What is the orbital velocity formula?

Orbital velocity for a circular orbit is v = √(GM/r), where G = 6.674 × 10⁻¹¹ N·m²/kg², M is the central body mass, and r is the orbital radius from the center. The ISS orbits at about 7.66 km/s; geostationary satellites at 3.07 km/s.

How is orbital velocity related to escape velocity?

Escape velocity = √2 × orbital velocity ≈ 1.414 × orbital velocity. An orbiting object needs to increase its speed by ~41.4% to escape the gravitational pull entirely. ISS at 7.67 km/s needs to reach ~10.85 km/s to leave Earth.

Why don't satellites fall down?

They are falling — constantly! But they're also moving sideways fast enough that they miss the ground. Orbital velocity is precisely the speed where downward fall exactly matches Earth's curvature. The ISS falls ~5 m every second but Earth curves away ~5 m per 7.67 km of horizontal travel.

Why is GEO at exactly 35,786 km?

It's the altitude where orbital period equals one sidereal day (23.93 hours). Solving v = √(GM/r) with T = 2πr/v = 86,164 s gives r = 42,164 km from center = 35,786 km altitude. Only this specific altitude works for geostationary orbit.

How fast do satellites travel?

Depends on altitude: LEO (200-2,000 km): 7-8 km/s. GPS (20,200 km): 3.87 km/s. GEO (35,786 km): 3.07 km/s. Moon (384,000 km): 1.02 km/s. Lower orbits = faster speeds. All Earth orbits are between 1 and 8 km/s.

Does orbital velocity depend on satellite mass?

No. Like free-fall, orbital velocity depends only on the central body's mass and the orbital radius. The ISS and a paperclip floating beside it have exactly the same orbital velocity at the same altitude. The satellite's mass cancels in v = √(GM/r).

What is Kepler's third law?

T² ∝ r³ for any orbits around the same body. The square of the orbital period is proportional to the cube of orbital radius. Discovered empirically by Kepler in 1619, derived from Newton's gravity in 1687. Used in everything from satellite design to exoplanet detection.

Why is LEO so much faster than GEO?

Gravity is stronger at lower altitudes, requiring more centripetal force, which requires higher velocity. v = √(GM/r) → as r decreases, v increases. LEO at ~7.7 km/s vs GEO at 3.1 km/s. The lower velocity at GEO allows the satellite to keep pace with Earth's rotation.

Orbital Velocity Calculator

Calculate the velocity needed for a stable circular orbit around a celestial body. Uses v = √(GM/r), where G is the gravitational constant, M is the central body mass, and r is the orbital radius.

Last updated: May 30, 2026

About this calculator

Orbital velocity is the speed an object needs to maintain a stable circular orbit around a celestial body. It comes from balancing two forces: gravity pulling the object toward the central body, and the centrifugal effect of circular motion. For a circular orbit, these forces must be equal, giving the elegant result v = √(GM/r).

The formula reveals an important principle: orbital velocity depends only on the mass of the central body and the orbital radius — not the satellite's mass. The International Space Station and a paperclip floating beside it both need exactly the same orbital velocity to stay at the same altitude. This is the same mass-independence as free-fall on Earth (Galileo's discovery).

Orbital velocity decreases with altitude. Low Earth orbit (LEO, ~400 km): ~7.66 km/s. Geostationary orbit (GEO, 35,786 km): ~3.07 km/s. Moon's orbit around Earth: ~1.02 km/s. This is why higher orbits take longer to complete — orbital period scales with r^(3/2) (Kepler's third law).

For any orbit, escape velocity is √2 times orbital velocity (~1.414×). A satellite in LEO at 7.66 km/s needs to reach 10.83 km/s to escape Earth entirely. Real spacecraft don't typically need to launch from Earth's surface to escape — instead they reach LEO first, then perform a "trans-Earth injection" or "trans-lunar injection" burn to gain the additional ~3.2 km/s.

Common applications: satellite mission planning (LEO, MEO, GEO), space station operations, lunar and interplanetary trajectory design, exoplanet orbital characterization, and physics education on celestial mechanics.

Formula

**Orbital velocity (circular orbit):** v = √(GM/r) Where: - v = orbital velocity (m/s) - G = 6.674 × 10⁻¹¹ N·m²/kg² (gravitational constant) - M = central body mass (kg) - r = orbital radius from center (m) **Worked example: ISS orbit** ISS altitude ~400 km → orbital radius ~6,771 km = 6.771 × 10⁶ m. Earth M = 5.972 × 10²⁴ kg. v = √(6.674e-11 × 5.972e24 / 6.771e6) v = √(3.986e14 / 6.771e6) v = √(5.886e7) v ≈ 7,672 m/s ≈ 7.67 km/s **Orbital period (Kepler's third law):** T = 2π × √(r³/(GM)) = 2πr/v For ISS: T ≈ 2π × 6.771e6 / 7672 ≈ 5,544 s ≈ 92.4 minutes. **Orbital velocities for Earth orbits:** | Altitude | r (km) | v (km/s) | Period | |---|---|---|---| | 200 km (LEO) | 6,571 | 7.79 | 88 min | | 400 km (ISS) | 6,771 | 7.67 | 92 min | | 800 km (sun-sync) | 7,171 | 7.45 | 101 min | | 1,000 km | 7,371 | 7.35 | 105 min | | 2,000 km | 8,371 | 6.90 | 127 min | | 20,200 km (GPS) | 26,571 | 3.87 | 12 hr | | 35,786 km (GEO) | 42,164 | 3.07 | 24 hr | | Moon (~384,000 km) | 384,400 | 1.02 | 27.3 days | **Solar orbital velocities:** | Body | Distance (AU) | v (km/s) | Period | |---|---|---|---| | Mercury | 0.39 | 47.4 | 88 days | | Venus | 0.72 | 35.0 | 225 days | | Earth | 1.00 | 29.8 | 365 days | | Mars | 1.52 | 24.1 | 687 days | | Jupiter | 5.20 | 13.1 | 11.9 yr | | Saturn | 9.54 | 9.7 | 29.5 yr | | Uranus | 19.2 | 6.8 | 84 yr | | Neptune | 30.1 | 5.4 | 165 yr | | Pluto | 39.5 | 4.7 | 248 yr | **Connection to escape velocity:** v_escape = √2 × v_orbital ≈ 1.414 × v_orbital For ISS: escape from that altitude needs 7.67 × √2 ≈ 10.85 km/s. **Origin of formula:** Centripetal force = gravitational force. m × v²/r = G × M × m / r² v² = GM/r → v = √(GM/r) The satellite's mass m cancels — orbits don't depend on satellite size. **Energy in circular orbit:** KE = ½mv² = ½ × GMm/r PE = −GMm/r Total E = −½ × GMm/r (negative, bound) Total energy is half the magnitude of PE — characteristic of inverse-square forces. **Elliptical orbits (general):** For an elliptical orbit with semi-major axis a: v² = GM × (2/r − 1/a) This is the "vis-viva equation". At any point in the orbit, v depends on current distance and orbit size. At perihelion (closest): v_max At aphelion (farthest): v_min For Earth's orbit (a = 1 AU, e = 0.017): v_perihelion ≈ 30.3 km/s (at January 4) v_aphelion ≈ 29.3 km/s (at July 4) **Kepler's third law (general):** T² = (4π²/(GM)) × r³ T² ∝ r³ for any orbits around the same body. **Hohmann transfer orbit:** Moving between two circular orbits (radii r₁ and r₂): Δv₁ = √(GM/r₁) × (√(2r₂/(r₁+r₂)) − 1) Δv₂ = √(GM/r₂) × (1 − √(2r₁/(r₁+r₂))) Total Δv minimized by elliptical transfer ellipse. Used for most interplanetary missions. **Geostationary orbit:** For an orbit synced with Earth's rotation (T = 23.93 hr sidereal): r_GEO = (GM × T² / (4π²))^(1/3) ≈ 42,164 km from center Altitude: 42,164 − 6,378 = 35,786 km v_GEO ≈ 3.07 km/s Only one altitude works for geostationary orbit. Used for communications, weather, broadcast satellites. **Sun-synchronous orbit:** Specific inclination (~98°) and altitude (~600-800 km) that maintains constant Sun angle. Used for Earth observation (always crosses equator at same local time). **Common orbital terminology:** | Term | Meaning | |---|---| | LEO | Low Earth Orbit (< 2,000 km) | | MEO | Medium Earth Orbit (2,000-35,786 km) | | GEO | Geostationary (35,786 km equatorial) | | GSO | Geosynchronous (35,786 km any inclination) | | HEO | Highly Elliptical Orbit | | Polar | Inclination ~90° | | Sun-synchronous | Special polar orbit | | Lagrange (L1-L5) | Special points around Earth-Sun, Earth-Moon | **Real mission examples:** - **ISS**: 400 km, 7.67 km/s, 92 min period, 51.6° inclination. - **GPS**: 20,200 km, 3.87 km/s, 12 hr period (semi-synchronous). - **GEO comm satellites**: 35,786 km, 3.07 km/s, 24 hr period. - **Hubble**: 540 km, 7.59 km/s. - **JWST**: L2 point, 1.5 million km from Earth.

How to use this calculator

Enter central body mass in kg (Earth: 5.972 × 10²⁴).
Enter orbital radius (from center of body) in meters.
For altitude, add Earth's radius (6,378 km) to get total r.
Calculator returns orbital velocity in m/s.
Multiply by √2 (≈1.414) to find escape velocity at same altitude.
Period T = 2πr/v gives time for one orbit.

Worked examples

International Space Station

**Scenario:** ISS orbits at 400 km altitude. Velocity and period? **Calculation:** r = 6,378 + 400 = 6,778 km. v = √(6.674e-11 × 5.972e24 / 6.778e6) ≈ 7,668 m/s = 7.67 km/s. T = 2π × 6.778e6 / 7,668 ≈ 5,555 s ≈ 92.6 min. **Result:** ISS orbits at 7.67 km/s (17,150 mph) — fast enough to circle Earth in ~93 minutes. Crew sees 16 sunrises per day. Drag from upper atmosphere requires periodic reboost.

Geostationary satellite

**Scenario:** Satellite must stay over the same point on Earth's equator. Required altitude? **Calculation:** Period T = 23.93 hr = 86,164 s (sidereal day). r = (GM × T² / (4π²))^(1/3). r = (3.986e14 × 7.42e9 / 39.48)^(1/3) ≈ 4.216 × 10⁷ m ≈ 42,164 km from center. Altitude = 35,786 km. **Result:** Must orbit at 35,786 km altitude, traveling 3.07 km/s. Only one specific altitude works. Used by ~500 communication and weather satellites. Slot allocation managed internationally (ITU).

Mars orbital velocity

**Scenario:** Mars orbits Sun at 1.524 AU. Orbital velocity? **Calculation:** r = 1.524 × 1.496e11 = 2.279e11 m. M_Sun = 1.989e30 kg. v = √(6.674e-11 × 1.989e30 / 2.279e11) ≈ 24,131 m/s ≈ 24.1 km/s. **Result:** Mars orbits Sun at 24.1 km/s — about 80% of Earth's 29.8 km/s. Slower because farther. Period: 687 Earth days. This explains why Mars launch windows occur every ~26 months when Earth catches up.

When to use this calculator

**Use orbital velocity for:**

- **Satellite design**: matching velocity to desired altitude. - **Mission planning**: launch and orbit insertion analysis. - **Space station operations**: rendezvous and reboost. - **Planetary science**: deriving body masses from orbit measurements. - **Exoplanet detection**: radial velocity method. - **Physics education**: gravity and circular motion problems.

**Real spacecraft don't just achieve v:**

To reach LEO from Earth surface requires ~9.4 km/s total Δv: - 7.8 km/s orbital velocity. - ~1.5 km/s gravity losses (rocket fights gravity while accelerating). - ~0.1-0.3 km/s atmospheric drag losses. - ~0.05-0.5 km/s steering losses.

Launching east near the equator saves ~0.4 km/s (Earth's rotation).

**Orbital decay:**

LEO satellites encounter residual atmosphere → slow drag → spiral inward.

Without reboost: - 400 km: decay in ~5-15 years. - 300 km: decay in months. - 200 km: decay in weeks.

ISS reboosts every few weeks to maintain altitude.

**Orbital perturbations:**

Real orbits affected by: - **Earth's oblateness (J₂)**: causes precession of orbital plane. - **Atmospheric drag**: lowers low orbits. - **Lunar/solar gravity**: affects high orbits. - **Solar radiation pressure**: minor for normal satellites. - **General relativity**: tiny correction for GPS satellites.

GPS satellites would drift ~30 km/day without GR corrections.

**Hohmann transfer mathematics:**

Most fuel-efficient transfer between two circular orbits: 1. Burn at lower orbit → enter elliptical transfer. 2. Coast to higher orbit altitude. 3. Burn at higher orbit → circularize.

Total Δv between LEO (200 km) and GEO (35,786 km): ~3.9 km/s. Direct GEO insertion from LEO is 30-40% more fuel.

**Inclination changes:**

Plane changes are expensive. Δv to change inclination Δi at velocity v: Δv = 2v × sin(Δi/2)

For 90° change at LEO: Δv ≈ 11 km/s (more than launching!).

Better to launch directly into desired inclination if possible.

**Common applications:**

- **Communications**: GEO for fixed coverage; LEO constellations (Starlink) for low latency. - **Earth observation**: sun-synchronous orbits for consistent lighting. - **Navigation**: GPS at MEO (12 hr period). - **Science**: HEO (Molniya), L1/L2 (JWST, Gaia, Euclid). - **ISS**: 51.6° inclination so Russian Soyuz can reach it from Baikonur.

**Software:**

- **STK (Systems Tool Kit)**: industry-standard mission design. - **GMAT**: NASA's open-source mission design tool. - **NASA SPICE Toolkit**: precise ephemerides and trajectories. - **MATLAB Aerospace Toolbox**: educational and engineering. - **Python (poliastro)**: open-source orbital mechanics.

**Pitfalls:**

- **Using altitude vs distance from center**: must use full r (Earth radius + altitude). - **Circular vs elliptical**: formula only for circular orbits. - **Ignoring perturbations**: real orbits drift due to non-spherical Earth, drag. - **Confusing orbital and escape**: v_escape = √2 × v_orbit. - **Forgetting inclination matters**: same altitude, different inclinations need different launch sites and Δv. - **Treating LEO as friction-free**: orbital decay is real and significant.

Common mistakes to avoid

Using altitude instead of radius from center.
Confusing orbital velocity with escape velocity (escape = √2 × orbital).
Applying circular formula to elliptical orbits (use vis-viva instead).
Forgetting orbital decay for low orbits.
Ignoring inclination changes (very expensive in Δv).
Treating satellites as massless point (mass cancels but spacecraft is real).
Forgetting Earth's oblateness and other perturbations for precision missions.
Using m_satellite in formula (orbital velocity doesn't depend on satellite mass).

Frequently Asked Questions

Sources & further reading

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