Escape to circular speed ratio calculator

You do not need mass or radius: they cancel when you form v_esc over v_circ.

Why escape speed is √2 times circular speed

Teachers use the ratio as a quick sanity check: if a student claims an Earth-surface circular speed near seven point nine kilometres per second, the corresponding surface escape figure should land near eleven point two kilometres per second without reopening the full √(2GM/r) derivation. The same factor applies at altitude: LEO circular speed near seven point eight kilometres per second implies an escape speed at that altitude near eleven kilometres per second, still far below interplanetary transfer budgets once you account for Oberth effects and plane changes. This tool exists so that conversion is a single validated multiply rather than a copy-paste of √2 from a calculator app with inconsistent rounding. For a test mass in a circular orbit of radius r about a central mass M, centripetal balance gives v_circ squared equals G M over r, so the circular speed is the square root of G M over r. Escape speed at the same radius follows from setting total mechanical energy to zero at infinity: one half m v_esc squared minus G M m over r equals zero, which rearranges to v_esc equals the square root of two G M over r. Dividing the two expressions cancels G, M, and r entirely and leaves v_esc over v_circ equal to the square root of two, approximately one point four one four. That identity is exact only in the ideal two-body Newtonian model with a spherical non-rotating primary and no atmosphere; real launches still face drag, gravity losses, and the need to climb out of a gravity well on a powered trajectory rather than coasting from a circular orbit.

How to call the API and read the response

Because mass and radius never appear in the input, this landing is the right tool when you already know circular speed and only need the escape counterpart—not when you must first compute circular speed from mass and radius, which belongs to the orbital velocity calculator, or absolute escape from mass and radius alone, which belongs to the planet escape velocity calculator. Send a JSON object with one required strictly positive finite number under circular_speed, or any of the aliases v_c, v_circular, circular_velocity, or orbital_speed. Absolute magnitude must stay within published limits so that the product with √2 remains finite in IEEE double precision on both the Worker and the browser. On success the response echoes the rounded circular speed under circular_speed, v_c, and v_circular; sets escape_speed and v_esc to √2 times that value; sets ratio to the rounded square root of two and ratio_exact to the string √2; and provides formula, equation, units, message, and an ordered steps array that restates the GM/r derivation and the substitution. Missing fields, non-finite values, values outside absolute limits, or non-positive circular speed return invalid_input and are never charged. Pricing is a flat $0.002 per successful request with unit item. Prefer numeric JSON numbers when you control the client; numeric strings that match a strict finite pattern are accepted for spreadsheet-style paste workflows.

Practical tips, limits, and related searches

Keep units consistent: if circular_speed is in metres per second, escape_speed is in metres per second; if you enter kilometres per second, the result stays in kilometres per second. Failed tasks that return invalid_input do not consume prepaid balance, so exploratory scripts that probe zero or negative inputs stay cheap. Round-trip checks are trivial: multiply your input by Math.SQRT2 or by one point four one four two one three five six two three seven and confirm the published escape_speed matches after twelve-decimal rounding.