Exercises: Orbit Determination

Work these with a calculator, $g_0 = 9.81\ \text{m/s}^2$, Earth's gravitational parameter $\mu_\oplus = 3.986\times10^{5}\ \text{km}^3/\text{s}^2$, and mean radius $R_\oplus = 6{,}371\ \text{km}$. Difficulty: ⭐ foundational, ⭐⭐ intermediate, ⭐⭐⭐ challenging. Worked solutions to the daggered (†) and odd-numbered problems are in the appendix answers-to-selected.md — try each one cold before you peek. For the "simulate it" problems, do not run the code: hand-trace it and record the result in an # Expected output: comment, exactly as the chapter does.

Part A — Observations, states, and TLEs (⭐)

13.1 † A spacecraft's state is its position and velocity. How many independent scalar measurements does it take, at minimum, to determine a full orbit, and why exactly that number?

13.2 A tracking radar reports, at one instant, range, range-rate, azimuth, and elevation. How many of the six state numbers is that in one look, and what is the simplest way to get the rest?

13.3 † A two-line element set lists a mean motion of $n = 14.00$ revolutions per day. Find the orbital period in minutes and the semi-major axis $a$ in km. What altitude is that (using $R_\oplus = 6{,}371\ \text{km}$)?

13.4 True or false, and justify in one sentence: a single optical (telescope) observation includes the object's range.

Part B — Lambert's problem (⭐⭐)

13.5 † Compute the minimum-energy (Hohmann-like, $180^\circ$) transfer time from a $300\ \text{km}$ parking orbit ($r_1 = 6{,}671\ \text{km}$) out to a circular orbit of radius $r_2 = 20{,}000\ \text{km}$. Give the transfer semi-major axis and the time of flight in hours.

13.6 Two position fixes place an object on opposite sides of Earth, each at radius $10{,}000\ \text{km}$ (a $180^\circ$ separation). What are the chord length $c$ and the minimum-energy transfer semi-major axis $a$?

13.7 † Explain in physical terms why two position vectors alone are consistent with infinitely many orbits, but adding the time of flight between them selects a unique one.

Part C — Simulate it in Python (⭐⭐)

Write each function, then hand-trace it for the given inputs and record the result in an # Expected output: comment. Do not run it.

13.8 † Write tle_semimajor(n_rev_per_day) that returns the semi-major axis (km) from a TLE mean motion, using $a = (\mu/n^2)^{1/3}$ with $n$ converted to rad/s. Trace it for tle_semimajor(15.0).

13.9 Write kalman_gain(var_pred, var_meas) returning $\sigma_{\text{pred}}^2/(\sigma_{\text{pred}}^2+\sigma_{\text{meas}}^2)$. Trace it for kalman_gain(9.0, 3.0).

13.10 † Write ls_constant(measurements) that returns the least-squares best constant fit to a list of noisy measurements (which, minimizing $\sum r_i^2$, is just their mean). Trace it for ls_constant([10.2, 9.8, 10.1, 9.9, 10.0]).

Part D — Find the error (⭐⭐)

13.11 † A student reads a TLE, sees mean anomaly $M = 45^\circ$, and reports "the satellite is $45^\circ$ past perigee — the true anomaly is $45^\circ$." For an orbit with $e = 0.5$, why is this wrong, and is the true anomaly larger or smaller than $45^\circ$?

13.12 A student claims: "The Kalman-filtered estimate always lies between the prediction and the measurement, and its uncertainty likewise lies between the two input uncertainties." One half of this is right and one half is wrong. Which is which, and why?

Part E — Design it (⭐⭐ / ⭐⭐⭐)

13.13 † Design the observation strategy for a new $6\ \text{U}$ CubeSat in a $550\ \text{km}$ sun-synchronous orbit. Which observation types would you plan on, and would you rely on an onboard GNSS receiver or on ground tracking for routine orbit determination? Justify each choice.

13.14 (Mission project) Write the Tracking, Navigation & Orbit Determination note for your mission (Track A/B/C/D): the observation types, the estimation approach (batch, Kalman, or both), and the accuracy you need and why. Save it to your MDR.

13.15 † You have two independent range measurements of the same quantity, with standard deviations $\sigma_1 = 1.5\ \text{km}$ and $\sigma_2 = 2.0\ \text{km}$. Using inverse-variance fusion, what is the standard deviation of the optimally combined estimate? Comment on how it compares to the two inputs.

Part F — Back of the envelope & "why can't you just…" (⭐⭐⭐)

13.16 † Why can't you just determine an orbit from a single, extremely precise optical image? Frame your answer in terms of how many state numbers one image supplies.

13.17 A skin-tracking radar's received power falls as $1/\rho^4$. If it can track a given target at $\rho = 2{,}000\ \text{km}$ (LEO), by roughly what factor is the received power weaker for the same target at GEO ($\rho \approx 36{,}000\ \text{km}$)? What does the answer say about which orbits skin-track radar suits?

13.18 † Why can't you just navigate a Mars mission with onboard GPS the way a LEO satellite does? What do you use instead, and why?

Part G — Interleaved & synthesis (from Chapters 8 and 12) (⭐⭐ / ⭐⭐⭐)

13.19 † (Ch. 8) A TLE gives eccentricity $e = 0.6$ and mean anomaly $M = 30^\circ = 0.5236\ \text{rad}$. Solve Kepler's equation $M = E - e\sin E$ for the eccentric anomaly $E$ by Newton's method (two iterations from $E_0 = M + e\sin M$), then determine whether the true anomaly $\nu$ is larger or smaller than $M$.

13.20 (Ch. 12) Why must the propagator used inside a precise least-squares orbit fit include J2 and atmospheric drag, and what appears in the residuals if it does not?

13.21 † (Ch. 8 / 12 synthesis) A batch fit's post-fit residuals show a slow, growing along-track trend over a week rather than structureless noise. Give two plausible physical causes and say how you would tell them apart.

13.22 (Ch. 8) Two satellites share all six orbital elements except the true anomaly $\nu$. When you determine the orbit of one, which five quantities are identical between them, and what physically distinguishes the two?

13.23 † (Lambert, design) For the LEO→GEO transfer ($r_1 = 6{,}771\ \text{km}$, $r_2 = 42{,}164\ \text{km}$), the minimum-energy transfer time is $5.29\ \text{h}$ (from §13.3). If a customer needs the payload delivered to GEO in $4\ \text{h}$ instead, will the required departure delta-v be higher or lower than the Hohmann value, and why?

13.24 Compute the root-mean-square of the residual set $[-2, +1, 0, +1, -1, +1]$ (units km). Show the sum of squares and the RMS.

13.25 † (Synthesis) You have a rough IOD orbit from Gauss's method and $200$ subsequent radar observations of the same object. Describe the full pipeline that turns this into a precise, maintained orbit, naming the method used at each stage.

13.26 Back of the envelope: a Kalman filter predicts a satellite's position with $\sigma_{\text{pred}} = 5\ \text{km}$; an incoming GNSS fix has $\sigma_{\text{meas}} = 0.01\ \text{km}$. What is the Kalman gain on that fix, and roughly what is the updated uncertainty? Interpret the result.

Solutions to † and odd-numbered exercises are in appendices/answers-to-selected.md. Reference code for the "simulate it" problems is in code/exercise-solutions.py. For design problems, the rubric rewards: observation types matched to the orbit regime, an estimation method matched to the operational need (real-time vs. after-the-fact), explicit units, and a sanity check on every number.