Mangalyaan reaches Mars
Kamal Lodaya, The Institute of Mathematical Sciences, Chennai

India's Mars orbiter mission Mangalyaan reached its destination on September 24 and soon sent back pictures of the orange-red coloured planet. Even after getting there, it was difficult for Mangalyaan to get into orbit and the Indian Space Research Organization is being complimented by everyone for getting it done at the first attempt. What is so difficult about this?

Kepler's Laws
Perhaps you may have heard of Kepler's laws. You have learned in school that the planets go around the Sun because of the Sun's gravitational attraction. Kepler discovered that the closer the planet is to the Sun, the faster it moves. Earth takes 365.2422 days to go around, and its orbital speed is nearly 30 kilometres per second (kmps). Mars takes 687 days to go around the Sun, and its orbital speed is just above 24 kilometres per second.

Escape Velocity
On the other hand, escaping from a planet's gravity depends only on that planet itself and does not have anything to do with the Sun. That is, in order to overcome the gravitational attraction of a planet and escape from it, you need a minimum velocity called the escape velocity. Thus the escape velocity is 11.2 kilometres per second relative to Earth while that from Mars is only 5 kilometres per second (relative to Mars). This means that you need to have only half the velocity required on Earth for an object to escape from Mars.
The figure shows a(n imaginary) cannonball being launched from the top of a mountain. When it is slow, it simply falls in an arc. As it speeds up, and reaches a velocity called orbital velocty, it goes into orbit around the Earth, and as it gets even faster it escapes from the Earth.
In comparison an aeroplane travels at about 0.3 kilometres per second. So a rocket engine has to fire really hard so that the spacecraft goes faster than the escape velocity from Earth. As you can imagine this takes enormous amounts of fuel. 

Orbital Velocity
Now what is the speed with which the craft is hurled out into space? Just around 30 kilometres per second relative to the Sun! Where did this large velocity come from? The space craft was earlier on the Earth, but the Earth goes around the Sun with this velocity. So when the space craft leaves the Earth, it carries the Earth's orbital velocity with it. Remember that the rocket engines stop firing once it is out of Earth so it will not keep the extra speed of 11 kmps which it acquired relative to Earth, but it will still have the large orbital velocity.
At this stage the skill of the mission controllers is to put the spacecraft onto an elliptical path which takes it from Earth to Mars. Now Mars is farther from the Sun than the Earth, and so this is a path which takes it farther from the Sun than the Earth. According to Kepler, therefore, the orbital speed will have to be brought down below 30 kmps relative to the Sun. 

Capture by Mars
When it reached Mars, Mangalyaan was travelling at a speed relative to the Sun greater than Mars's orbital speed of 24 kmps. So its engines had to fire to slow it down to that speed, so they are actually acting as brakes. Not only that, it had to be captured by Mars, so its speed relative to Mars had to be brought down to below 5 kmps, that is, less than the escape velocity. This is what the mission controllers did.
Now Mangalyaan is in an elliptical orbit around Mars which takes around 73 hours. The closest point to Mars, its periareon, is about 420 kilometres. The farthest point, the apoareon, is about 77,000 kilometres. 
Soon after reaching there, Mangalyaan and all the spacecraft already there (Mars Express, Mars Odyssey, Mars Reconnaissance Orbiter and Maven are all in orbit around the planet and the rovers Curiosity and Opportunity are on the surface) have to face a big challenge.

Comet Siding Spring
On October 19, around midnight Indian time, the comet Siding Spring will pass just around 1.3 lakh kilometres from Mars. Although a comet's nucleus is quite small its coma (head) consists of dustgrains spread over thousands of kilometres, and its tail has gas, ions and dust over lakhs of kilometres. From the orbital calculations done by space scientists all over the world, Mars will pass through the comet's coma about 20 minutes after the nucleus has gone by. No doubt the comet and its tail will still be a glorious sight seen from Mars.
Even tiny dustgrains, travelling at several kilometres per second, can irreparably damage instruments on a spacecraft. So all the spacecraft on Mars are taking precautionary measures to avoid being hit. Mangalyaan with its elliptical orbit may be more at risk, since it may actually come closer to the comet. ISRO scientists are busy calculating exactly where it will be at that time and how to protect its instruments.
On the other hand, being so close to a comet gives Mangalyaan a chance to photograph it, and perhaps to try and detect what compounds the gases escaping from it have. So we must keep its instruments safe from being hit, but once the danger is past, use those instruments to take pictures and do analysis. So Mangalyaan's first bit of space science is already going to be exciting.