Kepler's Laws: Ellipse Law, Equal-Area Law, and Harmonic Law (Part 1)

Before jumping into Kepler’s laws, let me lay down some background on universal gravitation.

If we talk a bit about Newton for a moment, the cartoon most often used to depict this fellow is this one. (Warning: contains profanity.)

Newton, lost in thought like this,

supposedly thought “if I fire an apple-cannon like this, it wouldn’t fall~~~”????

But have you also heard the saying that Newton was merely “a small boy standing on the shoulders of giants”????

What that means is, Galileo had pretty~~~ much established the motions that happen on Earth (including inertia)

before Newton’s time.

And the motions of celestial bodies beyond Earth, as everyone knows,

were elegantly established mathematically by Kepler, apparently? (Also before Newton.)

So what Newton did was take these two seemingly disparate things and

slick~~ly combine them into universal gravitation, apparently hehehehe.

Alright, now I need to talk about “reductionist thinking.”

Because we’re about to enter Section 2, “Gravity between a uniform sphere and a particle”!!!

To unpack what “reductionist thinking” means a bit —

“If you break everything down, you can find out it’s all a conclusion following scientific cause and effect. There is a scientific cause everywhere!!!!!”

Well lol, apparently this reductionist way of thinking was philosophically in vogue in Newton’s era??

(Could we say the phrase “Love is nothing but a hormonal prank happening in the brain!!!!!!!!!!!!!!” also followed that way of thinking???)

Alright, what I’m trying to say now is,

I’m trying to verify this.

r is the distance between the centers of the two celestial bodies, and M and m are the total masses!!!

Is it really okay to write it like that!!

What Newton thought was,

taking Earth and the Moon ~~

chopping them up into stupidly tiny pieces like that, and then the total force is the sum of the gravitational attraction between each little chunk’s mass and every other little chunk’s mass,

this~ is universal gravitation~~ — that’s what he said,

but we just go, F is great-mama over r-squared, and move on, don’t we???

Is that really okay!!!! Can we treat it as a point mass!!!!!!!!!!!!!!!!! We need to simply verify this,

and since it’s exactly the same as when we derived the electric force, let’s each verify this simply on our own!!!! heh — the conclusion is ‘yes, we can treat it as a point mass!!’

(I’ll post this as an appendix next time!)

Let’s move on now to Kepler’s laws.

What are Kepler’s laws, which nearly perfectly proved celestial motion mathematically before Newton~~!!!!

Law 1. The law of ellipses (1609): The orbit of each planet is an ellipse with the Sun at one focus!

Law 2. The law of equal areas (1609): The straight line connecting the Sun and a planet sweeps out equal areas in equal times as the planet orbits around the Sun.

Law 3. The harmonic law (1618): The square of a planet’s period (the time it takes for the planet to go around the Sun once) is proportional to the cube of the semi-major axis of its orbit.

Now, proving these Kepler’s laws with Newtonian mechanics becomes our job!!!

First, let me derive the 2nd law!!!!!!

Alright, we know that the Earth orbits around the Sun, but apparently we didn’t really know if it goes around in a circle, or an ellipse, or in a spiral drifting farther and farther away.

But we do know this much!!! The force applied from outside is Zero!! heh

The force applied from outside is Zero….

If the externally applied force is 0, then the torque that affects the rotational motion — which is r cross F — is also 0.

If the torque is 0, then angular momentum L is constant!!!!

Then let me write it this way!!!!

Now I’m going to calculate the areal velocity during the planet’s motion. First let me calculate the area!!!!

I’m going to calculate roughly this much area~~~

Pretty easy to understand, right??

Dividing both sides by dt and taking the limit,

Oh my god!!!! There’s r-squared-theta!!!!

Since we assumed angular momentum would be constant!!

the conclusion comes out that the areal velocity is constant!!!!!~~~~~

And just like that, Kepler’s 2nd law has been proved.

Now I’m going to derive Kepler’s 1st law, the law of ellipses,

but first, instead of writing universal gravitation as

for now let’s write it as

Something that depends only on r~~~~~~ is how we’ll see it~~

Then the equation of motion becomes

which looks like this, and I’ll solve this equation of motion! hehe

So the double-dot of r was done in Chapter 2, but let me write it once more and proceed!

Alright alright alright, so the equation of motion above becomes

What the bottom equation means is,

Alright, now let’s

focus on this.

What we’re going to do here is a substitution,

As for why we substitute like this, I’ll reveal it a bit later,

we need to express this in terms of the substituted u, but it’s kind of a headache…. hah;-_-

What just happened is, the information about time got hidden on the right-hand side. The

that contained information about time joined forces with

and transformed into angular momentum, became a constant, and just like that the time-dependence vanished,,,T_T

To give you a little spoiler of where we’re heading now,,,, we’re heading toward the orbit equation!!!

Alright, since we’ve done r-dot, let’s head to the double-dot too!!!!

Okay!!!! We’ve expressed the double-dot of r in terms of l, theta, and u!!!

Then let’s go back to the equation of motion!!!

For theta-dot, just like before,

I used this logic~~~~~

Keep Going~~~~

This is apparently called the “orbit equation.”

It’s expressed as a relation between r and theta.

But there’s a crucial premise!!!! f(r) has to be a central force field (or an isotropic force, in the r direction).

Now, within a central force field, once we see r(θ), we can figure out the force!~ Let’s keep going!!

Wanna solve an example problem and move on?????????? Ah, nope.

We now understand how that orbit equation we derived turns out that way,

but then we need to figure out why on earth the Earth just happe~~~ns to be not a circle,

not just some squiggly mess, but of all things orbits in an elliptical orbit — that should come first, I think!!!! hehe

(Actually it’s apparently very~~~~ close to a circular orbit hahaha You’ll find out later on.)

Then now I’ll substitute the real universal gravitation into f(r)~~~ (since we’re going to look at the real Earth now!)

Whoa, what this implies is that universal gravitation,

“of all things,” was inversely proportional to r squared?!??!~~?~? — apparently we’re supposed to feel this hahaha.

Our creator made universal gravitation, of all things, proportional to the inverse of the square of the distance,,,

and made us physics majors suffer……darn….

Ah, or maybe not? Maybe if it weren’t proportional to the inverse of r squared,,,

it would’ve been even harder??????????hehehehehe thanks thanks thanks thanks.

Then let’s solve the differential equation above!!!!

Let’s knock out this differential equation with simple intuition!!!

For the double derivative plus the thing itself to equal a constant???????

Ah, first let’s see what happens if the constant is 0.

If u were sin or cos, it would work out….??!

Then let me set up the trial solution like this:

Oh~! Doing it this way works!!!

Then the general solution that satisfies the differential equation with a constant on the right-hand side is

if it’s this, then the differential equation with a constant on the right-hand side holds, right????

Okay~~~!!~

We’ve now succeeded in expressing u in terms of theta!!

I’ve expressed u(theta) that satisfies the orbit equation above hehehehe hehe.

Now let me express it as r(theta).

This thing is supposed to become an ellipse now…………………..

but since it’s in spherical coordinates it doesn’t quite click~~~!!!

Why~~~~this expression is an ellipse —

I’ll derive a Real real ellipse in spherical coordinates!!! and compare it with that, and

“ahhh~~ so that expression is an ellipse~~~~~~” — let’s head in the direction of that kind of exclamation!!!

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

Let me derive the locus equation of an ellipse from this definition!!

We did this back in high school in the exam-prep hours too, the only difference is doing it in spherical coordinates!! hehe

The post has gotten too long, so I’ll continue in the next one!!!!

Continuing right after!!!! hehe