Stable Configurations of Planetary Systems Profielwerkstuk (N&T) Muriel van der Laan, het 4e gymnasium, Amsterdam begeleider: Sven Aerts February 2015 Contents Preface . 3 Introduction . 4 Introductie . 6 1 Kepler's Laws of Planetary Motion 9 1.1 Newton's Law of Gravitation . 9 1.2 Some Definitions . 10 1.2.1 Momentum . 10 1.2.2 Angular Momentum . 10 1.2.3 Total Energy . 10 1.3 Kepler's First Law . 11 1.3.1 A Closer Look at the Total Energy . 11 1.3.2 The Laplace-Runge-Lenz Vector . 11 1.3.3 Kepler's First Law . 12 1.4 Kepler's Second Law . 15 1.5 Kepler's Third Law . 16 1.5.1 The Area of the Ellipse . 16 1.5.2 The Semi-minor Axis . 17 1.5.3 Kepler's Third Law . 18 2 Chaos in Planetary Motion 19 3 Leapfrog Integration 21 3.1 Euler Integration . 21 3.2 Leapfrog Integration . 22 3.3 Other Benefits of Leapfrog Integration . 24 3.4 Implementation . 24 3.5 Numerical Analysis of Accuracy . 26 1 4 Determining the Starting Values 30 4.1 The Mean Anomaly . 31 4.2 The Eccentric Anomaly . 33 4.3 The True Anomaly . 35 4.4 The Distance from the Star . 37 4.5 Rotating the Orbit . 38 5 Determining Lyapunov Exponents 40 5.1 Implementation . 41 5.2 Numerical analysis of the results . 42 6 Monte Carlo Optimization 47 6.1 Theory . 47 6.2 Implementation . 47 7 Results 51 7.1 Gliese 777 . 51 7.2 23 Librae . 52 7.3 HD 155358 . 52 7.4 Kepler 68 . 52 7.5 47 Ursae Majoris . 53 8 Conclusions & Discussion 59 8.1 The extent of similarity to the initial systems . 59 8.2 Possible improvements . 60 Acknowledgements 72 A Properties of ellipses 73 B Other Formulae 74 C The Complete Program 75 2 Preface My interest in the stability of planetary systems was first sparked while I was reading '17 equations that changed the world' by Ian Stewart. The fourth equation was Newton's law of gravitation. The book went on to explain that Newton had used it to prove Kepler's laws of planetary motion, which were quite familiar to me, but did not give the proof. I was quite frustrated (if you are like me and are frustrated by proofs which are mentioned, but not given, I would like to point you towards chapter 1 and advise you to skip chapter 2). A little later the book mentioned multi-planetary dynamics and I was intrigued. Skip forward a few months and I'm trying to find a good topic for my profielwerkstuk. While leafing through every single book in my room (that's a lot of books) I came across this chapter again and figured the stability of planetary systems would be a 'pretty cool' topic and would finally give me a good excuse to take a look at that proof of Kepler's laws. My first plan was rather vague: I wanted to make a program to determine the stability of a system and calculate ’stuff' with it. I talked to Simon Portegies Zwart about it, and he pointed me in the direction my profielwerkstuk took in the end and suggested some methods I could use, such as leapfrog integration. After I'd done some research on these methods (and some others) and the theoretical background of the problem I started to write the program. This was a lengthy and at times incredibly frustrating process, but the program worked in the end. I proceeded to write a report of all the interesting things I'd done, which is what you are currently holding in your hands. 3 Introduction There currently are two main ways of detecting exoplanets (planets that orbit around other stars than our sun). Firstly, there is the transit method. This method depends on the planet in question blocking the light that we receive from the star. There are a number of constrictions on what planets we can detect using this method: the planet has to pass directly in front of the star, the planet has to be big enough to reduce the amount of light significantly and the planet has to pass in front of the star often enough for us to observe a transit in a limited timespan. Secondly, astronomers can use a phenomenon called a Doppler shift: when an object that emits light, like a star, moves towards or away from us the wavelength of the light that we receive changes. If this wavelength is measured carefully the velocity of the object can be determined. This effect can be used to determine the motion of a star under the influence of an exoplanet's gravity. This method also has some restrictions: the planet has to be massive enough and close enough to the planet to exert a significant force on the planet. In addition to these methods exoplanets can also be observed directly, although this rarely happens. Using these methods a large number of exoplanets has been found: the numbers are currently running in the thousands. These confirmed exoplanets mainly belong to a very specific sub- set of planets, however: they are mostly large, massive and orbit very close to their host stars. Smaller planets on larger orbits are rarely found. This is unfortunate, because these would be very in- teresting to study. A fair number of these planets lie in the so-called habitable zone of their host star: not so close to the star that water evaporates, but not so far away from the star that water freezes. On these planets liquid water (believed to be essential for extraterres- trial life) can exist. Except for finding exoplanets these methods can also be used to 4 determine several characteristics of the exoplanets and their orbits. This gets a lot harder when there are multiple planets in the system, because the dynamics of the system become a lot more complicated. If the system consists of one star and one planet their relative mo- tion obeys Kepler's laws of planetary motion and the orbit of the planet is periodic and predictable. When there are more planets in the system the gravitational forces from the other bodies in the sys- tem disturb these orbits and the motion becomes chaotic and thus unpredictable. Usually this effect is small: the mass of the other planets is tiny compared to that of the star, so the gravitational force from those planets is nearly negligible. There are, however, some configurations of systems in which this is not the case. This is the upside of chaos: by ruling out these configurations we can learn a lot about what configurations are possible and likely to occur. The goal of this project was to find the most stable configura- tion of a planetary system. More concretely: given the mass of a star and the orbital elements of a planet (which remain fixed), what planet(s) should be added to the system to maximize the stability? The first two chapters of this report give (some of) the theoreti- cal background of the problem. The stability was determined by finding lyapunov exponents (chapter 5). The initial conditions of the system were found using an algorithm described in chapter 4. To find the time evolution of the systems (required for determining the lyapunov exponent) the leapfrog algorithm (chapter 3) was used. The lyapunov exponent was minimized using a Monte Carlo method (chapter 6). The results of these calculations are given in chapter 7 and discussed in chapter 8. The orbital elements were taken from five real planetary systems. In these planetary systems the most massive planet has the largest influence on the other objects in the system. For this reason the most massive planet was chosen to act as the fixed planet in the system. Real systems were used so that the results could be compared to reality, which is done in chapter 8. 5 Introductie Er zijn op het moment twee belangrijke manieren om exoplaneten (planeten in een baan om een andere ster dan de zon) te detecteren. Als eerste is er de transit-methode. Deze methode maakt ervan ge- bruik dat een planeet op een baan rond de ster voor de ster langs beweegt en daarbij een deel van het licht van de ster blokkeert, wat kan worden waargenomen. De planeten die we kunnen detecteren met deze methode moeten wel aan een aantal eisen voldoen: de planeet moet recht voor de ster passeren, de planeet moet groot genoeg zijn om een meetbare hoeveelheid licht te blokkeren en de planeet moet vaak genoeg voor de ster langs bewegen om de transit te kunnen waarnemen in een beperkte periode. Als tweede kunnen sterrenkundigen gebruik maken van een fenoneem dat dopplerver- schuiving heet: als een object dat licht afgeeft, zoals een ster, naar ons toe of van ons af beweegt verandert de golflengte van het licht dat we ontvangen. Als deze golflengte precies genoeg wordt geme- ten kan de snelheid van het object worden bepaald. Dit effect kan gebruikt worden om de beweging van de ster onder invloed van de zwaartekracht van de exoplaneet te bepalen. Ook de planeten die met deze methode gedetecteerd kunnen worden moeten aan een aan- tal eisen voldoen: de planeet moet zwaar genoeg zijn en dicht genoeg bij de ster staan om deze zoveel te laten bewegen dat we het kun- nen waarnemen. Hiernaast kunnen exoplaneten ook direct worden waargenomen, hoewel dit weinig voorkomt. Door gebruik te maken van deze methodes is er een groot aantal exoplaneten gevonden: het aantal loopt in de duizenden.