C3 Departure Energy

C3 Departure Energy (space-systems/mission-design/c3-departure-energy)

Use when the task is the departure-side energy of an interplanetary mission leaving a circular parking orbit on an escape hyperbola: C3 characteristic energy, hyperbolic excess speed, the injection speed at the parking radius from the vis-viva integral, the injection delta-v above the circular parking speed, the parking period, and the declination of the outgoing asymptote. This leaf implements the classical two-body departure model in pure Python, stdlib only, deterministic. It covers the departure only: the parking orbit to the escape hyperbola asymptote. The encounter at the destination body belongs to space-systems/orbit-mechanics/gravity-assist-swingby, the transfer leg after escape to hohmann-transfer, and the launch geometry of the ascent to launch-window-analysis in this pack.

Domain quick reference

• Characteristic energy: C3 = v_inf^2, the square of the hyperbolic excess speed (m2/s2). C3 = 9 km2/s2 corresponds to v_inf = 3000 m/s.
• Excess speed from C3: v_inf = sqrt(C3). The two conversions form the round trip C3 -> v_inf -> C3, exact to float precision.
• Circular parking speed: v_c = sqrt(mu / r) at the parking radius.
• Injection speed on the departure hyperbola (vis-viva evaluated at the hyperbola periapsis, which is the parking radius): v_p = sqrt(v_inf^2 + 2mu / r). For v_inf = 0 the hyperbola degrades to a parabola and v_p is the local escape speed sqrt(2mu / r).
• Injection delta-v: dv = v_p - v_c, the burn that raises the circular parking speed to the hyperbolic injection speed; always positive for a positive excess speed.
• Parking orbit period: T = 2pisqrt(r^3 / mu).
• Outgoing asymptote declination: dec = asin(vz / |v|) in degrees, for the excess velocity vector (vx, vy, vz).
• Convenience bundle: departure_energy_assessment returns c3_m2_s2, c3_km2_s2, excess_speed_m_s, circular_speed_m_s, injection_speed_m_s, injection_delta_v_m_s, parking_period_s and asymptote_declination_deg (None when no velocity components are given).
• Units are SI throughout: m, m/s, m2/s2, s, deg. MU_EARTH = 3.986004418e14 m3/s2 and G0 = 9.80665 m/s2 are the module constants.
• ECSS E-ST-10C frames the mission analysis context; the relations above are standard engineering methodology, summary-only.

Workflow

1. Fix the central body gravitational parameter mu (MU_EARTH default) and the circular parking orbit radius r in meters.
2. Set the departure energy level: target hyperbolic excess v_inf with c3_from_excess_speed, or target C3 with excess_speed_from_c3; the round trip identity excess_speed_from_c3(c3_from_excess_speed(v)) equals v within 1e-6.
3. Compute the circular parking speed with circular_speed.
4. Compute the injection speed on the departure hyperbola at the parking radius with injection_speed (vis-viva), then the burn with injection_delta_v, which equals v_p - v_c exactly.
5. Get the parking orbit period with parking_period (about 90 min for a low Earth parking orbit).
6. When the outgoing excess velocity vector is known, convert it to the asymptote declination with asymptote_declination.
7. Bundle the assessment with departure_energy_assessment into one dict and report C3, excess speed, injection speed, injection delta-v, parking period and asymptote declination.
8. Confirm the deterministic checks with the contract test scripts/test_c3_departure_energy.py.

Worked example

Earth mu = 3.986004418e14 m3/s2, parking radius 6578 km (300 km circular), target hyperbolic excess 3000 m/s (C3 = 9 km2/s2). Real module outputs (deterministic, printed by the module):

• C3 = 9.0 km2/s2 exactly (9000000.0 m2/s2).
• Excess round trip: excess_speed_from_c3(c3_from_excess_speed(3000)) returns 3000.0 m/s.
• Circular parking speed: 7784.3428 m/s (spec bound 7700-7900).
• Injection speed on the departure hyperbola: 11410.1703 m/s (spec bound 11200-11700).
• Injection delta-v: 3625.8275 m/s (spec bound 3400-3900).
• Parking period: 5309.4775 s, about 88.5 min (spec bound 5300-5600).
• Asymptote declination for v = (2000, 2000, 1000) m/s: 19.4712 deg (spec bound 16-20); the vector magnitude is exactly 3000 m/s, so it matches C3 = 9 km2/s2.

Pitfalls

• Mixing parking orbit radius with planet radius: every speed and period here is evaluated at the parking radius r you pass in, so injecting at the wrong altitude (or using the surface radius) shifts C3, v_p, dv and T together.
• Confusing C3 with injection speed: C3 = v_inf^2 is the energy at infinity; the injection speed at the parking radius is the larger vis-viva value v_p = sqrt(v_inf^2 + 2*mu/r), and the burn is the difference v_p - v_c, not v_inf.
• Quoting dv as the excess speed: injection delta-v always exceeds the hyperbolic excess (3625.8 m/s dv for a 3000 m/s excess in the worked example); the escape-speed floor appears only when v_inf = 0.
• Treating the asymptote declination as optional input-agnostic: asymptote_declination needs the actual excess velocity components and returns None when none are given; the zero vector raises ValueError, and a vector whose magnitude does not match sqrt(C3) points at an inconsistent energy state.
• Skipping the round-trip check: excess_speed_from_c3(c3_from_ excess_speed(v)) must return v within 1e-6; a mismatch means a unit or conversion slip in the calling code.
• Forgetting the far-radius behavior: v_p approaches v_inf only as r grows (residual 130 m/s even at 1e9 m for a 3000 m/s excess), so low parking orbits never see the excess speed at injection.

Verification

• Deterministic: no RNG anywhere; repeated runs return bit-identical floats for the full assessment chain.
• Round trip: excess_speed_from_c3(c3_from_excess_speed(v)) equals v within 1e-6 for a sweep of physical excess speeds.
• Identities: injection_delta_v equals injection_speed minus circular_speed exactly; v_p^2 - v_inf^2 = 2mu/r holds at any radius; zero excess gives the local escape speed sqrt(2mu/r); a GEO parking radius reproduces the sidereal day (86164.09 s).
• Far-radius approach: v_p approaches v_inf as the radius grows; at 1e9 m the residual v_p - v_inf is 130.05 m/s for the 3000 m/s excess (2mu/r is still about 8.9% of v_inf^2), shrinks monotonically and drops below 1 m/s by 1e12 m. The contract test asserts the sub-50 m/s bound at 1e9 m for the 12 km/s excess, where 2mu/r is under 1% of v_inf^2 and the exact residual is 33.2 m/s.
• ValueError rejection: negative excess speed, negative C3, mu <= 0, radius <= 0 in every function, and the zero velocity vector in asymptote_declination.
• The convenience dict contains exactly the eight documented keys.

Related leaves

• space-systems/mission-design/mission-delta-v-budget: consumes the injection delta-v inside a full mission delta-v budget.
• space-systems/mission-design/launch-window-analysis: launch site geometry and ascent timeline for the same interplanetary mission.
• space-systems/orbit-mechanics/hohmann-transfer: sizes the transfer leg that follows the departure escape.
• space-systems/orbit-mechanics/gravity-assist-swingby: the encounter phase after the cruise leg, outside this leaf's departure-side claim.
• space-systems/mission-design/entry-descent-landing: the arrival end of the mission.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_c3_departure_energy.py

Run from the repo root:

python3 skills/space-systems/mission-design/c3-departure-energy/scripts/test_c3_departure_energy.py

The test covers the worked example outputs against their spec magnitude bounds and the module's real values, the C3/excess round trip, the circular and injection speeds, the injection delta-v identity, the vis-viva far-radius approach, the parking period, the asymptote declination, the convenience dict keys, determinism, and ValueError rejection of every non-physical input (35 tests).

Compliance

• Standards referenced, not reproduced: ECSS E-ST-10C (space engineering) frames the mission analysis context; the C3 and vis-viva relations above are standard two-body engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.