Archive for the ‘Lagrange’ Category
Apologies for this meandering piece of …
from Wikipedia – ‘Lagrange points in the Sun-Earth system (not to scale). This view is from the north, such that Earth’s orbit is anti-clockwise’
Living in Australia is such a relief, for someone who can’t afford to choose where in the world he might live, and happily being far from the madding action. I virtually never look at Aussie politics, being strangely and unhealthily drawn to more dysfunctional (IMHO) regions, such as the USA, Putinland, China, the Middle East etc. From my safe perch I can lecture these dysfunctional and brutal regions without fear of any blowback, and yet I fear, somehow, that nobody’s listening…
Global politics is a bit of a mess these days, to state the bleeding obvious. The USA in particular has gotten what it deserves, in boasting that any citizen can become President. Of course it isn’t true, but it’s just another claim used to make those citizens feel superior, even if they’re living in a prison cell.
Lately I’ve been somewhat pre-occupied with the artificiality of nations – of the nation concept. We make things up, and then come to believe in them as real. There are many examples – back in my university days (I was a late-goer to university, commencing in my thirtieth year and hanging around almost to my fortieth) I had a conversation with a bright young friend who pointed out that human rights were a made-up thing, and so…. Of course this is true, but, as I didn’t point out to him, because I’m rather slow-witted – and that’s the advantage of writing, based on reflections in tranquility – the human world is full of made-up things, like groups of buildings we designate as forming a university, or like formations made of wood and other materials that we call tables and chairs and houses.
So what is real? The land beneath our feet. The planet that gravity pushes us into (so gravity is real, it seems). The sun, the moon and the stars, which we now know are also suns. The mysterious universe that we’ve just begun to explore. The world revealed to us by microscopes and so many other technologies – genes, neurons, hormones, an enormous variety of cells and sub-cellular organelles, and of course myriad bacteria and archaea never known to exist until relatively recently. The near-infinite number of simple and complex organisms we share the planet with, including huge numbers that had their heyday, or relatively brief period on earth, long ago, like the Ediacaran biota (first found here in South Australia – and not so brief, they flourished between 600 and 540 million years ago, while we have so far managed well under half a million years, and we surely won’t manage much longer).
And then there are things we’re not quite sure are real (though some are surely more sure than others), such as photons, which are apparently massless particles, surely a contradiction in terms. Let me see, light comes in wave and also particle forms? For mass and energy are related according to Einstein’s equation, which is godlike in its omnipotence. Waves have energy, doubtless, I’ve felt that energy while floating about in the sea. Energy is, apparently, mass multiplied by the constant c, which is the speed of light, multiplied by itself, though I’ve heard that nothing can move faster than the speed of light, so…we can convert energy into mass only by means of an equation which is…. impossible, sort of? It would be a matter (bad choice of words) of a teeny-tiny mass being converted, who knows how, into a super-duper quantity of energy, and maybe even vice versa, and I’m not sure if knowing this, if we really do know it, helps us to understand – stuff. But everybody who’s very smart says that it helps us very much to understand stuff, like how space and time are intimately related, because… well, maybe because of that equation, which perhaps also explains how particles can get around without having any mass. And I don’t know if all this is very exciting or just mind-numbing or what. And then there are muons, gluons, colliding hadrons, quarks, neutrinos, bosons and maybe even god particles, for those who are still religious, for god knows what it’s all about. But I’m assured that it’s all more or less calculated so I should just shut up.
So, is all this real? Is dark matter really dark? Is dark energy a matter of expansion? Is the multiverse a bad joke? Is quantum tunnelling just a bore? Is theory just stringing us along?
But seriously, we really are clever – or okay, they, the clever ones – because they’ve landed people on the moon and roving machinery on Mars and placed other machinery at Lagrange points…
Now that’s a subject I want to get clear about. There are five Lagrange points created by the gravitational forces of the sun and our planet – the balance of those gravitational interactions. I’ll quote the clever ones from the European Space Agency to be clear:
There are five other locations around a planet’s orbit where the gravitational forces and the orbital motion of the spacecraft, Sun and planet interact to create a stable location from which to make observations.
But not-so-clever me, when he first conceptualised this, assumed there would be only one of those points, much closer to Earth than the Sun of course, due to the Sun’s bigness. How did they manage to locate/calculate five? The ESA’s description and explanation of these five points (L1 to L5) is intriguing, though it doesn’t answer my question. Here’s their description of L1:
The closer an object is to the Sun, the faster it will move. So, any spacecraft going around the Sun in an orbit smaller than Earth’s will soon overtake our planet. However, there is a loophole: if the spacecraft is placed directly between the Sun and Earth, Earth’s gravity pulls it in the opposite direction and cancels some of the Sun’s pull. With a weaker pull towards the Sun, the spacecraft needs less speed to maintain its orbit, so it can slow down.
I do actually get this, I think, but what about the other Lagrange points (named, by the way, after the Italian turned French astronomer, mathematician and physicist Joseph-Louis Lagrange (1736-1813)? And also by the way, Lagrangian mechanics is a whole nother field worth exploring, or not).
So, in my own words (sort of), every two-body gravitational system like that of Earth and Sun has five of these points. L1, L2 and L3 are called unstable points, while L4 and L5 are stable. So, those first three are gravitationally unstable, and they align with the Earth and the Sun. L1, being between Earth and Sun, offers an unimpeded view for solar observations. L2, being behind the Earth vis-a-vis the Sun (by about 1.5 million kilometres), is good for viewing deep space, in a direction away from the Sun-Earth-Moon system. So that’s where the Just Wonderful Space Telescope (JWST) is, and where the Nancy Grace Roman will be. Plenty of space apparently (haha), and other probes are stationed there too.
So, onto L3, L4 and L5, and I’m simplifying everything massively here of course. L3 is behind the Sun, from our perspective, and its orbit is just a bit wider than Earth’s. Any probe in that position can observe the far side of the Sun… which makes me wonder, does the Sun spin? Well, according to AI (never lies) it most certainly does:
Yes, the sun rotates on its own axis.However, because it is a giant ball of plasma and not a solid object like Earth, it spins at different speeds depending on the latitude and depth.
- The Equator: Spins the fastest, taking about 24.5 Earth days to complete one rotation.
- The Poles: Spin the slowest, taking over 34 days to complete one rotation.
- The Core: Rotates as a solid body much faster, turning about once a week.
Also, Lagrange points have much to do with the three body problem, which I may or may not get into. I can guess at least what the three bodies would be, re Lagrange points that is, for our region – the Sun, the Earth, and, say, JWST….
Anyway, I think I’ve meandered long enough. Maybe next time I’ll discombobulate myself with further details.
References
https://www.esa.int/Enabling_Support/Operations/What_are_Lagrange_points
https://www.space.com/does-the-sun-rotate
https://en.wikipedia.org/wiki/Lagrange_point#/media/File:Lagrange_points_simple.svg
on Lagrange points…
The five Lagrange points in the Earth-Sun system (not to scale obviously). I can only understand L1
So sometimes I just want to understand things – and not just advocate for female domination. For example, what exactly are Lagrange points, why are they important, and who was Lagrange, when he wasn’t Laplace?
First the easy stuff. Joseph-Louis Lagrange (1736-1813) was an Italian-born French naturalist (mathematician/astronomer/physicist). He also has an Italian name, and note that Italy wasn’t a country in his day, and France had quite flexible boundaries. In fact he was born in Turin, which then belonged to the kingdom of Sardinia. Most of his best work was produced in a Prussian city called Berlin. So much for the enduring permanence of nations.
The list of Lagrange’s mathematical contributions is long, and my general mathematical understanding is minuscule, but my fascination with the very sensible notion that there should be a point or region between two massive, gravitationally attracting bodies, such as, say, two planets, where an object would be ‘suspended’ between those two bodies, as their opposite forces (but gravity isn’t a force, they keep telling me), are counter-balanced – that fascination has brought me to attempt to understand, to know more…
So here’s a Wikipedia quote on Lagrange:
He studied the three-body problem for the Earth, Sun and Moon (1764) and the movement of Jupiter’s satellites (1766), and in 1772 found the special-case solutions to this problem that yield what are now known as Lagrangian points.
I’m thinking maybe that my description of a body in a space between two other bodies exerting a more or less equal and opposite gravitational attraction upon it has something to do with this ‘three body problem’ that I’ve heard about only recently. And again, looking at Wikipedia, that magical resource, this seems to be the case:
In celestial mechanics, the Lagrange points… also Lagrangian points or libration points) are points of equilibrium for small-mass objects under the gravitational influence of two massive orbiting bodies. Mathematically, this involves the solution of the restricted three-body problem.
And this is where it gets very complicated, at least for me. The restricted three-body problem seems to be, in essence, a two-body problem, due to the third body’s mass being negligible in the Newtonian scheme of things, such as in the case of a satellite or small ‘planetoid’. In such a situation, at such points, the two large gravitational forces and the centrifugal force are in balance. The centrifugal force is a type of inertial force in Newtonian mechanics. But how can a force be inert? When it’s not a force, obviously. It’s also called a fictitious or pseudo force, but such forces appear to act when viewed in a ‘rotating frame of reference’. And it must be hard to dismiss such rotating frames when we consider that our Earth rotates on its axis, our solar system rotates around its sun and our galaxy rotates around its black hole. And maybe our universe rotates around its centre, if it has one.
But I’m only writing this to avoid the mathematics. Anyway the point about rotating frames of reference is that, if that frame is regular or constant, as is the Earth’s rotation, it will appear to be stationary, and ‘the standard’, which can lead to confusion about other observable bodies, a confusion that lasted for millennia before the likes of Galileo and Newton began to question what had hitherto seemed obvious.
So, Newton’s second law of motion can’t be avoided. I’ll first state it in English words, then… I’m not sure how much further I’ll get:
At any instant of time, the net force on a body is equal to the body’s acceleration multiplied by its mass or, equivalently, the rate at which the body’s momentum is changing with time.
Apparently the dummy’s version of this is F = ma (force equals mass times acceleration), and the more sciencey versions are:
F = m.dv/dt = ma
F = d/dt.(mv)… where d stands for derivative, v for velocity and t for time.
And there are other versions, I think. It’s this second law that has proved the most controversial and it seems the most fruitful for further research and analysis. But don’t trust me on any of this. What is most interesting is that this classical description of forces has been fruitful enough for later (but not much later!) physicists like Lagrange to work out mathematically certain points in space where satellites and telescopes can hover or circulate well beyond Earth’s atmosphere. We now know of five Lagrange points within the Earth-Sun gravitational system, and another five within the Earth-Moon system. To explain why there are so many would be beyond my current level of competence, but I intend to try an online course in classical mechanics, to get me up to speed, or up to equilibrium.
References
https://en.wikipedia.org/wiki/Lagrange_point
https://en.wikipedia.org/wiki/Joseph-Louis_Lagrange