Specific orbital energy calculator
A low Earth orbit with a 6778 km semi-major axis has a specific energy of roughly minus twenty-nine million joules per kilogram, so the satellite is bound. Enter μ and the semi-major axis to get the energy per kilogram and the orbit type. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Gravitational parameter
- 398,600,441,800,000
- Semi major axis
- 6,778,000
- Specific orbital energy
- -29,403,986.5595 J/kg
- Orbit class
- elliptical
Specific orbital energy is the mechanical energy per unit mass that classifies a two-body trajectory as elliptical, parabolic, or hyperbolic under Newtonian gravity. For any Keplerian conic the conserved quantity is epsilon equals minus the standard gravitational parameter mu over twice the semi-major axis a.
What specific orbital energy means and how ε = −μ/(2a) arises
In the two-body problem the total mechanical energy of an orbiting mass m is conserved. Dividing by m yields the specific orbital energy epsilon, which no longer depends on the satellite mass and depends only on the central body through the standard gravitational parameter mu equals G times M and on the size of the conic through the semi-major axis a. The vis-viva equation relates speed and radius, and integrating the energy budget shows that for any closed or open Keplerian path epsilon equals minus mu over two a.
When a is positive the orbit is elliptical (including circular as the eccentricity-zero special case) and epsilon is negative: the body is gravitationally bound. When a is negative the path is hyperbolic and epsilon is positive: excess energy remains at infinity. The parabolic limit is a tending to infinity with epsilon tending to zero from below on the bound side.
Zero semi-major axis is not a physical conic parameter for this energy formula; the denominator vanishes and the calculator rejects that input rather than inventing a number. Keep units consistent: with mu in cubic meters per second squared and a in meters, epsilon is in joules per kilogram, equivalently square meters per second squared.
Where specific orbital energy proves useful
Classrooms and tutoring platforms use a specific orbital energy calculator so learners can check epsilon for arbitrary mu and a without sign errors when rearranging the energy integral by hand. Exam-prep generators need a deterministic oracle that marks zero semi-major axis as invalid rather than silently returning infinity or NaN that a buggy grader might treat as success.
Early mission concept tools compare binding energy per kilogram across candidate semi-major axes—low Earth orbit, geostationary radius near forty-two thousand kilometers, or interplanetary hyperbolic escapes—without installing Orekit, GMAT, or a local MATLAB toolbox. Distinguish this capability from total orbital energy calculators that multiply by satellite mass or from circular-orbit potential tools that take radius and masses separately: here the search intent is the mass-specific Keplerian energy from mu and a alone.
FAQ
What happens if the semi-major axis is zero?
Zero semi-major axis is rejected because ε = −μ/(2a) divides by a and is undefined for a vanishing axis.
What units should I use for μ and a?
Any consistent pair works. The calculator does not convert units for you.
Is the specific orbital energy calculator free?
The browser tool runs free on your device.
What does a negative specific orbital energy mean?
It means the orbit is bound: an ellipse, or a circle as a special case. A positive value means a hyperbolic path that keeps spare energy far away, and zero is the parabolic limit between the two.
Why does the satellite's mass not matter?
Because the energy is given per kilogram of the orbiting body. Dividing by its mass leaves a value that depends only on the central body, through μ, and on the size of the orbit.
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/mech/specific-orbital-energy \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"gravitational_parameter":398600441800000,"semi_major_axis":6778000}'const res = await fetch("https://api.kit.forhosting.com/mech/specific-orbital-energy", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"gravitational_parameter": 398600441800000,
"semi_major_axis": 6778000
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/specific-orbital-energy",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"gravitational_parameter": 398600441800000,
"semi_major_axis": 6778000
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/specific-orbital-energy", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"gravitational_parameter":398600441800000,"semi_major_axis":6778000}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"gravitational_parameter":398600441800000,"semi_major_axis":6778000}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/specific-orbital-energy", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"gravitational_parameter": 398600441800000,
"semi_major_axis": 6778000
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.specific_orbital_energy",
"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
Students, mission-design sketches, and homework checkers need that single number without standing up a full astrodynamics stack: given mu and a, this specific orbital energy calculator returns epsilon in joules per kilogram when SI units are used, labels the orbit class from the sign of a, and fails cleanly with invalid_input when the semi-major axis is zero because the formula divides by a. No network, no random seed, and no wall clock—identical inputs always produce identical JSON.
What specific orbital energy means and how ε = −μ/(2a) arises
Send gravitational_parameter as a finite strictly positive number—for example Earth mu near 3.986 times ten to the fourteen cubic meters per second squared—and semi_major_axis as a finite non-zero length in the same unit system. Aliases mu, mu_grav, and GM are accepted for the parameter; a, sma, and semi_major for the axis. The response echoes the cleaned inputs, reports specific_orbital_energy, sets unit to J/kg, sets orbit_class to elliptical or hyperbolic from the sign of a, and includes the formula string epsilon equals minus mu over two a so documentation and UI labels stay aligned with the physics.
Input contract, response shape, and zero semi-major axis errors
Negative semi-major axis is allowed and yields a positive energy for hyperbolic trajectories. Example: mu equals 3.986004418 times ten to the fourteen and a equals 6.778 times ten to the six meters (a low-Earth circular-scale axis) yields a negative specific energy of magnitude mu over two a, confirming a bound orbit. Pair the free in-browser specific orbital energy calculator with your own conversion of altitude to semi-major axis when the source data only quotes height above a planetary surface—you must add the body radius first. POST a JSON object with two required fields. gravitational_parameter must be a finite number strictly greater than zero; missing, empty, non-numeric, zero, or negative values fail with invalid_input naming the field. semi_major_axis must be a finite non-zero number; missing or non-finite values fail the same way, and the explicit product rule is that a zero semi-major axis is invalid_input because specific orbital energy divides by two a. On success the body includes gravitational_parameter and semi_major_axis rounded to twelve significant digits for stable goldens, specific_orbital_energy with the same rounding, unit equal to the string J/kg, orbit_class equal to elliptical when a is positive or hyperbolic when a is negative, method equal to specific_orbital_energy, and formula equal to the compact identity string. Example: the same mu with a negative a of similar magnitude yields a positive energy and orbit_class hyperbolic. Validation failures are not billed; successful API items use the published prepaid base price of $0.002.
Integration use cases
Homework and exam verification
Students and graders compute ε = −μ/(2a) for given gravitational parameter and semi-major axis, confirm bound versus hyperbolic sign, and get a clear invalid_input error when a is zero by mistake.
Tutoring and curriculum generators
Lesson builders produce worked specific-energy examples with structured JSON for flashcards, step sheets, and auto-graded orbital-mechanics drills without embedding a full astrodynamics library.
Early mission concept sketches
Lightweight pipelines that need a stable ε check from μ and a call a prepaid endpoint instead of reimplementing −μ/(2a) in every language for LEO, GEO, and escape comparisons.
Technical questions
How do I calculate specific orbital energy?
Send JSON with gravitational_parameter (μ, positive) and semi_major_axis (a, non-zero). The response includes specific_orbital_energy equal to −μ/(2a), unit J/kg for SI inputs, and orbit_class from the sign of a.
Can semi-major axis be negative?
Yes. Negative a describes a hyperbolic trajectory; the calculator returns positive specific orbital energy and orbit_class hyperbolic.