The Vis-Viva Equation

Why the vis-viva equation matters

• It closes the two-body problem's speed. Kepler's laws describe the shape and timing of an orbit; vis-viva delivers the missing kinematic quantity — how fast you are moving right now — without integrating the equations of motion.
• It is the algebra of spaceflight. Every impulsive maneuver — launch, orbit raising, transfer, capture, escape — is planned by subtracting one vis-viva speed from another to get a Δv budget.
• It unifies the whole conic family. The same expression covers circles, ellipses, parabolas and hyperbolas, just by changing the sign or magnitude of the semi-major axis a.
• It makes energy visible. Because ε = −GM/(2a), a spacecraft's total energy is encoded entirely in the size of its orbit — a deep and useful fact that vis-viva exposes in one line.
• It scales across the cosmos. The same formula works for a probe leaving Earth, a comet plunging past the Sun, and a star whipping around Sagittarius A* — you just swap in the right GM.

How it works, step by step

Vis-viva is really conservation of energy in disguise. For a light body (mass negligible compared with the central mass M), the specific orbital energy — total mechanical energy per unit mass — is

ε = v²/2 − GM/r

the sum of specific kinetic energy (v²/2) and specific gravitational potential energy (−GM/r). Two facts make this powerful:

1. ε is constant along the orbit. Gravity is conservative, so as the body climbs outward it trades speed for height and vice versa — but the sum never changes.
2. ε depends only on the semi-major axis. Evaluating the energy at the ends of the major axis and using the geometry of an ellipse gives the clean result ε = −GM/(2a). A bigger orbit (larger a) is a less-negative, higher-energy orbit.

Set the two expressions for ε equal, v²/2 − GM/r = −GM/(2a), and solve for v²:

v² = GM(2/r − 1/a)

That is the whole derivation. To use it in practice:

1. Pick your central body and its standard gravitational parameter μ = GM (for the Sun, μ = 1.327×10²⁰ m³/s²; for Earth, μ = 3.986×10¹⁴ m³/s²).
2. Find the semi-major axis a = (r_peri + r_apo)/2, the average of closest and farthest distance.
3. Insert the current radius r and read off v = √[μ(2/r − 1/a)].

Circular and escape speed as limits

The two most-quoted orbital speeds are just special cases of vis-viva:

• Circular orbit (r = a). A circle is an ellipse of zero eccentricity, so the radius never varies and r = a everywhere. Then 2/r − 1/a = 1/r and v² = GM/r, i.e. v_circ = √(GM/r). In Low Earth Orbit (r ≈ 6,778 km) this is about 7.67 km/s; at geostationary radius (42,164 km) it drops to about 3.07 km/s.
• Escape (a → ∞). A parabolic escape trajectory has infinite semi-major axis, so ε = 0 and the 1/a term vanishes: v² = 2GM/r, i.e. v_esc = √(2GM/r). At Earth's surface this is 11.19 km/s.

Comparing the two at the same radius gives a result worth memorising: escape speed is exactly √2 ≈ 1.414 times circular speed. Raising a circular-orbit spacecraft to escape therefore costs only about 41% more speed — not double.

Key numbers: speeds from one formula

Scenario	Radius r	Semi-major axis a	Speed v (vis-viva)
Low Earth Orbit (circular)	6,778 km	6,778 km	7.67 km/s
Geostationary orbit (circular)	42,164 km	42,164 km	3.07 km/s
LEO→GEO transfer, at perigee	6,778 km	24,471 km	10.07 km/s
LEO→GEO transfer, at apogee	42,164 km	24,471 km	1.62 km/s
Escape from Earth's surface	6,371 km	∞ (parabola)	11.19 km/s
Earth around the Sun, perihelion	1.471×10⁸ km	1.496×10⁸ km	30.29 km/s
Earth around the Sun, aphelion	1.521×10⁸ km	1.496×10⁸ km	29.29 km/s

Worked example: the LEO-to-GEO Hohmann transfer

Suppose we want to move a satellite from a 400 km circular Low Earth Orbit (r₁ = 6,778 km) up to geostationary orbit (r₂ = 42,164 km), using a Hohmann transfer — the two-burn ellipse that touches both circles. Earth's μ = 3.986×10¹⁴ m³/s². Vis-viva does all four evaluations:

1. Starting circular speed: v₁ = √(μ/r₁) = 7.67 km/s.
2. Transfer-ellipse semi-major axis: a_t = (r₁ + r₂)/2 = 24,471 km.
3. Speed at perigee of the transfer (r = r₁): v_p = √[μ(2/r₁ − 1/a_t)] = 10.07 km/s. First burn: Δv₁ = 10.07 − 7.67 = 2.40 km/s.
4. Speed at apogee of the transfer (r = r₂): v_a = √[μ(2/r₂ − 1/a_t)] = 1.62 km/s.
5. Destination circular speed: v₂ = √(μ/r₂) = 3.07 km/s. Second burn: Δv₂ = 3.07 − 1.62 = 1.46 km/s.

Total Δv ≈ 2.40 + 1.46 = 3.85 km/s, and the coast takes half the transfer-ellipse period — about 5.3 hours. Every geostationary launch, from the first Syncom to modern communications satellites, is planned with exactly this arithmetic. The transfer ellipse spends most of its time near apogee moving slowly, which is precisely why the second, circularising burn is comparatively cheap.

A short history of "living force"

The term vis viva ("living force") was coined by Gottfried Wilhelm Leibniz in the 1680s–90s for the quantity mv², which he argued was the true conserved measure of motion, against the Cartesian momentum mv. Johann Bernoulli and later Leonhard Euler developed the idea, and it survived into the 19th century until Gaspard-Gustave Coriolis and others rescaled it to ½mv² and the modern word "kinetic energy" (coined by William Thomson, Lord Kelvin) took over. In celestial mechanics the "vis-viva integral" is simply the energy integral of the two-body problem, and the name stuck to the orbital-speed formula even though the physics is standard energy conservation. Newton's Principia (1687) already contained the geometry that makes ε depend only on a; the algebraic one-liner we use today is the 18th- and 19th-century distillation of it.

The equation and every symbol

v² = GM ( 2/r − 1/a ) = μ ( 2/r − 1/a )

• v — orbital speed of the body relative to the central mass, in m/s.
• G — Newton's gravitational constant, 6.674×10⁻¹¹ m³ kg⁻¹ s⁻².
• M — mass of the central body, in kg (Sun 1.989×10³⁰ kg; Earth 5.972×10²⁴ kg).
• μ = GM — the standard gravitational parameter, in m³/s²; known far more precisely than G or M separately.
• r — instantaneous distance from the center of the central body, in m.
• a — semi-major axis of the orbit, in m. Positive for ellipses (bound), infinite for parabolas, negative for hyperbolas (unbound).

Two companion relations complete the picture. The specific orbital energy is

ε = v²/2 − GM/r = −GM/(2a)

and Kepler's third law fixes the period of a bound orbit from the same semi-major axis,

T = 2π √(a³ / GM)

For a hyperbolic flyby, a is negative, so −1/a > 0 and the speed at infinity — the hyperbolic excess — is v∞ = √(−GM/a); mission planners quote its square as the characteristic energy C3 = v∞².

Common misconceptions

• "Vis-viva is a separate law of nature." It isn't — it is conservation of energy for the inverse-square force, rearranged. No new physics.
• "Speed depends on where you are, not on the orbit's size." Both matter. Two spacecraft at the same radius r move at different speeds if their orbits have different semi-major axes a.
• "Escape velocity is a direction you fire in." Escape speed is a speed threshold, not a direction; √(2GM/r) is the same in any outward direction because energy has no direction.
• "A higher orbit is faster." The opposite. A larger circular orbit is slower (v_circ = √(GM/r) falls with r), even though its energy is higher — the extra energy is stored as gravitational potential, not speed.
• "The formula needs the orbiting body's mass." For the restricted (light-body) case it doesn't; v depends only on GM, r and a. For comparable masses you replace M with the total mass M₁+M₂ and treat the relative orbit.
• "It only works for planets." It works for any inverse-square two-body system — satellites, comets, binary stars, and stars orbiting a galactic black hole.