Orbital Velocity Calculator (Vis-Viva)

Background.

Orbital speed is not a single number. A body on an elliptical orbit is moving fastest at periapsis, slowest at apoapsis, and at every intermediate distance it takes an intermediate value — which is why a calculator that returns only one speed answers only one question. This page implements the vis-viva equation, v = √(μ(2/r − 1/a)), the general relation that gives the speed at any radius r on an orbit of semi-major axis a around a body of standard gravitational parameter μ = GM. It reports the speed at your chosen point and, alongside it, the periapsis speed, the apoapsis speed, the circular speed, the local escape speed, the specific orbital energy and the specific angular momentum, so the whole energy budget of the orbit is visible at once.

The name is Latin — vis viva, 'living force' — and the equation is simply conservation of energy rearranged. The specific orbital energy ε = v²/2 − μ/r is constant along the orbit, and for a bound ellipse it also equals −μ/2a. Setting those equal and solving for v gives vis-viva directly. Every familiar orbital speed formula is a special case: put r = a and you get the circular speed √(μ/a); let a go to infinity and you get the escape speed √(2μ/r). This page is the general relation those two are corners of, and it is the only calculator in this category that takes an eccentricity.

That distinction is worth stating plainly, because two neighbouring pages cover the corners. The Escape Velocity calculator gives √(2GM/r), the unbound limit. The Orbital Period calculator emits √(GM/a) as a secondary output, the circular case. Neither can tell you how fast Halley's Comet is moving at perihelion, or how fast a Molniya satellite crosses apogee, because neither accepts an eccentricity. If your orbit is circular, either of those pages will serve; if it is not, this is the tool.

Units are declared and fixed throughout. Masses are kilograms, distances metres, speeds metres per second with a kilometres-per-second conversion beside them, specific energy joules per kilogram (which is the same thing as m²/s²), and specific angular momentum m²/s. Eccentricity is dimensionless. The word 'specific' in both energy outputs means per unit mass of the orbiting body — multiply by the satellite's mass in kilograms to get joules and kg·m²/s. The default configuration is Earth's heliocentric orbit, using the solar mass implied by JPL's DE440 value of GM_Sun and CODATA 2022's gravitational constant, with the orbital elements taken from JPL's J2000 Keplerian element table.

Two limits deserve to be read before the number, not after it. First, the specific orbital energy is negative for every bound orbit; a minus sign in that field is the physics working correctly, and a value of zero would mean a parabolic escape trajectory. Second, this is a Newtonian, non-relativistic, two-body, point-mass model. It neglects the orbiting body's own mass, planetary oblateness, atmospheric drag, third-body perturbations and radiation pressure. When the periapsis speed climbs past one percent of the speed of light — which for a solar-mass primary happens inside a semi-major axis of about 14 800 km — the page says so explicitly, because at that point a Newtonian answer is only an order-of-magnitude guide and a relativistic treatment is required.

Eccentricity is also constrained to the bound case, 0 ≤ e < 1. At e = 1 the trajectory is parabolic and at e > 1 hyperbolic: neither has an apoapsis, neither is periodic, and rather than return an infinity the calculator refuses the input and says why. The same applies to the specific-radius mode, where the radius must lie between periapsis and apoapsis, since a body on a given ellipse simply never visits any other distance.

What is orbital velocity calculator?

The vis-viva equation relates a body's speed at any point on a Keplerian orbit to just two properties of that orbit: its current distance from the focus, r, and its semi-major axis, a. Written out, v² = μ(2/r − 1/a), where μ = GM is the standard gravitational parameter of the central body.

It is conservation of energy in orbital clothing. The specific orbital energy — total energy per unit mass — is ε = v²/2 − μ/r, and it does not change as the body moves. For a closed ellipse that same energy equals −μ/2a, a quantity fixed by the size of the orbit alone. Equating the two and solving for v produces vis-viva in one line. The negative sign is the definition of a bound orbit: escape corresponds to ε = 0.

The practical consequence is that orbital speed depends on where you are, not on where you are heading. Two spacecraft passing the same point on the same orbit at the same moment have the same speed regardless of direction of travel. It is also why the semi-major axis, rather than the instantaneous distance, is the quantity that fixes the period and the energy: change a and you change the orbit's energy; change e alone and you only redistribute the same energy between fast and slow parts of the same period.

How to use this calculator.

1. 01Choose the point of the orbit you want the headline speed for: periapsis, apoapsis, the circular case, or a specific radius.
2. 02Enter the mass of the central body in kilograms. The Sun's value is pre-filled; for a binary star, enter the sum of the two masses.
3. 03Enter the semi-major axis in metres — half the long axis of the ellipse, which equals the mean of the periapsis and apoapsis distances.
4. 04Enter the eccentricity between 0 and 1. Use 0 for a circular orbit; the calculator refuses e ≥ 1 because those trajectories are unbound.
5. 05If you chose 'at a specific radius', enter that radius. It must lie between periapsis and apoapsis.
6. 06Read the speed in m/s and km/s, and check the escape speed beside it — if your speed exceeds it, the orbit is not bound.
7. 07Read the regime note before quoting the result: it flags relativistic speeds and highly eccentric orbits, where a single 'orbital velocity' is a misleading summary.

The formula.

The calculator first forms μ = G·M from the mass you supply, using G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² (CODATA 2022). It then derives the two turning-point radii, r_p = a(1 − e) and r_a = a(1 + e), and evaluates v = √(μ(2/r − 1/a)) at the radius your chosen point corresponds to. Periapsis, apoapsis and circular speeds are computed in every mode, because all three depend only on μ, a and e, which are required inputs regardless of which point you select.

A practical note on precision that matters more than it looks. The product μ = GM is measured astronomically, by watching orbits, and for the Sun it is known to about fifteen significant figures. The mass M on its own is only known to about five, because extracting it requires dividing by G — the least precisely measured constant in physics, at 22 parts per million. This page takes a mass in kilograms because that is what people have, but if precision matters, enter M = μ/G computed with the same G used here, which is exactly how the default solar mass of 1.98840987132 × 10³⁰ kg was produced from JPL's GM_Sun.

ROUNDING STAGE. Nothing is rounded part-way through. All arithmetic runs at 40 significant digits in a dedicated high-precision decimal context and rounding happens exactly once, at the moment a result is returned: 10 decimal places for speeds, and 12 significant digits for the radius, the specific energy and the specific angular momentum, since those span many orders of magnitude. The displayed radius is therefore a rounded echo of the exact value used in the computation.

SIGNIFICANT FIGURES. Your answer is only as good as your inputs. JPL's orbital elements carry eight or nine significant figures, so solar-system results are meaningful to roughly seven; anything beyond that is display precision. If you enter a mass in kilograms rather than a gravitational parameter, the 22-ppm uncertainty of G puts a floor of about five significant figures on the absolute speed, though ratios between speeds on the same orbit stay far more precise because μ cancels.

SIGN CONVENTION. All speeds are positive magnitudes; vis-viva has no direction. Specific orbital energy is negative for bound orbits, zero for a parabolic escape and positive for hyperbolic. Specific angular momentum is reported as a positive magnitude; its direction, which fixes prograde against retrograde motion, is not modelled here.

INVALID DOMAIN. A zero or negative central mass leaves no gravitational parameter and no orbit. A zero semi-major axis is the singularity of both μ/a and 2/r − 1/a. An eccentricity of 1 or more describes a parabolic or hyperbolic trajectory, which has no apoapsis and is not periodic — the calculator refuses it rather than returning infinity. In the specific-radius mode, a radius outside the interval [a(1 − e), a(1 + e)] is not a point on the orbit, and beyond r = 2a the quantity 2/r − 1/a turns negative and the speed would be imaginary; the calculator rejects those radii and names the valid interval in the error message. A radius typed back in from the displayed periapsis or apoapsis is accepted within a relative tolerance of 10⁻⁹ and clamped to the exact boundary, so re-entering a displayed value never produces NaN.

APPROXIMATION REGIME. Two-body, point-mass, Newtonian, non-relativistic. The orbiting body's mass is neglected; oblateness (J2), drag, third bodies and radiation pressure are not modelled. Above 1 % of the speed of light the regime note warns explicitly that the Newtonian result should be treated as an order-of-magnitude estimate.

A worked example.

Frequently asked questions.

References& sources.

1. [1]NIST, CODATA 2022 recommended value of the Newtonian constant of gravitation: G = 6.674 30(15) × 10⁻¹¹ m³ kg⁻¹ s⁻², relative standard uncertainty 2.2 × 10⁻⁵. Independent, open access. Retrieved 2026-07-29.
2. [2]NASA JPL Solar System Dynamics, 'Astrodynamic Constants': astronomical unit 149 597 870 700 m; heliocentric gravitational constant GM_Sun = 1.327 124 400 412 794 19 × 10²⁰ m³ s⁻²; geocentric gravitational constant GM_Earth = 398 600.435507 km³ s⁻²; ephemeris DE440 (Park et al. 2021). Independent, open access. Retrieved 2026-07-29.
3. [3]NASA JPL Solar System Dynamics, 'Approximate Positions of the Planets' (Standish & Williams), Table 1: Keplerian elements referred to the mean ecliptic and equinox of J2000, valid 1800–2050. Source of Earth's a = 1.00000261 au and e = 0.01671123. Independent, open access. Retrieved 2026-07-29.
4. [4]NASA JPL Solar System Dynamics, 'Planetary Physical Parameters': Earth mass 5.97217 × 10²⁴ kg, used to cross-check GM_Earth/G. Independent, open access. Retrieved 2026-07-29.
5. [5]Bate, R.R., Mueller, D.D. and White, J.E. (1971). Fundamentals of Astrodynamics. New York: Dover. Chapter 1, §1.6 — derivation of the vis-viva (energy) equation from the specific orbital energy integral. Print/bibliographic reference; no free authoritative full text is linked.
6. [6]Vallado, D.A. (2013). Fundamentals of Astrodynamics and Applications, 4th ed. Hawthorne, CA: Microcosm Press. §1.3 — vis-viva, specific angular momentum h = √(μa(1 − e²)), and the periapsis/apoapsis speed forms. Print/paywalled reference.
7. [7]BIPM, 3rd CGPM (1901) Resolution 2 — 'Declaration on the unit of mass and on the definition of weight; conventional value of gₙ', which fixes gₙ = 980.665 cm/s². Cited for the definition of weight used in the related surface-gravity pages. Independent, open access. Retrieved 2026-07-29.