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.
2009 年,兩位研究者 做了一項簡單的實驗。 他們用上了我們 對太陽系所知的一切, 去計算五十億年後 每一顆行星的所在。 為了做到這一點,他們進行了 超過兩千次的數值模擬, 每一次的初始條件都相同, 除了一個差異: 從一次模擬進入到下一次模擬時, 就把水星和太陽之間的 距離增或減一公釐。 驚人的是,大約 1% 的模擬中, 水星的軌道大大改變, 大到有可能會衝進太陽 或撞上金星。 更糟的是,在一次模擬中, 它讓整個內太陽系變得很不穩定。 這不是錯誤;結果會有 這麼驚人的多樣性, 表示我們的太陽系事實上 可能沒有看起來這麼穩定。
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.
天體物理學家把這種 重力系統的驚人特質 稱為「N 體問題」。 雖然我們有方程式可以完全預測 兩個互相受引力作用的 質量會如何運動, 但面臨更多物體的系統時, 我們的分析工具就有所不足了。 事實上,不可能寫出一條通式 來精準描述互相受引力作用的 三個(或以上)物體如何運動。
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.
為什麼? 問題在於 N 體系統中 有多少個未知的變數。 因為牛頓的功勞, 我們可以寫出一組方程式 來描述兩個物體之間的引力作用。 然而,當試圖為 這些方程式中的未知變數 找出通解時, 我們面臨一個數學限制: 凡是有一個未知變數, 就必須要有至少一條 獨立的方程式來描述它。
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.
最初看似兩體系統未知的 位置和速度變量的數目 多於運動方程式的。 然而有一招: 考量兩個物體相對於 系統引力中心的位置和速度。 這樣就能減少未知變數的數目, 讓它變成有解的系統。
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.
若系統中有三個以上的繞行物體, 情況就會更亂了。 即使採用同樣的數學招式 去考量相對運動, 未知變數的數目仍多於 描述它們的方程式數目。 簡單來說就是這個 方程式系統有太多變數, 因此無法用一個通解來解決。
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.
但我們宇宙中的物體 根據無解的運動方程式運轉, 實際上看起來會是什麼模樣? 三個恆星的系統—— 比如南門二—— 有可能會撞上彼此, 或更有可能的情況是, 在經過長時間明顯的穩定之後, 有些恆星可能會被拋出軌道。 除了少數極不可能發生的 穩定組態之外, 幾乎每一個可能的情況 在長期來看都是無法預測的。 每一個情況在天文學上 都有廣泛的可能結果, 會根據位置及速度的 微小差距而有所不同。 物理學家將這種行為視為「混亂」, 是 N 體系統的重要特徵之一。 這種系統仍是確定性的系統, 意即它並不隨機。 如果有多個系統 都從同樣的條件開始, 它們一定會達到同樣的結果。 但把初始條件稍微改變一點點, 原本的預測就都不準了。 這很顯然會影響到人類的太空任務, 因為需要非常精確地 計算複雜的軌道。
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.
謝天謝地,電腦模擬的持續進步 提供了數種避免大災難的方式。 透過使用越來越強大的 處理器來找出近似解, 我們便能更有信心地預測 N 體系統的長期運動。 如果三個物體中有一個特別輕, 輕到它對其他兩個物體 不會產生明顯的引力, 這個系統的行為就會 非常近似二體系統。 這個方法就是所謂的 「設限三體問題」。 它被證明相當有用, 適用的例子包括 描述在地球太陽重力場中的小行星, 或者在黑洞與恆星力場中的小行星。
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.
至於我們的太陽系, 你會很高興聽到, 我們可以合理地肯定 它在接下來的數億年都會是穩定的。 但如果有另一顆恆星 從銀河系的另一端出發, 朝我們前來, 原本的預測就都不準了。