Tool 1 · Orbital Mechanics

Hohmann Transfer Calculator

Compute the two impulsive burns and the transfer time of flight required to move between two circular coplanar orbits around a central body. This tool is intended as a simple interactive learning module for first-order orbital maneuver design.

What this tool computes

A Hohmann transfer is the classical minimum-\(\Delta v\) two-burn maneuver used to move between two circular coplanar orbits in the ideal two-body model. For a given gravitational parameter \(\mu\), an initial orbit radius \(r_1\), and a final orbit radius \(r_2\), the calculator determines:

  • Initial circular velocity \(v_{c1}\)
  • Final circular velocity \(v_{c2}\)
  • Transfer orbit semi-major axis \(a_t\)
  • Transfer speed at departure \(v_{t1}\)
  • Transfer speed at arrival \(v_{t2}\)
  • First burn \(\Delta v_1\)
  • Second burn \(\Delta v_2\)
  • Total maneuver cost \(\Delta v_{total}\)
  • Transfer time of flight \(t_{TOF}\)

Core physics

For circular orbits, the orbital speeds are:

\[ v_{c1} = \sqrt{\frac{\mu}{r_1}}, \qquad v_{c2} = \sqrt{\frac{\mu}{r_2}} \]

The transfer ellipse semi-major axis is:

\[ a_t = \frac{r_1 + r_2}{2} \]

Using the vis-viva equation, the transfer-orbit speeds at departure and arrival are:

\[ v_{t1} = \sqrt{\mu \left(\frac{2}{r_1} - \frac{1}{a_t}\right)} \] \[ v_{t2} = \sqrt{\mu \left(\frac{2}{r_2} - \frac{1}{a_t}\right)} \]

Therefore, the required burns are:

\[ \Delta v_1 = \left|v_{t1} - v_{c1}\right| \] \[ \Delta v_2 = \left|v_{c2} - v_{t2}\right| \] \[ \Delta v_{total} = \Delta v_1 + \Delta v_2 \]

The time of flight is half the transfer-orbit period:

\[ t_{TOF} = \pi \sqrt{\frac{a_t^3}{\mu}} \]

Orbit sketch

r₁ r₂ Transfer ellipse Final orbit Initial orbit Body

The initial circular orbit, transfer ellipse, and final circular orbit are shown conceptually. For an orbit-raising transfer, the first burn accelerates the spacecraft into the ellipse, and the second burn circularizes at the higher orbit.

Example validation case

A standard check is a transfer from a 300 km low Earth orbit to geostationary orbit.

\[ r_1 = 6378 + 300 = 6678\ \text{km} \] \[ r_2 = 42164\ \text{km} \] \[ \mu_{Earth} = 398600.4418\ \text{km}^3/\text{s}^2 \]

The expected results are approximately:

  • \(\Delta v_1 \approx 2.43\ \text{km/s}\)
  • \(\Delta v_2 \approx 1.47\ \text{km/s}\)
  • \(\Delta v_{total} \approx 3.90\ \text{km/s}\)
  • \(t_{TOF} \approx 5.27\ \text{hours}\)

Interactive calculator

Assumptions and limitations

This calculator assumes:

  • Impulsive burns
  • Circular initial and final orbits
  • Coplanar geometry
  • Two-body dynamics
  • No drag, no perturbations, no finite-burn losses, and no plane change

That makes it ideal for teaching, first-order mission design, and quick orbital mechanics checks.

Why this tool matters

This is one of the most important introductory calculators in orbital mechanics because it connects orbital energy, circular speed, transfer ellipses, vis-viva, and maneuver cost into one compact design problem.

In a later version, this page can be extended with:

  • Bi-elliptic transfer comparison
  • Plane change coupling
  • Departure and arrival orbit plots
  • \(\Delta v\) trade studies versus altitude
← Back to Tools