Characteristic energy C3 calculator
A hyperbolic excess speed of 3 km/s means a characteristic energy C3 of 9 km²/s², since C3 is that speed squared. Enter the excess speed in your unit and get the departure energy per unit mass. Free in your browser.
- Free
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- Your data stays on your device
Run — free
Worked example
- V infinity
- 3
- C3
- 9
- Characteristic energy
- 9
- Equation
- C3 = (3)² = 9
Characteristic energy, written C3 in launch vehicle and interplanetary trajectory literature, is the specific energy (energy per unit mass) of a hyperbolic departure relative to the central body. This characteristic energy C3 calculator takes the hyperbolic excess velocity v∞ and returns C3 equal to v∞ squared in matching energy-per-unit-mass units—for example square kilometres per second squared when v∞ is given in kilometres per second. Students use the same identity to check homework that converts between excess speed and energy.
What characteristic energy C3 means for launch and escape
In the two-body problem a spacecraft on a hyperbolic trajectory relative to a planet has positive specific energy. The excess kinetic energy per unit mass at infinity is exactly one half v∞ squared; convention in launch design, however, quotes the full square C3 equals v∞ squared as the characteristic energy parameter that labels the departure energy state.
Positive C3 means a hyperbolic escape; C3 equals zero is the parabolic boundary (escape with zero residual speed); elliptical parking orbits have negative specific energy and are not described by a real v∞ of this form. Interplanetary porkchop plots colour contours of C3 at Earth departure and at target arrival; designers pick launch dates where both C3 values stay within vehicle and arrival-delta-v budgets. The identity is unit-consistent: if you measure v∞ in kilometres per second, C3 is in km²/s²; if you use metres per second, C3 is in m²/s².
This tool does not convert units for you—it squares the number you supply—so keep one consistent system end to end. Sign of the input speed is squared away: a negative numeric v∞ yields the same C3 as its absolute value, which matches the physical reading that only speed magnitude enters the energy.
Practical tips, worked checks, and related searches
A common textbook check is v∞ equal to three kilometres per second, which yields C3 equal to nine km²/s²; five km/s yields twenty-five km²/s². Earth escape from a circular low orbit first costs the circular-to-escape delta-v, then additional energy for residual v∞; this calculator only performs the C3 equals v∞ squared step once you already know the excess speed. Prefer SI or kilometre-second units consistently; mixing kilometres per second for input with an expected m²/s² result will off-by-one-million in energy units.
This landing targets people searching for a characteristic energy C3 calculator, C3 from hyperbolic excess velocity, or launch energy per unit mass online. It is not an orbital period tool, not a Tsiolkovsky rocket-equation solver, not a full Lambert or porkchop plot engine, and not a circular-orbit energy calculator (those use negative specific energy formulas).
Prefer this capability when you already have v∞ and only need C3; if you have semi-major axis of a hyperbola, first convert to v∞ using the standard two-body relations for your central body, then call here.
FAQ
What is the characteristic energy C3 formula?
C3 equals the square of the hyperbolic excess velocity: C3 = v∞². The result is energy per unit mass in units consistent with the square of your input speed unit (for example km²/s² when v∞ is in km/s).
What units should I use for v∞?
Any consistent speed unit. Common practice is kilometres per second, so C3 comes out in km²/s². Metres per second yield m²/s². The tool does not convert units; it only squares the number you provide.
What happens if the excess velocity is missing or non-numeric?
C3 is only computed for a finite numeric hyperbolic excess velocity; text, empty fields, NaN, and infinity are rejected.
Does a negative v∞ give a negative C3?
No. Speed is squared, so a negative numeric input produces the same positive C3 as its absolute value. Physically only the magnitude of hyperbolic excess speed enters the energy.
Is the characteristic energy C3 calculator free?
The browser tool is free and runs locally.
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/characteristic-energy-c3 \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"v_infinity":3}'const res = await fetch("https://api.kit.forhosting.com/astro/characteristic-energy-c3", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"v_infinity": 3
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/astro/characteristic-energy-c3",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"v_infinity": 3
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/astro/characteristic-energy-c3", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"v_infinity":3}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"v_infinity":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/astro/characteristic-energy-c3", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"v_infinity": 3
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "astro.characteristic_energy_c3",
"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
Mission designers use C3 to size upper stages, compare launch windows, and quote payload capability curves published by vehicle providers as mass versus C3. Non-numeric or missing excess velocity is rejected with invalid_input rather than a silent NaN.
What characteristic energy C3 means for launch and escape
Far from the body, residual speed settles to the hyperbolic excess velocity v∞ (also written v_infinity). Launch vehicle performance charts often list delivered payload mass as a function of C3 for a given launch site and upper-stage stack: a higher required C3 leaves less mass on the escape asymptote.
How to call the API and interpret the JSON result
The value is the hyperbolic excess speed in whatever speed unit you have chosen for the rest of your analysis. Send a JSON object with one required finite number: v_infinity (aliases v_inf, vinf, excess_velocity, v_excess). On success the response returns v_infinity rounded for stable display, c3 and characteristic_energy both set to the same rounded square, formula equal to the string C3 equals v∞ squared, equation as a human-readable substitution line, units_note explaining that C3 inherits energy-per-unit-mass units from the square of the input speed, and method set to characteristic_energy_c3. Failed validation—missing field, empty string, non-numeric text, NaN, or infinity—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item. Prefer JSON numbers when your client can emit them; numeric strings that match a strict finite pattern are also accepted for convenience when forms post text fields. Extremely large speeds that would overflow double precision when squared are rejected with a clear finite-range message rather than returning Infinity in the payload.
Practical tips, worked checks, and related searches
Do not feed orbital speed, escape speed, or launch azimuth into v_infinity unless you have already converted those quantities into true hyperbolic excess. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts and student trial-and-error cheap.
Integration use cases
Homework checks for C3 from excess speed
Students enter textbook hyperbolic excess velocities and compare hand-written v∞² against deterministic JSON before submitting orbital mechanics problem sets.
Unit tests for mission-design libraries
Generate known-good C3 values and deliberate non-numeric rejections as expected fixtures for coding katas and regression suites.
Quick launch-window energy labels
Analysts convert a planned departure v∞ into C3 to read payload-versus-C3 curves from vehicle manuals without a full trajectory stack.