Free Hohmann Transfer Calculator

Understanding the Hohmann Transfer

The Hohmann transfer is a classic orbital maneuver that enables a spacecraft to move between two circular orbits using the minimum possible amount of propellant. An Orbital Transfer Calculator based on this principle can quickly compute the delta‑v requirements, transfer orbit geometry, and fuel needed for the journey. This Delta V Calculator is essential for mission designers seeking a fuel‑efficient orbit transfer.

To use this tool, you provide the mass and radius of the primary celestial body (e.g., the Sun for interplanetary missions, Earth for satellite transfers). The Hohmann Transfer Orbit calculator then outputs key parameters such as specific angular momentum, velocities at periapsis and apoapsis, and the two delta‑v impulses required. If you also enter your rocket engine’s specific impulse ($I_{sp}$) and initial spacecraft mass, the tool applies the Tsiolkovsky rocket equation to determine the required propellant mass.

How to Use the Calculator

1. Specify the primary body – Enter the mass and radius of the Sun, Earth, or another celestial body. You can switch mass units to “Weights of Earth” or “Weights of Sun” for convenience.
2. Define the two circular orbits – Provide the altitudes (distance from the body’s surface) for the initial and destination orbits. The calculator automatically computes the orbital radii.
3. (Optional) Rocket data – If you know your engine’s $I_{sp}$ and the spacecraft’s initial mass $m_0$, the tool will estimate the propellant mass needed. The “Further properties” sections can reveal additional details like specific angular momentum and geometric parameters.

Core Astrodynamic Concepts

Two‑Body Motion and Kepler Orbits

When a spacecraft moves under the influence of a single massive body, its path is a conic section. Trajectories are either:

• Open: hyperbola or parabola (escape trajectories).
• Closed: circle or ellipse (bound orbits).

For closed orbits, the velocity is purely tangential only at the extreme points – periapsis and apoapsis. This property is directly used by the Hohmann transfer.

Specific Angular Momentum

The specific angular momentum vector $\vec{h} = \vec{r} \times \vec{v}$ is conserved in two‑body motion. Its magnitude is:

• Elliptical orbit: $h = \sqrt{\mu r_a r_p}$, where $r_a$ and $r_p$ are apoapsis and periapsis radii.
• Circular orbit: $h = \sqrt{\mu r} = r v$. Here $\mu = GM$ is the gravitational parameter of the primary.

Specific Impulse ($I_{sp}$)

Specific impulse measures rocket engine efficiency:

$$I_{sp} = \frac{T}{\dot{m}_p g_0}$$

where $T$ is thrust, $\dot{m}_p$ is propellant mass flow rate, and $g_0 = 9.80665\ \text{m/s}^2$. It is expressed in seconds. A higher $I_{sp}$ indicates more impulse per unit propellant weight.

The Rocket Equation and Delta‑V

The Tsiolkovsky rocket equation relates delta‑v to engine performance and mass ratio:

$$\Delta V = I_{sp} g_0 \ln\left(\frac{m_0}{m_f}\right)$$

where $m_0$ is the initial mass (including propellant) and $m_f$ is the final mass after burn‑out. This holds for impulsive burns, where the thrust duration is much smaller than the total transfer time.

If the required $\Delta V$ is known, the propellant mass $m_p$ can be solved:

$$m_p = m_0 \left(1 - e^{-\Delta V / (I_{sp} g_0)}\right)$$

The calculator uses this equation to compute propellant needs after deriving the total $\Delta V$ from the Hohmann model.

Orbital Transfer Types

• Impulse transfer: Thrusting time is negligible compared to time of flight. Hohmann transfers are the classic example, using chemical propulsion.
• Low‑thrust transfer: Thrusting time is significant (e.g., electric propulsion). They save propellant but require longer travel times. This calculator focuses on impulse transfers.

Hohmann Transfer in Detail

A Hohmann transfer connects two circular orbits with an elliptical transfer orbit whose periapsis equals the inner orbit radius ($r_1$) and apoapsis equals the outer orbit radius ($r_2$). It consists of two impulsive burns.

Burn 1 (Periapsis)

Velocity on the initial circular orbit:

$$v_1 = \sqrt{\frac{\mu}{r_1}}$$

Specific angular momentum of the transfer ellipse:

$$h_t = \sqrt{\mu r_1 r_2}$$

Velocity on the transfer ellipse at periapsis:

$$v_{t,p} = \frac{h_t}{r_1} = \sqrt{\frac{\mu}{r_1}} \sqrt{\frac{2 r_2}{r_1 + r_2}}$$

First delta‑v:

$$\Delta V_1 = v_{t,p} - v_1 = \sqrt{\frac{\mu}{r_1}} \left( \sqrt{\frac{2 r_2}{r_1 + r_2}} - 1 \right)$$

Burn 2 (Apoapsis)

Velocity on the transfer ellipse at apoapsis:

$$v_{t,a} = \frac{h_t}{r_2} = \sqrt{\frac{\mu}{r_2}} \sqrt{\frac{2 r_1}{r_1 + r_2}}$$

Velocity on the destination circular orbit:

$$v_2 = \sqrt{\frac{\mu}{r_2}}$$

Second delta‑v:

$$\Delta V_2 = v_2 - v_{t,a} = \sqrt{\frac{\mu}{r_2}} \left( 1 - \sqrt{\frac{2 r_1}{r_1 + r_2}} \right)$$

Total Delta‑V

$$\Delta V_{\text{total}} = |\Delta V_1| + |\Delta V_2|$$

For a transfer from a lower to a higher orbit both burns are positive; for higher to lower the signs reverse but the magnitudes remain the same.

Conditions for a Valid Hohmann Transfer

• Orbits must be circular (or nearly circular).
• Orbits must be coplanar (zero or negligible inclination difference).
• Propulsion must be impulsive (high thrust, short burn duration).

Example: Earth to Mars

The primary body is the Sun. Earth and Mars orbits are nearly circular (eccentricities 0.0167 and 0.0934) and nearly coplanar, so a Hohmann transfer is a good approximation.

Input data:

• Sun mass: $1.989 \times 10^{30}$ kg
• Sun radius: 696,340 km
• Earth orbital altitude: 146,403,660 km
• Mars orbital altitude: 206,003,660 km

Computed transfer orbit properties:

Property	Value
Semi‑major axis	176,900,000 km
Eccentricity	0.16846
Specific angular momentum	40,776,647,037 km²/s
Velocity at periapsis (Earth)	32.47211 km/s
Velocity at apoapsis (Mars)	23.10908 km/s

Delta‑V and travel parameters:

Parameter	Value
$\Delta V_1$	2.4318 km/s
$\Delta V_2$	2.2329 km/s
Total $\Delta V$	4.6647 km/s
Transfer time (ToF)	7.7145 months (~235 days)

Propellant mass (with $I_{sp}=400$ s, $m_0=200,000$ kg):

$$m_p = 200,000 \left(1 - e^{-4.6647/(400 \times 0.00980665)}\right) \approx 139,105\ \text{kg}$$

About 69 % of the initial mass is propellant, showing the high cost of interplanetary travel.

Launch Windows

A Hohmann transfer between Earth and Mars is feasible only when the planets are aligned. The Earth–Mars synodic period is about 779.9 days (2.14 years). Missions are launched during these windows to approximate a Hohmann transfer.

Summary

The Hohmann Transfer Calculator is a practical Spacecraft Trajectory Calculator that provides delta‑v budgets, transfer orbit characteristics, and propellant estimates for fuel‑efficient orbit transfers. Whether planning satellite orbit raising or an interplanetary mission, this Orbital Transfer Calculator brings classic astrodynamics to your fingertips.