Aphelion distance from eccentricity calculator
An orbit with a semi-major axis of 10 and an eccentricity of 0.2 reaches an aphelion of 12, in the same unit. Enter both orbital elements to get the farthest distance from the focus for homework or orbit comparisons. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Semi major axis
- 149,597,870.7
- Eccentricity
- 0.0167
- Aphelion distance
- 152,096,155.14068997
This aphelion distance calculator finds the farthest point of an elliptical orbit from its focus. Enter the semi-major axis and orbital eccentricity, and it applies Q = a(1 + e) to return the aphelion distance in the same unit used for the semi-major axis. The calculation is deterministic and needs no ephemeris, object mass, or gravitational parameter. It is useful for astronomy exercises, preliminary orbit comparisons, educational tools, and software pipelines that need a small, dependable orbital calculation with clear validation for missing or non-numeric input.
Understand the aphelion formula
Aphelion is the point where an object in an elliptical heliocentric orbit is farthest from the Sun, which occupies one focus of the ellipse. More generally, the same geometry describes the farthest orbital radius from a focus, although the term apoapsis is used when the central body is not specifically the Sun. The formula is Q = a(1 + e), where Q is aphelion distance, a is the semi-major axis, and e is eccentricity.
The semi-major axis sets the overall size of the ellipse, while eccentricity describes how stretched it is. As e increases toward one, the ellipse becomes more elongated and the farthest distance grows toward twice the semi-major axis. This calculator implements that formula directly.
It does not propagate an orbit through time, account for perturbations, or infer orbital elements from observations. It answers one focused geometric question using the two elements that determine the farthest radius of a bound Keplerian ellipse.
Enter values and interpret the result
Units are not converted or assumed. If the semi-major axis is supplied in astronomical units, the aphelion distance is in astronomical units; if it is supplied in kilometers, the answer is in kilometers. For example, a semi-major axis of 10 and eccentricity of 0.2 produce an aphelion distance of 12.
Keep more precision in the inputs when the result feeds another calculation, and apply display rounding only at the edge of your application according to its scientific or educational needs.
Use the calculator responsibly in orbital work
The result belongs to the ideal two-body Keplerian model represented by the supplied orbital elements. Real trajectories can change because of gravitational perturbations, atmospheric drag, radiation pressure, maneuvers, or updated observations. For operational navigation, collision assessment, spacecraft commanding, or an epoch-specific position, use current elements and a validated orbit propagator instead of treating this single formula as a complete flight-dynamics solution.
Also distinguish distance from the focus from altitude above a body's surface: this capability returns a focal radius in the same unit as a, not surface clearance. Because the implementation performs only validation and arithmetic, repeated valid inputs return the same structured result and can be pinned safely in tests or reproducible data-processing workflows.
What you can do with it
Astronomy coursework
Check the farthest solar distance for an elliptical orbit from orbital elements supplied in a lesson or exercise.
Orbit catalogue enrichment
Add a consistent aphelion-distance field to records that already contain semi-major axis and eccentricity.
Software validation
Use deterministic structured output in tests for forms, SDKs, and small orbital-mechanics pipelines.
FAQ
What formula does the calculator use?
It uses Q = a(1 + e), where a is the semi-major axis and e is orbital eccentricity.
What units does the result use?
The result uses the same distance unit as the semi-major axis. No unit conversion is performed.
Can eccentricity be zero?
Yes. An eccentricity of zero represents a circle, so the aphelion distance equals the semi-major axis.
Why must eccentricity be less than one?
This capability calculates aphelion for a bound elliptical orbit. Parabolic and hyperbolic trajectories do not have a finite aphelion.
Is the aphelion distance the same as altitude above a surface?
No. The result is the distance from the focus of the ellipse, which is the center of the Sun or central body, not the clearance above its surface. Subtract the body's radius yourself if you need an altitude.
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/aphelion-distance \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"semi_major_axis":149597870.7,"eccentricity":0.0167}'const res = await fetch("https://api.kit.forhosting.com/astro/aphelion-distance", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"semi_major_axis": 149597870.7,
"eccentricity": 0.0167
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/astro/aphelion-distance",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"semi_major_axis": 149597870.7,
"eccentricity": 0.0167
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/astro/aphelion-distance", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"semi_major_axis":149597870.7,"eccentricity":0.0167}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"semi_major_axis":149597870.7,"eccentricity":0.0167}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/astro/aphelion-distance", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"semi_major_axis": 149597870.7,
"eccentricity": 0.0167
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "astro.aphelion_distance",
"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
Understand the aphelion formula
A circular orbit has e = 0, so Q equals a.
Enter values and interpret the result
Provide semi_major_axis as a positive finite number and eccentricity as a finite number from zero inclusive to one exclusive. The shorter aliases a and e are also accepted. Numeric strings are converted deliberately, so values received from form fields can be used without a separate parsing step, but blank strings, words, infinity, arrays, and other non-numeric values produce an invalid_input error. The response echoes both normalized inputs, reports aphelion_distance, and includes the formula label for auditability.
Use the calculator responsibly in orbital work
The browser version is convenient for interactive checks, while automated API use costs $0.002 per request.
Technical questions
What does the API cost?
The browser calculator is free to run. Automated API use costs $0.002 per request.