The first step towards resolving everything was gathering precise data and
the name most commonly associated with this new data gathering is Tycho Brahe
who in 1580 makes very, very detailed observations of planetary motion with an
unprecedented accuracy, and these are very useful
Interestingly Brahe got the brilliant equipment and the patronage he needed to
make these observations by making a failed attempt to prove that comets and
other such unpredictable changes in the sky are in fact terrestrial events.
Things that happen on earth, not far like say meteors or shooting stars.
they're not, but this didn't sit well with current world views.
The important thing for us is that he made these very, very precise
measurements and some of them he shared with Johannes Kepler, who came to work
for him as an assistant. And in 1609 after playing with the data
and playing with the data, this was very precise data, close good enough that you
could take into account the angular measurements including the position of
earth relative to the planets in this heliocentric model that we're that Kepler
was pursuing, precise enough that you could use these,
these angular measurements to actually not just make a 3-dimensional model with
ordering of the planets, but actually find predict the ratios of their sizes in
other words you could tell whether Mars' orbit was seven times that of Earth or
three times that of Earth in radius rather than just that it's larger.
And with this precise data Kepler discovers much to his surprise that three
very, very simple rules describe all of the motions of all of the planets and
these rules are important so let's go through them.
Kepler's First Law says, that the orbits of planets, the trajectories that planets
follow are not circles. They're not circles upon circles.
They're a simple geometric figure, he found this very surprising.
They are ellipses. So what is an ellipse?
Well, an ellipse, is an oval, squashed circle type thing, and it's characterized
by two special points called the focii, the focus.
then plural, the full side of the ellipse.
And an ellipse, is a sum, is a set of all the points in the plane such that the sum
of their distances from the two full side, is the same.
So if I take any point on this ellipse, it's distance from this point, plus it's
distance from that point are always the same and if I take these two points to be
on top of each other then I am thus designing a circle because then those are
just twice the distance from that one point, so if the two focal points on a
top of each other then the ellipse is in fact a circle.
So this is what an ellipses and Kepler first law says the orbits of all planets
or ellipses such that one of the focal points is in fact the Sun.
What's in the other focal point? Well, the simulation here gives you a
hint. The other focus is the empty focus.
There is nothing there. It has no physical significance.
It's just a punked in space. It's not even the same for all the
planets. So, the empty focus of Mercury's orbit is
nowhere near Venus' orbit but they both share one focal point and that is where
the sun is. So we need to know a little bit about the
geometry of an ellipse. So again an ellipse is characterized you
see as I run here by this effect, the sum of these two distances is always the same
and I get some particular size and we characterize an ellipse by the size of
its. Semi minor axis to the half of the
diameter if you will, of the ellipse. Perpendicular, aligned through the
center. Perpendicular to the line between the
focii, that's the shorter axis. So that's why it's a minor axis and this
is half of it. And then the semi-major axis, which is
the longer radius. If this is a circle, they'd both be the
radius. The longer one is the semi-major axis.
That's along the line between the focii. And an eclip, ellipse is characterized,
either by giving these two numbers, or alternatively, giving one of them.
Typically we give the semi-major axis. And the parameter called eccentricity
that measures how different the two axises are from each other.
So eccentricity zero is when the focii are on top of each other, and you have a
circle in fact, an eccentricity one which this won't allow me to achieve, is the
case where the eclip, ellipse is so smashed down that it becomes a straight
line, and the semi minor axis vanishes. the eccentricities of, the planets, you
can reproduce the most eccentric planetary orbit is Mercury's, and that it
has an eccentricity of about 0.2. The eccentricity of the earth is closer
to 02. which means that it's very close to a circle indeed.
This is roughly what the earth's orbit looks like.
So this is Kepler's first law, the orbit of any planet is an ellipse, and the sun
is at one of the focal point of that ellipse.
The second law states that as a planet, as a planet that we again exaggerate the
electricity of this, as a planet orbits the Sun Kepler says it sweeps out, the
line between the planet and the Sun sweeps out equal areas in equal times, so
this will give us a demonstration. This turns on a sweep for
particular amount of time and I can start it whenever I want and the point is that
what this means is the areas of all these triangular shapes have to be the same,
the implication is that when the planet is near the Sun, it needs to be
travelling faster so the stubby triangles are abroad whereas when the planet is
farther from the Sun, it can traverse slower because the longer triangles can
be kind of narrow. So not only do Kepler's laws tell you the
shape of the orbit but also the relative speed with which the planet goes,
it goes fastest when it's nearest the sun, this point is called perihelion, the
point of nearest approach to the sun. And it's always along the major axis you
can see, and at the other end of the major axis is the point of farthest
distance from the sun, aphelion, which is where the planet moves most slowly.
So these are Kepler's first two laws. And let's go back and summarize that.
So, we said Kepler's First Law was that the orbit of every planet is an ellipse.
And these ellipses for the actual planets in the Solar System are very close to
being circles. the second law tells us how a, planet,
moves faster or slower as it orbits the sun.
This is this law of equal areas. The implication is that the planet moves
fastest near perihelion and slowest near aphelion when it's far from the sun.
This is not very dramatic for, say the earth.
For objects that might orbit the sun in for more eccentric orbit, which are
sometimes very near the sun and sometimes very far from the sun, say comets this
effect is much more dramatic. So when we get to, more eccentric orbits,
which exist in the universe, we will have to consider this, aspect of
Kepler's second law more seriously. And this is enough to determine all the
orbits of the planets. And this leads to agreement with
planetary positions to a precision, by now there were precise observations, that
neither the Copernician model as it was stated nor the Ptolemaic model could
achieve. And this is a very simple, very
straightforward geometric construction. So Kepler was sure that he was on to
something. Now the third law is different in the
sense that it does not address the motion of a planet.
It's a relation between the orbits of the, the motions of the different
planets. Remember that Kepler by now had enough
information to measure the actual size of the semi-major axes of the planets, at
least their ratios. So one could decide, let us call the
semi-major axis of earth's ellipse one something.
We call it one astronomical unit. So by convention, the definition of
something called an astronomical unit is that the semi-major axis of an ellipse is
typically called a. We will often call this R because
whenever possible, we'll be talking about circles, and for a circle, the semi-major
axis as we saw, is the radius. The semi-major axis of Earth's orbit
defines a unit of distance called an astronomical unit.
And so we could always measure distances in astronomical units, and later figure
out how many meters are in an astronomical unit.
Something that was difficult for Kepler to do, though he had some measurements,
we'll talk later about how precisely we know what an AU is.
And what Kepler knows is he then knows the semi-major axes for all of the
planets in units of an astronomical unit, or at least he knows their ratios.
And what he also knows, their sidereal periods, because he has the motion down,
and because if you want we can get it from the synodic period using the
Copernican calculation, although since things are not circles you have to be a
little more careful. And he notices the following relation.
The square of the period, for any planet, and the cube, of, the semi major axis,
are related by a constant of proportionality.
Now I can always find a number, such that P squared is K times A cubed.
Just divide P squared by A cubed. The point is that if you divide P squared
by A cubed, for each of the five planets that Kepler knows about, you get the same
answer. And this is very important because this
is then not a property of the planet, some of the planet, property of the solar
system, there is something that all these planets share and it's the value of this
ratio. There might be something interesting here
and it's worth expressing the same perhaps a different way.
So remember that we can write the period of a planet that it the, of the planet's
orbit, the sidereal period can be related to let me imagine that we are only doing
circular orbit so R and a are the same thing and I will call AR.
And so I can write the period, relate the period to the radius of the orbit and the
speed, which I will assume uniform because it's a circle with which, the
planet orbits the sun. So the period is simply the circumference
of the orbit, two pi R divided by the speed with which the planet moves.
So this is my expression for the period and A is simply R in this case so
Keppler's Law becomes, well, I'd have to square this thing so 4
pi squared R squared over V squared is K times R cubed.
And I have some enticing cancellations here.
I can cancel R squared and leave just one power of R over here.
And what I can find is an equation that we might find use for later that says, if
you know how far from the sun a planet orbits, from this you can figure out its
period. From this, you can figure out the speed
with which it's moving multiplying by v squared and dividing by KR, I find that V
squared is 4. pi squared over KR, a relation, to which
we will have a time to return. And, what is the numerical value of K?
Well, K, is a kind of funny physical constant the first that, of its kind but
we'll meet many of them. Remember that A is a length in, meters or
astronomical units or whatever it is, P is a length of time, so K is some number
of perhaps seconds squared per meters cubed.
It is a completely different number of hours where per mile cubed, K is the
physical constant which depends on the units you are using, you can always pick
your units of distance and time to make K equal to 1 if you so desire.
it natural units it, that of that we will try to use, seconds and meters it most
certainly is not one. But the important thing for us is that K
whatever it is, is the same for all the planets and that is hinting at some deep
underlying physics. Because it's something that is true for
all of them. Setting this aside for a minute we have
other progress. Galileo about a year after Kepler obtains
his laws, gets his hands on the newest technology, a telescope, a spyglass.
And we don't know if he was the first to do it.
But he was certainly the first to publish this and to become known for it.
He used a telescope to look up in the sky.
And, he finds many, many, many, many, exciting things when he looks up at the
sky with this newfangled invention. the first thing he finds is he finds that
Venus, the planet Venus, goes through phases.
So it's not surprising to us. It's a chunk of rock.
It's illuminated by the sun. It's angled to the sun's changes.
So Venus goes through phases, but in particular the relation between the
phases of Venus and its position in the sky show that sometimes Venus is closer
to Earth than the Sun is, and we get the effect, effectively a new Venus, just
like the new moon works. But on the other hand, unlike the full
moon, which happens when the moon is on the other side of Earth from the sun
Venus is never that twelve hours away from the sun in right ascension, Venus is
always near the sun remember. A full Venus happens when Venus is on the
other side of the sun from Earth. And what this means is this does not fit
in with the precise parameters of the Ptolemaic Model, but it fits very nicely
into the, Keplerian or even the Copernician model.
Also, he discovers that Jupiter has satellites.
He sees the four Galilean Moons, or as he, seeking patronage, calls them, the
Medician Stars. moving back and forth around the image of
Jupiter and he realizes quickly that what he is seeing is something, a collection
of bodies orbiting Jupiter. Well they're clearly not orbiting the
Earth so this is an important. A piece of breaking down the cultural
objection to the earth. Moving, he sees various exciting things,
he sees mountains on the moon, he sees sun spots on the sun, he sees some weird
structure on Saturn that later Huygens discovers are the rings of Saturn.
so Galileo finds many things with his telescope and in particular the evidence
of the faces of Venus is really the smoking gun that
Dooms the geocentric picture. Or the Ptolemaic Geocentric picture as
we, we will talk about later. Whether you think that the whole, the
Earth is stationary or the whole universe is doing a complicated motion, or
something else. At the end of the day, that's very,
difficult and almost philosophical question.
We'll see what we can have to say about why we prefer one description over the
other, but in particular the fact that Venus is sometimes behind the Sun and
sometimes in front of the Sun, that's now a fact.
and this does not fit with the Ptolemaic Model.
Galileo also, applied himself to beginnings of the science of mechanics,
the study of motion, and is well known for, formulating the, the, what is called
the Principle of Inertia. As he put it, an object will retain its
state of motion unless disturbed externally.
Now, we are all familiar with this in the following painful sense.
If you have a refrigerator that's not moving.
And you want to set it in motion, you want to change its state of motion from
rest to motion, you have to expend quite a bit of effort
to move that refrigerator. But Galileo reminds us is but then if you
let it go the refrigerator will stop. What Galileo realized is that this is an
artifact of rubbing against the floor and if you put the refrigerator on ice or on
roller skates and started it moving it would be just as hard to stop it or
change its direction of motion as it was to get it started in the first place.
And that is an abstraction that is, cannot be understated.
Galileo made many, many, many Brilliant deductions.
But he never really got mechanics right, what was missing for Galileo.
Among many things but the technical ingredient that was missing was that the
mathematics required to discuss mechanics correctly hadn't been invented yet.
real progress await Isaac Newton. And Newton also didn't find the right
mathematics, it hadn't been invented yet so Newton just ahead and invented it,
that was calculus and we will have to work around it because we don't use
calculus. So we've made incredible progress.
Look, the planets have and, and in general the heavens have been removed
from some our spiritual realm of spheres and in all have objects that are a given
distance and a given position in the universe, its a 3D world out there, there
are moons that orbit the planets. This is all a physical system.
We can study it. We can observe it.
Galileo observes the moon. As a.
Moon, something to be observed, not some heavenly object.
we have Kepler's laws, and the predictions of Kepler's laws from such
simplicity, such precision, is extremely compelling.
Moreover, it turns out that these laws are a lot more universal than Kepler
could have imagined. In fact they govern orbiting, they govern
any kind of orbiting system. They govern the Solar System.
They govern the various moons of Jupiter, as was discovered not long after he
published his book. They govern every orbiting system, in
some sense electrons in an atom. Each system is characterized perhaps by a
different Kepler constant K. That was a property of our solar system
but their relation, the fact that objects are ellipses and the central objects is
at one of the focal points. And that the semi-major axis and the
periods are related by P squared equals a constant times A cube for object,
different electrons orbiting a nucleus, or different moons orbiting the same
planet. This, it turns out, goes way beyond what
Kepler put into his model comes out of it.
And in physics, when something is that universal, it tells you that there's some
fundamental law underlying it. There's a reason why all of these thing
share a behavior and again, these underlying,
physical laws of which Kepler was just finding the manifestation in the Solar
System, were, waiting discovery by Newton, so we're now ready to discuss Sir
Isaac.