Circular Orbits

Speed-Square Closes a Circular Orbit

A circular orbit closes when the required centripetal acceleration matches the gravitational field, giving v^2 = mu / r for the speed-square that holds the body on its circle. That ledger is exact only for the stated gravity parameter mu, radius, and speed-square in an ideal two-body, circular orbit. Real orbits add perturbations, non-circularity, finite body sizes, reference-frame choices, and measurement uncertainty.

A circular orbit closes when v-square over radius equals the field. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The circular ledger starts with a speed-square

The candidate speed-square is 4 at radius 4.

v2=4 m2/s2v^{2}=4\ \mathrm{m^{2}/s^{2}}
Circular speed-square candidateThe acceleration arrow is anchored at the orbiting body.mu=16 m^3/s^2radius=4 mradiusSquared=16 m^2field=1 m/s^2speedSquared=4 m^2/s^2centripetalAcceleration=1 m/s^2acceptedBit=1 bit

Speed-square over radius equals the gravity field

Dividing 4 by radius 4 gives the same inward acceleration 1 as the field.

ac=v2r=44=1 m/s2a_c=\frac{v^{2}}{r}=\frac{4}{4}=1\ \mathrm{m/s^{2}}
Circular closureThe accepted bit is checked from acceleration equals field.mu=16 m^3/s^2radius=4 mradiusSquared=16 m^2field=1 m/s^2speedSquared=4 m^2/s^2centripetalAcceleration=1 m/s^2acceptedBit=1 bit