Kepler third law
A satellite 42,164,000 m from the centre of Earth, whose mass is 5.9722 × 10^24 kg, takes 86163.3432753 seconds per orbit: one sidereal day. Enter the semi-major axis in metres and the central mass in kilograms for Earth, Moon or Sun orbits. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Semi major axis
- 42,164,000
- Mass
- 5,972,200,000,000,000,000,000,000
- G
- 0.0000000001
- Mu
- 398,602,544,600,000
- Period
- 86,163.3432753
This Kepler third law calculator converts a satellite's semi-major axis and the mass of the body it orbits into the sidereal orbital period in seconds. The Newtonian two-body form used here is T equals two pi times the square root of a cubed divided by G times M, which is the engineering statement of Kepler's third law when the satellite is light compared with the central mass.
Kepler's third law for a satellite and the period formula
Kepler's third law is the closed-form relation that ties the sidereal period of a bound orbit to the size of that orbit and the mass of the body being orbited. In the two-body problem, when the satellite mass is negligible compared with the central body, the Newtonian statement is T squared equals four pi squared times a cubed over G times M, which rearranges to T equals two pi times the square root of a cubed over G M.
Here T is the orbital period, a is the semi-major axis of the ellipse (the orbital radius if the path is circular), M is the central mass, and G is Newton's gravitational constant. The original astronomical form compared two planets around the same star and said T squared over a cubed is the same for both; this calculator uses the equivalent engineering form with an explicit mass so a mission designer can enter Earth, Moon, or solar values without a second reference orbit.
The result is a single primary number: the period in seconds. The model is Keplerian: it ignores atmospheric drag, third-body perturbations, Earth oblateness, radiation pressure, and thrust. Those effects matter for high-precision ephemerides, but they are not Kepler's third law.
If you need a first-look period from altitude and a planetary mass, this is the formula to use. Convert altitude to semi-major axis by adding the central body's mean radius before you call, because the law is written in a, not in height above the surface.
Where satellite period from Kepler's third law is used
CubeSat teams and amateur radio operators convert a target altitude into a first-look period before they run a numerical propagator or schedule ground-station passes. Secondary-school and introductory university physics courses assign Kepler drills that rearrange the same formula for Earth satellites, lunar orbiters, and simplified exoplanet years; a deterministic oracle speeds grading and demo setup without standing up a computer-algebra system. Mission concept studies compare circular parking-orbit periods around Earth, the Moon, and Mars from tabulated masses without re-deriving constants each time.
Science communicators turn a semi-major axis into a human-scale period for exhibits that answer how long a satellite takes to go around. Prefer this landing when you already have a in metres and M in kilograms and only need T in seconds; if your source problem gives two planets around the same star without an explicit mass, that ratio form is a different search intent. Do not use this tool for n-body ephemerides, general-relativistic precession, drag-affected lifetimes, or Hohmann transfer durations: those are neighbouring questions with their own inputs.
Round-trip checks are easy: compute a cubed, divide by G times M, take the square root, multiply by two pi, and confirm the published period within the same twelve-significant-digit rounding used for goldens. Keep requests small, pin the rounded period in tests, and you get reproducible seconds for homework, LMS plugins, and edge workers.
What you can do with it
CubeSat and LEO period sketches
Convert a planned circular altitude into semi-major axis, pair it with Earth mass, and read the Keplerian period in seconds before running a full propagator or pass planner.
GEO and parking-orbit sanity checks
Mission studies compare Keplerian periods at geostationary-class radii and low parking orbits around Earth, Moon, or Mars using tabulated central masses in kilograms.
FAQ
What units should I send for semi-major axis and mass?
Convert kilometres, AU, Earth radii, solar masses, or Earth masses before calling so the period is in seconds.
What happens if the semi-major axis or central mass is not positive?
Zero or negative a cannot define a closed Keplerian ellipse, and zero or negative M is non-physical under Kepler's third law. Those failures are not billed.
Does the satellite's own mass enter the formula?
Not in this model. The two-body central-mass approximation assumes the satellite is light compared with M, or that any reduced mass is already folded into the mass you send.
How do I use an orbit altitude instead of the semi-major axis?
Add the mean radius of the central body to the altitude before entering it. The law works with the distance from the centre of the planet, not with the height above its surface.
Does the period include drag or other disturbances?
No. It is the ideal two-body period. Atmospheric drag, the pull of third bodies, the flattening of the Earth, radiation pressure and thrust all change real orbits but are left out.
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/phys/kepler-third-law \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"semi_major_axis":42164000,"mass":5.9722e+24}'const res = await fetch("https://api.kit.forhosting.com/phys/kepler-third-law", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"semi_major_axis": 42164000,
"mass": 5.9722e+24
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/kepler-third-law",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"semi_major_axis": 42164000,
"mass": 5.9722e+24
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/kepler-third-law", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"semi_major_axis":42164000,"mass":5.9722e+24}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"semi_major_axis":42164000,"mass":5.9722e+24}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/kepler-third-law", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"semi_major_axis": 42164000,
"mass": 5.9722e+24
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.kepler_third_law",
"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
Enter semi_major_axis in metres and mass in kilograms; the tool returns period in seconds using a fixed CODATA value of G so homework, mission sketches, and API tests share one constant. Semi-major axis and central mass must both be strictly positive: zero or negative values are non-physical for a closed Keplerian orbit and fail with invalid_input instead of a silent NaN. Successful API calls are billed at $0.002 per item; failed validation is not charged.
Kepler's third law for a satellite and the period formula
That is the quantity satellite operators convert into minutes, hours, or revolutions per day, and it is the quantity this endpoint returns as the field period.
Inputs, SI units, the gravitational constant, and errors
Semi-major axis must be strictly greater than zero: a zero or negative axis cannot define a closed Keplerian ellipse, and the square root of a cubed would not be a real positive period. Central mass must also be strictly greater than zero: zero or negative M is non-physical under Newtonian gravity and would divide by zero or produce an imaginary period. Classic checks include a geostationary-class radius of about forty-two thousand one hundred sixty kilometres about Earth mass recovering a period near one sidereal day, and a low-Earth radius of about six thousand seven hundred seventy-eight kilometres recovering a period near ninety-three minutes. Keep units consistent: convert kilometres, astronomical units, Earth radii, solar masses, and Earth masses to metres and kilograms before calling so the numeric product matches the SI interpretation your report claims. Send a JSON body with two required finite numbers: semi_major_axis (aliases a, sma) in metres, and mass (aliases central_mass, M) in kilograms. Each field may arrive as a JSON number or as a numeric string; missing fields, empty strings, NaN, and Infinity fail with invalid_input naming the field. On success the response echoes cleaned semi_major_axis and mass, exposes G as the fixed CODATA two thousand eighteen constant six point six seven four three times ten to the minus eleven, mu as the gravitational parameter G times M, period as the primary result in seconds, formula as the closed-form string, and a short units object so clients can label SI without guessing. Pricing for successful API calls is the published base of $0.002 per item; validation failures do not consume prepaid balance.
Where satellite period from Kepler's third law is used
Because this capability is a pure parse engine, API pricing is the published base of $0.002 per successful item and failed validation does not consume balance.
Integration use cases
Homework and exam answer keys
Students submit a and M for T = 2π√(a³/(GM)); graders compare the returned period in seconds against the same closed form, including deliberate non-positive mass rejections.
Technical questions
How does this Kepler third law calculator get the orbital period?
It evaluates T = 2π √(a³ / (G·M)) with a in metres, M in kilograms, and a fixed CODATA 2018 G. The primary result is period in seconds.
What does the Kepler third law calculator cost?
The browser tool is free. API requests use prepaid balance at $0.002 per successful call; invalid input is not billed.