Good morning. For thousands of years the geocentric model of the solar system
was considered to be correct. Bo, what is the geocentric model of the solar system?
♪ Flipping Physics ♪ The Earth would be at the center
of the solar system with the sun and the other planets revolving in circles around the Earth.
Thank you Bo. The geocentric model was formalized by the Greek astronomer Claudius
Ptolemy during the first century of the common era. Roughly 1400 years later, in 1543
a Polish astronomer named Nicolaus Copernicus proposed a heliocentric model with the Sun at the
center of the solar system instead of the Earth. A few decades later a Danish astronomer named Tycho
Brahe made accurate and comprehensive observations of the locations of planets and hundreds of stars
in the night sky. He did these observations for 20 years before his death in 1601 and they were all
done without a telescope because the telescope was not invented until the early 1600s. After Tycho
Brahe’s death, his assistant, Johannes Kepler, a German astronomer, analyzed Tycho Brahe’s data
for roughly 15 years and derived mathematical descriptions of planetary motion which are now
known as Kepler’s Three Laws of Planetary Motion. Today we are going to discuss Kepler’s First
Law of Planetary Motion. Bobby, please read it. Kepler’s First Law of Planetary Motion states that
“Planets move in elliptical orbits with the Sun at one focus.” That seems kinda straightforward.
Yeah, to us, more than 400 years later, the fact that planets move in elliptical orbits with the
sun at one focus does seem rather obvious, however, considering that people thought
the Earth was the center of the solar system for thousands of years, it was a remarkable
conclusion based on observation. Absolutely. Yeah. Sure.
Alright. Let’s draw an ellipse. We start with a piece of cardboard, two thumbtacks, a piece of
string tied in a loop, and a writing utensil. We can place the two thumbtacks which become
the foci of the ellipse. And, by looping the string around the two foci, we can use a marker,
along with the looped string, to draw an ellipse. And then redraw the ellipse to make it darker.
And, there you have it. An ellipse with the two foci where the thumbtacks are. This represents the
elliptical orbital paths of all satellites. Where the satellites follow the path of the ellipse,
the central body, or the object the satellite is orbiting, is located at one focus. And there
is nothing located at the other focus. And now let’s switch to a more accurate ellipse I used the
computer to create. I have added an xy coordinate system to help identify parts of the ellipse.
The two foci are labelled F sub 1 and F sub 2. The elliptical curve is drawn such that the
sum of r sub 1 and r sub 2, the distances from the respective foci to the elliptical curve,
is constant. In other words, r 1 plus r 2 equals a constant number. That was the purpose of the
string; that is how we physically made r 1 plus r 2 equal a constant number because the length of
the string was constant. The major axis is the longest distance through the center which also
passes through both foci and is the distance 2a; where the distance a is called the semimajor axis.
The minor axis is the shortest distance through the center and is the distance 2b;
where the distance b is called the semiminor axis.
And each focus is located a distance c from the center of the ellipse. Unfortunately,
not all textbooks agree on which letters to use for these distances in an ellipse, so …
Fun. (Yeah. )
So, r 1 plus r 2 equals the major axis, 2a, right, mr.p?
Billy, that is correct. Now that we have defined the major axis, we can identify
that the sum of the two distances from each focus any point on the ellipse equals the major axis.
I also want to point out that, at the location where the ellipse intersects the y-axis,
the two distances from the foci to the ellipse are equal in value and are also equal to a,
the semimajor axis. That means, for an ellipse, c squared plus b squared equals a squared.
I thought a squared plus b squared equaled c squared.
This is not the Pythagorean theorem. It is the definition of an ellipse.
Those are two different things. They sure look similar.
Yeah. Yeah, that is unfortunate. For an ellipse,
c squared plus b squared equals a squared. Let’s define the eccentricity of an elliptical orbit.
Eccentricity, lowercase e, equals c, the distance from the center to one focus, divided by a,
the semimajor axis. Eccentricity basically describes how circular an ellipse, or an orbit is.
Bobby, what is the eccentricity of a circle? The eccentricity of a circle. Well, for a circle,
the distance c equals zero because the two foci are both in the same location in the middle of
the circle. That means the eccentricity of a circle equals zero over a or zero.
Correct. The eccentricity of a circle equals zero. And if c equals a, which is an eccentricity of 1,
then b equals zero and the ellipse is actually just a line segment. Therefore, the range
of eccentricities for an ellipse are between zero and 1. And here are the eccentricities of
all 8 planets. (I miss Pluto.)
As you can see the planet with the largest eccentricity is Mercury at 0.2056.
To understand what an ellipse with an eccentricity of 0.2056 looks like. This is a circle.
And this is an ellipse with an eccentricity of 0.2056. Clearly,
that is nearly the same as a circle which is why we often approximate planetary orbits as circular.
Uh. How can that illustrate an ellipse with an eccentricity of 0.2056? It
should be squished and more like an oval, right? Actually, we could figure out if that makes sense.
How? Well, we know
eccentricity equals c over a, so, with an eccentricity of 0.2056, c equals 0.2056 times a.
We also know a squared equals b squared plus c squared. So, b squared equals a squared minus c
squared. Substitute 0.2056 times a in for c. 0.2056 squared equals 0.042271.
So, b squared equals 0.95773 a squared. And we get that b equals roughly
0.98 times a. In other words, for the elliptical orbit of Mercury, the semiminor axis b is 98%
the size of the semimajor axis a. That’s why it pretty much looks like a circle.
Oh. Yeah.
Thanks Bo. So, that is Kepler’s First Law of Planetary Motion.
In our next lesson, we will learn about Kepler’s Second Law of Planetary Motion.
(That seems logical.)
Thank you very much for learning with me today. I enjoy learning with you.