Refining the Orbits of the Planets in HD 207832

Home , Eccentric Jupiter, HD 207832

Refining the orbits of the planets in HD 207832

Emil Zadera

Lund Observatory Lund University 2017-EXA119

Degree project of 15 higher education credits June 2017

Supervisor: Alex Mustill

Lund Observatory Box 43 SE-221 00 Lund Sweden

Abstract

The star, HD 207832, has two Jupiter-like planets on orbits with poorly constrained ec- +0.22 +0.18 centricities. The eccentricities are 0.27−0.10 and 0.13−0.05 respectively. Notably, the two sigma error allows eccentricities up to 0.71 for one of the planets. Due to the large error bars, one aim of this project is to refine them. This is done by simulating the system for different initial eccentricities within the two sigma error bars for both of the planets. If a simulated system is shown to be unstable, the initial eccentricities used in the simulation, can not describe the observed system, HD 207832. In this project, it has been shown that the outer planet, in HD 207832, can not exceed an initial eccentricity of 0.6 in order for the system to remain stable. Furthermore, The level of chaos of the two planets, in each simulated system, is investigated with the use of Fourier analysis. A code is written which calculates the Fourier transform of the eccentricities. The code then counts the number of peaks in the spectrum which determines the level of chaos in the system. In this project, the use of Fourier analysis, to determine the level of chaos, is shown to be useful when comparing the chaos between simulations that have similar integration times. It is also shown that the outcome in each simulation is very sensitive to the fixed timestep used. It is highlighted that small changes in the timestep can change the outcome of the simulation in the sense of making a stable system, unstable. HD 207832, further, has a habitable zone, where a planet can support liquid water on its surface, that is located between the two Jupiter-like planets. Radial velocity measurements have yet not been able to detect any planet within this zone. In this project, stable orbits for a small planet, within the habitable zone, are thus searched for. This is done for the nominal system of HD 207832, and for the case when one sigma has been subtracted from the eccentricities of the two Jupiter-like planets. In this project, by the use of test particles, a few orbits are shown to be stable over at least 250 Myr in both of the simulated systems. It is thus possible that HD 207832 has a habitable planet that has not yet been detected.

Key words: HD 207832 – eccentricity space – chaos – habitable zone – stability Popul¨arvetenskaplig beskrivning

Vad Klarar Dessa Exoplaneterna Av?

V˚ardatabas av dokumenterade exoplaneter äruppe i tusental och nya planeter upptäks varje dag. Ofta ärdessa planeterna bundna till en stjärna,precis som planeterna i v˚arat solsystem ärbundna till solen. Vissa av de planetiska systemen ärgoda kandidater till att inneh˚allaplaneter som liknar jorden, medan andra har försträngaförh˚allanden.Oavsett mängdenav exoplaneter, ärdet viktig att följaupp och kontrollera hur väldessa planet- systemen ärbeskrivna av parametrarna som ärgivna i v˚arandatabas föratt kunna analy- sera dem vidare. I detta projekt bidrar vi med information till databasen, av de oräkneligt m˚angaexoplanet-systemen, genom att titta p˚aett av dem. Detta system best˚arav tv˚a planeter, stora som Jupiter, som circulerar omkring en stjärnavid namn HD 207832. Vi inspekterar systemets parametrar och även möjlighetenav att en oidentifierad, beboelig planet skulle kunna befinna sig i systemet.

VarförHD 207832? Mätningarnasom togs föratt verifiera detta systemet, tyder p˚aatt de tv˚abefintliga planeterna har en mycket stor osäkerhet i sina parametrar. Enligt mätningarnaskulle planeterna kunna befinna sig p˚amycket exotiska banor. Med exotisk i detta samanhanget menar vi att skillnaden mellan den högstaoch den lägstahastigheten fören planet i sin om- loppsbana skulle kunna vara stor. Hastigheten hoss en planet ärkopplad till planet-banans eccentricitet, som ären parameter som avgörplanetens närmasteposition till stjärnan,och det längsta avst˚andetfr˚an stjärnan. Härundersöker vi vilka möjligaeccentriciteter som planeterna klarar av föratt fortfarande holla sig stabila i systemet! Vi simulera systemet med olika eccentriciteter p˚aplaneterna och ser hur banorna evolverar. Det visar sig att m˚angaav de eccentriciteterna som osäkerheten i parametrarna i v˚ardatabas till˚ater,rent sagt leder till kaotiska eller helt ostabila banor förplaneterna. Vi har därförlyckats minska felet i parameterarna som beskriver HD 207832.

Kan en habitabel planet befinna sig i HD 207832? Stjärnan,HD 207832, delar m˚angaegenskaper med v˚aransol närdet kommer till dess storlek. Detta innebäratt en habitabel planet, som liknar Jorden, skulle befinna sig p˚a ungefärsamma avst˚andfr˚anstjärnansom Jorden ärfr˚ansolen. Detta omr˚adetr˚akar vara precis mellan de tv˚aplaneterna vi kännertill i HD 207832. Fr˚aganärom den Jord-lika planeten klarar av att h˚allasig kvar i systemet mellan de tv˚agiganterna, utan att kastas ut p˚agrundav gravitationen som drar planeten mellan de andra himlakropparna. I detta projekt har vi i simuleringar satt in en tredje planet i systemet. Planeten är satt p˚aolika avst˚andfr˚anstjärnaninnom den beboeliga zonen runt HD 207832. I de flesta fallen klara sig den tredje planeten inte mellan de tv˚aJupiter-liknande planeterna. Den brukar allts˚akollidera eller utlösasfr˚ansytemet. Men det finns n˚agrafall, därvi under vissa förutsättningarkunnat beh˚alladen tredje planeten. Detta innebäratt vi har hittat banor dären beboelig planet skulle kunna befinna sig runt HD 207832. Acknowledgements

This project has been a great experience and I would like to thank all the people who helped me to get where I am. Firstly, I want to thank Dr. Alexander James Mustill, who gave me the opportunity to take the project under his great supervision, and who introduced me to the tools used in planetary science. It was always exciting to bring my results to our meetings. Thank you for the great explanations, and for your great feedback and guidance. I also want to give my thanks to friends I spent so much time discussing with. You have been a great inspiration for my work. Lastly, I want to thank my parents and siblings, who motivated me all the way up to this point. Thanks for keeping up the contact. Especially thanks to Veronika, for always being there. Contents

1 Introduction 5 1.1 HD 207832 ...... 6 1.1.1 The Habitable Zone ...... 7 1.1.2 The Semi-Amplitude of a Low-Mass Planet ...... 8 1.2 The Orbital Elements ...... 8 1.2.1 The Conserved Quantities and Two Coordinates systems ...... 9 1.3 Planetary Evolution ...... 10 1.3.1 The Chaos in Planetary systems ...... 11 1.3.2 The Stability of Planetary systems ...... 12 1.3.3 The Hill Stability ...... 13

2 Method 15 2.1 The Hybrid Symplectic Integrator ...... 15 2.1.1 Working with Mercury6 ...... 15 2.2 The Stability Analysis ...... 16 2.2.1 The Nominal System with Variable Eccentricities ...... 16 2.2.2 Exploring the Eccentricity Space ...... 17 2.2.3 The Analysis ...... 18 2.3 The Search for a Habitable Planet ...... 19 2.3.1 Simulating with Test Particle ...... 19

3 Results 21 3.1 Exploring the Eccentricity Space ...... 21 3.1.1 100 Myr Simulations ...... 23 3.2 The Possibility of a Habitable Planet ...... 24

4 Conclusions 26 4.1 The Eccentricity Space of HD 207832 ...... 26 4.2 The Level of Chaos ...... 27 4.3 The Habitable Planet ...... 27

A The Principle used in Mercury6 31

1 List of Figures

1.1 The nominal planetary system in HD 207832 (Haghighipour et al., 2012) and its habitable zone deﬁned by the standard model (Kasting et al., 1993), with a radius between 0.85 AU and 1.65 AU. The inner and the outer planet are HD 207832b and HD 207832c respectively...... 7 1.2 Some of the basic parameters used to describe a planetary orbit. The shadowed area corresponds to the part of the orbit below a reference plane. The planet is moving in the direction of the arrow, away from the ascending node.9 1.3 The plot shows a time interval from a simulation of HD 207832, where the inner and the outer planet have the initial eccentricities 0.20 and 0.57 respectively. Notably, the amplitudes in the oscillations are found to be large. 11 1.4 The Fourier transform of the eccentricity for the outer planet in HD 207832. The simulations are for the nominal system but with the initial eccentricities, 0.40, for the inner planet, and 0.55, 0.56, and 0.57 for the outer planet respectively. The three systems are classiﬁed as fairly non-chaotic with a number of peaks in their power spectrum within [1-50), see Section 2.2.3. . 12 1.5 The eccentricity space of the planet HD 207832c versus HD 207832b. The Hill stability limit divides the region where collisions between planets are allowed, from where they are not (Gladman, 1993). The parameters used for this plot can be found in Table 1.1 & 1.2. The nominal system with the eccentricities 0.13 and 0.27 for the respective planet, and the ellipses spanned by its upper one sigma and two sigma error bars, have been plotted. 14

2.1 The level of chaos is determined by the number of peaks that intersect the limit at 1% of the strongest peak. The Fourier transform in the ﬁgure corresponds to the system with the initial eccentricities (ec, eb) = (0.20, 0.62), and has 59 peaks (denoted as a black dot in Figure 3.1)...... 18

3.1 The plot shows the eccentricity space spanned by the error bars of the outer versus the inner planet in HD 207832. The dots represents the systems which are stable during 20 Myr and their level of chaos (see Section 2.2.3). The unstable systems are denoted as, × and ◦, for a collision between the planets and an ejection of a planet, respectively...... 22

2 LIST OF FIGURES LIST OF FIGURES

3.2 The unstable systems in the first part of the project (Figure 3.1), are divided into time intervals depending on their life-time. A power law is then fitted to the data in order to confirm that most of the unstable systems undergo their close encounter before 20 Myr...... 22 3.3 The life-time of each test particle as a function of the initial semi-major axis. The upper plot corresponds to the nominal system, and the lower plot corresponds to the minus one sigma system. For the nominal system, the region between 0.85 AU and 1 AU is analysed in greater detail due to the longer life-times. The green lines mark the boundaries of the habitable zone. The arrows indicate that the life-time of the test particles exceeds 250 Myr which is the integration time...... 25

3 List of Tables

1.1 The parameters of the star (Haghighipour et al., 2012)...... 6 1.2 The orbital solution for the two planets (Haghighipour et al., 2012). . . . .7

3.1 The points that were simulated over 100 Myr with the same timesteps used in the 20 Myr simulations...... 23 3.2 The 100 Myr simulations with slightly diﬀerent initial conditions. (0.60, 0.10) is a continuation of the 20 Myr simulation. (0.61, 0.30), (0.59, 0.40), and (0.63, 0.50), are simulated with smaller timesteps compared to in Table 3.1. For (0.59, 0.40) the fractional energy change due to the integrator becomes larger than 10−5 and is thus rejected...... 23

4 Chapter 1

Introduction

Successfully confirming thousands of exoplanets, mainly with the radial velocity and the transit method1, has led to a large amount of statistical data which is to be explained with the theories of planetary evolution. This includes, for example, the distribution of planets that are stable for long periods of time and how they evolve (Davies et al., 2014). However, the measured data may contain large error bars, e.g. Mann et al. (2017), where their relative error bars in eccentricities are large. Thus, refining the parameters is necessary in some cases (e.g. done by Anderson et al. (2011), where they combine existing data with new measurements; Lecavelier des Etangs & Vidal-Madjar (2016), where they refine the data for β Pic b, assuming it is a transiting planet). In this project, the planetary system HD 207832, discovered by Haghighipour et al. (2012), is analysed. It was discovered in 2012 with the radial velocity (RV) method, using the High-Resolution Echelle Spectrometer (HIRES) at Keck observatory. The RV method, in simple terms, is to trace the velocity of the star along the line of sight, by measuring the Doppler shift of the emitted light. If the star possesses any planets, they will induce oscillations in the measured velocity curve (Mayor & Queloz, 1995). The RV method works if the planets are massive enough to induce oscillations that are not drowned in the noise of the measurements. In the paper by Haghighipour et al. (2012), they successfully fit a two- planet model to the RV curve of HD 207832. The basic parameters of the star and the two planets are presented in Table 1.1 and Table 1.2 respectively. Notably, the eccentricities +0.22 +0.18 are loosely constrained with large error bars. The eccentricities are 0.27−0.10 and 0.13−0.05, respectively. The project is divided into two parts with separate aims. In the first part of this project, the aim is to refine the eccentricities of the planets in HD 207832. For the second part, the aim is to find orbits, within the habitable zone, which are stable for low-mass planets that have escaped detection in the RV measurements. By simulating the system, HD 207832, with different initial eccentricities for the two planets within their error bars, the outcome tells whether the system is stable within the simulation time or not (Davies et al., 2014). The hypothesis is that the highly eccentric cases are less likely to be stable, because, the planets may have more close encounters

1The NASA Exoplanet Archive (Akeson et al., 2013) contains the data of the currently conﬁrmed exoplanets, including the method which was used to ﬁnd each planet, http://exoplanetarchive.ipac. caltech.edu.

5 1.1. HD 207832 CHAPTER 1. INTRODUCTION

(Chatterjee et al., 2008). Assuming that the observed planets in HD 207832 are stable, the initial eccentricities that lead to unstable systems, can not describe the observed planetary system. By this method, the error bars of the eccentricities can be reﬁned. For the second aim of the project, the habitable planet is assumed small, about an Earth mass. This is justiﬁed, because, the radial velocity measurements, of HD 207832, were not able to detect such a planet (Section 1.1.2). In this project, test particles, which have no mass, are used to represent the small planet within the habitable zone. If they are proven stable within the integrated time, HD 207832 may become a candidate for future measurements, when searching for habitable planets.

1.1 The System of HD 207832

HD 207832 is a Sun-like star, meaning, it has approximately the same mass, radius, and eﬀective temperature as the Sun. Its basic parameters are represented in Table 1.1. There are two known eccentric, Jupiter-like, planets orbiting the star. The basic planetary parameters are presented in Table 1.2, (see Section 1.2 for an explanation of the orbital elements). Jupiter-like exoplanets have a large scatter in the eccentricity space2, where the planets in HD 207832 are two of the more eccentric planets with eccentricities of 0.13 and 0.27 for the inner and outer planet, respectively. In Table 1.2, the one sigma error bars are shown, meaning, the interval includes the true value with 68% certainty. However, in this project the two sigma error bars, with 95% conﬁdence interval, are considered. This leaves the eccentricity of the inner and the outer planet in poorly constrained. Note, that the inclinations of the planetary orbits are unknown. The mass shown is the minimal mass of the planets. It is the true mass, multiplied with sinus of the unknown angle i. In this project the angle is assumed to be 90◦, meaning the minimal mass equals to the true mass of the planet. An overhead 2D plot of the nominal system of HD 207832 is given in Figure 1.1.

Table 1.1: The parameters of the star (Haghighipour et al., 2012). HD 207832 Spectral type Mass[M ] Radius[R ] Teﬀ[K] distance[pc] [Fe/H] Age[Gyr] G5V 0.94±0.10 0.901±0.056 5710±81 54.4±2.7 0.06 <4.5

2The Extrasolar Planets Encyclopaedia (Schneider et al., 2011), contains a representation of the distribution of diﬀerent parameters among the conﬁrmed exoplanets and stars, (exoplanet.eu). The planets with Jupiter-like masses are found to have a fairly large scatter in their eccentricity (with eccentricities ranging from zero to about 0.8).

6 1.1. HD 207832 CHAPTER 1. INTRODUCTION

Table 1.2: The orbital solution for the two planets (Haghighipour et al., 2012). ◦ ◦ Planet P[days] e $[ ] MA[ ] M sin i[Mjup] a[AU] +0.97 +0.18 +23.9 +83.2 +0.06 +0.02 HD 207832b 161.97−0.78 0.13−0.05 130.8−83.4 243.3−30.1 0.56−0.03 0.57−0.02 +71.9 +0.22 +32.4 +114.4 +0.18 +0.087 HD 207832c 1155.7−37.0 0.27−0.10 121.6−76.5 211.9−0.0 0.73−0.05 2.112−0.045 Epoch = 2453191.07306 JD

1.1.1 The Habitable Zone The habitable zone, around a star, is where a planet can support liquid water on its surface. However, the boundaries of the habitable zone are not strictly defined, because, of their strong dependence on the atmospheric parameters of the planet. The parameters are, for example, the pressure and the elements present in the planetary atmosphere which are not known to great extent (Seager & Deming, 2010). In this project, the habitable zone is defined by the standard model by Kasting et al. (1993), which assumes an Earth-like planet around a Sun-like star. The inner boundary of the habitable zone is determined by the distance from the star, which is close enough for a runaway greenhouse effect on an Earth-like planet, due to the high temperatures. The outer boundary of the habitable zone is defined as the distance from the star, where an Earth-like planet needs a maximal greenhouse effect to keep a habitable temperature. The two limits are illustrated in Figure 1.1 for the nominal system of HD 207832.

Figure 1.1: The nominal planetary system in HD 207832 (Haghighipour et al., 2012) and its habitable zone deﬁned by the standard model (Kasting et al., 1993), with a radius between 0.85 AU and 1.65 AU. The inner and the outer planet are HD 207832b and HD 207832c respectively.

7 1.2. THE ORBITAL ELEMENTS CHAPTER 1. INTRODUCTION

1.1.2 The Semi-Amplitude of a Low-Mass Planet The semi-amplitude, of the oscillations in the radial velocity measurements, is denoted K. For a planet with a ﬁxed period, the semi-amplitude is linearly dependent on the planetary mass, K ∝ mp (Perryman, 2011). The linearity can be used to estimate the semi-amplitude for an Earth-like planet in HD 207832, by comparing the masses of the planets in the system. The ﬁtted radial velocity semi-amplitudes for the two planets, HD 207832b and HD +5.2 −1 +2.7 −1 207832c, are 15.3−1.0 ms and 22.1−1.3 ms respectively (Haghighipour et al., 2012). The masses and the semi-amplitudes of the two planets can be approximated to 200 M⊕, and K = 20 ms−1, respectively. By using the linearity of K, an estimate for the semi-amplitude 20 −1 −1 of an Earth-like planet (1 M⊕) gives K⊕ = 200 ms = 0.1 ms . In Haghighipour et al. (2012), the median uncertainty, in the radial velocity measurements, is 1.95 ms−1. An Earth-like planet in HD 207832 is estimated to have semi-amplitudes which are an order of magnitude smaller. Thus the planet would not be detected in the measurements, which motivates the search for a small planet in HD 207832.

1.2 The Orbital Elements

The two following subsections are based on de Pater & Lissauer (2001). By Kepler’s ﬁrst law, planets bound to a star are described by an ellipse where the star is in one of the focal points. The parameters that describe the shape of the ellipse are the semi-major axis a, eccentricity e, and the true anomaly f. The distance r to each point on the ellipse is then given by a(1 − e2) r = , (1.1) ∗ 1 + e · cos f

Here r∗ indicates the distance from the star (focal point) to the planet. f is the angle between the perihelion and the position of the planet. a is the mean distance between q the planet and the star, and e ≡ 1 − b2/a2, where b is the shortest distance between the center of the ellipse and the planet, see Figure 1.2.

8 1.2. THE ORBITAL ELEMENTS CHAPTER 1. INTRODUCTION

Figure 1.2: Some of the basic parameters used to describe a planetary orbit. The shadowed area corresponds to the part of the orbit below a reference plane. The planet is moving in the direction of the arrow, away from the ascending node.

Kepler’s third law relates the semi-major axis to the period for each planet as, 2 3 2 4π a Porb = (1.2) G(m1 + m2) In case the ellipse is tilted with an angle i with respect to a reference plane, a point where the planet crosses the reference plane has been defined as the ascending node. By defining a zero longitude towards a fixed point in space (Figure 1.2), Ω is the longitude to the ascending node; ω is the angle between the ascending node and the pericenter. The longitude of the pericenter is then given by $ = Ω + ω, which is the zero point of the true anomaly f. From the period, Porb, of a planet, a mean motion (angular velocity) is defined 2π as n ≡ .A mean anomaly (MA), has then been introduced as, MA = n(t − t$) + $, Porb where t$ is the time when the planet passes the pericenter. The MA is not to be confused with f.

1.2.1 The Conserved Quantities and Two Coordinates systems The gravitation is the only force considered in this project, because, it is the dominating force that aﬀects the motion of the planets and the star. The energy for a planet in a planetary system, Ep, is given by the sum of the kinetic and the potential energy of the planet. This is done in equation 1.3 below, where the ﬁrst term, on the left-hand side, is the kinetic energy of the planet, and the second term is the potential energy of the planet due to the gravitational pull towards the star. G is the gravitational constant, vp is the velocity, rp∗ is the distance between the planet and the star, and M∗; mp are the masses of the star and the planet respectively. 2 mpvp GM∗mp GM∗ mp − = − = Ep. (1.3) 2 rp∗ 2 ap

9 1.3. PLANETARY EVOLUTION CHAPTER 1. INTRODUCTION

In equation 1.3, it is assumed that the energy and the angular momentum are conserved in every point in the planetary orbit. The sum of the kinetic and potential energy can then be rewritten to be dependent semi-major axis ap which is the middle term in equation 1.3. The potential energy due to the interaction between the planets is neglected. This is justiﬁed, because, the planets are about three orders of magnitude less massive than the star. From the conservation of energy and angular momentum in every point of the orbit, the angular momentum for a planet is given by equation 1.4. However, for a dynamical system where the orbits do change with time, it is the sum of the energies of all the planets, as well as the sum of their angular momentum, that is conserved (Section 1.3).

q 2 |Lp| = mp GM∗ap(1 − ep) (1.4)

The energies and the angular momentum can be well approximated by the use of heliocentric coordinates. This means the origin is defined at the position of the star, where the focal points of the planetary orbits are assumed to be located. Though, the star is not fixed but has an orbital motion, due to the gravitational pull of the planets. It is thus not the star, but the center of mass of the all the bodies in the system that is located at the focal point. Since the star is always at the origin when using the heliocentric coordinates, the kinetic energy of the star is neglected. In this project, the heliocentric approximation is used to calculate the energies due to practical reasons discussed in Section 2.1.1. However, the heliocentric coordinates of the planets do also, due to the motion of the star, induce oscillations in the orbital parameters of the planets, which are non-physical. This is, because, the whole coordinate frame moves with the star. To avoid the non-physical oscillations when analysing the motion of the planets, barycentric coordinates are used in the analysis. The barycentric coordinates have the origin defined at the center of mass, which is a fixed point in the system compared to the star.

1.3 The Evolution of planetary systems

Each orbital parameter, for the bodies in the system, has a time dependence. As the parameters change, the energy and the angular momentum of the whole system has to be conserved. If a planetary system is isolated from other nearby stars, the evolution of the planetary orbits depends on the exchange in angular momentum and the energy between all the bodies in the system, including the host star (Davies et al., 2014). This project takes the planet-planet and the planet-star interactions into account in the simulations. A body in a planetary system with a given semi-major axis has its maximum angular momentum when its orbital eccentricity equals to zero (equation 1.4). An angular momentum deficit has been defined as the sum of the angular momentum of all the planets if they had no eccentricity, subtracted by their true angular momentum. If the angular momentum deficit of the system is high, e.g. it has highly eccentric orbits, the system gets interesting in the sense of increasing the probability for close encounters (Chatterjee et al., 2008).

10 1.3. PLANETARY EVOLUTION CHAPTER 1. INTRODUCTION

In a well behaving system (e.g. low exchange of angular momentum between the planets), the oscillations in the orbital parameters of the planets can, to the first order, be described by a pendulum with a frequency ω (Davies et al., 2014). In a system with greater eccentricities, such as in HD 207832, the exchange of angular momentum is allowed to be large between the planets due to a high angular momentum deficit. This means that the amplitudes of the oscillations of the orbital parameters can have a large diffusion as the planets explore the phase space which conserves the energy and angular momentum (Laskar, 1994). In HD 207832, large amplitudes in the oscillations can be seen (Figure 1.3 & 3.1). The Figure shows how the angular momentum is conserved, because, when the eccentricity of one planet increases, the second planet decreases its eccentricity.

Figure 1.3: The plot shows a time interval from a simulation of HD 207832, where the inner and the outer planet have the initial eccentricities 0.20 and 0.57 respectively. Notably, the amplitudes in the oscillations are found to be large.

1.3.1 The Chaos in Planetary systems A chaotic system means that the evolution of the planetary orbits is very sensitive to the initial conditions of the system (Liao, 2012). For example, a small change in the initial eccentricity of a planet can have a great impact on its long-term evolution. The chaos in a planetary system can be described by analysing the power spectrum of the orbital elements (Guzzo et al., 2002). The number of peaks in a power spectrum, that is required to describe the system, estimates the sensitivity of the orbital elements. Thus, the number of peaks indicates how chaotic the system is (Michtchenko & Ferraz-Mello, 2001). The power spectrum of three diﬀerent simulations, from this project, are shown in Figure 1.4.

11 1.3. PLANETARY EVOLUTION CHAPTER 1. INTRODUCTION

Figure 1.4: The Fourier transform of the eccentricity for the outer planet in HD 207832. The simulations are for the nominal system but with the initial eccentricities, 0.40, for the inner planet, and 0.55, 0.56, and 0.57 for the outer planet respectively. The three systems are classiﬁed as fairly non-chaotic with a number of peaks in their power spectrum within [1-50), see Section 2.2.3.

1.3.2 The Stability of Planetary systems A planetary system can either be stable or unstable. If a system is unstable, it undergoes a radical change. In this project, an unstable system means, that a planet collides, or gets ejected. If no collision between planets or an ejection of a planet occurs in the system for all time, it is called Lagrange stable. However, there is no mathematical criterion to prove Lagrange stability and it is thus not practical to work with. It can only be said that, if a system is predicted to have no collisions or ejections within a time interval, it is Lagrange stable within the predicted time. Another type of stability is the Hill stability. If a system is Hill stable, collisions can not occur between the planets. Note, that an ejection of a planet is still possible and thus a Hill stable system does not necessarily have to be Lagrange stable. The Hill stability is practical in the sense of having a mathematical criterion, which tells that the planetary orbits can not cross each other. In this project, a stable system is referring to the Lagrange stability within a time interval. The Hill stability is then used for convenience to divide the phase space into two regions. One region, where collisions may happen, and a second region, where the planets can not collide. In the next section, the Hill stability is discussed in more detail.

12 1.3. PLANETARY EVOLUTION CHAPTER 1. INTRODUCTION

1.3.3 The Hill Stability If a system is Hill stable, the planets will not undergo collisions. An ejection of a planet may still occur though. The Hill stability does also not take the chaos of the system into account. Chaotic systems may thus also be Hill stable. Hill stable systems, do not have to be Lagrange stable (Section 1.3.2), but it is possible that they are. The Hill stability is a widely used limit, due to its mathematical criterion. In this project, the criterion for a two-planet system, derived by Gladman (1993), is used. In the paper, he gives the equation of the Hill stability limit as,

p ! m m m m (11m + 7m ) = 1 + 34/3 1 2 − 1 2 1 2 + ..., (1.5) a 2/3 4/3 3m (m + m )2 crit m3 (m1 + m2) 3 1 2 where m1 > m2, are the masses of the more and the less massive planets respectively. m3 is the mass of the star. p is the semi-latus rectum deﬁned as p ≡ a(1 − e2), and a, is the semi-major axis. For a two-planet system, with the total initial energy, h, and angular momentum, c, the ratio is expressed in the following equation (Gladman, 1993).

p ! 2M = c2h, (1.6) a G2M 03 where M is the sum of the mass for all the bodies in the system, including the star, G is 0 the gravitational constant, and M ≡ m1m2 + m1m3 + m2m3. With equation 1.3 and 1.4 giving the energy and the angular momentum respectively for the initial system, the limit can be utilized to sort out the hill stable systems. By Calculating the limit for the two-planet system, HD 207832, and letting the eccentricities of the planets vary, a two-dimensional space that illustrates the hill stability limit has been made in Figure 1.5. The lower eccentric region contains the hill stable systems, where ejections of planets is the only type of instability that can occur. Hence, in this project, the Hill stable systems are numerically analysed, before, going to the Hill unstable systems. In Figure 1.5, the one sigma error bars have been plotted, as well as the ellipse spanned by the upper,one sigma, and two sigma limits.

13 1.3. PLANETARY EVOLUTION CHAPTER 1. INTRODUCTION

Figure 1.5: The eccentricity space of the planet HD 207832c versus HD 207832b. The Hill stability limit divides the region where collisions between planets are allowed, from where they are not (Gladman, 1993). The parameters used for this plot can be found in Table 1.1 & 1.2. The nominal system with the eccentricities 0.13 and 0.27 for the respective planet, and the ellipses spanned by its upper one sigma and two sigma error bars, have been plotted.

14 Chapter 2

Method

In this chapter, the integrator used is brieﬂy explained as well as the procedure to produce the ﬁnal results. The project is divided into two parts. First, the planetary stability (see Section 1.3.2) of HD 207832, within the two sigma error bars for the eccentricities of the planets, is looked at. Secondly, the possibility of a small, undetected, habitable planet in the system is investigated.

2.1 The Hybrid Symplectic Integrator

In order to draw conclusions about the stability in HD 207832, the evolution of the planetary systems has to be simulated. In this project, the boundaries in eccentricity space, deﬁned by the two sigma error bars, are analysed. In the highly eccentric systems, close encounters are expected to happen, which the integrator has to be able to handle while still conserving the energy and the angular momentum. The hybrid symplectic integrator, contained in the package, mercury6 (Chambers, 1999), is used in this project. Symplectic integrators solve the Hamiltonian equation of the system, which is the sum of the kinetic and potential energy of the planets and the star. Due to the Hamiltonian for the whole system being dividable into multiple analytically solvable Hamiltonians (Chambers, 1999), the symplectic integrators are superior to normal N-body integrators (e.g. the Runge–Kutta method) in the sense of conserving the energy of the system. However, the symplectic integrators do have problems to conserve the energy if close encounters occur. This is because the interactions between the planets are treated as small perturbations to the Hamiltonian that describes the interaction between each planet and the star. The hybrid symplectic integrator, in mercury6, overcomes the problem with the close encounters by switching from a symplectic integrator to the Bulirsch–Stoer method (Stoer & Bulirsch, 1966), whenever a close encounter happens. An overview of how the hybrid symplectic integrator in mercury6 works can be found in Appendix A.

2.1.1 Working with Mercury6 This section includes practical information about mercury6 which motivates some choices in the methods of this project.

15 2.2. THE STABILITY ANALYSIS CHAPTER 2. METHOD

Coordinate Frames When using Mercury6, the code only reads heliocentric input parameters for the planets and satellites. When working with the initial parameters in this project, it is thus easier and more practical to work in the heliocentric frame. This includes the initial energy and angular momentum calculated with equation 1.3 and 1.4 which is justiﬁed in Section 1.2.1. The input parameters used are asteroidal, meaning the initial orbits are deﬁned with [a,e,i,$,Ω,MA]. For the orbital elements, see Section 1.2. Even though, the Mercury6 code has the option to provide the output data of the integrations in other coordinate systems, e.g. the barycentric frame. In this project, the barycentric coordinates have been used for all output data used in the analysis in order to avoid non-physical oscillations in the planetary motions, (see Section 1.2.1 and 1.3).

The Timestep The hybrid symplectic integrator, in Mercury6, uses a fixed timestep when integrating. This may become problematic if the eccentricities of a planets become high. The planet will travel a longer path during a timestep when it is close to the star, because of the increased velocity. On another hand, the planet will travel a much shorter path during timesteps when it is far away from the star. Fewer points will thus evaluate the position of the planet close to the pericenter which induces larger errors in the energy due to the integrator itself. For each integration, Mercury6 calculates the energy fraction difference due to the integrator itself. An upper limit should be set on the energy fraction difference due to the integrator, to evaluate how good the resulting output data is, e.g. for this project, 10−5 is used as the upper limit.

2.2 The Stability Analysis of HD 207832

This Section describes the ﬁrst part of the project. The large error bars in the eccentricities of HD 207832, allows a large variety of both Hill stable, and unstable systems (Figure 1.5). The observed system is assumed to be stable. A simulation which indicates an unstable system can, thus, not describe the observed system. In the following subsection, the parameters for the simulations, done in the ﬁrst part of this project, are presented.

2.2.1 The Nominal System with Variable Eccentricities The eccentricity space (Figure 1.5), spanned by the planets, HD 207832b and HD 207832c, is to be explored. This is done by integrating the evolution of the planetary orbits using the hybrid symplectic integrator, in Mercury6. For each integration, a set of asteroidal parameters (a,e,i,$,Ω,MA) deﬁnes the shape of the initial orbits of the planets. The semi- major axis, a, the longitude of the pericenter $, and the mean anomaly, MA, for the two planets, are given in Table 1.2. Since the inclination, i, is not known, it is assumed to be

16 2.2. THE STABILITY ANALYSIS CHAPTER 2. METHOD

90◦, as this is likely due to geometrical reasons1. Following, the longitude of the ascending node, Ω, does not aﬀect the initial orbit and is also set to zero. The eccentricities, e, are then left as free parameters within the two sigma error bars in HD 207832. Further quantities which are the same for all simulations are: the masses of the three bodies (Table 1.1 & 1.2), the densities of the planets which are set to Jupiter-like (ρJ ≈ 1.33 g cm−3) (de Pater & Lissauer, 2001), assuming they are gas giants, and lastly the radius of the star (Table 1.1). The distance between the planets before Mercury6 switches from a symplectic integrator, to the Bulirsch–Stoer method, is set to 3 hill radii for both planets which is roughly the recommended distance by Duncan et al. (1998)2. For simplicity when interpreting the output data, the epoch input for each planet is set to zero.

2.2.2 Exploring the Eccentricity Space The eccentricity range, within the two sigma error bars, for the two planets are,

• HD 207832b [0.03 < eb < 0.49]

• HD 207832c [0.07 < ec < 0.71]

The nominal system is then given by an integration which has the initial eccentricities at a point, (ec, eb) = (0.27, 0.13). Using the equation 1.5 and 1.6 with the energy and the angular momentum given by equation 1.3 and 1.4, a hill stability limit is constructed in the eccentricity space, (Figure 3.1). Each point in the plot corresponds to a simulation with the initial parameters given in Section 2.2.1, but with different initial eccentricities. The integration time for each point is 20 Myr, where the maximal energy fraction difference due to the integrator accepted has been set to 10−5. The nominal system, with the initial eccentricities (0.27, 0.13), is firstly simulated in order confirm its stability over at least 20 Myr. The eccentricities are then increased, towards the two sigma limits. The two sigma error bar for the inner planet HD 207832b, are within the hill stability zone, while the hill stability limit intersects the two sigma interval for the outer planet, HD 207832c. The hill stability limit is then used as a baseline for systematic set of integrations, where close encounters are expected, (Section 1.3.3). For the eccentricities 0.1, 0.2, 0.3, 0.4, and 0.5, for the inner planet, HD 207832b, the eccentricity of the outer planet, HD 207832c, is increased stepwise by 0.01 (see Figure 3.1). The eccentricity of HD 207832c is increased until at least four systems have shown to be Lagrange unstable.

1There are more ways to rotate an edge on system than a head on system while still keeping the properties. It is thus more likely to observe an edge on system. 2 Mp1+Mp2 1/3 ap1+ap2 Duncan uses the mutual hill radii, deﬁned as RH,m = ( ) . The hill radii used 3(Mp1+Mp2+M∗) 2 Mp 1/3 in this project is deﬁned for each planet as RH = ( ) ap which, for the planets in question, roughly, Mp+M∗ gives the same distances.

17 2.2. THE STABILITY ANALYSIS CHAPTER 2. METHOD

2.2.3 The Analysis A code is written, which can read the output data files from mercury6 and take the Fourier transform of the orbital elements. The code then estimates the number of peaks in the power spectrum by counting the number of peaks that intersects a line at 1% of the maximal peak value in the power spectrum (Figure 2.1). For each simulated system, which does not suffer any instability in the form of a collision or an ejection, the Fourier Transform is obtained from the oscillations in the eccentricities3 for both of the planets. The number of peaks above %1 of the strongest spectral line defines the level of chaos for each planet. The chaos of the whole system is then given by the planet which is more chaotic. This method has been adopted from Michtchenko & Ferraz-Mello (2001). The simulated systems are then divided into three subgroups with [1-50), [50-100), and (>100) number of peaks. Each group is represented with a colour in the eccentricity space (Figure 3.1). An example of this method, for the point (ec, eb) = (0.20, 0.62), is shown in Figure 2.1. Lastly, the systems, which are unstable within 20 Myr, are inspected. They are divided into groups, depending on their life-time (the time before the collision or the ejection occurred). A plot of the number of simulations versus their life-times is made. This type of plot are well fitted with power-law functions (Shikita et al., 2010). This is done in order to see if most of the unstable systems are covered within 20 Myr. Most instabilities do occur below a time-scale of 101 Myr (Figure 3.2). However, the stable cases with the highest eccentricities on the outer planet (Figure 3.1), are afterwards re-integrated. This time, the simulations are over 100 Myr. This, in order to confirm whether they become unstable within the same order of magnitude in time as the majority of the unstable systems.

Figure 2.1: The level of chaos is determined by the number of peaks that intersect the limit at 1% of the strongest peak. The Fourier transform in the ﬁgure corresponds to the system with the initial eccentricities (ec, eb) = (0.20, 0.62), and has 59 peaks (denoted as a black dot in Figure 3.1). 3The Fourier transform is calculated for the semi-major axis as well. However, the power spectrum of the semi-major axis does not show a clear peak structure, thus the eccentricity is used in the analysis.

18 2.3. THE SEARCH FOR A HABITABLE PLANET CHAPTER 2. METHOD

2.3 The Search for a Habitable Planet

This Section describes the second part of the project, which is to explore the possibilities for a habitable planet in HD 207832. The radial velocity measurements of HD 207832 (Haghighipour et al., 2012), are not able to show if a small planet exists in the system due to the semi-amplitudes induced by the small planet being too small (An estimation of the semi-amplitude is made in Section 1.1.2). Thus, when searching for a habitable planet, a small planet, which could not be detected in the observations, is searched for. A Habitable planet follows if the planet remains stable within the habitable zone in order to support liquid water on its surface. The habitable zone is dependent on many factors as discussed in Section 1.1, and thus the standard model by Kasting et al. (1993) is used which has its boundaries of the habitable zone at 0.85 AU and 1.65 AU.

2.3.1 Simulating with Test Particle By simply inserting an Earth-like planet, has in this project shown not to be an time efficient method to find a stable orbit for the small planet. Instead, test particles4 are introduced. Compared to objects with mass (e.g. an Earth mass planet), the test particles do no interact with the two giant planets. Instead of running multiple simulations with an Earth mass planet, multiple test particles can be used in one simulation. This makes the test particles more time efficient. Since the two confirmed planets in HD 207832 are Jupiter-like (Haghighipour et al., 2012), and the habitable planet in this project is assumed to be Earth-like, the habitable planet has about two orders of magnitude less mass (de Pater & Lissauer, 2001) than the two confirmed planets. Thus it can be treated as a test particle.

The Nominal System A set of test particles with e, i, $, and Ω put to zero, but with the semi-major axis a = [0.85AU, 0.90AU, ..., 1.65AU], are introduced into the nominal system5. This means that the test particles are coplanar with the two giant planets and initially on circular orbits. The focus is put on the initial semi-major axis of the test particles as the number of variables for each test particle is too high to be taken within the scope of this project. However, the integrations are run with four diﬀerent values on the mean anomaly to see how the life-times of the test particles depend on their initial position. The mean anomalies of the test particles are for no speciﬁc reason set to MA = [0◦, 90◦, 180◦, 270◦]. Those simulations are run for 250 Myr.

4Test particles are massless points which do not aﬀect the bodies in the system. The evolution of a test particle is determined by calculating the sum of the forces that the physical bodies in the system exert at the position of the test particle. 5The nominal system is referring to a simulated system, which has the initial eccentricities of HD 207832b and HD 207832c given by 0.13, and 0.27 respectively, presented by Haghighipour et al. (2012). The other parameters for the two Jupiter-like planets are explained in Section 2.2.1

19 2.3. THE SEARCH FOR A HABITABLE PLANET CHAPTER 2. METHOD

When plotting the life-times versus the semi-major axis for the diﬀerent mean anomaly, a pattern can be seen, in Figure 3.3, where the long-lived systems are in the range between 0.85 AU and 1.00 AU. This region is thus simulated with a ﬁner sampling on the semi-major axis, a = [0.85AU, 0.86AU, ..., 1.00AU].

Minus One Sigma The procedure in the previous Section is repeated for the nominal system but with minus one sigma error in the eccentricities of the two Jupiter-like planets (Haghighipour et al., 2012). The two planets, HD 207832b and HD 207832c, then have the initial eccentricities (ec,eb) = (0.17,0.08). For this system, the test particles are expected to have longer life- times as close encounters are less likely in non-eccentric systems (Chatterjee et al., 2008).

20 Chapter 3

Results

In this Chapter, the details in plots and tables are brought up to light. It is divided into the two Sections corresponding to the ﬁrst and the second part of the project.

3.1 Exploring the Eccentricity Space

The result in the first part of the project is represented in a plot (Figure 3.1) of the chaotic behaviour1 and the outcome of each simulation, versus the initial eccentricities used in the simulation for the two planets in HD 207832. Each point corresponds to a simulation which has an integration time of 20 Myr, where the dots represent the stable systems and their level of chaos. × and ◦ denotes the unstable systems that undergo collisions or ejections respectively. For the stable systems, the level of chaos is represented with different colours on the dots. The colours are cyan, black, and red, corresponding to [1-50), [50-100), and (>100) peaks in their power spectrum, respectively. The nominal system is specially marked with a star. The ellipses spanned by the upper one and two sigma error bars are then shown in order to illustrate the phase space of interest in this project. The exploration of the eccentricity space shows a diffuse limit between the stable and unstable systems as the eccentricities of the two planets are changed. However, the systems within the two sigma error bars, where the outer planet, HD 207832c, has an eccentricity above 0.6, are not likely to describe the observed planets in HD 207832. At this region above 0.6, most of the systems become unstable for nearly all eccentricities on the inner planet, HD 207832b. Notably, the systems are behaving relatively non-chaotic, until they are just about to become unstable. The unstable systems, shown in Figure 3.1, are divided into time intervals, depending on their life-time. In Figure 3.2, the number of unstable systems versus their life-time is plotted, where most of the systems undergo instabilities within 2 Myr. The data is then well fitted with a power law with χ2 = 3.15. This supports the work of Shikita et al. (2010), where they show that the life-times indeed can be described well by power laws. The fit tells that simulating the systems, for example, for another 10 Myr (30 Myr), would not lead to a significant increase of unstable systems. In the next Section, the results of

1The behaviour tells if the system ends up unstable or stable. For the stable systems, the chaos is quantiﬁed, see (Section 1.3).

21 3.1. EXPLORING THE ECCENTRICITY SPACE CHAPTER 3. RESULTS the 100 Myr simulations of the most eccentric, stable (Figure 3.1), systems are shown.

Figure 3.1: The plot shows the eccentricity space spanned by the error bars of the outer versus the inner planet in HD 207832. The dots represents the systems which are stable during 20 Myr and their level of chaos (see Section 2.2.3). The unstable systems are denoted as, × and ◦, for a collision between the planets and an ejection of a planet, respectively.

Figure 3.2: The unstable systems in the first part of the project (Figure 3.1), are divided into time intervals depending on their life-time. A power law is then fitted to the data in order to confirm that most of the unstable systems undergo their close encounter before 20 Myr.

22 3.1. EXPLORING THE ECCENTRICITY SPACE CHAPTER 3. RESULTS

3.1.1 100 Myr Simulations For the systems with the outer planet, HD 207832c, having the highest initial eccentricity for a systematic set of eccentricities on the inner planet, HD 207832b, the systems is integrated up to 100 Myr. Firstly, the same timestep as used for the 20 Myr simulations is used to simulate 100 Myr, see Table 3.1. Afterwards, some of the simulations in Table 3.1 are simulated again, but with slightly diﬀerent conditions. E.g. the timestep is smaller or the simulation is a continuation of a 20 Myr simulation, see Table 3.2. It should be noted that even though the energy conservation is acceptable in most of the simulations, the outcome is very diﬀerent, comparing Table 3.1 and 3.2.

Table 3.1: The points that were simulated over 100 Myr with the same timesteps used in the 20 Myr simulations.

(ec, eb) (0.60, 0.10) (0.62, 0.20) (0.61, 0.30) (0.59, 0.40) (0.63, 0.50) Outcome Stable Collision Stable Stable Collision Life-time [Myr] >100.00 55.30 >100.00 >100.00 64.98 #Peaks 1315 255 1495 335 79 Timestep [days] 0.5 0.1 0.5 0.5 0.1

Table 3.2: The 100 Myr simulations with slightly diﬀerent initial conditions. (0.60, 0.10) is a continuation of the 20 Myr simulation. (0.61, 0.30), (0.59, 0.40), and (0.63, 0.50), are simulated with smaller timesteps compared to in Table 3.1. For (0.59, 0.40) the fractional energy change due to the integrator becomes larger than 10−5 and is thus rejected.

(ec, eb) (0.60, 0.10) (0.61, 0.30) (0.59, 0.40) (0.63, 0.50) Outcome Collision Collision - Collision Life-time [Myr] 61.41 12.71 - 4.86 #Peaks 177 69 - 21 Timestep [days] 0.5 0.05 0.05 0.05

Notably, because of the smaller timesteps, two of the simulations in Table 3.2 result in collisions before 20 Myr, and the simulation, based on the point (0.59, 0.40), has a bad energy conservation. An acceptable can easily be obtained by changing the timestep and re-simulating the system. However, one simulation can take several days which would not ﬁt within the schedule of this project. Furthermore, for (0.60, 0.10), the fact that it is a continuation of a previous simulation, the system goes from a stable system (Table 3.1), to an unstable system (Table 3.2). Chaos is strongly dependent on small changes in the initial parameters (Liao, 2012) as mentioned in Section 1.3. These are examples of the sensitivity of chaos in simulated planetary systems. Comparing long simulations with short, e.g. 100 Myr versus 20 Myr, the number of peaks in the power spectrum increases with the integration time. This is a consequence of the method used (Michtchenko & Ferraz-Mello, 2001), which is veriﬁed in this project. The

23 3.2. THE POSSIBILITY OF A HABITABLE PLANET CHAPTER 3. RESULTS

resolution in the power spectrum goes as 1/T , where T is the integration time. More peaks are thus resolved with a longer integration time. The longer integrated systems will thus appear more chaotic. The level of chaos, determined by the method used in this project, can then only be used to compare between systems that have about the same simulation time.

3.2 The Possibility of a Habitable Planet

Simulating the small planet in the form of test particles for a set of different semi-major axes, and with four different mean anomaly, the two plots, shown in figure 3.3, are obtained. The upper plot shows the case when the test particles are put in the nominal system ((ec, eb) = (0.27, 0.13)). The lower plot corresponds to the case when the same set of test 2 particles are put into the minus one sigma system ((ec, eb) = (0.17, 0.08)). For the nominal system, only one value on the semi-major axis (a = 0.99 AU) gives a stable system over 250 Myr, which is the integration time. This point is shown to be stable for all the four mean anomalies used. For the minus one sigma system, many more initial conditions result in systems, stable over at least 250 Myr. The stable systems are also more spread out over the habitable zone. It can be noted that for some cases, the differences in the mean anomalies can change the life-time of the test particles with over five orders of magnitude. This shows the importance of the initial position for the habitable planet when placed in the system. The results shown in Figure 3.3 indicates that within the space, spanned by the semi- major axis in the habitable zone, there are orbits where a habitable planet could be located. However, this is only in the case of a non-eccentric planet with a negligible mass compared to its two Jupiter-like neighbours.

2The minus one sigma system is identical to the nominal system but with minus one sigma in the eccentricities of the two planets.

24 3.2. THE POSSIBILITY OF A HABITABLE PLANET CHAPTER 3. RESULTS

Figure 3.3: The life-time of each test particle as a function of the initial semi-major axis. The upper plot corresponds to the nominal system, and the lower plot corresponds to the minus one sigma system. For the nominal system, the region between 0.85 AU and 1 AU is analysed in greater detail due to the longer life-times. The green lines mark the boundaries of the habitable zone. The arrows indicate that the life-time of the test particles exceeds 250 Myr which is the integration time.

25 Chapter 4

Conclusions

The aim of the first part of this project is to refine the error bars of the planetary system in HD 207832. The aim of the second part of the project is to find stable planetary orbits within the habitable zone, where a small mass planet (e.g. Earth-like) could be located. In this chapter, the results are discussed followed by the conclusions which are set out as bullet points.

4.1 The Eccentricity Space of HD 207832

The simulations clearly show that the planets in HD 207832 destabilize as the Hill unstable region is analysed (Figure 3.1). Most systems are shown unstable above an eccentricity on the outer planet, HD 207832c, of 0.6. This is below the two sigma limit which stretches to an eccentricity of 0.71. If the systems were simulated over longer periods of time, an even lower limit on the eccentricity boundaries would probably be obtained.

• Assuming the planetary system is Lagrange stable over at least 20 Myr, the eccentricity of HD 207832c can not exceed 0.60 in order to describe the planetary system.

The Hill stable systems have, in this project, shown to be Lagrange stable for at least 20 Myr. The chaos of these systems is relatively low with [1-50) peaks in their power spectrum. Notably, the ejections, which are allowed in Hill stable systems, have in this project shown to occur only in the Hill unstable region. Longer simulations, e.g. for the life time of the star, are needed to confirm this. In this project, the timestep dependence of the outcome has been noticed. The small change between two different timesteps used in the simulations, which both conserve the energy, can cause the difference between a stable and an unstable system. The question to ask is which timesteps should be used in order to represent the observed system. A suggestion is to plot the outcome as a function of timestep for a given fraction of energy difference due to the integrator. The plot could give an idea if it is randomly scattered outcomes, or which outcome is more probable.

26 4.2. THE LEVEL OF CHAOS CHAPTER 4. CONCLUSIONS

4.2 The Level of Chaos

The method to determine the level of chaos in the systems is heavily dependent on the resolution in the power spectrum, which has also been pointed out by Michtchenko & Ferraz-Mello (2001). Comparing the 100 Myr simulations with 20 Myr is thus not reason- able. Furthermore, there is yet no deﬁnition for the level of chaos that corresponds to a speciﬁc number of peaks in a fully resolved power spectrum. This means that the method can only tell if a system is more, or less, chaotic compared to another system with the same integration time. An example, when the level of chaos is interesting for this matter, is in the unstable systems that have low integration times before they undergo the instability. Such systems may not give a resolved power spectrum. In order to compare the spectrum of an unstable with a stable simulation, the resolution of both spectra should be about the same.

• The level of chaos determined by counting the number of peaks in the power spectrum is heavily dependent on the resolution. Thus it can only be used to compare the chaos of systems which have about the same integration time.

4.3 Small Habitable Planets in HD 207832

In this project, the semi-major axis of the habitable planet is investigated. There are several more orbital elements which should be explored in order to ﬁnd all possible orbits for a habitable planet. The mean anomaly can itself change the life-time of a small habitable planet by several orders of magnitude as shown in Figure 3.3. The nominal system in the upper plot in Figure 3.3, shows that the test particles are unstable at 1 AU, whereas for 0.99 AU, the test particles are Lagrange stable for 250 Myr, for all four mean anomalies. We point out that the even if the mean anomalies are in some cases mirror images of each other, e.g. 0◦ and 180◦, the life time of the mirrored test particles in our simulations have shown to vary by orders of magnitude. The mirror images do not show to have the same outcome. However, for a better result, the whole mean anomaly space should be explored in a future project. This is an example of the sensitivity, that the life-time of the test particles has on the initial parameters. This indicates a chaotic system which is very sensitive to the initial conditions. Commenting on the minus one sigma system when the planets HD 207832b and HD 207832c have lower eccentricities, (lower plot, Figure 3.3), it can be concluded that the amount of Lagrange stable orbits, for the simulated time, increases.

• The survival of a small planet between the two giant planets in HD 207832 is very sensitive to the initial parameters. In the nominal system, for a semi-major axis of 0.99 AU, with the mean anomalies 0◦, 90◦, 180◦, and 270◦, at least four orbits for test particles, which are Lagrange stable for at least 250 Myr, can be found. See Section 2.3.1 for the other orbital parameters.

27 4.3. THE HABITABLE PLANET CHAPTER 4. CONCLUSIONS

The Lagrange stable simulations should be followed up by exploring more parameters in the phase space for the small habitable planet. Reminding that the error bars of the eccentricities for HD 207832b and HD 207832c are large, the phase space of this planetary system is very large. It is worth mentioning that Haghighipour et al. (2012) simulate the nominal system of HD 207832, with a 1 M⊕ planet with zero eccentricity on the initial semi-major axis 0.05-0.4 AU, and 0.75-1.25 AU. They only found stable orbits, for the duration of their integration time, between 0.05 and 0.3 AU. This project suggests that they did not ﬁnd any Lagrange stable systems due to the sensitivity of the initial parameters. For example, the importance of the mean anomaly and possibly the other orbital elements can have a large impact on the outcome.

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30 Appendix A

The Principle used in Mercury6

The hybrid symplectic integrator in Mercury6 allows close encounters between bodies (Chambers, 1999). It solves the Hamiltonian of the system, which is the sum of the kinetic and the potential energy of the whole planetary system. Compared to equation 1.3, the following equation generalizes the expression for the energy, taking all the interaction between the bodies in the system into count. N 2 N N X Pi X X mj H = − G mi , (A.1) i=1 2mi i=1 j=i+1 rij

Together with Hamilton’s equation of motion dxi = ∂H ; dpi = − ∂H , where x is the position dt ∂pi dt ∂xi and p is the momentum, the change in any parameter, q, with respect to time, t, can be written as, dq n ∂q dx ∂q dp n ∂q ∂H ∂q ∂H = X( i + i ) = X( + ) = F q, (A.2) dt i=1 ∂xi dt ∂pi dt i=1 ∂xi ∂pi ∂pi ∂xi where F is an operator on q, The general solution is then given by, q(t) = eτF q(t − τ), (A.3)

where τ is the time step. However, the general solution is not solvable in an analytic way for a single Hamiltonian that takes all the interactions in the system into count (Chambers, 1999). Though, the Hamiltonian can be divided into a sum of Hamiltonians which deal with diﬀerent interaction, in the planetary system, which are solvable. In hybrid symplectic integrator in Mercury6, the Hamiltonian is divided into two parts. One, which includes the kinetic energy of the system and the gravitational potential for each planet due to the star, and the second, which contains the interaction between the planets. This is shown in

the following equations, N 2 X pi Gm∗mi HA = ( − ), (A.4) i=1 2mi ri∗ N N X X mimj HB = −G (A.5) i=1 j=i+1 rij

The advantage of choosing one Hamiltonian with the dominant (HA) and one with the minor energies (HB), is due to the error in the total energy being proportional to a parameter, , which is deﬁned as HB ∼ HA (Chambers, 1999). Equation A.4 and A.5 are

31 APPENDIX A. THE PRINCIPLE USED IN MERCURY6

analytically solvable two body problems. The solution of each Hamiltonian can then be combined to approximate the real solution. To ﬁrst order the solution of a parameter, q, which can be the position or momentum of a planet, is given by, q(t) = eτAeτBq(t − τ), (A.6)

where A and B are operators of the same type as F for HA and HB respectively, e.g. n ∂ ∂H ∂ ∂H A = X( + ), i=1 ∂xi ∂pi ∂pi ∂xi compare with equation A.2. The problem with symplectic integrators is when close encounters between planets occur, such that the distance, rij, becomes small. In this case HB becomes comparable to HA, meaning the error in the calculations of energy become large. In the hybrid symplectic integrator in mercury6, this has been taken care of by introducing a third term in equation A.4 which includes the interaction between the planets. This term is then multiplied with a function, 1−K(rij), where K(rij) approaches a value of zero, when two planets are about to undergo a close encounter (when r is small), and approach a value of one, when there is no close encounter (when r is large). Equation A.4 and A.5 are then replaced with the following Hamiltonians. N 2 N N X pi Gm∗mi X X mimj HA = ( − ) − G [1 − K(rij)], (A.7) i=1 2mi ri∗ i=1 j=i+1 rij

N N X X mimj HB = −G K(rij). (A.8) i=1 j=i+1 rij During the close encounters, the Hamiltonian becomes a three-body problem which is not analytically solvable. In this case, the hybrid symplectic integrator in Mercury6 have been coded to go, from the use of the symplectic integrator, to the use of the Bulirsch–Stoer method (Stoer & Bulirsch, 1966), which is independent on the size of the perturbations from the diﬀerent interactions. Despite not being a symplectic integrator, when close encounters occur, the Bulirsch–Stoer method conserves the energy better. The change between the two integration methods during simulations makes integrator hybrid. For a detailed description see (Chambers, 1999) with references.