Synodic Launch Window

Synodic Launch Window (space-systems/mission-design/synodic-launch-window)

Use when you must determine the interplanetary launch-window timing between two planets on near-circular coplanar orbits. This leaf computes the synodic period of the launch opportunity recurrence, the heliocentric departure phase angle required for a Hohmann window, the recurrence epochs and the phase progression between windows, in pure Python stdlib. It sits between the Earth-orbit daily launch geometry leaf (space-systems/mission-design/launch-window-analysis) and the departure-energy analysis of this pack that sizes the hyperbolic escape, and it hands its window epochs to the transfer design leaves in space-systems/orbit-mechanics. Does NOT do: Earth-orbit daily launch geometry, hyperbolic departure-energy sizing, transfer orbit design, or multi-planet trajectory design.

Domain quick reference

• Synodic period of the launch opportunity recurrence: T_syn = (T_in * T_out) / (T_out - T_in), with T_in the inner orbital period and T_out the outer orbital period. It is the beat period of the two orbital frequencies, defined only when the outer period exceeds the inner one, and it always exceeds both orbital periods.
• Heliocentric departure phase angle for a Hohmann window: alpha_dep = pi * (1 - ((a_in + a_out) / 2 / a_out)^1.5) radians, with a_in and a_out the inner and outer semi-major axes. For Earth to Mars it is about 0.774 rad, or 44.34 degrees.
• Recurrence epochs: t_k = t_0 + k * T_syn, one launch opportunity per synodic period starting at the launch epoch t_0.
• Phase progression between windows: phi(t) = 2 * pi * (((t - t_0) / T_syn) mod 1) radians in [0, 2pi). The phase returns to zero modulo 2pi at every recurrence epoch and rises monotonically between them.
• Window summary report: synodic_report bundles synodic_period_days, departure_phase_angle_deg, window_epochs and phase_at_first_window, with the phase at the first recurrence epoch near zero modulo 2*pi.
• Units: days for all periods and epochs, AU for the semi-major axes, radians and degrees for the angles as documented per output.
• ECSS standards frame the space mission design context; the relations above are standard astrodynamics methodology, summary-only.

Workflow

1. Fix the two planets and their orbits: the inner orbital period T_in and semi-major axis a_in, the outer orbital period T_out and semi-major axis a_out. Earth to Mars defaults sit in the module constants EARTH_YEAR_DAYS (365.25), MARS_YEAR_DAYS (686.98), EARTH_SMA_AU (1.0) and MARS_SMA_AU (1.523679).
2. Compute the synodic period of the launch opportunity recurrence with synodic_period(inner_period_days, outer_period_days). The function rejects reversed or non-positive orbital periods with ValueError.
3. Compute the required heliocentric departure phase angle for the Hohmann window with hohmann_departure_phase_angle(inner_sma_au, outer_sma_au), returned in radians; convert with math.degrees when the plan needs degrees.
4. List the recurrence epochs with window_epochs(t0_days, synodic_days, count), returning t_0 + k * T_syn for k in 0..count-1. Reject count below 1 and non-positive synodic periods with ValueError.
5. Track the phase progression between windows with phase_progression(t_days, t0_days, synodic_days), the synodic cycle fraction since t_0 as an angle in [0, 2pi); zero modulo 2pi at every recurrence epoch, monotonically rising in between.
6. Bundle the window geometry with synodic_report(inner_period_days, outer_period_days, inner_sma_au, outer_sma_au, t0_days, count) and read the synodic period, the departure phase angle in degrees, the window epochs and the phase at the first recurrence epoch.
7. Confirm the deterministic checks with the contract test scripts/test_synodic_launch_window.py.

Worked example

Earth to Mars: T_in = 365.25 days, T_out = 686.98 days, a_in = 1.0 AU, a_out = 1.523679 AU, t_0 = 0 days.

• Synodic period: T_syn = (365.25 * 686.98) / (686.98 - 365.25) = 779.9069 days (779.9 days to one decimal), exceeding both orbital periods.
• Departure phase angle: alpha_dep = pi * (1 - (1.26184 / 1.523679)^1.5) = 0.773952 rad = 44.3442 degrees (44.34 degrees to two decimals).
• Recurrence epochs from t_0 = 0: 0.0, 779.9069, 1559.8138 days (0, 779.9, 1559.8 days to one decimal), window_epochs(0.0, 779.9069, 3).
• Phase check: the phase progression at t_0 + T_syn is 0.0 modulo 2*pi, so the next window opens exactly one synodic period later; at half a synodic period the phase reaches pi.
• Report: synodic_report(365.25, 686.98, 1.0, 1.523679) returns synodic_period_days 779.9069, departure_phase_angle_deg 44.3442, window_epochs [0.0, 779.9069, 1559.8138] and phase_at_first_window 0.0.

Verification

• Confirm synodic_period(365.25, 686.98) returns 779.9069 days, within 0.5 days of the 779.9 day anchor, and that it equals the beat period 1 / (1/365.25 - 1/686.98) of the two orbital frequencies.
• Confirm hohmann_departure_phase_angle(1.0, 1.523679) returns 0.773952 rad, 44.3442 degrees, within 0.1 degrees of the 44.34 degree anchor.
• Confirm window_epochs(0.0, 779.9, 3) returns 0.0, 779.9 and 1559.8 days within 0.1, and that the epochs are spaced by exactly one synodic period.
• Confirm the phase progression returns zero modulo 2*pi at every recurrence epoch and stays monotone between epochs, with the half synodic point at pi.
• Confirm the synodic_report dict exposes exactly the four documented keys with values matching the individual functions.
• Confirm ValueError rejection: outer period not exceeding the inner period, non-positive orbital periods or semi-major axes, window count below 1, and non-positive synodic periods.
• Run the contract test offline: python3 scripts/test_synodic_launch_window.py (29 tests, deterministic).

Related leaves

• space-systems/mission-design/launch-window-analysis: the Earth-orbit daily launch geometry that this leaf does not cover.
• space-systems/orbit-mechanics/hohmann-transfer: the transfer orbit flown between the departure and arrival planets once the window epoch is set.
• space-systems/mission-design/mission-delta-v-budget: the mission-level delta-v context around the interplanetary window planning.

Pitfalls

• Confusing the synodic period with an orbital period: T_syn = 779.9 days for Earth to Mars exceeds both the 365.25 day inner period and the 686.98 day outer period, so launch opportunities recur less often than either planet completes an orbit.
• Reporting radians where the plan expects degrees: the departure phase angle is 0.773952 rad but 44.3442 degrees, and synodic_report stores degrees under departure_phase_angle_deg; convert explicitly in the other functions.
• Treating every planetary pair as outward: the synodic period and departure phase angle formulas require the outer orbit to exceed the inner one, and reversed or equal inputs raise ValueError rather than silently returning a negative recurrence.
• Reading the phase progression as an instantaneous geometric angle: it is the synodic cycle fraction since t_0, zero modulo 2*pi only at the window epochs, so evaluating it halfway between windows gives pi, not a departure condition.
• Calling window_epochs with a zero or negative count: at least one window is required, and count below 1 raises ValueError.
• Rounding the synodic period before spacing epochs: window_epochs must use the full precision value (779.9069 days, not 779.9) or the epoch drift accumulates over many recurrences.

Behavior contract (gate 3)

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

python3 scripts/test_synodic_launch_window.py

The test covers the Earth to Mars synodic period anchor (779.9 days within 0.5), the beat-period identity, the departure phase angle anchors (0.7739 rad and 44.34 degrees within the spec bounds), the recurrence epoch anchors (0, 779.9, 1559.8 days within 0.1), the zero phase at every recurrence epoch, the monotone phase progression with the half synodic point at pi, the exact synodic_report keys and values, run determinism, and ValueError rejection of reversed or non-positive orbital periods, non-positive semi-major axes, window counts below 1 and non-positive synodic periods.

Compliance

• Standards referenced, not reproduced: ECSS mission design standards (ecss.nl) frame the space mission design context; the synodic window relations above are standard astrodynamics methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.