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Lagrange Points Joseph Louis Lagrange was an 18th century mathematician who tackled the famous "three-body problem" in the late 1700s. The problem cannot be solved exactly, but he found that in the case where the third body is very small compared to the other two, some useful approximate solutions could be found. The Earth, Sun, Moon system is an example of a “three- body problem.”

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Lagrange Points A mechanical system with three objects constitutes a three-body problem. The three- body problem is famous in both mathematics and physics circles, and mathematicians in the 1950s finally managed an elegant proof that it is impossible to solve. However, approximate solutions can be very useful, particularly when the masses of the three objects differ greatly.

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Lagrange Points For the Sun-Earth-Moon system, the Sun's mass is so dominant that it can be treated as a fixed object and the Earth-Moon system treated as a two-body system. 18th century mathematicians discovered that there were five special points where gravitational equilibrium could be maintained. An object placed at any one of these five points would stay there.

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The Lagrange points L1, L2 and L3 are not as useful because they are unstable equilibrium points. Keeping a satellite there is possible, but any perturbing influence will drive it out of equilibrium. However, in practice these Lagrange points have proven to be very useful since a spacecraft can be made to execute a small orbit around one of these points with a very small expenditure of energy. They have provided useful places to "park" a spacecraft for observations. The orbits around L1 and L2 are often called "halo orbits." L3 is on the opposite side of the Earth from the Moon is not easy to use.

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If the spacecraft is placed between Sun and Earth, the Earth's gravity pulls it in the opposite direction and cancels some of the pull of the Sun. With a weaker pull towards the Sun, the spacecraft then needs less speed to maintain its orbit. If the distance is just right--about 4 times the distance to the Moon or 1/100 the distance to the Sun--the spacecraft will need just one year to go around the Sun, and will keep its position. That position is point L1.

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L1 is a very good position for monitoring the solar wind, which reaches it about one hour before reaching Earth. In 1978 the "International Sun-Earth Explorer-3" (ISEE-3) was launched towards L1, where it conducted such observations for several years. In November 1994 a new spacecraft, WIND, was launched towards that position. More recently the solar wind at L1 has been monitored by the solar observatory SOHO and by ACE.

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DSCVR launches later this month to study the solar wind. It will be placed at L1.

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L2 is at about the same distance from Earth as L1 but on the night side, away from the Sun. The L2 point has been chosen by NASA as the future site of a large infra-red observatory, the "Next Generation Space Telescope," renamed in honor of a late NASA director The James Webb Observatory. Launch set for 2018.

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The Lagrange points L4 and L5 constitute stable equilibrium points, so that an object placed there would be in a stable orbit with respect to the Earth and Moon. Because of small departures, a satellite needs an effective restoring force to bring it back to the stable point.

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Attention has been given to two stable points L4 and L5, located in the Moon's orbit. These positions have been studied as possible sites for artificial space colonies in some very distant future. As seen from the Sun, the L4 and L5 points lie at 60 degrees ahead of and behind Earth. L4 and L5 are resistant to gravitational perturbations. Because of this stability, objects such as dust and asteroids tend to accumulate in these regions. At L4 or L5, a spacecraft is truly stable.

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© Oxford University Press 2009 12.3 Circles in space The planets follow circular paths around the Sun. The centripetal force needed to keep the planets.

© Oxford University Press 2009 12.3 Circles in space The planets follow circular paths around the Sun. The centripetal force needed to keep the planets.

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