Tycho Brahe (1546-1601)

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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 worlds best observatory on
it.
Designed, built and used very accurate
instruments for measuring sky positions. 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.
Uraniburg
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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!).
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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).
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Keplers 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.
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Keplers Second Law Animated
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Keplers 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 masses in Astronomy is
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?
Kepler didnt understand the physical basis of
these laws (though he suspected they arose
because the Sun attracted the planets, perhaps
through magnetism he speculated.
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Isaac Newton (1642-1727)
One of worlds 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
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Newtons Three Laws
1) The Law of Inertia objects will move at a
constant velocity unless acted upon by forces.
(really Galileos 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.
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Newtons Law of Gravitation
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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 Earths gravitational pull is
  negligible.
• 3) The Shuttles engines keep it on a path that
  matches the Earths curve, and there is no air
  resistance.

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Newton explains Keplers Laws
Newton was able to show mathematically (using his
calculus), that for inverse square forces, the
orbits are ellipses and obey Keplers 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
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 Keplers Third Law) can be
used to find the mass of any body in which an
orbiting bodys period and distance can be
measured (starting with the Earth-Moon system).
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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.
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 Keplers 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).
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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.
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Orbital and Escape velocity
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.