (refer back to Section 1.2.1 for a discussion of the regeneration of the solar dynamo). Figure 6.2, reproduced from [Nagy et al., 2017], displays the insertion of the large, rogue active region in the declining phase of a solar cycle and the resultant grand minimum.

The ability to replicate this result with our SFT treatment would provide an important test of this concept that large, anomalous active regions can trigger long-term solar variability and provide feedback on dynamo mechanisms. As discussed in Section 99

Figure 6.2 : The insertion of a large anomalous active region in a solar dynamo simulation ‘killed’ the dynamo and triggered a grand minimum akin to the Maunder Minimum observed in archival solar data. This result could explain the causes of solar cycle amplitude variations [Nagy et al., 2017].

2.1, the SFT that we employ incorporates a more detailed treatment of small-scale

flux than the dynamo models of [Nagy et al., 2017, Lemerle and Charbonneau, 2017], and the ephemeral region flux can contribute significantly to the overall flux and dynamics of the photospheric field distribution. A similar experiment with the SFT of this anomalous active region mechanism for triggering grand minima will be an important test of the role of ephemeral region flux in the time evolution of the solar magnetic field and solar dynamo feedback. Preliminary work in adapting the SFT to allow for the insertion of individual active regions of user-defined flux strength and tilt angle is currently underway.

6.2 Sub-Alfv´enicRegime Characterization

We have discussed in Chapter 4 an investigation into the magnetic and plasma environments of very close-in exoplanets orbiting M dwarfs. A typical M dwarf hab- 100 itable zone extends only hundredths of an AU outward from the star, meaning that habitable-zone exoplanets around M stars can have orbits on the order of only 10 days

(e.g., [Anglada-Escudéet al., 2016, Bonfils et al., 2018, Grimm et al., 2018]). Previ- ous work by other modeling groups [Garraffo et al., 2016, Garraffo et al., 2017], as well as the work discussed in Chapter 4, have suggested that the M dwarf circumstellar habitable zone is coincident with or entirely within the stellar Alfvénsurface, the causal plasma boundary inside of which the stellar wind is sub-Alfvénic.The Alfvén surface can be thought of as the boundary beyond which the stellar plasma leaves the star’s sphere of influence and behaves as ordinary stellar wind. Inside the Alfvén surface, stellar plasma flows comprise the corona and inner asterosphere. Habitable- zone planets orbiting around M dwarfs are likely to reside within the Alfvénsurfaces of their host stars and interact directly with their stellar coronae (see Chapter 4, published as [Farrish et al., 2019]).

The success of the SFT model in producing faithful representations of solar and stellar magnetic field distributions for user-defined activity levels makes it an excellent tool for examination of the Sun and other stars as drivers of magnetic phenomena in their asterospheric environments, including the variation of Alfvénsurface average radius, RA, as a function of Rossby number. The Alfvénsurface is not static but rather fluctuates with changes in the stellar photospheric magnetic field as new magnetic flux emerges and evolves on the star’s surface. Changes in the physical extent of the Alfvén surface occur on the timescales both of individual active region lifetimes as well as over the years- or decades-long magnetic activity cycles of stars. Recent and ongoing observation campaigns [Wargelin et al., 2017, Couperus et al., 2020] have revealed that

M dwarf stars have stellar activity cycles akin to the solar cycle, and therefore the

Alfvénsurfaces of M stars may fluctuate in extent by up to ∼50% in distance from the 101 stellar surface over their activity cycles (see Figure 4.4). Habitable-zone terrestrial planets around M stars may traverse their star’s Alfvénboundary periodically on timescales of their orbits or over the course of their stars’ activity cycles. Neither the sub-Alfvénicplasma environment itself, nor the planetary effects of crossing in and out of the Alfvénsurface over the course of days or years, are well-constrained in the current literature. Periodic crossings of a planet through the Alfvénboundary could lead to increased potential for violent disruptions at the exoplanet’s magnetosphere, resulting in strong atmospheric response.

Very little is known about the plasma environment inside a star’s Alfvénsurface or the effects of crossing the Alfvénboundary because there have never been in-situ mea- surements of such an environment. That will change with Parker Solar Probe’s close approaches to the Sun in the next four years. The solar Alfvénsurface has been estimated to be within about 20 RSun from the solar surface. Parker Solar Probe (PSP) will approach within 20 RSun at its 8th perihelion in April of 2021, and its subsequent perihelia throughout 2021-2025 will be well within this distance, reaching as close as about 10 RSun by Fall 2023 [Chhiber et al., 2019]. The unprecedented look at the very inner heliosphere and solar corona provided by upcoming PSP encounters will revolutionize the capabilities of space weather modelers to constrain the space plasma environments of terrestrial exoplanets in analogous sub-Alfvénicregimes around M stars, providing great potential for detailed characterization of these exoplanets.

Future work, most likely in collaboration with a team of researchers led by Dr.

Shannon Curry of the UC Berkeley Space Sciences Laboratory, will be able to interpret

PSP data in the exotic sub-Alfv´enicregime, providing novel understanding of this previously unexplored solar plasma environment. In addition, the team can extend this understanding to the exoplanet context via scaling of solar activity to more 102 active exoplanet host stars. This research plan relies on data from PSP gathered in

Spring 2021 and beyond through 2023-2024 as PSP makes closer approaches down to .10 RSun, well within the solar Alfv´ensurface. In-situ exploration of the very inner heliosphere with PSP provides many avenues of future work in characterizing the near-star environment of terrestrial planets around M dwarfs.

6.3 Potentially Habitable Exoplanet Systems of Interest

The James Webb Space Telescope’s anticipated launch date in late 2021 is expected to usher in a new era of JWST observations of exoplanet atmospheres [Bean et al., 2018,

Kilpatrick et al., 2018]. The current dearth of observations leaves heliophysicists and planetary scientists with little knowledge of the true breadth of exoplanet processes such as atmospheric loss. With JWST observations of the presence and composition of exoplanet atmospheres, there will be an important new observational constraint on exoplanet systems.

In a proposal to the NASA Postdoctoral Program, in collaboration with Dr.

Katherine Garcia-Sage of Goddard Space Flight Center (GSFC), we have discussed a possible future project in coupling the SFT to the WSA model for the solar wind discussed in Section 2.4. With a full suite of coupled models from the stellar surface to the planetary magnetosphere, mediated by the stellar wind, we will be well-placed to model speciﬁc exoplanet systems of interest. Using as input simple observed stellar parameters such as age, rotation period, stellar type, and/or X-ray and extreme ultraviolet (EUV) luminosities where available, we can produce detailed photospheric

ﬂux distributions modeling the host star’s magnetic activity via SFT. With observed planetary atmospheric conditions available from projected JWST campaigns, there may be several exoplanet systems of interest where our coupled star-planet modeling 103 framework can be compared to both star and planet observed characteristics. This will open a new ﬁeld of study in bringing comparative heliophysics modeling more closely into alignment with observations of exoplanet systems. 104

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