Tools / Physics · Astrodynamics
100% client-side · v = √(μ/r) · vis-viva · two-body modelHow fast must a satellite fly? Compute orbital velocity for circular orbits from the orbit radius, use the vis-viva equation for elliptical orbits, convert altitude to velocity with an escape-velocity comparison, or look up the first cosmic velocity at the surface of any of 5 bodies.
v = √(μ/r): the speed needed to stay in a circular orbit of radius r. Pick the central body — its μ (standard gravitational parameter) is used automatically.
v = √(μ(2/r − 1/a)): the speed at radius r on an ellipse with semi-major axis a. Special case: r = a gives exactly the circular velocity of Mode 1.
r = R + h, then v = √(μ/r). The table also shows the escape velocity at that radius — exactly √2 × the circular velocity (see the Escape Velocity tool).
v₁ = √(μ/R): the speed of a hypothetical circular orbit skimming the surface — the minimum speed to "orbit" a body. Earth's first cosmic velocity is ≈ 7.91 km/s; escape velocity is √2 higher.
| Body | μ (km³/s²) | R (km) | First cosmic v₁ (km/s) |
|---|---|---|---|
| Sun | 132,712,440,018 | 695,700 | 436.762 |
| Earth | 398,600.4418 | 6,371 (mean) | 7.90979 |
| Moon | 4,902.8000 | 1,737.4 | 1.67986 |
| Mars | 42,828.375214 | 3,389.5 | 3.55466 |
| Jupiter | 126,686,534 | 69,911 | 42.5689 |
μ values are the standard gravitational parameters (GM) consistent with our Orbital Period and Hohmann Transfer tools; 1 AU = 149,597,870.7 km (IAU 2012, exact).
For a circular orbit: v = √(μ/r), where r is the orbit radius and μ = GM is the central body's standard gravitational parameter. Example: the ISS flies at r = 6779 km (h = 408 km above Earth's mean radius); with Earth's μ = 398600.4418 km³/s² the velocity is v = 7.66807 km/s (7,668.07 m/s). Double the radius and the speed drops by a factor of √2 — velocity falls off as 1/√r.
μ = G·M is the product of the gravitational constant and the central body's mass — the single number that sets how fast anything orbits it. This calculator uses: Sun 1.32712440018×10¹¹, Earth 398600.4418, Moon 4902.8000, Mars 42828.375214, Jupiter 126686534 km³/s², with mean radii 695700 / 6371.0 / 1737.4 / 3389.5 / 69911.0 km. The μ values match our Orbital Period and Hohmann Transfer tools so results are consistent across all three pages.
v = √(μ(2/r − 1/a)): the speed at radius r on an elliptical orbit with semi-major axis a. Use Mode 2 for anything non-circular — e.g. the Hohmann transfer from ISS altitude (r = 6779 km) to GEO needs v = 10.06530 km/s at perigee (a = 24471.58 km), which is Δv₁ = 2.39723 km/s above the circular 7.66807 km/s. Special case: set r = a and vis-viva collapses exactly to the circular formula v = √(μ/r) — the calculator flags this in the result table (Δv vs circular = 0).
v₁ = √(μ/R): the speed of a hypothetical orbit skimming the surface — the minimum speed to stay in orbit around a body. Mode 4 computes it per body: Earth 7.90979 km/s, Moon 1.67986 km/s, Mars 3.55466 km/s, Jupiter 42.5689 km/s, Sun 436.762 km/s. Note this is a surface-skimming idealization: no real orbit can fly at r = R through atmosphere or terrain.
Use Mode 3: the orbit radius is r = R + h (body radius plus altitude), then v = √(μ/r). Examples: ISS at h = 408 km around Earth gives 7.66807 km/s; h = 300 km above Mars gives 3.40708 km/s; geostationary altitude h = 35793.17 km gives 3.07466 km/s. Altitudes below 160 km around Earth trigger a drag warning — drag there decays an orbit in days to weeks.
Scope boundary: our Orbital Period tool (#1753) focuses on the period T = 2π√(a³/μ); there velocity appears only as a derived number. This page focuses on the velocity v — the circular formula, vis-viva for elliptical orbits, altitude mode, and first cosmic velocity. Same physics, different question. Cross-check: v = 2πr/T with the ISS period T = 92.578 min gives exactly 7.66807 km/s — both tools agree.
They differ by exactly √2: escape velocity vₑ = √(2μ/r) is the speed needed to leave forever, while circular orbital velocity is v = √(μ/r) — so vₑ = √2 × v_orb at the same radius. Mode 3 shows both side by side; at Earth's surface the first cosmic velocity 7.90979 km/s becomes an escape velocity of 11.1861 km/s. For a full escape-velocity solver, see our Escape Velocity tool (#1613).
The calculator covers bound elliptical orbits only, which requires a > 0 and 2/r − 1/a > 0 — for example r = 6779 km with a = 3000 km is rejected (a < r/2 cannot describe a bound ellipse through r). Hyperbolic flybys use the negative-a form of vis-viva and are not covered here. Negative altitudes are rejected too: r = R + h must stay positive.
Engine-verified examples: the ISS (r = 6779 km) flies at 7.66807 km/s — about 15.55 orbits per day. The Moon orbits Earth (r = 384400 km) at 1.01830 km/s. Earth orbits the Sun (r = 1 AU) at 29.7847 km/s. A geostationary satellite (r = 42164.17 km) crawls at 3.07466 km/s — less than half the ISS speed.
A circular two-body model around a point mass: no atmospheric drag (except the explicit sub-160 km Earth warning), no J₂ oblateness, no third-body or radiation-pressure perturbations. Vis-viva adds elliptical orbits but stays inside the same two-body model. Treat results as design reference, not a substitute for a real propagator.
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