Orbit Calculator

Calculate circular, elliptical, or hyperbolic Keplerian orbits, including orbital periods, apoapsis, and periapsis about the Earth, Moon, Sun, and others — This Keplerian orbit model is made by joining a model of conic sections, a model of motion in polar coordinates, and a Kepler solver. The perifocal (PQW) coordinate system‘s 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 . The model infers orbital speed v and period T. Earth escape velocity can be found in the orbital invariants I.circ .
  2. To see how far the satellite moves along this orbit in 1 minute, type t = 1 . The model infers that the satellite moves 4° in this time.

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 a highest altitude (apoapsis) of 2,000 km ha = 2000  and lowest altitude (periapsis) of 300 km hp = 300 . The model infers orbit eccentricity e and orbital period T. Specific orbital energy, C₃, velocity at periapsis and apoapsis, as well as other temporal invariants of the orbit can be seen by typing .

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.

Hyperbolic flyby

  1. An asteroid enters Earth’s sphere of influence at a relative speed of 5 km/s I.vi = 5 . 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 has been 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 .