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Tour de Force – (part 1)

Posted in: Gravity Simulator, Lesson Time! | June 21, 2014 | 1 Comment


Another week — another Gravity Simulator Lesson!  This week: what happens when you change the laws of gravity?  In the past, I’ve touched on this topic a bit, but this time we’re really gonna dive in.

 

The Baseline: 1/r^2

Ah yes, Newtonian gravity.  The perfect place to start.  This is gravity as we know it in our universe (ignoring pesky things like general relativity for the moment).  With the “one-over-r-squared” law, gravity decreases inversely as the square of the distance between the two objects.

That’s a mouthful, but really it’s quite simple: If we get twice as far away from the center of the earth, it’ll pull on us with one quarter of the force it does now.  If we triple the distance, gravity will pull just one-ninth as hard.

And with this wonderful, happy force law, we get orbits that look like this:

 

Classic Inverse Square Law, what you'd find by default the Gravity Simulator

Classic Inverse Square Law, what you’d find by default the Gravity Simulator

(As usual, all the gifs you’ll see today are taken right from the Gravity Simulator.  Feel free to play around, yourself!)

 

Experiment 1: 1/r^2.1

In the Gravity Simulator, we can do something you can’t do in the real world: we can change the force of gravity and see what happens!  So let’s increase that exponent over ‘r’.  So now the force goes as “one-over-r-to-the-2.1.”  Doesn’t quite roll off the tongue as easy, eh?

Well, let’s see what happens:

 

The tiniest of changes -- a 1/r^2.1 force law.  Try it out in the Gravity Simulator with this code:  [ForceGr: r^(-2.1),Qual: 1,Zoom: 1,xSet: 0,ySet: 0], [x0: 62.5,y0: 17.5,vx: 0,vy: 2.7,t0: 0,who: 3,m: 0,c: 0], [x0: 0,y0: 17.5,vx: 0,vy: 0,t0: 0,who: 1,m: 1000,c: 1]

The tiniest of changes — a 1/r^2.1 force law. Try it out in the Gravity Simulator with this code: [ForceGr: r^(-2.1),Qual: 1,Zoom: 1,xSet: 0,ySet: 0], [x0: 62.5,y0: 17.5,vx: 0,vy: 2.7,t0: 0,who: 3,m: 0,c: 0], [x0: 0,y0: 17.5,vx: 0,vy: 0,t0: 0,who: 1,m: 1000,c: 1]

Huh, the orbits no longer match up.  That’s called precession — basically the planet is nearly making an ellipse as it goes around the star, but not quite.  The path keeps twisting around and around.  Turns out 1/r^2 is special — very few force laws will make orbits that don’t precess.

Why does making it 2.1 instead of 2 change things? Well, ‘r’ is raised to a slightly higher power, which means as two planets get further away, the force of gravity drops off faster than it would in our world.  And as they get closer, the force of gravity increases faster, too!

So when the planet comes in close, gravity pulls on it stronger (than in our universe at least), deflecting its path even more, which gives it a tighter curve.  That tighter curve when it’s close to the star means that the planet swings around faster than we’d expect.  So by the time it comes back out to its furthest distance again, it’s gone more than 360 degrees.  Matches with what we see above.  (Go, science!)

 

Experiment 2: 1/r^3

Let’s keep going!  Let’s raise ‘r’ to an even higher power!  We should expect even stronger precession, right?  Well, let’s see:

 

1/r^3 force law.  Try it in the Gravity Simulator with: Gravity Fun at TestTubeGames.com: [ForceGr: r^(-3),Qual: 1,Zoom: 1,xSet: 0,ySet: 0], [x0: 62.5,y0: 17.5,vx: 0,vy: 2.7,t0: 0,who: 3,m: 0,c: 0], [x0: 0,y0: 17.5,vx: 0,vy: 0,t0: 0,who: 1,m: 1000,c: 1]

1/r^3 force law. Try it in the Gravity Simulator with: Gravity Fun at TestTubeGames.com: [ForceGr: r^(-3),Qual: 1,Zoom: 1,xSet: 0,ySet: 0], [x0: 62.5,y0: 17.5,vx: 0,vy: 2.7,t0: 0,who: 3,m: 0,c: 0], [x0: 0,y0: 17.5,vx: 0,vy: 0,t0: 0,who: 1,m: 1000,c: 1]

Huh, now we’re getting something different.  When an object gets close, gravity starts pulling so strongly that the planet just spirals inward until it collides.  And if the object is too far away, it spirals outward… and since gravity gets a lot weaker the further out you go, it keeps spiraling out more and more, never to return.

 

Experiment 3: 1/r^10

What if we went really crazy with this?  (I always go really crazy with this.)  Let’s try a force law that decreases extremely fast.  1/r^10!!!! (Excitement, there, not factorials)  Now if two objects get twice as far apart, the force between them is about 1000 times smaller!  That makes for this weird world:

 

1/r^10 force law!  Play around with it in the Gravity Simulator with this code: Gravity Fun at TestTubeGames.com: [ForceGr: r^(-10),Qual: 1,Zoom: 1,xSet: 0,ySet: 0]

1/r^10 force law! Play around with it in the Gravity Simulator with this code: Gravity Fun at TestTubeGames.com: [ForceGr: r^(-10),Qual: 1,Zoom: 1,xSet: 0,ySet: 0]

Objects really don’t notice each other in that weird universe, they mostly travel in straight lines… at least until they get juuuust close enough to the star.  And once they do, they’re pulled quickly and without remorse into a collision.  Boy, I’m glad we don’t live there!  You can imagine how hard it would be to create galaxies in that universe, let along solar systems with stable orbits.

Tune in next week when we take things the other way… what happens for 1/r^1.9?  Or 1/r?  In the meantime, play around with your own force laws in the Simulator!

-Andy



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One Comment

  1. By exfret

    Posted July 5, 2014 at 2:13 pm | Permalink

    I tried ~1/(10!!!!). GSim just replaced it with 0. ):
    I’d be okay with that if I could just use Graham’s number instead…

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