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Celestial Mechanics I Introduction Kepler’s Laws Goals of the Course • The student will be able to provide a detailed account of fundamental celestial mechanics • The student will learn to perform detailed calculations in the two-body problem • The student will be able to provide a detailed description of basic perturbation theory and the N-body problem • The student will have penetrated several examples of how modern celestial mechanics research is conducted Planning of the course • Lectures, following B.J.R. Davidsson’s Compendium (course literature) • Problems with solutions • Computer Exercises • Introduction to research problems and Seminars on selected research papers First part of the course TWO-BODY PROBLEM Laws of Kepler Geometry & Position vs. Energy & Time Calculating Ephemerides Central Orbits Orbit Determination Orbit Improvement Solar System inventory • The Sun is >1000 times as massive as any other object • The planets span a range of ~6000 in mass • Dwarf planets are (so far) >10 times less massive • Small bodies are even less massive Centers of motion • • • • the Sun: Heliocentric motion the Earth: Geocentric motion Jupiter: Jovicentric motion the Center of mass: Barycentric motion • The orbits of planets, asteroids, comets, etc. are usually referred to the Sun: heliocentric orbits • The orbits of satellites are referred to the planet in question, e.g. geocentric orbits What is Celestial Mechanics? • The study of the motions of heavenly (“celestial”) objects (basically, within the Solar System): - The properties of orbits, and how to relate orbital properties to astrometric observations (;) at different times - The evolution of orbits, when the effect of the central object is “perturbed” by other effects (e.g., gravity of other objects, nongravitational forces): e.g., lunar theory, origin of comets What is it good for? • CM is an integral part of modern planetary science including the study of exoplanets • Space research: satellite launch and orbit insertion, satellite navigation, interplanetary space flight including gravity assists • Dealing with the “impact hazard”; predicting close encounters and judging the risk of collision with NEOs • Fundamental physics: natural laboratory for e.g., general relativity, gravitational waves • Archaeoastronomy: e.g., dating of eclipses Approaches to Celestial Mechanics • Analytic solutions - successive approximations (2-body problem, 3-body problem, perturbation theory, series developments) • Numerical integrations - may give accurate answers, but do not provide physical understanding of the output Kepler’s Laws (1609-1619) The first nearly correct general description of planetary motions Empirical fit to the observations, no causal explanation Kepler I Kepler II Kepler III P²a³ The Ellipse Definitions and relations “Proving” Kepler’s laws • This means showing that Newton’s Laws of Motion and Law of Gravity imply that planets should move in accordance with Kepler’s laws • Thus, Newton is consistent with Kepler – otherwise Newton would have been wrong! • But in addition it means that we have a theory that serves to give a physical significance to Kepler’s laws How to prove Kepler I There are several options, but here we follow the example From Davidsson’s compendium Equations of motion Newton’s 2nd Law Hence: Law of Gravity This holds if O is at rest in an inertial frame Equation of relative motion In the present case, the heliocentric motion of the planet G is the gravitational constant; m1 and m2 are the masses Motion in a plane Angular momentum per unit mass: This is perpendicular to the instantaneous motion Time derivative; the second term vanishes Hence: Since h is constant, the motion is confined to the plane perpendicular to this“Orbital vector plane” Some vector algebra The eccentricity vector From the previous page: Integrate: This is a constant vector in the orbital plane. It provides two constant quantities: its direction and its magnitude. The shape of the orbit Let be the angle between r and e; then: But: Hence: cf. the equation of an ellipse: Important consequences • The length of the e vector is the eccentricity of the orbit • The direction of the e vector is that of perihelion; thus is the true anomaly • The semilatus rectum of the orbit is proportional to the square of the angular momentum • Kepler I has been proved, but the orbit does not have to be an ellipse. It can also be a parabola (e=1) or a hyperbola (e>1) Proof of Kepler II The area of an infinitesimally small sector of the ellipse: Areal velocity & Angular momentum Thus: Conservation of the areal velocity follows from conservation of the angular momentum Proof of Kepler III Note on Kepler III • The square of the period is not only proportional to the cube of the semi-major axis but also inversely proportional to the quantity (1+m2/m1) • This slight inaccuracy was not noticeable to Kepler • Kepler III is a rather good approximation, but we have derived the correct form that includes the planetary masses