Orbital Mechanics Calculator | Period, Velocity & Altitude for Satellites
Compute orbital period, circular velocity, escape speed, orbital radius, and altitude for satellites around Earth, Moon, Mars, Jupiter, or custom bodies. Supports Hohmann transfer delta-v calculation between two circular orbits. Shows orbital energy and angular momentum.
What Is the Orbital Mechanics Calculator | Period, Velocity & Altitude for Satellites?
Orbital mechanics uses Newton's law of gravitation and conservation of energy to describe satellite motion. For a circular orbit of radius r around a body of mass M: the orbital velocity v = √(GM/r) and period T = 2π√(r³/GM). Escape velocity is √2 times orbital velocity. A Hohmann transfer between two circular orbits uses two engine burns at periapsis and apoapsis of an elliptical transfer orbit, minimising total Δv.
Formula
v = √(GM/r) · T = 2π√(r³/GM) · v_e = v·√2 · Δv₁ = vc₁·(√(2r₂/(r₁+r₂))−1)
How to Use
- 1
Select the central body: Earth, Moon, Mars, Jupiter, Sun, or Custom.
- 2
For Custom body, enter the mass M (kg) and mean radius R (km).
- 3
On the Circular Orbit tab, select an altitude preset (LEO, GPS, GEO) or enter altitude in km.
- 4
Click "Compute Orbit" to see orbital radius, velocity, period, escape velocity, energy, and angular momentum.
- 5
Switch to the Hohmann Transfer tab and enter both altitudes r₁ and r₂ in km.
- 6
Click "Compute Hohmann Transfer" to see Δv₁, Δv₂, total Δv, and transfer time.
- 7
Use body presets to compare how the same altitude behaves around different planets.
Select a central body (or enter custom mass and radius). For circular orbits, enter the altitude and click 'Compute Orbit'. Switch to the Hohmann Transfer tab to compute Δv and transfer time between two altitudes.
Example Calculation
Example 1 — ISS orbit: Earth, h = 400 km. r = 6371 + 400 = 6771 km. v = √(3.986×10¹⁴/6.771×10⁶) ≈ 7.67 km/s. T ≈ 92.6 min. v_e ≈ 10.85 km/s. Example 2 — LEO to GEO Hohmann: r₁ = 6771 km, r₂ = 42157 km. Δv₁ ≈ 2.45 km/s, Δv₂ ≈ 1.48 km/s, total Δv ≈ 3.93 km/s, transfer time ≈ 5.25 h.
Understanding Orbital Mechanics | Period, Velocity & Altitude for Satellites
Common Earth orbits
| Orbit | Altitude (km) | Period | Velocity (km/s) | Use case |
|---|---|---|---|---|
| LEO (ISS) | 400 | 92.6 min | 7.67 | Space station, imaging |
| Sun-synchronous | 550 | 95.6 min | 7.60 | Earth observation |
| MEO (GPS) | 20 200 | 11.97 h | 3.87 | Navigation satellites |
| GEO | 35 786 | 23.93 h | 3.07 | Communications, weather |
| Lunar orbit | 384 400 (from Earth) | 27.3 days | 1.02 | Moon orbit |
Hohmann transfer delta-v budget
| Transfer | Δv₁ (m/s) | Δv₂ (m/s) | Total Δv (m/s) | Transfer time |
|---|---|---|---|---|
| LEO → MEO (GPS) | ~2 420 | ~1 480 | ~3 900 | ~5.2 h |
| LEO → GEO | ~2 450 | ~1 480 | ~3 930 | ~5.25 h |
| LEO → Moon | ~3 140 | ~860 | ~4 000 | ~5 days |
| GEO → escape | ~1 466 | 0 | ~1 466 | — |
Kepler's laws and orbital mechanics
- ›Kepler's First Law: Orbits are ellipses with the central body at one focus. Circular orbits (e = 0) are a special case used in this calculator.
- ›Kepler's Third Law: T² ∝ r³. Doubling the orbital radius multiplies the period by 2^(3/2) ≈ 2.83. This is encoded in T = 2π√(r³/μ).
- ›Vis-viva equation: v² = μ·(2/r − 1/a). For a circular orbit a = r, giving v = √(μ/r). For escape, a = ∞, giving v_e = √(2μ/r) = v_c·√2.
- ›Hohmann efficiency: The Hohmann transfer is the most fuel-efficient two-burn maneuver between two coplanar circular orbits. It uses an elliptical transfer orbit tangent to both.
Frequently Asked Questions
Why does orbital velocity decrease with altitude?
v = √(GM/r) decreases as r increases. At higher altitudes, gravity is weaker, so less centripetal force (and thus less speed) is needed to maintain a circular orbit. However, a higher orbit takes longer to complete one revolution because both the circumference and the time per orbit increase.
What is escape velocity and how does it relate to orbital velocity?
Escape velocity v_e = √(2GM/r) = v_c·√2, exactly √2 ≈ 1.414 times the circular orbital velocity at the same radius. It is the minimum speed needed to escape gravitational attraction without further propulsion. At LEO altitude, v_e ≈ 11.2 km/s vs v_c ≈ 7.9 km/s.
What makes a Hohmann transfer optimal?
The Hohmann transfer minimises total Δv for transfers between coplanar circular orbits when the ratio r₂/r₁ < 11.94. For larger ratios, a bi-elliptic transfer uses three burns and less total Δv. The Hohmann is commonly used for LEO-to-GEO satellite insertion.
What is orbital energy?
The specific orbital energy E = −GM/(2r) = −GM/(2a) is negative for bound orbits. It equals the sum of kinetic (½v²) and potential (−GM/r) energy per unit mass. A more negative value means a tighter, lower orbit. At escape, E = 0.
What does angular momentum represent?
Specific angular momentum L = v·r (for a circular orbit) is conserved in any orbit per Kepler's second law. For an ellipse, L = v_p·r_p = v_a·r_a. Higher angular momentum means a larger, slower orbit. It is the reason Earth moves faster when closer to the Sun (perihelion) in January.
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