Proper motion calculator
A star moving sideways at 20 km/s at a distance of 10 parsecs drifts 0.42194092827 arcseconds per year across the sky. Enter any tangential speed and distance to get the proper motion. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Tangential velocity
- 20
- Distance pc
- 10
- Proper motion arcsec yr
- 0.4219409283
- Constant
- 4.74
Proper motion is the angular rate at which a star drifts across the sky, usually quoted in arcseconds per year. Introductory astronomy links that angular rate to the star's true sideways speed through space with the classic four-point-seven-four relation: tangential velocity in kilometres per second equals four point seven four times proper motion in arcseconds per year times distance in parsecs. Inverting the same identity lets you recover the proper motion when you already know the transverse space velocity and the distance—exactly the situation in many homework drills, planetarium demos, and kinematic sanity checks. This calculator accepts a finite tangential velocity in kilometres per second and a strictly positive distance in parsecs, returns proper motion in arcseconds per year, and fails with invalid input when distance is zero or negative so a bad scale never becomes Infinity.
How the 4.74 proper-motion relation works
Start from the small-angle geometry that connects a transverse linear speed to an angular rate on the sky. When velocity is in kilometres per second, distance is in parsecs, and proper motion is in arcseconds per year, the conversion factors that turn AU, arcseconds, and the length of a year into consistent SI units collapse into the familiar textbook constant four point seven four.
Distance must be a finite strictly positive number because division by zero is undefined and a non-positive scale is non-physical under this geometric formula; zero, negative, and near-zero values fail with invalid input instead of returning Infinity. Tangential velocity may be signed so that a signed proper-motion component can be recovered when your pipeline tracks direction along a sky axis.
Keep units honest: if your source quotes mas per year, multiply the output by one thousand after the fact, and if distance is in light-years convert to parsecs first so the numeric scale matches the formula your report claims.
Where proper-motion calculators show up in real work
Secondary-school and introductory university astronomy courses assign drills that convert a known tangential speed and distance into an expected sky drift rate; a single trusted oracle speeds grading and demo setup. Science communicators turn linear speeds from popular science articles into human-readable arcseconds per year without re-deriving unit conversions each time. Data pipelines that pre-filter high-proper-motion candidates sometimes need a deterministic geometric μ column before heavier kinematic or selection-function modules run.
Research notebooks comparing raw inverted-4.74 estimates against more sophisticated estimators can hold velocity and distance fixed and inspect the baseline result side by side. If you later need stellar parallax distance from an arcsecond angle, Hubble-law galaxy distances from recession velocity, or catalogue lookup of Gaia proper motions, treat those as separate search intents and landings rather than overloading this proper-motion-from-velocity calculator page.
What you can do with it
Planetarium and outreach demos
Turn published transverse speeds and parsec distances into human-readable proper motions for exhibits and classroom widgets.
FAQ
Can tangential velocity be negative?
Yes. A signed velocity is allowed so that signed proper-motion components can be recovered when your data track direction along a sky axis. Only distance must be strictly positive.
What does it cost?
The browser tool is free and runs locally.
Where does the 4.74 in the formula come from?
It gathers the conversions between astronomical units, arcseconds and the length of a year into one number. With speed in kilometres per second and distance in parsecs, dividing by 4.74 times the distance gives arcseconds per year.
How do I get the result in milliarcseconds per year?
Multiply the answer by 1000, because the calculator always gives arcseconds per year. Many catalogues, including Gaia, list proper motions in milliarcseconds per year, so this step keeps your numbers comparable.
What if my distance is in light-years?
Convert it to parsecs before entering it, so the scale matches the formula. The distance must also be greater than zero; zero or negative values are rejected instead of producing an infinite drift.
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/proper-motion-from-velocity \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"tangential_velocity":20,"distance_pc":10}'const res = await fetch("https://api.kit.forhosting.com/astro/proper-motion-from-velocity", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"tangential_velocity": 20,
"distance_pc": 10
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/astro/proper-motion-from-velocity",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"tangential_velocity": 20,
"distance_pc": 10
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/astro/proper-motion-from-velocity", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"tangential_velocity":20,"distance_pc":10}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"tangential_velocity":20,"distance_pc":10}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/astro/proper-motion-from-velocity", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"tangential_velocity": 20,
"distance_pc": 10
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "astro.proper_motion_from_velocity",
"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
How the 4.74 proper-motion relation works
A star at distance d moving sideways at speed v_t sweeps out an angle whose rate of change is the proper motion μ. The forward form is therefore v_t equals four point seven four times μ times d; rearranging for the unknown angular rate gives μ equals v_t divided by four point seven four times d. This capability evaluates that closed form in deterministic floating-point arithmetic: no random seeds, no wall-clock timestamps, no network calls, and no model drift, so identical inputs always produce identical JSON. This page intentionally covers only the inverted 4.74 relation from already-reduced v_t and d; converting light-years to parsecs, splitting a full space-velocity vector into radial and tangential parts, applying solar-motion or galactic-rotation corrections, and looking up catalogue μ from Gaia or Hipparcos are separate search intents and separate landings.
What you send and what you get back
Classic textbook checks include twenty kilometres per second at ten parsecs recovering roughly zero point four two two arcseconds per year, because twenty divided by four point seven four times ten equals twenty over forty-seven point four. If your notes list distance in light-years, convert with the usual factor of about three point two six light-years per parsec before calling. Because the engine is pure parse arithmetic, latency stays sub-millisecond and results never depend on external catalogues or clocks. Send two required finite numbers: tangential_velocity as the transverse space velocity v_t in kilometres per second, and distance_pc as the heliocentric distance d in parsecs. Missing fields, non-numeric strings, NaN, Infinity, zero, and negative distance fail with invalid_input and are not charged. On success the response returns proper_motion_arcsec_yr as the primary result, echoes cleaned tangential_velocity and distance_pc, exposes constant as four point seven four, includes a units hint object documenting arcseconds per year, kilometres per second, and parsecs, sets method to proper_motion_from_velocity, and sets formula to the explicit inverted relation string for audit trails. Pricing for successful API calls is the published base of $0.002 per item; validation failures do not consume prepaid balance.
Where proper-motion calculators show up in real work
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
Compute μ = v_t / (4.74 · d) for textbook tangential velocities and distances so students and auto-graders share one closed-form ground truth in arcsec/yr.
Kinematic pipeline fixtures
Generate deterministic proper-motion columns from reduced v_t and d before applying galactic solar-motion corrections or catalogue cross-matches in research notebooks.
Technical questions
What formula does this proper motion calculator use?
The inverted textbook relation μ (arcsec/yr) = v_t (km/s) / (4.74 · d_pc). The forward form is v_t = 4.74 · μ · d with the same units.
What happens if the distance is zero or negative?
The request fails with invalid_input. Proper motion μ = v_t / (4.74 · d) requires a strictly positive distance in parsecs; zero or negative d is undefined or non-physical and is never turned into Infinity.
Is this the same as a stellar parallax calculator?
No. Stellar parallax converts a parallax angle into distance (d = 1/π). This tool converts tangential velocity and distance into proper motion. Different inputs, units, and search intents.