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Hohmann departure speed calculator

Get the speed that starts a Hohmann transfer: leaving a 6678 km orbit for one at 42164 km around Earth, with μ of 398600.4418 km³/s², takes 10.1516085074 km/s at the start of the ellipse. Free in your browser.

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Worked example

If you enter

Inner radius: 6,678
Outer radius: 42,164
Mu: 398,600.4418

You get

Semi major axis: 24,421
Departure speed: 10.1516085074

A Hohmann transfer is the classic two-impulse ellipse that carries a spacecraft from a circular orbit at one radius to a coplanar circular orbit at another with the least ideal propellant under two-body dynamics. Units of radius and μ must be consistent so the returned speed is length per time.

What the departure speed of a Hohmann transfer means

A Hohmann transfer is the classic two-burn ellipse that carries a spacecraft from a circular orbit at one radius to a circular orbit at a larger radius in the same plane, using the least propellant in the ideal two-body picture. The ellipse touches the inner orbit at its nearest point and the outer orbit at its farthest point. The departure speed is how fast the spacecraft must travel at that nearest point to coast all the way out to the higher orbit.

How it is calculated and what to watch for

The semi-major axis of the transfer ellipse is the average of the two radii, and the vis-viva relation then gives the speed at the inner radius. With 6678 km, 42164 km and Earth's μ of 398600.4418 km³/s², the ellipse has a semi-major axis of 24421 km and the departure speed is 10.1516085074 km/s. Keep units consistent, because kilometres with cubic kilometres per second squared give kilometres per second. Enter distances from the planet's centre, not altitudes, and make the outer radius at least as large as the inner one.

FAQ

What units should I use for radii and μ?

Any consistent system. The calculator does not convert between unit systems for you.

Is the transfer orbit departure speed calculator free?

The browser tool runs free on your device.

Can I enter altitudes instead of orbit radii?

Not directly. Both radii are measured from the centre of the central body, so add the planet's radius to each altitude first. Using raw altitudes would give a speed for a very different, much smaller ellipse.

What if the outer radius is smaller than the inner one?

The calculation is refused. The inner radius must be the nearest point of the transfer ellipse, so the outer radius has to be at least as large. Swap the two values if you entered them the wrong way round.

Does it only work for Earth?

No. Nothing about Earth is built in: you supply the gravitational parameter μ of whatever body you are orbiting, along with radii in a matching unit, and the speed comes out in that same length per second.

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

POSThttps://api.kit.forhosting.com/astro/transfer-departure-speed

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/transfer-departure-speed \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"inner_radius":6678,"outer_radius":42164,"mu":398600.4418}'

const res = await fetch("https://api.kit.forhosting.com/astro/transfer-departure-speed", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "inner_radius": 6678,
    "outer_radius": 42164,
    "mu": 398600.4418
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/astro/transfer-departure-speed",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "inner_radius": 6678,
    "outer_radius": 42164,
    "mu": 398600.4418
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/astro/transfer-departure-speed", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"inner_radius":6678,"outer_radius":42164,"mu":398600.4418}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"inner_radius":6678,"outer_radius":42164,"mu":398600.4418}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/astro/transfer-departure-speed", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)

Example request

{
  "inner_radius": 6678,
  "outer_radius": 42164,
  "mu": 398600.4418
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "astro.transfer_departure_speed",
  "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

Per request$0.002

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

What Hohmann periapsis speed means and how to choose r1, r2, and μ

Vis-viva then gives the speed anywhere on that ellipse as the square root of μ times two over r minus one over a. All three must be finite and strictly positive. Outer radius must be at least inner radius so that r1 truly is periapsis of the transfer ellipse. The calculator does not convert altitude above a planetary surface into center-to-body radius; if your source quotes height above Earth, add the planetary radius first. Send the required field inner_radius as a JSON number or numeric string for r1; aliases r1, r_inner, and periapsis_radius are accepted. Send mu for the gravitational parameter; aliases gravitational_parameter and GM are accepted. Either radius non-positive fails with invalid_input by product contract. Identical inputs always produce identical JSON because the path is pure floating-point arithmetic with no network, no random seed, and no wall clock.

Reading the response fields and keeping units consistent

The engine does not assume Earth constants; it multiplies pure numbers. Units are your responsibility and must be consistent: if radii are in kilometers and μ is in cubic kilometers per square second, departure_speed is in kilometers per second; SI meters with m³/s² yield meters per second. Missing fields, non-positive radii, non-positive μ, outer radius smaller than inner radius, and non-finite numbers all fail with invalid_input and are not charged after the engine rejects the request. Use departure_speed alone when a dashboard only needs the absolute speed after the first burn; keep semi_major_axis when documenting the transfer ellipse for students; keep formula and method when logging which algorithm produced a mission sketch. Pair this endpoint with circular orbital velocity tools when the next question is the first Δv as transfer speed minus circular speed at r1, remembering that this capability returns absolute transfer speed, not the burn magnitude itself.

API usage, pricing, classroom workflows, and practical limits

Call the free in-browser transfer orbit departure speed calculator for homework checks, lecture demos, and quick concept verification without installing Orekit, GMAT, or a local MATLAB toolbox. Extremely large or tiny magnitudes are bounded for isolate safety; stay within the published numeric range for reliable results. Send a JSON body with required positive inner_radius, required positive outer_radius at least as large as the inner radius, and required positive mu. Use the prepaid API when a tutoring platform, coding-assessment sandbox, curriculum generator, or early mission-design pipeline needs a stable contract under KIT balance at $0.002 per successful item; validation failures never consume balance after rejection. Authors shipping SDKs can map departure_speed into tables that compare transfer periapsis speeds across destination radii while holding r1 and μ fixed, teaching how raising the target orbit increases the speed needed at departure on the ellipse—a companion lesson to period, energy, and circular-velocity formulas when building multi-step orbital-mechanics lesson flows.

Integration use cases

Homework and exam verification

Students and graders compute Hohmann transfer periapsis speed from r1, r2, and μ, compare against textbook answers, and get a clear invalid_input error when either radius is non-positive by mistake.

Tutoring and curriculum generators

Lesson builders produce worked Hohmann departure-speed examples with structured JSON for flashcards, step sheets, and auto-graded drills without embedding a full astrodynamics library.

Early mission concept sketches

Lightweight pipelines that need a stable transfer periapsis speed from radii and μ call a prepaid endpoint instead of reimplementing the vis-viva periapsis formula in every language.

Technical questions

How do I calculate Hohmann transfer departure speed from r1, r2, and μ?

Send JSON with positive inner_radius (r1), positive outer_radius (r2 ≥ r1), and positive mu. The response includes departure_speed equal to sqrt(2 μ r2 / (r1 (r1 + r2))), plus semi_major_axis, formula, and method labels.

What happens if either radius is non-positive?

The request fails with invalid_input. Both inner_radius and outer_radius must be strictly positive; non-positive values are rejected by product contract and are not charged.

Is departure_speed the first Δv or the absolute speed on the transfer orbit?

It is the absolute speed at periapsis of the transfer ellipse after the first burn. Circular-orbit speed at r1 is lower; the first Δv is departure_speed minus that circular speed.

Read the full KIT documentation →

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