Abstract
In this chapter, we obtain the equations of motion that govern the simplest, non-trivial, deformable dynamics of an ellipsoidal body. We will employ volume averaging (also known as the method of moments or the virial method) to this end.
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Notes
- 1.
Further support for analyses that assume that granular minor planets deform in an affine manner will be found from results in plasticity theory (Sect. 4.4), and when comparing the predictions of such investigations with simulations and observation in the coming chapters.
- 2.
In fact, if we had retained \({\mathsf {B}}_S\) and \({\mathsf {B}}_P\) in their entirety when computing \({\mathbf {F}}_S\) and \({\mathbf {F}}_P\), then the formulae derived in Sect. 3.4 for \({\mathbf {F}}_S\) and \({\mathbf {F}}_P\) obey Newton’s third law, i.e., yield forces that are equal and opposite. This is proved as follows. From Maciejewski (1995), the total force exerted by the primary on the secondary, \({\mathbf {F}}_S = -\rho ''\rho 'G\int _{V''}\int _{V'}\frac{\overline{\mathbf {x}}_S - \overline{\mathbf {x}}_P}{\left| \overline{\mathbf {x}}_S - \overline{\mathbf {x}}_P\right| ^3}\mathrm{{d}}V' \mathrm{{d}}V''\), where \(\overline{\mathbf {x}}_S = {\mathbf {r}}_S + {\mathbf {x}}_S\) and \(\overline{\mathbf {x}}_P= {\mathbf {r}}_P + {\mathbf {x}}_P\) locate with respect to the mass center C material points in the primary and the secondary, respectively; see Fig. 3.1. The force \({\mathbf {F}}_P\) has the same formula, but with minus sign, so that \({\mathbf {F}}_S = -{\mathbf {F}}_P\). From Chandrasekhar (1969, p. 48, Theorem 7), \(-\rho 'G\int _{V'}\frac{\overline{\mathbf {x}}_S - \overline{\mathbf {x}}_P}{\left| \overline{\mathbf {x}}_S - \overline{\mathbf {x}}_P\right| ^3}\mathrm{{d}}V' = -2\pi \rho 'G{\mathsf {B}}_P\cdot {\mathbf {d}}_S\) , as it is the force per unit mass exerted at \({\mathbf {d}}_S\) by the primary; see Fig. 3.1. Similarly, \(-\rho ''G\int _{V''}\frac{\overline{\mathbf {x}}_P - \overline{\mathbf {x}}_S}{\left| \overline{\mathbf {x}}_P - \overline{\mathbf {x}}_S\right| ^3}\mathrm{{d}}V'' = -2\pi \rho ''G{\mathsf {B}}_S\cdot {\mathbf {d}}_P\). Thus, \({\mathbf {F}}_S = \rho ''\int _{V''}\left( -2\pi \rho 'G{\mathsf {B}}_P\cdot {\mathbf {d}}_S\right) \mathrm{{d}}V'' \) and \({\mathbf {F}}_P = \rho '\int _{V'}\left( -2\pi \rho ''G{\mathsf {B}}_S\cdot {\mathbf {d}}_P\right) \mathrm{{d}}V'\), which are exactly the formulae obtained in Sect. 3.4; see equations before (3.39) and (3.44). Clearly, \({\mathbf {F}}_S = -{\mathbf {F}}_P\).
- 3.
In a contact binary, the primary and the secondary touch each other. There may or may not be significant contact forces present. In this book, we will assume that the contact forces, if any, are insignificant compared to other forces present. The extent to which this assumption is valid will often be revealed by the mismatch of our predictions with observations; see, e.g., applications to binary near-Earth binaries in Sect. 10.8.
References
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965)
J.P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd edn. (Dover, New York, 2001)
S. Chandrasekhar, Ellipsoidal Figures of Equilibrium (Yale University Press, New Haven, 1969)
C. Chree, Further applications of a new solution of the equations of an isotropic elastic solid, mainly to various cases of rotating bodies. Quart. J. Math 23, 11–33 (1889)
D.T. Greenwood, Principles of Dynamics (Prentice-Hall, Englewood Cliffs, 1988)
O.D. Kellogg, Foundations of Potential Theory (Dover, New York, 1953)
Z. Kopal, Dynamics of Close Binary Systems (D. Reidel Publishing Company, Dordrecht, 1978)
D. Lai, F. Rasio, S.L. Shapiro, Ellipsoidal figures of equilibrium: compressible models. Astrophys. J. Suppl. S. 88, 205–252 (1993)
D. Lai, F. Rasio, S.L. Shapiro, Equilibrium, stability, and orbital evolution of close binary systems. Astrophys. J. 423, 344–370 (1994)
A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th edn. (Dover, New York, 1946)
A.J. Maciejewski, Reduction, relative equilibria and potential in the two rigid bodies problem. Celest. Mech. Dyn. Astr. 63, 1–28 (1995)
C.D. Murray, S.F. Dermott, Solar System Dynamics (Cambridge University Press, Cambridge, 1999)
D.J. Scheeres, Stability of relative equilibria in the full two-body problem. Ann. NY Acad. Sci. 81–94, 1017 (2004)
D.J. Scheeres, Orbital Motion in Strongly Perturbed Environments: Applications to Asteroid, Comet and Planetary Satellite Orbiters (Springer, New York, 2012)
I. Sharma, The equilibrium of rubble-pile satellites: The Darwin and Roche ellipsoids for gravitationally held granular aggregates. Icarus 200, 636–654 (2009)
I. Sharma, Equilibrium shapes of rubble-pile binaries: the Darwin ellipsoids for gravitationally held granular aggregates. Icarus 205, 638–657 (2010)
I. Sharma, J.T. Jenkins, J.A. Burns, Tidal encounters of ellipsoidal granular asteroids with planets. Icarus 183, 312–330 (2006)
I. Sharma, J.T. Jenkins, J.A. Burns, Dynamical passage to approximate equilibrium shapes for spinning, gravitating rubble asteroids. Icarus 200, 304–322 (2009)
S. Sridhar, S. Tremaine, Tidal disruption of viscous bodies. Icarus 95, 86–99 (1992)
S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, 3rd edn. (Mc-Graw Hill, New York, 1970)
C. Truesdell, R.A. Toupin, The Classical Field Theories, in Encyclopedia of Physics Vol. III/I: Principles of Classical Mechanics and Field Theory, ed. by S. Flügge (Springer, Berlin, 1960), pp. 226–793
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Sharma, I. (2017). Affine Dynamics. In: Shapes and Dynamics of Granular Minor Planets. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-40490-5_3
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