Newton’s three-body problem explained - Fabio Pacucci

譯者: Lilian Chiu 審譯者: Pui-Ching Siu

In 2009, two researchers ran a simple experiment.

2009 年，兩位研究者 做了一項簡單的實驗。

They took everything we know about our solar system

他們用上了我們 對太陽系所知的一切，

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

去計算五十億年後

每一顆行星的所在。

To do so they ran over 2,000 numerical simulations

為了做到這一點，他們進行了 超過兩千次的數值模擬，

with the same exact initial conditions except for one difference:

每一次的初始條件都相同， 除了一個差異：

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

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。

from one simulation to the next.

Shockingly, in about 1 percent of their simulations,

驚人的是，大約 1% 的模擬中，

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

水星的軌道大大改變，

大到有可能會衝進太陽

or collide with Venus.

或撞上金星。

Worse yet,

更糟的是，在一次模擬中，

in one simulation it destabilized the entire inner solar system.

它讓整個內太陽系變得很不穩定。

This was no error; the astonishing variety in results

這不是錯誤；結果會有 這麼驚人的多樣性，

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

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

天體物理學家把這種 重力系統的驚人特質

as the n-body problem.

稱為「N 體問題」。

While we have equations that can completely predict

雖然我們有方程式可以完全預測

the motions of two gravitating masses,

兩個互相受引力作用的 質量會如何運動，

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

但面臨更多物體的系統時，

我們的分析工具就有所不足了。

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

事實上，不可能寫出一條通式

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

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.

為什麼？

問題在於 N 體系統中

有多少個未知的變數。

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

因為牛頓的功勞， 我們可以寫出一組方程式

to describe the gravitational force acting between bodies.

來描述兩個物體之間的引力作用。

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

然而，當試圖為 這些方程式中的未知變數

in these equations,

找出通解時，

we’re faced with a mathematical constraint:

我們面臨一個數學限制：

for each unknown, there must be at least one equation

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。

that independently describes it.

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

最初看似兩體系統未知的

位置和速度變量的數目

for position and velocity than equations of motion.

多於運動方程式的。

However, there’s a trick:

然而有一招：

consider the relative position and velocity of the two bodies

考量兩個物體相對於

with respect to the center of gravity of the system.

系統引力中心的位置和速度。

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

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.

若系統中有三個以上的繞行物體，

情況就會更亂了。

Even with the same mathematical trick of considering relative motions,

即使採用同樣的數學招式 去考量相對運動，

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

未知變數的數目仍多於 描述它們的方程式數目。

There are simply too many variables for this system of equations

簡單來說就是這個 方程式系統有太多變數，

to be untangled into a general solution.

因此無法用一個通解來解決。

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

但我們宇宙中的物體

根據無解的運動方程式運轉，

to move according to analytically unsolvable equations of motion?

實際上看起來會是什麼模樣？

A system of three stars— like Alpha Centauri—

三個恆星的系統—— 比如南門二——

could come crashing into one another or, more likely,

有可能會撞上彼此， 或更有可能的情況是，

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

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。

Other than a few highly improbable stable configurations,

除了少數極不可能發生的 穩定組態之外，

almost every possible case is unpredictable on long timescales.

幾乎每一個可能的情況

在長期來看都是無法預測的。

Each has an astronomically large range of potential outcomes,

每一個情況在天文學上 都有廣泛的可能結果，

dependent on the tiniest of differences in position and velocity.

會根據位置及速度的 微小差距而有所不同。

This behaviour is known as chaotic by physicists,

物理學家將這種行為視為「混亂」，

and is an important characteristic of n-body systems.

是 N 體系統的重要特徵之一。

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

這種系統仍是確定性的系統，

意即它並不隨機。

If multiple systems start from the exact same conditions,

如果有多個系統 都從同樣的條件開始，

they’ll always reach the same result.

它們一定會達到同樣的結果。

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

但把初始條件稍微改變一點點，

原本的預測就都不準了。

That’s clearly relevant for human space missions,

這很顯然會影響到人類的太空任務，

when complicated orbits need to be calculated with great precision.

因為需要非常精確地 計算複雜的軌道。

Thankfully, continuous advancements in computer simulations

謝天謝地，電腦模擬的持續進步

offer a number of ways to avoid catastrophe.

提供了數種避免大災難的方式。

By approximating the solutions with increasingly powerful processors,

透過使用越來越強大的 處理器來找出近似解，

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

我們便能更有信心地預測

N 體系統的長期運動。

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

如果三個物體中有一個特別輕，

it exerts no significant force on the other two,

輕到它對其他兩個物體 不會產生明顯的引力，

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

這個系統的行為就會 非常近似二體系統。

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

這個方法就是所謂的 「設限三體問題」。

It proves extremely useful in describing, for example,

它被證明相當有用， 適用的例子包括

an asteroid in the Earth-Sun gravitational field,

描述在地球太陽重力場中的小行星，

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

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

至於我們的太陽系， 你會很高興聽到，

that we can have reasonable confidence in its stability

我們可以合理地肯定

它在接下來的數億年都會是穩定的。

for at least the next several hundred million years.

Though if another star,

但如果有另一顆恆星

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

從銀河系的另一端出發， 朝我們前來，

all bets are off.

原本的預測就都不準了。