Mean-Motion Resonances in Exoplanetary Systems

Home , 47 Ursae Majoris c, 55 Cancri, 61 Virginis, HD 133131, HD 20781, HD 215152

ARISTOTLE UNIVERSITY OF THESSALONIKI

DEPARTMENT OF PHYSICS

BACHELOR OF SCIENCE IN PHYSICS

Konstantinos Foutzopoulos

Thesis Supervisor George Voyatzis, Associate Professor

October, 2019 ii iii

Abstract

In this thesis we perform a statistical and dynamical study of exoplanetary systems, focusing on the mean-motion resonances among the planets. We start by providing a background on mechanics, detection methods as well as numerical methods and symplectic integration. Then we proceed to analyze the characteristics of all detected exoplanets as well as the distribution of the commensurable period ratios in them. In this part we also make a catalog of systems with planets having a small integer period ratio. Finally, we perform a dynamic evolution for a handful of them for whether they are in mean-motion resonance.

Περίληψη

Σε αυτήν την εργασία πραγματοποιούμε μια στατιστική και δυναμική ανάλυση εξωπλανητικών συστημάτων, επικεντρόμενοι στους συντονισμούς μέσης-κίνησης μεταξύ των πλανητών. Ξεκινάμε παρέχοντας ένα υπόβαθρο στη μηχανική, μεθόδους ανίχνευσης όπως και αριθμητικές μεθόδους και συμπλεκτική ολοκλήρωση. Ύστερα προχωράμε σε ανάλυση των χαρακτηριστικών όλων των ανιχνευμένων εξωπλανητών όπως και τη κατανομή των σύμμετρων λόγο περιόδων σε αυτούς. Σε αυτό το μέρος φτιάχνουμε και ένα κατάλογο των συστημάτων με πλανήτες που έχουν μικρό ακέραιο λόγο περιόδων. Τέλος, πραγματοποιούμε μια δυναμική εξέλιξη για κάποια από αυτά για το αν είναι σε συντονισμό μέσης-κίνησης. iv

Εκτενής περίληψη Η πρώτη ανίχνευση εξωπλανήτη έγινε πριν 30 χρόνια. Από τότε ένας μεγάλος αριθμός τους έχει βρεθεί, με τις αποστολές Kepler και Spitzer να έχουν παίξει σημαντικό λόγο σ´ αυτό. Ο ορισμός του τι αποτελεί εξωπλανήτης στο Ηλιακό σύστημα έχει δωθεί από την IAU. Σύμφωνα με αυτόν, πλανήτης είναι ένα σώμα (α) σε τροχιά γύρω από τον ήλιο, (β) αρκετή μάζα ώστε η ιδιοβαρύτητα του να υπερνικήσει τις δυνάμεις άκαμπτου σώματος ώστε να αποκτήσει ένα υδροστατικά ευσταθές (σχεδών σφαιρικό) σχήμα, και (γ) έχει καθαρίσει τη γειτονιά γύρω της τροχιάς του. Μια θέση για το για τι αποτελεί εξωηλιακός πλανήτης δίνεται από την IAU. Πλανήτες είναι αντικείμενα με ελάχιστη μάζα όπως αυτή ορίζεται για εντός του Ηλιακού συστήματος, και μέγιστη μάζα αυτή για τη θερμοπυρηνική σύντηξη δευτέριου (περίπου 13 μάζες Δία για ηλιακή μεταλικότητα). Η κίνηση των πλανητών κυβερνάται από τους νόμους της μηχανικής, ενώ η τροχιά αυτών από τους νόμους του Κέπλερ οι οποίοι εξάγονται από τους προηγούμενους. Διάφοροι μέθοδοι ανίχνευσης εξωπλανητικών συστημάτων έχουν αναπτυχθεί, με τις πιο επιτη- χυμένες από αυτές να είναι αρχικά η μέθοδος ακτινικών ταχυτήτων (φασματοσκοπία) και τελευταία χρόνια η μέθοδος των διαβάσεων βάση δεδομένων από τις προαναφερθέντες αποστολές. Σημαντική είναι η αστρομετρία η οποία μαζί με τις προηγούμενες επιτρέπει την εξαγωγή των στοιχείων της τροχιάς. Οι νόμοι της δυναμικής μπορεί να γραφούν πέρα από τη διανυσματική μορφή (Νευτώνεια μη- χανική), και ως συναρτήσεις μέσω του φορμαλισμού Χάμιλτον (Χαμιλτονιανή μηχανική). Ο φορ- μαλισμός Χάμιλτον αποτυπώνει μια γεωμετρία (συμπλεκτική) στα μηχανική συστήματα. Αυτό έχει ως αποτέλεσμα των φυσικών ιδιοτήτων (α) μη-εκφυλισμού, (β) αντιστρεπτότητα και (γ) διατήρηση της δομής του φασικού χώρου. Αριθμητικές μεθόδοι μπορεί να αναπτυχθούν πάνω σε αυτές τις ιδιότητες μέσω του φορμαλισμού. Τροχιακοί συντονισμοί συμβαίνουν όταν σώματα σε τροχιά ασκούν μια κανονική, περιοδική βαρυτική επιρροή το ένα στο άλλο. Αυτό συμβαίνει όταν οι τροχιακοί περίοδοι σχετίζονται με ένα λόγο μικρών ακεραίων. Τότε οι βαρυτικές δυνάμεις που ασκούνται προστίθενται με συνεκτικό τρόπο. Δηλαδή όταν οι λόγοι περιόδου είναι σύμμετροι (en: commensurable).

n1/n2 = k1/k2 Η δυναμική ενός ζεύγους σε συντονισμό χωρίζεται στην αιώνια ή αέναη (en: secular) κίνηση και στη δυναμική του συντονισμού. Η θεωρία Laplace-Lagrange μπορεί να δώσει αναλυτικά τις τροχιακές παραμέτρους με χρήση μερικών όρων στη παρελκτική συνάρτηση. Παρόλο που είναι κομψή θεωρία δεν είναι απόλυτα ακριβής. Απλουστευμένα μοντέλα της αέναης κίνησης μπορούν να κατασευαστούν με τη μέσων όρων μέθοδο. H αέναη εξέλιξη ορίζεται από την αέναη (αψιδική) γωνία

∆¯ω =ω ¯2 − ω¯1 Αν λυκνίζει (en: librates), δηλαδή εκτελεί μικρού πλάτους ταλαντώσεις, τότε το σύστημα είναι σε συμπεριφορά που αποκαλείται αέναος συντονισμός. Αν ∆¯ω = 0 το σύστημα βρίσκεται βρίσκεται σε αψιδική ευθυγράμμιση, ενώ αν ∆¯ω = π το σύστημα βρίσκεται σε αψιδική αντιευθυγράμμιση. H δυναμική του τροχιακού συντονισμού περιγράφεται από την εξέλιξη της κρίσιμης γωνίας που είναι γραμμικός συνδυασμός των γωνιακών μεταβλητών. Αν η κρίσιμη γωνία κυκλοφορεί ή περιστρέφεται (en: circulates) τότε το ζεύγος είναι κοντά αλλά όχι σε συντονισμό. Δηλαδή αν καμιά κρίσιμη γωνία δεν λικνίζει τότε το σύστημα βρίσκεται σε διατάξη μη-συντονισμού. Οι γωνίες αυτές ορίζονται

ϕi = (p + q)λ2 − pλ1 − qω¯i Κάθε συντονισμός μέσης κίνησης έχει κάποιο καλά καθορισμένο πλάτος στο οποίο τα δύο σώματα κάνουν ένα λυκνισμό γύρω από ένα σημείο ισορροπίας. Ο συντονισμός μέσης κίνησης επεκτείται πέρα από δύο σώματα. Ένας συντονισμός τριών-σωμάτων συμβαίνει όταν μια αλυσίδα συντονισμών υπάρχει για τρία σώματα. Η αντίστοιχει κρίσιμη γωνία (μεταβλητή Laplace) δίνεται

ϕL = qλ1 − (p + q)λ2 + pλ3 v

Ταυτόγχρονη λύκνιση των δύο ζευγών σημαίνει πως η γωνία Laplace λυκνίζει επίσης. Ωστόσο συντονισμός τριών-σωμάτων μπορεί να υπάρξει ακόμα και αν τα ζεύγη δεν είναι σε συντονισμό μεταξύ τους. Καθώς το πρόβλημα των n-σωμάτων δεν λύνεται αναλυτικά, κατεφεύγουμε στη χρήση αριθ- μητικών μεθόδων. Συγκεκριμένα οι αναγκαίες είναι μια μέθοδος εύρεσης ρίζας για την εξίσωση Kepler (όπως η Newton-Raphson), και μια μέθοδος λύσης συστήματος διαφορικών εξισώσεων για ολοκλήρωση των τροχιών. Οι συμπλεκτικοί ολοκληρωτές, μια υποκατηγορία των γεωμετρικών ολοκληρωτών, για τα χαμιλτονιανά συστήματα εκμεταλλεύονται τις ιδιότητες της συμπλεκτικής δο- μής αυτών ώστε να οριοθετήσουν τη μεταβολή της ολικής ενέργειας. Οι μέθοδοι χωρίζονται στις έμμεσες όπου βρίσκεται μια νέα Χαμιλτονιανή, και στις άμεσες όπου η Χαμιλτονιανή χωρίζεται σε δυο ακριβώς ολοκληρώσιμα μέρη. Ο γενικός χωρισμός για μηχανική συστήματα είναι T + V , δηλαδή την κινητική και τη δυναμική ενέργεια. Αν είναι γνωστός ένας ολοκληρωτής 2n τάξης ένας συμπλεκτικός ολοκληρωτής (2n+2) τάξης, δημιουργείται ως γινόμενο των προηγούμενων. Στα πλανητικά, όπου η δυναμική κυριαρχείται από ένα σώμα (τον αστέρα), μπορεί να χωριστεί και ως ′ H0 +H , δηλαδή του αδιατάραχτου συστήματος (Κεπλεριανή κίνηση) και των εν-μεταξύ πλανητικών αλληλεπιδράσεων. Η Κεπλεριανή κίνηση μπορεί να εξελιχθεί αναλυτικά μέσω των συναρτήσεων f και g. Η συνθήκη διατήρησης της στροφορμής μπορεί να χρησιμοποιηθεί στις f και g, ώστε να οριεθετηθεί και η μεταβολή της ολικής στροφορμής. Από τον αριθμό των ανακαλυμένων πλανητών ανά έτος, βλέπουμε πως υπάρχουν αιχμές μετά το 2012 λόγο ανακαλύψεων μέσω διαβάσεων που αντιστοιχούν στις τελευταίες διαστημικές αποστολές. Τα περισσότερα συστήματα έχουν ένα μόνο πλανητή, ενώ ο μέγιστος αριθμός σε ένα σύστημα ανέρχεται στους 8. Βρίσκοντας όλους του σύμμετρους λόγους περιόδων, βρίσκουμαι πως ένας σημαντικός αριθμός ζευγαριών έχει λόγους 1:2 και 1:3. και οι πιο εμφανιζόμενες τάξεις είναι 1 και 3. Χωρίζοντας τις κατατάξεις σε εσωτερικούς και εξωτερικούς συντονισμούς βρίσκουμαι πως οι περισσότεροι συντονισμοί είναι εσωτερικοί, δηλαδή η κυρίαρχη μάζα βρίσκεται εξωτερικά του ζεύγους. Επαληθεύουμε τους ήδη γνωστούς συντονισμούς των συστημάτων HD 82943, HR 8799 και TRAPPIST-1. Χρησιμοποιήθηκαν αστροκεντρικά τροχιακά στοιχεία από δυναμικες προσαρμογές των δεδομένων. Το πρώτο σύστημα, του HD 82943, είναι δυο πλανητών σε συντονισμό μέσης κίνησης. Επιπλέον εμφανίζει αέναο συντονισμό και άρα βρίσκεται σε αψιδική ευθυγράμμιση. Στον ΗR 8799 βλέπουμε ένα συντονισμό τεσσάρων-σωμάτων. Τόσο οι γωνίες Laplace των δύο τριάδων ακόλουθων πλανητών, όσο και η γωνία συντονισμού τεσσάρων σωμάτων εμφανίζουν λύκνιση. Ο TRAPPIST-1 αναλύθηκε στο τέλος και εμφανίζει μια εκτενή σειρά συντονισμών. Επιπλέον έγινε ολοκλήρωση των συστημάτων Kepler-11 και GJ 9827. Αν και εμφανιζούν πολύ κοντινούς σύμ- μετρους περιόδους στους πλανήτες τους, με δοκιμές διάφορων απλών λογικών διατάξεων, άλλα όχι εκτενή αναζήτηση όλων των κοντινών αρχικών συνθηκών, δεν βρέθηκε κάποιος συντονισμός. Αυτό μπορεί να οφείλεται σε ελλειπή ή αδύναμα δεδομενα. Σημειώνεται όμως πως οι μάζες των πλανητών σε αυτά τα συστήματα, άρα και οι αλληλεπιδρώντες δυνάμεις, είναι χαμήλες και άρα τα συστήματα αυτά μπορεί να παραμείνουν ευσταθή σε σχεδών-συντονισμού κατάσταση χωρίς να είναι σε συντονισμό μέσης κίνησης. vi Contents

1 Background 1 1.1 History of discoveries ...... 1 1.2 Planet deﬁnition ...... 1 1.2.1 Solar system ...... 1 1.2.2 Extrasolar systems ...... 2 1.3 Planetary classiﬁcation ...... 4 1.3.1 Giants of ice and gas ...... 4 1.3.2 Terrestrials ...... 4 1.4 Classical mechanics ...... 5 1.4.1 Newtonian dynamics ...... 5 1.4.2 Hamiltonian mechanics ...... 6 1.5 Celestial mechanics ...... 8 1.5.1 Orbits ...... 8 1.5.2 N-body problem ...... 11 1.6 Detection ...... 13 1.7 Resonances ...... 17 1.7.1 Mean-motion ...... 18 1.7.2 Two-planet dynamics ...... 18 1.7.3 Classes ...... 23 1.8 Numerical methods ...... 24 1.8.1 Generic ...... 25 1.8.2 Symplectic integration ...... 26

2 Data analysis 31 2.1 Planets ...... 31 2.2 Stars ...... 34 2.3 Resonances ...... 35

3 Numerical integrations 43 3.1 HD 82943 ...... 43 3.2 GJ 9827 ...... 45 3.3 61 Vir ...... 46 3.4 HR 8799 ...... 47 3.5 TRAPPIST-1 ...... 50

A Transformations 57 A.1 Orbital elements ...... 57 A.2 Coordinate systems ...... 58 A.3 Gauss’ method ...... 60

vii viii

B Sources 63 B.1 Chapter 2 ...... 63 B.2 Chapter 3 ...... 69

C Catalog 77

Bibliography 93 List of Figures

1.1 Planets threshold using IAU requirement (Margot, 2015)...... 2 1.2 Planets thresholds using Margot criterion (Margot, 2015)...... 3 1.3 Orbital elements shown schematically...... 9 1.4 Schematics of Kepler’s ﬁrst and second law...... 10 1.5 Characteristic pictures for detection via astrometry and imaging...... 14 1.6 Characteristic pictures for detection via microlensing and pulsar timing...... 16 1.7 Characteristic pictures for detection via transit and radial velocities...... 17 1.8 Nominal resonances locations and approximate strengths...... 20 1.9 Paths in rotating frame for interior resonances...... 21 1.10 Paths in rotating frame for exterior resonances...... 22 1.11 Schematic geometry of inner Galilean moons...... 23 1.12 Schematic geometry of GJ 876 outer planets...... 24 1.13 Relative variations for T + V and DH integrator ...... 29 1.14 Absolute variations for T + V and DH integrator ...... 30

2.1 Histograms related to discoveries...... 32 2.2 Histograms on detection and distribution...... 33 2.3 Mass to period plot and period histogram...... 34 2.4 Mass to radii plot and radii histogram...... 35 2.5 Scatter plot and histogram for distances...... 36 2.6 Plot of star mass to semimajor axis and planetary masses...... 37 2.7 Histograms for masses and host stars masses...... 37 2.8 Eccentricities to inclinations plot and histogram for inclinations...... 38 2.9 Eccentricities to periods plot and histogram for eccentricities...... 38 2.11 Histograms for host star characteristics...... 39 2.12 Histograms for commensurable period ratios...... 40 2.13 Histograms for commensurable period ratios (dominant mass in)...... 40 2.14 Histograms for commensurable period ratios (dominant mass out)...... 41

3.1 Evolution of a, e and period ratio for HD-82943 planets b and c...... 44 3.2 Evolution of ∆¯ω, ϕ and ϕ′ for HD-82943 planets b and c...... 44 3.3 Evolution of a, e and period ratio for 61-Vir planets c and d...... 46 3.4 Evolution of ∆¯ω, ϕ and ϕ′ for 61-Vir planets c and d...... 46 3.5 Evolution of a, e and period ratio for HR-8799 planets e and d...... 47 3.6 Evolution of ∆¯ω, ϕ and ϕ′ for HR-8799 planets e and d...... 47 3.7 Evolution of a, e and period ratio for HR-8799 planets d and c...... 48 3.8 Evolution of ∆¯ω, ϕ and ϕ′ for HR-8799 planets d and c...... 48 3.9 Evolution of a, e and period ratio for HR-8799 planets c and b...... 48 3.10 Evolution of ∆¯ω, ϕ and ϕ′ for HR-8799 planets c and b...... 49 3.11 Evolution of Laplace and four-planet resonant arguments for HR-8799...... 49 3.12 Evolution of a, e and period ratio for TRAPPIST-1 planets b and c...... 50 3.13 Evolution of ∆¯ω, ϕ and ϕ′ for TRAPPIST-1 planets b and c...... 51

ix x

3.14 Evolution of a, e and period ratio for TRAPPIST-1 planets c and d...... 51 3.15 Evolution of ∆¯ω, ϕ and ϕ′ for TRAPPIST-1 planets c and d...... 51 3.16 Evolution of a, e and period ratio for TRAPPIST-1 planets d and e...... 52 3.17 Evolution of ∆¯ω, ϕ and ϕ′ for TRAPPIST-1 planets d and e...... 52 3.18 Evolution of a, e and period ratio for TRAPPIST-1 planets e and f...... 52 3.19 Evolution of ∆¯ω, ϕ and ϕ′ for TRAPPIST-1 planets e and f...... 53 3.20 Evolution of a, e and period ratio for TRAPPIST-1 planets f and g...... 53 3.21 Evolution of ∆¯ω, ϕ and ϕ′ for TRAPPIST-1 planets f and g...... 53 3.22 Evolution of a, e and period ratio for TRAPPIST-1 planets g and h...... 54 3.23 Evolution of ∆¯ω, ϕ and ϕ′ for TRAPPIST-1 planets g and h...... 54 3.24 Evolution of ϕL for TRAPPIST-1 inner planets...... 55 3.25 Evolution of ϕL for TRAPPIST-1 outer planets...... 55

A.1 Coordinate systems: heliocentric, barycentric and Jacobi...... 59 A.2 Coordinate systems: democratic heliocentric ...... 60 Chapter 1

Background

1.1 History of discoveries

The first extrasolar planet was detected in 1988 through the radial velocity variations of Gamma Cephei, by Campbell, Walker and Yang. The planet confirmation though came in 2002. The first confirmed detection came in 1992 by Aleksander Wolszczan, with the discovery of several terrestrial-mass planets orbiting the pulsar PSR B1257+12. The first confirmed detected exoplanet orbiting a main-sequence star came in 1995, when a giant planet, now known as Dimidium, was found in a four-day orbit around 51 Pegasi. In 1996 a Jupiter-like planet around 47 Ursae was the first long-period planet to be discovered. The first planet discovered around the giant star Iota Draconis, an orange giant in 2002. See Perryman[8] for an extended list of important discoveries accompanied with the respective papers. Moving forward, worth mentioning are Kepler and Spitzer missions which through the usage of the transit method uncovered numerous exoplanets. Of those, we note that in 2017, using Spitzer Space Telescope, four additional planets where found orbiting TRAPPIST-1, which had three already known planets.

1.2 Planet deﬁnition

1.2.1 Solar system A formal deﬁnition of a planet in our Solar System is given by IAU1. According to it, planets and other bodies, except satellites, in our Solar System can be deﬁned into three distinct categories.

1. A “planet” is a celestial body that: (a) is in orbit around the Sun, (b) has suﬃcient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and (c) has cleared the neighborhood around its orbit.

2. A “dwarf planet” is a celestial body that: (a) is in orbit around the Sun, (b) has suﬃcient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, (c) has not cleared the neighborhood around its orbit, and (d) is not a satellite.

3. All other objects, except satellites, orbiting the Sun shall be referred to collectively as “Small Solar System Bodies”.

Though another proposed statement is used for extrasolar systems it is of interest to closer analyze the first category shown here, that is a planet. Overall an ambiguity concerning formation and composition is being made. The first two points can easily be fulfilled by other smaller

1Deﬁnition of a Planet in the Solar System. International Astronomical Union, 2006. URL: https://www.iau. org/static/resolutions/Resolution_GA26-5-6.pdf (visited on 2018).

1 2 bodies (dwarf planets and even lesser asteroids). The important point is the third one which allows us to quantitatively assert whether an object is a planet or not. “Cleared the neighborhood” means it has become gravitational, or else dynamical, dominant in its orbit. That means there are no other bodies of comparable size other than its own satellites or those otherwise under its gravitational inﬂuence. The expression giving the minimum orbit-clear mass is given by Margot2

( ) ( )− ( ) 3/4 1/2 3/4 Mp ≥ M∗ t∗ ap M⊕ M⊙ 109y 1au where Mp the mass, ap the orbital radius, M∗ the star mass, t∗ the age of the planetary system, M⊕ and M⊙ the Earth’s and Sun’s mass respectively. Knowing the mass and semimajor axis of our six planets alongside (once planet; now dwarf) Pluto and Demetra (or Ceres), the plot shown on ﬁgure 1.1 was obtained.

Jupiter

102 Saturn

Neptune 101 Uranus

100 Earth Venus

10-1 Mars Mercury

-2

Mass (Earth masses) 10

Eris 10-3 Pluto

Ceres 10-4 100 101 102 Semi-major axis (au)

Figure 1.1: Planets threshold using IAU requirement (Margot, 2015).

As seen all planets lie above the “planetary thresholds” that were speciﬁed with Pluto and Demetra are below them.

1.2.2 Extrasolar systems On broad terms it can be said that planet is a “large” celestial body moving in an elliptical orbit around a star. A position statement on what constitutes an extrasolar planet is given by IAU3. According to it celestial bodies fall under the following categories.

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

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

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

2Jean-Luc Margot. “A quantitative criterion for deﬁning planets”. In: The Astronomical Journal 150.6 (2015), p. 185. 3Alan P Boss et al. “Working Group on Extrasolar Planets”. In: Proceedings of the International Astronomical Union 1.T26A (2005), pp. 183–186. 3

This definition creates ambiguity by making location, rather than formation or composition, the determining characteristics for planethood. By using all previous, we say that a planet is a celestial object that (a) orbit stars or stellar remnants, (b) has true mass below the limiting mass for thermonuclear fusion of deuterium, (c) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and (d) has cleared the neighborhood around its orbit. The last requirement is used only inside the Solar System and hasn’t been accepted yet for exoplanets. Worth mentioning is a, based on this, proposed criterion by Margot. Margot proposed a simple metric that could allow for quantification of the third requirement. The expression giving the minimum orbit-clear mass is

( ) ( )− ( ) 5/8 3/4 9/8 Mp 3/2 M∗ t∗ ap ≥ Mclear = C M⊕ M⊙ 105y 1au where Mp the mass, ap the orbital radius, M∗ the star mass, t∗ the age of the planetary system, M⊕ and M⊙ the√ Earth’s and Sun’s mass respectively. C is a numerical constant and its value must exceed 2 3 to ensure that the planet clears its feeding zone. Certain stability criteria and the observed dynamical spacing between exoplanets would impose C = 5. Then representing the mass of a planetary body in terms of the corresponding orbit-clear mass as Π, the following simple criterion can assert whether a body is a planet

Π = Mbody/Mclear ≥ 1

Knowing the mass and semimajor axis of our six planets alongside Pluto and Demetra, plot√ shown on ﬁgure 1.2 was obtained. The size of a planet’s feeding zone was adopted to C = 2 3, as the minimum extent of the orbital zone to be cleared.

105 Jupiter

104 Saturn

3 10 Earth Venus Neptune Uranus 2 Mercury 10 Mars

101

100

10-1 Ceres

Planet mass in units of orbit-clearing Eris 10-2 Pluto 100 101 102 Semi-major axis (au)

Figure 1.2: Planets thresholds using Margot criterion (Margot, 2015).

As pointed in the position statement, not all objects orbit a star. During planet system formation, planet-like objects could conceivably be thrown out by some instability, drifting since through the interstellar space. Those, otherwise similar to a planet, are commonly called “free-ﬂoating” or “rogue planets”[8]. To the best of our knowledge though, the vast majority of exoplanets exist as companions of stars. For planetary dynamics true planets, those objects part of a system, are more interesting. An extrasolar planet is designated by the name or designation of its host star and a lower case letter. The ﬁrst planet discovered in a system is getting the letter b and later planets are given subsequent letters. When more than one planets are discovered simultaneously then the planets take their letters in order of orbital size, from the innermost to the outermost. 4

1.3 Planetary classiﬁcation

A number of classiﬁcation systems for exoplanets have been proposed. Right now there is no all-inclusive planetary classiﬁcation scheme. Short descriptions are provided for some frequently used categories based on (Spiegel et al, 2014)4, as well as some characteristics of gas and ice giants and so-called Earth analogs that will be of use later.

1.3.1 Giants of ice and gas Gas giants are essentially giant spheres of hydrogen and helium (H/He), whereas ice giant are composed of heavy elements, rather than H/He gas. Ice giants are classed as such since much of their mass is made up of water and other ﬂuid ices. A better description of them as well as subcategories follows below.

• Brown dwarfs They are objects at the boundary between stars and planets. Most often considered by convention as such if their mass is larger than the lowest value required for deuterium fusion.

• Jupiter-like Jupiter and Saturn as said are composed primary from H/He with smaller contributions from heavier elements. Many of the known exoplanets are similar to them, and probably have similar structure. There is a thin outer region which is their atmosphere, and below there is a deep envelope extending almost the entire radius.

• “Hot Jupiters” Those are Jupiter-like planets in close-in orbits around their stars. The distances make their surface temperature in the order of hundreds of thousands Celsius degrees. Many mechanisms have been suggested to explain the inﬂated radii of the hottest of them.

• Neptune-like Neptune and Uranus are composed of heavy elements, rather H/He gas. Those planets still have a thick H/He envelope but it doesn’t comprise the majority of mass. Instead their planetary mass is mostly in a deep ﬂuid ionic sea. This class of exoplanets harbors tremendous diversity. Whether Neptune-like exoplanets are true ice giants remains unknown. If they form beyond the ice line, where water has condensed then much of mass is water. If they form within they would have rocky interiors with a gas envelope.

• “Hot Neptunes” Those are Neptune-like planets in close-in orbit around their stars. The distances make their surface temperature in the order of hundreds of thousands Celsius degrees. They have substantial H/He envelope and are not composed only of water.

• Mini-Neptune They are gas dwarfs with a mass comparable with that of terrestrial planets but with a low density. If their atmosphere is also of low density, then this could be attributed to a signiﬁcant water layer and they can be classiﬁed as ocean planets.

1.3.2 Terrestrials Terrestrial-type planets are rocky objects surrounded by a thin atmosphere, with masses up to an order of Earths. Those planets are known for having few or no moons and no ring systems.

4David S Spiegel, Jonathan J Fortney, and Christophe Sotin. “Structure of exoplanets”. In: Proceedings of the National Academy of Sciences 111.35 (2014), pp. 12622–12627. 5

Blue-dot clones Earth-analogs, also called Earth-twin or Earth-like, are terrestrial planets that, as the name implies, are similar to our own. Speciﬁcally, this is how are called planets with environmental conditions similar to those found on Earth. Those are of interest as they can potentially host and sustain complex extraterrestrial life, or be a prospective colony target for humanity in the far away (or not) future. Their size is similar to Earth’s as those are thought more capable for retaining an Earth-like atmosphere. They have mass within the range of 0.8–1.9 M⊕, below which are classed as sub-Earth and above classed as super-Earth, and radii within the range of 0.5–2.0 R⊕. The host star they orbit should be a solar analog, that is, a star much like our Sun. An exact solar twin would be a G2V star with a 5778 K temperature, age 4.6 billion years old, with similar metallicity and a 0.1% solar luminosity variation. So it should be similar photometrically or in terms of spectral type. Finally, they should lie within the host star’s habitable zone, that is the region where liquid water can exist on the planet’s surface.

1.4 Classical mechanics

1.4.1 Newtonian dynamics In modern terms, the three laws of motion can be formulated[7] as:

• First Law: Bodies remain in a state of rest or uniform motion in a straight line unless acted upon by a force.

• Second Law: The rate of change of momentum is equal to the force impressed and in the same direction as that force.

• Third Law: To every action there is an equal and opposite reaction.

The first law requires the identification of an inertial system with respect to which it is possible to define the absolute motion of the object. The second law can be expressed mathematically as dp F = dt where F is the resultant of the forces acting on the object and p = mr˙ is its momentum. For a constant mass m, it takes the famous form

F = m¨r where ¨r is the acceleration of the mass measured in an inertial reference frame. Newton set this equation as the basis of mechanics. By the theorem of existence and uniqueness of solutions to ordinary diﬀerential equations the function F and the initial conditions (positions and velocities) uniquely determine a motion. The third law permits to deal with a dynamical problem by using an equilibrium equation. If an object i applies a force Fij to an object j, it can be mathematically written as

Fij = −Fji Equal in importance to the three laws of motion, is the law of universal gravitation stating that two bodies, with masses mi and mj, (1) attract one another along the line joining them and (2) with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. m m F = G i j ij r2 6

Rewriting the previous as vector to account for direction as well as magnitude, we have

mimj − Fij = G 3 rij, rij = rj ri rij Though applies only to point masses, it can be extended to bodies with spherical symmetry. It is fairly accurate to object of arbitrary size when those objects are suﬃciently far away. The reasoning behind this being that when size is negligible to distance, mass distribution can be approximated by a delta function that peaks at the center of mass. This inverse square law of force governs the motion of celestial bodies. Newton showed that the Kepler’s laws are a natural consequence of this force. For a system of gravitational interacting objects it turns into

∑N m m F = G i j r , r = r − r i r3 ij ij j i j=1,j≠ i ij By dividing with mass we get the more useful expression for numerical calculations

∑N ... m r = G j r , r = r − r i r3 ij ij j i j=1,j≠ i ij The problem is thus a set of non-linear 3N 2nd-order ordinary differential equations relating the acceleration with the position of all bodies in the system. Once a set of initial condition is specified a unique solution exists. This can be found analytically (by first integral approach) only for up to two bodies, while for more bodies (n ≥ 3) require numerical integration. The vector field describing the gravitational force is called the gravitational field and equals the acceleration field. This allows us to rewrite the gravitational law according to that field as m′ F(r) = mg(r), g(r) = −G r r3 Gravitational fields are conservative and they can be written as the gradient of a potential. Thus F(r) = −m∇V (r),V = −Gm′/r We call V the gravitational potential, and it depends only on the distance. The negative sign is based on the convention that V (∞) = 0. The total energy of a body in such potential is

E = mv2/2 + mV (r), v = |v| and it is constant of motion, as

E˙ = mr˙ · a¨ + m∇V · r˙ = m(g + ∇V ) · r˙ = 0

Angular momentum is a constant of motion as well, as mm′ L˙ = r˙ × p + r × p˙ = mr × g = −G r × r = 0 r3 Angular momentum plays a key role in a planetary systems stability, as it constrains the semimajor axis, eccentricities and inclinations of the planets of the systems[2].

1.4.2 Hamiltonian mechanics As Newton’s laws involve forces which are vectors, in order to simplify calculations, a reformulation of said laws happened in terms of functions, particularly for systems of interacting objects for which energy is conserved. This reformulation deﬁnes a function called the Hamiltonian of the system. It expresses the energy E in terms of position and momentum. 7

Algebraic formulation Let H(q, p, t) be the Hamiltonian equation of the system, where r = (q, p) is a set of canonical coordinates. The time evolution of the system is uniquely deﬁned by the equations

q˙ = ∂pH, p˙ = −∂qH

If H does not contain t explicitly (the system is autonomous), H is a constant of motion.

H˙ = 0 ⇒ H(q, p) = H(q0, p0)

Thus it depends only on the initial state of the system. Suppose the equations are mechanical, so that the kinetic energy is a quadratic form with respect to q˙ . Then the Hamiltonian is the total energy of the system and has the form

∑N p2 H = T (p) + V (q) = i + V (q) 2m i=1 i and the Hamilton equations become

q˙ = p, p˙ = −∂qV (q)

Geometric formulation The phase space of a mechanical system has symplectic geometry as structure. The Hamiltonian H(q, p) is a function on phase space that governs the dynamics of the system. That is, it describes time evolution by specifying their equations of motion. The Hamiltonian system can be written using the more compact notation ( ) 0 I z˙ = J∇H(z), z = (p, q),J = −I 0

Any smooth real-valued function H on a symplectic manifold can be used to define a Hamiltonian system. The function H is known as the Hamiltonian and the symplectic manifold is then called the phase space. This function induces a vector field on the symplectic manifold known as the Hamiltonian vector field. The flow of the vector field is the one-parameter group of transformations ϕt :(p0, q0) → (p(t; p0, q0), q(t; p0, q0)). According to Poincare the flow ϕt(z) is symplectic. And thus the exact flow of a Hamiltonian differential equation is a symplectic transformation. The physical properties offered by using a symplectic form are the following.

• Non-degeneracy: The vector ﬁeld’s XH evolution is determined by the Hamiltonian H. • Reversibility: The system is ϕ-reversible if the ﬂow is ϕ-reversible.

• Closeness: The system preserves the symplectic structure. Each symplectomorphism (a transformation preserving the symplectic structure) preserves the volume on the phase space. That implies volume preservation which is equivalent to area preservation. This means, what is known as Louville theorem, that

dp × dq = dp′ × dq′

Its meaning being that the symplectic ﬂow is conserved. We can develop numerical methods exploiting the previous properties allowing us to perform long term integrations that provide insight in the dynamics of the system even though energy cannot be maintained constant. Those so-called symplectic methods have the energy ﬂuctuating thus removing the energy drift found in non-symplectic methods. 8

1.5 Celestial mechanics

Celestial mechanics is the branch of astronomy dealing with the motions of celestial objects using principles of classical mechanics. Traditionally it has been used to produce ephemeris data for astronomical objects. The birth of celestial mechanics came with the laws of planetary motion published by Johannes Kepler. Those laws though were originally derived empirically from observations of his professor, Tycho Brahe, were later proved to be theoretically correct from Isaac Newton using his laws of motion.

1.5.1 Orbits The general parameters derived from mechanics characterizing the orbital motion are

• Position (r) is the vector of the orbiting body’s position relative to the inertial frame and point of origin.

• Velocity (v) is the vector of the orbiting body’s velocity relative to the inertial frame and point of origin.

• Speciﬁc angular momentum (h = r × v) is the body’s angular momentum divided by its mass.

• Speciﬁc energy (ϵ) is the total energy of the system divided by the reduced mass.

• Period (P ) is the time it takes for the body to make a complete closed trajectory around the main body.

• Gravitational parameter (µ) is simply deﬁned as µ = G(m0 + m) ≈ Gm0.

The Cartesian vectors of position r and velocity v uniquely determine the trajectory of an orbiting body. Those vectors are called orbital state vectors and are defined with respect to a reference frame. The orbit that coincides with the current orbital state vectors is called osculating orbit. Keplerian orbits follow conic sections. Thus it is more convenient to characterize the motion of celestial bodies by quantities describing the geometrical properties of the orbit and the position on it. Orbital elements are the parameters that uniquely identify a specific orbit and the instantaneous position of the body on it. Those are shown on figure 1.3 and explained below. Formally, an orbital elements ϕ is a linear function of time in the unperturbed case. So their usefulness comes from the fact that in absence of perturbations the five out of six parameters are time independent (constant) and only the sixth depends on time. In the perturbed case, orbital elements vary non-linearly with time[7]. Perturbations in real astronomical orbits can evolve the osculating elements very quickly.

• Semimajor axis (a) is the sum of the periapsis and apoapsis distances divided by two.

• Eccentricity (e) deﬁnes the shape of the ellipse, describing how much it is elongated compared to a circle. A circle can be considered a special case of an ellipse with e = 0.

• Inclination (i) is the vertical tilt of the ellipse with respect to the reference plane, measured at the ascending node where the orbit passes upward through the reference plane.

• Longitude of the ascending node (Ω) horizontally orients the ascending node of the ellipse, measured where the orbit passes upward through the reference plane with respect to the reference frame’s vernal point. 9

hˆ

ν Ecliptic Plane ω

Ω y

Line of Nodes

Figure 1.3: Orbital elements shown schematically. The Euler-like angles i, Ω, ω describe the orientation. The true anomaly speciﬁes the position of the orbiting body.

• Argument of periapsis (ω) deﬁnes the orientation of the ellipse in the orbital plane, as an angle measured from the ascending node to the periapsis.

• True anomaly (ν = ν(t)) is the position of the orbiting body along the ellipse at the epoch.

In summary, the first two parameters specify the shape of the ellipse, the next three the orientation, corresponding to the three Euler angles, and the final one the location of the body in the orbit. Action variables related to the first three are associated with action angles related to the later three. Tilt angle is measured perpendicular to line of intersection between orbital plane and reference plane. The plane and the ellipse are two-dimensional objects defined in the three-dimensional space. Also, the following anomalies and longitudes are used.

• Eccentric anomaly (E = E(t)) is the angle subtended at the center of the ellipse by the projection of the position of the body on the circle with radius a and tangent to the ellipse at pericenter and apocentre.

• Mean anomaly (M = M(t)) gives an angular distance from the pericenter at arbitrary time. Geometrically can be considered as the angular distance from the periapsis for an imaginary body orbiting on a circular trajectory and with the same orbital period as the actual body. It is deﬁned as M = n(t−τ), where n = 2π/P is the mean-motion. The mean and eccentric anomaly can geometrically be shown that are connected via the relation

M = E − e sin E

which is known as Kepler’s equation. At time τ it is M = E = 0 and thus the body is situated in its pericenter.

• Longitude of pericenter (ω¯) is deﬁned as ω¯ = Ω + ω.

• Mean longitude (λ = λ(t)) is deﬁned as λ =ω ¯ + M.

The movements of celestial objects are governed by Kepler’s laws. In modern terms, the three Kepler laws can be formulated[7] as follows: 10

• First Law: The planets move in ellipses with the sun at a focus (see ﬁg. 1.4a). Then the mathematical model of the orbit which gives the distance between a central body and an orbiting body can be expressed as

r = a(1 − e2)(1 + e cos ν)−1, e < 1

• Second Law: A line segment joining a planet and the sun sweeps out equal areas during equal intervals of time (see ﬁg. 1.4b).

A˙ = r2θ˙/2 = h/2 = const.

• Third Law: The square of the orbital period of a planet is proportional to the cube of its semi-major axis. P 2 ∝ a3

It’s exact form may be found using the law of gravitation. A derivation for a circular orbit is as follows 2 2 2 Gm0m mv m(2πr/P ) ⇒ 2 4π 3 2 = = P = a r r r Gm0 A more correct derivation is by considering the reduced mass, deﬁned as β = Mm/(M + m), of the system, as the planet doesn’t orbit the star but the center of mass of the planet-star system.

2 2 2 Gm0m βv β(2πr/P ) ⇒ 2 4π 3 2 = = P = a r r r G(m0 + m) Using the gravitational parameter and the mean-motion it simpliﬁes to

µ = n2a3

Setting µ = 1 and a = 1 we deﬁne a time unit such that n = 1 or P = 2π. Although this expression was derived for the case of a circular orbit, exactly the same expression results for an ellipse. This is justiﬁed by considering the circle as a special case of an ellipse. A derivation for elliptic orbits can be done by using the second law.

P P

S F2 S

(a) First law: Sun at the focus of the ellipse. (b) Second law: Shaded areas have same size.

Figure 1.4: Schematics of Kepler’s ﬁrst and second law.

The orbital elements a, e, i, Ω, ω, M completely deﬁne the position and velocity of the secondary (orbiting) body with respect to the central one. The one-to-one correspondence, considering the relation between M and E, is given by the relationship

r = R(Ω, i, ω)q, r˙ = R(Ω, i, ω)q˙ 11 where [ √ ] q = a cos E − e, 1 − e2 sin E, 0 and [ √ ] na q˙ = − sin E, 1 − e2 cos E, 0 1 − e cos E as well as the Eulerian rotation matrix

R(Ω, i, ω) = Rz(Ω)Rx(i)Rz(ω) where Rz,Rx the clockwise rotation matrices for z, x respectively. Using the previous relation- ships it is possible to translate between the orbital elements and the coordinates in the reference frame.5

1.5.2 N-body problem The n-body problem is the problem of predicting the individual motions of a group of objects interacting with each other gravitationally. This is a problem usually not analytically solvable. Specifically a general, classical solution by first integral approach, is known to be impossible for n ≥ 3. There is an acceptable exact solution in a closed power form (in convergent power series6), which isn’t very usable as it has very slow convergence. Numerical methods are instead a good choice for the n-body problem. Nevertheless some qualitative results may be obtained analytically. The potential field is given, in barycentric coordinates, as

∑N mimj Vi(q) = −G qij j=1,j≠ i

The Hamiltonian can then be written

∑N p2 ∑N m m H = T (p) + V (q) = i − G i j 2m q i=1 i i=1,j>i ij where is G is the gravitation constant. Then the Hamilton equations take the form

∑N m m q˙ = p , p˙ = −G i j q i i i q3 ij j=1,j≠ i ij

The energy and the angular momentum are constants as

d ∑ H˙ = 0, L˙ = q × p = 0 dt i i i

We consider now this problem for a planetary system, that is a system consisting of few small objects in orbit around a large one, the host star. The motion of a body, in astrocentric coordinates, is described[6] ( ) µ ∑N r r r¨ = − i r + G m ij − j i r3 i j r3 r3 i j=1,j≠ i ij j

5The procedure is given as algorithm in sec. A.1 6Wang Qiu-Dong. “The global solution of the n-body problem”. In: Celestial Mechanics and Dynamical Astron- omy 50.1 (1990), pp. 73–88. 12

This force cannot be written as the gradient of a potential. By considering a body with mass negligible compared to the star’s, we get the restricted problem, described ( ) m ∑N r r r¨ = −G 0 r + G m ij − j i r3 i j r3 r3 i j=1,j≠ i ij j

The potential of the previous is ( ) ∑N · − m0 − 1 − ri rj Vi(r) = G G mj 3 ri rij r j=1,j≠ i j and the Hamiltonian can then be written ( ) 2 ∑N r · r vi − m0 − 1 − i j Hi = G G mj 3 2 ri rij r j=1,j≠ i j

Here mi are the planet masses with positions ri. The central mass has coordinate r0 and mass m0. As rj = rj(t) the Hamiltonian has the form H(r, p, t). Though nonautonomous, it can transformed into autonomous by extending the phase space. In this problem using Delaunay variables the Hamiltonian can be rewritten[6]

G2m2 H = − 0 + H′ i 2L2 i √ ′ where L = Gm0a and Hi the interactions with the other bodies. In the three body case the expression takes the form, with prime referring to the other body ( ) v2 m 1 r · r′ H = − G 0 − Gm′ − 2 r |∆r| r′3 or else H = H0 − R The R is called the disturbing function and is used to calculate the perturbations to an object’s elliptical orbit by the acceleration caused from a perturbing potential, as

V = U − R where U the potential due to two-body planet-star interactions, and R the disturbing function for the third body interaction, given by[7]

1 r · r′ R = µ′ − µ′ , µ′ = Gm′ |∆r| r′3 Consequently the disturbing function represents the additional terms in the potential due to the other body. The second term is the indirect term as it is considered in astrocentric coordinates. The disturbing function can be expanded in terms of standard orbital elements to an inﬁnite series ∑ R = µ′ S(a, a′, e, e′, i, i′) cos ϕ where, with p, q, p′, q′, l, l′, m, m′ integers

ϕ = (l − 2p′ + q′)λ′ − (l − 2p + q)λ − q′ω¯′ + qω¯ + (m − l + 2p′)Ω′ − (m − l + 2p)Ω or ∑ ′ ′ ϕ = j1λ − j2λ − j3ω¯ + j4ω¯ + j5Ω − j6Ω, jk = 0 13 and, with s = sin(i/2) and s′ = sin(i′/2)

f(α) ′ ′ S ≈ e|q|e′|q |s|m−l+2p|s′|m−l+2p | a′ The Hamiltonian can then be expressed

G2m2 ∑ H = − 0 − µ′ S(a, a′, e, e′, i, i′) cos ϕ 2L2 The Laplace-Lagrange secular theory is a formulation where the orbital parameters are described analytically by utilizing few terms in the disturbing function. Then the variation of the orbital elements of two planets in orbit of a star, which determine the system’s motion over time, for a given potential are described by the Lagrange equations[7]. The theory, though elegant, fails to reproduce the exact phase space of extrasolar planetary systems. Nevertheless, classical Laplace-Lagrange theory remains useful to qualitatively describe secular motion. It is useful to describe the complete system as a sum of integrable two-body unperturbed and interaction parts. This is not possible in relative coordinates[4], as the kinetic energy is not a diagonal sum of the squares of the momenta. Let Xi be the position vector of the three bodies in an inertial frame. Following Poincaré (see sec. A.2), we can consider for the planets the astrocentric position vectors ri = Xi − X0 and the barycentric momenta pi = miX˙ i. The transformation is canonical and by direct substitution in the barycentric expression, the Hamiltonian of the system can then be written as[10],

′ H = H0 + H

′ where H0 is the dominant and H the interaction term. By using the reduced masses, deﬁned βi = m0mi/(m0 + mi), those are written ( ) ( ) ∑N p2 m m ∑N m m p p H = i − G 0 i ,H′ = −G i j + i j 0 2β r r m i=1 i i i=1,j>i ij 0

The term H0 describes the evolution of the planets in the framework of the two body problem (unperturbed star-planet system). The second term, H′, includes the planetary interactions and is a perturbation term for the integrable part H0. Using Delaunay variables the Hamiltonian can be rewritten[6] ∑N G2(m + m )β3 H = − 0 i i + H′ 2L2 i=1 i The mutual planetary interactions are small compared to their interaction with the star, but they are not negligible. Their eﬀect becomes important in studying the orbital evolution for long time intervals[10].

1.6 Detection

A number detection methods have been developed for discovering exoplanets. Short reviews are provided based on (Wright et al, 2013)7 and Perryman[8]. What is shown here shall provide insight in the later data analysis. This section includes the most important methods used. That is the ones used to ﬁnd the bulk majority of exoplanets until now.

7Jason T WrightDr et al. “Exoplanet detection methods”. In: Planets, Stars and Stellar Systems. Springer, 2013, pp. 489–540. eprint: arXiv:1210.2471. 14

Astrometry A conceptually very simple technique. The variations in a star’s position provide the inclination and orientation of a planetary orbit. For exoplanet detection, when compared to well-separated binary stars of similar magnitude, is to detect motions around a star of an unseen companion with respect to stable background stars. Astrometry is sensitive and thus suited for detecting planets with long orbital periods. It does not depend on the distant planet being in near-perfect alignment with the line of site from the Earth. Therefore it can be applied to a great number of stars. Moreover, it provides an accurate estimate of a planet’s mass. Astrometry can also determine the relative inclinations between pairs of orbits. An example can be seen on fig. 1.5a, where the movement of a star can be seen. The (straight) dashed line is the barycentric motion viewed from the system’s barycenter, the dotted line is the effect of parallax (Earth’s orbital motion around the Sun), and the solid line is the motion of the star as a result of the orbiting planet (effect magnified). This movement is characteristic in an exoplanet’s presence.

Imaging The direct imaging is the most straightforward method of detection. The obstacle is the prox- imity to a much brighter stellar sources. The distangling of stellar and planetary photons is an imperfect process. Therefore the most important parameters to the difficulty of direct detection are the planet/star flux ratio fp and the angular separation of planet-star. That angular separation is given by ∆θ = rd/d where rd is the projected planet-star separation and d the system’s distance. The emission from an exoplanet is separable to two sources. Those are stellar emission reflected from the planet surface or/and atmosphere, and thermal emission, which is either intrinsic or reprocessed stellar luminosity. An example of directed imaged planets can be seen on fig. 1.5b. Four planets are directly visible in this near-infrared image.

(a) The movement of a star orbited by an (b) Near-infrared Keck adaptive optics images exoplanet (Perryman, 2011). The (straight) of the HR 8799 system (Marois et al, 2010)a. dashed line is the barycentric motion viewed Four giant planets, 3 to 7 times the mass of from the system’s barycenter. The dotted line Jupiter, are visible in near-infrared emission. is the effect of parallax (Earth’s orbital motion around the Sun). The solid line is the motion of aChristian Marois et al. “Images of a fourth the star as a result of the orbiting planet (effect planet orbiting HR 8799”. In: Nature 468.7327 magnified). (2010), p. 1080.

Figure 1.5: Characteristic pictures for detection via astrometry and imaging. 15

Microlensing

The gravitational microlensing method detects planets via the gravitational perturbation of a background light source due to a foreground planet. When a foreground compact object passes close to our line-of-sight to a distant star, the light of the background star splits into two images. These are unresolved and magnified by an amount dependent on the lens-source angular separation. This separation is a function of time and thus the background source exhibits a time-variable magnification. This is called a microlensing event. The rise and fall of the source brightness must be monitored over time using photometry. A typical microlensing light curve is given on fig 1.6b. The plot measures the change in brightness of the system as a function of time, where t = 0 is the time of closest (projected) approach between the lens and the source objects. Specifically, y equals the magnification of the source brightness; y = 1 is the original brightness.

Transit (photometry)

A planet transiting in front of its host star’s disk, will result in the observed visual brightness of the star be dropped by a small amount. This amount depends on the star-planet relative sizes (star and planet radii). The photometric method can determine the planet’s radius. This method has two major disadvantages. First, planetary transits are observable only when the planet’s orbit happens to be perfectly aligned from our perspective. The probability of a planetary orbital plane being directly on the line-of-sight to a star is the ratio of the diameter of the star to the diameter of the orbit. In small stars, the radius of the planet is also an important factor. The second disadvantage of this method is a high rate of false detections. A schematic of a planetary transit can be seen on fig. 1.7a. The important point is when the planet passes in front of the star disk. There a drop in brightness happens depending on the planet radius, and the total time it takes shows the planet’s orbital period. The rest of the schematic is explained by the fact that the planet reflects some of the host’s star light. Thus when it moves towards the back the flux increases but when it goes behind the star a drop happens. Measuring variations of the timing of transits it is possible to reveal the existence of additional planets, not necessarily transiting In such a case, the gravitational perturbations of the additional planet on the orbit of the transiting planet forces the transit to happen earlier or later. This distortion is called transit timing variations or TTVs. These perturbations can be large if the perturbing body is near a mean-motion resonance. TTV is an example of the more generalized method of timing periodic variations.

Timing

A planetary companion can be found to stars or stellar remnants that exhibit regular periodic photometric variability through timing variations of those periodic phenomena. It has been most successfully employed with pulsars and eclipsing binary systems. Based on the regular periodic rotation of a pulsar, the pulsar’s motion can be tracked through anomalies in the timing of the observed radio pulses. A pulsar’s orbit will be perturbed if it has a planet. The parameters of that orbit can the be calculated based on pulse-timing observations. The change in measured period has an amplitude given as

1 a sin iMp τp = c M⋆

The high accurate pulsar timing allow for low mass bodies to be detected. Let the orbit be circular, have period P and suppose pulsar has mass 1.32M·. Then the previous equation may 16 be written ( )( ) 2/3 MP P τp ≃ 1.2 M⊕ 1yr It shows that terrestrial planets should be detectable around normal slow pulsars. Masses down to the Moon, could be recognized in pulsars with period in milisecs.

(b) Data on the microlensing event OGLE-2005- BLG-390 together with a model light curve, showing the planetary deviation on its falling part, lasting about a day. Also shown are best- (a) The average pulse proﬁle of PSR 1257+12 ﬁtting models with a single lens and a binary a at 430 MHz (Wolszczan and Frail, 1992) . source and single-source-single-lens light curve (ESO)a. aAleksander Wolszczan and Dail A Frail. “A planetary system around the millisecond pulsar aESO. Light curve of OGLE-2005-BLG-390. PSR1257+ 12”. In: Nature 355.6356 (1992), p. 145. 2006. URL: https : / / www . eso . org / public / images/eso0603b/.

Figure 1.6: Characteristic pictures for detection via microlensing and pulsar timing.

Radial velocity (spectroscopy) The radial reflex motion of a star in response to an orbiting planet may be measured through the displacement in the host star’s spectral lines due to the Doppler effect. This motion reveals the period, distance and shape of orbit, as well as information about the mass. The parameters that determine the functional form of the periodic radial velocity variations are: P, K, e, ω, T0, and γ. Vr = K[cos(ν + ω) + e cos ω] + γ The semi-amplitude in units of velocity is K and the bulk velocity of the mass-center is given by γ. In circular orbits, having e = 0, there is no periastron approach, and the parameters T0 and γ are undefined. In such cases a nominal value of ω sets T0. Another approach is specifying the value of one of the angles at a given epoch. The variables P,T0 and K set the period, phase and amplitude of an RV curve. The variables ω and e determine the shape of the RV signature of an orbiting companion. The last orbital parameters necessary for the description of a planet’s orbit cannot be determined with radial velocity observations alone. Those are the inclination i and the longitude of the ascending node Ω. Those can only be measured through astrometry. While the radial velocity technique measures the m sin i value where is m the planet mass and i is the orbital angle, it would only provide the distribution of m sin i, but because this distribution for a random set of planetary systems is very close to the m distribution we can make a statistical analysis alongside the other methods using just the mass parameters. An example of a radial velocity curve is given on fig. 1.7b. This sinuoid is characteristic of a radial velocity graph. The peg velocities are plotted as a scatter and and upon it a Keplerian fit is being put. From it we derive P,K, and e. 17

(b) Peg velocities of 51 Pegasi phased with a Keplerian ﬁt (Marcy et al, 1997)a. The sinusoid is the characteristic shape of the radial velocity (a) Schematic of a transit (Perryman, 2011 af- graph of a star rocking to the tug of an orbiting ter Winn, 2009). planet. aGeoﬀrey W. Marcy et al. “The Planet around 51 Pegasi”. In: The Astrophysical Journal 481.2 (June 1997), pp. 926–935.

Figure 1.7: Characteristic pictures for detection via transit and radial velocities.

1.7 Resonances

Resonances occur when orbiting bodies exert a regular periodic gravitational influence on each other. They happen when some orbital variables are related by a ratio of small integers; or else when those variables are commensurable. In that case the gravitational forces exerted by the motions of the bodies add up over time in a coherent manner. A secular resonance involves a commensurability amongst the (slow) frequencies of orbital precession, whereas a mean motion resonance is a commensurability amongst the (fast) frequencies of orbital revolution[4]. Resonances are important as they can either provide stability to the system, allowing it to survive for eons, or they can set off destability, quickly disbanding it. Specifically, when eccentricities are quite high, the elliptical orbits can be close to each other or even intersect. In those cases the gravitational interactions may be very strong. Resonances provide a phase protection that prevent the close encounter of planets. Thus planetary systems with high eccentric orbits can only survive if they are trapped in a resonance, and of course have appropriate phases (initial conditions). Besides their paramount importance to the system evolution, they provide crucial information to the origins of systems. Current research shows that resonant pairs aren’t being formed as that but instead they arise either from capture, migration or other external forces. The can also drastically alter the orbits of bodies under the resonant bodies gravitational influence. The “Nice model”, a dynamical scenario for the Solar system early evolution, presumes that at one time Jupiter and Saturn passed through an orbital resonance resulting in large perturbations which destabilized many asteroids and ejected Uranus and Neptune to their current orbits8. These ice giants by advancing into the planetesimal disk, scattered the planetesimals which were in stable orbit in the outer Solar System, causing a sudden massive delivery of planetesimals to the inner Solar System9. Resonances are thought to explain some regularities in the orbital distribution of the planets in multiplanetary systems[3].

8Kleomeris Tsiganis et al. “Origin of the orbital architecture of the giant planets of the Solar System”. In: Nature 435.7041 (2005), p. 459. 9Rodney Gomes et al. “Origin of the cataclysmic Late Heavy Bombardment period of the terrestrial planets”. In: Nature 435.7041 (2005), p. 466. 18

Besides the two major categories described below, there are other orbital resonances[4]. Examples are apsidal resonances, when the angular velocity of the apsidal precessing is commensurable with an orbital mean motion, and super or secondary resonances, when there are commensurabilities between the libration frequency of a resonant angle and the circulation frequency of a diﬀerent one.

1.7.1 Mean-motion Instead of orbital period P we prefer using the mean-motion deﬁned as

n = 2π/P

Then, we say that two planets have commensurable mean-motions, when n p 2 = , p, q ∈ N, p > 0, q ≥ 0 (1.1) n1 p + q

Of course this usually cannot happen precisely and instead the equation becomes an approximation.

n2 p n2 k2 ≈ ⇒ − = d (1.2) n1 p + q n1 k1 where k2 = p, k1 = p + q two integers and d > 0 a small number. These commensurabilities can arise from dynamical considerations and are not a mean-motion resonance condition per se.

1.7.2 Two-planet dynamics

A two-planet system can be modeled as a three-point system with masses m0, m1 and m2 representing the star, the inner and the outer planet, respectively. As already mentioned, with a suitable choice of coordinates, the Hamiltonian can be written as

′ H = H0 + H

′ with H0 and H representing the Keplerian and interaction terms repectively. Thus the dynamics of a resonant pair may be separated into its own secular movement and the resonant dynamics.

Secular Simpliﬁed models can be constructed through the averaging method. The averaging principle states that most terms average to zero over a few orbital periods. So those can be ignored by using the averaged disturbing function. We can obtain the secular model by averaging the fast motion on the ellipse, given by the angles λi. The phase space structure up to second order in masses is found that it depends on the planetary mass ratio m1/m2[10]. Also, the semimajor axes remain almost constant and in a coplanar system whereas the eccentricities e1 and e2 oscillate slowly with opposite phases[10]. The secular evolution is deﬁned by the secular (apsidal) angle

∆¯ω =ω ¯2 − ω¯1 (1.3)

If ∆¯ω librates, that is oscillates with a certain amplitude around an equilibrium value, the system is said to be in a secular resonance. This is usually a purely kinetical phenomenon. As such, the circulatory and libration regions aren’t divided by a separatrix. True secular resonances though do exist. For symmetric periodic orbits, we have ∆¯ω = 0 referred as apsidal alignment or ∆¯ω = π referred as apsidal anti-alignment. 19

Resonant Suppose the general form of an argument in the expansion of the disturbing function.

ϕ = j1λ2 + j2λ1 + j3ω¯2 + j4ω¯1 + j5Ω2 + j6Ω1

The time derivative of it, using the variation of the orbital elements from Lagrange equations, is ϕ˙ = j1(n2 +ϵ ˙2) + j2(n1 +ϵ ˙1) + j3ω¯˙ 2 + j4ω¯˙ 1 + j5Ω˙ 2 + j6Ω˙ 1

The new angle ϵ denotes the mean longitude at epoch and is given ϵi = λi − nit. The perturbed body is in exact resonance when the time variation of the argument is zero. For mean-motion resonances the dynamics are dominated by the ﬁrst four terms. In case of coplanar orbits, we neglect the contributions from variations of the longitudes. Thus[7]

j1n2 + j2n1 ≈ 0 where we can let j1 = p+q and j2 = −p, recovering the expression 1.1. The nominal resonances locations are found using Kepler’s third law.

2/3 α = a2/a1 = (|j1|/|j2|)

In case of e1 = 0, e2 ≠ 0, ω¯˙ 2 ≠ 0, the resonant relation is

(p + q)n2 − pn1 − qω¯˙ 2 ≈ 0

In the case where there is a near commensurability (the relation 1.1 holds), the arguments containing such expressions, have slow variance and long-period large-amplitude variations. If the resonant relation holds, then

n − ω¯˙ p 2 2 = n1 − ω¯˙ 2 p + q where n2 −ω¯˙ 2 and n1 −ω¯˙ 2 can be considered as the mean motions in a reference frame corotating with the pericenter of the outer planet. From its viewpoint the orbit of the outer planet is stationary, and q-th conjunction takes place at the same point in the orbit of the outer planet[7]. In the action-angle variables for the Keplerian potential, the dynamics of a resonant system is described by the evolution of the critical argument which is a linear combination of the angle variables. If the critical argument circulates, then the planet pair is said to be near but not in resonance. That is, if neither angular variable librates, the system is in a non-resonant conﬁguration. This angle is deﬁned as[7][2]

ϕ = ϕ1 = (p + q)λ2 − pλ1 − qω¯1 or ′ ϕ = ϕ2 = (p + q)λ2 − pλ1 − qω¯2 Thus the critical arguments are

ϕi = (p + q)λ2 − pλ1 − qω¯i (1.4)

Any mean-motion resonance has a well-deﬁned width inside which the two orbiting bodies can have a libration movement around the equilibrium point. In a resonant conﬁguration, the longitude of the planets at every q-th conjunction librates slowly about a direction determined by the lines of apsides and nodes of the planetary orbits. Systems with small amplitude librations are in deep resonance. 20 1:6 1:5 1:4 1:3 2:5 1:2 2:3 3:4 1:1 4:3 3:2 2:1 5:2 3:1 4:1 5:1 6:1

1.00

0.75

q 0.50 e

0.25

0.00

1 2 3 α

Figure 1.8: Nominal resonances locations and approximate strengths for e = 0.4 and p + q ≤ 7.

The case q = 0 is sometimes called a corotation or co-orbital resonance. When q > 0, q represents the order of the resonance. As the order q increases the strength of the resonance decreases. Specifically the strength of a resonance is proportional to eq (for low eccentricities). For large integers resonance is not dominating the dynamics of the system. Using the relation for the nominal locations, the resonances for p + q ≤ 7 are shown in fig. 1.8 with eq, as an illustration for their relative strengths. The corotation resonance can be extended beyond the narrow case given. Resonance can exist when both ϕ and ϕ′ are trapped in resonance. In the previous case the periastron continues to rotate, and as such ∆¯ω varies monotonically. In this case ∆¯ω no longer circulates, but now librates. Thus, the simultaneous libration of ∆¯ω and ϕ is synonymous of corotation resonance, and the system is said to be in apsidal corotation resonance[2]. Planetary mean-motion resonances are classified into internal and external. The classifica- tion is done as a function of the dominant mass position compared relative to the reference body. Internal resonances are those with the dominant mass on an external orbit, whereas the external resonances have the dominant mass in an internal orbit. The internal resonances have analytical solutions in some specific problems, while the external resonances are described numerically or using the Hamiltonian approach[7]. An illustration of the induced geometry is seen on fig. 1.9 for interior and on fig. 1.10 for exterior resonances, where the paths in the rotating frame are drawn for a variety of resonances and eccentricities. The paths illustrate the relationship between resonance and frequency of conjunctions with the external and internal object respectively. Using the pendulum model approach, an analytical solution can be derived for an interior resonance. Specifically the second time derivative of the argument

ϕ¨ = j1(n ˙ 2 +ϵ ¨2) + j2(n ˙ 1 +ϵ ¨1) + j3ω¯¨2 + j4ω¯¨1 + j5Ω¨2 + j6Ω¨1 in case of resonance can be expressed in the form | | ¨ − 2 2 − 2 j4 ϕ = ω0 sin ϕ, ω0 = 3j2 Crne where ω0 is a parameter dependent on the elements and the masses. The libration width of a test particle can be calculated directly[7]. The relations are considered in units where m0 = µ = 1. For a q-order resonance the variation is ( ) δa 16eq 1/2 max = |C |a3/2 a 3 r 21

Figure 1.9: Paths in rotating frame for interior resonances (from top to bottom) 2:1, 3:1 and 3:2 for e (from left to right) 0.1, 0.2, 0.3 and 0.4. The positions are drawn at equal time intervals. (Figure is based on Murray and Dermott, ﬁg 8.4.)

Cr is described by ′ −3/2 Cr = m a αfd(α) ′ where m the mass of the perturber. The function fd(a) is the disturbing function, with α the semimajor axis ratio for two bodies in resonance and it can be expanded as a combination of Laplace coeﬃcients depending on the resonance order. Speciﬁcally the expansions corresponding to cosine arguments

jλ2 + (q − j)λ1 − qω¯1, jλ2 + (q − j)λ1 − qω¯2

For ﬁrst-order resonance the variation is calculated ( ) ( ) δa 16e 1/2 1 1/2 2 max = |C |a3/2 1 + |C |a3/2 + |C |a3/2 a 3 r 27p2e3 r 9pe r

Here a is the semimajor axis and e is the orbit eccentricity. According to this relation, in case of ﬁrst-order resonances the maximum libration width decreases as the eccentricity decreases until the width starts increasing again. A more simpliﬁed method can be used for orders higher than 4, with the libration width estimated from ( ) δa 16eq 1/2 max = |D |a3/2 a 3 r

The Dr is empirically calculated to be

′ Dr = m exp(c0 − A · r),A(r) = exp(c1 − c2 · r) + c3 where[3] (c0, c1, c2, c3) = (6, 1.122, 1.1239, 1.674) and r the resonance number. If the semimajor axis or mean motion relative diﬀerence is within the libration width, one can estimate the two planets are in mean-motion resonance. During the motion out of the resonance range, the libration transforms into circulation with the two modes separated by an apsidal separatrix[3]. Since their derivation is based by expanding the disturbing function and the lowest order eccentricity was used, the libration widths will not be accurate at large eccentricity[7]. 22

Figure 1.10: Paths in rotating frame for exterior resonances (from top to bottom) 1:2, 1:3 and 2:3 for e (from left to right) 0.1, 0.2, 0.3 and 0.4. The positions are drawn at equal time intervals. (Figure is based on Murray and Dermott, ﬁg 8.4.)

The notion of mean-motion resonance can be extended to more than two bodies. A resonant variable can be found by combining lower-order resonant angles. A many-body resonance can be a chain of lower-order resonances or a true n-body resonance with none of the lower-order critical angles librating. In the three-body case, a three-body resonance occurs when a resonant chain exists for three bodies. The resonant variables are the resonant angles of the two-body resonances taken separately. A combination of two of the resonant angles deﬁnes the Laplace variable. Let n1, n2, n3 be the mean motions of three bodies at increasing distance of a central one, hence it will be a1 < a2 < a3 or correspondingly n1 > n2 > n3.A Laplace relation exists between the three orbiting masses if the following relationship holds constant over time.

qn1 − (p + q)n2 + pn3 ≈ 0

The corresponding resonant argument is given

ϕL = qλ1 − (p + q)λ2 + pλ3 (1.5) and it can be shown that it has the form of pendulum equation

ϕL = c sin ϕL

A simultaneous libration of subsequent pairs means that the Laplace angle is also librating. But the existence of a Laplace relation does not imply resonant relationship between pairs. An example in our system will be the three-body resonance in Galilean moons, Io, Europa and Gamiaede. The three inner moons are in a 1:2:4 orbital resonance with each other, and the following argument remains constant.

ϕL = λI − 3λE + 2λG ≈ π

The previous relation locks the orbital phase of the moons and makes a triple conjunction impossible. Geometrically, this means a cycle of conjunctions is maintained. This can be seen illustratively in ﬁgure 1.11. Every three-body resonance with same ratios is called Laplace resonance. Three-body resonances involving other simple integer ratios are called Laplace-like. 23

Figure 1.11: Schematic geometry of inner Galilean moons. Considering (Ω, ω) = (0, 0), the angles will represent the mean anomaly and thus the mean longitude. Notice how the angle ϕL = λ1 − 3λ2 + 2λ3 (inner to outer) remains constant and that the three moons are never in conjunction together. (Figure is based on Murray and Dermott, ﬁg 8.30.)

An extrasolar example will be the three-body resonance in GJ 876 outer planets c, b and e. The three planets are in 1:2:4 orbital resonance with each other, and the following argument remains constant.

ϕL = λc − 3λb + 2λe ≈ 0

This can be seen illustratively in ﬁgure 1.12. The resonances mean that the periapses are nearly aligned and conjunctions occur when both planets are near periapse.

1.7.3 Classes

By having deﬁned MMR, we can classify those interactions as it is described in the review papers by Baauge, Ferraz-Mello et al[2]. Though those were deﬁned when a vast smaller number of exoplanets was known, the characteristics they mention are still relevant.

Class Ia: Mean-motion resonant planet pairs

Planets with large masses and eccentricities orbiting in relativity close orbits, are liable to strong gravitational interaction, and are unable to remain stable if not tied by a mean-motion resonance (MMR). Large eccentricities will make close encounters, and subsequent destabilization, unavoidable. Resonance oﬀers a phase protection and so stable (almost) periodic orbits may exist. A region of stability can be located around stable resonant periodic orbits. However, as eccentricities increase, the stability domain shrinks[10].

Class Ib: Low-eccentricity near-resonant planet pairs

A special class including systems with low-eccentricity planets which some can have a small period ratio. The gravitation interaction between planets are less important though. The systems can show long-term stability even if the planets have low period ratios and are not in MMR. Characteristic of this class is the presence of numerous near-resonant pairs. 24

Figure 1.12: Schematic geometry of GJ 876 outer planets. Considering (Ω, ω) = (0, 0), the angles will represent the mean anomaly and thus the mean longitude. Notice how the angle ϕL = λ1 − 3λ2 + 2λ3 (inner to outer) remains constant and that the three planets are in conjunction at epoch. (Figure is based on Murray and Dermott, ﬁg 8.30.)

Class II: Signiﬁcantly secular dynamical non-resonant planets Planets that may not be in MMR, but have strong gravitational interactions between them. As the angular momentum conservation limits the eccentricity variations, they can remain stable even if not in a MMR. They present long-term variations which primarily described by secular perturbations and large variation of the eccentricities. Also they show interesting dynamical eﬀects such as alignment and anti-alignment of the apsidal lines.

Class III: Hierarchical planet pairs Hierarchical planet pairs are those that have no strong interactions between them. That is they follow a more unperturbed orbit. The hierarchical term indicates pairs with large period ratios. Large period ratios means that the probability of MMR capture is small. The interactions lead to long-period eccentricity variations, precise or almost uniform variation of apsidal angle and variations in the osculating orbital elements. The variation of ∆¯ω can be used to distinguish this class and the previous[2]. Specifically if ∆¯ω is oscillating in the range (0, π), or if it varies in a toryuous way, the system belongs to class II. If ∆¯ω varies almost uniformly the system belongs to class III. Another difference is the sensitivity to element variations. The orbit stability of class II is critically dependable on the parameters. Different choices of elements lead to either very unstable orbit or a stable one.

1.8 Numerical methods

In planetary systems the dynamics is that of a Hamiltonian system. But, as already shown the n-body problem cannot be solved analytically. An alternative method is solving the problem numerically. That is given some initial conditions, the orbit to be found algorithmically by continuous application of a step. Two generic methods and symplectic integration for planetary systems are presented. Be- sides them, other well known methods are Runge-Kutta, used for short integrations and Bulirsch- Stoed, used for longer integrations. Though those can be accurate when suitable steps are used, for predicting the stability or resonance of a system, the symplectic integrators presented are 25 preferred. That said when accuracy is needed or for when close encounters take place, such as in highly elliptic orbits, those can be used.

1.8.1 Generic Newton-Raphson Newton-Raphson is a root-finding algorithm, not an integrator, that is, a method for finding successively better approximations to the roots of a real-valued function. It requires though an analytical expression of the first derivative of the function. If a function f satisfies the assumptions made in the derivation of the formula and the initial guess is close, then the root can be approximated using the relation ′ xn+1 = xn − f(xn)/f (xn) until a sufficiently accurate value is reached. It is used for solving the Kepler’s equation for low elliptic orbits (e < 1). We define

f = E − e · sin E − M

Then ′ En+1 = En − f(En)/f (En)

A frequently used initial approximation is E0 = M + e sin M. Thus the Kepler’s equation may be solved as En − e · sin En − M En+1 = En − ,E0 = M + e sin M (1.6) 1 − e · cos En which can alternatively be written

M + e · (sin En − En cos En) En+1 = ,E0 = M + e sin M 1 − e · cos En For a review of methods and approximations used in solving Kepler’s equation the interested reader may refer to (Esmaelzadeh and Ghadiri, 2014)10.

Euler Euler method is a first-order procedure for solving ordinary differential equation given an initial value. Most ordinary differential equation integrators follow similar principles albeit with different procedure. Suppose the differential equation system

x˙ = f(x, y), y˙ = g(x, y)

An approximate solution then is

xn+1 = xn + τf(xn, yn), yn+1 = yn + τg(xn, yn) For a gravitational n-body system we can express position and velocity as

r˙i = vi, v˙i = g(ri) Given certain initial conditions and the previous method, we have

(n+1) (n) (n) (n+1) (n) j (n) 3 ri = ri + τvi , vi = vi + τGm rij /rij Although Euler’s method is computationally cheap, it doesn’t perform good for practical pur- poses, due to its inaccuracy compared to other methods.

10Reza Esmaelzadeh and Hossein Ghadiri. “Appropriate Starter for Solving the Kepler’s Equation”. In: Interna- tional Journal of Computer Applications 89.7 (2014). 26

1.8.2 Symplectic integration A symplectic integrator is a numerical integration scheme for Hamiltonian systems. They are a subcategory of geometric integrators; integrators that preserve some of the geometric properties (structure) of the phase-space ﬂow exactly. Conventional integrators try to make the position in phase space at the end of one timestep as close as possible to the true position; geometric integrators give up some of this accuracy to ensure that the overall properties of the ﬂow in phase space are the same in the integrated and the true trajectory11. Though there is no energy conserving methods applicable to non-integrable Hamiltonian systems, symplectic integrators exploit the phase space conservation and the energy error is bound.

Explicit The explicit method is based on the separation of Hamilton function. In the general case the Hamilton function H is the sum of two separable parts, T (p) and V (q). If we ignore the second part then ′ ′ q = q, p = p − τ∂qV And if we ignore the ﬁrst part then

′ ′ p = p, q = q + τ∂pT

Each of those maps is exact and symplectic. Our interest lies in their sum. We provide some necessary theory in order to find whether it is symplectic as well[9]. By defining the operator DS· = {·,S} where {·, ·} the (antisymmetric) Poisson bracket, the equations of motion, corresponding to Hamiltonian flow, are written and solved as

τDH z˙ = {z,H} = DH z ⇒ z(t) = e z0

Let A = DT and B = DV be the integrable parts of the Hamiltonian, and DH = DT + DV the sum of them. Then, the exact time evolution with step τ is given

z(t + τ) = eτDH z(t) = eτ(A+B)z(t)

Thus, the representation of the solution as as a sum of elementary maps doesn’t give the exact solution of the system. The exponential operator exp{τ(A + B)} can be approached by an integrator of k steps. The two elementary algebraic maps are[9] { } ciτA ′ ′ e = q = q + ciτp, p = p and { } diτB ′ ′ e = q = q, p = p − diτ∂qV (q) These maps give the exact solution for canonical equations generative of the function H˜ = H + O(τ 2). This result changes the study from numerical analysis to dynamics. The integrator shown preserves the energy of this new function but not the old. Thus the integrator bounds the energy error for the H to O(τ 2).

Modiﬁed Euler Euler’s modiﬁed method corresponds to the sequential application of aforementioned operators. The algorithm is the following

xn+1 = xn + τf(xn, yn), yn+1 = yn + τg(xn+1, yn)

11Scott Tremaine. ISIMA lectures on celestial mechanics. 2. 2014. URL: http://isima.ucsc.edu/2014/ presentations/lectures/Tremaine2.pdf. 27

For a gravitational interacting system, this turns into

q′ = q + τp′, p′ = p − τ∇V (q′)

The diﬀerence is that we’re using the newly computed value. This new map is symplectic as it can deduced from theory. An advantage for symplectic methods as can be seen is that they require no additional memory for storing intermediate results. This is very important for large n-body simulations, albeit not so much for planetary integrations. A limitation is that they work well only with ﬁxed timesteps.

Higher-order Because the symplectic integrator is symmetric, it is possible to construct higher order symplectic integrators using lower order ones as seen in (Yoshida, 1990)12. If a 2nd-order symmetric integrator, S2n(τ) is known, a (2n+2)-order integrator is obtained by the product

S2n+2(τ) = S2n(z1τ)S2n(z0τ)S2n(z1τ) where z0 and z1 satisfy 2n+1 2n+1 z0 + 2z1 = 1, z0 + 2z1 = 0 As such, a 6th order integrator can be written as the product of 4th-order integrators or nine 2nd-order ones, for a total of twenty-seven steps. The procedure can be simplified reducing the steps to fifteen. The first of the three coefficients sets is computed to be[9] ci≤4 = (0.3922568052387809, 0.5100434119184590, −0.4710533854097560, 0.0687531682525208)

ci>4 = c9−i (i = 5 ... 8) di≤4 = (0.7845136104775619, 0.2355732133593562, −1.1776799841788772, 1.3151863206839183)

di>4 = c8−i (i = 5 ... 7)

Mixed-variable and democratic heliocentric An alternative symplectic method was developed by taking advantage of the analytical solution in the unperturbed Keplerian problem. Those so-called MVS (mixed variable symplectic) methods are eﬃcient yet retain symplectic advantages. Based on theory presented in sec. 1.5.2, the Hamiltonian in Jacobi coordinates explained in A.2 is written[1]

′ ′ p 2 ∑N p 2 ∑N m m H(q, p) = 0 + i − G i j 2m′ 2m′ q 0 i=1 i i=0,j>i ij The center of mass moves as a free particle and its contribution can be ignored. Thus the Hamiltonian is written, by adding and subtracting an interaction quantity

H(q, p) = HK + HI where ( ) ′ ( ) ∑N p 2 m m ∑N m m m m ∑N m m H = i − G i 0 ,H = G i 0 − i 0 − G i j (1.7) K 2m′ q′ I q′ q q i=1 i i i=1 i i0 i=1,j>i ij For this Hamiltonian, a second-order single time step integrator can be constructed.

EI (τ/2)EK (τ)EI (τ/2)

12Haruo Yoshida. “Construction of higher order symplectic integrators”. In: Physics letters A 150.5-7 (1990), pp. 262–268. 28 with EK representing the Keplerian motion, and EI gravitational interactions. Since the coordinates constitute a canonical conjugate pair, the integrator is symplectic. The Keplerian motion can be evolved using the f and g functions given in sec. A.3. After Wisdom & Holman, the ones who ﬁrst constructed it in 1991, this is called the WH91 method. Unfortunately this method requires that the order of the bodies is maintained, and as such larger eccentricities cannot be handled. Moreover the implementation has a slight complexity requiring conversions between ordinary and Jacobi coordinates. By splitting the Hamiltonian diﬀerently we can lift this limitation, and produce an arguably easier to handle algorithm. Based on theory presented in sec. 1.5.2, the Hamiltonian in democratic heliocentric coordinates explained in sec. A.2 is written[1]

( ) 2 ∑N p2 m m p2 1 ∑N ∑N m m H(q, p) = i − G i 0 + 0 + p − G i j 2m q 2m 2m i q i=1 i i0 t 0 i=1 i=1,j>i ij

The center of mass moves as a free particle and its contribution can be ignored. Thus the Hamiltonian is written H(q, p) = HK + HS + HI where

( ) 2 ∑N p2 m m 1 ∑N ∑N m m H = i − G i 0 ,H = p ,H = −G i j (1.8) K 2m q S 2m i I q i=1 i i0 0 i=1 i=1,j>i ij

That is it is separated in astrocentric coordinates and barycentric momenta. For this Hamil- tonian, a second-order single time step integrator can be constructed. This integrator has the form ED(τ/2)EI (τ/2)EK (τ)EI (τ/2)ED(τ/2) with EK representing the Keplerian motion, ED the linear drift in position (corresponding to HS), and EI gravitational interactions among the bodies sans the central one. Since the coordinates constitute a canonical conjugate pair, the integrator is symplectic. As before, the Keplerian motion can be evolved using the f and g functions given in sec. A.3. As {HS,HI } = 0 the ordering of ED and EI does not aﬀect the result. Similarly for EK and EI as {HK ,HI } = 0. The Keplerian advance requires O(n) computations but interaction term requires O(n2) computations. As such we can reduce the total number of computations per timestep, by reversing the order of EK and EI . The integrators then take the form

EK (τ/2)EI (τ)EK (τ/2) and ED(τ/2)EK (τ/2)EI (τ)EK (τ/2)ED(τ/2)

Democratic heliocentric vs 6th-order T+V We compare the democratic heliocentric with the 6th-order T + V algorithm by integrating the HD 82943 two-planet system which we analyze in third chapter. We integrate the system using the democratic heliocentric algorithm, where we modify the Keplerian step, to constrain the angular momentum, as explained in sec. A.3. The relative error in energy remains less than 2e − 5 with average being 1e − 6, and in angular momentum less than 3e − 13 with average being 9e − 14. The absolute errors are less than 8e − 8 with average being 5e − 9 and less than 2e − 15 with average being 8e − 16 respectively. Then we integrate the system without aforementioned modiﬁcation. The relative error in energy remains less than 2e − 5 with average being 1e − 6, and in angular momentum less than 29

1e − 12 with average being 5e − 13. The absolute errors are less than 8e − 8 with average being 5e − 9 and less than 1e − 14 with average being 5e − 15 respectively. And finally we integrate with the 6th-order T + V algorithm. The relative error in energy remains less than 8e − 8 with average being 6e − 9, and in angular momentum less than 2e − 13 with average being 6e − 14. The absolute errors are less than 3e − 10 with average being 2e − 11 and less than 2e − 15 with average being 6e − 16 respectively. We plot the energy and angular momentum variation for the three methods. The relative errors can be seen in figures 1.13a and 1.13b and the absolute errors in figures 1.14a and 1.14b respectively. As expected the 6th-order algorithm has smaller energy variations. DH integrator’s (first variant) runtime and allocations were 16.5% and 5.7% respectively less13 than the 6th-order T+V integrator. It is reminded that computing the interaction forces is a O(n(n − 1)) algorithm. As the number of bodies increases this difference will increase as well, since the later (being 6th- order) computes more times the interaction forces and also (being T+V) includes yet another body (the star itself). Thus the 6th-order integrator runs slower but with potentially better quantitative results. Nevertheless, both show the same dynamic evolution and effects, that is the qualitative results remain unchanged. As such we have preferred to use the dynamic heliocentric integrator, since those variations can be neglected.

(a) Relative energy variation. (b) Relative angular momentum variation.

Figure 1.13: Relative variations for (blue) 6th-order T + V , (orange) democratic heliocentric integrator with no modiﬁed and (red) with modiﬁed Keplerian step.

13Loading the program in clean Julia session and timing using BenchmarkTools package. 30

(a) Absolute energy variation. (b) Absolute angular momentum variation.

Figure 1.14: Absolute variations for (blue) 6th-order T +V , (orange) democratic heliocentric integrator with no modiﬁed and (red) with modiﬁed Keplerian step. Chapter 2

Data analysis

We initially perform a simple statistical interpretation of the orbits of all found exoplanets. The following analysis is based on Voyatzis review article[10]. A number of similar studies have been done over the years1,2,3,4,5, each showing newer discoveries and their deviations from current models, allowing us to constraint planetary formation further. We will try to provide an updated report of the previous results alongside some new insights from recent discoveries. Using the catalog provided by Extrasolar Planets Encyclopedia6, up to 2 Sep 2019, a total of 4109 conﬁrmed exoplanets have been found and 2493 candidates. Our analysis will be constrained only on conﬁrmed planets.

2.1 Planets

A histogram of their discovery year is given on figure 2.1a. A histogram of the total number of known exoplanets is given on figure 2.1b. The color as shown in legend represents the detection method. The importance of transit discoveries becomes evident with the histogram on figure 2.2a. A large spike of transit discoveries is obvious on recent years. Until around 2012, the radial-velocity method was by far the most productive technique used to hunt planets. Now, most of discovered exoplanets have been detected through primary transit. Considering the recent successful space-based missions this should come as no surprise. The discovered planets are arranged in 3059 systems. The distribution of them is shown on the histogram of figure 2.2b. A number of 2392 systems contain only one planet, while the rest 1717 are arranged in 667 multi-planet systems. Of those 444 consist only of two planets. We plot the mass to period (days), and According to Kepler’s law similar distribution applies to semimajor axis. Also, we plot the planets mass to their radius. Those are shown on figure 2.3a and 2.4a. A relationship between mass and radius can be seen. We split the masses 0.62 into two regimes. For masses < 0.8MJ a power law fit gives R ∼ M , whereas for masses −0.01 2 < M < 13MJ a power law fit gives R ∼ M . The M-R relation has appeared in a number

1Geoﬀrey Marcy et al. “Observed properties of exoplanets: masses, orbits, and metallicities”. In: Progress of Theoretical Physics Supplement 158 (2005), pp. 24–42. 2Stephane Udry and Nuno C Santos. “Statistical properties of exoplanets”. In: Annu. Rev. Astron. Astrophys. 45 (2007), pp. 397–439. 3Isabelle Baraﬀe, Gilles Chabrier, and Travis Barman. “The physical properties of extra-solar planets”. In: Reports on Progress in Physics 73.1 (2009), p. 016901. 4Scott Tremaine and Subo Dong. “The statistics of multi-planet systems”. In: The Astronomical Journal 143.4 (2012), p. 94. 5Andrew W Howard. “Observed properties of extrasolar planets”. In: Science 340.6132 (2013), pp. 572–576. 6Jean Schneider et al. The Extrasolar Planets Encyclopaedia. 2018. URL: http://exoplanet.eu.

31 32

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Astrometry Microlensing Primary Transit TTV Astrometry Microlensing Primary Transit TTV Method Method Imaging Other Radial Velocity Timing Imaging Other Radial Velocity Timing

(a) Exoplanet discoveries per year. (b) Total discovered exoplanets per year.

Figure 2.1: Histograms related to discoveries. of studies the recent years7,8. Also figures 2.5a and 2.6a show the mass of found planets to their distance from Earth and the mass of the star to the semimajor axis of the planet. The colors represents the detection method as shown. The histogram of distances and star masses are given alongside the previous. Most of the massive exoplanets have been detected in orbit of stars with mass similar to our own. We notice the absence of long-period planets with small mass. All planets with small mass are in close distance to their star. Though a bias exists because of our methods, those results make a case that there could be a physical mechanism behind them, such that the creation of large planets is favored on long distances. That said, we also see that the vast majority of planets with large radii appear in similar distance (that is same period) from their star. Interesting is the map of planet size distribution between the transit and radial velocities methods. The Jupiter-sized discoveries are more through radial velocities and the Earth-sized discoveries are mainly through transit. The transit method favours exoplanets with short orbital periods and orbiting relatively close to their host star. As total brightness only needs to be observed, it is possible to detect planets far away with this method. The radial velocity signal is distance independent, but for high precision a high signal-to-noise ratio spectra is required. So it is generally used only for relatively nearby stars. The same can be said for the imaging method which detects massive exoplanets in close distance from Earth. In contrast the microlensing method detects far distant planets. Radial velocities works best when the exoplanet is large, the central star is medium sized and are close together. It is easier to detect planets around low-mass stars with this method, as (a) those stars are more affected by gravitational tug from planets and (b) low-mass main-sequence stars generally rotate relatively slowly. Fast rotation makes spectral-line data less clear and so spectroscopy more difficult. For that reason it very good on finding so-called ”hot Jupiters”; large planets close to the home star. Those giants weren’t naturally made there but instead moved there (migrated) during the initial stages of the creation of the system. Microlensing predominantly detects planets around low-mass stars. Giants are more rare to have short

7Sean M Mills and Tsevi Mazeh. “The Planetary Mass–Radius Relation and Its Dependence on Orbital Period as Measured by Transit Timing Variations and Radial Velocities”. In: The Astrophysical Journal Letters 839.1 (2017), p. L8. 8Dolev Bashi et al. “Two empirical regimes of the planetary mass-radius relation”. In: Astronomy & Astrophysics 604 (2017), A83. 33

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Radial Velocity 1 2 3 4 5 6 7 8 Primary Transit Method Planets (a) Exoplanets method of detection. (b) Planets distribution.

Figure 2.2: Histograms on detection and distribution. periods, that is smaller planets are more abundant for short periods. Long-period gas giants are also mostly discovered through radial velocity. Giants on far distance from their parent star are being detected through imaging. Those could have been formed around another star and moved in their present orbits at a later time. We plot the frequency distribution of the masses and periods, seen in figure 2.7a and 2.3b respectively. As it can be seen there are verified exoplanets that fall above the upper proposed limit. Those over 13MJ lie in the so-called “brown dwarf zone”. We note the sudden increase of planets after a specific period (accordingly the semimajor axis), whereas before there are nearly none. The exact reason behind this remain unknown. It could be because of tidal interactions or a large in radius Roche zone. Another explanation could be that solar winds force matter farther from the star. Considering planets ≤ 13MJ we find that occurrences vary inversely with mass. The count data specifically fits a model N ∼ exp(−1.5M) or N ∼ M −0.99, where M was the mean mass in each of n = 20 bins. The periods seem to have a peak at 18 JD. By splitting the periods to two regimes, for P < 18JD the counts follow the relation N ∼ P whereas for P > 18JD the counts follow the relation N ∼ 1/ log P . We also make histograms of radius and inclination, seen in figure 2.4b and 2.8b. The vast majority appears with small radii. Nevertheless this is biased, as most planets with known radius are detected through transit, which as said favors smaller planets. We plot a distribution of P − e of planets for which an estimation of those parameters has been found. The plot may been seen on figure 2.9a. The majority appear with short periods and so small semimajor axes. An obvious reason is that short orbital periods allow a planet to be more quickly detected. Moreover planets farther from their orbiting star have higher eccentricity. That makes their observation more difficult. It also explains the vastly higher number of observed single systems as only the planet closest to star is being observed. A plausible explanation of the previous could be the strong tidal effects that planets close to their stars experience. Those tend to make the orbit cyclic; that is one with low eccentricity. Farther away planets do not have such issues and instead follow a more unperturbed orbit. For that reason, giants are associated with a broad eccentricity distribution. By making a histogram of eccentricities seen on figure 2.9b, it can be deduced that most planets tend to have low eccentric orbits. Various theories have been proposed to explain the orbital eccentricities, but none is defini- tive. Mechanisms used are gravitational scattering or interplanetary perturbations, Those per- 34

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Figure 2.3: Mass to period plot and period histogram. turbations can arise from resonances or interactions with the protoplanetary disk. The occurrence of circular orbits may require special initial conditions. Common initial conditions may lead to interplanetary perturbations or the protoplanetary disk, leading to orbital ellipticities or even ejection. The exoplanets show that coplanar nearly circular orbits could perhaps represent an unperturbed low-entropy state for a planetary system. Nevertheless there is a a strong anticorrelation of orbital eccentricity with the number of planets (multiplicity) in a system, as seen in ﬁg. 2.10a and explained in depth in (Limbach and Turner, 2015)9. The distribution of orbital eccentricities as a function of multiplicity provides an important constraint for planetary system formation and evolution models.

2.2 Stars

A Hertzsprung–Russell diagram of host stars can been seen on figure 2.10b. As seen most of stars with a planetary system lie on the main sequence. Moreover most stars with detected exoplanets are with mass about that of Sun. An obvious bias exist here, as observations are made around stars that are considered to have a planetary system basing the existence of one on our own. Also most detected planets are in orbit of fairly young stars, which have a wider mass distribution than older stars which have mass around Sun’s. Another interesting note is that exoplanets are more frequent around stars with higher metallicity. To put it differently, the occurrence of planets is a sensitive function of metallicity of the host star. We may see this result by plotting the frequency of planets to the metallicity of the stars. This is given on 2.11b and was noticed by (Fischer et al, 2004)10 as well. There could be an observation bias as planets around lower metallicity stars have shallower spectra lines, so radial velocity, one of the most frequent used techniques, isn’t very effective.

9Mary Anne Limbach and Edwin L Turner. “Exoplanet orbital eccentricity: Multiplicity relation and the Solar System”. In: Proceedings of the National Academy of Sciences 112.1 (2015), pp. 20–24. 9A scatter plot of luminosity to temperature. 10Debra Fischer, Jeﬀ A Valenti, and Geoﬀ Marcy. “Spectral analysis of stars on planet-search surveys”. In: Symposium-International Astronomical Union. Vol. 219. Cambridge University Press. 2004, pp. 29–40. 35

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● Astrometry Microlensing Primary Transit TTV Method 1e−05 1e−03 1e−01 1e+01 Imaging Other Radial Velocity Timing Radii (RJup) (a) Plot of mass to radius. (b) Histogram distribution of radii.

Figure 2.4: Mass to radii plot and radii histogram.

2.3 Resonances

We now compute the orbital period ratio of all planet pairs for every system and compare them to ratio k2/k1, with coprime integers k1 + k2 ≤ 7, such that the divergence

d = |P1/P2 − k2/k1| is minimized. That is we find planet pairs with commensurable mean-motions. For d less than 5%, 3% and 1% we find respectively 839, 534 and 183 commensurable planet pairs. Their distribution to each value k2/k1 is shown in figure 2.12a, where we have order all values to be k1 > k2. Moreover, we find the order of ratio for every pair, derived from the following simple relationship k2 p = ⇒ q = k1 − k2 k1 p + q The distribution of each order for every divergence is shown in figure 2.12b. We see that the vast majority of pairs has a period ratio between 1 and 3 for their adjacent orbits. The most common ratios encountered for d < 5% and d < 3% were 1:2 and 1:3. For d < 1% all ratios between 1 and 3 for their adjacent orbits have same appearance frequency, but those in a higher order have a sharp decline. We also see that most of pairs with commensurable period ratios are in order 1 followed by 3, and the result is similar for every divergence. The low number of occurrences for large orbital period ratios suggest that most planets tend to have stable orbits for low ratios of adjacent orbits[3]. For d < 5% there is a planetary pair with 1:1 ratio (order 0). This is planets Kepler-132 b and c, with semi major axis 0.67 and 0.68 AU respectively. That is the semimajor axes of the two planets are almost equal, but the eccentricities and the position of each planet on its orbit at a certain epoch take different values. An 1/1 resonant system with large eccentricities and drag force included, migrates due to the drag force and finally can be trapped in a satellite orbit11. Unfortunately the system lacks mass and eccentricity data for numerical analysis. The multiple-planet systems discovered by the Kepler mission show an excess of planet pairs with period ratios close in commensurability for first-order resonances. The width of first-order

11John D. Hadjidemetriou and George Voyatzis. “The 1/1 resonance in extrasolar systems”. In: Celestial Me- chanics and Dynamical Astronomy 111.1 (Oct. 2011), pp. 179–199. 36

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1 10 100 1000 10000 Star distance (pc) 0 ● Astrometry Microlensing Primary Transit TTV Method 1 10 100 1000 10000 Imaging Other Radial Velocity Timing Star distance (pc) (a) Plot of mass to distance from Earth. (b) Histogram of planets distance from Earth.

Figure 2.5: Scatter plot and histogram for distances. resonances diverges in the limit of vanishingly small eccentricity12. As such, these planet pairs could have both resonance angles librating if the orbital eccentricities are sufficiently small. We now split the previous distributions in two cases, where the orbit of the dominant body in the pair (a) being ”in” and (b) being ”out”, where ”in” and ”out” denote whether is ”inner” or ”outer” in the orbit, respectively compared to the perturbed body. The dominant body in the pair is being considered the ”perturber”. In both cases the orbital period of the outer body is To and of the inner body is Ti. The distribution of orbital ratios may be seen on 2.13a and 2.14a and that of ratio orders on on 2.13b and 2.14b, for the first and second case respectively. As it can be seen the majority of planet lack mass information and as such the data shown of these figures is far less than previously. From what we have we see that most commensurable ratios are found with the dominant body being out. Also the most frequent commensurabilities now are 1 : 2 and 2 : 3, with 1 : 3 and 1 : 4 following. Most frequent orders seen still remain 1 and 3. The abundance of 2:1 and 3:1 commensurabilities can perhaps be explained dynamically. Specifically13, the dynamics of 2:1 and 3:1 resonant capture are related with the periodic orbits of the three-body problem. When the system is captured in resonance, the migration continues and the phase space evolution is driven by families of stable periodic orbits. Finally, the migration ends in a stable resonant planetary configuration. As a result both 2:1 and 3:1 are very probable resonances. The multi-planet systems for d ≤ 3% percent and small integer ratio (k1 + k2 ≤ 7) are given as table at the end of the thesis. The presence of a near resonance may reflect that a perfect resonance existed in the past, or that the system is evolving towards one in the future. This table will be of use in the next part of the thesis.

12Man Hoi Lee, D Fabrycky, and DNC Lin. “Are the kepler near-resonance planet pairs due to tidal dissipation?” In: The Astrophysical Journal 774.1 (2013), p. 52. 13George Voyatzis. “Resonant capture of multiple planet systems under dissipation and stable orbital conﬁgura- tions”. In: The European Physical Journal Special Topics 225.6-7 (2016), pp. 1071–1086. 37

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(a) Plot of star mass to semimajor axis. (b) Plot of star mass to planet mass.

Figure 2.6: Plot of star mass to semimajor axis and planetary masses.

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Figure 2.7: Histograms for masses and host stars masses. 38

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Figure 2.8: Eccentricities to inclinations plot and histogram for inclinations.

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Figure 2.9: Eccentricities to periods plot and histogram for eccentricities. 39

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Figure 2.11: Histograms for host star characteristics. 40

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Figure 2.12: Histograms for commensurable period ratios.

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Figure 2.13: Histograms for commensurable period ratios (dominant mass in). 41

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Figure 2.14: Histograms for commensurable period ratios (dominant mass out). 42 Chapter 3

Numerical integrations

In the previous chapter we ended by making a catalog of all those planetary systems that have pairs whose orbital periods have been found to be close to an integer ratio. That orbital periods have been found to closely match a integer ratio though doesn’t imply resonance. For whether a pair is in resonance, or not, a dynamical evolution of the entire system must be made. That is what we are doing below using a symplectic integrator developed according to theory presented. Note that in our integrations, we are making use of canonical units. That is, the masses of the star and the planets of the system are normalized so as the mass of the star will be equal to 1 and the semi-major axes a of the planets are also normalized so as the distance from the star of a reference planet will be equal to unity. The reference planet is always chosen to be the innermost one in the system. Also the gravitational constant is fixed to unity, which results in the gravitational parameter be fixed to unity. This defines a unit of time where one period of the reference planet is 2π.

3.1 HD 82943

HD 82943 is a G0 star of mass M = 1.18 solar masses at distance d = 22.46 pc from us. This stars harbors a two-planet system which is a well-known resonant pair studied the recent decade. The planets and their corresponding orbital elements are given in the following table, taken from a coplanar inclined dynamical ﬁt1.

Planet m (MJup) a (AU) e i (deg) Ω (deg) ω (deg) M (deg) b 4.78 0.7423 0.425 19.4 0 133 256 c 4.8 1.1866 0.203 19.4 0 107 333

Table 3.1: Orbital elements for HD 82943.

The periods are 217.95 and 440.51 JD respectively. Time is set to 1e5 years and time step to τ = 0.05 the period of the inner planet. We plot ai, ei, ∆¯ω and ϕi for planets

• b and c, with Pb/Pc = 1/2, which can been seen on ﬁgures 3.1 and 3.2. It is reminded that the critical arguments using eqn. 1.4 are

′ (ϕ, ϕ ) = (ϕ1, ϕ2) = ϕi = 2λ2 − λ1 − ω¯i

We observe libration of the resonant angles ∆¯ω and ϕi about 0°, with amplitudes δ∆¯ω ≈ ′ 50°, δϕ ≈ 23°, δϕ ≈ 45°. As both ϕi librate the planets are in mean-motion resonance, and moreover that is a corotation resonance. Thus the system can be considered as stable. 1Xianyu Tan et al. “Characterizing the Orbital and Dynamical State of the HD 82943 Planetary System With Keck Radial Velocity Data”. In: (2013). eprint: arXiv:1306.0687.

43 44

Figure 3.1: Evolution of a, e and period ratio for HD-82943 planets b and c.

Figure 3.2: Evolution of ∆¯ω, ϕ and ϕ′ for HD-82943 planets b and c. 45

3.2 GJ 9827

GJ 9827 is a K6V star of mass M = 0.65 solar masses at distance d = 30.3 pc from us. The planets and their corresponding orbital elements are given in the following table, determined with joint analysis of photometric and RV data2. The system was only recently discovered and as such e, Ω, ω, M are still not known.

Planet m (MJup) a (AU) e i (deg) Ω (deg) ω (deg) M (deg) b 0.011767386 0.021 0.01 88.33 NA NA NA c 0.0046251492 0.0439 0.01 89.07 NA NA NA d 0.0074883368 0.0625 0.01 87.703 NA NA NA

Table 3.2: Orbital elements for GJ 9827.

The periods are 1.3974, 4.2235 and 7.1746 JD respectively. We analyze a near-circular case, setting for all planets (e, Ω) = (0.01, 0). Circularization isn’t arbitrarily chosen, as short-period multi-planetary systems tend to have circular orbits3. We integrate all (ω, M) conﬁgurations such that ω, M ∈ {0, π}. Considering the geometry of the system, we set (ω, M) = (0, 0) for planet b, and then we permute the rest two. Changing the (ω, M) variables of planet b will result in conﬁguration with symmetry identical to one already computed. So we reduce the 3 2 number of integrations from n3 = 4 = 64 to n2 = 4 = 16. Time is set to 1e5 years and time step to τ = 0.05 the period of the inner planet.

(ωc,Mc)(ωd,Md) Pb/Pc ϕbc,i Pc/Pd ϕcd,i ϕL ∆¯ωbc ∆¯ωcd (0,0) (0,0) const. circ. const. circ. circ. libr. libr. (0,0) (0,π) const. circ. const. circ. circ. libr. libr. (0,0) (π,0) const. circ. const. circ. circ. circ. circ. (0,0) (π,π) const. circ. const. circ. circ. circ. libr. (0,π) (0,0) const. circ. const. circ. circ. libr. libr. (0,π) (0,π) const. circ. const. circ. circ. libr. libr. (0,π) (π,0) const. circ. const. circ. circ. circ. circ. (0,π) (π,π) const. circ. const. circ. circ. circ. libr. (π,0) (0,0) const. circ. const. circ. circ. circ. circ. (π,0) (0,π) const. circ. const. circ. circ. circ. libr. (π,0) (π,0) const. circ. const. circ. circ. circ. circ. (π,0) (π,π) const. circ. const. circ. circ. circ. circ. (π,π) (0,0) const. circ. const. circ. circ. circ. circ. (π,π) (0,π) const. circ. const. circ. circ. circ. libr. (π,π) (π,0) const. circ. const. circ. circ. circ. circ. (π,π) (π,π) const. circ. const. circ. circ. circ. circ.

For every conﬁguration the system remains stable and period ratios constant (Pb/Pc = 1 : 3,Pc/Pd = 3 : 5). For some cases secular resonances are found. We see though that none of those pairs are in mean-motion resonance, as the resonant arguments ϕi and ϕL (periods 1:3:5) circulate. So the system appears to be in a near-resonant state. It also appears stable regard- less on initial conditions. Something to be expected as small eccentricities and masses mean small gravitational interactions among the planets, and the system’s dynamics are completely dominated by the star.

2J. Prieto-Arranz et al. “Mass determination of the 1:3:5 near-resonant planets transiting GJ 9827 (K2-135)”. In: (2018). eprint: arXiv:1802.09557. 3Vincent Van Eylen and Simon Albrecht. “Eccentricity from transit photometry: small planets in Kepler multi- planet systems have low eccentricities”. In: The Astrophysical Journal 808.2 (2015), p. 126. 46

3.3 61 Vir

61 Vir is a G5V star of mass M = 0.95 solar masses at distance d = 8.52 pc from us. The planets and their corresponding orbital elements are given in the following table, taken from a coplanar inclined Newtonian ﬁt4.

Planet m (MJup) a (AU) e i (deg) Ω (deg) ω (deg) M (deg) b 0.016 0.0502 0.12 90.0 0 105 166 c 0.0576 0.2175 0.14 90.0 0 341 177 d 0.0721 0.476 0.35 90.0 0 314 56

Table 3.3: Orbital elements for 61 Vir.

The periods are 4.2719, 38.52 and 124.73 JD respectively. Time is set to 1e6 years and time step to τ = 0.05 the period of the inner planet. We plot ai, ei, ∆¯ω and ϕi for planets

• c and d, with Pc/Pd = 1/3, which can been seen on ﬁgures 3.3 and 3.4.

Figure 3.3: Evolution of a, e and period ratio for 61-Vir planets c and d.

Figure 3.4: Evolution of ∆¯ω, ϕ and ϕ′ for 61-Vir planets c and d.

We observe libration of the secular angle ∆¯ω about 0.7°, with amplitude 38.3°. The resonant angles ϕi circulate and thus the pair is not in mean-motion resonance. A noticeable variation of eccentricities can also been seen. Nevertheless the period ratios seem constant and the system appears stable in the studied time age.

4Steven S Vogt et al. “A super-Earth and two Neptunes orbiting the nearby Sun-like star 61 Virginis”. In: The Astrophysical Journal 708.2 (2009), p. 1366. 47

3.4 HR 8799

HR 8799 is a A5V star of mass M = 1.56 solar masses at distance d = 39.4 pc from us. The planets and their corresponding orbital elements are given in the following table, taken from a four-planet coplanar ﬁt5. It is an observationally well-studied system of directly-imaged planets6.

Planet m (MJup) a (AU) e i (deg) Ω (deg) ω (deg) M (deg) e 9 15.4 0.13 25 64 176 326 d 9 25.4 0.12 25 64 91 58 c 9 39.4 0.05 25 64 151 148 b 7 69.1 0.02 25 64 95 321

Table 3.4: Orbital elements for HR 8799.

The periods are 17912 (5̃0yr), 37942 (1̃05yr), 73302 (2̃05yr) and 170251 (4̃70yr) JD respectively. Time is set to 1e5 years and time step to τ = 0.05 the period of the inner planet. We plot ai, ei, ∆¯ω and ϕi for planets

• e and d, with Pe/Pd = 1/2, which can been seen on ﬁgures 3.5 and 3.6.

Figure 3.5: Evolution of a, e and period ratio for HR-8799 planets e and d.

Figure 3.6: Evolution of ∆¯ω, ϕ and ϕ′ for HR-8799 planets e and d.

• d and c, with Pd/Pc = 1/2, which can been seen on ﬁgures 3.7 and 3.8. 5Krzysztof Gozdziewski and Cezary Migaszewski. “Multiple mean motion resonances in the HR 8799 planetary system”. In: (2013). eprint: arXiv:1308.6462. 6Thayne Currie. HR 8799: The Benchmark Directly-Imaged Planetary System. 2016. eprint: arXiv:1607. 03980. 48

Figure 3.7: Evolution of a, e and period ratio for HR-8799 planets d and c.

Figure 3.8: Evolution of ∆¯ω, ϕ and ϕ′ for HR-8799 planets d and c.

• c and b, with Pc/Pb = 1/2, which can been seen on ﬁgures 3.9 and 3.10.

Figure 3.9: Evolution of a, e and period ratio for HR-8799 planets c and b.

We see that the period for subsequent pairs remain constant, and the respective resonant ar- ′ guments librate. Speciﬁcally the arguments ϕ(ed), ϕ(ed), ϕ(dc), librate about 342.3°, 72.5°, 1.6°, with amplitudes 10.5°, 14.9°, 17.9° respectively. We plot the Laplace argument for the three ﬁrst (e,d,c) and last (d,c,b) planets. In both cases MMR 1:2:4, and thus the Laplace arguments are given

ϕL = λ1 − 3λ2 + 2λ3

ϕL = λ2 − 3λ3 + 2λ4 49

Figure 3.10: Evolution of ∆¯ω, ϕ and ϕ′ for HR-8799 planets c and b. given in ﬁgures 3.11a and 3.11b. Both arguments librate showing a Laplace resonance for those planets. Speciﬁcally they librate about 289.2°, 78.4° with amplitudes 15.9°, 21.2° respectively. We consider the four-body resonant argument for a four-planet 1:2:4:8 MMR

ϕ = λ1 − 2λ2 − λ3 + 2λ4

A plot over the integration duration is given in 3.11c. The critical argument librates about 7.5° with amplitude 19.8°. As such the system is locked deeply in a four-body mean-motion resonance.

Figure 3.11: Evolution of Laplace and four-planet resonant arguments for HR-8799.

We explore whether an non-fit aligned four-planet resonance using the same elements a, e, i is possible. Specifically by setting (the inner) planet e as reference choosing (¯ωe,Me) = (0, 0), we integrate all cases (¯ωj,Mj) ∈ {0, π} such as λ1 − 2λ2 − λ3 + 2λ4 = 0 (32 in total). For all cases the system is unstable and quickly disbands. We also explore whether an anti-aligned four-planet resonance using the same elements a, e, i is possible. Specifically by setting (the inner) planet e as reference choosing (¯ωe,Me) = (0, 0), we integrate all cases (¯ωj,Mj) ∈ {0, π} such as |λ1 − 2λ2 − λ3 + 2λ4| = π (32 in total). For all cases the system is unstable and quickly disbands. We conclude that the stability of HR 8799 is very dependent on the initial conditions. This shows that the resonance of the previous fit provides a phase protection to the four planets. We note though the used parameters are given with a very high error. 50

3.5 TRAPPIST-1

Next we perform the dynamical evolution of a very interesting system. TRAPPIST-1, first discovered in 1999, is an M8V star of M = 0.09 solar masses and distance d = 12.1 pc from us. Seven exoplanets have been found to orbit this ultra-cool red dwarf, with five of them to potentially be in the optimistic habitable zone. It should be noted though that all seven planets are likely tidally locked and thus life development is challenging. Besides the system’s interest in planetary science, it is also interesting from dynamical prospective. The orbital motions form a chain of near- and mean-motion resonances, as we shall shortly see. The planets and their corresponding orbital elements are given in the following table, computed with an evolution method7. We consider a coplanar case by setting (i, Ω) = (90, 0). Coplanarity isn’t arbitrarily chosen, as it is found the seven planets deviate by the sky plane by < 0.1 degrees8. Thus neglecting the mutual inclination is justified.

Planet m (MJup) a (AU) e i (deg) Ω (deg) ω (deg) M (deg) b 0.0031998481 0.01154775 0.00622 90.0 0 336.86 203.12 c 0.0036371922 0.01581512 0.00654 90.0 0 282.45 69.860 d 0.0009344689 0.02228038 0.00837 90.0 0 351.27 173.92 e 0.0024289899 0.02928285 0.00510 90.0 0 108.37 347.95 f 0.0029387002 0.03853361 0.01007 90.0 0 368.81 113.61 g 0.0036120213 0.04687692 0.00208 90.0 0 191.34 265.08 h 0.0010414452 0.06193488 0.00567 90.0 0 338.92 269.72

Table 3.5: Orbital elements for TRAPPIST-1.

The periods are 1.6242, 2.6032, 4.3529, 6.5586, 9.9003, 13.284 and 20.174 JD respectively. Time is set to 1e6 years and time step to τ = 0.05 the period of the inner planet. From the computed values in our catalog, we have the ratios for Pd/Pe,Pe/Pf ,Pf /Pg and from the others we derive the ratios for Pb/Pc,Pc/Pd,Pg/Ph. We plot ai, ei, ∆¯ω and ϕi for planets

• b and c, with Pb/Pc = 5/8, which can been seen on ﬁgures 3.12 and 3.13.

Figure 3.12: Evolution of a, e and period ratio for TRAPPIST-1 planets b and c.

• c and d, with Pc/Pd = 3/5, which can been seen on ﬁgures 3.14 and 3.15.

• d and e, with Pd/Pe = 2/3, which can been seen on ﬁgures 3.16 and 3.17.

• e and f, with Pe/Pf = 2/3, which can been seen on ﬁgures 3.18 and 3.19.

7Simon L. Grimm et al. “The nature of the TRAPPIST-1 exoplanets”. In: (2018). eprint: arXiv:1802.01377. 8Michaël Gillon et al. “Seven temperate terrestrial planets around the nearby ultracool dwarf star TRAPPIST-1”. In: Nature 542.7642 (2017), p. 456. 51

Figure 3.13: Evolution of ∆¯ω, ϕ and ϕ′ for TRAPPIST-1 planets b and c.

Figure 3.14: Evolution of a, e and period ratio for TRAPPIST-1 planets c and d.

• f and g, with Pf /Pg = 3/4, which can been seen on ﬁgures 3.20 and 3.21.

• g and h, with Pg/Ph = 2/3, which can been seen on ﬁgures 3.22 and 3.23.

We see that the period for subsequent pairs remain constant. The resonant arguments for the first three planets circulate, whereas the resonant arguments for the last five planets librate. ′ ′ ′ Specifically the arguments ϕ(de), ϕ(ef), ϕ(ef), ϕ(fg),ϕ(fg), ϕ(gh) librate about 180.4°,29.3°,167.2°, 9.1°,175.6°,354.8°. The amplitudes are 72.8°,62.4°,48.8°, 49.2°,80.4°,73.7° respectively. So the last five planets seem to form a resonant chain. We plot the temporal evolution of the three-body resonant angles for consecutive triples of

Figure 3.15: Evolution of ∆¯ω, ϕ and ϕ′ for TRAPPIST-1 planets c and d. 52

Figure 3.16: Evolution of a, e and period ratio for TRAPPIST-1 planets d and e.

Figure 3.17: Evolution of ∆¯ω, ϕ and ϕ′ for TRAPPIST-1 planets d and e. planets. The coeﬃcients (p, q) are (3,2), (2,1), (3,2), (2,1) and (1,1) respectively9, and thus the resonant angles, using eqn. 1.5, are

ϕL(bcd) = 2λ1 − 5λ2 + 3λ3

ϕL(cde) = λ1 − 3λ2 + 2λ3

ϕL(def) = 2λ1 − 5λ2 + 3λ3

ϕL(efg) = λ1 − 3λ2 + 2λ3

ϕL(fgh) = λ1 − 2λ2 + λ3

9Rodrigo Luger et al. “A seven-planet resonant chain in TRAPPIST-1”. In: (2017). eprint: arXiv:1703.04166.

Figure 3.18: Evolution of a, e and period ratio for TRAPPIST-1 planets e and f. 53

Figure 3.19: Evolution of ∆¯ω, ϕ and ϕ′ for TRAPPIST-1 planets e and f.

Figure 3.20: Evolution of a, e and period ratio for TRAPPIST-1 planets f and g.

It can been seen that all consecutive triples of planets are in three-body resonance. Speciﬁcally the arguments librate about 179.3°,7.9°,208.9°,280.9°,179.8°. The amplitudes are 57.8°,86.5°, 19.5°,5.5°,5.4° respectively. It is of interest to see that b, c, d constitute a resonant triple, considering that b,c and c,d are not resonant pairs. As we already mentioned in 1.7.2, three-body resonances exist even if there are no individual two-body resonant pairs involved. From the amplitude variation we conclude that the outer planets showing small variations are deeper in resonance than the inner planets that show large variations.

Figure 3.21: Evolution of ∆¯ω, ϕ and ϕ′ for TRAPPIST-1 planets f and g. 54

Figure 3.22: Evolution of a, e and period ratio for TRAPPIST-1 planets g and h.

Figure 3.23: Evolution of ∆¯ω, ϕ and ϕ′ for TRAPPIST-1 planets g and h. 55

(a) For planets b,c και d. (b) For planets c,d and e. (c) For planets d,e and f.

Figure 3.24: Evolution of ϕL for TRAPPIST-1 inner planets.

(a) For planets e,f and g. (b) For planets f,g and h.

Figure 3.25: Evolution of ϕL for TRAPPIST-1 outer planets. 56 Appendix A

Transformations

A.1 Orbital elements

The procedure is based on methods found in (Franz, 2002)1 and (Schwarz, 2017)2. The orientation of an orbit to the ecliptic and origin at the orbital focus can be deﬁned by the inclination, the longitude of the ascending node and argument of periapsis. The orbital frame has its X-axis from the focus to the periapsis and its Z-axis perpendicular to the orbital plane. In this system the position and velocity vectors are given √ r = r(cos ν, sin ν, 0), v = µ/p(− sin ν, e + cos ν, 0)

The following algorithm may be used to convert the orbital elements to position and velocity vectors. Let a, e, i, Ω, ω, M be the orbital elements deﬁning the orbit. We start by solving the Kepler’s equation numerically using 1.6.

En − e · sin En − M En+1 = En − ,E0 = M + e sin M 1 − e · cos En Then we ﬁnd the distance to the central body.

r = a(1 − e cos E)

Then the coordinates in the orbital frame. The equation given previously can be expressed directly by the eccentric anomaly. [ √ ] 2 rof = a cos E − e, 1 − e sin E, 0 √ [ √ ] −1 2 vof = r µa − sin E, 1 − e cos E, 0

We transform those coordinates to the reference frame with the Eulerian rotation.

(r, v) = R(Ω, i, ω)(rof, vof),R(Ω, i, ω) = Rz(Ω)Rx(i)Rz(ω) where     1 0 0 cos ϕ − sin ϕ 0     Rx(ϕ) = 0 cos ϕ − sin ϕ Rz(ϕ) = sin ϕ cos ϕ 0 0 sin ϕ cos ϕ 0 0 1

1M Fränz and D Harper. “Heliospheric coordinate systems”. In: Planetary and Space Science 50.2 (2002), pp. 217–233. 2René Schwarz. Cartesian State Vectors → Keplerian Orbit Elements. Oct. 2017. URL: https://downloads. rene-schwarz.com/download/M002-Cartesian_State_Vectors_to_Keplerian_Orbit_Elements.pdf.

57 58

The following algorithm may be used to translate those vectors back to to the orbital elements. We start by calculating the orbital angular momentum vector and the vector n pointing towards the ascending node. h = r × r˙, n = [−hy, hx, 0] Then we obtain the eccentricity vector. The eccentricity is the magnitude of it. r˙ × h r e = − ⇒ e = |e| µ r The semi-major axis is found right away from the vis viva equation. ( ) −1 a = 2/r − |r˙|2/µ

And the orbit inclination, from the angular momentum vector, is

i = acos(hz/h).

The following apply only in the case where e > 0 and n > 0. For our usage they are suﬃcient. We obtain the longitude of the ascending node, the argument of periapsis and the true anomaly. e · n e · r Ω = acos(n /n), ω = acos , ν = acos x en er The following corrections are performed if applicable.

′ ′ ′ ny < 0 → Ω = 2π − Ω, ez < 0 → ω = 2π − ω, v · r < 0 → ν = 2π − ν

Then the eccentric anomaly is tan ν/2 E = 2 atan √ (1 + e)/(1 − e) and the mean anomaly can be computed using the the Kepler’s equation.

M = E − e sin E

A.2 Coordinate systems

In celestial mechanics there are various coordinate system in use. We provide a short review based on (Bazsó, 2016)3. For each system there are associated generalized orbital elements. Supposedly a system of N + 1 masses mi, i = 0 ...N. The inertial system has position and velocity vectors (Xi, X˙ i). We deﬁne the generalized linear momentum as pi = miX˙ i, and the generalized coordinates as qi = Xi. The Hamiltonian system has an equation H(q, p), and the canonical variables are (p, q).

Barycentric They are used for extra-solar system studies. Speciﬁcally upon those coordinates are based the radial velocity and astrometry detection methods. We deﬁne the barycenter or center of mass of the system as 1 ∑N ∑N X = m X ,M = m bc M i i i i=0 i=0

3Ákos Bazsó. On the use of diﬀerent coordinate systems in Celestial Mechanics. 2016. URL: https://www. univie.ac.at/adg/Teaching/astsemws16/bazso.pdf (visited on 2018). 59 where M is the the total mass. Then the barycentric vectors are the inertial vectors with origin shifted to Xbc. bi = Xi − Xbc Thus, the Hamiltonian of the system is written as

∑N |p |2 ∑N m m H = T (p) + V (q) = i − G i j 2m q i=0 i i,j=0,j>i ij

where is G is the gravitation constant and qij = qi − qj .

Astrocentric (or Heliocentric)

Those are typically used for solar system studies. They can be found in classical perturbation theory (Laplace-Lagrange). The heliocentric vectors are the inertial vectors with origin shifted to X0, the position of the host star. hi = Xi − X0

It should be remarked that (pi, qi) = (mir˙i, ri) do not constitute a canonical set of variables. The Hamiltonian of the system is written as

′ H = H0 + H

′ where H0 is the integrable 2-body part, and H a small perturbation.

Jacobi

They are used for studying the hierarchical 3-body problem[5], and are also used in mixed variable symplectic integration methods. The coordinates are given

− 1 ∑i 1 ∑i ji = Xi − mjXj, σi = mj σ − i 1 j=0 j=0 where σi is the partial sum of masses. That is they make a tree, where the reference point for each body is the barycenter of the interior (inner) to it bodies. It is like the mi body moving around a body of mass σn−1. The Hamiltonian of the system is written as

′ H = H0 + H

′ where H0 is the unperturbed part and H the interaction part.

c c c

d d r2 d 2 3 r r2 r3 r r3 r1 r1 b b r1 b a a a

Figure A.1: Coordinate systems: heliocentric (left), barycentric (middle) and Jacobi (right). (Figure is based on Bazsó.) 60

Poincaré Their main use comes in symplectic integration methods. They are also called democratic or canonical heliocentric coordinates, Though they are not conceptually based on elliptical motion, the analytical studies can be cleaner as the Hamiltonian has a clear symmetry[8]. The positions are in astrocentric and the velocities are in barycentric coordinates respectively. Speciﬁcally ˙ ˙ qi = Xi − X0, pi = Xi − Xbc The Hamiltonian of the system is written as ′ H = H0 + H ′ where H0 is the unperturbed part and H the interaction part.

d r2 v3 v2

r3 v1 b

a r1 Figure A.2: Coordinate systems: democratic heliocentric, where v is the radius upon which velocities are regarded.

A.3 Gauss’ method

Any planar motion that is any vector in the plane can be written as the linear combination of r0 and v0. r = fr0 + gv0, v = f˙r0 +g ˙v0 The meaning being that if positions and velocities of a body in a Keplerian field are known, then the positions and velocities in a later time can be found in term of initial values. The f and g are known as Gauss’ functions. They’re also referred to as Lagrange’s or Gibb’s coefficients. It can be proved that their analytical form is √ ( ) ( ) a a3 f = 1 − 1 − cos E˜ , g = (t − t0) + sin E˜ − E˜ r0 µ and respectively √ ( ) µa a f˙ = − sin E,˜ g˙ = 1 − 1 − cos E˜ rr0 r with √ r = a + (r0 − a) cos E˜ + aσ0 sin E˜ ˜ − where E = E E0, the change in eccentric anomaly, µ the gravitational√ parameter, a the semimajor axis, r and r0 the normal of the radial vector. Also, σ0 = r0 · v0/ µ. The previous forms are in terms of eccentric anomaly, but they are frequently given in terms of true anomaly as well. Those analytical solutions can be used to advance a Keplerian orbit. The E˜ is found by numerically solving the (modified Kepler’s) equation given4

−1/2 E˜ − (1 − r0/a) sin E˜ − a σ0(cos E˜ − 1) = n(t − t0)

4Puneet Singla. Lagrange/Gibbs F and G solutions. 2006. URL: http://www.eng.buffalo.edu/~psingla/ Teaching/CelestialMechanics/Lectures_ClassicalF&GSoln.pdf (visited on 2018). 61

The NR method converges rapidly and as such an initial guess can be E˜ = π. The method given is for bound orbits, that is elliptical motion. An alternative method, involving a solution of Kepler’s equation in universal variables, lifts this limitation. This is slightly more complex and isn’t required in our problem. The Keplerian evolution step can introduce precision errors. This is a source of error that can cause a systematic radial drift that does not average to zero. To minimize errors during the Kepler advances we can chose f and g functions that maintain angular momentum conservation across each evolution step. Speciﬁcally the angular momentum after a step is

L = r × v = (fg˙ − gf˙)(r0 × v0) = (fg˙ − gf˙)L0

Utilizing the conservation of angular momentum we can reduce the inward radial drift.

L = L0 ⇒ fg˙ − gf˙ = 1 ⇒ g˙ = (1 + gf˙)/f 62 Appendix B

Sources

B.1 Chapter 2

The source code of the script used to ﬁnd the orbital ratios can be found here. This includes the part which made our catalog. The program is written in R, a programming language for statistical computing1, and the program was used with version 3.5.2.

Listing B.1: R program used for ﬁnding and printing orbital resonances. 1 library("compiler") 2 enableJIT(3) 3 4 data <- read.csv('data/exoplanet.eu_catalog.csv') 5 df <- subset(data , planet_status == 'Confirmed') 6 d <- droplevels(subset(df, star_name != '')) 7 8 ## options 9 pqmax <- 7 10 dvp <- c(0.01, 0.03, 0.05) 11 12 ## functions 13 make.fracs.max <- function(pqmax) { 14 fracs <- do.call(rbind , sapply(1:(pqmax - 1), 15 function(i) 16 t(sapply(1:(pqmax - i), 17 function(j) 18 c(i, j, i / j))))) 19 fracs[!duplicated(fracs[, 3]),] 20 } 21 make.fracs.per <- function(op) { 22 lst <- sapply(1:(length(op) - 1), 23 function(i) 24 t(sapply((i + 1):length(op), 25 function(j) 26 c(i, j, op[i] / op[j])))) 27 if (length(op) == 2) { 28 t(lst) 29 } 30 else { 31 do.call(rbind , lst) 32 } 33 }

1R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing. Vienna, Austria, 2018. URL: https://www.R-project.org/.

63 64

34 make.fracs.all <- function(dat, stars){ 35 dat <- dat[!is.na(dat$orbital_period), ] 36 stsys <- table(dat$star_name) 37 fraca <- list() 38 for (star in names(stsys[stsys > 1])) { 39 op <- dat[dat$star_name == star, ]$orbital_period 40 fr <- make.fracs.per(op) 41 fraca[star] <- list(data.frame(na.omit(fr))) 42 } 43 fraca 44 } 45 check.fracs <- function(fr, dv, system){ 46 for (i in 1:nrow(system)){ 47 dvn <- dv 48 for (echa in as.numeric(fr[, 3])) { 49 cn <- as.numeric(system[i, 3]) - echa 50 if (abs(cn) < dvn) { 51 system[i, 3] <- echa 52 dvn <- abs(cn) 53 } 54 } 55 system[i, 3] <- 56 paste0(fr[fr[, 3] == system[i, 3]][1:2], collapse = ":") 57 system[i, 4] <- dvn 58 } 59 system[!system[, 3] == "NA:NA",] 60 } 61 find.pairs <- function(maxfr, dv, fraca) { 62 pairs <- list() 63 for (star in names(fraca)) { 64 fr <- check.fracs(maxfr, dv, fraca[[star]]) 65 if (nrow(fr) > 0) { 66 pairs[star] <- list(data.frame(na.omit(fr))) 67 sys.x <- pairs[[star]] 68 sys.d <- d[d$star_name == star, ] 69 for (i in 1:nrow(sys.x)) { 70 pairs[[star]][i, 5] <- sys.d[sys.x[i, 1], ]$semi_major_axis 71 pairs[[star]][i, 6] <- sys.d[sys.x[i, 2], ]$semi_major_axis 72 pairs[[star]][i, 7] <- sys.d[sys.x[i, 1], ]$mass 73 pairs[[star]][i, 8] <- sys.d[sys.x[i, 2], ]$mass 74 } 75 } 76 } 77 pairs 78 } 79 print.pairs <- function(dat, dv, pairs){ 80 sink("out/catalog.csv") 81 for (star in names(pairs)){ 82 sys <- subset(dat, star_name == star) 83 tp <- nrow(sys) 84 dfs <- pairs[[star]] 85 dfs <- subset(dfs, dfs$V4 <= dv) 86 pn <- 87 sapply(1:tp, function(p) 88 trimws(gsub(star , '', sys$X..name[p], fixed = TRUE))) 89 if (nrow(dfs) == 0) 90 next 91 row <- paste0(star) 65

92 for (pi in 1:nrow(dfs)) 93 row <- paste0(row,",", 94 pn[dfs[pi, 1]], "/", 95 pn[dfs[pi, 2]], ",", 96 dfs[pi, 3]) 97 cat(paste0(row, '\n')) 98 } 99 sink() 100 } 101 form.round <- function(this , d) { 102 format(round(this, digits = d), nsmall = d) 103 } 104 make.catalog <- function(dat, dv, pairs){ 105 sink("out/catalog.tex") 106 for (star in names(pairs)){ 107 sys <- subset(dat, star_name == star) 108 tp <- nrow(sys) 109 sm <- sys$star_mass[1] 110 dfs <- pairs[[star]] 111 dfs <- subset(dfs, dfs$V4 <= dv) 112 if (nrow(dfs) == 0) 113 next 114 pn <- 115 sapply(1:tp, function(p) 116 trimws(gsub(star , '', sys$X..name[p], fixed = TRUE))) 117 row <- '' 118 for (i in 1:nrow(dfs)) 119 row <- paste0(row, 120 pn[dfs[i, 1]], '/', 121 pn[dfs[i, 2]], ' is ', 122 dfs[i, 3], ', ') 123 cat( 124 paste0( 125 '\\multirow{', 126 tp + 1, 127 '}{*}{', 128 star , 129 '} & \\multirow{', 130 tp + 1, 131 '}{*}{', 132 form.round(sm, 3), 133 '} &&&&&&& \\multirow{', 134 tp + 1, 135 '}{3cm}{', 136 gsub(', $','', row), 137 '}\\\\\n' 138 ) 139 ) 140 for (p in 1:tp) 141 cat( 142 paste0( 143 " && ", 144 pn[p], 145 " & ", 146 form.round(sys$mass[p], 4), 147 " & ", 148 form.round(sys$semi_major_axis[p], 4), 149 " & ", 66

150 form.round(sys$eccentricity[p], 2), 151 " & ", 152 form.round(sys$inclination[p], 2), 153 " & ", 154 form.round(sys$orbital_period[p], 2), 155 " & ", 156 "\\\\\n" 157 ) 158 ) 159 } 160 sink() 161 } 162 163 ## results 164 maxfr <- make.fracs.max(pqmax) 165 fraca <- make.fracs.all(d) 166 pdl <- find.pairs(maxfr , max(dvp), fraca) 167 print.pairs(d, dvp[2], pdl) 168 make.catalog(d, dvp[2], pdl) All plots were made in R using the ggplot2 data visualization package2. In particular the following script was used to plot the orbital resonances. Listing B.2: R program used for plotting the orbital resonances. 1 # after resonances.R 2 # in the same session 3 4 library("ggplot2") 5 theme_set(theme_bw()) 6 theme_replace(legend.position = "bottom") 7 8 check.axis <- function(x) { 9 x <- x[complete.cases(x[, 5:6]), ] 10 cond <- x$V5 > x$V6 11 cr <- strsplit(x[cond , ]$X3, ':') 12 x[cond , ]$X3 <- lapply(cr, function(y) 13 paste0(y[2], ':', y[1])) 14 tmp <- x[cond , ]$V7 15 x[cond , ]$V7 <- x[cond , ]$V8 16 x[cond , ]$V8 <- tmp 17 x 18 } 19 check.mass <- function(x) { 20 x <- x[complete.cases(x[, 7:8]), ] 21 cond <- x$V7 > x$V8 22 x[cond , 9] <- "dominant_in" 23 x[!cond , 9] <- "dominant_out" 24 x 25 } 26 get.ratios <- function(x) { 27 tmp <- do.call(rbind , strsplit(unlist(x$X3), ':')) 28 tmp <- t(apply(tmp, 1, sort)) 29 paste0(tmp[, 1], ':', tmp[, 2]) 30 } 31 get.orders <- function(x) { 32 tmp <- do.call(rbind , strsplit(x, ':'))

2Hadley Wickham. ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag, New York, 2016. ISBN: 978-3-319-24277-4. URL: https://ggplot2.tidyverse.org. 67

33 abs(as.numeric(tmp[, 1]) - as.numeric(tmp[, 2])) 34 } 35 barplot.ready <- function(list){ 36 mtl <- lapply(list , table) 37 unul <- unique(names(unlist(mtl))) 38 mtl <- lapply(mtl, "[", unul) 39 mat <- as.table(do.call(rbind , mtl)) 40 colnames(mat) <- c(unul) 41 as.data.frame(mat) 42 } 43 44 pdv <- do.call(rbind , pdl) 45 pdd <- lapply(dvp, function(dv) 46 subset(pdv, pdv$V4 <= dv)) 47 pdd <- lapply(pdd, check.axis) 48 pds <- lapply(pdd, get.ratios) 49 pod <- lapply(pds, get.orders) 50 br_pds <- barplot.ready(pds) 51 br_pod <- barplot.ready(pod) 52 53 pdd_ma <- lapply(pdd, check.mass) 54 br_pds_mr <- list() 55 br_pod_mr <- list() 56 for (c in c('dominant_in', 'dominant_out')) { 57 pdd_mr <- lapply(pdd_ma, function(x) 58 subset(x, x$V9 == c)) 59 pds_mr <- lapply(pdd_mr, get.ratios) 60 pod_mr <- lapply(pds_mr, get.orders) 61 br_pds_mr[[c]] <- barplot.ready(pds_mr) 62 br_pod_mr[[c]] <- barplot.ready(pod_mr) 63 } 64 65 res.plot <- function(name , data , xaxis) { 66 pdf(paste0("figs/", name, ".pdf")) 67 ggplot(data , aes( 68 x = as.character(Var2), 69 y = Freq , 70 fill = Var1 71 )) + 72 geom_bar(stat = "identity", position = position_dodge()) + 73 labs(x = xaxis, 74 y = "Pairs") + 75 scale_fill_discrete( 76 name = "d < x%", 77 breaks = c("A","B","C"), 78 labels = c("1", "3", "5") 79 ) 80 dev.off() 81 } 82 83 res.plot("pep", br_pds, "Ratio") 84 res.plot("pep_mri", br_pds_mr[['dominant_in']], "Ratio") 85 res.plot("pep_mro", br_pds_mr[['dominant_out']], "Ratio") 86 res.plot("pop", br_pod, "Order") 87 res.plot("pop_mri", br_pod_mr[['dominant_in']], "Order") 88 res.plot("pop_mro", br_pod_mr[['dominant_out']], "Order") Also we made of the use the following helpful script which can be used to ﬁnd systems with more than 2 planets that have at least one planet with mass equal or more than 5 Jupiter masses. 68

Listing B.3: R program that ﬁnds systems w. >2 planets having one w. m ≥ 5 MJ . 1 data <- read.csv('exoplanet.eu_catalog.csv') 2 d <- droplevels(subset(data , planet_status == 'Confirmed')) 3 stsys <- table(d$star_name) 4 mt1 <- subset(d, star_name %in% names(stsys[stsys > 2])) 5 res <- names(table(droplevels(subset(mt1, mass >= 5))$star_name)) 6 (res <- res[-1]) #drop empty 69

B.2 Chapter 3

The source code of the program used to perform the numerical integration can be found here. The integrator is written in Julia, a dynamic programming language for technical computing3, and the program was used with version 1.1.0. The program can either be run as script feeding a ﬁle (example given after), or interactively in Julia session where data can entered manually. In belief that it may be found useful beyond this thesis, it is fully reproduced here. The usage of the various methods is documented in-program. Refer to introductory chapter and previous appendix for used theory.

Listing B.4: Julia program that performs the numerical integration. 1 #!/usr/bin/env julia 2 # 3 # usage: 4 # script...... plaint.jl /path/to/system.yml 5 # interactively.. julia -qi plaint.jl 6 7 using YAML #read data file 8 using DelimitedFiles #output results 9 using LinearAlgebra 10 11 """ 12 ksnr(e, M, eps, max_iterations) 13 14 Solve Kepler's equation with NR method. 15 """ 16 function ksnr(e::Float64, M::Float64, 17 eps::Float64 = 1e-16, nm::Int64 = 16) 18 x = M + e * sin(M) 19 d = (x - e * sin(x) - M) / (1 - e * cos(x)) 20 while abs(d) > eps && nm > 0 21 x -= d 22 d = (x - e * sin(x) - M) / (1 - e * cos(x)) 23 nm -= 1 24 end 25 return x 26 end 27 28 """ 29 euler_rotdΩ(, i, ω) 30 31 Return Eulerian rotation. 32 """ 33 function euler_rotdΩ(::Float64, i::Float64, ω::Float64) 34 return [ cosdΩ() -sindΩ() 0; sindΩ() cosdΩ() 0; 0 0 1 ] * 35 [ 1 0 0; 0 cosd(i) -sind(i); 0 sind(i) cosd(i) ] * 36 [ cosdω() -sindω() 0; sindω() cosdω() 0; 0 0 1 ] 37 end 38 39 """ 40 oe_to_xv!(mu, j, orbital_elements, positions, velocities) 41 42 Convert orbital elements to cartesian vectors. 43 """ 44 function oe_to_xv!(mu::Float64, j::Int64,

3Jeﬀ Bezanson et al. “Julia: A fresh approach to numerical computing”. In: SIAM review 59.1 (2017), pp. 65–98. DOI: 10.1137/141000671. 70

45 oe::Array{Float64,2}, 46 vr::Array{Float64,2}, vv::Array{Float64,2}) 47 a, e, i, Ω, ω, M = oe[:, j] 48 E = ksnr(e, M * π / 180) 49 r = a * (1 - e * cos(E)) 50 pof = [cos(E) - e, sqrt(1 - e^2) * sin(E), 0] * a 51 vof = [ - sin(E), sqrt(1 - e^2) * cos(E), 0] * (sqrt(mu * a) / r) 52 rot = euler_rotdΩ(, i, ω) 53 vr[:, j] = rot * pof 54 vv[:, j] = rot * vof 55 end 56 57 """ 58 xv_to_oe!(mu, j, positions, velocities, orbital_elements) 59 60 Convert cartesian vectors to orbital elements. 61 """ 62 function xv_to_oe!(mu::Float64, j::Int64, 63 rt::Array{Float64,2}, vt::Array{Float64,2}, 64 oe::Array{Float64,2}) 65 tiny = 1e-10 66 vr, vv = rt[:, j], vt[:, j] 67 r = norm(vr) 68 v = norm(vv) 69 vh = cross(vr, vv) 70 h = norm(vh) 71 vn = [-vh[2], vh[1], 0] 72 n = norm(vn) 73 ve = cross(vv, vh) / mu - vr / r 74 a=1/ (2 /r -v^2 / mu) 75 e = norm(ve) 76 i = acos(vh[3] / h) 77 if e > tiny 78 if n > tinyΩ 79 = acos(vn[1] / n) 80 if vn[2] < 0Ω 81 = π2 - Ω 82 endω 83 = acos(dot(ve, vn) / (e * n)) 84 if ve[3] < 0ω 85 = π2 - ω 86 end 87 else 88 i = Ω = .0ω 89 = atan2(ve[2], ve[3]) 90 if cross(ve, vr)[3] < 0ω 91 = π2 - ω 92 end 93 end 94 nu = acos(dot(ve, vr)/(e * r)) 95 if dot(vr, vv) < 0 96 nu = π2 - nu 97 end 98 else 99 #not implemented 100 end 101 E = 2 * atan(sqrt((1 - e) / (1 + e)) * tan(nu / 2)) 102 M = E - e * sin(E) 71

103 if M < 0 104 M += π2 105 end 106 i, Ω, ω, M = [i, Ω, ω, M] * 180 / π 107 oe[:, j] = [a, e, i, Ω % 360, ω % 360, M % 360] 108 end 109 110 """ 111 grav_a!(no_of_bodies, masses, positions, acceleration) 112 113 Compute gravitational acceleration. 114 !!! note 115 The `a` should be multiplied with the appropriate gravitational constant. 116 """ 117 function grav_a!(n::Int64, m::Array{Float64,1}, 118 x::Array{Float64,2}, a::Array{Float64,2}) 119 fill!(a, 0) 120 @inbounds for i = 1:(n - 1), j = (i + 1):n 121 R = x[:, i] - x[:, j] 122 Rd3 = R / norm(R)^3 123 a[:, i] -= m[j] * Rd3 124 a[:, j] += m[i] * Rd3 125 end 126 end 127 128 """ 129 mksnr(dt, α, n, σ0, r0, eps, max_iterations) 130 131 Solve modified Kepler's equation with NR method. 132 !!! note 133 Argument α`` is the inverse of semimajor axis. 134 """ 135 function mksnr(dt::Float64, α::Float64, n::Float64, s::Float64, r0::Float64, 136 eps::Float64 = 1e-16, nm::Int64 = 16) 137 r0a, s0a, ndt = 1 - r0 * α, s * sqrtα(), n * dt 138 x = π 139 d = (x - r0a * sin(x) - s0a * (cos(x) - 1) - ndt) / 140 (1 - r0a * cos(x) + s0a * sin(x)) 141 while abs(d) > eps && nm > 0 142 x -= d 143 d = (x - r0a * sin(x) - s0a * (cos(x) - 1) - ndt) / 144 (1 - r0a * cos(x) + s0a * sin(x)) 145 nm -= 1 146 end 147 return x 148 end 149 150 """ 151 gauss_step!(dt, mu, no_of_bodies, positions, velocities) 152 153 Keplerian drift using Gauss' f, g functions. 154 """ 155 function gauss_step!(dt::Float64, mu::Float64, np::Int64, 156 rt::Array{Float64,2}, vt::Array{Float64,2}) 157 @inbounds @simd for i = 1:np 158 vr0, vv0 = rt[:, i], vt[:, i] 159 r0, v0 = norm(vr0), norm(vv0)α 160 = 2 / r0 - v0^2 / mu 72

161 a, n = 1 / α, sqrt(mu * α^3) 162 s = dot(vr0, vv0) / sqrt(mu) 163 E = mksnr(dt, α, n, s, r0) 164 cose, sine = cos(E), sin(E) 165 r1 = a + (r0 - a) * cose + sqrt(a) * s * sine 166 a1ce = a * (cose - 1) 167 ivr0, ivr1 = 1 / r0, 1 / r1 168 ft = 1 + a1ce * ivr0 169 gt = dt + (sine - E) / n 170 fd = - a^2 * n * sine * ivr1 * ivr0 171 #gd = 1 + a1ce * ivr1 172 gd = (1 + gt * fd) / ft 173 rt[:, i] = ft * vr0 + gt * vv0 174 vt[:, i] = fd * vr0 + gd * vv0 175 end 176 end 177 178 """ 179 eam_dh(mu, no_of_bodies, masses, positions, velocities) 180 181 Compute energy and angular momentum, with democratic heliocentric vectors. 182 """ 183 function eam_dh(np::Int64, m::Array{Float64,1}, 184 x::Array{Float64,2}, v::Array{Float64,2}) 185 et, am = norm(sum(m' .* v, dims = 2))^2 / 2, zeros(3) 186 @inbounds @simd for i = 1:np 187 et += m[i] * norm(v[:, i])^2 / 2 188 et -= m[i] / norm(x[:, i]) 189 am += m[i] * cross(x[:, i], v[:, i]) 190 end 191 @inbounds for i = 1:(np - 1), j = (i + 1):np 192 et -= m[i] * m[j] / norm(x[:, i] - x[:, j]) 193 end 194 return et, norm(am) 195 end 196 197 """ 198 plaint_dhi(iostream, total_iterations, dt, step_iterations, 199 mu, masses, orbital_elements) 200 201 Perform planetary integration using democratic heliocentric integrator. 202 """ 203 function plaint_dhi(of::Array{IOStream, 1}, 204 kmax::Float64, dt::Float64, kpri::Float64, mu::Float64, 205 mp::Array{Float64,1}, oe::Array{Float64,2}) 206 mt = 1 / (sum(mp) + 1) 207 np = size(mp, 1) 208 an = Array{Float64}(undef, 3, np) 209 x = similar(an) 210 v = similar(an) 211 @inbounds @simd for j = 1:np 212 oe_to_xv!(mu, j, oe, x, v) 213 end 214 v .-= mt * sum(mp' .* v, dims = 2) #to_dh 215 el0 = eam_dh(np, mp, x, v) 216 println("Initial energy: ", el0[1], "\n", 217 "Initial angular momentum: ", el0[2], "\n") 218 d2 = dt / 2 73

219 @inbounds for k = 1:kmax 220 try 221 x .+= d2 * sum(mp' .* v, dims = 2) 222 gauss_step!(d2, mu, np, x, v) 223 grav_a!(np, mp, x, an) 224 v .+= dt * an 225 gauss_step!(d2, mu, np, x, v) 226 x .+= d2 * sum(mp' .* v, dims = 2) 227 if k % kpri == 0 228 eln = eam_dh(np, mp, x, v) 229 va = v .+ mt * sum(mp' .* v, dims = 2) #to_ac 230 writedlm(of[end], [collect(eln .- el0); collect(@. 1 - eln / el0)]') 231 @simd for j = 1:np 232 xv_to_oe!(mu, j, x, va, oe) 233 writedlm(of[j], oe[:, j]') 234 end 235 end 236 catch 237 println("Error: Exiting on ", k, "\n") 238 break 239 end 240 end 241 close.(of) 242 end 243 244 """ 245 data_get(file) 246 247 Get planetary system's data and integration options. 248 """ 249 function data_get(file::String) 250 data = YAML.load_file(file) 251 tmax::Float64 = data["options"]["tmax"] 252 step::Float64 = data["options"]["step"] 253 ever::Float64 = data["options"]["ever"] 254 unit = "solar" 255 try 256 unit = data["system"]["units"] 257 catch 258 end 259 ms = data["system"]["star_mass"] 260 datp = data["system"]["planets"] 261 sort!(datp, by = x -> x["orbital_elements"][1]) 262 np = size(datp, 1) 263 name = Array{String}(undef, np) 264 mp = Array{Float64}(undef, np) 265 oe = Array{Float64}(undef, 6, np) 266 @inbounds @simd for i = 1:np 267 name[i] = data["system"]["star_name"] * " " * datp[i]["id"] 268 mp[i] = datp[i]["mass"] 269 oe[:, i] = datp[i]["orbital_elements"] 270 end 271 return tmax, step, ever, ms, name, mp, oe, unit 272 end 273 274 """ 275 data_prep(tmax, step, ever, 276 star_mass, file_names, masses, orbital_elements, units) 74

277 278 Prepare data and options to feed to the integrator. 279 """ 280 function data_prep(tmax::Float64, step::Float64, ever::Float64, 281 ms::Float64, name::Array{String, 1}, 282 mp::Array{Float64, 1}, oe::Array{Float64,2}, 283 unit::String = "solar") 284 if unit == "solar" 285 mp = mp * 9.547919e-4 / ms 286 else #natural 287 mp = mp / ms 288 end 289 oe[1, :] /= oe[1, 1] 290 tp = π2 291 dt = step * tp 292 kmax = div(tmax * tp, dt) 293 kpri = div(ever * tp, dt) 294 of = map(f -> open("tmp/$f.txt", "w"), name) 295 push!(of, open("tmp/eamerr.txt", "w")) 296 println("Total simulation time is: ", tmax, " yr\n", 297 "Time of every step to be: ", step, " yr\n", 298 "Total number of steps is: ", kmax, " steps\n", 299 "Output will be every: ", ever, " yr\n", 300 "or differently every: ", kpri, " steps\n") 301 return of, kmax, dt, kpri, 1.0, mp, oe 302 end 303 304 """ 305 usage() 306 307 Return an examplotary session. 308 """ 309 function usage() 310 println("## Example session:\n", 311 "#set file containing data\n", 312 "file = \"data/TRAPPIST-1.yml\";\n", 313 "#get data from file\n", 314 "tmax, step, ever, ms, name, mp, oe = data_get(file);\n", 315 "#or enter manually\n", 316 "dgr = (tmax, step, ever, ms, name, mp, oe);\n", 317 "#prepare data\n", 318 "of, kmax, dt, kpri, mu, mp, oe = data_prep(dgr...);\n", 319 "#integrate, available: plaint_dhi, plaint_s6i\n", 320 "plaint_dhi(of, kmax, dt, kpri, mu, mp, oe);\n", 321 "#benchmark integration of a particular system\n", 322 "@time plaint_dhi(data_prep(data_get(file)...)...);") 323 end 324 325 ## Session 326 # Run as script or, interactively in a Julia session. 327 328 if ! isinteractive() 329 @time @fastmath plaint_dhi(data_prep(data_get(ARGS[1])...)...) 330 else 331 usage() 332 end 75

An example input ﬁle for a ﬁctional system4 is given, where tmax is the total simulation time (in yr), step is fraction of inner planet period, ever is every when to print state (in yr). Year here refers to one orbital period of the reference planet. The orbital elements are in order a, e, i, Ω, ω, M and the masses are given in Solar and Jupiter for star and planets respectively.

Listing B.5: Example input ﬁle for the integrator. 1 --- 2 options: 3 tmax: 1e6 4 step: 0.1 5 ever: 1e3 6 system: 7 cite: Spock et al 8 star_name: Nevasi 9 star_mass: 0.84 10 planets: 11 - id: Ket-Cheleb 12 mass: 0.001 13 orbital_elements: [ 0.224, 0.04, 0.42, 0.0, 0.0, 0.0 ] 14 - id: Vulcan 15 mass: 0.004 16 orbital_elements: [ 0.561, 0.02, 0.42, 0.0, 0.0, 0.0 ] 17 - id: T'Khut 18 mass: 0.004 19 orbital_elements: [ 0.562, 0.02, 0.42, 0.0, 0.0, 0.0 ] 20 - id: Delta Vega 21 mass: 0.1 22 orbital_elements: [ 3.142, 0.04, 0.42, 0.0, 0.0, 0.0 ]

4From the Star Trek franchise, the star is supposed to be 40 Eridani, which is pretty famous in ﬁctional settings though no planets had been found. Interestingly, a planet was recently discovered in orbit (see arXiv:1807.07098). 76 Appendix C

Catalog

In the period ratio column, “a/b is i : j” means the period of planet a to the period of b is (near) equal to i/j. Those pairs that have ratios with r are veriﬁed as resonant in literature whereas those with n as near- or non-resonant.

Exoplanet pairs with |P1/P2 − k2/k1| ≤ 3% and k1 + k2 ≤ 7 up to 2 Sep 2019.

Star M (MSun) Planet m (MJup) a (AU) e i (deg) P (JD) Period Ratios

24 Sex 1.540 b 1.3330 0.09 452.80 b/c is 1:2r 1 c 2.0800 0.29 883.00

b 2.1000 0.03 1078.00 47 Uma 1.030 c/d is 1:5 c 3.6000 0.10 2391.00 d 11.6000 0.16 14002.00

b 0.8400 0.1134 0.00 89.73 14.65 c 0.2374 0.07 44.37 55 Cnc 1.015 b/c is 1:3r 2, c/f is 1:5 d 5.4460 0.03 4867.00 e 0.0270 0.0154 0.03 90.36 0.74 f 0.7733 0.08 260.91

b 0.0502 0.12 4.21 61 Vir 0.950 c/d is 1:3 c 0.2175 0.14 38.02 d 0.4760 0.35 123.01

b 0.0428 0.08 3.87 BD-06 1339 0.700 c 0.4350 0.11 125.26 c/d is 1:4 d 1.0600 0.02 487.00 e 0.0300 0.03 2.39

CoRoT-7 0.930 b 0.0149 0.0172 0.12 80.10 0.85 b/c is 1:4 c 0.0427 0.0460 0.12 3.70

b 4.44 EPIC 203826436 0.900 b/c is 2:3, b/d is 1:3 c 6.43 d 14.09

EPIC 211331236 0.532 b 0.0188 1.29 b/c is 1:4 c 0.0491 5.44

b 0.0128 76.48 0.66 EPIC 248435473 0.670 c 0.0667 0.04 88.58 7.81 c/e is 2:5, d/e is 3:4 d 0.1017 0.04 89.73 14.70 e 0.1227 0.03 89.74 19.48

b 0.0044 0.0441 88.89 5.24 EPIC 248545986 0.400 b/c is 2:3, b/d is 1:2, c/d is 3:4 c 0.0028 0.0576 88.77 7.78 d 0.0041 0.0685 89.43 10.12

b 0.0210 0.31 3.20 GJ 1061 0.120 b/c is 1:2, b/d is 1:4 c 0.0350 0.29 6.69 d 0.0520 0.54 12.43

GJ 1132 0.181 b 0.0052 0.0154 0.00 1.63 b/c is 1:5 c 0.0476 0.27 8.93

b 0.0600 0.02 8.63 c 0.1240 0.03 25.64 GJ 163 0.400 b/c is 1:3, c/f is 1:4 d 1.0210 0.02 604.00 e 0.7000 0.03 349.00 f 0.3260 0.04 109.50

GJ 176 0.490 b/c is 1:3 1Rob Wittenmyer, Jonathan Horner, and Conor Tinney. “Resonances Required: Dynamical Analysis of the 24 Sex and HD 200964 Planetary Systems”. In: The Astrophysical Journal 761 (Nov. 2012). DOI: 10.1088/0004- 637X/761/2/165. 2Jianghui Ji et al. “Could the 55 Cancri planetary system really be in the 3: 1 mean motion resonance?” In: The Astrophysical Journal Letters 585.2 (2003), p. L139.

77 78

b 0.0660 0.08 8.77 c 0.1460 0.02 28.59

b 0.0900 0.03 18.64 GJ 273 0.290 c 0.0360 0.12 4.72 d/e is 3:4 d 0.7120 0.17 413.90 e 0.8490 0.03 542.00

b 0.1434 0.06 30.60 GJ 3293 0.420 c 0.3618 0.11 122.62 b/c is 1:4 d 0.1940 0.12 48.13 e 0.0821 0.21 13.25

GJ 3998 0.500 b 0.0290 0.00 2.65 b/c is 1:5 c 0.0890 0.05 13.74

b 0.0410 0.02 5.37 GJ 581 0.310 b/c is 2:5 c 0.0740 0.09 12.92 e 0.0290 0.12 3.15

b 0.0505 0.13 7.20 c 0.1250 0.02 28.14 b/c is 1:4, b/f is 1:5, c/d is 1:3, GJ 667 C 0.330 d 0.2760 0.03 91.61 c/f is 3:4, d/e is 3:2, d/g is 1:3, e 0.2130 0.02 62.24 e/g is 1:4 f 0.1560 0.03 39.03 g 0.5490 0.08 256.20

b 6.7000 1.8120 0.32 45.00 1052.10 GJ 676 A 0.710 c 6.8000 6.6000 0.20 7337.00 b/c is 1:6 d 0.0413 0.15 3.60 e 0.1870 0.24 35.37

GJ 682 0.270 b 0.0800 0.08 17.48 b/c is 1:3 c 0.1760 0.10 57.32

GJ 785 0.780 b 0.3200 0.13 74.72 b/c is 1:6 c 1.1800 0.32 525.80

GJ 849 0.490 b 2.3500 0.01 1914.00 b/c is 1:4 c 0.22 7049.00

b 1.9380 0.2083 0.00 84.00 61.03 b/c is 2:1r 3, b/e is 1:2, c/e is GJ 876 0.334 c 0.8560 0.1296 0.00 48.07 30.23 r 4 d 0.0220 0.0208 0.08 50.00 1.94 1:4 e 0.0450 0.3343 0.07 59.50 124.69

b 0.0258 0.0211 0.00 85.80 1.21 GJ 9827 0.659 b/c is 1:3, b/d is 1:5 c 0.0079 0.0440 87.80 3.65 d 0.0120 0.0627 0.00 87.39 6.20

c 0.0641 0.04 5.76 d 0.1286 0.09 16.36 c/d is 1:3, d/e is 1:3, e/f is 2:5, HD 10180 1.060 e 0.2699 0.03 49.74 g/h is 1:4 f 0.4929 0.14 122.76 g 1.4220 0.19 601.20 h 3.4000 0.08 2222.00

HD 102272 1.900 b 0.6140 0.05 127.58 b/c is 1:4 c 1.5700 0.68 520.00

HD 102329 1.950 b 2.0100 0.21 778.10 b/c is 2:3 c 0.21 1123.00

HD 108874 1.000 b 1.0510 0.07 395.40 b/c is 1:4 c 2.6800 0.25 1605.80

HD 109271 1.047 b 0.0790 0.25 7.85 b/c is 1:4 c 0.1960 0.15 30.93

HD 116029 1.580 b 1.7300 0.21 670.20 b/c is 3:4 c 0.04 907.00

HD 128311 0.840 b 1.0990 0.25 448.60 b/c is 1:2r 5 c 4.1900 1.7600 0.17 50.00 919.00

HD 133131 A 0.950 b 1.4400 0.32 649.00 b/c is 1:5 c 4.3600 0.47 3407.00

b 0.0933 0.18 11.58 HD 136352 b/c is 2:5, c/d is 1:4 c 0.1665 0.16 27.58 d 0.4110 0.43 106.72

HD 13808 b 0.1017 0.17 14.18 b/c is 1:4 c 0.2476 0.43 53.83

b 0.4150 0.04 94.44 HD 141399 1.070 c 0.6890 0.05 201.99 c/d is 1:5 d 2.0900 0.07 1069.80

3Man Hoi Lee and SJ Peale. “Dynamics and origin of the 2: 1 orbital resonances of the GJ 876 planets”. In: The Astrophysical Journal 567.1 (2002), p. 596. 4Eugenio J Rivera et al. “The Lick-Carnegie Exoplanet Survey: a Uranus-mass fourth planet for GJ 876 in an extrasolar Laplace conﬁguration”. In: The Astrophysical Journal 719.1 (2010), p. 890. 5Steven S Vogt et al. “Five new multicomponent planetary systems”. In: The Astrophysical Journal 632.1 (2005), p. 638. 79

e 5.0000 0.26 5000.00

HD 143761 0.889 b 0.2196 0.04 39.85 b/c is 2:5 c 0.4123 0.05 102.54

HD 1461 1.020 b 0.0634 0.11 5.77 b/c is 2:5 c 0.1117 0.15 13.51

HD 147873 1.380 b 0.5220 0.21 116.60 b/c is 1:4 c 1.3600 0.23 491.54

HD 155358 0.920 b 0.6400 0.17 194.30 b/c is 1:2r 6 c 1.0200 0.16 391.90

HD 1605 1.310 b 1.4800 0.08 577.90 b/c is 1:4 c 3.5200 0.10 2111.00

HD 176986 0.789 b 0.0630 0.07 6.49 b/c is 2:5 c 0.1188 0.11 16.82

b 0.0801 0.34 9.37 HD 181433 0.780 c/d is 1:6 c 1.8190 0.24 1014.50 d 6.6000 0.47 7012.00

HD 20003 b 0.0974 0.40 11.85 b/c is 1:3 c 0.1961 0.16 33.82

HD 200964 1.440 b 1.6010 0.04 613.80 b/c is 3:4r 7 c 1.9500 0.18 825.00

HD 20781 b 0.1690 0.11 29.15 b/c is 1:3 c 0.3456 0.28 85.13

HD 207832 0.940 b 0.5700 0.13 161.97 b/c is 1:6 c 2.1120 0.27 1155.70

b 0.0576 0.36 5.76 HD 215152 0.770 c 0.0674 0.16 7.28 b/e is 1:4, c/d is 2:3 d 0.0880 0.33 10.86 e 0.1542 0.17 25.20

HD 21693 b 0.1484 0.26 22.66 b/c is 2:5 c 0.2644 0.24 53.88

b 0.0149 0.0385 0.00 85.06 3.09 c 4.3600 0.0648 0.00 6.76 HD 219134 0.794 d 0.2351 0.00 46.71 c/d is 1:6, d/g is 1:2 e 2.5600 0.34 1842.00 g 0.3753 0.00 94.20 h 3.0640 0.37 2198.00

b 0.1250 0.05 18.02 HD 27894 0.800 b/c is 1:2 c 0.1980 0.02 36.07 d 5.4480 0.39 5174.00

b 0.1253 0.16 16.56 HD 31527 b/c is 1:3, c/d is 1:5 c 0.2665 0.08 51.20 d 0.8181 0.67 274.20

HD 33142 1.540 b 1.0700 0.07 326.00 b/c is 2:5 c 1.9600 0.16 809.00

b 2.0750 0.01 1056.70 c 0.7181 0.04 214.67 HD 34445 1.070 d 0.4817 0.03 117.87 b/g is 1:5, c/f is 1:3, d/f is 1:5 e 0.2687 0.09 49.17 f 1.5430 0.03 676.80 g 6.3600 0.03 5700.00

b 0.5336 0.05 154.38 HD 37124 0.830 b/c is 1:5r , c/d is 1:2r c 1.7100 0.12 885.50 d 2.8070 0.16 1862.00

b 0.0519 0.20 5.64 HD 39194 b/c is 2:5, c/d is 2:5 c 0.0954 0.11 14.03 d 0.1720 0.20 33.94

b 0.0475 0.12 4.31 c 0.0812 0.05 9.62 c/d is 1:2, c/e is 1:4, c/f is 1:5, HD 40307 0.770 d 0.1340 0.07 20.41 d/f is 2:5, e/f is 2:3, e/g is 1:5, e 0.1886 0.15 34.62 f/g is 1:4 f 0.2485 0.19 51.68 g 0.6000 0.29 197.80

HD 45364 0.820 b 0.6813 0.17 226.93 b/c is 2:3r 8 c 0.8972 0.10 342.85

HD 51608 b 0.1059 0.15 14.07 b/c is 1:6

6Paul Robertson et al. “The McDonald Observatory Planet Search: New Long-period Giant Planets and Two Interacting Jupiters in the HD 155358 System”. In: The Astrophysical Journal 749.1 (2012), p. 39. 7Robert A Wittenmyer, Jonathan Horner, and CG Tinney. “Resonances required: dynamical analysis of the 24 sex and HD 200964 planetary systems”. In: The Astrophysical Journal 761.2 (2012), p. 165. 8Alexandre CM Correia, Jean-Baptiste Delisle, and Jacques Laskar. “Planets in Mean-Motion Resonances and the System Around HD45364”. In: Handbook of Exoplanets (2017), pp. 1–19. 80

c 0.3791 0.41 95.42

HD 5319 1.560 b 1.7500 0.12 675.00 b/c is 3:4r 9 c 2.0710 0.15 886.00

HD 60532 1.440 b 9.2100 0.7700 0.28 20.00 201.83 b/c is 1:3r 10 c 21.8100 1.5800 0.04 20.00 607.06

HD 67087 1.360 b 1.0800 0.17 352.20 b/c is 1:6 c 3.8600 0.76 2374.00

b 0.1427 0.0785 0.07 13.00 8.67 HD 69830 0.860 b/c is 1:4 c 0.1649 0.1860 0.06 13.00 31.60 d 0.2529 0.6300 0.10 13.00 200.57

HD 73526 1.080 b 0.6500 0.29 188.90 b/c is 1:2r 11 c 1.0300 0.28 379.10

b 0.0566 0.06 5.40 HD 7924 0.832 b/c is 1:3, b/d is 1:4 c 0.1134 0.10 15.30 d 0.1551 0.21 24.45

b 14.5000 1.1900 0.20 19.40 442.40 HD 82943 1.180 b/c is 2:1r 12, b/d is 2:5 c 14.3900 0.7460 0.42 19.40 219.30 d 2.1450 0.00 1078.00

HD 95089 1.580 b 1.5100 0.16 507.00 b/c is 1:4 c 0.29 1860.00

HD 99706 1.720 b 2.1400 0.36 868.00 b/c is 3:4 c 0.41 1123.00

b 88.40 15.57 c 89.58 31.70 b/c is 1:2, c/e is 1:4, d/f is 1:2, HIP 41378 1.150 d 157.00 e/f is 2:5 e 131.00 f 324.00

HIP 54373 0.570 b 0.0630 0.20 7.76 b/c is 1:2 c 0.0990 0.20 15.14

b 0.0700 0.19 8.14 HIP 57274 0.730 b/c is 1:4 c 0.1780 0.05 32.03 d 1.0100 0.27 431.70

HIP 65407 0.930 b 0.1770 0.14 28.12 b/c is 2:5 c 0.3160 0.12 67.30

b 0.0480 0.30 85.50 3.59 HR 858 1.145 b/d is 1:3 c 0.0740 0.19 86.23 5.97 d 0.1027 0.28 87.43 11.23

b 7.0000 68.0000 0.00 28.00 164250.00 b/c is 2:1r 13, b/d is 4:1, c/d is HR 8799 1.560 c 8.3000 42.9000 0.00 28.00 82145.00 2:1r d 8.3000 27.0000 0.10 28.00 41054.00 e 9.2000 16.4000 0.15 25.00 18000.00

b 0.10 89.30 7.98 K2-136 0.740 b/d is 1:3, c/d is 2:3 c 0.13 89.60 17.31 d 0.14 89.40 25.58

b 0.0338 0.00 86.90 2.35 b/c is 2:3, b/f is 1:5, c/d is 2:3, c 0.0445 0.00 87.50 3.56 K2-138 0.930 c/f is 1:4, d/e is 2:3, d/f is 2:5, d 0.0588 0.00 87.90 5.40 e/f is 2:3 e 0.0781 0.00 88.70 8.26 f 0.1043 0.00 89.03 12.76

K2-16 0.680 b 0.0662 7.62 b/c is 2:5 c 0.1220 19.08

b 0.1870 0.0770 0.12 88.83 7.92 K2-19 0.930 b/c is 2:3 c 0.0310 0.1032 0.10 89.91 11.91 d 2.51

b 0.0081 0.0129 0.00 83.90 0.58 K2-229 0.837 c/d is 1:4 c 0.0670 0.0758 0.00 87.94 8.33 d 0.0790 0.1820 0.39 88.92 31.00

K2-24 1.120 b 0.0661 0.1540 89.25 20.89 b/c is 1:2 c 0.0850 0.2470 89.76 42.36

b 0.0305 0.0382 86.85 3.47 b/c is 1:2, b/d is 1:3, b/e is 1:4, K2-285 0.830 c/d is 2:3, c/e is 1:2 9Matthew J Giguere et al. “Newly discovered planets orbiting HD 5319, HD 11506, HD 75784 and HD 10442 from the N2K consortium”. In: The Astrophysical Journal 799.1 (2015), p. 89. 10Jacques Laskar and Alexandre CM Correia. “HD 60532, a planetary system in a 3: 1 mean motion resonance”. In: Astronomy & Astrophysics 496.2 (2009), pp. L5–L8. 11Chris G Tinney et al. “The 2: 1 resonant exoplanetary system orbiting HD 73526”. In: The Astrophysical Journal 647.1 (2006), p. 594. 12Man Hoi Lee et al. “On the 2: 1 orbital resonance in the HD 82943 planetary system”. In: The Astrophysical Journal 641.2 (2006), p. 1178. 13Krzysztof Gozdziewski and Cezary Migaszewski. “Multiple mean motion resonances in the HR 8799 planetary system”. In: (2013). eprint: arXiv:1308.6462. 81

c 0.0493 0.0824 89.86 7.14 d 0.0205 0.1178 89.64 10.46 e 0.0337 0.1804 89.80 14.76

K2-290 1.194 b 0.0664 0.0923 0.00 88.14 9.21 b/c is 1:5 c 0.7740 0.3050 0.00 89.37 48.37

K2-3 0.612 b 0.0204 0.0775 0.06 89.59 10.05 b/c is 2:5 c 0.0067 0.1405 0.04 89.70 24.65

b 0.0664 0.0804 0.21 89.00 8.99 K2-32 0.870 c 0.0381 0.1399 88.23 20.66 c/d is 2:3 d 0.1100 0.1862 88.40 31.71 e 0.0495 4.35

K2-35 0.663 b 0.0306 86.10 2.40 b/c is 2:5 c 0.0539 87.85 5.61

K2-36 0.800 b 0.0223 84.45 1.42 b/c is 1:4 c 0.0540 86.92 5.34

K2-5 0.530 b 0.0509 5.73 b/c is 1:2 c 0.0783 10.93

b 0.13 6.89 Kepler-100 1.109 b/d is 1:5, c/d is 1:3 c 0.02 12.82 d 0.38 35.33

b 0.0013 0.0550 5.29 b/c is 3:4, b/d is 1:2, b/e is 1:3, c 0.0670 7.07 Kepler-102 0.810 b/f is 1:5, c/d is 2:3, c/f is 1:4, d 0.0082 10.31 d/e is 2:3 e 0.0281 0.1170 0.00 89.56 16.15 f 0.0020 0.1650 27.45

b 0.0940 11.43 Kepler-104 0.810 b/c is 1:2, b/d is 1:4 c 0.1530 23.67 d 0.2570 51.76

Kepler-1050 1.090 b 15.38 b/c is 3:4 c 21.13

b 6.16 Kepler-106 1.000 c 0.0328 13.57 b/d is 1:4, b/e is 1:6, c/e is 1:3 d 0.0250 23.98 e 43.84

Kepler-1065 0.940 b 3.61 b/c is 3:2 c 2.37

b 0.0110 0.0454 0.00 89.05 3.18 Kepler-107 c 0.0295 0.0604 0.00 89.49 4.90 b/c is 2:3, b/d is 2:5, c/e is 1:3 d 0.0120 0.0838 0.00 87.55 7.96 e 0.0271 0.1264 0.00 89.67 14.75

Kepler-108 0.870 b 0.2920 0.22 49.18 b/c is 1:4 c 0.7210 0.04 190.32

Kepler-109 1.069 b 0.0041 0.21 6.48 b/c is 1:3 c 0.0070 0.03 21.22

Kepler-1093 1.130 b 25.08 b/c is 1:4 c 89.72

b 0.0060 0.0910 0.00 88.50 10.30 c 0.0091 0.1060 0.00 89.00 13.03 b/e is 1:3, b/f is 1:4, c/e is 2:5, Kepler-11 0.950 d 0.0230 0.1590 0.00 89.30 22.69 c/f is 1:4, d/f is 1:2, d/g is 1:5, e 0.0300 0.1940 0.00 88.80 32.00 e/f is 2:3, e/g is 1:4, f/g is 2:5 f 0.0063 0.2500 0.00 89.40 46.69 g 0.9500 0.4620 0.00 89.80 118.38

Kepler-110 b 0.1070 12.69 b/c is 2:5 c 0.1980 31.72

Kepler-1129 1.000 b 24.34 b/c is 1:3 c 76.54

b 0.0220 0.0520 87.66 5.19 Kepler-114 0.710 b/c is 2:3, c/d is 2:3 c 0.1259 0.0700 88.24 8.04 d 0.0151 0.0900 88.24 11.78

Kepler-115 1.000 b 0.0360 2.40 b/c is 1:4 c 0.0870 8.99

Kepler-120 b 0.0550 6.31 b/c is 1:2 c 0.0880 12.79

b 0.0640 5.77 Kepler-122 0.990 c 0.1080 12.47 b/d is 1:4, c/e is 1:3 d 0.1550 21.59 e 0.2270 37.99

Kepler-123 1.030 b 0.1350 17.23 b/c is 2:3 c 0.1810 26.70

b 0.0390 3.41 Kepler-124 b/c is 1:4 c 0.1000 13.82 d 0.1700 30.95

Kepler-1245 0.860 b 4.35 b/c is 3:2 82

c 2.94

Kepler-125 0.550 b 0.0410 4.16 b/c is 3:4 c 0.0510 5.77

b 0.0990 0.07 10.50 Kepler-126 b/c is 1:2 c 0.1620 0.19 21.87 d 0.4480 0.02 100.28

b 0.1250 0.47 14.44 Kepler-127 b/c is 1:2 c 0.2000 0.03 29.39 d 0.2800 0.03 48.63

Kepler-128 1.170 b 0.0941 15.09 b/c is 2:3 c 0.1040 22.80

Kepler-129 1.180 b 0.1310 0.01 15.79 b/c is 1:5 c 0.3930 0.20 82.20

b 0.0790 0.15 8.46 Kepler-130 1.000 b/c is 1:3, c/d is 1:3 c 0.1780 0.08 89.34 27.51 d 0.3770 0.80 87.52

b 0.0670 6.18 Kepler-132 c 0.0680 6.41 b/d is 1:3, c/d is 1:3 d 0.1360 18.01 e 110.29

Kepler-1321 0.540 b 11.13 b/c is 5:1 c 2.23

Kepler-133 b 0.0830 8.13 b/c is 1:4 c 0.2040 31.52

Kepler-1336 0.940 b 23.20 b/c is 4:1 c 5.78

Kepler-134 b 0.0600 5.32 b/c is 1:2 c 0.0920 10.11

Kepler-135 b 0.0670 6.00 b/c is 1:2 c 0.1030 11.45

b 2e-04 0.0746 0.00 89.95 10.31 Kepler-138 0.570 b/c is 3:4 c 0.0032 0.05 13.78 d 0.02 23.09

b 12.29 Kepler-1388 0.630 c 5.54 b/e is 1:3, c/d is 1:4, c/e is 1:6 d 20.96 e 37.63

Kepler-1398 1.130 b 2.79 b/c is 2:3 c 4.14

b 0.0320 2.02 Kepler-142 0.990 b/c is 2:5 c 0.0570 4.76 d 0.2420 41.81

Kepler-146 b 0.0182 0.0248 0.13 88.93 2.64 b/c is 2:3 c 0.0236 0.0327 0.08 87.30 4.00

b 0.0280 1.73 Kepler-148 b/c is 2:5 c 0.0500 4.18 d 51.85

b 0.1840 29.20 Kepler-149 b/c is 1:2, b/d is 1:5, c/d is 1:3 c 0.2810 55.33 d 0.5710 160.02

b 0.0440 3.43 c 0.0730 7.38 Kepler-150 b/d is 1:4, c/e is 1:4, d/e is 2:5 d 0.1040 12.56 e 0.1890 30.83 f 0.0283 1.2400 90.00 637.21

Kepler-153 b 0.1290 18.87 b/c is 2:5 c 0.2370 46.90

Kepler-1530 0.920 b 2.59 b/c is 1:2 c 5.32

b 0.1980 33.04 c 0.3030 62.30 Kepler-154 0.890 e/f is 2:5 d 20.55 e 3.93 f 9.92

b 3.95 Kepler-1542 0.940 c 2.89 b/d is 2:3, b/e is 3:4, c/d is 1:2 d 5.99 e 5.10

Kepler-156 b 0.0540 4.97 b/c is 1:3 c 0.1170 15.91

b 0.0280 1.73 Kepler-157 b/d is 1:4 c 0.1100 13.54 d 7.03 83

Kepler-159 b 0.0820 10.14 b/c is 1:4 c 0.2180 43.60

Kepler-160 b 0.0500 4.31 b/c is 1:3 c 0.1090 13.70

Kepler-161 0.770 b 0.0540 4.92 b/c is 2:3 c 0.0680 7.06

Kepler-162 b 0.0690 6.92 b/c is 1:3 c 0.1370 19.45

b 0.0580 5.04 Kepler-164 1.110 b/d is 1:5 c 0.0970 10.95 d 0.1870 28.99

b 0.0720 7.65 Kepler-166 b/c is 1:4 c 0.1950 34.26 d 1.55

b 0.0480 4.39 Kepler-167 0.760 c 0.0680 7.41 c/d is 1:3 d 0.0060 0.1405 0.00 89.35 21.80 e 4.0000 1.8900 0.06 89.98 1071.23

Kepler-168 b 0.0560 4.43 b/c is 1:3 c 0.1160 13.19

b 0.0400 3.25 c 0.0620 6.20 b/c is 1:2, b/d is 2:5, b/e is 1:4, Kepler-169 0.860 d 0.0750 8.35 c/d is 3:4 e 0.1050 13.77 f 0.3590 87.09

Kepler-170 b 0.0800 7.93 b/c is 1:2 c 0.1310 16.67

b 0.0400 2.94 Kepler-172 0.860 c 0.0680 6.39 c/e is 1:5, d/e is 2:5 d 0.1180 14.63 e 0.2110 35.12

b 0.1000 13.98 Kepler-174 b/c is 1:3, c/d is 1:5 c 0.2140 44.00 d 0.6770 247.35

Kepler-175 1.040 b 0.1050 11.90 b/c is 1:3 c 0.2130 34.04

b 0.0580 5.43 b/c is 2:5, c/d is 1:2, c/e is 1:4, Kepler-176 c 0.1020 12.76 d/e is 1:2 d 0.1630 25.75 e 51.17

Kepler-177 0.930 b 0.0054 35.85 b/c is 3:4 c 0.0198 49.41

Kepler-179 b 0.0360 2.74 b/c is 2:5 c 0.0640 6.40

b 0.0216 0.0447 84.92 3.50 Kepler-18 0.972 b/d is 1:4, c/d is 1:2 c 0.0541 0.0752 87.68 7.64 d 0.0513 0.1172 88.07 14.86

Kepler-180 0.840 b 0.1090 13.82 b/c is 1:3 c 0.2290 41.89

Kepler-181 b 0.0400 3.14 b/c is 3:4 c 0.0490 4.30

Kepler-182 1.140 b 0.0960 9.83 b/c is 1:2 c 0.1570 20.68

Kepler-183 b 0.0640 5.69 b/c is 1:2 c 0.1030 11.64

b 0.0920 10.69 Kepler-184 b/c is 1:2 c 0.1410 20.30 d 0.1790 29.02

b 0.0400 3.89 c 0.0610 7.27 Kepler-186 0.478 b/e is 1:5, c/e is 1:3, e/f is 1:5 d 0.0910 13.34 e 0.1290 22.41 f 0.3560 129.95

Kepler-188 b 0.0320 2.06 b/c is 1:3 c 0.0660 6.00

Kepler-189 0.790 b 0.0880 10.40 b/c is 1:2 c 0.1370 20.13

b 0.0260 0.0850 0.12 89.94 9.29 Kepler-19 0.936 b/c is 1:3, b/d is 1:6 c 0.0412 0.21 28.73 d 0.05 62.95

b 0.0870 9.94 Kepler-191 0.850 c/d is 3:1 c 0.1280 17.74 84

d 5.95

Kepler-193 b 0.1060 11.39 b/c is 1:4 c 0.2860 50.70

b 0.0320 2.09 Kepler-194 c/d is 1:3 c 0.1310 17.31 d 0.2750 52.81

Kepler-195 b 0.0770 8.31 b/c is 1:4 c 0.1970 34.10

b 0.0600 0.02 5.60 b/d is 1:3, b/e is 1:4, c/d is 2:3, Kepler-197 c 0.0170 0.0900 0.08 10.35 c/e is 2:5 d 0.1190 0.03 15.68 e 0.1640 0.38 25.21

b 0.1310 17.79 Kepler-198 0.930 b/c is 1:3 c 0.2590 49.57 d 1.31

Kepler-199 b 0.1580 23.64 b/c is 1:3 c 0.3160 67.09

b 0.0305 0.0463 0.03 87.36 3.70 c 0.0401 0.0949 0.16 89.81 10.85 b/c is 1:3, b/f is 1:5, c/d is 1:6, Kepler-20 0.912 d 0.0317 0.3506 0.60 89.71 77.61 c/g is 1:3, e/f is 1:3, e/g is 1:5 e 0.0097 0.0639 87.63 6.10 f 0.0450 0.1396 88.79 19.58 g 0.2055 0.15 34.94

Kepler-202 b 0.0450 4.07 b/c is 1:4 c 0.1130 16.28

b 0.0430 3.16 Kepler-203 0.980 b/d is 1:4, c/d is 1:2 c 0.0610 5.37 d 0.1000 11.33

b 0.0780 7.78 Kepler-206 0.940 b/d is 1:3 c 0.1110 13.14 d 0.1630 23.44

b 0.0290 1.61 Kepler-207 b/c is 1:2, b/d is 1:4, c/d is 1:2 c 0.0440 3.07 d 0.0680 5.87

b 0.0540 4.23 Kepler-208 1.030 c 0.0790 7.47 b/e is 1:4, c/d is 2:3, d/e is 2:3 d 0.1030 11.13 e 0.1320 16.26

Kepler-209 b 0.1220 16.09 b/c is 2:5 c 0.2310 41.75

Kepler-210 0.630 b 0.0142 0.44 77.86 2.45 b/c is 1:3 c 0.0370 0.50 85.73 7.97

Kepler-211 0.970 b 0.0480 4.14 b/c is 2:3 c 0.0620 6.04

Kepler-212 1.160 b 0.1330 16.26 b/c is 1:2 c 0.2070 31.81

Kepler-213 0.940 b 0.0360 2.46 b/c is 1:2 c 0.0570 4.82

b 0.0840 9.36 Kepler-215 0.770 c 0.1130 14.67 b/c is 2:3, b/e is 1:6, c/d is 1:2 d 0.1850 30.86 e 0.3140 68.16

b 0.0460 3.62 Kepler-218 c/d is 1:3 c 0.2480 44.70 d 124.52

b 0.0570 4.59 Kepler-219 c/d is 1:2 c 0.1650 22.71 d 0.2720 47.90

b 0.0460 4.16 Kepler-220 c 0.0760 9.03 b/d is 1:6, c/d is 1:3, c/e is 1:5 d 0.1630 28.12 e 0.2260 45.90

b 0.0370 2.80 Kepler-221 0.720 c 0.0590 5.69 b/c is 1:2, b/d is 1:4, c/e is 1:3 d 0.0870 10.04 e 0.1300 18.37

b 0.0480 3.94 Kepler-222 b/c is 2:5, b/d is 1:6, c/d is 1:3 c 0.0910 10.09 d 0.1800 28.08

b 0.0730 7.38 b/c is 3:4, b/d is 1:2, c/d is 2:3, Kepler-223 c 0.0880 9.85 c/e is 1:2, d/e is 3:4 d 0.1160 14.79 e 0.1400 19.72

b 0.0380 3.13 b/c is 1:2, b/d is 1:4, c/d is 1:2, Kepler-224 0.740 c/e is 1:3 85

c 0.0580 5.93 d 0.0890 11.35 e 0.1240 18.64

Kepler-225 b 0.0560 6.74 b/c is 1:3 c 0.1110 18.79

b 0.0470 3.94 Kepler-226 0.860 b/c is 3:4, b/d is 1:2, c/d is 2:3 c 0.0580 5.35 d 0.0760 8.11

Kepler-227 b 0.0900 9.49 b/c is 1:5 c 0.2900 54.42

b 0.0380 2.57 Kepler-228 b/d is 1:4 c 0.0520 4.13 d 0.1010 11.09

b 0.0620 6.25 Kepler-229 b/c is 2:5, c/d is 2:5 c 0.1170 16.07 d 0.2200 41.19

b 0.0478 0.0750 0.06 7.11 Kepler-23 1.110 b/c is 2:3 c 0.1890 0.0990 0.02 10.74 d 0.0550 0.1240 0.08 15.27

Kepler-230 b 0.1910 32.63 b/c is 1:3 c 0.3800 91.77

Kepler-231 0.580 b 0.0740 10.07 b/c is 1:2 c 0.1140 19.27

Kepler-232 b 0.0540 4.43 b/c is 2:5 c 0.1010 11.38

Kepler-233 b 0.0770 8.47 b/c is 1:6 c 0.2870 60.42

b 0.0370 3.34 Kepler-235 0.590 c 0.0650 7.82 b/c is 2:5, c/d is 2:5 d 0.1220 20.06 e 0.2130 46.18

Kepler-236 0.560 b 0.0650 8.30 b/c is 1:3 c 0.1320 23.97

b 0.0340 2.09 c 0.0690 6.16 Kepler-238 1.430 b/c is 1:3, c/e is 1:4, d/f is 1:4 d 0.1150 13.23 e 0.6260 23.65 f 0.0560 50.45

Kepler-24 1.030 b 1.6000 0.0800 8.15 b/c is 2:3 c 1.6000 0.1060 12.33

Kepler-240 b 0.0480 4.14 b/c is 1:2 c 0.0740 7.95

Kepler-241 b 0.0940 12.72 b/c is 1:3 c 0.1890 36.07

b 0.0500 4.31 Kepler-244 c/d is 1:2 c 0.0870 9.77 d 0.1400 20.05

b 0.0710 7.49 Kepler-245 0.800 c 0.1240 17.46 b/c is 2:5, c/d is 1:2 d 0.2020 36.28 e 3.22

Kepler-246 0.860 b 0.0520 4.60 b/c is 2:5 c 0.0950 11.19

b 0.0420 3.34 Kepler-247 0.884 b/c is 1:3 c 0.0840 9.44 d 0.1400 20.48

Kepler-248 b 0.0660 6.31 b/c is 2:5 c 0.1230 16.24

b 0.0280 0.0680 0.05 6.24 Kepler-25 1.220 b/c is 1:2 c 0.0450 0.1100 0.01 12.72 d 123.00

b 0.0480 4.15 Kepler-250 0.800 b/d is 1:4, c/d is 2:5 c 0.0690 7.16 d 0.1270 17.65

b 0.0640 5.83 Kepler-254 b/d is 1:3, c/d is 2:3 c 0.1050 12.41 d 0.1390 18.75

b 0.0270 1.62 Kepler-256 1.020 c 0.0450 3.39 b/c is 1:2, b/d is 1:4, c/e is 1:3 d 0.0640 5.84 e 0.0960 10.68

b 0.0340 2.38 Kepler-257 b/c is 1:3, c/d is 1:4 c 0.0660 6.58 86

d 0.1600 24.66

Kepler-258 0.800 b 0.1030 13.20 b/c is 2:5 c 0.1930 33.65

b 0.0088 0.0850 12.28 Kepler-26 0.650 b/e is 1:4 c 0.0195 0.1070 17.25 e 0.2200 46.83

Kepler-261 0.870 b 0.0880 10.38 b/c is 2:5 c 0.1560 24.57

Kepler-263 b 0.1200 16.57 b/c is 1:3 c 0.2420 47.33

b 0.0690 6.85 Kepler-265 1.030 c 0.1270 17.03 b/c is 2:5, c/d is 2:5, c/e is 1:4 d 0.2360 43.13 e 0.3190 67.83

b 0.0370 3.35 Kepler-267 0.560 b/c is 1:2, c/d is 1:4 c 0.0600 6.88 d 0.1540 28.46

Kepler-268 b 0.1800 25.93 b/c is 1:3 c 0.3910 83.45

Kepler-269 0.980 b 0.0610 5.33 b/c is 2:3 c 0.0810 8.13

Kepler-27 0.650 b 9.1100 0.1180 15.33 b/c is 1:2 c 13.8000 0.1910 31.33

b 0.0560 5.22 Kepler-271 b/d is 1:1 c 0.0710 7.41 d 5.25

b 0.0380 2.97 Kepler-272 0.790 b/c is 1:2, b/d is 1:4 c 0.0610 6.06 d 0.0910 10.94

Kepler-274 b 0.1010 11.63 b/c is 1:3 c 0.2040 33.20

b 0.0980 10.30 Kepler-275 1.240 b/c is 2:3 c 0.1320 16.09 d 0.2240 35.68

b 0.1190 14.13 Kepler-276 0.960 c/d is 2:3 c 0.0450 31.88 d 0.0440 48.65

Kepler-277 1.190 b 0.2800 17.32 b/c is 1:2 c 0.2100 33.01

b 0.1120 12.31 Kepler-279 1.230 b/c is 1:3, b/d is 1:4, c/d is 2:3 c 0.1710 0.2320 35.74 d 0.1300 0.3073 54.41

Kepler-28 0.750 b 1.5100 0.0620 5.91 b/c is 2:3 c 1.3600 0.0810 8.99

Kepler-281 0.980 b 0.1170 14.65 b/c is 2:5 c 0.2150 36.34

b 0.0820 9.22 Kepler-282 1.090 c 0.1060 13.64 b/c is 2:3, c/e is 1:3 d 0.2040 24.81 e 0.1920 44.35

Kepler-284 b 0.1040 12.70 b/c is 1:3 c 0.2130 37.51

Kepler-285 0.850 b 0.0360 2.63 b/c is 2:5 c 0.0640 6.19

b 0.0270 1.80 Kepler-286 c 0.0420 3.47 b/c is 1:2, b/d is 1:3 d 0.0610 5.91 e 0.1760 29.22

b 0.0650 6.10 Kepler-288 0.890 b/c is 1:3, c/d is 1:3 c 0.1400 19.31 d 0.2870 56.63

b 0.0230 0.2100 0.02 89.59 34.55 Kepler-289 1.080 b/c is 1:2, b/d is 1:4, c/d is 1:2 c 0.0130 0.3300 0.01 89.73 66.06 d 0.4150 0.5100 0.00 89.79 125.85

Kepler-29 1.000 b 0.0142 0.0900 10.34 b/c is 3:4 c 0.0130 0.1100 13.29

Kepler-290 b 0.1100 14.59 b/c is 2:5 c 0.2050 36.77

b 0.0350 2.58 c 0.0450 3.72 b/c is 2:3, c/d is 1:2, c/e is 1:3, Kepler-292 0.880 d 0.0680 7.06 c/f is 1:5, d/f is 1:3 e 0.0970 11.98 87

f 0.1410 20.83

Kepler-293 1.010 b 0.1440 19.25 b/c is 1:3 c 0.2860 54.16

b 0.0390 3.62 c 0.0540 5.84 Kepler-296 b/d is 1:5, c/e is 1:5, d/f is 1:3 d 0.1220 19.85 e 0.1740 34.14 f 0.2630 63.34

Kepler-297 b 0.2170 38.87 b/c is 1:2 c 0.3360 74.92

b 0.0400 2.93 b/c is 2:5, b/d is 1:5, c/e is 1:5, Kepler-299 0.970 c 0.0700 6.89 d/e is 2:5 d 0.1180 15.05 e 0.2200 38.29

b 0.0360 0.1800 89.82 29.33 Kepler-30 0.990 b/c is 1:2, c/d is 2:5 c 2.0100 0.3000 89.68 60.32 d 0.0730 0.5000 89.84 143.34

Kepler-300 0.940 b 0.0940 10.45 b/c is 1:4 c 0.2320 40.71

b 0.0360 2.51 Kepler-301 0.910 b/d is 1:5, c/d is 2:5 c 0.0600 5.42 d 0.1120 13.75

Kepler-302 b 0.1930 30.18 b/c is 1:4 c 0.5030 127.28

Kepler-303 0.590 b 0.0240 1.94 b/c is 1:4 c 0.0570 7.06

b 0.0390 3.30 Kepler-304 0.800 c 0.0540 5.32 b/d is 1:3 d 0.0800 9.65 e 1.50

b 0.0356 0.0586 5.49 Kepler-305 0.830 b/c is 2:3, b/d is 1:3, c/d is 1:2 c 0.0192 0.0771 8.29 d 0.1210 16.74

b 0.0500 4.65 b/c is 2:3, b/d is 1:4, c/d is 2:5, Kepler-306 0.820 c 0.0670 7.24 d/e is 2:5 d 0.1200 17.33 e 0.2270 44.84

b 0.1600 20.86 Kepler-31 1.210 b/c is 1:2, b/d is 1:4, c/d is 1:2 c 4.7000 0.2600 42.63 d 0.3900 87.65

b 0.1110 13.93 Kepler-310 0.850 b/c is 1:4, b/d is 1:6 c 0.2810 56.48 d 0.3920 92.88

Kepler-314 1.020 b 0.0350 2.46 b/c is 2:5 c 0.0640 5.96

Kepler-315 0.780 b 0.4020 96.10 b/c is 1:3 c 0.7910 265.47

Kepler-316 0.530 b 0.0270 2.24 b/c is 1:3 c 0.0580 6.83

Kepler-318 1.050 b 0.0560 4.66 b/c is 2:5 c 0.1050 11.82

b 0.0510 4.36 Kepler-319 1.290 b/d is 1:6 c 0.0690 6.94 d 0.1910 31.78

Kepler-32 0.580 b 4.1000 0.0500 5.90 b/c is 2:3 c 0.5000 0.0900 8.75

Kepler-323 1.090 b 0.0280 1.68 b/c is 1:2 c 0.0460 3.55

b 0.0530 4.54 Kepler-325 0.870 b/c is 1:3, c/d is 1:3 c 0.1050 12.76 d 0.2200 38.72

b 0.0320 2.25 Kepler-326 0.980 b/c is 1:2, b/d is 1:3, c/d is 2:3 c 0.0510 4.58 d 0.0660 6.77

b 0.0290 2.55 Kepler-327 0.550 b/c is 1:2, b/d is 1:5 c 0.0470 5.21 d 0.0900 13.97

Kepler-328 0.980 b 0.0690 34.92 b/c is 1:2 c 0.1000 71.31

Kepler-329 0.530 b 0.0610 7.42 b/c is 2:5 c 0.1130 18.68

b 0.0677 86.39 5.67 b/d is 1:4, b/e is 1:5, b/f is 1:6, Kepler-33 1.291 c/e is 2:5, c/f is 1:3, d/e is 2:3, e/f is 3:4 88

c 0.1189 88.19 13.18 d 0.1662 88.71 21.78 e 0.2138 88.94 31.78 f 0.2535 41.03

Kepler-330 0.780 b 0.0750 8.26 b/c is 1:2 c 0.1160 15.96

b 0.0650 8.46 Kepler-331 0.510 b/c is 1:2, b/d is 1:4 c 0.1050 17.28 d 0.1590 32.13

b 0.0700 7.63 Kepler-332 0.800 b/c is 1:2, b/d is 1:4 c 0.1140 16.00 d 0.1890 34.21

Kepler-333 0.540 b 0.0870 12.55 b/c is 1:2 c 0.1350 24.09

b 0.0610 5.47 Kepler-334 1.000 b/c is 2:5, c/d is 1:2 c 0.1070 12.76 d 0.1680 25.10

Kepler-337 0.960 b 0.0450 3.29 b/c is 1:3 c 0.0930 9.69

b 0.0963 0.1170 0.04 13.73 Kepler-338 1.100 c 0.1720 0.03 24.31 b/d is 1:3 d 0.2570 0.03 44.43 e 0.0270 0.05 9.34

b 0.0550 4.98 Kepler-339 0.840 b/d is 1:2, c/d is 2:3 c 0.0690 6.99 d 0.0910 10.56

Kepler-340 2.110 b 0.1340 14.84 b/c is 2:3 c 0.1780 22.82

b 0.0600 5.20 b/c is 2:3, b/d is 1:5, c/e is 1:5, Kepler-341 0.940 c 0.0800 8.01 d/e is 2:3 d 0.1820 27.67 e 0.2420 42.47

b 0.1280 15.17 Kepler-342 1.130 c 0.1850 26.23 b/d is 2:5, c/d is 2:3 d 0.2420 39.46 e 1.64

Kepler-343 1.040 b 0.0880 8.97 b/c is 2:5 c 0.1670 23.22

Kepler-344 0.900 b 0.1530 21.96 b/c is 1:5 c 0.4880 125.60

Kepler-346 0.970 b 0.0710 6.51 b/c is 1:4 c 0.1680 23.85

Kepler-348 1.150 b 0.0760 7.06 b/c is 2:5 c 0.1380 17.27

Kepler-349 0.970 b 0.0650 5.93 b/c is 1:2 c 0.1050 12.25

b 0.1040 11.19 Kepler-350 1.030 b/d is 2:5, c/d is 2:3 c 0.0176 17.85 d 0.0480 26.14

b 0.2140 37.05 Kepler-351 0.890 b/c is 2:3, b/d is 1:4, c/d is 2:5 c 0.2870 57.25 d 142.54

Kepler-353 0.540 b 0.0510 5.80 b/c is 2:3 c 0.0650 8.41

b 0.0540 5.48 Kepler-354 0.650 b/c is 1:3, b/d is 1:4 c 0.1150 16.93 d 0.1460 24.21

Kepler-355 1.050 b 0.1020 11.03 b/c is 2:5 c 0.1790 25.76

Kepler-356 0.970 b 0.0570 4.61 b/c is 1:3 c 0.1150 13.12

b 0.0630 6.48 Kepler-357 0.780 b/c is 2:5, c/d is 1:3 c 0.1200 16.86 d 0.2460 49.50

Kepler-358 0.950 b 0.2100 34.06 b/c is 2:5 c 0.3810 83.49

b 0.1780 25.56 Kepler-359 1.070 b/d is 1:3, c/d is 3:4 c 0.3070 57.69 d 0.3720 77.10

Kepler-362 0.770 b 0.0870 10.33 b/c is 1:4 c 0.2070 37.87

b 0.0480 3.61 Kepler-363 1.230 b/c is 1:2 89

c 0.0790 7.54 d 0.1070 11.93

Kepler-364 1.200 b 0.1780 25.75 b/c is 2:5 c 0.3120 59.98

Kepler-366 1.050 b 0.0450 3.28 b/c is 1:4 c 0.1100 12.52

Kepler-369 0.540 b 0.0300 2.73 b/c is 1:5 c 0.0940 14.87

b 0.0088 0.1003 88.63 13.37 Kepler-37 0.803 b/d is 1:3 c 0.0315 0.1368 89.07 21.30 d 0.0384 0.2076 89.33 39.79

Kepler-370 0.940 b 0.0540 4.58 b/c is 1:4 c 0.1400 19.02

Kepler-371 0.940 b 0.2000 34.76 b/c is 1:2 c 0.3130 67.97

b 0.0750 6.85 Kepler-372 1.150 b/c is 1:3, b/d is 1:4, c/d is 2:3 c 0.1540 20.05 d 0.2010 30.09

Kepler-373 0.870 b 0.0600 5.54 b/c is 1:3 c 0.1260 16.73

b 0.0290 1.90 Kepler-374 0.840 c/d is 2:3 c 0.0420 3.28 d 0.0560 5.03

Kepler-376 1.050 b 0.0570 4.92 b/c is 1:3 c 0.1150 14.17

Kepler-379 1.080 b 0.1520 20.10 b/c is 1:3 c 0.3260 62.78

Kepler-380 1.050 b 0.0500 3.93 b/c is 1:2 c 0.0780 7.63

Kepler-381 1.340 b 0.0660 5.63 b/c is 2:5 c 0.1170 13.39

Kepler-383 0.670 b 0.0950 12.90 b/c is 2:5 c 0.1720 31.20

Kepler-384 0.760 b 0.1480 22.60 b/c is 1:2 c 0.2360 45.35

Kepler-385 1.090 b 0.0970 10.04 b/c is 2:3 c 0.1270 15.16

Kepler-386 0.740 b 0.0960 12.31 b/c is 1:2 c 0.1550 25.19

Kepler-388 0.590 b 0.0360 3.17 b/c is 1:4 c 0.0930 13.30

Kepler-389 0.780 b 0.0410 3.24 b/c is 1:4 c 0.1100 14.51

Kepler-390 0.670 b 0.0650 6.74 b/c is 1:2 c 0.1010 13.06

Kepler-391 1.220 b 0.0820 7.42 b/c is 1:3 c 0.1610 20.49

Kepler-392 1.130 b 0.0590 5.34 b/c is 1:2 c 0.0930 10.42

Kepler-394 1.110 b 0.0830 8.01 b/c is 2:3 c 0.1100 12.13

Kepler-396 0.810 b 0.2200 0.3671 42.99 b/c is 1:2 c 0.0530 0.2266 88.50

b 0.0440 4.08 Kepler-398 b/c is 1:3 c 0.0870 11.42 d 6.83

b 0.1030 14.43 Kepler-399 b/d is 1:4 c 0.1550 26.68 d 0.2610 58.03

Kepler-400 b 0.0870 9.02 b/c is 1:2 c 0.1340 17.34

b 0.1220 14.38 Kepler-401 b/c is 1:3, c/d is 1:4 c 0.2690 47.32 d 184.26

b 0.0510 4.03 Kepler-402 c 0.0680 6.12 b/c is 2:3, b/e is 1:3, c/d is 2:3 d 0.0870 8.92 e 0.1020 11.24

b 0.0760 7.03 Kepler-403 b/d is 1:2, c/d is 4:1 90

c 0.2970 54.28 d 13.61

Kepler-405 b 0.0950 10.61 b/c is 1:3 c 0.1880 29.73

Kepler-406 1.070 b 2.43 b/c is 1:2 c 4.62

b 0.0380 0.00 88.27 3.01 Kepler-411 0.830 c 7.83 b/c is 2:5, c/e is 1:4 d 0.0478 0.2790 0.13 89.43 58.02 e 0.0340 0.1860 0.02 88.04 31.51

b 0.0090 0.0116 1.21 Kepler-42 0.130 b/d is 2:3, c/d is 1:4 c 0.0060 0.0060 0.45 d 0.0030 0.0154 1.86

Kepler-430 1.166 b 0.2244 35.97 b/c is 1:3 c 0.4757 110.98

b 0.0719 6.80 Kepler-431 1.071 c/d is 3:4 c 0.0847 8.70 d 0.1045 11.92

b 0.0418 0.08 88.00 3.60 c 0.0488 0.12 88.20 4.55 Kepler-444 0.758 c/d is 3:4 d 0.0600 0.18 88.16 6.19 e 0.0696 0.02 89.13 7.74 f 0.0811 0.58 87.96 9.74

b 0.0140 0.00 87.42 1.57 Kepler-446 0.220 b/c is 1:2, b/d is 1:3 c 0.0090 0.00 88.97 3.04 d 0.0100 0.00 88.72 5.15

Kepler-457 1.040 b 0.1610 89.30 31.81 b/c is 2:5 c 89.70 75.20

Kepler-460 1.070 b 89.90 440.78 b/c is 2:1 c 220.13

Kepler-462 1.965 b 0.06 89.34 84.69 b/c is 2:5 c 0.50 90.64 207.58

b 0.2956 89.59 49.51 Kepler-47 A 1.043 b/ d is 1:4 c 0.9890 89.83 303.15 d 0.0598 0.6992 0.02 90.00 187.35

b 0.0124 4.78 Kepler-48 0.880 c 0.0460 9.67 b/c is 1:2, c/d is 1:4 d 0.0250 42.90 e 982.00

Kepler-487 0.910 b 15.36 b/c is 2:5 c 38.65

b 7.20 Kepler-49 0.550 c 10.91 b/c is 2:3, b/e is 2:5, d/e is 1:6 d 0.0310 2.58 e 0.1160 18.60

b 0.0066 0.2514 0.04 45.16 Kepler-51 1.000 b/c is 1:2, b/d is 1:3, c/d is 2:3 c 0.0126 0.3840 0.01 85.31 d 0.0239 0.5090 0.01 130.19

b 8.7000 7.88 Kepler-52 0.540 b/c is 1:2 c 10.4100 16.39 d 0.1820 36.45

b 18.4100 18.65 Kepler-53 0.980 b/c is 1:2 c 15.7500 38.56 d 0.0910 9.75

b 8.01 Kepler-54 0.510 b/c is 2:3 c 12.07 d 0.1260 21.00

b 27.95 c 42.15 Kepler-55 0.620 b/c is 2:3, d/e is 1:2 d 0.0290 2.21 e 0.0480 4.62 f 0.0810 10.20

b 0.0700 0.1028 0.04 83.84 10.50 Kepler-56 1.320 b/c is 1:2 c 0.5690 0.1652 0.00 84.02 21.40 d 2.0000 0.20 1002.00

Kepler-57 0.830 b 18.8600 5.73 b/c is 1:2 c 6.9500 11.61

b 1.3900 10.22 Kepler-58 0.950 b/c is 2:3, b/d is 1:4, c/d is 2:5 c 2.1900 15.57 d 0.2360 40.10

Kepler-59 1.040 b 2.0500 11.87 b/c is 2:3 c 1.3700 17.98 91

b 0.0132 0.0750 7.13 Kepler-60 1.100 r 14 c 0.0121 0.0870 8.92 c/d is 3:4 d 0.0131 0.1055 11.90

b 0.0280 0.0553 89.20 5.71 c 0.0126 0.0929 89.70 12.44 b/d is 1:3, c/d is 2:3, d/e is 1:6, Kepler-62 0.690 d 0.0440 0.1200 89.70 18.16 e/f is 1:2r 15 e 0.1130 0.4270 89.98 122.39 f 0.1100 0.7180 89.90 267.29

b 0.0350 0.02 2.15 Kepler-65 1.199 b/d is 1:4, c/d is 3:4 c 0.0680 0.08 5.86 d 0.0840 0.10 8.13

b 12.11 Kepler-758 1.160 c 4.76 b/e is 3:2, c/d is 1:4, d/e is 5:2 d 20.50 e 8.19

b 18.93 Kepler-770 0.940 c/d is 1:3 c 1.48 d 4.15

b 0.0343 0.1170 0.02 88.78 13.48 b/c is 1:2, b/d is 1:4, c/d is 1:2, Kepler-79 1.165 c 0.0186 0.1870 0.03 89.48 27.40 c/e is 1:3, d/e is 2:3 d 0.0189 0.2870 0.02 89.93 52.09 e 0.0129 0.3860 0.01 89.13 81.07

b 0.0218 7.05 c 0.0212 9.52 b/c is 3:4, b/e is 3:2, b/g is 1:2, Kepler-80 d 0.0212 3.07 c/g is 2:3, d/e is 2:3, e/g is 1:3 e 0.0130 4.64 f 0.99 g 0.1400 89.35 14.65

Kepler-804 1.010 b 14.37 b/c is 3:2 c 9.65

b 0.0560 0.00 85.94 5.95 Kepler-81 0.648 b/c is 1:2 c 0.0890 0.00 87.66 12.04 d 0.1280 20.84

b 0.1690 0.00 89.38 26.44 c 0.2640 0.00 89.95 51.53 b/c is 1:2, b/f is 1:3, c/f is 2:3, Kepler-82 0.850 d 0.0340 2.38 d/e is 2:5 e 0.0630 5.90 f 0.0658 0.3395 0.00 86.30 75.73

b 0.0780 0.00 88.24 9.77 Kepler-83 0.664 b/c is 1:2 c 0.1260 0.00 88.81 20.09 d 0.0510 5.17

b 0.0830 0.00 88.24 8.73 c 0.1080 0.00 88.24 12.88 Kepler-84 1.022 b/c is 2:3, b/e is 1:3, b/f is 1:5 d 0.0520 4.22 e 0.1810 27.43 f 0.2500 44.55

b 0.0465 0.0789 8.31 Kepler-85 0.950 c 0.0590 0.1037 12.51 b/c is 2:3, b/e is 1:3, c/e is 1:2 d 0.1300 17.91 e 0.1630 25.22

Kepler-88 1.022 b 0.0274 0.06 89.06 10.95 b/c is 1:2 c 0.6255 0.1529 0.06 86.20 22.34

b 0.0330 0.0500 3.74 Kepler-89 1.250 c 0.0490 0.0990 10.42 b/c is 1:3, c/e is 1:5, d/e is 2:5 d 0.1640 0.1650 22.34 e 0.0410 0.2980 54.32

b 0.1369 0.1430 0.06 87.10 19.22 Kepler-9 1.000 b/c is 1:2n c 0.0941 0.2290 0.07 87.20 38.97 d 0.0165 0.0273 1.59

b 0.0740 89.40 7.01 c 0.0890 89.68 8.72 d 0.3200 89.71 59.74 b/i is 1:2, c/d is 1:6, d/e is 2:3, Kepler-90 1.130 e 0.4200 89.79 91.94 d/f is 1:2, d/h is 1:5, e/f is 3:4, f 0.4800 0.01 89.77 124.91 e/h is 1:4 g 0.7100 89.80 210.61 h 1.0100 89.60 331.60 i 0.2000 89.20 14.45

b 0.1890 0.17 13.75 Kepler-92 1.209 b/c is 1:2, b/d is 1:4 c 0.0179 0.04 26.72 d 0.07 49.36

Kepler-968 0.760 b 3.69 b/c is 2:3 c 5.71

L 98-59 0.320 c/d is 1:2 14K Goździewski et al. “The Laplace resonance in the Kepler-60 planetary system”. In: Monthly Notices of the Royal Astronomical Society: Letters 455.1 (2015), pp. L104–L108. 15Rajib Mia and Badam Singh Kushvah. “Orbital dynamics of exoplanetary systems Kepler-62, HD 200964 and Kepler-11”. In: Monthly Notices of the Royal Astronomical Society 457.1 (2016), pp. 1089–1100. 92

b 0.0227 0.09 2.25 c 0.0315 0.09 3.69 d 0.0500 0.18 7.45

b 0.0069 0.0330 87.56 3.07 LP 358-499 0.520 b/d is 1:4 c 0.0100 0.0450 88.65 4.87 d 0.0260 0.0770 89.52 11.02

b 0.0148 0.0562 88.30 6.34 LP 415-17 0.650 c/d is 1:3 c 0.0205 0.0946 88.96 13.85 d 0.0154 0.1937 89.61 40.72

LP 791-18 0.139 b 0.0097 87.30 0.95 b/c is 1:5 c 0.2939 89.55 4.99

NN Ser (AB) 0.646 c 6.9100 5.3800 0.00 5660.00 c/d is 2:1r 16 d 2.2800 3.3900 0.20 2830.00

OGLE-2006-109L 0.510 b 0.7270 2.3000 64.00 1790.00 b/c is 1:3 c 0.2710 4.5000 0.15 64.00 4931.00

b 1e-04 0.1900 0.00 25.26 PSR 1257 12 b/d is 1:4, c/d is 2:3n c 0.0130 0.3600 0.02 53.00 66.54 d 0.0120 0.4600 0.03 47.00 98.21

PSR B0943+10 1.500 b 1.8000 730.00 b/c is 1:2 c 2.9000 1460.00

TOI 125 0.871 b 0.0267 0.0521 0.18 88.99 4.65 b/c is 1:2 c 0.0271 0.0818 0.06 88.52 9.15

TOI-216 0.770 b 0.0820 0.1293 0.10 88.00 17.09 b/c is 1:2 c 0.5950 0.2068 0.08 89.89 34.56

b 0.0060 0.0306 0.00 88.65 3.36 TOI-270 0.400 c/d is 1:2 c 0.0208 0.0472 0.00 89.53 5.66 d 0.0170 0.0733 0.00 89.69 11.38

TOI-402 0.851 b 0.0227 0.0524 88.36 4.76 b/c is 1:4 c 0.0277 0.1235 88.41 17.18

b 0.0027 0.0111 0.00 89.65 1.51 c 0.0043 0.0152 0.00 89.67 2.42 b/e is 1:4, c/e is 2:5, c/f is 1:4, d 0.0013 0.0214 0.00 89.75 4.05 r 17 TRAPPIST-1 0.080 c/g is 1:5, d/e is 2:3 , d/g is e 0.0020 0.0282 0.00 89.86 6.10 1:3, e/f is 2:3r , e/g is 1:2, e/h is f 0.0021 0.0371 0.00 89.68 9.21 1:3, f/g is 3:4r g 0.0042 0.0451 0.00 89.71 12.35 h 0.0010 0.0630 89.80 20.00

TYC+1422-614-1 1.150 b 0.6879 0.07 198.44 b/c is 1:3 c 1.3916 0.05 569.20

WASP-134 1.131 b 1.4120 0.0956 0.14 89.13 10.15 b/c is 1:6 c 0.17 70.01

b 0.0375 0.15 4.89 Wolf 1061 0.250 b/c is 1:4 c 0.0890 0.11 17.87 d 0.4700 0.55 217.21

b 0.0156 0.00 1.97 YZ Cet 0.130 b/c is 2:3, b/d is 2:5, c/d is 2:3 c 0.0209 0.04 3.06 d 0.0276 0.13 4.66

e 0.5380 0.18 162.87 tau Cet 0.783 f 1.3340 0.16 636.13 e/f is 1:4, g/h is 2:5 g 0.1330 0.06 20.00 h 0.2430 0.23 49.41

b 0.6200 0.0590 0.01 90.00 4.62 ups And 1.270 c 9.1000 0.8610 0.24 11.35 240.94 c/d is 1:5r , d/e is 1:3 d 23.5800 2.5500 0.32 25.61 1281.44 e 5.2456 0.01 3848.86

Newer data may have been found for each planet but their entry hasn’t been updated. It is advised to check the planets pages for papers that provide newer insights. Alternatives to exoplanet.eu which is the used reference catalogue are

• Open Exoplanet Catalogue, openexoplanetcatalogue.com

• NASA Exoplanet Archive, exoplanetarchive.ipac.caltech.edu

• Exoplanet Data Explorer, www.exoplanets.org

16Jonathan Horner et al. “A detailed investigation of the proposed NN Serpentis planetary system”. In: Monthly Notices of the Royal Astronomical Society 425.1 (2012), pp. 749–756. 17Rodrigo Luger et al. “A seven-planet resonant chain in TRAPPIST-1”. In: (2017). eprint: arXiv:1703.04166. Bibliography

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