# Launch Window Analysis > Use when you must determine launch windows and launch geometry for orbital missions: compute the launch azimuth for a target inclination from cos(inc) = cos(lat) * sin(az), the daily launch window when the target orbit plane passes through the launch site (window center and half-width from the plane-crossing geometry), sun-synchronous LTAN to RAAN conversion, the plane change delta-v penalty for out-of-plane launches, window duration from the orbit plane regression rate versus Earth rotation, and pass elevation and lighting checks. Produces the azimuth, window open/close times, RAAN, delta-v penalty, and constraint verdicts for the launch. Trigger: launch window, launch azimuth, inclination, sun-synchronous, ltan, raan, plane change, delta-v, orbital plane, direct injection, ksc. - Skill: `ashfordeou/launch-window-analysis` (Agent Skill, multi-file: 4 files) - Install (CLI): `npx skillmds@latest add ashfordeou/launch-window-analysis` - Raw SKILL.md: https://api.skillmd.com/api/skills/ashfordeou/launch-window-analysis/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Coding & Dev Tools - License: Apache-2.0 - Author: ashfordeOU (https://skillmd.com/u/ashfordeou) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/ashfordeou/launch-window-analysis --- # Launch Window Analysis (space-systems/mission-design/launch-window-analysis) Use when the task is launch geometry for an orbital mission: finding the launch azimuth that direct-injects into a target inclination, locating the daily launch window when the target orbit plane sweeps through the launch site, converting a sun-synchronous local time of ascending node (LTAN) into a RAAN, sizing the plane change delta-v penalty for an out-of-plane launch, computing the window duration from the orbit plane regression rate versus Earth rotation, and checking pass elevation and lighting constraints. Units convention (stated once): angles in degrees, time in seconds, speed in km/s, altitude and radius in km. The geometry is classical two-body plus the rotating Earth; it is a mission design tool, not a flight dynamics product. A real program validates against the actual orbit determination and the launch vehicle ascent profile. ## Domain quick reference - Launch azimuth for direct injection: cos(inc) = cos(lat) * sin(az), so az = asin(cos(inc) / cos(lat)). Due east (az = 90 deg) gives inc = lat. Direct injection requires |lat| <= inc <= 180 - |lat|; outside that range the azimuth formula has no real solution and the function raises ValueError. Retrograde targets (inc > 90) launch westward: az = 180 - asin(cos(inc) / cos(lat)). From KSC (28.5 N): inc 28.5 gives az 90.0; inc 51.6 gives az 44.98; inc 98 gives az 189.11. - Daily window geometry: the launch site is in the orbit plane when its local sidereal time LST = GMST + site_lon equals the plane crossing right ascension. Ascending-side crossing at raan + asin(t), descending side at raan + 180 - asin(t), with t = tan(lat) / tan(inc). The window center is that instant; the half-width is the time the site stays within the out-of-plane tolerance. For KSC into 51.6 deg with raan 100 and a 5 deg tolerance: center 49326 s after the reference epoch, half-width 1865 s, window duration about 62 min. - Sun-synchronous LTAN to RAAN: raan = sun_ra + 15 * (LTAN - 12), mod 360. LTAN 10:30 gives raan 337.5 at sun_ra 0; LTAN 18:00 gives 90; LTAN 06:00 (dawn-dusk) gives 270; LTAN 12:00 gives sun_ra. - Plane change delta-v: dv = 2 * v * sin(di / 2). A 10 deg plane change at 7.8 km/s costs 1.360 km/s; a 20 deg change costs 2.709 km/s (the 2.72 km/s figure sometimes quoted for a "10 deg" change is 2 * v * sin(10 deg), which is the half-angle formula applied at 20 deg). - Window period and regression: successive windows recur every 360 / (earth_rate - node_regression) days with earth_rate 360.9856 deg/day (sidereal). A sun-synchronous orbit (node_regression +0.9856 deg/day) gives a period of exactly 1.0 day: the window falls at the same local solar time every day. With no regression the period is 0.99727 days. - Pass elevation: at the crossing instant the site lies on the ground track (a zenith pass), so the satellite elevation is 90 deg there. Near the crossing, e = atan2(cos(mu) - R / r, sin(mu)) with mu = v * t / r and v = sqrt(mu_earth / r). The horizon is at mu = acos(R / r), about 305 s either side for a 400 km orbit. - Lighting (beta angle): sin(beta) = n_hat . s_hat between the orbit normal and the sun. Dawn-dusk orbits sit near |beta| = 90 deg at equinox (sun in the plane); noon-midnight orbits near 0. For LTAN 10:30 at equinox, beta is about -22.3 deg. ## Workflow 1. State the target inclination, the launch site latitude and longitude, and (for sun-synchronous orbits) the LTAN or the RAAN. 2. Get the direct-injection azimuth with launch_azimuth_for_inclination, or check feasibility first with direct_injection_feasible. If the inclination is below the site latitude the function raises ValueError: the orbit cannot be direct-injected, and the alternative is a plane change (step 5). 3. Convert a sun-synchronous LTAN to the RAAN with sun_sync_ltan_to_raan (sun_ra 0 at the vernal equinox; use the actual sun right ascension otherwise). 4. Compute the window with daily_window_center_halfwidth or window_open_close from the inclination, site latitude/longitude, RAAN, GMST at the reference epoch, and the out-of-plane tolerance. Pass node_regression_deg_per_day = 0.9856 for sun-synchronous orbits. The center and half-width come back in seconds from the reference epoch; the open/close times and duration in window_open_close, with the repeat period. 5. If the launch is out of plane (dogleg or inclination below the direct-injection limit), size the penalty with plane_change_delta_v from the inclination change and the orbital speed. 6. Check the pass with elevation_angle_at_crossing (visibility around the crossing, horizon crossing time) and the lighting with beta_angle (sun relative to the plane). Compare against the mission constraints: elevation mask, beta limits for power and thermal. 7. Sanity-check: an eastward launch from KSC into 51.6 deg flies at about 45 deg azimuth; a sun-synchronous 10:30 window repeats at the same local time daily; a 10 deg plane change at LEO speeds costs about 1.36 km/s. ## Worked example Mission: launch from KSC (28.5 N, 80.6 W) into a 51.6 deg orbit with raan 100 deg, GMST 0 deg at the reference epoch, 5 deg out-of-plane tolerance, orbital speed 7.8 km/s. - launch_azimuth_for_inclination(51.6, 28.5) = 44.98 deg (about 45 deg, northeast). - window_open_close(51.6, 28.5, 100.0, -80.6, 0.0, 5.0): center 49326 s, open 47462 s, close 51191 s, duration 3729 s (about 62 min), period 0.99727 days (sidereal repeat). - With node_regression 0.9856 deg/day (a sun-synchronous-like plane) the period becomes exactly 1.0 day. - plane_change_delta_v(10.0, 7.8) = 1.360 km/s if the launch must be corrected by a 10 deg plane change. - elevation_angle_at_crossing(400.0, 0.0) = 90 deg at the crossing, falling to 24.97 deg at 100 s and below the horizon after about 305 s. Sun-synchronous variant: LTAN 10:30 gives raan 337.5 (sun_ra 0); with gmst 45 the window center is 2088 s after the reference epoch with a half-width of 1384 s (about 46 min total). ## Pitfalls - Asking for an unreachable inclination: direct injection requires |lat| <= inc <= 180 - |lat|; below the site latitude the azimuth function (and the window function, whose plane never crosses the site) raises ValueError instead of returning a number. - Forgetting the retrograde branch: targets above 90 deg launch westward with az = 180 - asin(cos(inc)/cos(lat)) (the 98 deg KSC case gives 189.11 deg), not with the eastward formula. - Mixing GMST and the site longitude sign: the crossing instant uses LST = GMST + site_lon, so a West-negative longitude convention must be applied consistently (KSC at -80.6 in the worked example). - Dropping the node regression when predicting window repeats: the repeat period is 360 / (earth_rate - node_regression); a sun-synchronous plane needs 0.9856 deg/day to give the exact 1.0 day solar repeat, while no regression gives the 0.99727 day sidereal period. - Using the sun right ascension sloppily: sun_sync_ltan_to_raan assumes sun_ra 0 at the vernal equinox; outside equinox the actual sun RA must be supplied or the RAAN and the whole window shift. - Quoting the plane change penalty with the doubled half-angle: the formula dv = 2 v sin(di/2) already contains the factor of two, so a 20 deg change is 2.709 km/s, not the 2.72 km/s figure that comes from misapplying 2 v sin(10 deg) to a 20 deg change. ## Verification checklist - Azimuth for inc equal to the site latitude is exactly 90 deg (due east), and inc 51.6 from KSC lands in 40 to 45 deg. - An inclination below the site latitude raises ValueError, for the azimuth function and for the window function (the plane never crosses the site). - plane_change_delta_v matches 2 * v * sin(di / 2): 1.360 km/s at (10 deg, 7.8 km/s) and 2.709 km/s at (20 deg, 7.8 km/s), within 1% of the 2.72 km/s reference. - LTAN conversions: 10:30 to 337.5, 18:00 to 90, 06:00 to 270 at sun_ra 0. - The sun-synchronous window period is exactly 1.0 day; the no-regression period is 0.99727 days. - Elevation at the crossing instant is 90 deg; the 400 km horizon crossing sits at about 305 s. - Run the gate 3 contract test: scripts/test_launch_window.py must pass offline with all assertions green. ## Scripts - scripts/launch_window_logic.py: the six core functions plus direct_injection_feasible and beta_angle. Pure Python 3, stdlib only. - scripts/test_launch_window.py: gate 3 behavior contract test (stdlib unittest, offline, deterministic, 22 tests). Run from the repo root: python3 skills/space-systems/mission-design/launch-window-analysis/scripts/test_launch_window.py ## References - references/launch-window-engineering-notes.md: derivations of the crossing geometry, the half-width expansion, the LTAN/RAAN relationship, and the ECSS framing of mission analysis. ## Related skills - mission-delta-v-budget (../mission-delta-v-budget): the plane change delta-v from this skill feeds the mission delta-v budget. - radiation-debris (../radiation-debris): environment assessment for the orbit the launch window targets. - entry-descent-landing (../entry-descent-landing): the other end of the mission, descent and landing of the spacecraft. ## Behavior contract (gate 3) The launch geometry logic (azimuth, window crossing, LTAN to RAAN, plane change penalty, elevation and beta angle) is exercised by the gate 3 contract test scripts/test_launch_window.py against scripts/launch_window_logic.py (stdlib unittest, offline, deterministic, 22 tests). Run: python3 scripts/test_launch_window.py ## Compliance - Standards referenced, not reproduced: ECSS-E-ST-10C (space engineering, system engineering general requirements) frames the mission analysis and the ECSS lifecycle around launch window work; the formulas above are common space engineering methodology, summary-only per standards-map.yaml. - compliance: STANDARDS-REF, gated: false.