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- Thread starter Czcibor
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Vanadium 50

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You would have libration - a rocking back and forth. The moon does this.

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- The long axis: the direction to the star

- The intermediate axis: the direction of motion in the planet's orbit

- The short axis: the orbit's pole

A good analogy to the late stages of spindown is an unbalanced wheel. Its equations of motion also closely resemble the equations of motion for the planet's longitude libration, and have much the same behavior. As the wheel spins down, it changes from circulation (complete rotations) to libration in pendulum fashion.

$$ \frac{d^2 \theta}{dt^2} = - (\omega_l)^2 \sin \theta ,\ \frac{d^2 \theta}{dt^2} = - \frac12 (\omega_l)^2 \sin (2\theta) $$

The unbalanced-wheel equation and the planet's libration equation, where θ is the orientation angle around the stable direction(s) and ω

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Libration: ## \theta = \arcsin ( \sqrt{m} \text{sn} (\omega_l t, m) ) ##

Transition (m = 1): ## \theta = \arcsin ( \tanh (\omega_l t) ) ##

Circulation: ## \theta = \arcsin ( \text{sn} ((\omega_l / \sqrt{m}) t, m) ) ##

They have limits:

Libration: ## \theta = \sqrt{m} \sin (\omega_l t) ##

Circulation: ## \theta = (\omega_l / \sqrt{m}) t ##

Here, m is the elliptic-function parameter. A common alternative is to use its square root as the parameter.

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