Two Particle Elastic Collisions

Discussion in 'Physics & Math' started by rpenner, Apr 8, 2015.

  1. danshawen Valued Senior Member

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    3,951
    Nice analysis, as always, rpenner.

    When I was studying physics in high school, I got ill while we were doing a unit on a Crooke's radiometer, which is great for discussing differences between elastic, inelastic collisions.

    I did manage to catch the beginning of the unit, in which we were shown an old Currier and Ivy film of the experiment where a Crookes radiometer runs in one direction under low vacuum, but in the other direction when the vacuum is taken to such an extreme that photons are finally able to drive it (instead of collisions with low pressure gas molecules).

    So here is my related inquiry, which thus far has not been addressed in your analysis.

    Starting from an even distribution of molecules of a mix similar to that of the early universe, how likely or unlikely is it that any of them will be able to aggregate by means of just gravity to form larger gravitating masses? Would the collisions of particles due to gravity be primarily elastic or inelastic? Would electrostatics or some other process also be needed to aggregate the particles initially so that their combined gravity is eventually large enough for gravitation to do the rest/make most collisions primarily inelastic? Has this calculation already been done?
     
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  3. RajeshTrivedi Valued Senior Member

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    Great question.....
     
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  5. rpenner Fully Wired Valued Senior Member

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    The primordial constituents of the universe at a time it was cool enough to have nuclei were Hydrogen and Helium and traces of other light elements.
    So clumping together atom-by-atom is not what the bulk of cosmological evolution was about, but rather about the general tendency of cold gas to collapse under self-gravitation.

    In a perfectly uniform universe without edge, the mean density is the same everywhere and there is no preferred location to clump to. The universe expands or contracts as a whole. However, since matter isn't continuous, there are always going to be some statistical expectation of density variation, and so the question in #101 has most basic applicability as:
    What are the conditions when a region of over-dense primordial gas is unstable against gravitational collapse.

    The math for this dates back to 1902 in the study of gravitational collapse -- the Jeans instability. It turns out to be a relation between how much kinetic energy each particle of a gas has at a certain temperature versus how much energy it takes for a cloud to expand against its internal gravity. Since about the time the universe became transparent, density anomalies of more than 10,000 solar masses have been cool enough to be unstable against gravitational collapse. At current CMB temperatures, an over-density region with a diameter of a light year would tend to collapse if its over density totaled more than a solar mass.

    Further reading (ranked in terms of approachability):
    http://en.wikipedia.org/wiki/Jeans_instability
    http://sgoodwin.staff.shef.ac.uk/jeans.pdf
    http://www.physics.drexel.edu/students/courses/physics-502/jeans_instability.pdf
    https://books.google.com/books?id=c5fvAAAAMAAJ (A college library might have it)
    https://books.google.com/books?id=2a1Dxo9HxE8C
     
    Last edited: Apr 17, 2015
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  7. danshawen Valued Senior Member

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    The Jeans instability link was most helpful. Thanks.

    Even an 'elastic' collision between gas molecules can be at least partially inelastic (consumes some energy for a particular combination of temperature, pressure, and gravity), based on thermodynamic principles which have been used to calculate the Jeans instability.

    Time dilation effects are one dynamic which seems to have been neglected here, unless I am misreading. A layer of atmosphere which is closer to the surface of the Earth would mean that the molecules of gas are moving slower (lower rms velocity = 460 m/s at STP, mean free path ~ 68 nm) than the molecules of gas in any layer above it, and so in order to come into thermal equilibrium with upper layers, these collisions would need to be more inelastic than with those coming from below or from the sides. This is a slight asymmetric effect, but since atomic clocks can measure differences in the passage of time where delta h is on the order of 1 meter, not altogether a negligible one.

    Differences in pi meson decay rates in the upper atmosphere was one of the first solid confirmations of General Relativity. This problem appears similar, if more mundane.
     
  8. rpenner Fully Wired Valued Senior Member

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    At low temperatures, I would expect most gas molecule collisions to be elastic. Due to quantization, there aren't a lot of degrees of freedom to lead to electronic excitation or chemistry such that the collision would be inelastic.

    General relativity may safely ignored in atmospheric modeling because unpredictable chaotic variation in temperature (i.e. "weather") swamps such tiny effects. Your claim about how this requires that the collisions be inelastic seems based on nothing.

    Do you have a reference for your pi meson claim?
     
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  9. danshawen Valued Senior Member

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    A description of the Rossi-Hall and Frisch-Smith experiments may be found here:

    http://en.wikipedia.org/wiki/Time_dilation_of_moving_particles

    But as you have suggested, these are mostly Special Relativistic, not gravitational or GR time dilations. The wiki article does make mention of length contraction as the mesons descend. For something traveling as slowly as a gas molecule, this would not produce very much relativistic length contraction or time dilation.

    Hydrogen is kind of a special case in gas interactions, and I doubt such collisions would be elastic if one of those molecules collided with anything other than another hydrogen molecule. They tend to get absorbed by all sorts of molecules. Time dilation or other cheats would not be needed for such interactions to be very 'sticky', and to quickly form much larger gravitating masses.

    That answers my question completely. Thanks again.
     
    Last edited: Apr 17, 2015
  10. brucep Valued Senior Member

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    Thanks for these interesting posts. I especially like the simple #2 reference on your list.
     
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