Kepler's Laws: Ellipse Law, Equal-Area Law, and Harmonic Law (Part 2)

I’ll pick up a bit from the previous post and go straight on.

We derived the orbit equation like this

The claim is that this thing is an ellipse…..

But it’s in spherical coordinates so it’s hard to get a feel for it~~!!!

To see why~~~~this equation is an ellipse, I’ll derive a Real, for-real ellipse in spherical coordinates!!! and compare it against that

and go in the direction of “Ahhhh~~ so that equation is an ellipse~~~~~~~” — that kind of admiration!!!

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

I’ll derive the locus equation of an ellipse from this definition!!

We did this back in high school math-study time too

the only difference is that we’re doing it in spherical-coordinates (said in a goofy voice)!!! hehehe

First, draw an ellipse and let the semi-major axis of the ellipse be a,

and let the distance of focus f from the origin be ε times the semi-major axis a

Then now let’s pick out the points where the sum of distances between the two foci is the same

the distance from one point f’ will be r'

and the distance from f will be r

And θ has to be set up like this.

ta-daaa~~~~

I’ll define all these various variables like this.~~

And by the definition of an ellipse

r’ can be expressed in terms of θ and r~~!

how you do it is — by Pythagoras-goras Mr. Pythagoras’s theorem

This is the ellipse equation expressed as r(θ)!

Let me just point out one thing!

What if ε=0???????

r=a. This represents a “circle”~!!!

ε is called the eccentricity!

ε>1 ☞ hyperbola

ε=1 ☞ parabola

0<ε<1 ☞ ellipse

ε=0 ☞ circle

Anyway, the r(θ) we got by applying universal gravitation to the orbit equation (in a central force field) has the same shape as above!!!

so it’s an ellipse!!!!!!!

chotto matte kudasai!!!!!!! (hold on a sec!!!>

The constant that plays the role of ε

we should check whether this is greater than 0 and less than 1!!!

As for A over there, there’s no way to even compute it… hehe, through observation…. we have no choice but to get it experimentally….

Let me show you that~ hehe

krarararahahahahahalalal

the Earth is doing a liiiiiiiike almost circular motion!!!! hehehe

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

Kepler’s 3rd law, the harmonic law, says that the square of the planet’s orbital period T and the cube of the semi-major axis a of the elliptical orbit are proportional, so

“wooo~~~wow~~~ how beautiful~~~~~~ the world is in such harmony~~~~~~”

……………………………..bullshit, beautiful my ass. T_T T_T T_T

Anyway, deriving the equation is simple. This also comes from conservation of angular momentum.

According to the law of constant areal velocity, the areal velocity is

this was it

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

I’ll express the semi-minor axis b in terms of a d-d-d-d-d-d-d!!!

so~

onto 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

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