Launch Window Calculator
Compute the required departure phase angle between two planets for a minimum-energy Hohmann transfer window, plus the synodic period determining how often that window repeats.
🪐 What is a Launch Window Calculator?
A launch window calculator determines the required angular alignment between a departure planet and a destination planet for a minimum-energy interplanetary transfer. For a Hohmann transfer (the standard two-burn, minimum-fuel trajectory), the spacecraft travels a half-ellipse from the source orbit to the target orbit in a fixed transit time. For the spacecraft to arrive at the target orbit at the same location as the target planet, the planet must be at a specific angular position relative to the spacecraft's departure point. That position is the departure phase angle, and this calculator computes it for any planet pair.
The concept governs every interplanetary mission. NASA's Mars Science Laboratory (Curiosity) launched on November 26, 2011 and arrived on August 6, 2012 because that departure date placed Mars at approximately 44 degrees ahead of Earth in its orbit, close to the Hohmann minimum. Voyager 1 and Voyager 2 exploited a rare outer planet alignment in the late 1970s that reduced the total delta-v required to visit Jupiter, Saturn, Uranus, and Neptune using gravity assists. Whether planning a simple two-planet hop or a gravity-assist tour, the phase angle determines whether a departure date is physically viable at the given delta-v budget.
The synodic period is equally important. It tells mission planners how long they must wait between successive launch windows. For Earth to Mars, the synodic period is approximately 780 days (about 26 months), meaning a missed launch window costs over two years of delay and often millions of dollars in standby and recertification costs. For Earth to Venus, the synodic period is about 584 days (19 months). For Earth to Jupiter it is only 399 days (about 13 months), making Jupiter transfers more forgiving in scheduling even though the journey takes nearly three years.
This calculator supports two modes. Phase Angle mode computes the required departure phase angle, Hohmann transfer time, total heliocentric delta-v, and synodic period for any source-to-target planet combination. Synodic Period mode focuses on launch window frequency, computing the synodic period in days and years and estimating how many windows occur per decade. Both modes accept custom AU and period inputs for asteroids, comets, Lagrange-point objects, and hypothetical orbits beyond Neptune.
📐 Formula
Phase Angle = π − ωtarget × tTOF
tTOF = π × √(a3 / μ) — Hohmann transfer time (half the ellipse period), seconds
a = (r1 + r2) / 2 — semi-major axis of transfer ellipse in metres
μ = 1.327 × 1020 m3/s2 — Sun gravitational parameter
ωtarget = 2π / Ttarget — angular velocity of target planet (rad/s)
Ttarget — orbital period of target planet in seconds
r1, r2 — orbital radii of source and target planets in metres (AU × 1.496 × 1011)
Synodic Period = 1 / |1/T1 − 1/T2|
T1, T2 — orbital periods of the two planets in the same units
Example: Earth (T=1 yr) and Mars (T=1.881 yr): S = 1/|1 − 0.532| = 2.135 yr = 780 days
📖 How to Use This Calculator
Phase Angle Mode
1
Select source and target planets from the two dropdowns. The default setup shows an Earth-to-Mars transfer. Select Custom from either dropdown to enter a specific orbit radius in AU and orbital period in years.
2
Read the departure phase angle. A positive angle means the target must be that many degrees ahead of the source at departure (standard for outer planet transfers). A negative angle means the target must trail (standard for inner planet transfers).
3
Check the transfer time and delta-v shown below the phase angle. Transfer time is the one-way Hohmann flight duration; delta-v is the heliocentric budget (does not include planetary departure or capture burns).
4
Note the Synodic Window Period to know how often this launch geometry repeats. Divide 365.25 days by the synodic period in years to see how many windows are available annually.
5
Switch to Synodic Period mode for a focused view of window frequency. Select any two planets (or enter custom periods) to see the synodic period in days, years, and the number of windows per decade.
💡 Example Calculations
Example 1 - Earth to Mars (Classic Interplanetary Window)
Earth (1.000 AU) departing for Mars (1.524 AU)
1
Transfer ellipse semi-major axis: a = (1.000 + 1.524) / 2 = 1.262 AU = 1.888 x 1011 m.
2
Transfer time: t = pi x sqrt(a3 / mu) = pi x sqrt((1.888e11)3 / 1.327e20) = pi x 7.12e6 s = 22.37 Ms = 258.9 days.
3
Mars angular velocity: omega = 2pi / (1.881 yr x 3.156e7 s/yr) = 1.059e-7 rad/s.
4
Phase angle: theta = pi - 1.059e-7 x 22.37e6 = pi - 2.369 = 0.773 rad = 44.3 degrees (Mars leads Earth).
Departure phase angle = 44.3 degrees | Transfer time = 258.9 days | Synodic period = 780 days (2.14 yr)
Try this example →Example 2 - Earth to Venus (Inward Transfer)
Earth (1.000 AU) departing for Venus (0.723 AU)
1
Transfer ellipse semi-major axis: a = (1.000 + 0.723) / 2 = 0.8617 AU = 1.289 x 1011 m.
2
Transfer time: t = pi x sqrt((1.289e11)3 / 1.327e20) = pi x 4.018e6 s = 12.62 Ms = 146.1 days.
3
Venus angular velocity: omega = 2pi / (0.615 yr x 3.156e7 s/yr) = 3.236e-7 rad/s.
4
Phase angle: theta = pi - 3.236e-7 x 12.62e6 = pi - 4.083 = -0.942 rad = -54.0 degrees (Venus trails Earth).
Departure phase angle = 54.0 degrees (target behind source) | Transfer time = 146.1 days | Synodic period = 584 days (1.60 yr)
Try this example →Example 3 - Earth to Jupiter (Outer Planet Grand Tour Entry)
Earth (1.000 AU) departing for Jupiter (5.203 AU)
1
Transfer ellipse semi-major axis: a = (1.000 + 5.203) / 2 = 3.101 AU = 4.639 x 1011 m.
2
Transfer time: t = pi x sqrt((4.639e11)3 / 1.327e20) = pi x 2.739e7 s = 86.05 Ms = 996.4 days (2.73 yr).
3
Jupiter angular velocity: omega = 2pi / (11.862 yr x 3.156e7 s/yr) = 1.678e-8 rad/s.
4
Phase angle: theta = pi - 1.678e-8 x 86.05e6 = pi - 1.443 = 1.699 rad = 97.3 degrees (Jupiter leads Earth).
Departure phase angle = 97.3 degrees | Transfer time = 996 days (2.73 yr) | Synodic period = 399 days (1.09 yr)
Try this example →❓ Frequently Asked Questions
What is a launch window in space mission planning?
A launch window is the period during which a spacecraft can depart Earth and reach a target planet on a minimum-energy trajectory. For a Hohmann transfer, the source and target planets must be at a specific angular separation (the phase angle) at departure so the spacecraft intercepts the target after completing the transfer ellipse. Windows are narrow near the optimum and widen as mission planners accept higher delta-v. Missing an Earth-to-Mars window means waiting approximately 26 months for the next alignment.
How do you calculate the phase angle for an interplanetary launch?
Phase angle = pi - omega_target x t_TOF, where t_TOF is the Hohmann transfer time and omega_target is the angular velocity of the target planet. The transfer time comes from t_TOF = pi x sqrt(a^3 / mu), where a = (r1 + r2) / 2 is the transfer ellipse semi-major axis. For Earth to Mars: a = 1.262 AU, t_TOF = 258.9 days, omega_Mars = 2pi / 686.97 days = 0.5240 deg/day, and phase angle = 180 - 0.5240 x 258.9 = 44.4 degrees.
Why does Earth to Mars have a 26-month launch window cycle?
The 26-month cycle is the Earth-Mars synodic period: the time for Earth and Mars to return to the same relative configuration as seen from the Sun. Earth's period is 365.25 days; Mars's is 686.97 days. Synodic period = 1/(1/365.25 - 1/686.97) = 779.9 days = 25.6 months, commonly rounded to 26 months. This means any given Mars alignment repeats roughly every 26 months. The actual launch window around the optimal alignment is typically 2 to 3 weeks wide at a given delta-v budget.
What is the synodic period formula?
S = 1 / |1/T1 - 1/T2|, where T1 and T2 are the sidereal orbital periods of the two bodies around the Sun in the same units. For Earth and Venus: S = 1/|1/365.25 - 1/224.70| = 583.9 days. For Earth and Jupiter: S = 1/|1/365.25 - 1/4332.6| = 398.9 days. The formula works because the synodic period is the inverse of the difference in angular speeds; when both planets orbit at nearly the same rate (e.g., Neptune vs Uranus), the synodic period is very long.
What does a negative phase angle mean for an inner planet transfer?
A negative phase angle means the target planet must be behind (trailing) the source planet at departure. For an Earth-to-Venus transfer, the phase angle is about minus 54 degrees: Venus must be 54 degrees behind Earth in heliocentric longitude at the moment of departure. This is because the spacecraft decelerates relative to Earth's orbital speed to fall inward, and Venus (moving faster in its inner orbit) needs time to lap Earth's departure angle and meet the spacecraft at arrival.
How often do Earth-to-Jupiter launch windows occur?
Earth-Jupiter synodic period is approximately 398.9 days (about 13 months). This means launch windows to Jupiter repeat roughly once per year, making Jupiter more accessible in scheduling than Mars. However, the transfer takes about 997 days (2.73 years), and the required departure phase angle is about 97 degrees, which is a fairly wide angle that means Jupiter must be nearly a quarter of its orbit ahead of Earth at departure.
Can I use this calculator for asteroid or comet missions?
Yes. Select Custom as the target planet and enter the asteroid's semi-major axis in AU and orbital period in years. If you only know the semi-major axis, use Kepler's third law: T = a^(3/2) years for a heliocentric orbit. For Ceres (a = 2.77 AU), T = 2.77^1.5 = 4.60 yr. For near-Earth asteroid Apophis (a = 0.922 AU, T = 0.886 yr), the phase angle to intercept on a Hohmann transfer from Earth would be about minus 128 degrees, though direct transfers to NEAs are rarely pure Hohmann due to orbital inclination and eccentricity.
What is the difference between a type I and type II interplanetary trajectory?
A Type I trajectory is a transfer that uses less than half an ellipse (heliocentric transfer angle less than 180 degrees). A Type II trajectory uses more than half an ellipse (transfer angle greater than 180 degrees). The standard Hohmann transfer is exactly a 180-degree (half-ellipse) Type I trajectory at minimum energy. Type II trajectories take longer but can be used when the planets are not at the Hohmann phase angle; they trade time for delta-v. Porkchop plots show both Type I and Type II opportunities as separate lobes for any given mission.
Why does Saturn have a shorter synodic period than Jupiter?
Jupiter's period is 11.86 yr and Saturn's is 29.46 yr. Earth-Jupiter synodic = 1/|1/1 - 1/11.86| = 1.092 yr = 399 days. Earth-Saturn synodic = 1/|1/1 - 1/29.46| = 1.035 yr = 378 days. Saturn moves so slowly (about 12 deg/yr) that Earth essentially laps it every year; the tiny difference in angular speeds gives a synodic period only slightly longer than one Earth year. Jupiter moves faster (about 30 deg/yr), creating a slightly larger angular speed differential and a somewhat longer synodic period.
How accurate is the Hohmann phase angle for real missions?
The Hohmann model assumes circular, coplanar orbits and gives the phase angle for the idealized minimum-energy transfer. Real planetary orbits are elliptical (Mars eccentricity 0.093) and inclined (Mars inclination 1.85 degrees relative to the ecliptic). These deviations shift the optimal departure date by a few days to a few weeks from the circular-orbit prediction and require slightly different delta-v. For preliminary mission design and conceptual studies, the Hohmann phase angle is the standard reference. For precise mission planning, porkchop plots using actual planetary ephemerides are used.
What was the departure phase angle for the Curiosity rover mission?
The Mars Science Laboratory carrying Curiosity launched on November 26, 2011 and arrived at Mars on August 6, 2012, a transit of 253 days, close to the 258.9-day circular-orbit Hohmann ideal. The departure phase angle was approximately 43 to 45 degrees (Mars leading Earth), consistent with the computed value of 44.3 degrees. The slight difference from the ideal reflects Mars's elliptical orbit and the actual trajectory optimization for arrival geometry and landing site access at Gale Crater.
How do I compute the phase angle for a Mars-to-Earth return mission?
For a Mars-to-Earth return, set Source Planet to Mars and Target Planet to Earth. The formula is the same: theta = pi - omega_Earth x t_TOF. Earth's angular velocity is 2pi/(365.25 days) = 0.9856 deg/day. Transfer time is the same 258.9 days. Phase angle = 180 - 0.9856 x 258.9 = 180 - 255.1 = -75.1 degrees. So Earth must be about 75 degrees behind Mars (trailing) at the moment of Mars departure for the crew to intercept Earth on arrival. This is why Mars surface stays for crewed missions must last about 500 days to wait for Earth to reach the correct position for the return window.