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Viscous disc paradox

  1. Nov 14, 2015 #1
    Which way does friction in a viscous disc operate?

    Imagine a ring consisting of ringlets. First consider a case of a pair of nearby ringlets - in the same plane, both circular orbits.
    If all particles of both ringlets are in circular orbits in the same plane, then they can never collide and therefore never exert any force. There is then zero temperature and zero viscosity.
    Now suppose the viscosity is nonzero. What then?
    By Third Law of Kepler, the inner ringlet should move faster.
    Therefore, the inner ringlet should propel the outer ringlet ahead: the outer ringlet should expand and the inner one shrink.
    But the problem is that the particles of rings, whether dust grains or gas molecules, are severally subject to Newton´s laws... and therefore also laws of Kepler. Including the Second.
    While particles of inner ringlets are indeed faster than outer ringlet, they are so while they are in the inner ringlet, and do not meet outer ringlet.
    The particles which can and do collide are those on elliptical orbit.
    Considering two neighbouring circular ringlets and an elliptical ringlet tangent to both at its apsides.
    The outer ringlet is slower than the inner, as per 3rd law - but the elliptical orbit at its apoapse is even slower, as per 2nd.
    Therefore, the outer ringlet should be slowed down and shrink. And inner ringlet, by the same reasoning, should encounter the faster part of elliptical ringlet at periapse, speed up and expand.

    So what´s the solution of the paradox? How should a viscous disc behave?
     
  2. jcsd
  3. Nov 14, 2015 #2
    I guess a viscous disk rotating in a Newton potential by friction would convert kinetic energy into heat.
    Thus it would heat up as its radius would shrink. Eventually it would be absorbed by the source of the Newton potential.
    However I expect the matter to condense into clumps in which no friction occurs as long as it is in a circular orbital (no tidal forces).
    I would also expect that the end result depends on how much matter is involved and on the size of the orbit.
    This will also set the time scale of the evolution.
    Of course it would be better to consult a text on planetary formation than to follow my guesses.
     
  4. Nov 14, 2015 #3

    mfb

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    @my2cts: Angular momentum is conserved (neglecting relativistic effects like the angular momentum of radiated light), it won't fall in.

    Angular momentum and energy scale with ##L^2 \propto E^3##. Putting objects in different orbits together in a common orbit releases energy, pushing them apart would require energy. Therefore, your objects should get closer together.


    Objects in different circular orbits will never collide, but you still get some interaction via gravity. It's not a simple "this is pushing this" however. See Saturn's rings for the complex interactions in such a disk.
     
  5. Nov 14, 2015 #4

    Chronos

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    Saturns rings are a good example. They are composed largely of water ice so any heating due to friction is not a major factor. The also orbit at a fairly leisurely pace the inside edge of the inner [C] ring has as orbital velocity of a little under 24 km/sec whereas the outside edge of the outer [A] ring has an orbital velocity of just over 16 km/sec. The ISS, by comparison circles the earth at a little less than 8 km/sec. The aggregate material that the rings are composed of vary in size, although not greatly. The smaller particle average less than a cm in size whereas the relatively rarer larger particle can be a meter or two across. It is beleived the larger variety are slowly broken apart or eventually migrate out beyond the rings. Orbital resonance with the moons of saturn are believed to protect the rings from any significant collapse or dispersal.
     
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