Transcript Kepler (PowerPoint)

Theme 3 Part 3 – Kepler
ASTR 101
Prof. Dave Hanes
Johannes Kepler
He wanted to use Tycho’s data to
work out the shapes and relative
sizes of all the planetary orbits.
He assumed a heliocentric model
(sun at center).
He also assumed that the Earth’s
orbit around the Sun is a circle.
(This can be justified by the fact
that the Sun seems to look roughly
the same size at all times of year.)
He Addressed Three Obvious Questions
1.
2.
3.
What are the shapes and sizes of the orbits followed
by the other planets? (Are they also circles, as he had
assumed for the Earth?)
Consider a particular planet. Does it move at constant
speed around its orbit?
Intercompare planets. How do the different planets
compare in their speeds and/or orbital periods?
These led to three important findings: Kepler’s Laws.
Kepler’s First Law (K-I):
The Shapes of Planetary Orbits
Any given planet moves in a path that is an ellipse.
(Note that this word has nothing to do with
eclipses or the ecliptic.)
Moreover, the sun is at one focus of the ellipse!
Drawing an Ellipse
The two focuses [foci] are marked with “F”
http://www.youtube.com/watch?v=29esLneio3o
Ellipses are‘Flattened Circles’
Note that a circle is itself an ellipse! It is
just a special sort: one that is‘not
flattened.’
Analogy: every square is itself a rectangle, a rather special
one! Other rectangles can be described as ‘flattened
squares.’
Every Ellipse Has Two Foci
(Focuses)
…plus an
‘eccentricity’
[don’t worry about
the math definition or
the actual values]
e = 0.0 not eccentric at all
e = 0.96 very eccentric
e = 0.017 Earth’s orbital eccentricity
(very close to circular!!)
The Relative Sizes of the Orbits
Kepler worked out not just the shapes of the
individual orbits, but also their relative sizes
e.g. Venus is abut 0.72 times as
far from the Sun as the Earth is;
Mars is about 1.52 times as far.
The Role of the Sun
As the planet moves in from
the right (the blue arrow),
the Sun’s gravity pulls on it
and whips it around, throwing
it back out to the right (orange
arrow).
[Note that Kepler did not know about gravity!]
But: a Puzzle?
Why does the planet turn around at the other end
of its orbit? There is no ‘Sun’ at the second focus.
Answer:
The Sun is Again Responsible!
As the planet moves
away (the blue arrow),
the sun’s gravity pulls
it back (the orange
arrow) like a stone
falling back towards
the ground.
K-II: No Individual Planet
Moves at Constant Speed
Instead, it speeds up and slows down
at various times.
How can this behaviour be simply
described? Is there some physical
principle we can appeal to?
“Sweeping Out Areas”
Imagine drawing a line from a planet to the Sun,
then seeing how much area it ‘sweeps out’ as the planet
moves. This is shown here a couple of times:
First, it moves from X to Y,
sweeping out a long skinny
area (shaded green).
Later, it moves from A to B,
sweeping out the dark blue area.
K-II: The “Law of Areas”
If the two orange swept-out areas are the same, then it
will take the same time to go from a to b as it does to go
from c to d.
Segment I is long and skinny,
so c-d is short. Segment II is
short and stubby, so a-b is
longer.
The implication is that a planet travels more slowly when it
is far from the Sun, but faster when it is close to it
Halley’s Comet: An Extreme Example
-it spends most of its time very far from the Sun, moving slowly!
Here is a link to an interactive animation that allows you to
try this out. Note that you can adjust the eccentricity
and other factors, and explore all three of Kepler’s laws.
http://astro.unl.edu/classaction/animations/renaissance/kepler.html
Does This Behaviour Sound Familiar?
When the planet gets closer to the sun in its orbit, it moves
faster. Where have we encountered this sort of thing?
(Remember the figure skater, who draws in her arms to
spin faster?)
Kepler’s law of areas (K-II) is actually a statement of the
conservation of angular momentum!
Moreover, Energy is Conserved!
Far away from the Sun, the planet has a lot of ‘potential
energy’
As it falls in, it picks up speed (‘kinetic energy’) as it loses
potential energy
It whips around the corner, climbs back away from the Sun,
losing speed and kinetic energy but regaining potential
energy.
The total energy remains the same. Conservation!
An Important Implication
An object falling in from a great distance will build up a lot
of speed -- enough, indeed, to allow it to climb away
again, retreating back out to the same large distance!
(This is analogous to a swinging pendulum which swings
back up to the same height from which it was released.)
It will not and cannot take up a new, close-in orbit. It will
not be ‘captured’ by the gravity of the central object!
Why This Matters
Imagine wanting to put a
space probe into close orbit
around Jupiter.
First, we launch it on its way.
As it nears Jupiter, it speeds up under the influence of Jupiter’s
gravity. We have to slow it down a lot (using rocket engines) so it
will take up the planned new orbit -- otherwise it will shoot right
past!
This means we have to carry extra fuel with us for this purpose – and
reduced payload!
Determining Orbital Periods of Planets
Mars is overhead at night
Exactly 1 year later: no Mars!
(It’s on the far side of the Sun.)
Two years later: Mars is again overhead at night
Conclusion: Mars takes about 2 years to
go around the sun. (We can be more precise.)
Put This Together (for K-III)
From earlier work (K-I) we know the relative
sizes of the orbits.
We now also know their different periods.
How are they related?
KIII: The Various Planets Move at
Different Speeds
Those farther from the sun take longer to go around. For
example, a ‘year’ on Jupiter [one orbit around the sun]
take 11.86 Earth years
But it is only 5x as far from the Sun as we are, so its orbital
path is only 5x as long. It must be moving more slowly
than we are here on Earth!
What is the exact relationship?
Some of Kepler’s Numbers
(no need to memorize!)
How are these related?
[remember, by the way, that 1 Astronomical Unit (AU) is
the average distance between the Earth and the Sun]
Planet
Distance (AU)
Period (Earth years)
Mercury
0.387
0.241
Venus
0.723
0.615
Earth
1.0 (by definition!)
1.0
Mars
1.524
1.88
Jupiter
5.203
11.86
Saturn
9.554
29.46
KIII Quantified
After a decade or more of analysis, Kepler worked it out.
(A modern physics student would do so in 5 minutes! We have better
analytic tools, and know ‘what to look for’.)
He discovered that:
(Period squared) is proportional to (distance cubed).
Consider Jupiter as an example:
P x P = 11.86 x 11.86 ~ 141
d x d x d = 5.203 x 5.203 x 5.203 ~ 141
What About Speeds?
Mathematical analysis reveals that the speed of a
planet depends on the square root of the distance
from the Sun (no need to memorize this!). Thus:
…a planet 4 x farther from the Sun than the Earth (rather like Jupiter!)
moves around its orbit at 1/2 the Earth’s speed
..a planet 9x farther out than the Earth (rather like Saturn!) moves
around its orbit at 1/3 of the Earth’s speed
..a planet 36x farther out than the Earth (rather like Pluto!) moves
around its orbit at 1/6 of the Earth’s speed - very slowly indeed
The Profound Importance
You do not need to know the precise form of this simple
dependence. The important point is that it exists, that
there is some fundamental relationship -- and presumably
a physical law that drives this.
Moreover, it implies predictability: if we find a new object
orbiting the Sun, we can work out its distance by
determining its orbital period (or vice versa). And we can
predict exactly how spacecraft will move, once in orbit.
The Keplerian Solar System
In the end, Kepler knew the shapes and relative sizes of all
the planetary orbits all the way out to Saturn, as shown
here:
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Remember: to him the planets were merely moving dots
of light: he had no telescopes, no images.
Nor did he know the actual true size of the Solar System.
So Much for Science!
Now Meet Kepler the Mystic
He asked: Why are there exactly six planets?
(Mercury, Venus, Earth, Mars, Jupiter, Saturn)
(Of course, we now know that there are more!!)
He came up with an ‘explanation.’
Regular Polyhedra
These are solids that have all faces exactly the
same. (For example, a cube has six identical
faces, each one a square.) There are only five
such solids.
Now imagine making
one of each, but of
different sizes, and
‘packing them’ one inside
another in various orders.
Packing Them Together
Kepler thought that this must
explain the spacing and number
of the planets – and was very
proud of this ‘discovery!’
Here is a link to a nice actor’s
rendition of Kepler himself
visualizing this ‘breakthrough’:
http://vimeo.com/44788828
Second: Music of the Spheres
Since the time of Pythagoras,
who studied stretched strings,
we have known that musical
notes are caused by vibrations,
and that higher frequencies
(more rapid vibrations)
create higher-pitched notes.
Here’s a link to a brief film of real vibrating guitar strings:
http://www.youtube.com/watch?v=ttgLyWFINJI
Kepler’s Interpretation
Kepler reasoned that since
the planets orbit the Sun at
different and varying rates
and frequencies, they must
produce “Music of the
Spheres.”
(Galileo, a real scientist, had
no patience with this!)
Something Essential is Still Missing
1. We Still Need Proof
Kepler had assumed the Sun was the center, and
derived some pleasingly simple behavioural laws. But
can we somehow prove that the Earth and planets are
truly orbiting the Sun?
2. Where’s the Physics?
What makes the planets move the way they do?
What are the Laws of Nature that govern the motions?
The answers were to come from Galileo and Newton.