Hi, welcome back. We ended last week with some success and
a promise of more challenges. We realize that no matter how much we've
added, we still have to make modifications to our little universe.
We discovered the planets, we have to add them, we're going to see that they move,
too. we're still pretty much on pace with
Aristotle, but we're going to take that little thread, these extra moving parts,
and we're going to pull on it. And by the time we're done unraveling it
or following the way scientists unraveled it, we would have
gone places that Aristotle could not even have begun to imagine.
And we're going to spend much of this week actually away from astronomy,
talking about fundamental physics. it'll be tough, but I promise you it will
pay off. Because what we learn this week when we bring up the understanding we
take away from it is going to help us understand everything we talk about
throughout the class. And hopefully, way beyond this particular
astronomy class. And, of course, the father of all of this is Sir Isaac
Newton. So, let's start with the beginning. We'll
start with the problem we found and Aristotle would have been perfectly aware
that there are five planets, planets derived from the word for wandering.
Because these were very bright stars, the five were Mercury, Venus, Mars, Jupiter
and Saturn. And they, as we saw in our image follow
paths very near to the ecliptic. We're beginning to sense a pattern here
and they move along the ecliptic, so that means we need planetary spheres in
addition to the lunar and solar and celestial spheres that we have.
So far so good. They move around the ecliptic at
different rates. Each planet moves at a different rate.
So certainly, you can't put them all on a planetary sphere, you need a venution
sphere separate from the Mercurean sphere.
so, we're up to seven moving spheres and a celestial sphere.
Their motion is not completely regular. Remember, even the sun doesn't completely
move uniformly around the ecliptic. But, the deviations for the planets are
far more serious. And, in fact, will end up telling us
something deep. So, let's look closely at some of these
deviations from regular motion that we see with planets but do not see with the
sun and the moon. And the simulation, we're looking at a
patch of sky near the constellation Gemini and Leo just to the east of Orion
that we were looking at before. And, we see the ecliptic and we're going
to focus on the behavior of the planet Mars.
And this simulation starts little over a year ago, in October 2011.
And its set up so that time advances by sidereal day so that the stars repeat
themselves. And we that Mars, like all the objects
that we've been encountered so far, is moving slowly to the east along the
ecliptic. We see the moon here moving much faster to the east along the
ecliptic. And as time rolls by, the moon will
quickly overtake Mars. But then, as the moon overtakes Mars, in
December last year, something weird happens.
Mars, in fact, stopped it's eastward progress.
And then, for about three months, Mars was actually moving west along the
ecliptic. And then, it recanted and continued its
normal eastward motion. This retrograde motion of planets happens
to most planets is a slightly more involved engineering challenge than what
we've met so far. Challenging?
For sure. It's not easy to construct a model that
leads to the apparent motions of the planets in the sky as we see them.
But, very brilliant people were working on this and they were very competent
geometers. And they came up with the solution this
is first the germ of the idea first proposed in an 150 BC by Aristocles.
And, the basic idea is that to understand how Mars moves, you imagine that there is
a circle around Earth called the deferent.
And essentially, Mars moves along this
deferent, but it's not on the deferent itself.
On the deferent rolls an epicycle which turns, itself.
and Mars is at a point, a fixed point, on this epicycle. So that as the animation
runs, you see that the motion of Mars rather
resembles what we saw in the picture, in the simulation. Mars is sometimes moving
faster, sometimes slower and sometimes even retrograde.
And so, along these principles by combining a deferent with an epicycle,
one could describe the motions of the planets as we see them.
Explaining how it is that sometimes a planet moves faster and sometimes slower
is one thing what we were after or what they were after was an actually precise
matching of the predictions to the data. And this leads to a rather elaborate
model that Ptolemy comes out with 300 years after Hipparchus floated the idea.
For example, in order to understand why the sun does not move at a fixed rate
around the celestial sphere the solar sphere is a little bit off center.
Its center is not exactly on the Earth and there are various such elaborations.
The end result is that the model is extremely successful.
It makes good predictions. Far into the future about, exactly where
in the sky, about where on the celestial sphere, you'll find which planets.
And because of these slight deviations of getting the precise agreement, you now
have an actual ordering of the sizes of the spheres.
the moon is on the inner most sphere of Mercury, Venus, the Sun, Mars, Jupiter,
Saturn, and then outside the celestial sphere of the fixed stars.
And an example of how this process works out would be the planet Venus.
Venus is a planet that as we observe it in the sky, is never very far from the
Sun. It's sometimes a little but the west of
the Sun, and then it's a little bit ahead of the Sun.
And so, we see it rising just before Sun rise, some times it's to the east of the
Sun then we see it in the west in the evening sky.
So then, it's the evening star or the morning star the way that's arranged in a
Ptolemaic model is that the deferent for Venus moves at about the same rate, as
the Sun does. In other words, the deferent orbits the
Earth once a year. And then, the epicycle rotation around
that, which is not too large, accounts for the periodic change in elongation.
In other words, the distance in right ascension between the Sun and the planet,
and Ptolemy builds in tilts, and the, the model as I said, is somewhat elaborate.
But at the end of the day, it's a successful mathematical model.
And the reason I'm bringing this up is because it's often presented as the wrong
model. It is not the way we think today.
But, it is certainly wonderfully successful mathematical model of how
things work. what was its competition?
Well, from early on, there was an, an idea first attributed
definitively to our star clusters as early as 270 BC which is that the
perceived motion of the planets along the celestial sphere follows from the fact
that both the planets and the Earth orbit the Sun. And what we see is basically the
result of the relative motion of the planets and the earth.
in other words, the solar system is now looking kind of like a race track along
which the planets race. One of the principles of this model is
that the planets closer to the sun move around the Sun, orbit the Sun, in shorter
periods, faster. And so, the inside lane always wins.
And as a result, when one planet is passing the other, you get the same
effect when you're looking at the planet that you get when you're looking out the
window of your car and it looks like relative to the distant mountains, the
trees near the road are moving backwards. Trees are not going anywhere, it's just
that your line of sight is changing. We'll see that in a second.
So, this has a nice, simple explanation for retro-grade motion.
Fine. the model that Aristocles proposed was
primitive. It needed elaboration,
and nobody paid it enough attention to bring it up to the level of
sophistication that it needed until Copernicus in the 16th century brings out
Heliocentric model that is as detailed and as predictive and as successful as
the Ptolemaic model. And the planets move along circles and
they move at uniform speed. And the way that this explains the
non-uniform motion along the sky is because of this aspect of relative
motion. We can do a nice calculation that might
help us get a handle on this racetrack story.
So, imagine that we have planets and they're all orbiting the Sun.
take some planet and let P, for period, be the time it takes that planet to
complete one circuit around the Sun because its one complete circuit relative
to the stars, we'll call this the sidereal period. Earth itself is a
planet, it has its own period. we'll distinguish that with the name E
for earth's period. That is exactly the sidereal year that we
discussed, the time it takes Earth to orbit around the Sun once with respect to
the stars. As with the moon, we'll define a synodic
period which we will call S, which is the time between consecutive occurrences of
some particular alignment between the Earth, the planet in question, and the
Sun, just as with the Lunar phases. And we are going to try to relate these
mathematically, something we sorted it for the moon but will be instructed to do
it here using this race track model. This is another one of those wonderful
animations from the University of Nebraska-Lincoln.
It's a very elaborate animation, you can do lots of things with it.
I'm going to do something very simple. I want to understand the relation between
synodic and sidereal periods. So, what do we have here?
Well, we have two planets. We have a blue planet,
that's this one. And, I'm going to assume that it has a
siderial period P1. That means, P1 is the time it takes the
blue planet to a 360 degree circumlocution of the Sun.
I have another planet, I will call it the green planet, or number two because I do
not have a working gray marker. And this one takes a period P2 to
circumnavigate the Sun to complete a full sidereal orbit of 360 degrees around the
sun in right ascension. And because of our assumptions, P1 being
the inner planet takes a shorter time to go around the sun.
So, P1 is less than P2. Okay.
Suppose that I know these or they are what they are, what I want to know is the
following thing, I'm going to understand the synodic period.
So, what is the synodic period? That's the time between repeated
alignments of the two planets and the Sun.
So, for example, I am starting my animation right here,
h, very nicely and the situation where the three bodies are aligned, they are in
a line. The synodic period will be the time that
needs to pass before this pattern is repeated.
And to see how long that takes, all we need to do is set the runners off and let
them go. So, I'm going to press the button and off
they're running. When you stop and see what's going on, we were write,
the blue planet is much faster than the green planet and is outrunning it.
And as I let them run indeed within a short time, the blue planet is back home.
What does that mean? Well, this means the time that we just
watched was precisely this period P1. The blue planet completed a 360 degrees
sidereal rotation. Of course, the green one is nowhere near home.
To keep track of what has gone on, let me give these guys trails, imagine they were
trailing smoke or something. This is what the blue planet has done so
far, and this is what the green planet has done so far.
but we're not back, as usual, we're not back to alignment.
Because in the time that it took the blue planet to get back where he was, the
green planet has run away. And as we're by now familiar to reproduce
the synodic alignment, we need to keep going until, boom.
There we go. Notice, we're back in alignment. So,
this, the time that passed until this happened, from here to here is the
synodic period. How do we compute it?
Well, let's finish our star trails here. So, this is a very ugly trail, but this
is how far the green planet has gone, and this is how far the blue planet has gone.
A full circle. And then ooh, and then the same amount.
Why? Because they started in the same place,
they ended in the same place, but the blue planet has done an extra lap.
This has taken a period S. Well, how much of a circle is this little
arc? We've played that game before.
This little arc, I will write it in green.
The green arc is simply the fraction that S is out of the period P2.
Because a time S has passed and it takes a longer time,
P2 for the green planet to complete a full circumlocution of the sun.
Now, because they started at the same place and ended at the same place, one
could be confused and imagine that this is the same as the fraction of the circle
that the blue planet has done because, after all, it is the same fraction.
However, the blue planet has gone through this arc but also a complete circle.
So, in fact, this number is much bigger than this number by how much?
By a full circle. In other words, this,
how much the green planet has done is how much the blue planet has done minus one.
And that is the equation we wanted. The three variables, synodic period, time
between alignments, sidereal periods for the two planets are
related by this relation. And then, if you know any two of them,
you can get the third. Hooray.
The actual resolution of the question, which of these two models describe the
universe? Was going to have to wait for better observations within the technology
they would definitively settled this it the question. Along the way,
understanding how the, the heliocentric model will turn out works out led to
insights that are much deeper than anything that you've been discussing now
and far more broad in their implications than just understanding the stars.
And it is following that trier that is going to occupy us for the rest of this
week.