DEGREE PROJECT IN TECHNOLOGY, FIRST CYCLE, 15 CREDITS STOCKHOLM, SWEDEN 2018 Discovery of Neptune GUSTAV ERIKSSON KEVIN GARCIA MARTIN KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES EXAMENSARBETE INOM TEKNIK, GRUNDNIVÅ, 15 HP STOCKHOLM, SVERIGE 2018 Upptäckten av Neptunus GUSTAV ERIKSSON KEVIN GARCIA MARTIN KTH SKOLAN FÖR TEKNIKVETENSKAP Abstract This project is an analysis of how a planet can be found in space with the aid of mathematics. This is based on the fact that in the 19th century two mathematicians John C. Adams and Urbain Le Verrier both independent of each other found Neptune, the 8th planet in the solar system, by calculating its location based on discrepancies between theoretical and observed longitudes. We recreate Adams’ problem and solve it with numerical analysis to see how one could improve this method of finding a planet using mathematics. We created a model of the solar system using Runge-Kutta 4 (RK4) to solve ODE’s explaining how the planets affect each other. We then created an inverse problem where we pretended that Neptune did not exist and tried to find its position and data using Gauss-Newton’s algorithm. Our method gives a better result than those of Adams, although we use a better start guess for the position of Neptune than he did. The important parameter to find is at what direction to look for the planet, also called the longitude angle. Both Adams and us get close to the correct longitude — Adams’ being 2:5◦ off and us within 1◦. This is especially interesting since without getting this parameter correct they would never have found the planet at that time. Sammanfattning Detta projekt är en analys om hur en planet kan hittas i rymden med hjälp av matematik. Det är baserat på två matematiker, John C. Adams och Urbain Le Verrier, som på 1800-talet oberoende av varandra hittade Neptunus, den åttonde planeten i solsystemet, genom att approximera dess position baserat på avvikelser mellan teoretiska och observerade longituder. Vi återskapar Adams problem och löser det med numerisk analys för att se hur man kan förbättra metoden att hitta planeter genom matematik. Vi skapade en modell av solsystemet med Runge-Kutta 4 (RK4) för att lösa ODE’s som beskriver hur planeterna påverkar varandra. Sedan skapar vi ett inverterat problem där vi låtsas om att Neptunus inte finns och försöker hitta dess position med Gauss-Newtons algoritm. Vår metod ger ett bättre resultat än Adams, vilket beror på att vi använder en bättre startgissning för Neptunus position. Den viktiga parametern att hitta är vid vilken vinkel man ska kolla efter planeten, även kallat longitudvinkeln. Både Adams och vi kommer nära det riktiga värdet — Adams är 2; 5◦ ifrån och vi är inom 1◦. Detta är särskilt intressant eftersom de aldrig skulle hittat planeten utan denna parameter. Contents 1 Introduction 1 1.1 Background.........................................1 1.2 Problem...........................................2 1.3 Result............................................3 2 Simulation of the solar system4 2.1 Newton’s law of universal gravitation...........................4 2.2 System of ODEs for the solar system............................5 2.3 Runge-Kutta 4........................................6 2.4 Acceleration.........................................7 3 Recreating Adams’ problem 12 3.1 Difference in longitude................................... 13 3.2 Changing Uranus’ initial values.............................. 14 3.3 Discussion.......................................... 15 4 Finding Neptune 16 4.1 Gauss-Newton........................................ 16 4.1.1 Stability depending on initial guess........................ 19 4.1.2 Convergence.................................... 19 4.1.3 Minimize residual for position vs angle...................... 20 4.1.4 Local minima.................................... 20 4.1.5 Problem with divergence.............................. 20 4.2 Monte Carlo method.................................... 21 4.3 Conclusions......................................... 26 Chapter 1 Introduction 1.1 Background In 1781 astronomer William Herschel observed and discovered a heavenly body he thought to be either a comet or a star. He later determined the body to be a comet since observations showed that the body was moving. Other astronomers started to research the discovery and Johann E. Bode came in 1783 to the conclusion that the body was a planet due to its nearly circular orbit [1]. This planet was named Uranus, being the 7th planet in the solar system and the first to be discovered with the aid of a telescope. In the aftermath of Uranus’ discovery, astronomers started to observe and record the arc of the planets orbit. The astronomer Alexis Bouvard found discrepancies between the observed and theoretical orbit of Uranus and came to the conclusion that an undiscovered planet beyond Uranus was the cause. In 1821 Bouvard compiled a table of the differences between theoretical and observed heliocentric longitude which later came to be very important for the discovery of the planet beyond Uranus, today known as Neptune. In 1843 the British mathematician John C. Adams made a first attempt at solving the problem with the positional discrepancies in Uranus’ orbit. Adams’ conclusion was that "The result showed [...] a good general agreement between theory and observation might be obtained." [2, p. 266] and he was determined to further investigate the reason for the differences in theory and observation for Uranus orbit. At the same time the French mathematician Urbain Le Verrier also tried to predict the undiscovered planets position using mathematics. Figure 1.1: John C. Adams and Urbain Le Verrier 1 Independent of each other’s work both Adams and Le Verrier presented predictions for Neptune’s position in 1846; Le Verrier on August 31st and Adams in the beginning of September [2, p. 267]. Here is a picture, figure 1.2, taken from Adams’ article presenting one of his predictions. Figure 1.2: Adams’ predictions presented in his article [2] Adams also writes that he had no intention of interfering with Le Verrier’s claims to the discovery of Neptune since the Frenchman presented his predictions first. The actual discovery of Neptune was made by Johann G. Galle on September 23rd in 1846. Using the predicted position calculated by Le Verrier, Galle found the planet within 1◦ and later observations showed that Adams’ prediction was off by 2.5◦. Both Adams and Le Verrier assumed Bode’s law (see Section 4.3) to apply for Neptune, an incorrect assumption, resulting in substantial error in the predicted data for the planet. The results of Adams’ and Le Verrier’s calculated predictions compared with Neptune’s true data is presented in table 1.1. Table 1.1: Calculated and real data for Neptune, values from [3] Neptune Adams Le Verrier Mean Distance [AU] 30.1 37.2 36.2 Longitude to Perihelon [◦] 44 299 284 Eccentricity 0.0113 0.121 0.108 Neptune’s Mass [1026 kg] 1,024 3,014 2,127 Heliocentric Longitude [◦] 328.4 330.9 327.4 The error in mean distance is a direct consequence of the assumption that Bode’s law applies for Neptune. Longitude to perihelion is the angle (longitude) to the planet’s position when it is closest to the sun (perihelion) in its orbit. Eccentricity e is a measurement of how much the orbit deviates from being circular; 0 < e < 1 gives an elliptic orbit and e = 0 makes it circular. Both the longitude to perihelion and eccentricity was determined after the actual discovery in an attempt to describe Neptune’s orbit. This involved extremely extensive calculations at the time which probably caused the errors [3]. The error in mass is a direct cause of trying to predict Neptune’s position assuming Bode’s law, where their increase of over 20% in mean distance results in a correspondingly much too large mass [4]. 1.2 Problem Both Adams and Le Verrier had to do their calculations and iterations by hand, without the aid of current technology and modern methods for solving differential equations. Their approach to predicting the position for Neptune was first to linearize the system of equations and then solve these with use of least squares. In this thesis the problem considered will be how to find Neptune using numerical analysis implemented in MATLAB using the same initial data as Adams and Le Verrier. The project will be split 2 into three parts; simulating the solar system, recreating Adams’ problem and to find Neptune using the Gauss-Newton algorithm and perturbation analysis. The simulation of the solar system was made by solving ODEs describing the planets motion with Runge-Kutta 4 in MATLAB. We decided to focus on Adams’ calculations since these are presented in English unlike Le Verrier’s which are in French. To recreate Adams’ inital problem we try to find values that satisfies the data Adams used. To find Neptune we decided to use the Gauss-Newton algorithm which solves a non-linear least square problem by minimizing the sum of the residuals. We also applied a perturbation analysis — the Monte Carlo method — on our Gauss-Newton solver to show the uncertainties in Neptune’s position due to uncertainty in observed data. 1.3 Result Here are our results compared with Le Verriers and Adams. Table 1.2: Calculated and real data for Neptune, values from [3], compared with our result Neptune Adams Le Verrier Eriksson / Garcia Martin Mean Distance [AU] 30.1 37.2 36.2 30.0± 0.06 Longitude to Perihelon [◦] 44 299 284 42.2± 5 Eccentricity 0.0113 0.121 0.108 0.0104± 0.0012 Neptune’s Mass [1026 kg] 1.024 3.014 2.127 1.024± 0.002 Heliocentric Longitude [◦] 328.4 330.9 327.4 329.0± 0.08 3 Chapter 2 Simulation of the solar system Before finding a way to calculate the location of Neptune we make a simulation of the solar system based on data from NASA [5] of the planets’ position, velocity and mass at a given date to act as a reference for the rest of the project.