Exercise set

Spacecraft Orbit, Delta-v, and Propellant Exercises

Worked spacecraft mission exercises for orbit speed, period, Hohmann transfer, plane change, rocket equation, propellant reserve and delta-v release checks.

Branch: Aerospace Engineering
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.1.0 · reviewed

These exercises focus on spacecraft mission mechanics: orbit speed, period, transfer delta-v, plane change, propellant mass, burn time, stationkeeping reserve and release gates. The goal is to make the mission budget traceable before it is coupled to power, communications, ADCS and operations constraints.

Use these calculations as screening models. Flight mission design still requires orbit propagation, perturbation modelling, finite-burn effects, navigation uncertainty, collision-risk policy, operations rules, propulsion qualification and configuration control.

How to use these exercises

Work through the set as a mission budget review. Exercises 1 to 6 establish the orbital mechanics basis: radius, period, transfer delta-v and plane-change cost. Exercises 7 to 10 convert maneuver demand into propellant and burn-time implications. Exercises 11 to 17 add recurring stationkeeping, protected reserve, uncertainty and mass-growth effects. Exercise 18 converts the numeric result into a conditional release decision.

Before each calculation, name the mission phase, mass state, propulsion mode, orbit epoch, delta-v owner and reserve policy. A commissioning burn, a collision-avoidance maneuver, an attitude desaturation dump and end-of-life disposal do not have the same release authority even if they use the same thruster. The engineering comment below each exercise identifies the missing evidence that would stop the number from becoming an approved mission budget.

Release Evidence Notes

Every delta-v value should belong to a named mission phase. Record the orbit, epoch, mass state, maneuver purpose, propulsion mode, uncertainty allowance and operational trigger. A propellant budget that lacks collision avoidance, disposal, attitude desaturation, navigation cleanup or failed-burn recovery is not a release budget.

The evidence package should keep ideal mechanics, operational losses and protected reserve separate. Ideal Hohmann or plane-change delta-v is only the starting point; finite burn duration, pointing error, thrust misalignment, navigation cleanup, missed maneuver recovery, residual propellant and gauging uncertainty all affect usable margin. If these items are hidden inside a single percentage, the review cannot tell which risk controls the mission.

Propellant evidence must be tied to the same mass configuration used by power, thermal, ADCS and communications analyses. Mass growth changes low-thrust acceleration, burn duration, pointing windows and sometimes power availability. A delta-v budget that closes before mass freeze should remain conditional until mass growth is bounded.

Engineering Boundary Notes

Orbit calculations are sensitive to units and reference radius. Delta-v margins are sensitive to wet mass, specific impulse, residual propellant, unusable propellant and operations policy. A narrow pass in a spreadsheet should trigger mission-analysis review, not launch approval.

The first boundary is reference frame and units. Altitude is not orbital radius, km/s is not m/s, and specific impulse in seconds must be converted with standard gravity before it is used as effective exhaust velocity. The second boundary is timing. Low-thrust burns can satisfy total delta-v but fail the mission if operations windows, attitude constraints, power availability or ground-contact rules do not support the burn timeline.

Common Release Mistakes

using altitude instead of orbital radius in velocity equations;
adding ideal impulsive maneuvers without finite-burn and pointing losses;
omitting collision avoidance, disposal or commissioning reserve;
applying specific impulse in seconds without multiplying by $g_0$ ;
treating propellant remaining as usable without residual and gauging uncertainty;
approving delta-v before mass growth is frozen.

Another common mistake is spending protected reserve as ordinary margin. Disposal and collision-avoidance reserves are mission rules, not convenient buffers for routine stationkeeping. If reserve can be reallocated, the approval authority and trigger condition should be explicit.

Do not treat low-thrust delta-v as an impulsive burn with a longer clock. Long burns interact with attitude control, power-positive pointing, communication windows, thermal constraints and navigation updates. A low-thrust budget needs both total impulse and timeline feasibility.

Scenario Map

Scenario	Main calculation	Release decision
Circular orbit	speed and period	Check mission timing and contact cadence.
Transfer maneuver	Hohmann and plane-change delta-v	Size mission maneuver budget.
Propellant	rocket equation and impulse	Check usable reserve.
Low-thrust operation	thrust, mass and burn time	Decide whether operations window closes.
End of life	disposal and collision avoidance	Hold or release mission reserve.
Guarded budget	planned delta-v plus uncertainty	Approve, derate or redesign mission plan.

Validation Package Checklist

orbit radius, gravitational parameter and epoch convention;
mass state before each maneuver;
ideal delta-v plus losses and uncertainty;
specific impulse, usable propellant and residual assumption;
stationkeeping, collision-avoidance and disposal reserve;
finite-burn duration, pointing compatibility and navigation cleanup policy;
mass-growth sensitivity and propulsion qualification boundary;
reserve authority, disposal rule and failed-burn recovery assumption;
operations rule for when reserve may be spent.

A complete validation package should make the mission budget auditable maneuver by maneuver. Another engineer should be able to see which delta-v is planned, which delta-v is protected, which propellant is usable, which mass state was assumed and which open item would force redesign, derating or reserve protection.

Exercise 1: Circular Orbit Speed

A spacecraft is in circular Earth orbit at altitude $h=500\ \text{km}$ . Use Earth radius $R_E=6378\ \text{km}$ and $\mu=398600\ \text{km}^3/\text{s}^2$ . Compute orbital speed.

Solution

r=R_E+h=6878\ \text{km}

\displaystyle v=\sqrt{\frac{\mu}{r}}=\sqrt{\frac{398600}{6878}}=7.61\ \text{km/s}

Engineering Comment

Use orbital radius, not altitude. A radius error would strongly bias speed, period and ground-track timing.

Plausibility Check

Low Earth orbit speed near $7.5\ \text{km/s}$ is expected.

Exercise 2: Orbit Period

Use $r=6878\ \text{km}$ . Compute circular orbit period.

Solution

\displaystyle T=2\pi\sqrt{\frac{r^3}{\mu}}

\displaystyle T=2\pi\sqrt{\frac{6878^3}{398600}}=5677\ \text{s}=94.6\ \text{min}

Engineering Comment

Period drives ground-station contact spacing, eclipse cadence and operations staffing.

Plausibility Check

An LEO period of about $90$ to $100$ minutes is plausible.

Exercise 3: Revolutions Per Day

Use $T=94.6\ \text{min}$ . Compute revolutions per day.

Solution

\displaystyle N=\frac{1440}{94.6}=15.2\ \text{rev/day}

Engineering Comment

This is an orbit count, not a guarantee of useful contacts. Ground-station geometry and payload duty cycle still matter.

Plausibility Check

LEO spacecraft commonly complete about fifteen orbits per day.

Exercise 4: Hohmann Transfer Delta-v

A spacecraft raises a circular orbit from $r_1=6878\ \text{km}$ to $r_2=7278\ \text{km}$ . Use:

\displaystyle \Delta v_1=\sqrt{\frac{\mu}{r_1}}\left(\sqrt{\frac{2r_2}{r_1+r_2}}-1\right)

Solution

m_f=m_0 e^{-\Delta v/(I_{sp}g_0)}

m_f=120e^{-90/(220(9.81))}=115.1\ \text{kg}

m_p=120-115.1=4.9\ \text{kg}

Engineering Comment

Usable propellant must exceed this ideal mass after residuals, gauging uncertainty and failed-burn reserve.

Plausibility Check

Small delta-v relative to exhaust velocity consumes only a few percent of mass.

Exercise 8: Usable Propellant After Residual

A tank contains $6.2\ \text{kg}$ propellant. Residual unusable propellant is $0.7\ \text{kg}$ and gauging uncertainty is $0.4\ \text{kg}$ . Compute conservative usable propellant.

Solution

m_{usable}=6.2-0.7-0.4=5.1\ \text{kg}

Engineering Comment

Compare planned maneuvers against usable propellant, not tank-book quantity.

Plausibility Check

Conservative usable mass is lower than measured tank inventory.

Exercise 9: Propellant Margin

Use required propellant $4.9\ \text{kg}$ and usable propellant $5.1\ \text{kg}$ . Compute margin.

Solution

M=5.1-4.9=0.2\ \text{kg}

\displaystyle M_\%=\frac{0.2}{4.9}=4.1\%

Engineering Comment

This margin is too narrow for flight release unless the maneuver is already highly certain and reserves are separately protected.

Plausibility Check

The percentage is small because usable and required masses are nearly equal.

Exercise 10: Ion Thruster Burn Time

An electric thruster produces $T=18\ \text{mN}$ . Spacecraft mass is $160\ \text{kg}$ . Estimate time to accumulate $\Delta v=12\ \text{m/s}$ .

Solution

Acceleration:

\displaystyle a=\frac{T}{m}=\frac{0.018}{160}=1.125\times10^{-4}\ \text{m/s}^2

Solution

Acceleration ratio:

\displaystyle \frac{a_2}{a_1}=\frac{160}{172}=0.930

Acceleration drops by:

1-0.930=7.0\%

Engineering Comment

Mass growth lengthens low-thrust burns and can change operations windows even if total propellant still appears adequate.

Plausibility Check

Higher mass lowers acceleration for the same thrust.

Exercise 17: Delta-v Guard Band

Available qualified delta-v is $240\ \text{m/s}$ . Guarded requirement is $223.4\ \text{m/s}$ . Compute margin.

Solution

M=240-223.4=16.6\ \text{m/s}

\displaystyle M_\%=\frac{16.6}{223.4}=7.4\%

Engineering Comment

The pass is moderate, not generous. Track mass growth, finite-burn losses and operations changes.

Plausibility Check

Positive margin means the budget passes the simplified gate.

Exercise 18: Mission Propellant Release Gate

A mission requires guarded delta-v $223.4\ \text{m/s}$ , has qualified delta-v $240\ \text{m/s}$ , protects disposal and collision avoidance reserve, and has a mass-growth open item of $3\ \text{kg}$ . Decide release status.

Solution

Nominal margin is:

240-223.4=16.6\ \text{m/s}

But the mass-growth item can reduce acceleration and usable delta-v margin.

Engineering Comment

Release should be conditional. The budget numerically passes, but the mass-growth item must be closed or bounded before baseline approval.

Plausibility Check

A positive delta-v margin does not automatically close a mission review when mass state is not frozen.

REF