Download Celestial Mechanics I

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
P²a³
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