Let's Go to Mars! Calculating Launch Windows

In this lesson, students will:

• Use algebraic computations to determine the relative positions of Earth and Mars during which an optimal (low-energy) transfer of a spacecraft can occur.
• Combine this information with planetary-position data to determine the next launch opportunity to Mars.

• This activity is designed for students familiar with advanced algebra concepts.
• Consider having students sit on a carpeted floor when using the pushpins and string to make an ellipse. The carpeted floor will absorb the tips of the pushpins that may extend past the thickness of the cardboard. Alternately, have students use more than one piece of cardboard to cushion against protruding pushpin tips.

When a spacecraft is launched from Earth, its forward velocity combined with the gravitational pull of Earth cause it to travel in a curved path. As the spacecraft heads toward another planet, the gravitational pull of that planet factors in to the path the spacecraft takes. The more a spacecraft can “coast” with engines off, the lower the cost of the mission (rocket fuel is not cheap!).

Think of a quarterback throwing a football to a receiver. The initial impulse (throw) is all the football gets as far as power is concerned. The football follows a curved path into the hands of the receiver. Likewise, the quarterback throws the football to where the receiver is going to be, not necessarily to where the receiver is currently. So, the quarterback throws the football downfield as the receiver is running in that direction. In a perfectly thrown pass, the receiver’s running speed will bring him or her to the exact spot where the football arrives at hand-level.

Launching to Mars is similar to this. A spacecraft is given an initial impulse (launch) toward Mars and then shuts off its engines and coasts (obeying Newton’s First Law) until it gets close to its target. Depending on the mission, the spacecraft may slow down – to get into orbit or land – by using the Martian atmosphere or retro-rockets that fire opposite to the direction of travel (obeying Newton’s Third Law).

Though a spacecraft could follow a variety of curved paths from Earth to Mars, one path called the Hohmann transfer orbit uses the least energy and is thereby considered to be the most efficient.

The Hohmann transfer is an elliptical orbit with the sun at one focus of the ellipse that intersects the orbit of the target planet. Launch occurs when Earth is at Hohmann perihelion (the point of the Hohmann orbit that is closest to the sun). Arrival occurs when Mars is at Hohmann aphelion (the point of the Hohmann orbit that is farthest from the sun).

Depending on mission objectives and spacecraft characteristics, engineers will use variations on the Hohmann transfer orbit to get spacecraft to Mars. These variations can make travel time more or less lengthy than a standard Hohmann transfer.

To make sure the spacecraft and Mars arrive at the same place at the same time, the spacecraft must launch within a particular window of time. This window is called the “launch window” and, depending on the target, can be a few minutes or as much as a few weeks in length.

If a spacecraft is launched too early or too late, it will arrive in the planet’s orbit when the planet is not there.

When launched within the proper launch window, the spacecraft will arrive in the planet’s orbit just as the planet arrives at that same place. At this point, the spacecraft is positioned for either going into orbit about the planet or landing on the planet.

Calculating orbit trajectories and launch windows is a complex task involving a variety of parameters that may or may not be constantly changing. In order to make this task accessible to high-school students, some variable parameters have been stabilized and some assumptions have been made. This problem, with these simplifications, allows students to generate an approximation of the launch window to Mars.

• Must a spacecraft be launched at an exact moment in the launch window? What happens if it is launched early or late?
• Research: What is the average length of a launch window to Mars?

• Approximately when was the most recent opportunity for a launch to Mars? What countries took advantage of that opportunity and launched to Mars at that time? What is the current status of those missions? Were they successful?
• Have students create a spreadsheet that will subtract heliocentric longitudes for Earth and Mars to simplify launch window calculations.
• Relative to Mars, where is Earth in its orbit when the spacecraft arrives?

• Meet JPL engineer Sue Finley – Finley started at JPL in 1958 as a human computer
• Women at JPL website
• JPL History
• JPL 80th Anniversary Article
• JPL Timeline
• JPL 80th Anniversary Video Playlist
• JPL 80th Anniversary Printable Calendar
• Mars in a Minute Video Series
• Stomp Rockets Activity
• Basics of Space Flight Tutorial