Chapter 13 — Key Takeaways (Orbit Determination)

A one-page reference. Reread this before an exam, or before you write a mission's tracking-and-navigation plan.

The one idea

Orbit determination is prediction run backward: recover the six-number state (or six orbital elements) of a spacecraft from indirect, noisy observations. You never measure the orbit — only shadows of it (an angle, a range, a range-rate), through a known observation model $\mathbf{z} = h(\mathbf{x}) + \text{noise}$. Because observations are indirect and noisy, OD is estimation, not exact solution, and every estimate carries an uncertainty.

Observation types — what each one measures

Type Measures Numbers/look Regime Key limit
Radar (skin track) range, range-rate, 2 angles 4 LEO power falls as $1/\rho^4$
Optical 2 angles (direction) 2 LEO→deep space no range
GNSS (onboard) position (± velocity) 3–6 LEO (below constellation) fails at/above GEO
DSN radiometric range, Doppler, delta-DOR — deep space line-of-sight strong, plane-of-sky weak (add delta-DOR)

A state has 6 numbers, so you need ≥ 6 independent measurements: optical needs ≥ 3 looks (2 angles each); one radar look gives 4.

Two-line element set (TLE)

  • Compact mean elements + a $B^\*$ drag term at an epoch; propagate with SGP4 (bakes in J2/drag).
  • Stores mean anomaly $M$ → must solve Kepler's equation $M = E - e\sin E$ for position.
  • Accuracy ~km at epoch, degrades over days (unmodeled perturbations, Ch. 12).
  • Decode mean motion: $n\,[\text{rad/s}] = n\,[\text{rev/day}]\cdot 2\pi/86400$, then $a = (\mu/n^2)^{1/3}$.

The four techniques

Technique Given Produces Note
Lambert's problem 2 positions + time of flight the connecting orbit ($\mathbf{v}_1,\mathbf{v}_2$) TOF depends only on $a$, $r_1+r_2$, chord $c$; Hohmann = min-energy $180^\circ$ case, $a=(r_1+r_2)/2$
Gauss's method (IOD) 3 angles-only observations a rough first orbit ranges unknown → 8th-degree polynomial; a lead, not a final orbit (Gibbs: 3 positions → orbit)
Batch least squares many noisy observations best-fit orbit over an arc minimizes $\sum r_i^2/\sigma_i^2$; iterate (differential correction); read residuals
Kalman filter a stream of observations running estimate + uncertainty recursive predict–update, real-time, onboard

Least squares & the residual

  • Residual $= z_{\text{observed}} - h(\mathbf{x}_{\text{est}})$ (observed − predicted).
  • Least squares minimizes $\sum r_i^2$ (weighted: $\sum r_i^2/\sigma_i^2$). Squared, because it is smooth, punishes big misses, and is the most probable orbit for Gaussian noise.
  • General solution (linear-algebra level, flagged): $\delta\mathbf{x} = (H^\mathsf{T} W H)^{-1} H^\mathsf{T} W\,\mathbf{r}$, $H = \partial h/\partial\mathbf{x}$.
  • Read the residuals: structureless noise = healthy fit; a trend/wobble/jump = an unmodeled force, a maneuver, or a bad point.

The Kalman filter — predict/update

Step Action Uncertainty
Predict propagate state with the dynamics (Ch. 8/12) grows (extrapolation + process noise)
Update fuse the measurement via gain $K$ shrinks (information added)

Scalar update (the whole idea, no matrices): $$K = \frac{\sigma_{\text{pred}}^2}{\sigma_{\text{pred}}^2 + \sigma_{\text{meas}}^2}, \qquad \hat{x}^+ = \hat{x}^- + K(z - \hat{x}^-), \qquad (\sigma^+)^2 = (1-K)\sigma_{\text{pred}}^2.$$

  • $K = 0$: ignore a worthless measurement; $K = 1$: jump to a perfect one.
  • Fusion beats both inputs: $1/\sigma_+^2 = 1/\sigma_{\text{pred}}^2 + 1/\sigma_{\text{meas}}^2$ (inverse variances add).
  • Real 6-D nonlinear orbits → the extended Kalman filter (EKF); it is the "N" in GN&C (Ch. 27).

Batch vs. Kalman — which when

Need Use
Precise, after-the-fact orbit from a whole arc batch least squares
Real-time, onboard, changing state, per-measurement Kalman filter
A first orbit from minimal data, no prior Gauss's method (IOD)
The orbit between two known positions in a set time Lambert's problem

Numbers & formulas worth memorizing

  • State $=$ 6 numbers; every estimate has an uncertainty.
  • Kalman gain $K = \sigma_{\text{pred}}^2/(\sigma_{\text{pred}}^2+\sigma_{\text{meas}}^2)$; update $\hat{x}^+ = \hat{x}^- + K(z-\hat{x}^-)$.
  • Lambert min-energy $180^\circ$: $a = (r_1+r_2)/2$, TOF $= \pi\sqrt{a^3/\mu}$ (= the Hohmann transfer).
  • Averaging $N$ independent measurements: error $\propto 1/\sqrt{N}$.
  • Mean motion → axis: $a = (\mu/n^2)^{1/3}$.

Common pitfalls

Pitfall Reality
"Three optical observations = three positions." They are three directions; ranges are unknown (the crux of Gauss).
"A TLE's mean anomaly is where it is now." Solve Kepler's equation for the true anomaly first.
"The best orbit passes through all observations." Noise means none does; find the least-squares best fit.
"More precise sensors always fix any error." Precision can't supply a missing dimension — angles never give range; you need varied observations.
"A determined orbit stays determined." Perturbations decay it; re-determine continuously (fresh TLEs).

Mission / project additions this chapter

  • MDR: added the Tracking, Navigation & Orbit Determination note — observation types, batch/filter approach, and required accuracy for your track.
  • Illustrative code: kalman_update(x_pred, var_pred, z, var_meas) — the scalar filter that seeds the Chapter 27 GN&C navigation filter. (Not a stable astrotools module; the operational filter is the 6-D EKF.)