Rayleigh angular resolution calculator
Green light at 550 nanometres through a 100 mm aperture gives a Rayleigh limit of 1.3840368499180167 arcseconds. Enter any wavelength and diameter in metres to compare instruments or observing bands. 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
- 0.1
- Angular resolution radians
- 0.00000671
- Angular resolution arcseconds
- 1.3840368499
- Rayleigh factor
- 1.22
The Rayleigh angular resolution calculator finds the smallest angular separation that an ideal circular aperture can resolve at a chosen wavelength. Enter the wavelength and aperture diameter in metres, and the calculator applies the standard expression 1.22 times wavelength divided by diameter. It returns the result in radians and converts it to arcseconds, the unit most often used for objects on the sky.
Understanding the Rayleigh diffraction limit
A circular aperture does not reproduce a distant point source as an infinitely small point. Diffraction spreads its light into an Airy pattern with a bright central disk and progressively fainter rings. The Rayleigh criterion provides a conventional boundary for deciding when two equally bright point sources are just resolved: the central maximum of one pattern lies at the first minimum of the other.
For an ideal circular opening, that angular separation is theta equals 1.22 times wavelength lambda divided by aperture diameter D. The ratio is dimensionless, so its natural angular unit is radians when wavelength and diameter use the same length unit. This calculator asks for both quantities in metres to make the contract explicit and reduce unit mistakes.
It also converts the result to arcseconds because astronomical separations are commonly reported that way. A smaller returned angle represents finer theoretical resolving power. The value is a diffraction limit, not a prediction of every detail visible through a real instrument.
Optical aberrations, imperfect focus, obstruction geometry, detector sampling, source contrast, tracking, and atmospheric turbulence can all make practical resolution worse. The result is therefore best understood as an ideal reference for a circular aperture at one wavelength.
Entering wavelength and diameter correctly
Provide wavelength as a finite positive number in metres. Visible green light near 550 nanometres, for example, is entered as 5.5e-7 metres. Provide diameter as the clear aperture diameter in metres, not its radius, area, focal length, or focal ratio.
A 100 millimetre objective is therefore entered as 0.1 metres. Keeping both fields in metres avoids hidden scaling and makes requests easy to reproduce in scripts. The diameter must be greater than zero.
The wavelength must also be positive and finite. Once validated, the engine multiplies wavelength by the fixed Rayleigh factor 1.22, divides by diameter, and converts radians to arcseconds using 180 times 3600 divided by pi. No network request, random value, clock, catalog lookup, or iterative approximation affects the answer.
If you compare instruments, hold wavelength constant so aperture is the changing variable. If you compare observing bands for one instrument, hold diameter constant and remember that shorter wavelengths produce smaller ideal angles.
Interpreting and applying the result
Radians are convenient for calculations, simulations, and dimensional analysis, while arcseconds make the scale easier to compare with star separations, angular diameters, seeing reports, and telescope specifications. Suppose two apertures observe at the same wavelength and one has twice the diameter. Its Rayleigh angle is half as large, so its theoretical resolving power is twice as fine.
Likewise, an instrument used at twice the wavelength has twice the diffraction-limited angle when its diameter stays fixed. Do not treat the answer as an atmospheric seeing forecast or as a complete optical transfer analysis. Ground-based observations may be dominated by turbulence, and real systems can include central obstructions, segmented pupils, aberrations, finite pixel sizes, and signal-to-noise constraints.
The criterion also describes separation of point sources under a particular convention; other definitions may use different constants or performance metrics. Use this result as a clean baseline, then compare it with the additional limits relevant to the actual instrument.
What you can do with it
Compare telescope apertures
Calculate how the ideal angular resolution changes when two instruments observe at the same wavelength with different clear diameters.
Prepare optics lessons
Create reproducible examples that connect Airy diffraction, wavelength, aperture size, radians, and arcseconds.
FAQ
Which units should I enter?
Enter both wavelength and diameter in metres. The output is returned in radians and arcseconds.
Does the result include atmospheric seeing?
No. It is the ideal diffraction limit for a circular aperture and does not model turbulence or optical imperfections.
What does the Rayleigh criterion mean?
It is the conventional point at which two equally bright stars are just resolved: the bright centre of one diffraction pattern falls on the first dark ring of the other. The angle is 1.22 times wavelength divided by diameter.
Should I enter the radius or the diameter?
The diameter of the clear aperture, in metres. Not its radius, area, focal length or focal ratio. A 100 millimetre objective is entered as 0.1 metres.
Does a shorter wavelength give finer resolution?
Yes. For the same instrument, shorter wavelengths produce smaller ideal angles. To compare telescopes fairly, keep the wavelength fixed so the aperture is the only thing that changes.
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/rayleigh-resolution \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"wavelength":5.5e-7,"diameter":0.1}'const res = await fetch("https://api.kit.forhosting.com/astro/rayleigh-resolution", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"wavelength": 5.5e-7,
"diameter": 0.1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/astro/rayleigh-resolution",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"wavelength": 5.5e-7,
"diameter": 0.1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/astro/rayleigh-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":0.1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"wavelength":5.5e-7,"diameter":0.1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/astro/rayleigh-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": 0.1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "astro.rayleigh_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.
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 calculation is deterministic, includes no atmospheric model, and is available through the API for $0.002 per request.
Entering wavelength and diameter correctly
A zero diameter would require division by zero, while a negative diameter has no physical meaning for this calculation, so either case produces an invalid_input error instead of a misleading numeric response. Always convert millimetres, micrometres, or nanometres to metres before submitting the request.
Interpreting and applying the result
The response includes angular_resolution_radians, angular_resolution_arcseconds, the Rayleigh factor, and a formula label. Interactive calculations can be run in the browser, while reproducible API requests cost $0.002 each and return the same deterministic computation.
Integration use cases
Build instrument test fixtures
Generate deterministic reference values for software that displays or validates diffraction-limited angular resolution.
Technical questions
What formula does the calculator use?
It uses the Rayleigh criterion theta = 1.22 times wavelength divided by aperture diameter.
What happens when diameter is zero or negative?
The request fails with an invalid_input error because the diameter must be strictly greater than zero.
How much does an API request cost?
Each API request costs $0.002. The calculation is also available interactively in the browser.