Self-Assessment Quiz: Orbit Determination

Twenty questions to check your grasp of the inverse problem, observation types, Lambert's problem, initial orbit determination, least squares, and the Kalman filter. Answer each before opening the key. Aim for 16 or more. Use $\mu_\oplus = 3.986\times10^{5}\ \text{km}^3/\text{s}^2$ where needed.

Question 1

Orbit determination is best described as:

A) predicting a satellite's future position from its known orbit B) the inverse of prediction — estimating the orbit from observations C) measuring a satellite's mass from its orbit D) computing the delta-v of a transfer

Question 2

How many independent scalar measurements are needed, at minimum, to determine a full orbital state?

A) 3 B) 4 C) 6 D) 9

Question 3

A single optical (telescope) observation of a spacecraft provides:

A) range and range-rate B) two angles (a direction), but no range C) the full position vector D) the six orbital elements directly

Question 4

A two-line element set (TLE) stores, among its elements, which anomaly?

A) true anomaly B) eccentric anomaly C) mean anomaly D) flight-path angle

Question 5

The inputs to Lambert's problem are:

A) two velocity vectors and a time B) two position vectors and the time of flight between them C) three angles-only observations D) a single position and the orbital energy

Question 6

The Hohmann transfer is, in the language of this chapter, the:

A) highest-energy solution of Lambert's problem B) minimum-energy, $180^\circ$ solution of Lambert's problem C) only solution of Gauss's method D) same as a bi-elliptic transfer

Question 7

Gauss's method of initial orbit determination uses:

A) one radar observation B) two GNSS fixes C) three angles-only (optical) observations D) a full month of continuous tracking

Question 8

A residual is:

A) the leftover propellant after a burn B) the observed value minus the value predicted by the estimated orbit C) the difference between two orbital elements D) the noise floor of a sensor

Question 9

Least-squares estimation chooses the state that minimizes:

A) the largest single residual B) the sum of the absolute residuals C) the sum of the squared residuals D) the number of observations used

Question 10

During the Kalman filter's predict step, the estimate's uncertainty:

A) grows B) shrinks C) stays exactly constant D) drops to zero

Question 11

During the Kalman filter's update step, the estimate's uncertainty:

A) grows B) shrinks C) stays exactly constant D) becomes infinite

Question 12

A Kalman gain of $K = 1$ means the filter:

A) ignores the measurement entirely B) trusts the measurement completely and jumps to it C) has diverged D) has zero process noise

Question 13

An onboard GNSS receiver is the natural primary navigation source for which orbit regime?

A) low Earth orbit B) geostationary orbit C) interplanetary cruise D) lunar orbit

Question 14

A skin-tracking radar's received power from a passive target falls with range $\rho$ as:

A) $1/\rho$ B) $1/\rho^2$ C) $1/\rho^3$ D) $1/\rho^4$

Question 15

Which estimation method is recursive and real-time, storing only the current estimate and its uncertainty?

A) batch least squares B) Gauss's method C) the Kalman filter D) Lambert's problem

Question 16 (True/False, justify)

"Three optical observations give you three position vectors." True or false? Justify in one sentence.

Question 17 (True/False, justify)

"When a Kalman filter fuses a prediction and a measurement, the resulting uncertainty can be smaller than both inputs." True or false? Explain briefly.

Question 18 (Short answer)

Give one practical and one statistical reason for minimizing the sum of squared residuals rather than the sum of absolute residuals.

Question 19 (Short answer)

A filter predicts a position as $100\ \text{km}$ with variance $\sigma_{\text{pred}}^2 = 4$; a measurement reads $106\ \text{km}$ with variance $\sigma_{\text{meas}}^2 = 4$. Compute the Kalman gain, the updated estimate, and the updated standard deviation.

Question 20 (Short answer)

In one or two sentences, why does a TLE's accuracy degrade over days rather than staying fixed?

Answer Key

Q Ans Note
1 B OD inverts prediction: observations → state.
2 C The state is six numbers (position + velocity, or six elements).
3 B Optical measures a direction (two angles) only; range is unknown.
4 C TLEs store mean anomaly $M$; solve Kepler's equation for the true anomaly.
5 B Two positions and the time of flight between them.
6 B It is the minimum-energy, half-revolution Lambert solution; $a = (r_1+r_2)/2$.
7 C Three angles-only observations (6 angles = 6 numbers).
8 B Residual = observed − predicted.
9 C Least squares minimizes $\sum r_i^2$ (weighted by measurement quality).
10 A Extrapolating with process noise grows the uncertainty.
11 B A measurement adds information, so uncertainty shrinks.
12 B $K = 1$: measurement perfectly trusted; $K = 0$: measurement ignored.
13 A GNSS works below/within the constellation; LEO is ideal, GEO marginal, deep space no.
14 D Beam spreads out ($1/\rho^2$) and the weak reflection spreads back ($1/\rho^2$).
15 C The Kalman filter is recursive and real-time; batch waits for a whole arc.
16 False They give three directions (six angles); the ranges are unknown — the crux of Gauss's method.
17 True Inverse variances add ($1/\sigma_+^2 = 1/\sigma_{\text{pred}}^2 + 1/\sigma_{\text{meas}}^2$), so the fused $\sigma_+$ is smaller than either input.
18 — Practical: squaring is smooth/differentiable (calculus finds the minimum) and penalizes large misses heavily. Statistical: for Gaussian noise the least-squares estimate is the most probable state.
19 — $K = 4/(4+4) = 0.5$; estimate $= 100 + 0.5(106-100) = 103\ \text{km}$; variance $=(1-0.5)(4)=2$, so $\sigma^+ = \sqrt{2} = 1.41\ \text{km}$.
20 — Real orbits are perturbed (J2, drag, third-body), so a mean-element snapshot drifts from reality; the TLE goes stale as those unmodeled effects accumulate.

Topics to review by question

Questions Topic Section
1, 2, 16 The inverse problem and the six-number state §13.1
3, 4, 13, 14, 20 Observation types and TLEs §13.2
5, 6 Lambert's problem §13.3
7 Gauss's method / IOD §13.4
8, 9, 18 Residuals and least squares §13.5
10, 11, 12, 15, 17, 19 The Kalman filter §13.6