Launch Window Calculator

A launch window is the period during which a spacecraft can depart Earth and reach another planet using a minimum-energy trajectory. For a Hohmann transfer, the planets must be at a specific angular separation (the phase angle) at departure so that the spacecraft arrives at the target planet's location after completing the transfer ellipse. If the planets are not aligned correctly, the spacecraft will arrive at the target orbit but the planet will be elsewhere. Launch windows for Earth-Mars missions occur about every 26 months; missing one means waiting over two years for the next opportunity.

For an Earth to Mars Hohmann transfer, Mars must be approximately 44.4 degrees ahead of Earth at the moment of departure. The transfer takes about 258.9 days, and during that time Mars travels 180 minus 44.4 = 135.6 degrees, while Earth (on the shorter inner orbit) completes a larger arc. When the spacecraft arrives at Mars's orbital radius, Mars is at the arrival point. This 44-degree geometry repeats every 779.9 days (about 26 months), which is the Earth-Mars synodic period.

The formula is theta = pi minus omega_target times t_TOF, where t_TOF is the Hohmann transfer time (pi times sqrt(a^3 / mu)) and omega_target is the angular velocity of the target planet (2pi divided by its orbital period). The derivation: the spacecraft must travel a half-ellipse (pi radians of true anomaly) in time t_TOF. During that same time, the target must travel from its initial position to the arrival point. So the target's angular travel equals pi minus theta, giving theta = pi minus omega_target times t_TOF. For an outer planet transfer this is positive (target leads); for an inner planet it is negative (target trails).

The synodic period S is the time between successive alignments of two planets as seen from each other. Formula: S = 1 / |1/T1 - 1/T2|, where T1 and T2 are the sidereal orbital periods in the same units. For Earth (T=1 yr) and Mars (T=1.88 yr): S = 1 / |1 - 0.532| = 2.135 yr = 779.9 days. For Earth and Venus (T=0.615 yr): S = 1 / |1 - 1.626| = 1.599 yr = 583.9 days. Planets with similar periods have very long synodic periods; those with very different periods align more frequently.

For an Earth to Venus Hohmann transfer, Venus must be approximately 54 degrees behind Earth at departure. The negative sign indicates Venus (inner planet) trails Earth rather than leading it. Transfer time is about 146 days. The synodic period is approximately 584 days (19.2 months), meaning Earth-Venus launch windows open roughly every year and a half. The negative phase angle means the spacecraft must decelerate relative to its heliocentric speed to fall inward to Venus's orbit.

A Hohmann transfer is a fixed half-ellipse: the transfer orbit's semi-major axis is fully determined by the source and target radii. Once the spacecraft departs, it follows a ballistic path and arrives at the target orbit after exactly the transfer time t_TOF. The target planet moves at a fixed angular rate, so there is exactly one departure-phase configuration where planet and spacecraft reach the same point in space simultaneously. Any other phase angle requires a non-Hohmann (higher delta-v) trajectory with a different transfer time to intercept the planet.

For outer planet destinations (Mars, Jupiter, Saturn), the target planet must be ahead of Earth at departure (positive phase angle) because the spacecraft travels a longer arc and the outer planet moves more slowly. For inner planet destinations (Venus, Mercury), the target must be behind the departure planet (negative phase angle) because the spacecraft must slow down and fall inward, and the inner planet moves faster and must have time to catch up. The synodic period formula S = 1 / |1/T1 - 1/T2| applies equally in both cases.

Yes, but the math becomes significantly more complex. Non-Hohmann transfers (Type I and Type II trajectories) allow departures when planets are not at the exact Hohmann phase angle by flying a different-shaped ellipse or hyperbola. Mission planners use porkchop plots, which graph C3 (hyperbolic excess energy) vs departure and arrival dates, to identify the range of acceptable launch dates around the minimum-energy window. The exact Hohmann phase angle computed here represents the center of the porkchop plot's minimum-energy region.

For an Earth to Jupiter Hohmann transfer, Jupiter must be approximately 97 degrees ahead of Earth at departure. Transfer time is approximately 997 days (2.73 years). Jupiter's synodic period with Earth is about 398.9 days (13.1 months), meaning Jupiter opposition windows open roughly every 13 months. The required phase angle of about 97 degrees is quite large, reflecting the long transfer time and the relatively slow angular motion of Jupiter at 5.2 AU.

Use the Custom option in Phase Angle mode. Enter the object's semi-major axis in AU (from ephemeris data) and compute the orbital period using Kepler's third law: T = a^(3/2) years for a heliocentric orbit around the Sun. For example, the main-belt asteroid Ceres has a = 2.77 AU, so T = 2.77^(3/2) = 4.60 years. Enter these values as the target. The calculator will compute the required departure phase angle and synodic period for a Hohmann transfer to that orbit.

This calculator is accurate for circular, coplanar orbits using the two-body Hohmann transfer model. Real planetary orbits are slightly elliptical (eccentricity 0.017 for Earth, 0.093 for Mars) and inclined, which shifts the optimal departure date by several days to weeks from the perfect circular-orbit prediction. Real mission launch windows span several weeks around the Hohmann optimum. For preliminary mission design and educational purposes, the circular-orbit phase angle is the standard starting point used in textbooks and mission feasibility studies.

The phase angle used here is the angle between the source and target planets measured from the central body (the Sun), equal to the difference in their heliocentric longitudes at departure. Elongation is a different quantity: the angle between a planet and the Sun as seen from Earth. At opposition (Mars elongation = 180 degrees), Earth and Mars are approximately aligned for a Hohmann departure only if Mars is also near the correct phase angle. The two concepts coincide at opposition for outer planets, but not in general.

The synodic period depends on the difference in angular velocities: S = 1 / |n1 - n2| where n = 1/T. Earth's period is 1 yr; Jupiter's is 11.86 yr; Saturn's is 29.46 yr. Jupiter synodic = 1/|1 - 1/11.86| = 1.092 yr = 398 days. Saturn synodic = 1/|1 - 1/29.46| = 1.035 yr = 378 days. Saturn's synodic is shorter because it moves so slowly that Earth essentially laps it every year; the tiny difference in rates gives a synodic period only slightly longer than Earth's year. Jupiter moves faster relative to Earth, so the synodic period is a bit longer.