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.