Complete Energy Analysis

For a satellite of mass m in circular orbit at radius r:

KE = ½mv_{0}^{2} = ½m × GM/r = GMm/(2r) [positive]

PE = −GMm/r [negative, exactly −2×KE]

Total E = KE + PE = GMm/(2r) − GMm/r = −GMm/(2r) [negative]

Virial Theorem for Circular Orbits

KE : PE : E = 1 : (−2) : (−1)

Or equivalently:

• |PE| = 2KE
• E = −KE
• PE = 2E

Memory: "One positive, two negative, one negative"

Physical Interpretation

The total energy is always negative for a bound orbit. This means:

• The satellite cannot escape to infinity (would require E ≥ 0)
• More negative E = more tightly bound = smaller orbit
• To raise orbit: add energy → E increases (less negative)

When energy is added:

• PE increases by 2$\Delta E$ (gain in altitude)
• KE decreases by $\Delta E$ (satellite slows down)
• Net: total energy increases by $\Delta E$

Orbital Energy Table

Escape Condition

A satellite escapes when E ≥ 0:

• E = 0: parabolic trajectory (minimum escape)
• E > 0: hyperbolic trajectory
• If E < 0 but orbit perturbed: elliptical with same total energy