Snapshots of modern mathematics № 9/2017 from Oberwolfach Computing the long term evolution of the solar system with geometric numerical integrators Shaula Fiorelli Vilmart • Gilles Vilmart 1 Simulating the dynamics of the Sun–Earth–Moon sys- tem with a standard algorithm yields a dramatically wrong solution, predicting that the Moon is ejected from its orbit. In contrast, a well chosen algorithm with the same initial data yields the correct behavior. We explain the main ideas of how the evolution of the solar system can be computed over long times by taking advantage of so-called geometric numerical methods. Short sample codes are provided for the Sun–Earth–Moon system. 2 1 Computing the trajectories Let us step back in time and imagine we are on the first of January of the year 1600, when Johannes Kepler (1571–1630) has just moved to Prague to become the new assistant of the astronomer Tycho Brahe. Kepler had to escape from persecution in Graz, particularly caused by his adhesion to the controversial Copernican theory, boldly saying that the planets revolve around the Sun. 1 Partially supported by the Swiss National Science Foundation grants 200020_144313/1 and 200021_162404. 2 An earlier version in French of this article first appeared in [11]. 1 How the strange motion of Mars inspired Kepler. Tycho Brahe is very interested in planetary motion and has already calculated very precisely the orbits of known planets. But Mars escapes comprehension: he cannot properly predict its trajectory. Without warning him of the difficulty, Brahe asks Kepler to calculate the precise orbit of Mars. It will take about six years for Kepler to complete this work. Indeed, while Venus has a nearly circular orbit, the trajectory of Mars is more complex: it turns out to be an ellipse, whose flattening 3 is, after Mercury’s, the second largest of all planets’ in the solar system. P b 1 P b 2 S C F b b b b b Semi-major axis a b P4 b P3 Figure 1: Illustration of Kepler’s laws. The Sun S and the point F are the foci of the elliptic trajectory (in red) of the planet P . By the law of equal areas, when the time elapsed between positions P1 and P2 is equal to the time elapsed between positions P3 and P4, then the domains SP1P2 and SP3P4 (in blue) have the same area. Kepler’s three laws. This takes Kepler to propose his three basic laws (com- pare Figure 1): 1. the planets describe elliptical orbits, with the Sun at a focus of each; 2. the segment connecting the Sun and a planet sweeps out equal areas during equal times; this so-called invariant of the problem is called the law of equal areas 4 ; 3 The flattening may be measured by the eccentricity of the ellipse, which is the ratio of the distance between C and F to the semi-major axis a, compare Figure 1. The smaller its eccentricity, the more an ellipse is shaped like a circle. 4 That the derivative with respect to time of the area swept out is zero turns out to be connected to the conservation of the planet’s angular momentum. 2 3. the square of the orbital period T of a planet is proportional to the cube of the semi-major axis a of the elliptical orbit: T 2 = Constant · a3. Newton’s laws of motion and the universal law of gravitation. In 1687, Isaac Newton (1642–1727) publishes his book Philosophiæ Naturalis Principia Mathematica. Inspired by Kepler’s work, he proposes the three laws of motion and the universal law of gravitation, stating that all cosmic objects attract each other mutually with equal forces (but in opposite directions), proportional to the product of their masses and inversely proportional to the square of the distance between them. Let mS and mP denote the masses of two bodies S and P , let D be the distance between S and P , and let −→u be a vector with unit length in the −→ direction from S to P . The gravitational force F S→P applied by S to P is then given by the formula −→ −→ m m F = −F = −G S P −→u , S→P P →S D2 where the constant of proportionality G is called the universal constant of gravitation. It is this law that we will use later to calculate the motion of the planets. Newton’s explanation of Kepler’s law of equal areas. Newton also pro- vides a justification for Kepler’s laws: they represent the motion of a body subject to a single gravitational force, that of the Sun. To explain the law of equal areas, he uses a method that can be interpreted as the first geometric numerical integrator, as presented in [3, 9]. The idea is to apply the Sun’s attractive gravitational force not continuously over time, but via a sequence of impulses: Suppose that S is the position of the Sun, and suppose that a planet is located initially at point A, see Figure 2. Let us first assume that the Sun applies no gravitational force at all; in this case, the planet moves during a certain time from A to a point B along a straight −−→ line, with a constant velocity in the direction AB (see Figure 2). Waiting for the same time again, the planet should continue on the same straight line until −−→ −→ it reaches the point c, with AB = Bc. However, let us apply an impulse of force from the Sun to the planet: this force adds a velocity component to the motion of the planet that Newton −−→ represents by the vector BV along the segment SB. The planet’s velocity is −→ −−→ now the sum of two components: the vector Bc and the vector BV , and the −−→ resulting vector is BC which defines the point C. The planet thus moves with this new constant velocity until it reaches C. 3 bc c C b V b B b Sb A (a) Figure from Newton’s (b) The planet’s motion subject Principia Mathematica (1687). to a single “force impulse”. Figure 2: Newton’s proof of Kepler’s second law. Iterating this process, the planet follows the path A, B, C, D, E, F, . .. New- ton proves that all the triangles SAB, SBC, SCD, SDE, SEF have the same area by the following argument: Newton’s geometric proof. We first note that the triangles SAB and SBc −−→ −→ have the same area, because they have the same basis AB = Bc and the same height issued from S. Next, observing that BcCV is a parallelogram, we deduce −→ −→ that Cc is parallel to SB. The triangles SBc and SBC thus have the same −→ basis SB and the same heights issued from c and C respectively; and hence they have the same area. Thus, the triangles SAB and SBC have the same area. Newton thus proves a discrete version of Kepler’s second law, meaning a motion with successive jolts. The process which permits to get from A to B, then B to C, and so forth, corresponds in fact to a geometric numerical scheme, known today as the symplectic Euler method, which we will present later on. We will also see that as the time interval between two force impulses tends to zero, the obtained approximation of the trajectory converges towards the exact solution of the problem. 2 What is a differential equation, and how can we solve it? Many physical phenomena can be modeled by differential equations, that is to say, equations in which the unknown is not a number but a function, and involving one or more derivatives of this function. 5 5 Note to advanced readers: in this snapshot, we are only concerned with the case where these functions depend on one variable only, so-called ordinary differential equations. 4 A differential equation example: de Beaune’s problem. As an example, let us consider a problem formulated by Florimond de Beaune in 1638: 6 Find a curve C in the plane, given by a function y(t), such that the tangent to C in any point M with abscissa t intersects the horizontal axis at the point u = t − 1 (see Figure 3). M b C u = t − 1 t b b D = 1 Figure 3: The problem of de Beaune (1638). We note that the slope of the tangent at M = (t, y(t)) to the curve C equals the quotient of the height y(t) over the width D = 1, which means the slope is equal to y(t). In addition, we recall that by definition the slope is the derivative of y at t. Together, this gives the differential equation y0(t) = y(t). The general solution of this equation is y(t) = C · et, where C is a constant that can be determined by adding an initial condition y(0) = y0. The differential t equation with a given initial value then possesses a unique solution y(t) = y0 · e . Numerical methods. For a general differential equation, however, it is in practice often difficult or even impossible to find a formula for the exact solution. Therefore a numerical integrator, which is an algorithmic method for calculating an approximate solution, must be used. 6 De Beaune (1601–1652) is famous mostly for the problem presented here. It is one of four problems that he submitted to René Descartes (1596–1650), who had just published La Géométrie in 1637. 5 There exists a wide range of algorithms for solving a differential equation of the form 0 y (t) = f(y(t)), y(0) = y0, where y0 is a given initial condition, f is a given function and y is an unknown function depending on t.