Calculating a Hohmann Transfer

A Hohmann Transfer is a very common orbital maneuver used by astrophysicists to send a spacecraft from a small circular orbit to a larger one. In this Instructable I will walk you, step by step, through calculating the Hohmann Transfer for sending a spacecraft from Earth to Mars.

Step 1: Gather Supplies

For this project you will need a pencil, paper, and a calculator.

Step 2: Find the Planets' Distances From the Sun

We will need to know how far the planets are from the sun. You can use the chart provided to get this information, use the distance in kilometers. We will denote these distances with the variables R1 and R2 where R1 equals Earth's distance from the sun and R2 equals Mars' distance from the sun

Step 3: Write Down Your Constants

Constants are unchanging values that will be repeatedly used in the problem, so it is helpful to write them down at the top of the page for easy access.

In our problem the necessary constants will be Earth and Mars' distances from the sun, R1 and R2, and what is called the standard gravitational parameter, which will be represented by GM = 1.327 x 10^11 km³/s². This is the gravitational constant times the mass of the sun.

You should have something like this on your paper:

R1 = 149,600,000 km
R2 = 227,920,000 km
GM = 1.327 x 10^11 km³/s²

Step 4: Convert the Planets Orbital Periods to Seconds

Using the information in the chart, convert the orbital periods of Earth and Mars from days to seconds. Do this by multiplying the number of days by 86,400.
The orbital period of Earth will be denoted by the variable P1 and the orbital period of Mars will be denoted by P2.

Step 5: Compute the Semi-major Axis of the Transfer Orbit

The semi-major axis of an ellipse is the distance from its center to its furthest side. In this Hohmann transfer the ellipse is the path the spacecraft will take from Earth to Mars. The semi-major axis will be denoted by the variable a(transfer) such that

a(transfer) = (R1 + R2 ) / 2

Step 6: Find the Period of the Transfer Orbit

The period of the orbit is found using Kepler's third law, which is shown in the picture. For the period of the transfer orbit, the variable a will be a(transfer) so that

P(transfer) =√(4π²·a³/GM )

Step 7: Find the Velocity of Earth's Orbit

Now we must find the velocity of Earth's orbit so we'll know how much we have to alter a spacecraft's velocity to enter the elliptical orbit that will get it from Earth to Mars. The velocity for Earth's orbit will be denoted by V1.

V1 = (2π x R1) / P1

Step 8: Find the Velocity of Mars' Orbit

Now we need to find the velocity for Mars' orbit, V2. This is done using the same formula, but substituting in the distance from the sun and period of Mars instead.

V2 = (2π x R2) / P2

Step 9: Find the Velocity of the Elliptical Orbit at Its Perihelion

Because the elliptical transfer orbit is closer to the sun at the end with Earth's orbit than it is at the end with the Mars' orbit, it will have a larger velocity near Earth than it will near your Mars. The end of the ellipse closest to the sun is called the perihelion. We need to find out how fast the orbit is at the perihelion in order to launch our spacecraft into the elliptical orbit from Earth's orbit and ensure that it makes it to our destination.

V(perihelion) = (2π x a(transfer) / P(transfer) ) x √( (2a(transfer) / R1) - 1)

Step 10: Find ΔV1

ΔV1 is how much the velocity of our spacecraft needs to change to switch from Earth's orbit to the transfer orbit that will take it to our destination planet. In order to start on the elliptical transfer orbit our spacecraft will need to speed up. This burst of velocity, ΔV1 is equal to the difference between the V(perihelion) and V1.

ΔV1 = V(perihelion) - V1

This Δv is crucial in the engineers' process of figuring out how much fuel a spacecraft will need.

Step 11: Find the Velocity of the Elliptical Orbit at Its Aphelion

Now we need to find the velocity the spacecraft will be traveling at the aphelion of the elliptical orbit. This is the end of the ellipse furthest from the sun, ergo, the end that lines up with the orbit of Mars.

V(aphelion) = (2π x a(transfer) / P(transfer) ) x √( (2a(transfer) / R2) - 1)

Step 12: Find ΔV2

Just like how ΔV1 was the change in velocity necessary to send the spacecraft from Earth's orbit into the elliptical transfer orbit, ΔV2 is the change in velocity necessary to send the spacecraft from the elliptical transfer orbit into Mars' orbit. The burst of speed needed is equal to the difference between the velocity of Mars' orbit and the velocity of the elliptical orbit at its aphelion.

ΔV2 = V2- V(aphelion)

Step 13: Calculate the Time of Flight

The time it will take your spacecraft to get from Earth to Mars is equal to half the period of the transfer orbit. This value will seem large because it is in seconds.

TOF = ½(P(transfer))

Step 14: Convert the Time of Flight to Days

To convert your time of flight to days, divide it by 86,400.

TOF(days) = TOF(seconds) / 86,400

Step 15: Review

Now you have successfully calculated the two changes in speed necessary to get your spacecraft into the orbit of Mars and the number of days it will take your spacecraft to get there.

These numbers are extremely important for the engineers building the spacecraft so that they can know exactly how much fuel the ship will need.