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Orbit Calculator

Calculate circular, elliptical, and hyperbolic Keplerian orbits, including orbital periods, apoapsis, and periapsis about the Earth, Moon, Sun, and others — This Keplerian orbit calculator is made by joining a model of conic sections, a model of motion in polar coordinates, and a Kepler solver. It uses a perifocal (PQW) coordinate system whose periapsis vector is aligned with the +x axis, so that true anomaly θ is in the trigonometric standard position (implicitly assumed by some forms of Kepler’s equation). By convention, time t=0 and angle (true anomaly) θ=0 correspond to periapsis. The default central body is Earth.

A family of circular, elliptic, parabolic, and hyperbolic Keplerian orbits about a common central body and having a common periapsis, shown to relative scale and according to the conventions of the orbit calculator model.

Press ⏎ to see the model. Try your own inputs or the examples below. Refresh your browser to clear results.

Circular orbit

  1. Calculate a circular orbit by asserting circularity isCircular = true ⏎ and 400 km altitude h = 400 ⏎, which infers orbital speed v and period T. Also, the inferred Earth escape velocity among in the orbital invariants in I.circ ⏎.
  2. To see how far the satellite moves along this orbit in 1 minute, type t = 1 ⏎. As a consequence, the model infers that the satellite moves 4° in this time.
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Aside from Earth (the default), the Sun, Moon, or other planets can be selected as the central body. To see a list of pre-configured central bodies type I.C ⏎ (tip: type wrap ⏎ to wrap subsequent output if the full list is truncated by your screen size). To swap the central body to the Moon, for example, type I.C = 10 ⏎. Given the inputs of the example above, the model infers a lower orbital speed v yet longer period T than for Earth.

Elliptical orbit: Calculate orbit eccentricity from highest and lowest altitudes

  1. To calculate an orbit’s eccentricity, specify the highest altitude (apoapsis) of 2,000 km ha = 2000 ⏎ and the lowest altitude (periapsis) of 300 km hp = 300 ⏎. This infers orbit eccentricity e and orbital period T. Also, specific orbital energy, C₃, velocity at periapsis and apoapsis, as well as other temporal invariants of the orbit can be seen by typing I ⏎.
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Elliptical orbit: A harder example

  1. A satellite with a periapsis of 400 km hp = 400 ⏎ is now measured to have an altitude of 500 km h = 500 ⏎ while at a speed of 10 km/s v = 10 ⏎ , what kind of orbit does that imply? Answer: The orbit has an apogee ha ⏎ of 37,000 km so it is a geosynchronous transfer orbit. 
  2. If the satellite altitude is increasing isOutbound = true ⏎, how long ago was it at periapsis, and how long until it reaches apoapsis on this orbit? Answer: The satellite passed perigee 3 minutes ago (because t=3 minutes was inferred) and has an orbital period T.h ⏎ of 11 hours, therefore it approaches apogee in 5 hours.
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Hyperbolic flyby

  1. An asteroid enters Earth’s sphere of influence at a relative speed of 5 km/s I.vi = 5 ⏎, but it misses Earth and is deflected by 90° I.d = 90 ⏎; how close did it get? Answer: The periapsis altitude hₚ shows the asteroid skimmed beneath 250 km altitude.
  2. The asteroid is detected as it passes the Moon at a distance of 238,000 miles h.mi = 238000 ⏎ from Earth, how long ago did it pass closest to Earth isOutbound = true ⏎? Answer: The asteroid skimmed the atmosphere 19 hours ago based on the time since periapsis t.h ⏎.
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See also

3D Orbit Calculator

Orbit Transfer Calculators