Physics HL · Chapter 17: Gravitation
17.2 Kepler's Laws and Orbital Motion from Newtonian Mechanics
Connect Kepler's laws to force and angular-momentum ideas, then derive the circular-orbit speed and period relations used in satellite calculations.
Estimated time: 40 minutes
Kepler's Three Laws as Orbit-Structure Rules
Kepler's first law states that bound planetary orbits are ellipses with the central attracting body at a focus. The second law states that equal time intervals sweep equal areas. The third law states that orbital period squared scales with semi-major axis cubed for bodies orbiting the same central mass. These are not independent curiosities; they are signatures of central inverse-square attraction.
The second law is especially useful for qualitative reasoning. If equal areas are swept in equal times, then the planet must move faster when it is closer to the focus and slower when it is farther away. This lets you infer speed ranking at periapsis and apoapsis without direct computation.
Circular-Orbit Speed and Period Relations
For circular orbit radius r, gravity provides the full centripetal force requirement. Equating Newton's gravitation formula to mv^2/r gives one of the most used orbit expressions in IB questions. Even when an orbit is not exactly circular, this relation remains a critical local scale reference.
v_{ ext{orbit}} = sqrt{ rac{GM}{r}},qquad T = 2pisqrt{ rac{r^3}{GM}}
For a fixed central mass, larger radius means lower speed but much longer period.
The period result directly yields Kepler's third-law form T^2 proportional to r^3 for circular or near-circular cases around one dominant mass. This is why period does not grow linearly with radius. Doubling radius multiplies period by 2^(3/2), not by 2.
Area Sweep Rate and Angular Momentum
A central force produces zero torque about the force center, so angular momentum about that center is conserved. The areal sweep rate dA/dt equals L/(2m), so if L is constant, areal rate is constant. That is the mechanics bridge from Newton's laws to Kepler's second law.
rac{dA}{dt} = rac{L}{2m}
Constant angular momentum implies equal areas are swept in equal times.
Simulation: Kepler Orbit and Area Sweep Studio
Tune semi-major axis, eccentricity, and phase to compare orbital speed, period scaling, and equal-time area sweeps.
Gravitation + Orbit Lab
Instant radius
2.53e+4 km
Instant speed
68.794 km/s
Orbital period
0.576 h
Area ratio (equal dt)
1.000
Elliptical orbit with area sweep in equal time windows
Speed vs radius
Current speed is 68.794 km/s; circular speed at a = 72.673 km/s.
Kepler scaling
For fixed central mass, T^2 scales with a^3. Expanding the semi-major axis stretches period quickly, even if instantaneous speed does not collapse linearly.
Test Yourself
Around the same star, Orbit B has radius four times Orbit A (assume circular). How does period change?
Hint: Use T^2 proportional to r^3 for the same central mass.