Hohmann second burn Δv calculator

No network, no random seed, and no wall clock—identical inputs always produce identical JSON.

What the Hohmann second burn is and how Δv2 is derived

Radius values are center-to-satellite distances, not altitudes above a surface—if you only know height, add the body radius first. The standard gravitational parameter mu equals G times central mass; with mu in cubic metres per second squared and radii in metres, delta-v two is in metres per second. Send gravitational_parameter as a finite strictly positive number—for example Earth mu near three point nine eight six times ten to the fourteen—and radius_inner plus radius_outer as finite strictly positive lengths in the same unit system.

Input contract, response shape, and non-positive radius errors

Example: Earth mu with an inner low-Earth scale radius near six point seven seven eight times ten to the six metres and an outer geostationary-scale radius near four point two one six four times ten to the seven metres yields a second-burn magnitude of roughly one point four seven kilometres per second—the classic LEO-to-GEO circularisation scale. Example: the same mu with two closer circular radii yields a smaller second burn, as expected when the transfer ellipse is less eccentric. Pair the free in-browser Hohmann second burn calculator with your own conversion of altitude to geocentric radius when the source data only quotes height above a planetary surface. POST a JSON object with three required fields. gravitational_parameter must be a finite number strictly greater than zero; missing, empty, non-numeric, zero, or negative values fail with invalid_input naming the field. radius_inner must be a finite number strictly greater than zero; radius_outer must likewise be finite and strictly greater than zero. The explicit product rule is that either non-positive radius is invalid_input because the Hohmann second-burn formula divides by the radii under square roots and is undefined for a vanishing or inverted radius. On success the body echoes gravitational_parameter, radius_inner, and radius_outer rounded to twelve significant digits for stable goldens, reports delta_v2 with the same rounding, sets unit to the string m/s, method to hohmann_deltav2, and formula to the compact identity string so documentation and UI labels stay aligned with the physics. Validation failures are not billed; successful API items use the published prepaid base price of $0.002.

Where a Hohmann second-burn Δv API earns its keep

Classrooms and tutoring platforms use a Hohmann second burn delta-v calculator so learners can check the circularisation burn for arbitrary planetary parameters and radius pairs without sign or unit mistakes when rearranging vis-viva by hand. Exam-prep generators need a deterministic oracle that marks zero or negative radii as invalid rather than silently returning Infinity or NaN that a buggy grader might treat as success. Early mission concept tools compare second-burn budgets across candidate outer targets—medium Earth orbits, geostationary radius, lunar-distance scales, and Martian circular paths—without installing Orekit, GMAT, or a local MATLAB toolbox.