Chapter 8 — Key Takeaways (Kepler's Laws and the Two-Body Problem)

A one-page reference. Reread this before an exam, or before you specify any orbit.

Kepler's three laws (and what each really is)

Law Statement Underlying principle
First (ellipses) Orbits are ellipses with the primary at one focus the conic solution $r = p/(1+e\cos\nu)$ of inverse-square gravity
Second (equal areas) The primary–body line sweeps equal areas in equal times conservation of angular momentum ($dA/dt = h/2$)
Third (periods) $T^2 \propto a^3$ area of ellipse $\pi ab = (h/2)T$ ⇒ $T^2 = (4\pi^2/\mu)a^3$

The big idea: Kepler's laws are theorems, not axioms — all three fall out of $\ddot{\mathbf{r}} = -\mu\mathbf{r}/r^3$. The ellipse is forced by the exponent $2$ in gravity.

The equations that run the chapter

Equation Use it to find
$r = \dfrac{p}{1 + e\cos\nu}, \quad p = a(1-e^2) = h^2/\mu$ the radius at any true anomaly (the orbit equation / conic)
$r_p = a(1-e), \quad r_a = a(1+e)$ apsidal radii from size and shape
$a = \dfrac{r_p + r_a}{2}, \quad e = \dfrac{r_a - r_p}{r_a + r_p}$ size and shape from the apsides
$T = 2\pi\sqrt{\dfrac{a^3}{\mu}}, \quad n = \dfrac{2\pi}{T} = \sqrt{\dfrac{\mu}{a^3}}$ period and mean motion (Kepler III)
$M = E - e\sin E$ Kepler's equation — link time ($M$) to geometry ($E$)
$M = n(t - t_p)$ mean anomaly from time since periapsis
$\cos\nu = \dfrac{\cos E - e}{1 - e\cos E}, \quad \tan\dfrac{\nu}{2} = \sqrt{\dfrac{1+e}{1-e}}\tan\dfrac{E}{2}$ true anomaly from eccentric anomaly
$r = a(1 - e\cos E)$ radius from eccentric anomaly
$\varepsilon = -\dfrac{\mu}{2a}$ specific energy from size (proved for any ellipse here; from Ch. 6)

Conic classification by eccentricity (= energy sign)

$e$ Conic $\varepsilon$ $a$ Bound?
$0$ circle $<0$ $=r$ yes
$0 ellipse $<0$ $>0$ yes
$1$ parabola $=0$ $\to\infty$ marginal escape
$>1$ hyperbola $>0$ $<0$ no (escapes)

The six orbital elements

Element Symbol Sets Constant as it coasts?
Semi-major axis $a$ size (⇒ period, energy) yes
Eccentricity $e$ shape yes
Inclination $i$ tilt of plane from equator yes
RAAN $\Omega$ swivel of plane about the pole yes
Argument of periapsis $\omega$ orientation of ellipse in plane yes
True anomaly $\nu$ where the body is now no — the only one that moves

Six numbers + a named primary (its $\mu$) = a complete orbit specification. Real "constants" drift under perturbations — Chapter 12.

The three anomalies (find position in time)

  • True $\nu$ — physical angle at the focus (what you want). Eccentric $E$ — geometric middleman, measured at the center. Mean $M$ — fictitious, grows uniformly with time ($M = n\,\Delta t$).
  • Ordering (perigee → apogee): $M \le E \le \nu$. All three equal at perigee ($0^\circ$) and apogee ($180^\circ$).
  • The near-circular shortcut: as $e \to 0$, the three anomalies converge — position advances almost uniformly. The divergence between them is a direct readout of $e$.

Procedure — time → position: (1) $M = n(t - t_p)$; (2) solve $M = E - e\sin E$ for $E$ (Newton, start at $E_0 = M + e\sin M$); (3) convert $E \to \nu$ and $r = a(1-e\cos E)$.

Decision aid — "which relation do I use?"

You know… You want… Use
$a$ (and $\mu$) period / mean motion $T = 2\pi\sqrt{a^3/\mu}$, $n = \sqrt{\mu/a^3}$
$T$ (and $\mu$) semi-major axis $a = (\mu T^2/4\pi^2)^{1/3}$
$n$ from a TLE semi-major axis $a = (\mu/n^2)^{1/3}$
$r_p, r_a$ $a, e$ $a=(r_p+r_a)/2$, $e=(r_a-r_p)/(r_a+r_p)$
$a, e, \nu$ radius $r = a(1-e^2)/(1+e\cos\nu)$
time since perigee true anomaly Kepler's equation, then $E\to\nu$ (3-step procedure above)
$a$ (and $\mu$) speed at radius $r$ vis-viva $v=\sqrt{\mu(2/r-1/a)}$ (🔗 Ch. 6)

Common pitfalls

Pitfall Reality
Primary at the center of the ellipse It sits at a focus, off to one side. The center is empty.
Confusing $M$ (or $E$) with the true anomaly $\nu$ They coincide only at perigee/apogee; for eccentric orbits $\nu$ runs well ahead of $M$.
Thinking period depends on eccentricity $T$ depends on $a$ only. Same $a$, any shape ⇒ same period.
Reading $\nu$ straight off the clock Only valid for near-circular orbits; solve Kepler's equation otherwise.
Using $24\ \text{h}$ for the GEO period It is the sidereal day, $23\ \text{h}\ 56\ \text{m}$ ($86{,}164\ \text{s}$).
Treating Kepler's laws as postulates They are consequences of inverse-square gravity (§8.2).

Numbers worth memorizing

  • Kepler constant (Earth): $T^2/a^3 = 4\pi^2/\mu_\oplus = \mathbf{9.90\times10^{-5}\ \text{s}^2/\text{km}^3}$.
  • $400\ \text{km}$ low orbit ($a=6{,}771\ \text{km}$): $T \approx \mathbf{92\ \text{min}}$ ⇒ ~16 orbits/day.
  • GEO ($a = 42{,}164\ \text{km}$): $T = \mathbf{23\ \text{h}\ 56\ \text{m}}$ (sidereal day, $86{,}164\ \text{s}$).
  • Molniya ($a \approx 26{,}560\ \text{km}$): 12-hour orbit, $e \approx 0.74$, apogee ~$40{,}000\ \text{km}$.
  • Doubling $a$ multiplies the period by $2^{3/2} = \mathbf{2.83}$.
  • Earth $\mu = 3.986\times10^{5}$; Moon $4.903\times10^{3}$; Mars $4.283\times10^{4}\ \text{km}^3/\text{s}^2$.

Mission / astrotools additions this chapter

  • MDR: recorded your target orbit's full six-element set $(a, e, i, \Omega, \omega, \nu)$ plus its period and mean motion — the definitive orbit spec you carry to the capstone.
  • orbits.py: added period(mu, a) (Kepler III) and a sketch of elements_to_rv(...) (the perifocal position/velocity; Chapter 9 adds the rotation to the inertial frame). Energy functions circular_velocity, specific_energy, vis_viva come from Ch. 6.