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Once our spacecraft has exceeded escape velocity it is leaving Earth in a hyperbolic orbit as we saw above. But that is with respect to Earth’s gravity. Now we must look at the bigger picture and see that the gravity field of the Sun becomes important because the spacecraft is actually now in an elliptic orbit around the Sun just like Earth itself. Actually for precise calculations we must also include the gravity fields of most of the planets, in particular the largest planet Jupiter. But we will ignore these effects here for simplicity and continue to work with Kepler orbits about one central body.

The simplest way to get e.g. to Mars is similar to the Hohmann transfer technique we discussed above.

Curiositys orbit showNOriginal image: NASA. Edited and animated by Ngā Whetū Resources  

The diagram shows the actual trajectory of the Curiosity mission (Mars Science Laboratory, MSL) that was launched in October 2011. You can see that MSL was launched into an elliptic orbit around the Sun, that has its aphelion just touching the orbit of Mars. Once the MSL got to that position Mars obviously needed to be there too. This sets critical conditions on the time of the launch from Earth and requires a just favourable common configuration of Earth and Mars. Such configuration happens only once in about every 2.1 years.

Once MSL reaches Mars, its aphelion velocity is slower than Mars in its orbit so the spacecraft must speed up. Then in order to go into orbit around Mars it needs to slow down to be caught in Mars’ gravity field.

It must be noted that the gravity field of the Sun is dominant during most of the trajectory, but as the spacecraft approaches Mars, the gravity field of that planet has increasing effect on the spacecraft’s orbit. Hence this seemingly simple trajectory is already highly complex and requires careful planning and calculations as well as in-flight orbital corrections at critical points. An overriding issue always is the fuel economy of inter-planetary travel as it is prohibitively expensive, and from some point impossible, to take large amounts of propellant into space.



Tip: If you really want to “get your hands dirty” try this practical lesson in calculating launch windows for Mars.


Gravity Assist
Many missions that have been carried out would never have been possible without an important technique for saving propellant. This is the technique of gravity assist that could be referred to as “stealing a bit of orbital energy from a natural Solar system object”.

Compare this in its simplest form with an elastic ball that bounces off a wall. If the collision is elastic the ball will bounce off with the same speed but opposite direction as when it came in. Now assume that the wall itself is moving towards the incoming ball. The ball now bounces off the wall with the sum of the incoming speed plus twice the speed of the wall. If the wall is very massive in comparison to the mass of the ball, there will be no noticeable change in the speed of the wall. When we do this with spacecraft passing by (of course not bouncing off) say a planet, the spacecraft can gain a lot of speed that in effect is “stolen” from the planet. The image shows the principle of a spacecraft performing a gravity assist at Jupiter.


 cassini at jupiterImage from


This technique is also used to slow a spacecraft down, which is important when travelling to the inner part of the Solar system or when going into orbit around a planet or moon.

Read more about this important technique here and here.



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