A specific-energy-sign cross-scan holds the radius fixed and varies speed-square across rows, moving total specific energy through bound (negative), escape (zero), and unbound (positive) values. The arithmetic is exact only for the stated speeds, radii, specific energies, and rows in the ideal two-body model. Real observations add noncircular orbits, perturbations, extended bodies, reference-frame choices, instrument calibration, finite precision, and measurement uncertainty.
Speed-square moves specific energy through bound, escape, and unbound rows. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Total specific energy -2
At radius four, potential stays negative four. Speed-square 4 gives kinetic 2 and total -2.
2−4=−2
Total specific energy 0
At radius four, potential stays negative four. Speed-square 8 gives kinetic 4 and total 0.
4−4=0
Total specific energy 1
At radius four, potential stays negative four. Speed-square 10 gives kinetic 5 and total 1.
5−4=1
Specific-energy sign separates bound, escape, and unbound
Kinetic energy rises with speed-square while the potential term stays fixed. Negative, zero, and positive totals mark the three statuses.