Hyperbolic semi-major axis calculator

This capability evaluates that closed form deterministically from two finite numbers, rejects zero excess velocity with a clear invalid input error, and returns structured JSON that homework checkers, mission concept tools, and lightweight flyby pipelines can pin in continuous integration without installing a full orbital mechanics stack.

What the negative semi-major axis means for a hyperbola

Under Newtonian two-body gravity, specific mechanical energy and the shape of the conic section are tightly linked. When energy is positive, the path is a hyperbola and the classical semi-major axis is defined as a equals minus μ over two times energy, or equivalently a equals minus μ over v-infinity squared when energy is expressed through the asymptotic excess speed. The result is always negative when μ is positive and v-infinity is non-zero: textbooks often plot the magnitude of a as a characteristic length, but the signed value is what appears in the orbit equation and in relations such as periapsis radius equals a times one minus eccentricity with a negative. Both must be finite. Mu must be strictly positive. Send the required field mu as a JSON number or numeric string for the central-body gravitational parameter GM—Earth in SI is about three point nine eight six times ten to the fourteen cubic meters per second squared, or about three hundred ninety-eight thousand six hundred cubic kilometers per second squared in common mission units. Send the required field v_inf for the hyperbolic excess velocity, the speed relative to the central body far from closest approach. Aliases gravitational_parameter, gm, and GM are accepted for mu; hyperbolic_excess_velocity, excess_velocity, vinf, and v_infinity for v_inf. V_inf must be non-zero: zero excess velocity is the parabolic boundary case, the formula divides by zero, and the engine returns invalid_input rather than infinity. Negative v_inf values are allowed numerically because only the square enters the formula; prefer a non-negative speed for clarity. Identical inputs always produce identical JSON because the path is pure floating-point arithmetic with no network, no random seed, and no wall clock.

Reading the response fields and keeping units consistent

The calculator does not convert units for you. On success the response echoes mu and v_inf, then reports semi_major_axis as the signed negative length a equals minus mu over v_inf squared, and abs_semi_major_axis as the positive magnitude mu over v_inf squared for displays that want a length without the sign. The formula field is always the string a equals minus mu over v_inf squared in compact form, and method is always hyperbolic_semimajor so downstream code can branch without sniffing shapes. Units are your responsibility: with μ in cubic kilometers per second squared and v_inf in kilometers per second, semi_major_axis is in kilometers; with SI cubic meters per second squared and meters per second, a is in meters. Missing mu, missing v_inf, zero v_inf, non-positive mu, and non-finite numbers all fail with invalid_input and are not charged after the engine rejects the request. Use semi_major_axis when you need the signed conic parameter for further Keplerian algebra; use abs_semi_major_axis when a dashboard only plots a characteristic scale; keep formula and method when documenting the physics step for students or when logging which algorithm produced a flyby sketch. Pair this endpoint with apoapsis-periapsis or orbital-energy tools when the next question is a different search intent—those landings serve bound or circular cases and should not be confused with hyperbolic sizing.

API usage, pricing, classroom workflows, and practical limits

Call the free in-browser hyperbolic orbit semi-major axis calculator for homework checks, lecture demos, and quick flyby concept verification without installing Orekit, GMAT, or a local MATLAB toolbox. Typical classroom examples include Earth gravity with a few kilometers per second of excess speed for an interplanetary departure hyperbola, a planetary flyby with a stated asymptotic speed, and simplified solar-system problems with solar μ and a chosen v-infinity—always with matching units. Extremely large or tiny magnitudes are bounded for isolate safety; stay within the published numeric range for reliable results. Send a JSON body with required positive mu and required non-zero v_inf. Use the prepaid API when a tutoring platform, coding-assessment sandbox, curriculum generator, or early mission-design pipeline needs a stable contract under KIT balance at $0.002 per successful item; validation failures never consume balance after rejection. Authors shipping SDKs can map semi_major_axis into tables that show how larger excess velocity shrinks the absolute size of the hyperbola while smaller excess velocity makes the path more open and the magnitude of a grow without bound as the parabolic limit is approached.