Neela wrote:BTW, why should we meet Mars when it is 300 million km away? Why not when it is "just" 54 million km away?
That would be the distance covered by the probe if it could fly at infinite speed, i.e. it leaves earth at the point it is nearest to Mars and reaches Mars at exactly the same time. But because the probe has a finite speed, the mathematics is slightly more complicated.
First, I would give a simplifying solution. Let us assume that the earth and Mars go around the sun in perfect circular paths. Also let us disregard the 'radius' of the orbits of the spacecraft around earth and Mars as insignificant to the interplanetary distance. Now, once you have fixed the speed at which you can eject the spacecraft in the helio-centric orbit, the time taken by the craft to cross the Mars's orbit is going to be fixed (say 'x' time units). The question is would Mars also be at the point of intersection of the both the orbits at that the same time? This is a simple equation to solve. Fix a initial point of reference. At this point, you know the initial angular displacement between earth and Mars around the sun wrt to an arbitrary a reference point. The angular velocity of the earth and Mars is also fixed. The spacecraft will be travelling in an elliptical path. However it's angular velocity follows simple properties. Now the question can be simply formulated as "at what angular displacement of the earth should you inject the spacecraft into it's helio-centric orbit, so that angular displacement of the spacecraft and the Mars is the same after 'x' units of time".
(Added later: pictorial description: via ISRO)
However this gets a little more complicated:
1. Because earth and Mars are not in circular orbit. The angular velocity around the sun is not fixed, but it is well defined. So what? The question that you would ask will still remain the same.
2. What if I can inject the spacecraft into its heliocentric orbit at different speeds, i.e. 'x' is no longer constant. Yes, so what? It is just one more level of differentiation to find the minima.
3. But it is not a superman who is launching the spacecraft from Earth to Mars. You have to reach the point of ejection into the heliocentric orbit. How to get there optimally?
Nothing very complex. It is lots of class XIth mathematics. Just lots of it