A Hohmann Transfer is an orbital maneuver that transfers a satellite or spacecraft from one circular orbit to another. It was invented by a German scientist in 1925 and is the most fuel efficient way to get from one circular orbit to another circular orbit. Because the Hohmann Transfer is the most fuel efficient way to move a spacecraft, it is a fairly slow process and is used mostly for transferring spacecraft shorter distances.

A Hohmann Transfer is half of an elliptical orbit (2) that touches the circular orbit the spacecraft is currently on (1) and the circular orbit the spacecraft will end up on (3). It takes two accelerations to get the original orbit to the destination orbit. To move from a smaller circular orbit to a larger one the spacecraft will need to speed up to get onto the elliptical orbit at the perigee and speed up again at the apogee to get onto the new circular orbit. To move from a larger circular orbit to a smaller one, the processes are reversed.

In order to understand the Hohmann Transfer, students must first know a little about orbits. The Hohmann Transfer consists of two circular orbits and one elliptical orbit.

For circular orbits students should know:

Discuss the radius of the orbit with students. Is the radius constant or does it change and why?

Students need to understand the difference between speed and velocity.

Speed is a scalar quantity which refers to “how fast an object is moving.” A fast-moving object has a highspeed while a slow-moving object has a low speed. An object with no movement at all has a zero speed.

Velocity is a vector quantity which refers to “the rate at which an object changes its position.” Imagine a person moving rapidly – one step forward and one step back – always returning to the original starting position. While this might result in a frenzy of activity, it would result in a zero velocity.

***** Speed is a scalar and does not keep track of direction; velocity is a vector and is direction-aware.

There are two types of forces involved in orbits. F1 is the force that pushes down on the Earth (in this case) from the other object and F2 is the force between two objects in general.

We define F1 and F2 as:

Discuss with students the speed of an object in a circular orbit (constant because the radius is constant) and the force of an object (also constant). F2 always equals a constant and is always equal to F1.

Now that students know this information, they can calculate the velocity of an object in a circular orbit.

Let F1 = F2 and solve for V. We get,

Also, discuss the inverse relationship between the radius of the circular orbit and the velocity of the object. Ask them what happens to the velocity as the radius increases and when it decreases.

For elliptical orbits students should know:

Again, disucss with the students the speed of an object in an elliptical orbit. The speed is not constant because the radius changes. When the radius is the greatest, at the Apogee, velocity will be at the minmum. When the radius is the smallest, at the Perigee, velocity is at the maximum.

All that is left is to find the velocity of an object in an elliptical orbit. This is where it gets a little tricky.

*** Teacher’s, from this point on use your discretion on how much information you want to give your studnets. How much you lead them will depend on the level of your students.

In order to find the velocity, students need to have a basic understanding of kinetic energy and potential energy. Potential energy is stored energy, while kinetic energy is energy of motion. It is the energy it possesses because of its motion. If we subtract an objects potential energy from its kinetic energy, we get the total amount of energy an object possesses. We know the formula for the potential energy, kinetic energy and the total energy of an object, so we can substitute those into the formula and solve for velocity.

To simplify things, substitute

where a is half the distance of the major axis or the distance from one vertex to the center

Now we know how to find the velocity of an object in a circular orbit and an elliptical orbit. Ask students what else they think they need to know to successfully make a Hohmann Transfer. Talk a little more about what it is and how it is done to lead them to the conclusion that they also need to know the velocity at the apogee and perigee in order to transfer the spacecraft from one orbit to the next.

In order to find the velocity at the apogee and perigee, it is important that students understand the eccentricity of ellipses and how to label a and c on an ellipse. Have students refer to the diagram below.

We already know that the velocity of an object in a elliptical orbit is

In order to find the velocity at A and P, we need to put the formula in terms of A and P. This is where eccentricity and our diagram come into play.

Talk about whether velocity is faster at the apogee or perigee. Students should come to the conclusion that Vp > Va because the radius is the shortest at the perigee, which means velocity is at its fastest.

Now that students can find the velocity of an object in a circular and elliptical orbit, and the velocity of an object at the apogee and perigee of an elliptical orbit, they can begin to explain how to move a spacecraft from one circular orbit to another.

### Earth to Mars via Least Energy Orbit

Getting to the planet Mars, rather than just to its orbit, requires that the spacecraft be inserted into its interplanetary trajectory at the correct time so it will arrive at the Martian orbit when Mars will be there. This task might be compared to throwing a dart at a moving target. You have to lead the aim point by just the right amount to hit the target. The opportunity to launch a spacecraft on a minimum-energy transfer orbit to Mars occurs about every 25 months.

To be captured into a Martian orbit, the spacecraft must then decelerate relative to Mars using an orbit insertion rocket burn or some other means. To land on Mars from orbit, as did Viking, the spacecraft must decelerate even further using a rocket burn to the extent that the lowest point of its Martian orbit will intercept the surface of Mars. Since Mars has an atmosphere, final deceleration may also be performed by aerodynamic braking direct from the interplanetary trajectory, and/or a parachute, and/or further retrograde burns.

### Inward Bound

To launch a spacecraft from Earth to an inner planet such as Venus using least propellant, its existing solar orbit (as it sits on the launch pad) must be adjusted so that it will take it to Venus. In other words, the spacecraft’s aphelion is already the distance of Earth’s orbit, and the perihelion will be on the orbit of Venus.

This time, the task is to **decrease the periapsis** (perihelion) of the spacecraft’s present solar orbit. Recall from Chapter 3…

**A spacecraft’s periapsis altitude can be lowered by decreasing the spacecraft’s energy at apoapsis**

To achieve this, the spacecraft lifts off of the launch pad, rises above Earth’s atmosphere, and uses its rocket to accelerate **opposite** the direction of Earth’s revolution around the sun, thereby **decreasing its orbital energy** while here at apoapsis (aphelion) to the extent that its new orbit will have a perihelion equal to the distance of Venus’s orbit. Once again, the acceleration is tangential to the existing orbit.

Of course the spacecraft will continue going in the same direction as Earth orbits the sun, but a little slower now. To get to Venus, rather than just to its orbit, again requires that the spacecraft be inserted into its interplanetary trajectory at the correct time so it will arrive at the Venusian orbit when Venus is there. Venus launch opportunities occur about every 19 months.

### Type I and II Trajectories

If the interplanetary trajectory carries the spacecraft less than 180 degrees around the sun, it’s called a Type-I Trajectory. If the trajectory carries it 180 degrees or more around the sun, it’s called a Type-II.

### Gravity Assist Trajectories

Chapter 1 pointed out that the planets retain most of the solar system’s angular momentum. This momentum can be tapped to accelerate spacecraft on so-called “gravity-assist” trajectories. It is commonly stated in the news media that spacecraft such as Voyager, Galileo, and Cassini use a planet’s gravity during a flyby to slingshot it farther into space. How does this work? By using gravity to tap into the planet’s tremendous angular momentum.

In a gravity-assist trajectory, angular momentum is transferred from the orbiting planet to a spacecraft approaching from behind the planet in its progress about the sun.

Experimenters and educators may be interested in the Gravity Assist Mechanical Simulator, a device you can build and operate to gain an intuitive understanding of how gravity assist trajectories work. The linked pages include an illustrated “primer” on gravity assist.

Consider Voyager 2, which toured the Jovian planets. The spacecraft was launched on a Type-II Hohmann transfer orbit to Jupiter. Had Jupiter not been there at the time of the spacecraft’s arrival, the spacecraft would have fallen back toward the sun, and would have remained in elliptical orbit as long as no other forces acted upon it. Perihelion would have been at 1 au, and aphelion at Jupiter’s distance of about 5 au.

However, Voyager’s arrival at Jupiter was carefully timed so that it would pass behind Jupiter in its orbit around the sun. As the spacecraft came into Jupiter’s gravitational influence, it fell toward Jupiter, increasing its speed toward maximum at closest approach to Jupiter. Since all masses in the universe attract each other, Jupiter sped up the spacecraft substantially, **and the spacecraft tugged on Jupiter**, causing the massive planet to actually lose some of its orbital energy.

The spacecraft passed on by Jupiter since Voyager’s velocity was greater than Jupiter’s escape velocity, and of course it slowed down again relative to Jupiter as it climbed out of the huge gravitational field. The speed component of its Jupiter-relative velocity outbound dropped to the same as that on its inbound leg.

But relative to the sun, it never slowed all the way to its initial Jupiter approach speed. It left the Jovian environment carrying additional orbital angular momentum stolen from Jupiter. Jupiter’s gravity served to connect the spacecraft with the planet’s ample reserve of angular momentum. This technique was repeated at Saturn and Uranus.

#### Voyager 2 Gravity Assist Velocity Changes

The same can be said of a baseball’s acceleration when hit by a bat: angular momentum is transferred from the bat to the slower-moving ball. The bat is slowed down in its “orbit” about the batter, accelerating the ball greatly. The bat connects to the ball not with the force of gravity from behind as was the case with a spacecraft, but with direct mechanical force (electrical force, on the molecular scale, if you prefer) at the front of the bat in its travel about the batter, translating angular momentum from the bat into a high velocity for the ball.

(Of course in the analogy a planet has an attractive force and the bat has a repulsive force, thus Voyager must approach Jupiter from a direction opposite Jupiter’s trajectory and the ball approaches the bat from the direction of the bats trajectory.)

See the vector diagram showing the spacecraft’s speed relative to Jupiter during a gravity-assist flyby. The spacecraft slows to the same velocity going away that it had coming in, relative to Jupiter, although its direction has changed. Note also the temporary increase in speed nearing closest approach.

When the same situation is viewed as sun-relative in the diagram below, we see that Jupiter’s sun-relative orbital velocity is added to the spacecraft’s velocity, and the spacecraft does not lose this component on its way out. Instead, the planet itself loses the energy. The massive planet’s loss is too small to be measured, but the tiny spacecraft’s gain can be very great. Imagine a gnat flying into the path of a speeding freight train.

Gravity assists can be also used to decelerate a spacecraft, by flying in front of a body in its orbit, donating some of the spacecraft’s angular momentum to the body. When the Galileo spacecraft arrived at Jupiter, passing close in front of Jupiter’s moon Io in its orbit, Galileo lost energy in relation to Jupiter, helping it achieve Jupiter orbit insertion, reducing the propellant needed for orbit insertion by 90 kg.

The gravity assist technique was championed by Michael Minovitch in the early 1960s, while he was a UCLA graduate student working during the summers at JPL. Prior to the adoption of the gravity assist technique, it was believed that travel to the outer solar system would only be possible by developing extremely powerful launch vehicles using nuclear reactors to create tremendous thrust, and basically flying larger and larger Hohmann transfers.

An interesting fact to consider is that even though a spacecraft may double its speed as the result of a gravity assist, it feels no acceleration at all. If you were aboard Voyager 2 when it more than doubled its speed with gravity assists in the outer solar system, you would feel only a continuous sense of falling. No acceleration. This is due to the balanced tradeoff of angular momentum brokered by the planet’s — and the spacecraft’s — gravitation.

### Enter the Ion Engine

All of the above discussion of interplanetary trajectories is based on the use of today’s system of chemical rockets, in which a launch vehicle provides nearly all of the spacecraft’s propulsive energy. A few times a year the spacecraft may fire short bursts from its chemical rocket thrusters for small adjustments in trajectory. Otherwise, the spacecraft is in free-fall, coasting all the way to its destination. Gravity assists may also provide short periods wherein the spacecraft’s trajectory undergoes a change.

But ion electric propulsion, as demonstrated in interplanetary flight by Deep Space 1 — and employed on the Dawn science mission to the asteroids — works differently. Instead of short bursts of relatively powerful thrust, electric propulsion uses a more gentle thrust continuously over periods of months or even years. It offers a gain in efficiency of an order of magnitude over chemical propulsion for those missions of long enough duration to use the technology.

Click the image below for more information about Deep Space 1. The Japan Aerospace Exploration Agency’s asteroid explorer HAYABUSA also employed an ion engine.

Even ion-electric propelled spacecraft need to launch using chemical rockets, but because of their efficiency they can be less massive, and require less powerful (and less expensive) launch vehicles. Initially, then, the trajectory of an ion-propelled craft may look like the Hohmann transfer orbit. But over long periods of continuously operating an electric engine, the trajectory will no longer be a purely ballistic arc.