Newton’s three-body problem explained - Fabio Pacucci

Reviewer: Nika Kotnik

In 2009, two researchers ran a simple experiment.

Leta 2009 sta dva raziskovalca izvedla preprost poskus.

They took everything we know about our solar system

Upoštevala sta vse, kar vemo o našem Sončnem sistemu,

and calculated where every planet would be up to 5 billion years in the future.

in izračunala, kje se bo vsak planet nahajal do pet milijard let v prihodnost.

To do so they ran over 2,000 numerical simulations

Za to sta izvedla preko 2000 numeričnih simulacij,

with the same exact initial conditions except for one difference:

pod popolnoma enakimi začetnimi pogoji, z eno samo razliko:

the distance between Mercury and the Sun, modified by less than a millimeter

razdaljo med Merkurjem in Soncem sta za vsako simulacijo

from one simulation to the next.

spremenila manj kot za milimeter.

Shockingly, in about 1 percent of their simulations,

Šokantno: pri približno odstotku teh simulacij

Mercury’s orbit changed so drastically that it could plunge into the Sun

se je Merkurjeva orbita spremenila tako zelo, da bi strmoglavil v Sonce

or collide with Venus.

ali trčil v Venero.

Worse yet,

Še huje,

in one simulation it destabilized the entire inner solar system.

v eni simulaciji je destabiliziral celoten notranji del Osončja.

This was no error; the astonishing variety in results

To ni bila napaka; izjemna razlika v rezultatih

reveals the truth that our solar system may be much less stable than it seems.

razkriva, da je naš Sončni sistem morda veliko manj stabilen, kot se zdi.

Astrophysicists refer to this astonishing property of gravitational systems

Astrofiziki tej neverjetni lastnosti gravitacijskih sistemov

as the n-body problem.

pravijo problem n-teles.

While we have equations that can completely predict

Medtem ko enačbe lahko popolnoma točno napovejo

the motions of two gravitating masses,

gibanje dveh teles pod vplivom težnosti,

our analytical tools fall short when faced with more populated systems.

naša analitična orodja pri sistemih z več telesi ne delujejo.

It’s actually impossible to write down all the terms of a general formula

Pravzaprav je nemogoče zapisati vse pogoje za splošno formulo,

that can exactly describe the motion of three or more gravitating objects.

ki bi opisala gibanje treh ali več teles pod vplivom težnosti.

Why? The issue lies in how many unknown variables an n-body system contains.

Zakaj? Težava je v številu neznank v sistemu n-teles.

Thanks to Isaac Newton, we can write a set of equations

Zaradi Isaaca Newtona znamo zapisati set enačb,

to describe the gravitational force acting between bodies.

ki opišejo silo težnosti med telesi.

However, when trying to find a general solution for the unknown variables

Vendar se pri iskanju splošne rešitve za neznanke

in these equations,

v teh enačbah

we’re faced with a mathematical constraint:

srečamo z matematično omejitvijo:

za vsako neznanko mora obstajati vsaj ena enačba,

for each unknown, there must be at least one equation

that independently describes it.

ki jo neodvisno opiše.

Initially, a two-body system appears to have more unknown variables

Na začetku se zdi, da ima sistem dveh teles več neznank

for position and velocity than equations of motion.

za položaj in hitrost, kot ima enačb za gibanje.

However, there’s a trick:

Toda trik je v tem:

consider the relative position and velocity of the two bodies

upoštevajte relativni položaj in hitrost teh dveh teles

with respect to the center of gravity of the system.

glede na težišče sistema.

This reduces the number of unknowns and leaves us with a solvable system.

To zmanjša število neznank in omogoči rešitev sistema.

With three or more orbiting objects in the picture, everything gets messier.

Težava pa nastane s tremi ali več telesi, ki medsebojno krožijo.

Even with the same mathematical trick of considering relative motions,

Tudi z istim matematičnim trikom uporabe relativnih gibanj

we’re left with more unknowns than equations describing them.

imamo več neznank kot enačb, ki jih opisujejo.

There are simply too many variables for this system of equations

Za tak sistem enačb je enostavno preveč spremenljivk,

to be untangled into a general solution.

da bi jih razvozlali v splošno rešitev.

But what does it actually look like for objects in our universe

Kaj pa pravzaprav pomeni, če se telesa v vesolju

to move according to analytically unsolvable equations of motion?

premikajo po analitično nerešljivih enačbah gibanja?

A system of three stars— like Alpha Centauri—

V sistemu treh zvezd, denimo Alfa Centauri,

could come crashing into one another or, more likely,

bi te lahko trčile ena v drugo, ali, kar je bolj verjetno,

some might get flung out of orbit after a long time of apparent stability.

katera od njih bi morda odletela iz orbite po dolgem času stabilnosti.

Other than a few highly improbable stable configurations,

Razen nekaj skrajno neverjetnih stabilnih konfiguracij

almost every possible case is unpredictable on long timescales.

je skoraj vsak mogoč primer na dolgi rok nepredvidljiv.

Each has an astronomically large range of potential outcomes,

Vsak ima astronomsko velik razpon možnih izidov,

dependent on the tiniest of differences in position and velocity.

odvisno od najmanjših sprememb položaja in hitrosti.

This behaviour is known as chaotic by physicists,

Takemu obnašanju fiziki pravijo kaotično

and is an important characteristic of n-body systems.

in je pomembna značilnost sistema n teles.

Such a system is still deterministic— meaning there’s nothing random about it.

Tak sistem je še vedno determinističen, se pravi, da na njem ni nič naključnega.

If multiple systems start from the exact same conditions,

Če več sistemov začne pod povsem enakimi pogoji,

they’ll always reach the same result.

bodo vedno prišli do istega rezultata.

But give one a little shove at the start, and all bets are off.

Ampak če enega na začetku le malo sunemo, se lahko zgodi karkoli.

That’s clearly relevant for human space missions,

To je seveda pomembno za odprave v vesolje,

when complicated orbits need to be calculated with great precision.

ko je treba zapletene orbite zelo natančno izračunati.

Thankfully, continuous advancements in computer simulations

Na srečo stalni razvoj računalniških simulacij

offer a number of ways to avoid catastrophe.

na več načinov pomaga odvrniti katastrofe.

By approximating the solutions with increasingly powerful processors,

S približnimi rešitvami s pomočjo vse močnejših procesorjev

we can more confidently predict the motion of n-body systems on long time-scales.

lahko z večjo gotovostjo napovemo gibanje sistemov n teles na dolgi rok.

And if one body in a group of three is so light

In če je eno telo v sistemu treh tako lahko,

it exerts no significant force on the other two,

da na ostali dve ne izvaja pomembne sile,

the system behaves, with very good approximation, as a two-body system.

se sistem obnaša zelo približno tako kot sistem dveh teles.

This approach is known as the “restricted three-body problem.”

Ta pristop je znan kot “omejen problem treh teles”.

It proves extremely useful in describing, for example,

Zelo uporaben je denimo pri opisovanju

an asteroid in the Earth-Sun gravitational field,

asteroida v težnostnem polju Zemlje in Sonca

or a small planet in the field of a black hole and a star.

ali majhnega planeta v polju črne luknje in zvezde.

As for our solar system, you’ll be happy to hear

Za naš Sončni sistem vam z veseljem povem,

that we can have reasonable confidence in its stability

da smo dokaj prepričani v njegovo stabilnost,

for at least the next several hundred million years.

vsaj za naslednjih nekaj sto milijonov let.

Though if another star,

Če pa je na poti nova zvezda,

ki prihaja z druge strani galakcije,

launched from across the galaxy, is on its way to us,

se lahko zgodi karkoli.

all bets are off.