Hohmann Transfer Orbit: DIY Inexpensive Journey from Earth to Mars

The incredible emotion that a human being experiences when viewing frames of space launches, or, if you are lucky, listening with delight as the rocket's combustion engine produces the energy necessary to take off, accompanied by a magnificent roar, is something that changes the perspective in a second. From that moment on, it is almost impossible not to wonder: and now ... How will it reach its destination? What route will it follow to get out of the atmosphere? How do they trace that trajectory?

What is the transfer orbit?

It is essentially known that to move in space it is necessary to use indirect impulses in order to achieve impulsive maneuvers. However, to travel from one planet to another, specific routes are required, which are not only safe and efficient, but also optimal in terms of fuel consumption.

The transfer orbit is the path to be followed by an object to move from a circular orbit of radius "x" to one of radius "y". The existence of this type of trajectory orbits, as can be deduced, is expressed in radius measurements.

Hohmann Transfer Orbit: Efficiency

At first glance, it could be said that the most efficient way to move from one place to another is the straight line, however, taking a closer look at this approach, we find that the spacecraft, when leaving the gravitational field of the Earth, do not travel in a straight line at constant speed; on the contrary, they are subjected to the gravitational laws imposed by solar gravity. This implies an elliptical motion, like the one followed by the rest of the celestial bodies in our neighborhood. If you wanted to go a straight-line route, it would probably take colossal (and costly) amounts of fuel to give the spacecraft enough momentum for it to achieve the inefficient trajectory.

Walter Hohmann was a German engineer who built a great legacy of knowledge in the aerospace field, and as it can be concluded, he is the creator of the “Hohmann transfer orbit”, born thanks to the enormous interest that the engineer had towards interplanetary space flight. This orbit traces a direction that begins with an initial impulse of the spacecraft, on the ground and a final impulse, upon arrival at our destination, which could be Mars, if we so desired. Along the way, the spacecraft draws a half ellipse with the Sun, which would be located as one of its foci and with Earth and Mars as vertices of the figure.

To form our trajectory, data such as initial and final velocity, impulse, numerical values and launch windows would be taken into account.

The initial and final speeds are defined in terms of the ratio between the distances to the Sun, from which the following expressions are derived:

In these expressions, v_T is the orbital speed of the Earth around the Sun, γ is the ratio of the radius of Mars's orbit to that of Earth's orbit, and v₁ and v₂ are the spacecraft's speeds relative to the Sun when leaving Earth's orbit and when reaching Mars's orbit.

Taking into account Earth's gravitational pull, the escape velocity and the initial velocity, the following expression is obtained for the impulse.

And regarding the launch windows, the Hohmann transfer orbit works if the positions of the Earth and Mars coincide as extreme points of the imaginary ellipse, therefore, the ideal launch position would occur roughly once every 780 days (about 26 months).

We hope that this article has been able to explain a little better how to travel economically from our home to Mars. Keep feeding your brain more Science Cookies!