Circular orbit speed calculator
With Earth's gravitational parameter of 398600441800000 m³/s² and a radius of 6778000 m, a circular orbit needs a little under 8 km/s. Enter the parameter and radius to get the speed. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Gravitational parameter
- 398,600,441,800,000
- Radius
- 6,778,000
- Circular orbit speed
- 7,668.6356752 m/s
Circular orbit speed is the tangential velocity that keeps a satellite on a constant-radius path around a central body under Newtonian gravity. In two-body mechanics that speed is the square root of the standard gravitational parameter mu divided by the orbital radius r: v equals sqrt of mu over r. Students, mission-concept sketches, and homework checkers often already know mu from planetary tables and only need the circular speed at a chosen radius—without multiplying G by mass separately or installing a full astrodynamics suite.
What circular orbit speed means and how v = √(μ/r) arises
For a circular Keplerian orbit the centripetal acceleration v squared over r is supplied entirely by the central gravitational acceleration mu over r squared. Setting them equal cancels one factor of r and yields v squared equals mu over r, so the required circular orbit speed is the positive square root of mu divided by r. The standard gravitational parameter mu equals G times the central mass M; once you have mu from a handbook or ephemeris product you never need G and M separately for this calculation.
Radius r is the distance from the center of mass of the primary to the satellite—not altitude above the surface. For low Earth orbit you must add the planetary radius to the altitude before calling this tool. Keep units consistent: with mu in cubic meters per second squared and r in meters, circular orbit speed is in meters per second.
What to enter and why the radius must be positive
Example: mu equals 3.986004418 times ten to the fourteen and r equals 6.778 times ten to the six meters (a low-Earth circular-scale radius near four hundred kilometers altitude above a six-thousand-three-hundred-seventy-eight-kilometer Earth radius) yields a circular speed of roughly seven point seven kilometers per second. Example: the same mu with a larger geostationary-scale radius near forty-two thousand one hundred sixty-four kilometers yields a slower circular speed near three kilometers per second, matching the well-known GEO velocity scale.
Pair the free in-browser circular orbit speed calculator with your own conversion of altitude to geocentric radius when the source data only quotes height above a planetary surface.
Who uses a circular orbit speed calculator
Classrooms and tutoring platforms use a circular orbit speed calculator so learners can check v equals sqrt of mu over r for arbitrary planetary parameters and radii without sign or unit mistakes when rearranging the force balance by hand. Exam-prep generators need a deterministic oracle that marks zero or negative radius as invalid rather than silently returning Infinity or NaN that a buggy grader might treat as success.
Early mission concept tools compare circular speeds across candidate altitudes—low Earth orbit, medium Earth orbit, geostationary radius, and lunar or Martian circular paths—without installing Orekit, GMAT, or a local MATLAB toolbox. Distinguish this capability from mass-and-G orbital velocity tools that take separate G and central mass: here the search intent is circular speed from the gravitational parameter mu and radius alone—the form preferred in astrodynamics tables and mu-based textbooks.
FAQ
What happens if the radius is zero or negative?
Non-positive radius is rejected because v = sqrt(μ/r) divides by r under a square root and is undefined for a vanishing or inverted radius.
Is radius altitude or distance from the center?
Radius is the distance from the central body's center of mass to the satellite. If you only know altitude, add the body radius first before calling the calculator.
How is this different from an orbital velocity calculator that takes mass and G?
This tool takes the standard gravitational parameter μ = GM as a single input plus radius. Mass-and-G tools expand μ into separate G and M; both recover the same circular speed when μ equals G times M.
How fast is a circular orbit at geostationary distance?
Much slower than in low orbit. With the same gravitational parameter and a geostationary-scale radius, the circular speed drops to roughly 3 km/s, because the speed falls with the square root of the radius.
Which units should I use?
Keep one system throughout. With the gravitational parameter in cubic metres per second squared and the radius in metres, the speed comes out in metres per second. The calculator does not convert units for you.
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/circular-orbit-speed \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"gravitational_parameter":398600441800000,"radius":6778000}'const res = await fetch("https://api.kit.forhosting.com/astro/circular-orbit-speed", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"gravitational_parameter": 398600441800000,
"radius": 6778000
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/astro/circular-orbit-speed",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"gravitational_parameter": 398600441800000,
"radius": 6778000
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/astro/circular-orbit-speed", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"gravitational_parameter":398600441800000,"radius":6778000}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"gravitational_parameter":398600441800000,"radius":6778000}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/astro/circular-orbit-speed", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"gravitational_parameter": 398600441800000,
"radius": 6778000
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "astro.circular_orbit_speed",
"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 circular orbit speed calculator accepts mu and r, returns the speed in meters per second when SI units are used, and fails cleanly with invalid_input when the radius is zero or negative because the formula divides by r under a square root. No network, no random seed, and no wall clock—identical inputs always produce identical JSON.
What circular orbit speed means and how v = √(μ/r) arises
When r is positive and mu is positive the formula always returns a finite positive speed; when r is zero or negative the geometry is not a circular orbit and the calculator rejects the request rather than inventing a complex or infinite result. Send gravitational_parameter as a finite strictly positive number—for example Earth mu near 3.986 times ten to the fourteen cubic meters per second squared—and radius as a finite strictly positive length in the same unit system. Aliases mu, mu_grav, and GM are accepted for the parameter; r and orbital_radius for the radius. The response echoes the cleaned inputs, reports circular_orbit_speed, sets unit to m/s, and includes the formula string v equals sqrt of mu over r so documentation and UI labels stay aligned with the physics.
Input contract, response shape, and non-positive radius errors
POST a JSON object with two 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 must be a finite number strictly greater than zero; missing or non-finite values fail the same way, and the explicit product rule is that a non-positive radius is invalid_input because circular orbit speed divides by r under a square root. On success the body includes gravitational_parameter and radius rounded to twelve significant digits for stable goldens, circular_orbit_speed with the same rounding, unit equal to the string m/s, method equal to circular_orbit_speed, and formula equal to the compact identity string. Validation failures are not billed; successful API items use the published prepaid base price of $0.002.
Integration use cases
Homework and exam verification
Students and graders compute v = √(μ/r) for a given gravitational parameter and orbital radius, confirm SI units yield m/s, and get a clear invalid_input error when radius is zero or negative by mistake.
Tutoring and curriculum generators
Lesson builders produce worked circular-orbit speed examples with structured JSON for flashcards, step sheets, and auto-graded orbital-mechanics drills without embedding a full astrodynamics library.
Early mission concept sketches
Lightweight pipelines that need a stable circular speed from μ and r call a prepaid endpoint instead of reimplementing sqrt(μ/r) in every language for LEO, MEO, GEO, and planetary comparisons.
Technical questions
How do I calculate circular orbit speed?
Send JSON with gravitational_parameter (μ, positive) and radius (r, strictly positive). The response includes circular_orbit_speed equal to sqrt(μ/r) and unit m/s for SI inputs.
What does the circular orbit speed calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.