In 2009, two researchers ran a simple experiment. They took everything we know about our solar system and calculated where every planet would be up to 5 billion years in the future. To do so they ran over 2,000 numerical simulations with the same exact initial conditions except for one difference: the distance between Mercury and the Sun, modified by less than a millimeter from one simulation to the next. Shockingly, in about 1 percent of their simulations, Mercury’s orbit changed so drastically that it could plunge into the Sun or collide with Venus. Worse yet, in one simulation it destabilized the entire inner solar system. This was no error; the astonishing variety in results reveals the truth that our solar system may be much less stable than it seems.
Leta 2009 sta dva raziskovalca izvedla preprost poskus. Upoštevala sta vse, kar vemo o našem Sončnem sistemu, in izračunala, kje se bo vsak planet nahajal do pet milijard let v prihodnost. Za to sta izvedla preko 2000 numeričnih simulacij, pod popolnoma enakimi začetnimi pogoji, z eno samo razliko: razdaljo med Merkurjem in Soncem sta za vsako simulacijo spremenila manj kot za milimeter. Šokantno: pri približno odstotku teh simulacij se je Merkurjeva orbita spremenila tako zelo, da bi strmoglavil v Sonce ali trčil v Venero. Še huje, v eni simulaciji je destabiliziral celoten notranji del Osončja. To ni bila napaka; izjemna razlika v rezultatih razkriva, da je naš Sončni sistem morda veliko manj stabilen, kot se zdi.
Astrophysicists refer to this astonishing property of gravitational systems as the n-body problem. While we have equations that can completely predict the motions of two gravitating masses, our analytical tools fall short when faced with more populated systems. It’s actually impossible to write down all the terms of a general formula that can exactly describe the motion of three or more gravitating objects.
Astrofiziki tej neverjetni lastnosti gravitacijskih sistemov pravijo problem n-teles. Medtem ko enačbe lahko popolnoma točno napovejo gibanje dveh teles pod vplivom težnosti, naša analitična orodja pri sistemih z več telesi ne delujejo. Pravzaprav je nemogoče zapisati vse pogoje za splošno formulo, 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. Thanks to Isaac Newton, we can write a set of equations to describe the gravitational force acting between bodies. However, when trying to find a general solution for the unknown variables in these equations, we’re faced with a mathematical constraint: for each unknown, there must be at least one equation that independently describes it.
Zakaj? Težava je v številu neznank v sistemu n-teles. Zaradi Isaaca Newtona znamo zapisati set enačb, ki opišejo silo težnosti med telesi. Vendar se pri iskanju splošne rešitve za neznanke v teh enačbah srečamo z matematično omejitvijo: za vsako neznanko mora obstajati vsaj ena enačba, ki jo neodvisno opiše.
Initially, a two-body system appears to have more unknown variables for position and velocity than equations of motion. However, there’s a trick: consider the relative position and velocity of the two bodies with respect to the center of gravity of the system. This reduces the number of unknowns and leaves us with a solvable system.
Na začetku se zdi, da ima sistem dveh teles več neznank za položaj in hitrost, kot ima enačb za gibanje. Toda trik je v tem: upoštevajte relativni položaj in hitrost teh dveh teles glede na težišče sistema. To zmanjša število neznank in omogoči rešitev sistema.
With three or more orbiting objects in the picture, everything gets messier. Even with the same mathematical trick of considering relative motions, we’re left with more unknowns than equations describing them. There are simply too many variables for this system of equations to be untangled into a general solution.
Težava pa nastane s tremi ali več telesi, ki medsebojno krožijo. Tudi z istim matematičnim trikom uporabe relativnih gibanj imamo več neznank kot enačb, ki jih opisujejo. Za tak sistem enačb je enostavno preveč spremenljivk, da bi jih razvozlali v splošno rešitev.
But what does it actually look like for objects in our universe to move according to analytically unsolvable equations of motion? A system of three stars— like Alpha Centauri— could come crashing into one another or, more likely, some might get flung out of orbit after a long time of apparent stability. Other than a few highly improbable stable configurations, almost every possible case is unpredictable on long timescales. Each has an astronomically large range of potential outcomes, dependent on the tiniest of differences in position and velocity. This behaviour is known as chaotic by physicists, and is an important characteristic of n-body systems. Such a system is still deterministic— meaning there’s nothing random about it. If multiple systems start from the exact same conditions, they’ll always reach the same result. But give one a little shove at the start, and all bets are off. That’s clearly relevant for human space missions, when complicated orbits need to be calculated with great precision.
Kaj pa pravzaprav pomeni, če se telesa v vesolju premikajo po analitično nerešljivih enačbah gibanja? V sistemu treh zvezd, denimo Alfa Centauri, bi te lahko trčile ena v drugo, ali, kar je bolj verjetno, katera od njih bi morda odletela iz orbite po dolgem času stabilnosti. Razen nekaj skrajno neverjetnih stabilnih konfiguracij je skoraj vsak mogoč primer na dolgi rok nepredvidljiv. Vsak ima astronomsko velik razpon možnih izidov, odvisno od najmanjših sprememb položaja in hitrosti. Takemu obnašanju fiziki pravijo kaotično in je pomembna značilnost sistema n teles. Tak sistem je še vedno determinističen, se pravi, da na njem ni nič naključnega. Če več sistemov začne pod povsem enakimi pogoji, bodo vedno prišli do istega rezultata. Ampak če enega na začetku le malo sunemo, se lahko zgodi karkoli. To je seveda pomembno za odprave v vesolje, ko je treba zapletene orbite zelo natančno izračunati.
Thankfully, continuous advancements in computer simulations offer a number of ways to avoid catastrophe. By approximating the solutions with increasingly powerful processors, we can more confidently predict the motion of n-body systems on long time-scales. And if one body in a group of three is so light it exerts no significant force on the other two, the system behaves, with very good approximation, as a two-body system. This approach is known as the “restricted three-body problem.” It proves extremely useful in describing, for example, an asteroid in the Earth-Sun gravitational field, or a small planet in the field of a black hole and a star.
Na srečo stalni razvoj računalniških simulacij na več načinov pomaga odvrniti katastrofe. S približnimi rešitvami s pomočjo vse močnejših procesorjev lahko z večjo gotovostjo napovemo gibanje sistemov n teles na dolgi rok. In če je eno telo v sistemu treh tako lahko, da na ostali dve ne izvaja pomembne sile, se sistem obnaša zelo približno tako kot sistem dveh teles. Ta pristop je znan kot “omejen problem treh teles”. Zelo uporaben je denimo pri opisovanju asteroida v težnostnem polju Zemlje in Sonca ali majhnega planeta v polju črne luknje in zvezde.
As for our solar system, you’ll be happy to hear that we can have reasonable confidence in its stability for at least the next several hundred million years. Though if another star, launched from across the galaxy, is on its way to us, all bets are off.
Za naš Sončni sistem vam z veseljem povem, da smo dokaj prepričani v njegovo stabilnost, vsaj za naslednjih nekaj sto milijonov let. Če pa je na poti nova zvezda, ki prihaja z druge strani galakcije, se lahko zgodi karkoli.