February 15 India sent a craft, Aditya-L1, hurtling through space a few months ago. It will study the sun. That's interesting by itself. But it also aims to settle at - or actually, in a small orbit around - a particular spot known as a Lagrangian point.
There are five such points. (Or strictly, five in relation to the Earth). What they are, why we would send a spacecraft to one of them - these are the kinds of questions that make astronomy so endlessly fascinating. That's what prompted my column for January 12: The point where Aditya stays put, almost, https://www.livemint.com/opinion/columns/the-point-where-aditya-stays-put-almost-11704969237649.html Please send me any reactions! cheers, dilip --- The point where Aditya stays put, almost Perhaps you've noticed: The word "Lagrangian" has been popping up in Indian news reports over the last several days. The last time it did so, but I suspect far less frequently, was in late 2021. That was prompted by the launch of the James Webb Space Telescope (JWST) in December that year. Today, the word is reappearing because India has just launched Aditya-L1, a spacecraft meant for the study of the sun and its atmosphere. Today it is reappearing because that "L1" means something - that Aditya will be stationed at the first Lagrangian point. JWST, for its part, is positioned at the second Lagrangian point, L2. Well, not quite in either case, but I'll come back to that. So to begin with, what are these Lagrangian points and how many are there? First of all, they are named for the 18th Century mathematician Joseph-Louis Lagrange. Born in Italy, he later became a naturalized Frenchman. He made a number of important contributions to mathematics - in calculus, solving differential equations and more. Famously, he proved his "four-square theorem": that every natural number - the non-negative integers - can be expressed as the sum of four squares. But Lagrange also applied his considerable mathematical talents to celestial mechanics. How do objects in space act on each other? What can we say about their motion relative to other objects, influenced by other objects? There is the "two-body" problem, which seeks to find answers to the questions above when it's just two massive objects we're concerned with. How do we determine how they will move as their gravities affect each other? If they are widely separated, can we predict the way they behave? If they come sufficiently close to one another, will they start orbiting each other? What if one is much heavier than the other - like our Earth and the nearby and much smaller Moon? The two-body problem, first solved by Isaac Newton in the 17th Century, seeks to answer such questions about these celestial behaviours. But while it can explain pretty accurately how the Earth and its Moon move, there are actually more bodies involved. The Sun, for example, is a large elephant in that particular room. What effect does it have? So if all that sounds complicated, Lagrange applied his mind to the possibly even-more complicated three-body problem - the Sun, Earth and Moon forming an obvious example. Specifically, he wanted to determine if three bodies of different masses, moving at different velocities, could orbit each other while also staying "stationary" relative to each other. In other words, in the same positions relative to each other. Meaning, can their respective centrifugal forces and the gravitational pulls they exert on each other cancel out so that they are stable in those positions? There are ways to understand what Lagrange was thinking about, even if they are rather different situations. For one, imagine three strong bar magnets that you place close together on a table, each South pole pointing at the other two. Naturally, the magnets will mutually repel: if you place them very close, they will rapidly move backward. Place them ever further apart, though, and eventually there is a point at which the repellent magnetic force is cancelled out by the friction between magnets and table. Or take the recent story about the iPhone that fell out of a Boeing aircraft at 16,000 feet - and survived. How did it survive? Wouldn't the phone keep accelerating and eventually hit the ground at a fearful speed that would smash it to smithereens? Not quite. As a Washington Post news report put it, "Any object falling toward Earth will reach a point, known as its terminal velocity, where the force of gravity can't accelerate it anymore because of resistance from the air in the atmosphere." That is, the force of gravity is cancelled out by the air resistance. (Yes, that the instrument landed in grass, and not on concrete, also helped it survive intact.) About three bodies in space, Lagrange was asking something analogous to these situations: essentially, how do you balance competing forces? For the purposes of JWST and Aditya, he solved a more restricted three-body problem, in which two of the bodies are much larger than the third. He wanted to know if there's a point where the gravitational attraction of the larger bodies is equal to the centripetal force that acts on a small object that is trying to move in tandem with them. I realize this talk of different forces may be getting obscure. Still, think of the Sun and Earth. Is there a point where the gravitational attraction from those two celestial bodies on a spacecraft such as Aditya act to keep Aditya there? Turns out, there is. Not just one, but five such points. At three of those points - called L1, L2 and L3 - the three bodies are in a straight line. At the other two, L4 and L5, they form an equilateral triangle. L1 is between the Sun and Earth, L2 is beyond the Earth and L3 is beyond the Sun. L4 and L5 are on the Earth's orbit around the Sun. L3 is always hidden behind the Sun, which makes communication with it impossible. It is also diametrically across the Earth's orbit from us, meaning several hundred million km away and thus much harder to reach. By contrast, L1 and L2 are each only about 1.5 million km from Earth. JWST is positioned at L2. Or actually, on a small orbit around L2, because it is near impossible to keep it steady exactly there. For the kind of astronomy that JWST is designed for, L2 is an ideal spot. It is close enough for regular communication with the Earth, yet allows for clear views of distant celestial objects. Aditya chose L1 - or again, a small orbit around L1 that takes about 178 days to complete. From there, Aditya will have unimpeded views of the Sun, which it is designed to observe. Incidentally, Aditya won't be alone at L1. The Solar and Heliospheric Observatory (SOHO) is there too, as also the LISA Pathfinder mission, which is intended to test the feasibility of detecting gravitational waves. Remember that Lagrange lived in the 18th Century. His paper solving the three-body problem is from 1772 ("Essai sur le Probleme des Trois Corps") and won a prize from the Paris Academy of Sciences. Prize notwithstanding, he knew he would not live to see the physical validation of the idea of his Lagrangian points. It's taken about 250 years, but so what? Missions like Aditya and JWST are a splendid marriage of Lagrange's theoretical calculations and 21st-Century technology and ingenuity. And that kind of marriage is enough to captivate me. You too, I trust. -- My book with Joy Ma: "The Deoliwallahs" Twitter: @DeathEndsFun Death Ends Fun: http://dcubed.blogspot.com -- You received this message because you are subscribed to the Google Groups "Dilip's essays" group. To unsubscribe from this group and stop receiving emails from it, send an email to dilips-essays+unsubscr...@googlegroups.com. To view this discussion on the web, visit https://groups.google.com/d/msgid/dilips-essays/CAEiMe8ptaDtvxxH%3DVnto2yqkbYw%3DxaVVQnLsVsaK%2BZPQxgWjSw%40mail.gmail.com.
