Orbital energy calculator

This capability evaluates both expressions deterministically from finite numeric inputs, rejects zero or non-positive radius with a clear invalid input error, and returns structured JSON that homework checkers, mission concept tools, and lightweight astrodynamics pipelines can pin in continuous integration without installing a full orbital mechanics stack.

What gravitational potential and total orbital energy mean

Potential is energy per unit mass: multiply by satellite mass m to recover potential energy U equals m Phi equals minus G M m over r. Send the required field satellite_mass as a JSON number or numeric string for the orbiting body, the required field central_mass for the body being orbited—for example Earth mass in kilograms—and the required field radius as the distance from the center of that body to the satellite, not merely the altitude above the surface. Aliases m_sat, sat_mass, and m are accepted for satellite mass; mass, M, and M_central for the central mass; r and orbital_radius for the path radius. The optional field G, with aliases gravitational_constant and grav_const, lets you inject a textbook approximation or a non-SI constant; when omitted the engine uses the fixed value 6.6743 times ten to the minus eleven so every environment agrees. Radius must be strictly positive as well: zero radius is invalid_input because both formulas divide by r and a vanishing orbit is not a physical circular path. 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 unit consistency

If you switch to astronomical unit systems, supply a matching G; the calculator does not convert units for you. On success the response echoes satellite_mass, central_mass, radius, and the G that was used, then reports gravitational_potential as Phi equals minus G M over r and total_orbital_energy as E equals minus G M m over two r. The formula field is always the string Phi equals minus G M over r semicolon E equals minus G M m over two r, and method is always circular_orbital_energy so downstream code can branch without sniffing shapes. Units are your responsibility and must be consistent: with SI masses in kilograms, radius in meters, and the default G, gravitational_potential is in joules per kilogram and total_orbital_energy is in joules. Missing satellite_mass, missing central_mass, missing radius, zero radius, negative masses, non-positive G, and non-finite numbers all fail with invalid_input and are not charged after the engine rejects the request. Use gravitational_potential alone when a dashboard only needs the field value at r; use total_orbital_energy when comparing bound-orbit budgets across satellite masses; keep formula and method when documenting the physics step for students or when logging which algorithm produced a mission sketch. Pair this endpoint with an orbital velocity tool when the next question is tangential speed, remembering that speed depends only on central mass and radius while total energy also scales with satellite mass under the two-body model.

API usage, pricing, classroom workflows, and practical limits

Call the free in-browser orbital energy calculator for homework checks, lecture demos, and quick concept verification without installing Orekit, GMAT, or a local MATLAB toolbox. Typical classroom examples include a one-tonne satellite in low Earth orbit at roughly Earth radius plus a few hundred kilometers, the same satellite at geostationary radius near forty-two thousand kilometers, and simplified solar orbits with solar mass and astronomical-unit scales—always with matching G and consistent 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 satellite_mass, central_mass, and radius, and optional positive G when you need a non-default constant. 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 total_orbital_energy into tables that compare binding energy across altitudes, teaching how the magnitude of E falls as one over r while potential scales as one over r—a companion lesson to circular speed falling as the square root of one over r when building multi-step lesson flows.