# The Earth has a Trojan

Friends,

The world is a beautiful place full of remarkable things. But when I use the word “world”, I did not just mean the earth. We all know that the earth is beautiful, but remember that it is just a little speck of dust in an infinite universe. To give you an idea of the size of the universe, there are more than 10 stars for every grain of sand on earth( including every beach, every desert and anything else that you can think of). And our sun is just one of these stars.

So we should not restrict our search for beauty to the earth alone. The power of science is that it lets you find laws which are not just applicable to our earth, but to the entire universe. Today I will tell you about a beautiful/interesting phenomenon, which probably occurs all over the universe. Today I will  tell you about Trojans. And the exciting breaking news is that The Earth also has a Trojan!

When we imagine a planet, we image a massive sphere revolving at an enormous speed around a star- through the void of space.  That is how I imagined a planet to move. But a few years back I learnt that in certain cases this picture may become even more interesting. There can be another body in the same orbit, always a certain distance ahead or behind the planet. So a planet may not be alone in its orbit, there may be other bodies ahead or behind it, revolving in such a way that a collision never happens. Such objects are known as Trojans.

Now let’s understand how this configuration can be stable.  The great French physicist and mathematician Joseph Louis Lagrange studied the three body problem( how three bodies move under the gravitational field of each other) and he found that there are 5 configurations of three bodies( one of them much smaller than the other two) which can be perfectly stable under each other’s gravity. In other words, if a planet is revolving around a star, there are 5 points at which a third much smaller body can be put so as to be gravitationally stable with respect to the other two larger bodies. These 5 points are known as Lagrangian points. Out of these 5 points, there are 3 points which lie on the orbit of the planet. A Trojan is a small body at any one of these three points.

So, if earth is revolving around the sun, there are three points on the earth’s orbit on which a much smaller body can exist and remain stable. The first one is the simplest to understand. The object can be diametrically opposite the planet. So if earth is on one side of the sun, this point is on the opposite side, and the three bodies( The Trojan, the Sun and The earth) form a straight line. In terms of angles, we say that the Trojan is placed at 180 degrees with respect to the planet(earth).  This point has been the subject of many science fiction and mythological stories and is typically known as anti-earth.

Now, lets look at the other two points at which Trojans can exist. One point is 60 degrees ahead of the planet and the other is 60 degrees behind. So the line connecting these Trojans to the sun would make an angle of 60 degrees with a line connecting the planet to the sun.  See the attached image from Wikipedia. The 5 lagrangian points are shown. The planet is shown in blue, the sun as yellow and the 5 Lagrangian points are numbered from 1 to 5. The Lagrangian point at 180 degrees is typically called L3. The points at 60 degree ahead and behind are known as L4 and L5 respectively. So a Trojan at L4 is always revolving the star in front of the planet. The Trojan at L5 is always behind the planet. But they never collide.

Trojans are pretty common in our solar system. Jupiter, Neptune and Mars have many Trojans. The first discovery of Trojans happened in the case of Jupiter. Astronomers follow a very interesting  naming convention for naming Jupiter’s Trojans. I am sure you all know about the Trojan war- the Greeks fought the state of Troy sometime between 1260 and 1240 BC. This war involved Helen( of Troy) and the Trojan horse. I am sure many of you would have seen the movie “Troy”. Anyway, so if a Trojan is found at point L4 of Jupiter( 60 degree ahead of Jupiter) it is named after a character on the Greek side( example Achilles or Agamemnon). If a Trojan is found at L5 of Jupiter ( 60 degrees behind Jupiter) it is named after someone on the Trojan camp( like Paris or Laocoon). Two Trojans have been placed on the wrong side. One is Hektor which is named after the character on the Trojan camp but it orbits at L4. There are more than a million Trojans in L4 and L5 of Jupiter.

Interestingly Saturn does not have a Trojan. However the moons of Saturn have a lot of Trojans. These are called the Trojan moons of Saturn. A Trojan moon and a “normal” moon share the same orbit around the planet. So a Trojan Moon is found at the lagrangian points of a “normal” moon in its orbit around the planet.

Also, interestingly, there are no known L3( 180 degrees opposite) Trojans in our solar system.  The reason is that L3 is relatively unstable and a small nudge will displace the Trojan from its orbit permanently. When I said that objects at Lagrangian points are stable, I said stable with respect to the sun and the planet. However if there is another gravitational influence, apart from the planet and the star then L3 is the most unstable so no Trojans are found there.

Typically Trojans are small objects, many less than a km across. The biggest Trojan in our solar system is Jupiter’s Trojan Hektor which is 370 Km across. So most Trojans are asteroids. However, theoretically it is possible for a small planet to be the Trojan of a much larger planet. These are called Trojan Planets. However no such planet has been found.

And now for the breaking news, in 2010, astronomers found the first Earth Trojan. It has been named 2010 TK7( a rather uninteresting name). It orbits at Lagrangian point L4( so it is 60 degrees ahead of earth). It is a small body( about 300 m in diameter).

Maybe there are more Trojans in the solar system. Maybe Earth has more Trojans at L4 or L5. Who knows. But what I do know is that the universe is a very interesting place and we still have a lot to learn.

And before we close, a short poem by Sterling Bunch:

In starry skies, long years ago,

I found my Science. Heart aglow

I watched each night unfold a maze

Of mystic suns and worlds ablaze,

That spoke: “Know us and wiser grow.”

And with each season’s ebb and flow,

My soul with faltering steps and slow,

Still wanders up far-glimmering ways,

In starry skies.

Nor do I heed Life’s gaudy show,

But onward, upward I shall go,

Until new star-lands meet my gaze,

And where, perhaps in after days,

I’ll learn the things I long to know

In starry skies.

Kanwar

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Go wondrous creature, mount where science guides

go measure earth, weigh air, state the tides,

instruct the planets in what orbs to run

correct old time, regulate the sun

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