Tour de Force – (part 1)
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:
(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:
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:
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:
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!
-AndyPost a Comment
Black Hole or Bust
How hard could it be to add in black holes, right?
We all know that black holes are extremely massive, extremely dense objects. (Mostly.) Get close enough to them, and gravity pulls so strong, that not even light can escape. Whoa! So we just need to make a big star in the Gravity Simulator — and we get a black hole!
Well, no. You’ll never get a black hole if you’re dealing with boring, old Newtonian gravity, though. In the old simulator, say, you could make a star bigger and bigger and bigger, and all you’d get is a bigger star. Any astroid or planet or star can escape its pull, so long as it’s moving fast enough.
Around black holes, there’s a line of no return called the Event Horizon. If you’re outside of this boundary — you could escape the black hole. But the moment you cross it, you’re sunk. You’ll get swept ever further into the black hole.
This happens because General Relativity contains our old friend Special Relativity. And, if you’ll recall, a key part of Special Relativity is that nothing can travel faster than light. The speed of light is the speed limit for everything. The Event Horizon represents the line near enough to the black hole where, if you wanted to escape, you’d have to travel at light speed. Fall in a bit closer, and gravity gets a bit stronger, and you’d need to go even faster than light to escape. No can do. You’re stuck.
In the pictures above, I draw where the Event Horizon would be on each the star. The smaller the star is, the weaker the gravity it, and the closer you’d have to get to reach the Horizon. In fact, most of the time, you’d actually have to go deep inside the star to find this line. Which means, it isn’t really an Event Horizon. The calculations I used to draw these assumed that all the mass of the star is inside the Horizon. As you can see above, that’s not the case. The stars aren’t dense enough, which means: no Event Horizon and no black hole.
But if we get enough mass in place, the Horizon grows big enough that it swallows up the whole star — and we finally get our black holes! Now let’s have some fun with them!
Black Holes have tons of neat properties, which you’ll all be able to check out in the next update to the Gravity Simulator. Stay tuned for more General Relativistic fun!
-AndyPost a Comment
Thought I’d keep you all apprised of the latest addition I’m working on:
RELATIVITY IN THE GRAVITY SIMULATOR.
At least, *mostly* relativity. You see, we’d talked about adding in Black Holes… which would be awesome. But not just *nom*nom*nom* generic sci-fi ‘sucks-stuff-in’ Black Holes. This is TestTubeGames, after all. So I wanted to at least get stuff approximately right. Maybe so ‘orbits’ become something like this:
A while back, in an earlier chat in the forums, we found a General Relativistic formula for the attraction between two objects. Seems reasonable that we could plug in that force law (after all, we’ve got change-able force laws already). It won’t be precisely right (there won’t be gravity waves…), but it’ll get us close.
Simple, then, slap on a GR Force Law and call it a day! Well, nope. Because Black Holes have this neat feature where once something gets too close — ~~inside the event horizon ~~ — it’ll *never* come back out. Mwahahaha.
Never, that is, unless it travels faster than light. Which, in the real world (as far as we know) nothing does. But in the Gravity Simulator, you can launch stuff at any speed! Black Holes would lose all meaning, objects could escape at will. They’d become just ‘really strong stars’ instead of ‘points of no return.’ Boo, hiss.
That means I need to add more relativity in the sim, to make objects obey the speed limit of light. Now when something accelerates, it can get close to, but never reach the speed of light. And, lo and behold, we get neat orbits like this:
Great. Can we just paint that star black and stop there?
Because once you have objects traveling near the speed of light, well, then E=mc^2 becomes important. Namely, mass is energy, energy is mass. So what? So EVERYTHING. Imagine two stars colliding. They rush inwards to meet one another, then *boom* they combine to form a single, stationary star.
The total energy has to remain the same, which means that Kinetic Energy had to go somewhere. In our sim, there’s only one place that energy can go: into rest mass. Just as two subatomic particles can combine to form something massive (wee protons crashing into each other to make the Higgs boson, anyone?), two stars can combine to make one *huge* star.
What other parts of relativity will come into play down the line? Well, the Schwartzschild radius is important. And relativistically slowed clocks are awesomely fun…
Where does this all end? With a bang? With a whimper? Will the simulation collapse into a singularity under all the weight of the new code? Stay tuned to find out!
-AndyPost a Comment