GG: Energy Gained by Whirling Motion as Fraction of Energy Lost by Spinning Rotor

A.M. Nobili, D. Bramanti and G. Catastini
Pisa 1997

Gruppo di Meccanica Spaziale, Dipartimento di Matematica, UniversitÓ di Pisa, Via Buonarroti 2, I-56127 Pisa, Italia

The total spin angular momentum of a rotor, made of two (mechanically coupled) equal masses m, each of dimension R, spinning at angular velocity tex2html_wrap_inline75 , is:

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The angular momentum of the whirl motion with angular velocity tex2html_wrap_inline79 in the inertial frame and with radius of whirl tex2html_wrap_inline81 is:

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Since the total angular momentum has to be conserved it must be:

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from which, since tex2html_wrap_inline79 is constant, it follows:

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The spin energy of the rotor is:

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The energy (kinetic plus elastic) of the whirl motion is:

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The time derivatives of (3) and (2) (using (1)) give:

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and therefore

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This is Eq. (32) of our Paper I. It is obviously a very important result: since in GG we have in the inertial frame the extremely slow whirl motion at angular frequency tex2html_wrap_inline105 (contrary to Schutz's example of the stellar case in which, in the inertial frame, tex2html_wrap_inline107 ) Eq (4) represents the very small fraction of energy lost by the rotor which goes into the destabilizing whirling motion. All the rest, that is tex2html_wrap_inline109 !!, is dissipated as heat inside the springs. So, the idea one might have that the faster the spin the higher the energy gained by the whirl motion (by which argument GG would be highly unstable), is proved to be incorrect.

The Celestial Analog

A \2body celestial model close to GG (with two masses only) is a binary system of two spinning bodies. Tidal friction is known to change the spin period of the bodies and their relative distance. For instance, in the Earth-Moon system the rotation rate of the Earth decreases and the relative distance increases. Tides raised by each body on the other produce a change in the orbital angular momentum through dissipation inside the bodies, and it is amazing that a relation just analog to (4) holds. Let m, I and tex2html_wrap_inline75 be the mass, moment of inertia and spin angular velocity of the bodies orbiting around one another on circular orbits at relative distance 2r and orbital angular velocity n. We refer here to the case of equal masses and parallel rotation and revolution vectors.

The total spin angular momentum of the binary system is:

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The orbital angular momentum of the system is:

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Their time derivatives are:

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The third law of Kepler for this \2body system reads:

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and it relates tex2html_wrap_inline131 to tex2html_wrap_inline133 :

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so that:

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Since the total angular momentum has to be conserved it must be:

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from which it follows:

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The total spin energy of the bodies is:

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The orbital energy of the \2body system is:

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The time derivatives of (7) and (6) (using (5) and third Kepler's law) give:

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and therefore

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which is perfectly analog to (4).


Research Papers Available Online

       (Anna Nobili- nobili@dm.unipi.it)