Hohmann Transfer Calculator

The Hohmann transfer is a two-impulse orbital maneuver that uses an elliptical orbit to transit between two circular orbits of different radii. The calculation relies on fundamental laws of gravitation and conservation of angular momentum. First, the gravitational parameter ($\mu$) of the primary body is computed using its mass and the gravitational constant. Next, the initial and destination orbital radii are determined from the planetary radius and respective altitudes. The semi-major axis ($a$) of the transfer ellipse is calculated as the average of the initial and final radii: $a = (r_1 + r_2) / 2$. Velocity updates ($\Delta v_1$ and $\Delta v_2$) are derived by finding the difference between circular orbital velocities and the velocities along the periapsis and apoapsis of the transfer ellipse. Finally, the total delta-v is used alongside the rocket's specific impulse to determine the propellant mass fraction via the Tsiolkovsky rocket equation.

Let us walk through a practical scenario for a satellite transferring from a lower Earth orbit to a higher circular orbit around Earth. Assume a primary mass (Earth) of $5.972 \times 10^{24}\text{ kg}$, a primary radius of $6,378\text{ km}$, an initial altitude of $300\text{ km}$, and a destination altitude of $1,000\text{ km}$. First, we calculate the initial radius ($r_1 = 6378 + 300 = 6678\text{ km}$) and destination radius ($r_2 = 6378 + 1000 = 7378\text{ km}$). Using Earth's gravitational parameter, the semi-major axis of the transfer orbit is $(6678 + 7378) / 2 = 7028\text{ km}$. The first velocity change ($\Delta v_1$) needed to enter the transfer ellipse at periapsis is calculated to be approximately $0.301\text{ km/s}$. The second burn at apoapsis ($\Delta v_2$) to circularize the orbit at the higher altitude is approximately $0.275\text{ km/s}$. This results in a total delta-v requirement of $0.576\text{ km/s}$. Using a rocket with a specific impulse of $300\text{ s}$ and an initial mass of $1,000\text{ kg}$, the required propellant mass is approximately $179\text{ kg}$, and the transfer time of flight (TOF) is roughly 47 minutes.

When planning orbital transfers, always verify that your input units are consistent—typically using kilometers, kilograms, and seconds. Keep in mind that the classic Hohmann transfer assumes coplanar, circular orbits; if your target orbit has inclination or eccentricity, additional delta-v budget must be accounted for. Furthermore, real-world maneuvers often employ finite-burn corrections rather than instantaneous impulses, so use these theoretical results as a baseline for high-fidelity mission simulations.