Using the concept of gravitational potential energy (potential), motion on a cosmic scale can be treated uniformly in terms of energy.
Basic Knowledge
Gravitational potential energy (with infinity as the reference U = 0):U = −GMm/r (a negative value; the smaller r is, the smaller U is, meaning a deeper bound state)
Conservation of mechanical energy: E = ½mv² − GMm/r = constant (without air resistance)
Relationship between the sign of mechanical energy and the orbit:
・E < 0: Bound orbit (circle or ellipse) → cannot escape
・E = 0: Parabolic orbit (exactly the escape velocity) → v = 0 at infinity
・E > 0: Hyperbolic orbit (still has speed after escaping)
Energy in a circular orbit:
From gravity = centripetal force, v² = GM/r
→ E = −GMm/(2r) (in a circular orbit, E = U/2 = −KE)
Potential: U = −GMm/r (negative value)
Mechanical energy: E = ½mv² − GMm/r = const
E < 0: Elliptical orbit (bound)
E = 0: Parabolic orbit (exactly escaping)
E in a circular orbit: E = −GMm/(2r) (relationship between KE and U: KE = −E)
Further Exploration (Background and Significance)
In a circular orbit, the interesting result E = −GMm/(2r) is obtained. The larger r is (the higher the orbit), the larger E is (its negative magnitude is smaller); in fact, it is a higher-energy state. Work must be done to raise a satellite to a higher orbit, which agrees with intuition.
However, as the orbit rises, the speed decreases (as r increases in v = √(GM/r), v decreases). KE becomes smaller, but PE increases greatly (toward larger values from a negative value); as a result, the total E increases.
Classifying orbits by the sign of mechanical energy is used in astronomy to determine the fate of actual celestial bodies—whether they remain in orbit or leave. For example, if a comet's orbit has E < 0, it is an elliptical orbit bound to the Sun and returns periodically; if E > 0, it is a one-time visit.
The virial theorem states that in a gravitational system, “the time average of KE = −E and the time average of U = 2E” (for a circular orbit, these are also the instantaneous values).
- U = −GMm/r (infinity is the reference; always negative)
- E = KE + U = constant (conservation of energy)
- E < 0 → ellipse or circle (bound)
- E = 0 → parabolic orbit (just enough to escape)
- E > 0 → hyperbola (still has speed after escaping)
- Circular orbit: E = −GMm/(2r) = U/2
- Higher orbit = higher mechanical energy (even though it is slower)
Points of Caution (Common Confusions)
① U = −GMm/r is negative. “Higher potential energy” means U is closer to zero (= r is larger). ② Even in a high orbit with low speed (small KE), E is larger because of PE (negative but smaller in magnitude). ③ “Conservation of energy” here means conservation of mechanical energy (without air resistance). It does not hold when friction is present. ④ The relationship E = −KE = U/2 is specific to circular orbits (for ellipses, it holds as a time average).
Practice
- Find the minimum initial speed required for an object launched from Earth's surface (r = R) to reach infinity (r → ∞). This is the second cosmic velocity.
- Express the mechanical energy of a satellite in a circular orbit of radius r in terms of G, M, m, and r.
- State the type of orbit for each case in which the sign of mechanical energy E is positive, negative, or zero.