Theme 3 Part 3 – Kepler

ASTR 101Prof. Dave Hanes

Johannes KeplerHe wanted to use Tycho’s data towork 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. 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?)

2. Consider a particular planet. Does it move at constant speed around its orbit?

3. 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 EllipseThe 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 aboutthe math definition or the actual values]

e = 0.0 not eccentric at alle = 0.96 very eccentrice = 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 movesaway (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 isshort 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 MattersImagine 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’sgravity. We have to slow it down a lot (using rocket engines)

so itwill take up the planned new orbit -- otherwise it will shoot

rightpast!

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 SystemIn the end, Kepler knew the shapes and relative sizes of all the planetary orbits all the way out to Saturn, as shown here:

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 mustexplain 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, hadno 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.