Binary Roche lobe radius calculator
Two equal-mass stars 10 units apart each have a Roche lobe radius of 3.78920518380456 units, 0.378920518380456 of the separation. Enter any separation and mass ratio to check whether a star may fill its lobe. Free in your browser.
- Free
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- No sign-up
- Your data stays on your device
Run — free
Worked example
- Separation
- 10
- Mass ratio
- 1
- Roche lobe fraction
- 0.3789205184
- Roche lobe radius
- 3.7892051838
- Mass ratio definition
- q = M_star / M_companion
- Radius unit
- same as separation
This binary Roche lobe radius calculator applies the Eggleton approximation to estimate the volume-equivalent radius of the gravitational region surrounding one star in a binary system. Enter the centre-to-centre separation and the mass ratio defined as the target star’s mass divided by its companion’s mass. The result includes the Roche lobe as a fraction of separation and as a radius in the same length unit you supplied. It is useful for checking whether a star may fill its lobe, comparing binary models, and producing reproducible classroom or simulation calculations.
Define the mass ratio for the star you are studying
The Eggleton approximation is directional: the mass ratio must describe the particular star whose effective Roche lobe radius you want. In this calculator, q equals the target star mass divided by the companion mass. If you want the companion’s lobe instead, invert the ratio and run the calculation again.
For example, equal masses give q equal to one, while a target star twice as massive as its companion gives q equal to two. This definition matters because reversing q changes the answer even though the physical binary is unchanged; you have merely selected the other lobe. Mass ratio is dimensionless and must be finite and strictly greater than zero.
A value of zero or below has no physical meaning in this approximation and is returned as an invalid-input error rather than being allowed to generate a misleading radius. Use component masses derived under the same convention and in the same mass unit before forming their ratio. Solar masses, kilograms, and other units all cancel correctly when both component masses use one unit.
The calculator needs only the resulting ratio, not the individual masses, so it does not infer which component is the primary or make assumptions about stellar type, luminosity, or evolutionary state.
Apply the Eggleton approximation to the binary separation
The calculator evaluates R L divided by a as zero point four nine times q to the two-thirds power, divided by zero point six times q to the two-thirds power plus the natural logarithm of one plus q to the one-third power. Here a is the centre-to-centre binary separation and R L is the radius of a sphere having approximately the same volume as the target star’s Roche lobe. The formula is deterministic and requires only a cube root, a logarithm, multiplication, and division.
Enter a finite separation greater than zero in any convenient length unit. If separation is in solar radii, the reported Roche lobe radius is in solar radii; if it is in kilometres or astronomical units, the result retains that unit. For q equal to one, the fraction is about zero point three seven nine, a useful sanity check for equal-mass binaries.
The approximation describes the equipotential geometry of a circular, synchronously rotating binary and is widely used because it remains accurate across a broad range of positive mass ratios. It computes an effective volume-equivalent radius, not the distance to every point on the non-spherical lobe surface.
Interpret the radius in stellar and orbital work
Compare the calculated effective Roche lobe radius with the radius of the same target star. A stellar radius well below the lobe radius suggests a detached component under the assumptions of the approximation. A radius close to the lobe radius indicates possible Roche-lobe filling, while a larger stellar radius signals overflow and possible mass transfer, although a full physical interpretation also depends on orbital evolution, stellar structure, eccentricity, spin, and how the input values were measured.
The calculator does not predict a mass-transfer rate, stability, common-envelope evolution, or the shape of an accretion stream. It gives a transparent geometric scale for preliminary analysis, validation, teaching, and software tests. For eccentric systems, deciding which instantaneous separation to use is a modelling choice; periastron may be relevant for a quick limiting comparison, but the standard Eggleton expression assumes a circular synchronous configuration and should not be presented as a time-dependent eccentric solution.
Run the tool twice with reciprocal mass ratios when you need both component lobes, always keeping the same separation. Store the fraction alongside the dimensional radius so later unit conversions can be checked without repeating the entire calculation.
What you can do with it
Check Roche-lobe filling
Compare a measured or modelled stellar radius with its effective lobe radius to screen a binary for possible overflow.
Teach close-binary geometry
Demonstrate how selecting a component and changing its mass ratio changes the volume-equivalent Roche lobe.
FAQ
How is mass ratio defined?
Use q = M_star / M_companion, where M_star is the mass of the star whose Roche lobe radius you want. Invert q to calculate the companion’s lobe.
Which unit does the radius use?
The Roche lobe radius uses the same length unit as separation. The reported Roche lobe fraction is dimensionless.
Is a Roche lobe spherical?
No. The result is an effective radius of a sphere with approximately the same volume as the non-spherical Roche lobe.
How do I get the companion star's Roche lobe?
Invert the mass ratio and run the calculation again. The ratio always means the target star's mass divided by its companion's, so swapping it selects the other lobe of the same binary.
What is the Roche lobe radius useful for?
It tells you whether a star is close to filling the region its own gravity controls. Comparing the star's radius with the lobe radius helps you check binary models and build reproducible classroom or simulation examples.
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/roche-radius-binary \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"separation":10,"mass_ratio":1}'const res = await fetch("https://api.kit.forhosting.com/astro/roche-radius-binary", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"separation": 10,
"mass_ratio": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/astro/roche-radius-binary",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"separation": 10,
"mass_ratio": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/astro/roche-radius-binary", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"separation":10,"mass_ratio":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"separation":10,"mass_ratio":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/astro/roche-radius-binary", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"separation": 10,
"mass_ratio": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "astro.roche_radius_binary",
"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
Apply the Eggleton approximation to the binary separation
The response also reports roche_lobe_fraction, which is dimensionless and can be multiplied by any consistently expressed separation.
Interpret the radius in stellar and orbital work
Browser use is free, while each successful API item costs $0.002; invalid mass ratios are rejected before a scientific result is returned.
Integration use cases
Build binary-star simulations
Generate a deterministic Roche radius and dimensionless fraction for initial conditions, regression fixtures, or population studies.
Technical questions
What happens when the mass ratio is zero or negative?
The request fails with invalid_input because the Eggleton approximation requires a strictly positive mass ratio.
What does an API calculation cost?
A successful API item costs $0.002. You can also run the same deterministic calculation free in the browser.