Lower Orbit Needs Larger Speed-Square - Rotational Dynamics and Orbital Mechanics Step by Step

Because the closing speed-square scales as v^2 = mu / r, dropping the radius raises the required speed-square, so a lower orbit must move faster. The 1 / r trend is the same inverse-square gravity pulling harder as you get closer. It is exact only for the stated gravity parameter mu and radius in an ideal circular two-body orbit; real orbits add perturbations, non-circularity, finite body sizes, reference-frame choices, and measurement uncertainty.

Smaller radius around the same source needs larger circular speed-square. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

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The lower orbit has smaller radius and stronger field

The upper radius is 4 while the lower radius is 1.

r_{\mathrm{upper}}=4\quad r_{\mathrm{lower}}=1

The lower orbit needs larger speed-square

The upper speed-square is 4, while the lower speed-square is 16.

v_{\mathrm{lower}}^{2}=16>4=v_{\mathrm{upper}}^{2}