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 Orbit Determination Orbit Improvement Central Orbits

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 ( ) The first nearly correct general description of planetary motions Empirical fit to the observations, no causal explanation

P²  a³ Kepler I Kepler II Kepler III

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 LawLaw of Gravity Hence: 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; m 1 and m 2 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 vector “Orbital 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+m 2 /m 1 ) 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