Transcript Tycho Brahe (1546-1601) - University of California, Berkeley

Tycho Brahe (1546-1601)
Danish nobility; lost nose in duel
(so had metal one). Got King
Frederick II to give him a little
island and build the world’s best
observatory on it.
Designed, built and used very accurate
instruments for measuring sky positions. Uraniburg
Kept voluminous records for years. Hired
Kepler to try to understand motion of
Mars. Had model with Sun going around
Earth, but planets orbit Sun. Found that
comets moved between planetary orbits
(not Ptolemaic). Motion of Mars still not
fully explained. Fell out of favor; moved.
Johannes Kepler (1571-1630)
Born sickly and poor. Smart: got scholarships.
Became Lutheran minister; learned Copernicus.
Went to work with Tycho to escape 30 Years
War. Tycho withheld important data until he died
in 1601. Kepler proposed geometrical
heliocentric model with imbedded polygons
(clever and aesthetic, but not better). With full
Mars data, Kepler found his laws of planetary
motion in 1605 and published in 1609. Had to
keep moving around, but kept publishing better
predictions of planetary positions, which were
confirmed observationally.
A recent note: it turns out that the 1609
publication did not contain real data, but data
generated using the laws (which constitutes no
independent support at all… Bad Science!).
Ellipses
An ellipse is an example of a
“conic section”. Circles and
hyperbolas are others. All are
possible forms for orbits.
You can make an ellipse with 2 tacks
and a string. The tacks are the “foci”,
and if you put them further apart, the
ellipse is more “eccentric” (one tack
makes a circle).
Kepler’s Laws of Planetary Motion
1) The planets move in elliptical orbits,
with the Sun at one focus.
2) A line between a planet and the
Sun sweeps out equal areas of the
ellipse in equal amounts of time.
Notes: There is nothing at the
other focus or in the center.
The Second Law means that
planets swing around the Sun
faster when they are closer to
it. These laws work for
anything orbiting around
anything due to gravity.
Kepler’s Second Law Animated
Kepler’s Third Law
3) The orbital period of a planet is proportional to its semi-major
axis, in the relation P2 ~ a3
The more general form of this law (crucial for determining all
3
masses in Astronomy is
a
2
P 
M central
For the planets (with the Sun as the central mass), you can take
the units to be AU for a (semi-major axis) and years for P (with M
in solar masses). Then all the numbers are “1” for the Earth.
Example: if Jupiter is at 5 AU, how long is its orbital period?
P2  53  125; P  125  11.2
Kepler didn’t understand the physical basis of these laws (though he
suspected they arose because the Sun attracted the planets, perhaps through
magnetism he speculated.
Isaac Newton (1642-1727)
One of world’s greatest scientists. Co-inventor of
calculus. Discovered the law of Universal
Gravitation. Newton's 3 laws of motion.
Corpuscular theory of light. Law of cooling.
Professor, Theologian, Alchemist, Warden of the
Mint, President of Royal Society, member of
Parliment. Personally rather obnoxious, poor
relations with women, lots of odd stuff with the
great stuff. Did most of it in before he turned 25!
Trinity
College,
Cambridge
Newton’s Three Laws
1) The Law of Inertia: objects will move at a constant velocity
unless acted upon by forces. (really Galileo’s law)
2) The Force Law: a force will cause an object to change its velocity
(accelerate) in proportion to the force and inversely in proportion to
the mass of the object. This can be expressed: F = m * a or a = F / m
3) The Law of Reaction: forces
must occur in equal and opposite
pairs – for every action there is an
equal and opposite reaction.
Newton’s Law of Gravitation
Astro Quiz
• Why are astronauts in the Space Shuttle weightless?
1) The extra inertial of the Shuttle just compensates for the extra
gravitational pull on it, so it falls at the same rate as the astronauts.
2) The Shuttle is sufficiently high in its orbit that the Earth’s
gravitational pull is negligible.
3) The Shuttle’s engines keep it on a path that matches the Earth’s
curve, and there is no air resistance.
Newton explains Kepler’s Laws
Newton was able to show mathematically (using his calculus),
that for inverse square forces, the orbits are ellipses and obey
Kepler’s laws. He realized this must apply to all celestial bodies.
In particular, he could show that the period and size of an orbit
are given by:
2 3
4 a
P 
GM
2
Where P is period, a is semi-major axis, M is central mass,
and G is the “gravitational constant” that expresses the
strength of gravity (in the right units, of course).
Thus, this law (or Kepler’s Third Law) can be used to find the
mass of any body in which an orbiting body’s period and
distance can be measured (starting with the Earth-Moon system).
Finding the mass of the Earth
We know that the Sun is about 400 times further away than the
Moon, and takes a month to orbit the Earth. Thus, its semi-major
axis is about 1/400 AU, and its period is about 1/12 years.
3
1
a3
144
6
400
M 2


2
.
25
x
10
2
P
64x106
1
12
Since we used AU and years, the mass is in solar masses. So the
Earth is about a million times less massive than the Sun. To find
out how many kilograms (or whatever) it has, we have to use
the form of Kepler’s Law given by Newton, and put in all the
physical units [like P(sec), a(meters), G (mks units)]. In this
class, we will always use ratios and avoid units (so we get
relative comparisons).
Orbital Motion
Gravity always makes
things fall. The question
is whether the path of the
fall intersects any
surface. The shape of the
orbit depends on the
velocity the body has at a
given point. Low velocity
will make the point the highest, high
velocity will make it the lowest
(circular orbits mean it has to be “just
right”). If the velocity is too high, the
orbit will be a hyperbola instead of an
ellipse, and the body will not return.
Orbital and Escape velocity
Vcircular
GM

R
Escape velocity depends
on the mass and size of
the body. It is about 11
km/s from the Earth. You
have a black hole when
it is the speed of light
(you need a lot of mass
in a little size).
Note these velocities do
not depend on the mass
of the escaping or
orbiting body.