Angular resolution calculator
A 200 mm telescope observing green light at 550 nm can separate two stars 0.692018 arcseconds apart, its diffraction limit. Enter the wavelength and aperture in metres to get the resolution in arcseconds, radians and degrees. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Wavelength
- 0.00000055
- Diameter
- 0.2
- Resolution arcsec
- 0.692018
- Resolution rad
- 0.000003355
- Resolution deg
- 0.00019223
- Wavelength m
- 0.00000055
Rayleigh criterion in arcseconds, free
What angular resolution means
Every telescope is limited by diffraction. Even a perfect mirror or lens cannot focus starlight to an infinitely small point, because light waves spread as they pass through a finite aperture. The result is the Airy disk: a bright central spot surrounded by faint rings.
When two stars are close together on the sky, their Airy disks overlap, and below some separation the eye or the camera can no longer tell them apart. That limiting separation is the angular resolution, and it depends only on the wavelength you observe at and the diameter of the aperture. Bigger apertures resolve finer detail; longer wavelengths blur it.
This is why radio telescopes need enormous dishes to match what a modest optical telescope achieves, and why aperture, not magnification, is the number that matters when choosing an instrument for double stars or planetary detail.
The Rayleigh criterion behind the number
The factor 1.22 comes from the first zero of the Bessel function that describes the Airy pattern — it marks the separation at which the centre of one star's disk falls on the first dark ring of the other, the conventional point at which two equal-brightness sources are just resolvable. The output converts radians to arcseconds, where one arcsecond is 1/3600 of a degree.
For context, a 200 mm telescope observing green light at 550 nanometres resolves about 0.69 arcseconds, while the human eye, with a 7 millimetre pupil, manages roughly 20 arcseconds under ideal conditions. Atmospheric seeing usually limits ground-based instruments before diffraction does, but the Rayleigh figure remains the hard physical floor that no observing condition can beat.
How to read the result
Send the wavelength and the aperture diameter, both in metres, and receive the resolution in arcseconds, radians and degrees in a single response. Wavelengths are tiny numbers, so scientific notation is your friend: 550 nanometres is 5.5e-7. An optional precision field controls the decimal places of the returned angles, between zero and fifteen.
What you can do with it
Compare telescope apertures before buying
Evaluate how much resolving power each extra centimetre of aperture buys at your favourite observing wavelength, before spending money on a larger instrument.
Plan a double-star observing session
Check whether a target pair's separation is above the diffraction limit of your telescope at the wavelength of your eyepiece or camera filter.
Teach diffraction limits with live numbers
Let students vary wavelength and diameter and watch the resolution in arcseconds respond, turning an abstract formula into intuition.
FAQ
Which units do the inputs use?
Both wavelength and diameter are in metres. A 550 nm wavelength is 5.5e-7 m and a 200 mm aperture is 0.2 m.
Why does my telescope not reach this resolution?
The Rayleigh limit assumes perfect optics and no atmosphere. Ground-based seeing typically blurs images to 1-2 arcseconds, so small telescopes are usually seeing-limited rather than diffraction-limited.
Can I use it for radio wavelengths?
Yes. The physics is identical; just express the wavelength in metres. A 21 cm line observation is wavelength 0.21 m, which shows why radio apertures must be so large.
Does a bigger telescope always resolve finer detail?
For resolution, yes: aperture, not magnification, sets the limit. A larger diameter resolves finer detail and a longer wavelength blurs it, which is why radio telescopes need enormous dishes.
How sharp is the human eye compared with a telescope?
With a 7 mm pupil, the eye manages roughly 20 arcseconds under ideal conditions. A 200 mm telescope observing at 550 nm resolves about 0.69 arcseconds, far finer detail.
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/angular-resolution \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"wavelength":5.5e-7,"diameter":0.2}'const res = await fetch("https://api.kit.forhosting.com/phys/angular-resolution", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"wavelength": 5.5e-7,
"diameter": 0.2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/angular-resolution",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"wavelength": 5.5e-7,
"diameter": 0.2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/angular-resolution", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"wavelength":5.5e-7,"diameter":0.2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"wavelength":5.5e-7,"diameter":0.2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/angular-resolution", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"wavelength": 5.5e-7,
"diameter": 0.2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.angular_resolution",
"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.
Limits
max_abs_wavelength | 1000 |
max_abs_diameter | 1000000 |
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 angular resolution API answers the most fundamental question about any telescope: how close can two point sources be before they blur into one? Given the observing wavelength and the aperture diameter, it applies the Rayleigh criterion — theta = 1.22 times the wavelength divided by the diameter — and returns the minimum resolvable angle in arcseconds, the unit astronomers actually quote, along with radians and degrees. It runs in one deterministic calculation, free in your browser and at $0.002 per request through the API, with no account needed for the web version.
The Rayleigh criterion behind the number
The calculator uses the Rayleigh criterion for a circular aperture: theta equals 1.22 times the wavelength divided by the aperture diameter, with theta in radians.
Using the API and reading the result
Both inputs must be positive and finite — a zero or negative aperture or wavelength is rejected with an invalid_input error that names the offending field, and a rejected request is never charged. Because the calculation is pure arithmetic, the response is fully deterministic: the same pair of numbers always yields the same resolution, which makes the endpoint safe to cache and easy to test against. The same code that answers the API also runs free in your browser on this page, so you can explore apertures and wavelengths interactively — from a small refractor to a hundred-metre radio dish at 21 centimetres — and only pay $0.002 per request when you automate the calculation inside a pipeline, a planning tool for an observing session, an educational dashboard, or a comparison across instrument designs.
Technical questions
What does it cost?
$0.002 per request via the API. It is also free to run in your browser on this page.
What is the Rayleigh criterion?
The rule that two point sources are just resolvable when the centre of one Airy disk falls on the first dark ring of the other: theta = 1.22 * wavelength / diameter, in radians.
What happens if I send a zero or negative value?
The request is rejected with an invalid_input error naming the field, and it is not charged. Wavelength and diameter must both be positive.