Orbits: understanding free fall, energy and manoeuvres
An orbit is not a place where gravity disappears. It is continuous fall that keeps missing the surface. This module builds that intuition and connects circular speed, period, energy, vis-viva, Hohmann transfers and rendezvous. Every formula includes units and a worked check.
1. Why does a satellite not hit the ground?
It is falling. Gravity continually curves the path while tangential velocity carries the spacecraft forward. For a circular orbit of radius r around a body with gravitational parameter μ, v = √(μ/r).
Exercise A
Use μEarth = 3.986 × 10¹⁴ m³/s² and r = 6.771 × 10⁶ m, roughly 400 km altitude.
v ≈ 7,670 m/s = 7.67 km/s. The quantity under the square root has units m²/s², so the result is m/s.
Energy and angular momentum provide a second language for orbit
In the two-body problem, an orbit can be described with two ideal invariants: specific mechanical energy and specific angular momentum. Specific energy is ε = v²/2 - μ/r, measured in J/kg, equivalent to m²/s². Here v is speed in m/s, μ gravitational parameter in m³/s² and r distance from the central body in metres. If ε < 0 the trajectory is bound; ε = 0 is the parabolic limit; ε > 0 is hyperbolic. Specific angular momentum h = |r × v| has units m²/s. These quantities explain why a tangential burn near periapsis strongly changes apoapsis: the spacecraft changes energy where its speed is already high. They also provide independent checks when orbital elements from software look suspicious.
2. Orbital period
For a circular orbit, T = 2π√(r³/μ). Using the same numbers gives about 5,545 s or 92.4 minutes.
3. Ellipses, apsides and energy
Periapsis is nearest to the central body and apoapsis farthest away. The semimajor axis a controls specific orbital energy: ε = −μ/(2a) for a bound Keplerian orbit. A prograde burn at periapsis raises the opposite side of the orbit; a retrograde burn lowers it.
Recover orbital period from Kepler’s third law
For a Keplerian orbit, T = 2π√(a³/μ). T is period in seconds, a semi-major axis in metres and μ gravitational parameter in m³/s². For a circular Earth orbit of radius 7,000 km, using μ = 398,600 km³/s² gives T ≈ 2π√(7000³/398600) ≈ 5,828 s, about 97.1 minutes. The dependence is proportional to a^(3/2), so doubling orbital radius does not simply double period. This relationship is a fast propagation check and also explains phasing: a small deliberate change in semi-major axis changes period, allowing relative angle to accumulate over several revolutions before the spacecraft returns to the target orbit.
4. Vis-viva
This relation connects position, speed and energy. For a circular orbit a = r and it reduces to v = √(μ/r).
Exercise B
For Earth, let a = 10,000 km, r = 7,000 km and μ = 398,600 km³/s².
v² ≈ 74.03 km²/s², so v ≈ 8.60 km/s.
Use vis-viva as a fast numerical cross-check
Vis-viva, v² = μ(2/r - 1/a), relates speed v to radius r and semi-major axis a. Distances must use one consistent unit and μ must match it. For a circular Earth orbit at r = 7,000 km, using μ = 398,600 km³/s² gives v = √(398600/7000) ≈ 7.55 km/s. If the same point becomes periapsis of an ellipse with a = 10,500 km, speed is √[398600(2/7000 - 1/10500)] ≈ 8.71 km/s. The ideal impulse is therefore about 1.16 km/s. This compact calculation reveals the order of magnitude of a transfer and is an excellent independent check on an astrodynamics program, especially when unit or radius-versus-altitude errors are possible.
5. Hohmann transfer
A Hohmann transfer between coplanar circular orbits uses a tangent half-ellipse. One impulse enters the transfer; another circularises. It is a minimum-energy model under simplified assumptions, not a universal mission optimum when flight time, plane changes or low thrust matter.
The same geometry provides first intuition for Earth-to-Mars transfer around the Sun. Departure timing matters because Mars must reach the intercept point when the spacecraft does.
6. Rendezvous requires phase, not just orbit
Two vehicles must occupy the same place at the same time with compatible velocity. Phasing uses a temporary orbit with a different period. A common intuition trap is to “speed up toward” the target; changing speed changes the orbit and can change average angular rate in the opposite way expected.
Hohmann is a benchmark, not a universal command
A Hohmann transfer between circular coplanar orbits uses two impulses and minimises Δv within that specific model. It does not automatically minimise time, operations risk or real mission cost. A mission may spend more Δv to shorten exposure, meet lighting constraints or arrive at a required rendezvous phase. The first burn enters an ellipse whose periapsis matches the lower orbit and apoapsis the upper orbit; the second circularises. A reliable hand calculation obtains circular and transfer speeds separately with vis-viva, then forms each velocity difference. Total Δv is the sum of the absolute burn magnitudes, in m/s or km/s. Treating Hohmann as a reference case rather than a rule makes later optimisation easier to understand.
7. What two-body theory omits
Real missions add oblateness, atmosphere, third bodies, solar radiation pressure, manoeuvre errors and navigation uncertainty. The simple model remains essential for intuition and for detecting impossible numerical results before high-fidelity propagation.
Two-body theory has a clearly defined boundary
The two-body model assumes point masses and one dominant central gravity field. Real trajectories also experience oblateness, third-body gravity, atmospheric drag in low orbit, solar radiation pressure and manoeuvre errors. The correct lesson is not that the simple model is useless; it is that each omitted effect has a time and accuracy scale. A minutes-long hand calculation may be excellent for checking an impulsive transfer, while months of precise orbit prediction require perturbations and numerical integration. Model fidelity should therefore be chosen from the decision being made. Before adding complexity, quantify the error allowed in position, velocity, event time or propellant and ask whether the simpler model already meets it.
8. Check yourself
- Explain orbit as free fall.
- State the units of μ.
- Why is speed higher at periapsis?
- Why is rendezvous also a timing problem?
9. Escape speed: leave a bound orbit
At distance r, ideal escape speed is vesc = √(2μ/r), √2 times circular speed at the same radius. Escape does not mean gravity disappears; it means specific orbital energy is non-negative in the two-body model.
Exercise D
At r = 6.771 × 10⁶ m around Earth and μ = 3.986 × 10¹⁴ m³/s², vesc ≈ 10.85 km/s. Compare with the 7.67 km/s circular speed.
10. Plane change prefers lower speed
For an instantaneous velocity-vector rotation Δi at constant speed, Δv = 2v sin(Δi/2). At v = 7.7 km/s and Δi = 10°, Δv ≈ 1.34 km/s. Launch inclination and where a plane change is performed therefore matter greatly.
Ellipse geometry and rendezvous belong in the same toolkit
For an ellipse, semi-major axis is a = (rₚ+rₐ)/2 and eccentricity can be written e = (rₐ-rₚ)/(rₐ+rₚ), where rₚ and rₐ are periapsis and apoapsis radii. These parameters define geometry but not where a spacecraft is at a particular time. Rendezvous adds phase: two vehicles can have identical orbital elements and still be separated by hundreds of kilometres. A practical design therefore moves between inertial orbital elements and a relative frame near the target. Far away, transfer energy and period dominate; close in, line-of-sight rate, closing velocity and collision geometry become critical. This transition explains why “same orbit” is only a prerequisite for rendezvous, not the rendezvous solution itself.
11. Phasing: change period to change angle
If a target is ahead on the same orbit, pointing thrust directly at it is wrong. A temporary semimajor-axis change modifies orbital period, allowing relative phase to evolve before final rendezvous manoeuvres remove position and velocity differences.
Plane change exposes why geometry and timing matter
For an instantaneous plane change of angle Δi at speed v, a useful relationship is Δv = 2v sin(Δi/2). Δv and v share the same velocity unit; Δi must be provided in the angular unit expected by the sine function, usually radians in software. At 7.5 km/s, a 10° plane change costs about 2 × 7.5 × sin(5°) ≈ 1.31 km/s. At 2.0 km/s the same angular change costs only about 0.35 km/s. Mission designers therefore try to combine plane change with another manoeuvre where speed is low. Rendezvous adds a further constraint: matching an orbital plane or altitude is insufficient unless the vehicles also reach the same place at the same time.
Phasing changes time to change angle
Phasing exploits the connection between semi-major axis and period. A vehicle that briefly moves to a slightly lower orbit has a shorter period and advances in phase relative to a target on the original orbit; a higher phasing orbit does the opposite. The design problem is to choose a temporary period such that, after an integer or selected fraction of revolutions, the relative angle closes at the desired point. Burns then enter and leave the phasing orbit. This is not free: Δv, minimum altitude, lighting, communications and conjunction risk constrain the solution. A spreadsheet can explore candidate periods, but a final rendezvous plan also needs relative navigation and bounded closing speed.
12. Mars orbits are service geometries
Relay, crew-staging and mapping orbits optimise different goals. Altitude and inclination change coverage, period, eclipse, radiation, insertion cost and site visibility. Highly elliptical orbits can provide long dwell over a region at the price of varying range.
13. Reference frames can simplify or break the problem
An orbital state is position and velocity in a defined frame at a defined time. Inertial frames support dynamics; rotating frames can simplify relative problems; local frames support operations. Transformations require explicit conventions and time handling.
A Mars orbit is selected for the service it provides
Low Mars orbit supports close observation and can reduce some surface-link distances, but it has high orbital speed and frequent occultations. Higher orbits cover more of the planet and can support relay service at the cost of greater range and delay. Highly elliptical orbits dwell near apoapsis for long periods but pass through a wider thermal and radiation environment. The correct choice therefore starts with the service—communications, mapping, rendezvous, vehicle staging or landing support—and then evaluates periods, eclipses, line-of-sight geometry and manoeuvre cost. Orbital mechanics defines the possible paths; mission architecture chooses which path makes the required operation reliable.
14. Mini-project
Two vehicles share a circular Mars orbit and are separated by 20°. Decide qualitatively whether the chaser needs a temporarily shorter or longer period to close the phase angle. Only after explaining the direction should you use a numerical propagator.
Mini-project: close geometry, timing and Δv together
A useful mini-project starts with two circular orbits, a target phase offset and a maximum manoeuvre budget. First compute circular periods. Then choose a transfer or phasing ellipse and use vis-viva to calculate burn magnitudes. Propagate the elapsed time to determine where the target will be when the chaser arrives. If the phase does not close, change the temporary semi-major axis rather than forcing a final large correction. Record every distance as radius or altitude explicitly and keep μ units consistent. The project is complete only when geometry, time and velocity all agree. This is the same discipline later required by numerical rendezvous tools: a trajectory is not valid because one Δv number looks plausible.
15. Energy tells you whether the path is bound
Specific orbital energy ε = v²/2 − μ/r is negative for an ellipse, zero for ideal escape and positive for a hyperbola. This single equation lets you classify a trajectory before calculating every orbital element. It is especially useful after a partial burn: engine status alone does not tell whether capture succeeded.
16. Why burns at periapsis are powerful
Because speed is highest near periapsis, a given prograde Δv there produces a large energy change. This is the intuition behind the Oberth effect. The effect does not create energy for free; the engine adds the same velocity increment, but performing it where the spacecraft already moves rapidly changes kinetic energy more strongly.
17. Plane change and rendezvous compete for Δv
A mission designer tries to combine manoeuvres when geometry permits, because a separate large plane change can be expensive. Rendezvous planning therefore begins with launch plane, phasing and arrival geometry, not only with the final close approach.
Plane change and rendezvous compete for the same budget
A mission rarely has separate propellant tanks labelled “plane change” and “rendezvous.” All manoeuvres consume the same Δv reserve. If the target orbit differs in inclination, performing the plane change where speed is lower can save substantial propellant, but the resulting geometry may complicate phasing or communications. Combining a plane change with another burn can reduce total vector change when the directions are chosen correctly. The trade is therefore multi-dimensional: total Δv, time, eclipse, navigation observability and operational complexity. A good design records not only the mathematically minimum manoeuvre but a robust sequence that can tolerate injection error and still preserve collision-avoidance authority.
18. Numerical-tool readiness check
Before using an orbit propagator, write down central body, μ, frame, epoch, position and velocity units. Then predict qualitatively what the trajectory should do. If the software produces a result with the opposite trend, investigate the inputs before trusting the plot.
Engineering studio — reconstruct an orbital manoeuvre
The scenario starts from a circular Mars orbit near 3,800 km from the planet centre and targets an orbit whose apoapsis reaches 6,000 km. Use the vis-viva relation v² = μ(2/r − 1/a), where v is speed in m/s, μ is the Martian gravitational parameter in m³/s², r is radius from the centre in m and a is semi-major axis in m. The task is not to memorise a final number: compute speed before and after the impulse, check units, and explain why an impulse near periapsis mainly raises apoapsis.
Now add a +1 m/s execution error. The student estimates its consequence for apoapsis and selects a correction policy: correct at once, wait for better observability, or retain the error if the orbit remains inside the accepted envelope. The choice depends on Δv cost, time, navigation accuracy, collision risk and eclipse geometry. Orbital mechanics therefore becomes an operational decision rather than a formula exercise.
Worked comparison — plane change now or after raising apoapsis?
For an instantaneous plane change of angle Δi at speed v, the required velocity change is Δv = 2v sin(Δi/2). The same angular change therefore costs less where orbital speed is lower. Compare a 10° plane change at 7.7 km/s with the same change at 3.0 km/s: the geometry is identical, but the velocity penalty is not. This is why mission designers often combine plane changes with burns near apoapsis or other low-speed points.
The comparison also shows a trap: raising apoapsis costs Δv too. A good solution adds the burn used to reshape the orbit, the cheaper plane change and any burn needed to restore the final orbit. “Do it where speed is low” is a useful principle, but the complete manoeuvre sequence decides whether the strategy actually saves propellant.
Sources and references
NASA STEMonstrations — Orbits · NASA Basics of Space Flight — Trajectories · NASA NTRS — Astrodynamics Convention and Modeling Reference for Lunar, Cislunar, and Libration Point Orbits.