How Do Grains Gravitate Toward Each Other in Zero-G?

[1d20]

I recently saw How the Universe Works, and was impressed by the video of astronaut Don Pettit’s sugar-and-salt experiment. (6 minutes into the show. Put sugar and salt in a plastic bag, and watch as the grains visibly gravitate toward each other in zero-G.) I’m trying to express this process with math, but the numbers I’m coming up with are way too big. Here’s the formula I derived:

Assuming two masses (m) separated by some distance (d) in zero-G conditions, how long does it take for the two masses to meet halfway?

First I take the distance equation and solve for time: t = \sqrt{2d/a}

Now I modify this equation to take into account the fact that each mass only has to move half of d: t = \sqrt{d/a}

Now I use the G-force equation to derive each mass’ acceleration: a = F/m = Gm^2/d^2 / m = Gm/d^2

Now I plug the second equation into the first: t = \sqrt{d/ (Gm/d^2)} = \sqrt{d^3/ Gm}

Being extremely liberal, I’ll assume each grain is 1 gram and the two are separated by 10 cm. The equation says it takes them 34 hours to drift together, but that's way too long. The video isn’t time-lapse, and it clearly takes only moments for gravity to pull the grains together.

I suspect that I need to do some integration, but I have no practice defining integrals. Obviously acceleration is the variable that has to be integrated, but with respect to what? I’m not sure I can even do set up the proper integration without going back a few steps. Anyway, help is appreciated!

 

[Nabeshin]

You are incorrect in assuming that it is gravity responsible for clumping. (Note: your analysis is incorrect, for it assumes constant acceleration. A more full analysis would involve solving a differential equation, as you note. It's more complex than I want to get into here.) What is important is that as you note the free-fall time for these two objects is really large. Another way of arriving at your result is using dimensional analysis, we have three quantities: G, m (the mass of the particles), and d (their separation). Forming a time out of us leads us to conclude t \sim \sqrt{\frac{d^3}{G m}}. The result is the same, but I haven't assumed anything. In reality, we're only missing a factor of pi/2!

At any rate, using my formula for d=1cm, m=1g, and the standard value for G, I get t~ 1hr. You can quibble about these values a bit, but the point is, it's not on the order of a few seconds.

What I suspect IS responsible, is simply the clumping due to friction between bodies when they meet. They are all floating around more or less in random motion, and they happen to collide every so often. When they do, they latch onto each other. Thus the object becomes larger and presents a larger cross section to all the other particles floating around, so it will tend to grow even without this gravitational interaction.

 

[Lsos]

As Nabeshin said it's not gravity that causes them to clump. I suspect it's electrostatic force due to, as again Nabeshin mentioned, the friction between the particles. They rub against each other, they gain different charges, and they attract (I hope my analysis there is correct). Once they get big enough (much MUCH bigger), gravity can take over.

 

[1d20]

That's disappointing. Thanks anyway.