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Kepler's Laws & Orbits

Source lecture(s): SC133 Lec 17

Intuition

Kepler spent a decade wrestling Tycho Brahe's planetary data into three empirical rules — and half a century later Newton showed all three tumble out of one inverse-square law. This is the template of theoretical physics: patterns first, then the single principle underneath. Bonus: Einstein's turn came when Mercury's orbit refused to follow Newton exactly.

The three laws

K1 — Ellipses. Planets move on ellipses with the Sun at one focus. (The circle is the special case of zero eccentricity; comets ride extreme ellipses.)

K2 — Equal areas. The Sun–planet line sweeps equal areas in equal times: planets move fastest at perihelion, slowest at aphelion.

This is nothing but conservation of angular momentum: gravity is central, exerts no torque about the Sun, so \(L = mvr_\perp\) is constant — and the swept-area rate is \(\frac{dA}{dt} = \frac{L}{2m}\).

K3 — Harmonic law. Period² ∝ semi-major-axis³:

\[\boxed{\,T^2 = \frac{4\pi^2}{GM}\,a^3\,}\]

Derivation for circular orbits (two lines): gravity supplies the centripetal force,

\[\frac{GMm}{r^2} = \frac{mv^2}{r} = m\omega^2 r \;\Rightarrow\; \omega^2 = \frac{GM}{r^3} \;\Rightarrow\; T^2 = \frac{4\pi^2}{GM}r^3.\]

Measure \(T\) and \(r\) for any satellite and you have weighed the central body — this is literally how we know the masses of the Sun, the planets, and the black hole at the galactic center.

Orbital energetics

Total energy of an orbit of semi-major axis \(a\):

\[E = K + U = -\frac{GMm}{2a}\]
  • \(E < 0\): bound (ellipse) · \(E = 0\): parabola (escape, exactly) · \(E > 0\): hyperbola (flyby)
  • For circular orbits \(K = -E = |U|/2\) — the virial relation, the same \(2\langle K\rangle = -\langle U\rangle\) that governs star-cluster simulations.
  • Counterintuitive gem: drag on a satellite makes it speed up (it falls to a lower, faster orbit — energy drops, kinetic energy rises).

Worked example: geostationary orbit

What radius keeps a satellite over one spot (\(T = 24\) h = 86 400 s)?

\[r = \left(\frac{GM_E T^2}{4\pi^2}\right)^{1/3} = \left(\frac{(6.67\times10^{-11})(5.97\times10^{24})(86400)^2}{4\pi^2}\right)^{1/3} \approx 4.2\times10^7\,\text{m}\]

— about \(6.6\,R_E\), the crowded ring where communication satellites live.

Einstein's postscript

Mercury's perihelion drifts 43″/century beyond Newtonian prediction. General relativity — gravity as spacetime curvature — accounts for it exactly, plus light bending and GPS clock corrections. Newton remains the working approximation everywhere weaker than these extremes.

Common mistakes

  • Sun at the center. It's at a focus; the ellipse's center is empty.
  • Constant orbital speed on an ellipse. K2 forbids it — speed varies, only the areal rate is constant.
  • Applying K3 across different central bodies. The constant contains \(M\): moons of Jupiter and planets of the Sun follow different \(T^2/a^3\) lines.
  • Confusing \(r\) with \(a\) for ellipses — K3 uses the semi-major axis.

Knowledge graph position

Prerequisites: Gravitation, Angular momentum. Leads to: astrophysics, orbital mechanics, N-body simulation.

Quiz

Q1 (computational). Mars orbits at \(a = 1.52\) AU. Its year, in Earth years?

Answer

\(T = a^{3/2} = 1.52^{1.5} \approx 1.87\) years — K3 with Sun-units, where \(T^2/a^3 = 1\) by construction.

Q2 (conceptual). A comet's speed at perihelion (0.5 AU) vs aphelion (50 AU)?

Answer

K2 / angular momentum: \(v_p r_p = v_a r_a\) (velocity ⊥ radius at both extremes), so \(v_p = 100\,v_a\) — a two-order-of-magnitude sprint past the Sun.

Q3 (multiple choice). To catch up with a space station ahead of you in the same orbit, you should briefly fire your engines: (a) forward (speed up) (b) backward (slow down) (c) toward Earth

Answer

(b). Slowing drops you to a lower, faster orbit; you overtake below, then re-raise. Orbital mechanics runs on energy logic, not car logic.