Orbital speed in kilometres per second calculator
A body of 1 solar mass orbited at 1 astronomical unit gives a circular speed of 29.7851421692 km/s, the Earth-like case. Enter any mass and radius to compare orbits around other stars. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Central mass solar masses
- 1
- Orbital radius au
- 1
- Orbital speed kms
- 29.7851421692
This calculator finds the ideal circular orbital speed around a central body when its mass is given in solar masses and the orbital radius is given in astronomical units. It applies the Newtonian circular-orbit equation, converts the supplied astronomical quantities through fixed SI constants, and returns the speed directly in kilometres per second. The calculation is deterministic and needs no network connection, random values, or current date. A radius of zero or any negative radius produces an invalid input error.
Choose the mass and radius correctly
A value of one represents one solar mass, while a value of two represents twice that mass. For a planet or star with a meaningful physical radius, convert altitude to centre-to-centre distance before using the calculator. The orbital radius must be strictly positive.
Zero and negative values are rejected because the circular-speed equation divides by radius and cannot describe an orbit at or inside a zero distance. The mass must be finite and non-negative; a zero mass returns a mathematical speed of zero, although it is not a useful physical orbit. Use decimal values when the body is lighter than the Sun or the orbit is smaller than one astronomical unit.
Keep the values within ordinary finite JavaScript numeric ranges.
Understand the calculation and result
Circular orbital motion balances gravitational acceleration against the centripetal acceleration required to continuously bend an object's path. Equating those terms produces v equals the square root of G times M divided by r. The calculator converts the central mass from solar masses to kilograms using a fixed solar-mass constant, converts the orbital radius from astronomical units to metres using the defined astronomical unit, and evaluates the equation with a fixed gravitational constant.
The response also echoes both inputs and includes a compact formula label, making the result easy to audit or store beside source data. The speed is an ideal magnitude for a circular two-body orbit. It does not include direction, orbital inclination, eccentricity, atmospheric drag, radiation pressure, relativistic corrections, or disturbances from additional bodies.
The orbiting object's own mass is assumed negligible compared with the central mass. For a binary system with comparable masses, a barycentric model using the combined gravitational parameter is more appropriate.
Use the calculator in practical workflows
The calculator is useful for quick astronomy checks, classroom exercises, catalogue enrichment, simulation setup, and automated comparisons. A learner can enter one solar mass and one astronomical unit to estimate the circular speed associated with an Earth-like distance around a Sun-like star. When comparing systems, remember the scaling relationship: speed grows with the square root of central mass and falls with the square root of orbital radius.
Doubling mass does not double speed, and doubling radius does not halve it. Validate units at the boundary of your application because a number expressed in kilometres or metres cannot be submitted unchanged as astronomical units. For elliptical orbits, use an appropriate vis-viva calculation instead, since speed changes around the path and a single circular radius does not capture that behaviour.
What you can do with it
Astronomy education
Calculate ideal circular speeds for Sun-like and stellar systems while teaching how mass and radius affect orbital motion.
Simulation validation
Generate deterministic reference speeds in kilometres per second for checking initial conditions in simple orbital simulations.
Exoplanet comparisons
Compare ideal circular speeds across candidate systems whose stellar masses and orbital distances are published in astronomical units.
FAQ
Which units should I use?
Supply the central mass in solar masses and the centre-to-centre orbital radius in astronomical units. The result is in kilometres per second.
Does radius mean altitude?
No. It is the distance from the central body's centre to the orbit. Convert any surface altitude to a centre-to-centre radius first.
Can this calculate an elliptical orbit?
No. It calculates the ideal speed for a circular orbit. Elliptical paths require a vis-viva calculation and additional orbital information.
How does orbital speed change with a heavier star?
Speed grows with the square root of the central mass, so a star four times heavier than the Sun gives twice the speed at the same distance. Moving the orbit farther out lowers it in the same way.
What happens if I enter a mass of zero?
You get a speed of zero, which is mathematically correct but does not describe a useful physical orbit. A radius of zero or below is rejected, because the formula divides by the distance.
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/orbital-velocity-kms \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"central_mass_solar_masses":1,"orbital_radius_au":1}'const res = await fetch("https://api.kit.forhosting.com/astro/orbital-velocity-kms", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"central_mass_solar_masses": 1,
"orbital_radius_au": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/astro/orbital-velocity-kms",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"central_mass_solar_masses": 1,
"orbital_radius_au": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/astro/orbital-velocity-kms", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"central_mass_solar_masses":1,"orbital_radius_au":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"central_mass_solar_masses":1,"orbital_radius_au":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/astro/orbital-velocity-kms", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"central_mass_solar_masses": 1,
"orbital_radius_au": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "astro.orbital_velocity_kms",
"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
The browser calculation is free, while an API request costs $0.002 per item.
Choose the mass and radius correctly
Enter central_mass_solar_masses as the mass of the object being orbited, expressed as a multiple of the Sun's mass. Enter orbital_radius_au as the distance from the centre of the central body to the orbit, expressed in astronomical units. This is the orbital radius, not the height above a visible surface. Numeric strings are accepted by the calculation core, but JSON numbers are clearer for API integrations and reduce accidental formatting errors.
Understand the calculation and result
It then divides metres per second by one thousand to report orbital_speed_kms in kilometres per second. The returned digits are stabilized with significant-digit formatting so the same valid input produces the same JSON result across repeated executions.
Use the calculator in practical workflows
API clients should treat invalid_input as a correction request, especially when orbital_radius_au is absent, zero, negative, or not finite.
Technical questions
What happens when the radius is zero?
The request fails with invalid_input because circular orbital speed is undefined at a zero radius. Negative radii fail for the same reason.
How much does the calculator cost?
The browser runner is free. Each successful API item costs $0.002; invalid input is returned as an error.