Sunday, July 01, 2018

Conditions for Contact Binary via collision

This post is to demonstrate that the conditions for CB via collision require highly balanced rotational and translational components in two similarly sized objects for them to retain their individuality as lobes in an otherwise combined object. This to demonstrate both that binary asteroids can not be a source for contact binary asteroids, and that random primordial collisions cannot be either.

1: Formerly connected bodies "reconfiguring"
As shown in a previous blog post, with the real case of couples skaters to demonstrate, bodies twist away from each other to retain angular momentum using their previous independent rotation as a dynamo (at the point of collision in the case of asteroidal bodies)

2: Independently rotating objects colliding from outside each other's hill spheres at "baby crawling" speed.

Inelastic collisions are of course possible for non-rotating (or trivially rotating bodies that collide in the perfect offset to cancel independent rotation). This is because the required collisional/deformation damping is obtained passively and gives a reaction force in the correct direction opposite to motion, and proportionally to motion so that the force stops when the relative velocity between bodies comes to zero.

A head on collision with a random (in observed ranges of possibilities) rotation or offset collision even with no initial rotation makes an inelastic collision impossibly rare given:

Coherent bodies - Mainly solid that can deform but not to the point of liquidity.
The two bodies are of similar size - smaller body is no less than about a third the radius, or no less than a tenth of the moment of inertia in rough figures.

Due to the laws of physics, forces that perform torque on each other must balance.
Overall angular momentum must be conserved.
If work is required to perform torque to change rotation that must be in the amount and direction made possible by friction, reaction force or gravity.

The following is a way to create model to test generated samples to verify this blog's order of magnitude analysis:

For a perfectly inelastic collision to ensue, not only relative speeds must be damped to zero, but relative rotation rates must also be damped to zero and still be touching. At the point of touching, the combined bodies' moment of inertia is at the minimum - Therefore, the rotational velocity required to maintain angular momentum is at its maximum. The kinetic energy associated with the required rotation is also at a maximum. There is work required to achieve this synchronised higher rotation unless the initial conditions are perfectly selected. The same fluidity that allows damping of the impact force allows shear perpendicular with the relative spin velocities which would throw the bodies apart further than reactive forces alone could do. In other words, a notionally inelastic collision would convert rotational velocity to inertial frame velocity in all except perfectly tuned cases which are extremely rare given the assumption of random initial spin state.

Friction under tensile stress or associated with outward movement (Stretch) is the only option to passively glue the bodies together keeping coherent orientation. With stretch (or expanding orbit) there is natural damping as the moment of inertia increases for the duo in proportion to its reduced velocity. Whether loosely connected at the neck or two bodies orbiting the barycentre, outward movement from previously connected bodies is the only passive way to stabilise mutual spin.

Pairs skaters use this particular rule of thumb to spectacular effect with the "spiral of death" spin. The technique spirals the (female) skater outwards with her head balanced inches above the ice. The gradual outward release damped with the action of muscles gives exceptional control that appears paradoxical and adds to the appearance of magic levitation of the head off the ice.

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