Kepler's Laws: Ellipse Law, Equal-Area Law, and Harmonic Law (Part 2) [Classical Mechanics I Studied #9]

I’ll pick up a bit from the previous post and go straight on. We derived the orbit equation like this:

The orbit equation from last time: u = A cos(θ − θ0) + k/(m l²), so r = (m l²/k) / ((m l²/k) A cos(θ − θ0) + 1)

This little guy is supposed to be an ellipse….. But it’s in polar coordinates, so I can’t really get a feel for it~~!!!

To see why~~~~ this equation is an ellipse, I’ll derive a Real, for-real ellipse in polar coordinates!!! and compare it against that one, and then we can go “Ahhhh~~ so that equation is an ellipse~~~~~~~” — that kind of admiring sigh is where we’re headed!!!

First, the definition of an ellipse is “the points where the sum of the distances from the two foci is the same!!! The set of all such points!!!!”

Let’s derive the locus equation of an ellipse from this definition!! We did this back in high school math class too, and the only difference is that we’re doing it in polar coordinates (said in a goofy voice)!!! hehehe

An ellipse centered on the origin of the x-y axes, with semi-major axis a (from −a to a on the x-axis) and its two foci f’ and f on the x-axis

Now, draw an ellipse and let its semi-major axis be $a$, and let the distance from the origin to the focus $f$ be $\varepsilon$ times the semi-major axis $a$:

Then let’s pick out the points for which the sum of the distances to the two foci is the same. Take one such point: its distance from $f'$ is $r'$ and its distance from $f$ is $r$. And $\theta$ has to be set up like this.

The same ellipse with a point on it: r’ is the distance to f’, r is the distance to f, θ is the angle at f measured from the x-axis, and the legs r sin θ and r cos θ are marked

ta-daaa~~~~ I’ll define all these various variables like this.~~ And by the definition of an ellipse, r + r' = 2a .

$r'$ can be expressed in terms of $\theta$ and $r$~~! How you do it is — by Pythagoras-goras Mr. Pythagoras’s theorem:

A right triangle with hypotenuse r’, vertical side r sin θ and horizontal side 2εa + r cos θ

Pythagoras gives r’² = (r sin θ)² + (2εa + r cos θ)²; with r’ = 2a − r this becomes (r sin θ)² + (2εa + r cos θ)² = (2a − r)², so r(θ) can be built

Expanding and tidying up gives r = a(1 − ε²)/(1 + ε cos θ), which has the form const’/(1 + const cos θ)

This is the ellipse equation expressed as $r(\theta)$! Let me just point out one thing!

What if $\varepsilon=0$???~?~?

It becomes $r=a$. This represents a “circle”~!!! And $\varepsilon$ is called the eccentricity!

$\varepsilon>1$ ☞ hyperbola
$\varepsilon=1$ ☞ parabola
$0<\varepsilon<1$ ☞ ellipse
$\varepsilon=0$ ☞ circle

Anyway, the $r(\theta)$ we got by applying universal gravitation to the orbit equation (in a central force field) has the same shape as the one above!!! So it’s an ellipse!!!

The orbit equation r = (m l²/k) / ((m l²/k) A cos(θ − θ0) + 1) has the same form const’/(1 + const cos θ)

chotto matte kudasai!!! (hold on a sec!!!) The constant that plays the role of $\varepsilon$ is A(m l²/k) .

Is it greater than 0 and less than 1? We should check, right!!!

As for that $A$ over there, there’s no way to even work it out from the equation itself… hehe. Only through observation…. we have no choice but to get it experimentally…. Let me show you that~ hehe

Here it is: a table of planetary data from a textbook (period, semi-major axis and eccentricity $\varepsilon$ for each planet), and Earth’s $\varepsilon$ is 0.017. Kra-ra-ra-ra-ha-ha-ha-ha! Earth is moving in alllllllmost a circle!!!! hehehe

Alright, now the 3rd!! Let’s get into the harmonic law!!!

Kepler’s 3rd law, the harmonic law, says that the square of a planet’s orbital period $T$ and the cube of the semi-major axis $a$ of its elliptical orbit are proportional, so everybody goes “wooo~~~wow~~~ how beautiful~~~~~~ the world is in such harmony~~~~~~~”

……bullshit, beautiful my ass. T_T

Anyway, deriving that formula is simple. This also comes from conservation of angular momentum. According to the law of constant areal velocity, the areal velocity was A-dot = L/(2m) .

If we integrate this over one period $T$, with respect to $dt$!!!

Integrating the areal velocity over one period T: (L/2m) T = πab, the area of the ellipse

I’ll express the semi-minor axis $b$ in terms of $a$ — drumroll: da-da-da-da-da-da-da!!!

A right triangle with legs b and εa and hypotenuse a

so~ (remember, $l = L/m$ from last time)

Into this~ now, into this thing made of ellipse info, we’re going to plug in the info about our Earth and the Sun~ We can bring the info from the orbit equation and plug it in, right?

This is what we’ll plug in hehehe

T² = (4/l²) π² a³ a(1 − ε²) = (4/l²) π² a³ (m l²/k) = 4π² a³ m/(GMm) = 4π² a³/(GM), so T² = (4π²/GM) a³

And that’s how we derived the harmonic law~~!!

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