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Saturday, February 7, 2015

Nulls in the Void ...... The Lagrange Points




Recall school physics classes on magnetism.
Figure shows how the undisturbed magnetic field of Earth ( shown in the leftmost part ) gets disturbed with a bar magnet.

Two cases are
1. With North pole of bar magnet pointing to Geographic North and
2. South Pole of bar magnet pointing to Geographic North.

We know that we get two NULL points in each case ,,  two points are perpendicular to magnet axis in first case  and in 2nd case these points are along the axis of magnet.

A somewhat similar situation creates what are called as Lagrange points  ( 5 points  ..  similar to 2 nulls in case of bar magnet ) when one steller body revolves around the other. For example  Moon revolving around Earth   have 5  Lagrange points while  Earth going round Sun has different 5 points.

We will study Earth's rotation around Sun to understand this phenomenon.

Earth is rotating around Sun at about 1AU distance ( 149 597 871 kms ) due the  gravitational field and can be analysed with Kepler's laws. It follows that any object at a distance less than 1 AU will move faster. But there is a small hitch in this argument.
Suppose there is a spacecraft between Earth and Sun. If Earth were not there then it would have rotated around Sun following exact rules of Kepler i.e. faster than Earth,
BUT,  the presence of Earth's gravity adds a pull to the spacecraft away from Sun and tries to slow down spacecraft a bit. If the spacecraft distance is such that the effective speed of spacecraft ( due to Sun's gravity and Earth's pull ) matches with that of Earth then the spacecraft ( although it is less than Earth's distance from Sun ) will move synchronously with Earth. It can be computed that if the spacecraft is about 1.5 million kms nearer to Sun ( than 1 AU ), the rotation period of spacecraft matches with that of Earth. This point between Sun and Earth is called as 1st Lagrangian point ( denoted by L1 ).
A similar situation is created at L2, where the spacecraft would have otherwise moved slower than Earth  but speeds up due to the slight addition of Earth's gravity to Sun's gravitational pull.
A third point L3 lies on the other side of Sun w.r.t. Earth.
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A slight deviation from subject:
L1  and L3 pose a  typical communications problem. If a spacecraft is placed at L1, and a antenna from Earth is pointed to it then most probably it will not receive any signal from satellite because the antenna sees the spacecraft embedded with Sun in the background and the solar noise overshadows any signal from spacecraft ( recall communications basic ), This problem is overcome by placing the spacecraft slightly away from L1 so that it is just outside the solar disk.

Similarly, a spacecraft at L3 will not be seen from Earth because it is beyond ( behind ) Sun.
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Back to main topic now.

Note that all the three points are a result of a precarious balance viz all these points are possible ONLY when the points lie on the same line joining Sun and Earth. So they are called Meta Stable points. A satellite kept at these points can easily 'loose' its synchronicity and wander about ..  so it requires constant station keeping to maintain their location.
The remaining two points L4 and L5 do not lie on the line joining the two main bodies.

These points lie on the apexes of two equilateral triangles formed by the base made up of the line joining Sun and Earth.

Why???  ..  for the simple reason that the equilateral triangle exerts equal gravitational force on the spacecraft ..  being pulled with equal force by Earth and Sun while the centrifugal force tries to revolve it around sun

Also ,  these points, unlike the earlier 3 points these points are 'stable'  i.e. a body kept here will try to fall back if it gets away by any perturbation . In fact a lot of asteroids and space dust 'falls' in these points and stays here permenantly. Note the kidney shape of these points in 2nd figure above.

One more condition has to be met for L4 and L5 to be possible ..  viz the mass ratio  ( M1/M2 )  of two bodies ( Sun and Earth in our case ) has to be greater than ( 25 + sqrt( 621 ) ) /2 i.e. greater than 24.96 .


Lagrange points of various pair systems (  Like Sun - Mars, Sun - Earth, Earth - Moon, Sun Jupiter etc )    have been studied in detail by space scientists.
 Several spacecraft have been placed at L1  points of Sun-Earth pair like SOHO missions to study solar phenomenon. Other space crafts are WIND, ACE.

Space crafts using Sun Earth L2 points are : WMAP, Herschel, Euclid, GALA etc.

ISRO has planned for ADITYA mission for solar studies  at L1 of Sun- Earth system in 2016-17.



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