Orbit total energy calculator
Get the total energy of an orbit from its size alone: a 1000 kg craft on an orbit with a 7,000,000 m semi-major axis has -28,471,460,128.57143 joules, so it stays bound. A positive result means an escape path. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Mass
- 1,000
- Gravitational parameter
- 398,600,441,800,000
- Semi major axis
- 7,000,000
- Total mechanical energy j
- -28,471,460,128.57143
The orbit total mechanical energy calculator applies the standard two-body energy relation to an orbiting body's mass, the central body's gravitational parameter, and the orbit's semi-major axis. It returns a signed result in joules, making the physical character of the trajectory immediately visible: bound elliptical orbits have negative total energy, while hyperbolic escape trajectories have positive total energy. The calculation is deterministic, accepts no hidden defaults, and rejects a zero semi-major axis because the defining expression would divide by zero.
Understand the energy equation and its sign
Total mechanical orbital energy combines the orbiting body's kinetic energy and gravitational potential energy. In the ideal Newtonian two-body model, those quantities can be reduced to one compact expression: E equals negative mu times m divided by two times a. Here, m is the mass of the orbiting body, mu is the standard gravitational parameter of the central body, and a is the semi-major axis of the conic trajectory.
The result does not require the body's instantaneous position or speed because semi-major axis already identifies the orbit's specific energy. For an elliptical orbit, a is positive and the returned total mechanical energy is negative. That negative sign indicates a bound system: energy must be added before the body can escape to infinity.
A larger positive semi-major axis produces an energy value closer to zero, describing a more weakly bound ellipse. For a hyperbolic trajectory, the conventional semi-major axis is negative, so the same equation returns positive energy and correctly identifies an unbound passage. This capability deliberately preserves that sign convention rather than rejecting every negative axis.
A semi-major axis of exactly zero is rejected, since division by zero is undefined and a parabolic trajectory is represented by zero specific orbital energy rather than by inserting zero into this semi-major-axis formula.
Enter compatible quantities and interpret joules
The familiar symbol for gravitational parameter is mu, and it equals the gravitational constant multiplied by the central body's mass. Use a published mu value when possible because it is generally measured more accurately than the two factors are known separately. The mass field refers only to the orbiting body under the usual approximation that its mass is negligible compared with the central body.
Mass and gravitational parameter must be strictly positive. Semi-major axis may be positive for an ellipse or negative for a hyperbola, but it cannot be zero. Convert the semi-major axis to metres or convert mu into the matching kilometre-based unit system before calculating.
The calculator does not infer units, add a planetary radius to an altitude, or substitute a default gravitational parameter, because each of those choices could silently change the physical problem.
Use the result in analysis and automation
Use the browser calculator for a quick engineering check, classroom exercise, or comparison between candidate orbits. Since the implementation uses only a fixed algebraic expression, identical inputs always produce identical outputs; there is no network lookup, random value, current date, or mutable state involved. When comparing two elliptical orbits for the same spacecraft and central body, the orbit with the smaller semi-major axis has the more negative total mechanical energy and therefore requires energy to move outward.
The difference between two returned energies can serve as an idealized first check on a maneuver's required change in mechanical energy, although it is not itself a complete propulsion calculation and does not provide delta-v, burn direction, losses, or transfer timing. For escape studies, a positive result produced from a negative hyperbolic semi-major axis represents excess mechanical energy in this sign convention. Real mission design may also require perturbations, finite-body corrections, relativistic effects, or multi-body dynamics; this capability intentionally answers the standard deterministic two-body question only.
What you can do with it
Compare bound Earth orbits
Calculate and compare the negative total mechanical energy of the same spacecraft at different elliptical semi-major axes.
Check an escape trajectory
Use a negative hyperbolic semi-major axis to confirm that an unbound trajectory has positive total mechanical energy.
Automate physics exercises
Generate deterministic answer keys from explicitly supplied masses, gravitational parameters, and semi-major axes.
FAQ
Why is the energy of an elliptical orbit negative?
Negative total mechanical energy indicates a bound orbit. Energy must be supplied to reach the zero-energy escape boundary.
Can the semi-major axis be negative?
Yes. A negative semi-major axis follows the standard hyperbolic convention and produces positive total mechanical energy.
What units should I use?
Use kilograms, cubic metres per square second, and metres to receive joules. Other coherent unit systems work mathematically, but the returned field is specifically labelled joules, so SI inputs are expected.
What does a positive orbital energy mean?
The body is on a hyperbolic escape path rather than a closed orbit. It has enough energy to leave the central body for good, while every bound ellipse gives a negative total.
Do I need the body's position or speed?
No. The semi-major axis already fixes the total energy of the orbit, so the mass of the body, the central body's gravitational parameter and the semi-major axis are all you need.
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/orbital-energy-total \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"mass":1000,"gravitational_parameter":398600441800000,"semi_major_axis":7000000}'const res = await fetch("https://api.kit.forhosting.com/astro/orbital-energy-total", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"mass": 1000,
"gravitational_parameter": 398600441800000,
"semi_major_axis": 7000000
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/astro/orbital-energy-total",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"mass": 1000,
"gravitational_parameter": 398600441800000,
"semi_major_axis": 7000000
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/astro/orbital-energy-total", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"mass":1000,"gravitational_parameter":398600441800000,"semi_major_axis":7000000}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"mass":1000,"gravitational_parameter":398600441800000,"semi_major_axis":7000000}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/astro/orbital-energy-total", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"mass": 1000,
"gravitational_parameter": 398600441800000,
"semi_major_axis": 7000000
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "astro.orbital_energy_total",
"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
It runs free in the browser or costs $0.002 for each API item.
Enter compatible quantities and interpret joules
Supply mass in kilograms, gravitational_parameter in cubic metres per square second, and semi_major_axis in metres to obtain total_mechanical_energy_j in joules. All three inputs must be finite JSON numbers. Unit consistency matters: if values from kilometres are mixed with an SI gravitational parameter expressed in metres, the numerical result will be wrong by powers of one thousand even though the request remains syntactically valid. The response echoes the accepted inputs, provides the signed joule result, and includes the formula label so stored results remain understandable.
Use the result in analysis and automation
Each successful API item costs $0.002. A parabolic case should be treated analytically as zero energy rather than approximated with a zero semi-major axis, which this endpoint correctly reports as invalid input.
Technical questions
What formula does the calculator use?
It uses E = -mu*m/(2*a), where m is orbiting-body mass, mu is the central body's gravitational parameter, and a is semi-major axis.
What happens when the semi-major axis is zero?
The request returns invalid_input because the equation divides by twice the semi-major axis and is undefined at zero.