Equal Areas, Constant Angular Momentum and T² Proportional to a³
Planets and satellites on ellipses obey Kepler's laws. They all come from gravity being a central force: angular momentum is conserved, so speed changes with distance, and energy is conserved, so the total never varies around the orbit.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
Kepler's three laws
First law: orbits are ellipses with the Sun (or Earth) at one focus. Second law: the line from the Sun to the planet sweeps equal areas in equal times, so an area twice as large takes twice the time.
Speeds at perihelion and aphelion
At the closest and farthest points the velocity is perpendicular to the radius, so v1 r1 = v2 r2 and v1 / v2 = r2 / r1: the planet moves fastest when closest. For three points A, B, C with the Sun at the focus nearest A, the speed falls in the order V_A > V_B > V_C. The relation Lₐ = Lₚ is true for the vector, and not for r x L.
What stays constant
For a satellite on an elliptical orbit around the Earth, the acceleration always points toward the centre of the Earth, the angular momentum is constant, the total mechanical energy is constant (it does not vary periodically), and the linear momentum changes in magnitude since the speed changes.
OlympIQ exam lens
- ✓Equal areas in equal times; twice the area takes twice the time
- ✓T² is proportional to a³ with a = (r_A + r_P)/2
- ✓Angular momentum is conserved: v_P r_P = v_A r_A, so the speed is highest at perihelion
- ✓Total mechanical energy and angular momentum are constant; linear momentum is not
- ✓Acceleration always points toward the central body
Do not use the distance between the planet and the Sun's position as the semi-major axis in Kepler's third law. The semi-major axis is half the sum of the nearest and farthest distances, (r_A + r_P)/2.
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