This question was posed to me back in the '60s, either in a high school or college physics class. It wasn’t particularly difficult, but my knowledge of physics has become somewhat rusty.
Suppose two grains of sand (mass of 1.11*10^-8 kg each) are alone in the universe (other universes are irrelevant, as are quantum theory, dark matter, etc.). They are 1,000,000,000 light years apart. The only force acting on them is each other’s gravitational attraction. How long until they collide?
I’m not sure it’s possible to answer the question because you haven’t specified the angular momentum of the system. They will only collide if it’s zero. If it’s not – and that is a much more likely situation – then they will orbit each other instead.
I don’t think they’d collide regardless, as the protons should decay within about 10^40 years, which would be sooner than the amount of time required to gravitationally accelerate them together. At that point it would just be electrons, positrons, and photons.
Though my impression is that proton decay and the halflife thereof is not a totally settled question.
I’ve tinkered with the derivation, and if I’m not mistaken, the formula works out to:
t = π√(r[SUP]3[/SUP]/(8G(m[SUB]1[/SUB] + m[SUB]2[/SUB])))
So: m[SUB]1[/SUB] = m[SUB]2[/SUB] = 1.11 × 10[SUP]−8[/SUP] kg r = 9.4605284 × 10[SUP]24[/SUP] m G = 6.67384 × 10[SUP]−11[/SUP] m[SUP]3[/SUP] kg[SUP]−1[/SUP] s[SUP]−2[/SUP]
which works out to, unless I made a mistake somewhere:
26552999503835112058013175142506730307724161111 s
But this assumes constant masses, doesn’t it? For the long periods under discussion proton decay needs to be accounted for. Even if the grains don’t completely decay, the gradual loss in mass may still significantly retard the collision.
It’s been more than 45 years since I took a relevant class, so I tried my hand at it too. Acceleration increases as the particles approach, but if a crude ballpark estimate is good enough we needn’t worry about that! I get a much shorter time than others, quite possibly due to my arithmetic error, but will post anyway.
Let’s start with Newton’s gravitational constant.
G = 6.7*10^-11 meter^3/kilogram/sec^2
It’s convenient to work with years, light-years and sand-grain’s mass, rather than seconds and meters.
second = 3.2 * 10^-8 year
meter = 3.3 * 10^-7 ly
kilogram = .9 * 10^8 grain
which allow the constant G to be expressed as
G = 27*10^-24 ly^3/grain/year^2
Newton’s formula for gravitation is ma = Gmm/rr where
r = 10^9 ly initially, or
a = Gm/rr = 2710^-24 ly^3 / year^2 / 10^18 ly^2
= 2710^-42 ly / year^2
We want the sand to travel 500 million light-years. Just to first get a ballpark estimate let’s first travel just 100 million light-years and ignore the increasing acceleration:
s = att/2 = 1310^-42 ly / year^2 * tt = 100 million ly
which solves to t ~= 310^24 years
The sand grains are much closer together by then, but still travelling very slowly compared with speed of light. So the total time elapsed till collision will be in the ballpark of 10^25 years. A fairly long time, but less than others’ estimates.
[septimus goes to hide in the corner, waiting to be embarassed when his arithmetic error is exposed]
I’m hesitant to reply - however, I believe the conversion of meters to light years used above is incorrect. From Google, I see a conversion of 1 meter = 1.1x10^-16 light years. I haven’t worked through the rest of the math to see what impact this might have though.
Under Newtonian calculations, if the grains had started moving in the empty universe, at the time that our universe was undergoing the Big Bang, how far would they have moved to date?
If there are only two grains of sand in the universe can there be any angular momentum under General Relativity? There can be in Newtonian mechanics which assumes the existence of a fixed space. But General Relativity does not if I recall correctly so I’m not sure how you determine if there is any revolution. Either grain of sand could be spinning on it’s own axis, but that would be irrelevant.
I don’t see why not. You can still have the particles, with velocity in a direction not parallel to the vector connecting them. THis will produce angular momentum of the system. However if this velocity is larger than an almost negligible quantity it will be large enough for them to achieve escape velocity from each other.
Definitely, in fact I wouldn’t be surprised if random molecular motion or quantum effects are enough to cause them to miss each other. The only way this is going to work at all is if they start out absolutely still an no forces outside of gravity affect them
If their journey lasts 10^38 years, its clear that they just started their journey, so we can just assume that their current acceleration was equal to their total acceleration of the first 13.8*10^9 years. From Radical P’s calculations I calculate it to be around 6x10^-50 m/s
To determine to velocity at impact, we can just use the escape velocity. since energy involved in launching a particle to infinity is that same a catching one from infinity.
Based on the density of silicon, I calculate that the radius of the sand grain is going to be about 5x10^-4 meters. So at impact they should be going about 5*10^-15 m/s at impact. I’m not sure that I calculated this right given that there are two bodies, but it should be correct within an order of magnitude.Also note, most of the acceleration will happen in the last centimeter.
