Orbital mean motion calculator
Find the average angle an orbiting body sweeps each second: with a gravitational parameter of 1 and a semi-major axis of 1, it advances 1 radian per second and completes a lap in 6.28318530718 seconds. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Gravitational parameter
- 1
- Semi major axis
- 1
- A cubed
- 1
- Mean motion
- 1
- Period
- 6.2831853072
Mean motion is the average angular rate of a Keplerian orbit: how many radians the body advances per unit time if motion were uniform around a circle of radius equal to the semi-major axis. In the two-body model it is exactly n equals the square root of μ over a cubed, where μ is the standard gravitational parameter of the central body and a is the semi-major axis of the relative orbit. The same pure arithmetic powers every path, so classroom demos and automated pipelines agree byte-for-byte without installing a full astrodynamics stack.
What orbital mean motion is and how n equals square root of μ over a cubed
Under the two-body Keplerian model a small satellite orbiting a fixed central body has a mean anomaly that advances at a constant rate called the mean motion. That rate is the positive square root of the standard gravitational parameter divided by the cube of the semi-major axis: n equals the square root of μ over a cubed. Equivalently, the sidereal orbital period is T equals two pi over n, so mean motion is simply the average angle per unit time around one full revolution of two pi radians.
When a is the circular orbital radius the same formula gives the angular rate of circular motion; when a is the semi-major axis of an ellipse the formula still holds for the mean (not true) angular rate. Units of the inputs must be consistent: with SI choices, μ in cubic metres per second squared and a in metres yield n in radians per second. The calculator does not convert altitude above a planetary surface into center-to-body distance, and it does not fold in higher-order perturbations, drag, or third-body effects.
Classroom use and practical limits
Call the free in-browser orbital mean motion calculator for homework checks, lecture demos, and quick concept verification without installing Orekit, GMAT, or a local MATLAB toolbox.
Typical classroom examples include a unit-system check where μ equals one and a equals one so n equals one, a low-Earth-orbit sketch with Earth μ about three point nine eight six times ten to the fourteen and semi-major axis near six thousand seven hundred seventy-one kilometres, and a solar-system check where solar μ and one astronomical unit recover the tiny mean motion of Earth about the Sun. Extremely large magnitudes are bounded for isolate safety; stay within the published numeric range for reliable results.
What you can do with it
Homework and exam answer keys
Compute Keplerian mean motion n from μ and semi-major axis a so students and auto-graders share one closed-form ground truth in rad/s with convenience period.
Satellite pass and mean-anomaly clocks
Convert target circular or elliptical orbit size about Earth into mean motion for ground-station planning and mean-anomaly time scales.
Planetarium and curriculum widgets
Turn tabulated semi-major axes and gravitational parameters into average angular rates for exhibits and multi-step orbital-mechanics demos.
FAQ
What formula does this orbital mean motion calculator use?
Keplerian mean motion n = √(μ / a³), where μ is the standard gravitational parameter and a is the semi-major axis.
What happens if semi-major axis is zero or negative?
Non-positive semi-major axis cannot define a real positive mean motion under n = √(μ/a³).
What units should I use?
SI convention: gravitational parameter in m³/s² and semi-major axis in metres yield mean motion in radians per second and period in seconds. Any consistent unit system works if μ and a stay aligned.
How is this different from an orbital period calculator?
Period tools expand a and mass (or μ) into T. This page’s search intent is mean motion n from μ and a directly; period is returned only as a convenience field equal to 2π/n.
What does it cost?
The browser tool is free and runs locally.
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/mean-motion \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"gravitational_parameter":398600441800000,"semi_major_axis":6771000}'const res = await fetch("https://api.kit.forhosting.com/astro/mean-motion", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"gravitational_parameter": 398600441800000,
"semi_major_axis": 6771000
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/astro/mean-motion",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"gravitational_parameter": 398600441800000,
"semi_major_axis": 6771000
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/astro/mean-motion", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"gravitational_parameter":398600441800000,"semi_major_axis":6771000}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"gravitational_parameter":398600441800000,"semi_major_axis":6771000}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/astro/mean-motion", 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,
"semi_major_axis": 6771000
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "astro.mean_motion",
"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
What orbital mean motion is and how n equals square root of μ over a cubed
If your source quotes mass instead of μ, form μ equals G times M with a fixed gravitational constant first, then call this endpoint. Identical finite positive inputs always produce identical JSON because the path is pure floating-point arithmetic with no network, no random seed, and no wall clock—ideal for pinning goldens in continuous integration and for matching free browser widgets to paid API responses.
Input contract, response shape, and error behaviour
Both values must be strictly greater than zero. POST a JSON body with two required finite numbers: gravitational_parameter for μ equals GM, and semi_major_axis for the orbital size parameter a. Each field may arrive as a JSON number or as a numeric string. Aliases mu, mu_grav, and GM are accepted for gravitational_parameter; aliases a, sma, and radius are accepted for semi_major_axis. Missing fields, empty strings, NaN, Infinity, and non-numeric types fail with invalid_input messages that name the field. A zero or negative semi-major axis also fails with invalid_input by product contract, because a non-positive a cannot define a real positive mean motion under the square root of μ over a cubed. A zero or negative gravitational parameter likewise fails with invalid_input, because μ is positive for real central bodies and the expression would not yield a physical angular rate. Those validation failures are not billed after the engine rejects the request. On success the body echoes cleaned gravitational_parameter and semi_major_axis, reports a_cubed as a to the third power, mean_motion as the square root of μ over a cubed rounded for stable cross-runtime comparison, period as two pi over mean motion for convenience, sets formula to the string n equals square root of μ over a cubed, sets method to mean_motion, and includes a units hint object. Example: gravitational_parameter equal to one and semi_major_axis equal to one yields mean motion one and period two pi in the same unit system. Pair the result with period or energy tools when the next question needs dynamical scale; this endpoint focuses on n from μ and a.
API usage, pricing, classroom workflows, and practical limits
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 mean_motion into tables that recover n from published μ and a pairs, teaching that average angular rate follows from orbit size once the central body is fixed—a natural companion lesson to tools that expand a and mass into period when building multi-step orbital-mechanics lesson flows.