The motion of the sky, read from gravity alone

Two masses pull with F = GMm/r². A circular orbit appears when that pull exactly matches the centripetal need, at v = √(GM/r). The Moon is not failing to fall, it is falling sideways forever.

A planet traces an ellipse with the Sun at one focus. It sweeps equal areas in equal times, so it moves faster at perihelion. Period and semi-major axis lock together as T² ∝ a³.

The orbital energy is E = ½mv² − GMm/r. If E is negative the orbit is an ellipse, zero a parabola, positive a hyperbola. The vis-viva equation v² = μ(2/r − 1/a) ties speed to position.

Escape speed is √2 times the circular speed. To move from a low circle to a high one you ride a transfer ellipse tangent to both, burning once at perigee and once at apogee. The total Δv is the cost.

In the planet's frame the probe's speed is unchanged and only its direction bends. But switch to the Sun's frame and the planet's velocity adds on, so the speed grows or shrinks. That is the secret of the free boost.

The difference in gravity between the near and far sides raises two tidal bulges. Those bulges brake the spin until one face stays locked toward the primary, and integer-ratio resonances between periods hold orbits in place.