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Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
4. How Orbits Work
Astronomy 1001, Sept 2007 – Prof. K. Davidson
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
KEPLER’S LAWS (c. 1610)
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
1. Each planet follows an elliptical orbit,
with
the Sun at one focus*
2. Equal areas in equal times
3. Period squared = radius cubed
* (Note: the orbits aren’t aligned)
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
KEPLER’S FIRST LAW (except that planets’ orbits are usually more circular than this)
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
KEPLER’S SECOND LAW:“EQUAL AREAS IN EQUAL TIMES”
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
KEPLER’S THIRD LAW relates the speeds of different planets; those farther out move slower.
If a = average distance from the Sun* and P = orbital period, then
P 2 = (constant) x a 3 .
* (Distance is reckoned from the center of the Sun.)
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
Kepler’s third law -- the numbers
a (AU) P (yr) a 3 P 2
MERCURY 0.387 0.241 0.058
0.058
VENUS 0.723 0.615 0.378
0.378
EARTH 1.000 1.000 1.00
1.00
MARS 1.524 1.881 3.54 3.54
JUPITER 5.203 11.86 141 141
SATURN 9.54 29.46 868 868
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
Isaac Newton (1642 – 1727)
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
Newton’s theory of dynamics, also called mechanics
Motions of objects impelled by forces
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
Brief digression ...
A VECTOR IS A QUANTITY THAT HAS A DIRECTION IN SPACE.
examples: ** POSITION ** ** VELOCITY ** ** ACCELERATION **
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
“vector addition”
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
VELOCITY tells us how POSITION changes with time (in all directions);
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
VELOCITY tells us how POSITION changes with time (in all directions);
ACCELERATION tells us how VELOCITY changes with time (in all directions).
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
VELOCITY tells us how POSITION changes with time (in all directions);
ACCELERATION tells us how VELOCITY changes with time (in all directions).
So: If we know an object’s three-dimensional acceleration at all times, then we can predict its motion.
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
“If we know an object’s acceleration at all times, then we can predict its motion.”
( Newton invented a new kind of math specifically for this purpose: “differential calculus”, which concerns rates of change. )
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
Newton also invented the concept of force in the modern sense.
FORCE = MASS x ACCELERATION, or
F = m a , or
a = F / m .
( F and a are vectors; they have directions.)
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
Newton’s
crucial
“thought experiment”
(1665)
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
The Moon’s orbit around Earth
r = 384000 km = about 60 x (radius of Earth),
v = 3700 km/hr = 1.02 km/s.
So the required acceleration toward earth is
a = v 2 / r = 0.27 cm / s / s.
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
Moon’s acceleration toward Earth isabout 0.27 cm / s / s.
So what? -- Our acceleration towardEarth is g = 980 cm / s / s. Newtonnoticed that these have the ratio3600.
60 x farther makes gravity weaker by a factor of 3600 x.
This is obviously 60 x 60 = 60 2 !
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
Newton’s law of gravity:
( attractive force between M and m )
= G x M x m / (distance) 2 .
For instance, 3 x farther makes it 9 x
weaker;
10 x farther makes it 100 x weaker.
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
THE “INVERSE SQUARE LAW” OF GRAVITY WAS ENOUGH TO EXPLAIN KEPLER’S LAWS!
1. Orbits are ellipses, Sun at one focus
2. Equal areas in equal times
3. Period squared = orbit radius cubed
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
NEWTON’S REVOLUTION WAS AS IMPORTANT AS COPERNICUS’.
There are “laws of physics” that apply
everywhere, from this room to the
edge
of the universe. In 1680 this was a
breathtaking new idea!
It led to modern physical science.
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
Newton’s influence on the following century -- “the age of reason”, at least for philosophers.
His historical importance was recognized in his own time -- arguably “the most important man in the world”, outranking even Louis XIV.
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
ORBIT SHAPES: ELLIPSE, PARABOLA, HYPERBOLA
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
“CONIC SECTIONS”:
CIRCLE
ELLIPSE
PARABOLA
HYPERBOLA
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
ENERGY IN ORBITS
-- KINETIC ENERGY -- POTENTIAL ENERGY
K.E. + P.E. = constant
The Roller Coaster analogy
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
CELESTIAL MECHANICS
(ORBITAL DYNAMICS)
Kepler’s laws are not exactly true ...
Ast 1001 lecture 4 -- 2007 Sept 13 (kd)
