Good morning. We have already learned about Kepler’s First and Second Laws
of Planetary Motion. Bobby, please read Kepler’s Third Law of Planetary Motion.
Ya know, other people can read too.
(I know.) ♪ Flipping Physics ♪
“The square of the orbital period of any planet is proportional to the cube of the semimajor axis
of the elliptical orbit; in other words, capital T squared, or period squared, is proportional to the
cube of the semimajor axis, or a cubed.” Thank you very much Bobby. You did an
excellent job. (Thanks?)
To clarify Kepler’s Third Law, it means the square of the orbital period,
or the time it takes for a satellite to complete one revolution around the sun,
is proportional to the cube of the semimajor axis “a” as described in Kepler’s First Law. Billy,
please remind me of what the semimajor axis is. The semimajor axis, a, is the longest distance
from the center of the ellipse to the curve of the ellipse. You can see it labelled in the figure.
Thank you Billy. More specifically, Kepler’s Third Law, an equation we will derive in a
future lesson, states that orbital period squared equals the quantity four pi squared
over the quantity the universal gravitational constant times the mass of the primary
all times the cube of the semimajor axis. “Primary”? What is the primary?
The Sun is the “primary” of the solar system. The primary is the object at one of the foci
of the ellipse which each of the satellites trace when they orbit around the sun.
Thanks. You’re welcome.
Notice all the quantities in the parenthesis in Kepler’s Third Law are constants, therefore,
the orbital period squared is proportional to the cube of the semimajor axis.
To illustrate Kepler’s Third Law let’s look at data for the planets in our solar system.
The units on the semimajor axis are Astronomical Units or AUs. Where one astronomical unit is
the average distance between the Earth and the Sun. AUs are a useful distance measurement when
discussing celestial objects in our solar system. (Hold up mr.p)
The orbital … The semimajor axis is the average
distance between the satellite and the primary (Yeah, sorry. Have I not mentioned that before?)
No. You have not mentioned that the semimajor axis is the average distance between
a satellite and the primary. That’s cool. I agree. And, back to the table. The orbital
periods are in Earth years, a logical unit of time in our solar system. And we can cube the semimajor
axis, and square the orbital period. And we can add orbital period squared over semimajor axis
cubed. Hopefully you can see that, for all the planets, they each have roughly the same value for
orbital period squared over semimajor axis cubed. Woah! That is amazing! They all equal almost one!
That IS amazing! I guess the stars really do align!
Yeah. Magic. Uh We based all
our numbers on the Earth, so of course period squared over semimajor axis cubed equals one.
What? Huh? The semimajor axis of the Earth is defined
to be exactly 1 astronomical unit. The orbital period of the Earth is exactly 1 year. 1 squared
over 1 cubed equals 1. And Kepler’s Third Law states that all the planets should have the same
constant value for the numbers in the parenthesis. So, yeah, nothing magical. We just used units to
make it so the period squared over semimajor axis cubed for every planet should equal 1.
Oh. yeah.
And we can graph orbital period squared versus semimajor axis cubed and the best
fit line equation for all of these data works out to be orbital period squared equals 1.01
times semimajor axis cubed. The slope equals roughly 1,
as we would expect it to because we chose units of astronomical units for the semimajor axis and
Earth years for orbital period, as Bobby just explained. Next time we are going to derive
Kepler’s Third Law of Planetary Motion. (Of course we are.)
Thank you very much for learning with me today, I enjoy learning with you.