Vis-viva equation¶
The Keplerian two-body relation v² = μ(2/r − 1/a) linking orbital speed, radius, and semimajor axis.
Core Idea¶
The vis-viva equation states v²=μ(2/r−1/a) for an ideal Keplerian two-body orbit. Here v is the instantaneous relative speed, r the separation from the attracting center, μ the gravitational parameter, and a the orbit's semimajor-axis energy parameter. It follows from conserving specific mechanical energy, ε=v²/2−μ/r=−μ/(2a), while the body moves under central inverse-square gravity. On one ellipse, a is fixed but r varies, so the relation predicts faster motion at periapsis than at apoapsis.
The same form can describe other two-body conics when their a sign or parabolic limit is handled correctly. It does not turn every gravitational flight into an unperturbed conic: thrust, drag, major third-body perturbations, and finite-body complexities may require additional equations or segmenting approximations. NASA has used vis-viva to obtain transfer-orbit endpoint speeds, then combined those values with angular-momentum relations. The formula supplies speed magnitude, not velocity direction or a complete maneuver schedule; a numerical answer is meaningful only with a specified μ, r, and conic convention.
Scope of Application¶
The equation supplies ideal Keplerian scalar speed, not an entire flight path.
- Elliptic orbit analysis. Compare periapsis and apoapsis speeds from one energy parameter.
- Transfer design. Estimate ideal endpoint speeds before separate maneuver modeling.
- Escape limit. Interpret the parabolic case through 1/a approaching zero.
- Assumption audit. Mark perturbations and nonconservative effects outside the formula.
Clarity¶
Given μ, separation r, and conic semimajor-axis parameter a, vis-viva yields the scalar speed of an ideal two-body orbit. Specific energy is the nearest miss until its constant is expressed through a. Elliptic, hyperbolic, and parabolic cases require the correct sign or limit. Thrust, drag, third-body effects, direction, and actual maneuver cost are not supplied by this one equation.
Manages Complexity¶
One compact relation suppresses the derivation from conserved energy and the geometry of a conic. That compression makes endpoint-speed comparisons easy, but it also tempts users to infer more than speed: full trajectory, burn cost, and perturbation response require independent information. The a convention is especially important when leaving the familiar ellipse case.
Abstract Reasoning¶
- Establish the two-body gravitational regime for the interval being modeled.
- Choose μ for the relevant relative orbit.
- Identify separation r at the orbital point and conic a with correct sign or limit.
- Evaluate v²=μ(2/r−1/a) for scalar speed.
- Use separate geometry/dynamics for direction, transfer impulses, and perturbations.
Knowledge Transfer¶
The μ–r–a relation can be carried from a satellite ellipse to a solar-system transfer conic if the central parameter and conic convention are reset for the new pair. Earth-orbit numbers do not transfer as heliocentric numbers, nor does an ideal speed difference equal an actual burn budget without further modeling. General energy-conservation reasoning travels farther, but outside Keplerian two-body dynamics it is no longer the vis-viva equation.
Relationships to Other Abstractions¶
Current abstraction Vis-viva equation Domain-specific
Parents (1) — more general patterns this builds on
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Vis-viva equation is a kind of Conservation Laws Prime
Vis-viva expresses conserved specific mechanical energy as a speed–radius relation on a Keplerian orbit.
Hierarchy path (1) — routes to 1 parentless root
- Vis-viva equation → Conservation Laws → Invariance
Neighborhood in Abstraction Space¶
Vis-viva equation sits in a sparse region of the domain-specific corpus (63rd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Barycenter — 0.88
- Heliocentrism — 0.85
- Sidereal year — 0.85
- Space travel under constant acceleration — 0.84
- Selberg zeta function — 0.84
Computed from structural-signature embeddings · 2026-10-08