Orbital radius from vis-viva calculator
Find where on its orbit a spacecraft must be at a given speed, for example 7668.6356752 m/s on an orbit with a semi-major axis of 6778000 m around Earth, which puts it at 6778000 m, the circular case. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Speed
- 7,668.6356751977
- Semi major axis
- 6,778,000
- Gravitational parameter
- 398,600,441,800,000
- Speed
- 7,668.6356752
- Orbital radius
- 6,778,000
The vis-viva equation is the energy integral of the two-body problem written as a relation among instantaneous orbital speed, radial distance from the focus, semi-major axis of the conic, and the central body's standard gravitational parameter. Mission analysts, students, and lightweight flight-dynamics tools often know the current speed and the planned orbit size and need the radius that is consistent with that energy state.
Vis-viva meaning and which inputs to send
Under Newtonian two-body gravity the specific mechanical energy of a Keplerian trajectory is conserved and equals minus mu over two a, where mu is the standard gravitational parameter G times M of the central body and a is the semi-major axis of the conic. Combining that energy constant with the definitions of kinetic and potential terms produces the classic vis-viva relation: v squared equals mu times the quantity two over r minus one over a.
Here v is the speed relative to the central body, r is the current radial distance from the center of mass of that body to the spacecraft, and a is positive for elliptical bound orbits including the circular special case, negative for hyperbolas, and formally infinite in the parabolic limit which this calculator does not model as a separate mode. Earth mu is near three point nine eight six times ten to the fourteen cubic meters per second squared when you work in SI meters and seconds.
Keep units consistent: with mu in cubic meters per second squared, a and r in meters, and v in meters per second, the algebra is homogeneous. Zero semi-major axis is rejected because vis-viva is undefined when a vanishes. Negative speed is rejected because the equation uses speed magnitude.
How the radius is solved and how to read the response
Starting from v squared equals mu times two over r minus one over a, isolate the two over r term: two over r equals v squared over mu plus one over a. Inverting yields r equals two divided by that sum, which rearranges to the numerically convenient product form r equals two mu a divided by mu plus a times v squared.
For a circular orbit the circular speed satisfies v squared equals mu over a, and substitution recovers r equals a, which is a useful sanity check when you pick textbook LEO numbers. For elliptical orbits with the same a, higher speed corresponds to a smaller instantaneous radius toward periapsis, and lower speed to a larger radius toward apoapsis, as energy conservation demands. Units remain your responsibility: the calculator does not convert kilometers to meters or Earth radii to SI for you.
FAQ
What happens if the resulting radius is non-positive?
Non-positive r means the speed and conic parameters are inconsistent with a physical Keplerian point under vis-viva.
Can semi-major axis be negative?
Yes. Negative a denotes a hyperbolic trajectory. Positive a is elliptical (including circular). Zero a is rejected because vis-viva is undefined.
Is speed the same as circular orbital velocity?
Only for circular orbits. Vis-viva accepts any non-negative speed consistent with the energy of the given a; circular speed is the special case v = sqrt(μ/a) where r equals a.
Is the orbital radius from vis-viva calculator free?
The browser tool runs free on your device.
Why does a higher speed give a smaller radius?
For a fixed orbit size, energy is conserved, so a spacecraft moves fastest near its closest point and slowest near its farthest. Entering a higher speed with the same semi-major axis therefore returns a radius closer to periapsis.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/astro/vis-viva-radius \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"speed":7668.64,"semi_major_axis":6778000,"gravitational_parameter":398600441800000}'const res = await fetch("https://api.kit.forhosting.com/astro/vis-viva-radius", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"speed": 7668.64,
"semi_major_axis": 6778000,
"gravitational_parameter": 398600441800000
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/astro/vis-viva-radius",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"speed": 7668.64,
"semi_major_axis": 6778000,
"gravitational_parameter": 398600441800000
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/astro/vis-viva-radius", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"speed":7668.64,"semi_major_axis":6778000,"gravitational_parameter":398600441800000}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"speed":7668.64,"semi_major_axis":6778000,"gravitational_parameter":398600441800000}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/astro/vis-viva-radius", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"speed": 7668.64,
"semi_major_axis": 6778000,
"gravitational_parameter": 398600441800000
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "astro.vis_viva_radius",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
This capability rearranges vis-viva algebraically for the orbital radius r, evaluates the closed form with deterministic floating-point arithmetic, and rejects non-positive radii with a clear invalid_input error so pipelines never pin a non-physical distance. Identical inputs always yield identical JSON: no network calls, no random seeds, and no wall-clock timestamps. API usage is billed at $0.002 per successful item; validation failures do not consume balance.
Vis-viva meaning and which inputs to send
Send the required field speed as a non-negative finite number or numeric string for the current speed magnitude; aliases v, velocity, and orbital_speed are accepted. Send semi_major_axis as a finite non-zero number for a; aliases a, sma, and semi_major work. Send gravitational_parameter as a strictly positive finite number for mu; aliases mu, mu_grav, and GM are accepted. Missing fields and non-finite values fail with invalid_input naming the problem so clients can correct the form without guessing.
How the radius is solved and how to read the response
On success the JSON echoes speed, semi_major_axis, and gravitational_parameter after stable rounding, then reports orbital_radius as the positive solved r, formula as the string r equals two mu a over mu plus a times v squared, and method as vis_viva_radius so downstream code can branch without sniffing shapes. If the denominator is zero or the computed r is not strictly positive, the capability fails with invalid_input rather than inventing a distance; that product rule catches inconsistent combinations such as a hyperbolic semi-major axis paired with a speed that is too small for any real point on that conic.
API usage, pricing, classroom workflows, and practical limits
Call the free in-browser orbital radius from vis-viva calculator for homework checks, lecture demos, and quick concept verification without installing Orekit, GMAT, or a local MATLAB aerospace toolbox. Typical classroom examples include low Earth orbit with Earth mu and a near six thousand seven hundred seventy-eight kilometers expressed in meters with the matching circular speed, geostationary-scale semi-major axes with slower speeds at apoapsis of a transfer ellipse, and simplified hyperbolic flyby sketches with negative a and large v. Extremely large magnitudes may overflow intermediate products; stay within ordinary solar-system SI ranges for reliable results. POST a JSON body with required speed, semi_major_axis, and gravitational_parameter as finite numbers, using aliases if your client already speaks mu and a. 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 orbital_radius into tables that compare radius versus speed at fixed a, teaching how vis-viva encodes the trade between kinetic energy and gravitational potential along a single Keplerian path without requiring a full numerical integrator.
Integration use cases
Homework and exam verification
Students and graders solve for orbital radius from given speed, semi-major axis, and μ under vis-viva, and get invalid_input when the combination yields a non-positive r.
Tutoring and curriculum generators
Lesson builders produce worked vis-viva radius examples with structured JSON for flashcards, step sheets, and auto-graded drills without embedding a full astrodynamics library.
Early mission concept sketches
Lightweight pipelines that know cruise speed and target orbit size call a prepaid endpoint for a consistent radius check instead of reimplementing the vis-viva rearrange in every language.
Technical questions
How do I calculate orbital radius from the vis-viva equation?
Send JSON with speed (v ≥ 0), semi_major_axis (a ≠ 0), and gravitational_parameter (μ > 0). The response includes orbital_radius r = 2μa/(μ + a v²), plus formula and method labels.