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3D Orbit Calculator

Orient 2D Keplerian orbits in 3D space using orbital inclination, longitude of the ascending node, and argument of periapsis, about the Earth, Moon, Sun, and others. Sun-synchronous, critically inclined, and Molniya type Earth orbits are calculated using nodal and apsidal precession models.

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

Circular equatorial low Earth orbit

  1. Calculate the position of a satellite in circular orbit at 1 minute past periapsis t = 1 ⏎ by asserting zero eccentricity 2D.e = 0 ⏎ and 400 km altitude 2D.h = 400 ⏎. The ‘2D’ object is the 2D Keplerian orbit model. This step mirrors the first example on the 2D Keplerian orbit calculator page.
  2. Assert zero inclination i = 0 ⏎ for an equatorial orbit. This special case fully orients the 2D orbit in 3D space; the general case requires additional angles.
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Sun-synchronous Earth orbit (SSO)

  1. A sun-synchronous Earth orbit isSunSynchronous = true ⏎ has a nodal precession of 360° over the course of an Earth year. The model infers from this that the orbit is bounded and that its secular nodal precession rate Ω′ is 0.986°/day, corresponding to a ground track repetition period of exactly 1 Earth day.
  2. Following Vallado‘s RADARSAT example, a circular orbit 2D.e = 0 ⏎ at 800 km altitude 2D.h = 800 ⏎ is calculated to have 98.6° inclination (i ⏎) and a 7,178 km semimajor axis (2D.a ⏎).
  3. But what would the altitude need to be for an eccentricity of 0.02 and the same inclination? Un-asserting the altitude 2D.h = ⏎ then asserting eccentricity 2D.e = 0.02 ⏎ and inclination i = 98.6 ⏎, the model infers a 7,179.8 km semimajor axis (2D.a ⏎), 945 km apoapsis (2D.ha ⏎) and 658 km periapsis (2D.hp ⏎).

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Critically inclined Earth orbit

  1. A critically inclined orbit has zero secular nodal precession «ctrl+g» w’ = 0 ⏎. Earth orbits satisfy this condition at two particular inclinations, one prograde and the other retrograde. Asserting the retrograde case isRetrograde = true ⏎ infers 116.6° inclination.
  2. Asserting the prograde case isRetrograde = false ⏎ infers 63.4° inclination.
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Molniya orbit

  1. A Molniya orbit has zero nodal precession «ctrl+g» w’ = 0 ⏎ isRetrograde = false ⏎ and high eccentricity so that it loiters over far northern latitudes.
  2. For northernmost apoapsis, the orbit’s periapsis is southernmost «ctrl+g» w = 270 ⏎ (use ω=90° for an inverted Molniya orbit that loiters over far southern latitudes).
  3. Periapsis altitude is chosen to be high enough to minimize atmospheric drag but low enough to be expeditious, 2D.hp = 450 ⏎.
  4. The ratio of ground track to nodal period Tg/Tn needs to be a whole number for the ground track to exactly repeat, which is desirable for the satellite to repeatedly loiter over the same place. Choosing Tg/Tn = 1 ⏎ implies an 0.84 eccentricity orbit with 24-hour Keplerian period 2D.T.h ⏎ and apoapsis at 71,000 km! An orbit with such a distant apoapsis might take a larger rocket to reach and the satellite itself would need larger apertures for imaging and communications and more power for the latter.
  5. Changing to Tg/Tn = 2 ⏎ implies an 0.74 eccentricity orbit with 12-hour Keplerian period and apoapsis at a somewhat more modest 40,000 km. The price that is paid is that only every second apoapsis is over the desired region, with the other one occurring on the opposite side of the planet. That being said, the period is half what it was in the previous case, so the desired region is visited just as often but the loiter time is reduced.
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3D orbit orientation

Keplerian orbits are inherently flat — they are trajectories that follow a conic section, forming a 2D orbit plane in 3D space. Keplerian orbits are uniquely defined by 2 pieces of information, such as their semi-major axis a and eccentricity e (the 2D orbit calculator does this). Orienting such orbits relative to known references in 3D space at a particular time takes 4 more pieces of information: 3 sequential rotations relative to a reference plane and a reference direction plus a reference time at which the orbiting body is at periapsis. These 6 pieces of information constitute the traditional orbital elements.

To visualize an orbit based on the traditional orbital elements it is crucial to understand that the rotations are performed in a particular sequence about particular axes. If the sequence or axes are wrong, then the orbit’s orientation is generally wrong, so the present model rotates by the argument of periapsis ω, orbital inclination i, and longitude of the ascending node Ω, in that order (corresponding to the quaternion named R3(ω)R1(i)R3(Ω).inv on the left screen). Here is a visualization of how the rotations work, starting from the reference orbit for which these angles are zero (press the play button toward the top right of the animation to restart it):

  1. A 2D elliptical orbit having 2D.e=0.5 and 2D.I.a/RE=4 appears and is tilted for perspective.
  2. 50° argument of periapsis ω rotates the periapsis about the Z-axis (note that the play button can be used to pause and resume animation).
  3. 40° orbital inclination i rotates the orbit’s plane about its X-axis, defining its node line.
  4. 277° longitude of the ascending node Ω splits the apsis line from the node line, rotating periapsis out of the reference plane (about the orbit’s Z-axis).
  5. The resulting orbit, drawn in black, is now uniquely oriented in its 3D frame.
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Equivalently, the 6 orbital elements could be represented by a 3D position and velocity vector, though this form conveys none of what is temporally invariant about the orbit. Either way, the orbital elements only have meaning within the context of the astronomical reference system in which they are defined.

For example, the Geocentric Celestial Reference Frame (GCRF) is an Earth-centered inertial (ECI) frame whose origin is Earth’s center of mass and whose reference plane is Earth’s equatorial plane, with the X-axis pointing toward the first point of Aries ♈︎ as its reference direction and Z-axis being the North Pole, all at a given snapshot in time (epoch) because over long timescales planets change shape and measurable references move. This is the frame that is generally used for Earth satellite orbits, and it is closely aligned with the older WGS-84 frame in which GPS coordinates are defined.

Other reference systems are employed when navigating about the Sun or other central bodies. When orbiting the Sun, for example, the International Celestial Reference System (ICRS) has been established by the international astronomical community in a series of agreements made from 1997 to 2006. The origin of the ICRS is at the barycenter of the solar system and the orientation of its axes is “space fixed” (kinematically non-rotating) with respect to distant objects in the universe, making it suitable for Solar System interplanetary patched conic trajectories. Within the ICRS, several International Celestial Reference Frames (ICRFs) have been realized so far, with new and more accurate realizations being made possible as observations are made and measurement techniques improve.

These orbital elements, reference frames, and reference systems provide a standardized way to describe 3D orbits, which is crucial for satellite navigation and space exploration. The detailed specification of these rotations and their sequence is crucial because any error in sequence or axis of rotation will misalign the intended orbit if more than one of the rotation angles is nonzero. Understanding this precise orientation process is fundamental in astrodynamics and celestial mechanics.

See also

2D Orbit Calculator

Preliminary Orbit Determination

Orbit Transfer Calculators