矮人国的孩子们 Children of the Dwarfs

近年来，随着越来越多的系外行星被发现，我们对行星系的形成与演化的认识也有了显著的发展。不过离我们最近的‘地球’ （类地行星）还是4光年……

With more and more discoveries of new exoplanets in the recent years, our knowledge of the formation and evolution of planetary systems have also developed greatly. Even though the next closest Earth-like planet is still more than 4 light years away …

但这并不是今天的重点。目前发现的大部分系外行星围绕的都是类太阳恒星，而我们又一次忘记了宇宙中‘小众’群体——红矮星（低质量的恒星）周围的行星。或者说，我们不应该仅仅局限于研究我们当前处于的系统，当我们回过头看看那些不一定绕着‘太阳’运行的星星时，说不定会发现一个全新的世界。

However, that’s not the main point here. With the bulk of those discovered ones orbiting around Sun-like stars, we have again forgotten about the minatory groups in the universe - the planets around red dwarfs (the low-mass stars). In another word, we should not be restricted to studying only our solar system’s set up, there is a whole new world out there when we turn around and look at some of the others that doesn’t necessary orbit a ‘Sun’.

例如，2017年观测到的TRAPPIST-1就是一个紧凑的恒星系统，自从被发现以后，它因系统中宜居带有多个类地行星而闻名。而且这并不是一个特例，银河系中80%的恒星（近千亿颗）都是红矮星，所以有非常高的概率找到更多像TRAPPIST-1这样的。因此，在寻找宜居行星时，观察红矮星们也非常重要。红矮星是主序列中最小且温度最低的恒星类型。

TRAPPIST-1, for example, is a compact star system observed in 2017, and since then have made itself famous by having a number of Earth-like planets in the system’s habitable zone. And this is not just a single special case, around 80% of all stars in our Milky Way (～100 thousand million) are red dwarfs. A very high chance of finding more like TRAPPIST-1. Therefore, it is extremely important to look at the red dwarfs as well, when in search for habitable planets. Red dwarf are the smallest and coolest types of stars in the main sequence.

图为TRAPPIST-1（上半） 与太阳系（下半）的比较，图中TRAPPIST-1在被按照比例被放大25倍前，是太阳系中蓝色圆圈的大小，可见现实中它有多紧凑 The TRAPPIST-1 System (top) and Solar System (bottom) in comparison. Before TRAPPIST-1 was scaled up 25 times. it was the size of blue circle in the Solar System, this shows how compact it is in real life

尽管这项额外的任务听起来挺容易的，但由于这些宜居系外行星的低亮度，它们实际上是非常难被探测到的。所以这里NASA新的凌日系外行星勘测卫星（TESS）就能被派上用场啦。它的高灵敏度仪器可以帮助我们探测到环绕红矮星的上百颗系外行星（以下简称为M矮行星）。

Though this extra task sounds easy, due to the low luminosities of those habitable exoplanets, they are actually extremely difficult to detect. And this is where NASA’s Transiting Exoplanet Survey Satellite (TESS) comes into use, with its highly sensitive instruments, it can help us to detect hundreds of exoplanets in orbit around red dwarfs (hereafter M dwarf planets).

现在，让先我们跟随美国一个研究小组的步伐，了解这些M矮行星最初是如何形成的吧。

For now, let us now follow the steps of a research group in the US to learn how these M dwarf planets were formed in the first place.

一颗假想的系外行星围绕着一颗红矮星运行的艺术图像 An illustration of a red dwarf star orbited by a hypothetical exoplanet

成长的路

Growing Up

正如之前几篇文章所描述的那样，宇宙中基本一切都是由夸克和轻子组成的，物体都是由它们组成的块块。原行星盘中的尘埃由就是由分子组成，然后聚拢、凝结，形成星子（由尘埃、岩石和其他物质形成的固体物体），最后相互碰撞形成胚胎——行星的开始。

As explained in a few articles before, everything in this universe is made of the basic quarks and lepton, objects are just big lumps of them. With dust in the protoplanetary disc made of molecules, their accumulation and coagulation form planetesimals  (solid object formed from dust, rock and other materials), which then collide into each other to form embryos – the beginning of a planet.

原行星盘就像行星的子宫，由气体和尘埃组成，为行星的形成提供了最理想的初始孕育环境。值得注意的是，这些原行星盘的性质因光谱类型而异，光谱类型取决于母恒星的温度。由于这里讨论的恒星（红矮星）比太阳小得多，它们的温度也大不相同，所以关于太阳的规则在这并不适用。因此，必须分别研究这些行星的演化。

Protoplanetary discs are like the wombs of planets, consisting ideal gases and dust that provide the initial conditions for planet formation. It is important to note that the properties of these protoplanetary discs vary according to the spectral type, which is dependent on the parent star’s temperature. Since the stars discussed here are much smaller than the Sun, their temperatures are also quite different, the rules for the Sun do not necessarily apply. Therefore, the evolution of these planets has to be studied separately.

胚胎形成后，成为最后的恒星前还要经历许多物理过程，包括潮汐力、力矩、空气阻力和吸积。为了建立一个完整的模型，包括行星形成时发生的所有这些因素，研究小组使用了一个标准n体模拟，以及一些额外的‘力’来涵盖所有物理过程。

After the embryos are formed, they are subject to many physical processes, including tidal forces, torques, aerodynamic drag and accretion. In order to have a complete model with all these factors included for the formation of planets, the team used a standard N-body simulation with some additional forces to cover the range of physical processes.

图为一颗被原行星盘包围的恒星的艺术图像 An illustration of a star surrounded by a protoplanetary disk

最后总共使用了10种不同的模型，每个模型包含147个胚胎，以及不同初始条件的原行星盘和气体含量。气体比例作为一个变量在这些模型中是至关重要的，因为它的存在会导致共振力矩，从而影响胚胎围绕恒星的运动轨迹。

Totally, 10 different models were used, each with 147 embryos as well as different initial conditions of the protoplanetary disc and the amount of gas there. Including the gas proportion as a variable is hugely important, its presence leads to resonant torques, which effects embryo’s motion around the star.

模拟结果表明，大部分M矮行星最终会成为‘超级地球’，质量在地球的2.5 - 5.2倍之间。在所有初始147个胚胎的模拟中，平均96%的碰撞发生在椎间盘形成最初的一百万年。在包含气体的模型中，99%的碰撞发生在气体盘消散之前。因此，大多数行星系统都在原行星盘生命的早期阶段就稳定下来了，那是气体盘也还在。

The simulation shows that the majority of the M dwarf planets formed would be Super-Earths, having masses between 2.5 - 5.2 Earth’s mass. Averaged over all simulations of the initial 147 embryos, 96% of collisions would occur in the first 1 million year of the disc formation. In models that included gas, 99% of collisions occur before the gas disc dissipates. Thus, most planets systems stabilized in the presence of the gas disc, at early stages of protoplanetary disc’s lifetime.

图为所有胚胎的半长轴（以天文单位表示，离母星的最大半径）作为模拟时间（年为单位）的函数。阴影部分代表原行星盘。胚胎颜色是根据其最初的半长轴分配的 Plot of the semi-major axis (maximum radius from the parent star) of all embryos (in AU) as a function of simulation time (in years). The shaded region represents the protoplanetary disc. Embryos are coloured according to their initial semi-major axis

出生地的差异

The Different Hometowns

就像人从出生开始，命运就是不公平的，有些比其他的更精彩。对于行星来说，它们出生的原行星盘环境便决定了它们一生的起跑线。而不可多得的高光时刻便是那些“巨型撞击”，例如那些被认为创造了月球的碰撞。

As we say, 'life is unfair', some more spectacular than others. For planets, the protoplanetary disc environments determines thier starting point in life. And the rare highlights would be those ‘giant impacts’, for example the collisions that are believed to have created the Moon.

研究小组将这些命中注定的相遇定义为两个足够大小的胚胎之间的碰撞。它们的总质量约为地球的一半，其中一个胚胎的重量不超过另一个的5倍。

The team defined these love matches as impacts between two sufficiently sized embryos. With the combined mass around 0.5 Earth masses, one of embryo needs to be no more than 5 times heavier than the other.

模拟显示，绝大部分碰撞发生在原行星盘消散之前，而晚期的巨型撞击发生的频率明显低很多。然而，这并不说明迟来的真爱不会发生，原行星盘外，甚至星盘消散之后都还是有可能的。在含有气体的模拟中，大约四分之一的最终撞击发生在星盘内。这也使得这些行星更有可能保留大气，因为它们能够重新吸积星盘中的气体。

Simulations show that more of these crashes happen before the disk dissipates, while late giant impacts occur significantly rarer. However, that does not stop those late-stage true love happening when being outside the disks or even after the disk dissipates. With the gas contained simulations, about a quarter of the final impacts happened within the disc. There planets are hence more likely to retain atmospheres, since they are able to re-accrete gas from the disc.

图为假想的系外行星围绕着一颗恒星运行的艺术图像 An illustration of a star orbited by a hypothetical exoplanet

找呀 找呀 找朋友

Where’s My Friend

为了更好地了解行星系统，除了模拟外，还需要将模拟结果与观测数据进行比较。这里，研究小组计算了系统中有多于一颗凌日行星的概率，以及它们的轨道周期的比率，称为周期比率。凌日行星是观测时经过恒星和观测者之间的行星。

To get to know the planetary system better, apart from modeling with the simulations, their results should also be compared with observational data. Here, the team calculated the probability of have more than one transiting planet in the system and the ratio between their orbital periods, called the period ratio. A transit planet is one that passes between the star and observer when observing.

假如一颗恒星有一颗凌日行星，研究小组发现存在另外一颗凌日行星的平均概率为64.4%。然而，在无气体模型中，那里往往有较大的半长轴和相对较高的相互倾斜度，所以看到另一颗凌日行星的平均概率直接下降为5.0%。

Given that one planet transits, the team found the average probability of seeing an additional transiting planet has a 64.4% chance. However, gas-free models tend to have larger semi- major axes and relatively high mutual inclinations, so the average probability of seeing an additional transiting planet decreases to only 5.0%.

图为由NASA的TESS发现的第一颗环双星的轨道。这颗土星大小的系外行星名为TOI1338b，距地球1300光年，围绕两颗恒星 (TOI1338A和TOI1338B) 运行 The orbit of the first circumbinary planet found by NASA’s TESS. The Saturn-size exoplanet called TOI1338b orbits two stars (TOI1338A & TOI1338B) 1,300 light-years away

把观测数据和模拟数据画在直方图上后，小组对每个系统的行星对周期进行了比较。这里的观测数据来自开普勒望远镜。

Histograms were then used for the observational and simulated data, to compare the differences between in the period ratio of pairs of planets for each system. The observational data were from the Kepler space telescope.

模拟数据显示，最常见的比例是2左右，也就说这对行星中的一颗行星围绕恒星运行的时间是另一颗行星的两倍。而开普勒样本的峰值则在1.5左右。研究小组将这个差距归因于大多数开普勒观察到的行星，它们围绕的恒星质量更大。

Simulation data shows that the most common ratio is around 2 for the simulated data, meaning one planet in the pair takes twice as long to orbit the star compared to the other one in the pair. While for the Kepler sample, the peak is more around 1.5. The team attribute this difference to the fact that most Kepler planets orbit higher mass stars.

图为三组数据的直方图：基础模拟周期比(粉色)，无气体模拟周期比(蓝色)，开普勒样本的周期比(黑色) The orbit of The histograms of the period ratios for the base simulation (pink), the simulation with no gas (blue) and the sample of planetary pairs found by Kepler (black)

未来可期

Hopes for the Future

从这些模拟中我们可以得出几个关键结论。首先，M矮行星的形成很快，而且随着时间的推移倾向于向内迁移，并因为碰撞而导致几乎所有初始条件“记忆”被破坏。它们的形成周期相对于它们所来自的圆盘的生命周期，是相对较短的。

Several key conclusions can be drawn from these simulations. The first being that M dwarf planets form rapidly and tend to migrate inwards over time, which results in collisions and destroys any ‘memories’ of the initial conditions. Their formation period is relatively short compared to the lifetime of the disc they came from.

其次，行星在一个系统中的位置和密度很大程度上取决于气体盘在形成阶段的存在。有气体的系统往往会是个更小、更紧密的家庭，而没有气体的模型则会产生更多行星，但更分散。在气体盘中进行最后一次碰撞的行星往往更有可能保留大气层。

这就是所谓的‘一团和气’吗 

Secondly, the location and density of planets in a system is largely dependent on the presence of gas disc during the formation stage. Systems brought up with gas tend to have a smaller and closer family, while models without gas produces more planets but more separated. Planets that have their final collisions in the gas discs tends to also retain an atmosphere.

最后，在TESS探测了数百颗M矮行星，并给出了一系列丰富的参数值样本后（如质量、半径等），我们将在未来的模拟中获得更精准的数据条件。

Finally, with TESS detecting hundreds of M dwarf planets, and giving samples of wide range of parameter values (e.g. masses, radii, etc.), it will help to provide tighter constraints on the outputs of future simulations.

不久的未来，

我们将解开更多

红矮星系外行星形成过程的秘密，

以及我们的未来“地球”的位置！

Soon in the future, we will be able to unlock 

more secrets of how red dwarf exoplanets form and where our next ‘Earth’ is!

图为NASA的开普勒任务发现的两颗土星大小的行星的艺术图像 An artist illustration of the two Saturn-sized planets discovered by NASA's Kepler mission

图片来自 NASA, MIT 官网以及以下文章

文中部分英文信息参考来自 B Zawadzki, D Carrera, E Ford 的 ‘Rapid Formation of Super-Earths Around Low-Mass Stars' 文章

其余中英文内容为原创

Images from official website of NASA, Chandra and the below articles

Parts are sited from ‘Rapid Formation of Super-Earths Around Low-Mass Stars' from  B Zawadzki, D Carrera, E Ford

The rest of the Chinese and English content are original