---
title: "Hohmann Transfer Planner Interactive Lab | Sandbox Physics"
description: "Can two burns change an orbit and meet a moving target? They can, provided you leave at the right moment. Two ideal burns carry the spacecraft out to the target's orbit along half of a transfer ellipse. The target must also reach that meeting point when the spacecraft arrives, or the two miss. The right radius, the right position along the orbit and the right velocity are separate arrival conditions."
canonical_url: "https://sandboxphysics.com/labs/hohmann-transfer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/hohmann-transfer/"
last_modified: "2026-10-22"
---

M074 · Orbit transfers and launch windows

# Hohmann Transfer Planner

**Can two burns change an orbit and meet a moving target?** They can, provided you leave at the right moment. Two ideal burns carry the spacecraft out to the target's orbit along half of a transfer ellipse. The target must also reach that meeting point when the spacecraft arrives, or the two miss. The right radius, the right position along the orbit and the right velocity are separate arrival conditions.

Design a transfer between two circular orbits in the same plane. Trim both burns, delay the departure and drag the arrival circle, then compare the radius, specific energy, arrival position and velocity, and the fuel and time of a bi-elliptic alternative.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The model uses an ideal spherical Earth, kilometre–second units, instantaneous burns and two-body gravity. It leaves out the atmosphere, Earth's oblateness, finite thrust and third bodies. Both circular orbits lie above the reference surface, and the trajectory ends if it touches the surface. The largest sampled coast errors and the anomaly residuals are numerical checks, not mission certification.

Physics tutorial

## An orbit transfer is also a timing problem

**Background** A short prograde burn raises the opposite side of the orbit. The spacecraft then coasts under gravity, and a second burn brings it to circular speed at the destination. A target on that destination circle keeps moving during the whole coast.

**Why it matters** The vis-viva relation and the half-ellipse travel time come from two-body energy and Kepler motion. The launch-phase calculation was inspired by NASA’s educational Hohmann construction. The bi-elliptic costs here are derived by joining two tangent half-ellipses and adding up three signed burns.

Start with the essentials

- **Focus question**: Can a perfect change of orbital radius still miss the target?
- **One-sentence intuition**: Yes: radius, angular position and velocity must each be matched on arrival, and waiting shifts the phase even when both burns are perfect.

### Core mathematical model

#### Circular and elliptic speeds

$$
v_c(r)=\sqrt{\frac{\mu}{r}},\qquad v^2=\mu\left(\frac{2}{r}-\frac{1}{a}\right)
$$

The specific orbital energy sets the speed at each radius. Gravity is a fixed central field, and there is no thrust while coasting.

#### Hohmann coast time

$$
a_H=\frac{r_1+r_2}{2},\qquad t_H=\pi\sqrt{\frac{a_H^3}{\mu}}
$$

Both burns are scheduled at the ideal apsides. Trimming the first burn changes the actual conic, but the second burn stays at its scheduled time.

#### Signed burn sizes

$$
\Delta v_{1,H}=\sqrt{\mu\left(\frac{2}{r_1}-\frac{1}{a_H}\right)}-v_c(r_1),\quad\Delta v_{2,H}=v_c(r_2)-\sqrt{\mu\left(\frac{2}{r_2}-\frac{1}{a_H}\right)}
$$

Positive means prograde and negative means retrograde. The total cost shown adds up the sizes of the burns, not their signed values.

#### Phase needed at launch

$$
\phi_{0,H}=\pi-n_2t_H\pmod{2\pi},\qquad \phi_{\mathrm{burn}}=\phi_0+(n_2-n_1)t_{\mathrm{wait}},\quad n_i=\sqrt{\frac{\mu}{r_i^3}}
$$

The initial phase is set for departing right away. If departure is delayed, both objects keep moving on their circles before the first burn.

#### Energy and angular momentum change at each burn

$$
\Delta\varepsilon=\mathbf v^-\!\cdot\!\Delta\mathbf v+\frac{\lVert\Delta\mathbf v\rVert^2}{2},\qquad\Delta h=\mathbf r\times\Delta\mathbf v
$$

Energy and angular momentum jump at each burn and are conserved separately on each coast. For planar motion, the cross product is the single angular-momentum component.

#### The three-burn bi-elliptic alternative

$$
\Delta v_B=|\Delta v_{1,b}|+|\Delta v_{b,1}+\Delta v_{b,2}|+|\Delta v_{2,b}|,\quad t_B=\pi\sqrt{\frac{a_{1b}^3}{\mu}}+\pi\sqrt{\frac{a_{b2}^3}{\mu}}
$$

The intermediate radius lies outside both circular orbits. For large radius ratios, raising it can lower the cost, but the coast takes much longer.

### Common difficulties

#### Reaching the right radius is not a rendezvous

**Typical misconception** Once the spacecraft reaches the destination circle, docking is guaranteed.

**Better mental model** Compare where the target actually is and the relative velocity after the burn. The launch phase matters.

#### A burn has a direction

**Typical misconception** Lowering an orbit takes the same positive burns as raising it.

**Better mental model** When the transfer lowers the orbit, both ideal Hohmann burns change sign.

#### Hohmann is cheapest only under its assumptions

**Typical misconception** Hohmann is the cheapest of every imaginable transfer.

**Better mental model** The comparison covers only coplanar transfers between circular orbits under two-body gravity. At large radius ratios, a three-burn bi-elliptic transfer can cost less.

#### Orbits cannot pass through the Earth

**Typical misconception** A conic can simply pass through Earth.

**Better mental model** The trajectory ends where the conic first touches the reference surface, and any later burns are not carried out.

### Run the experiment

1. 01

   #### Match the target

   Acquire the nominal transfer and check the arrival distance and relative speed.

   **What to observe:** After the second ideal burn, both are close to zero.

2. 02

   #### Miss even with perfect burns

   Use the delayed-departure preset, then remove the delay.

   **What to observe:** The orbital radius still matches; what changes is whether the angular positions line up.

3. 03

   #### Trim the burn

   Use the underburn preset and look at the radius and the specific energy.

   **What to observe:** The second burn still fires at its planned time, but now on a different actual conic.

4. 04

   #### Trade fuel for time

   Use the large-radius-ratio preset and move the intermediate apoapsis.

   **What to observe:** The bi-elliptic point can sit below the Hohmann cost and far to its right on the time axis.

Physical conclusion

Weigh the total velocity change, the flight time and the match at arrival together.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)