SETI and SEH (Statistical Equation for Habitables)

Claudio Maccone

摘要

The statistics of habitable planets may be based on a set of ten (and possibly more) astrobiological requirements first pointed out by Stephen H. Dole in his book "Habitable planets for man" (1964). In this paper, we first provide the statistical generalization of the original and by now too simplistic Dole equation. In other words, a product of ten positive numbers is now turned into the product of ten positive random variables. This we call the SEH, an acronym standing for "Statistical Equation for Habitables". The mathematical structure of the SEH is then derived. The proof is based on the central limit theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be arbitrarily distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov form of the CLT, or the Lindeberg form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that (1) The new random variable N_(Hab), yielding the number of habitables (i.e. habitable planets) in the Galaxy, follows the lognormal distribution. By construction, the mean value of this lognormal distribution is the total number of habitable planets as given by the statistical Dole equation. But now we also derive the standard deviation, the mode, the median and all the moments of this new lognormal N_(Hab) random variable. (2) The ten (or more) astrobiological factors are now positive random variables. The probability distribution of each random variable may be arbitrary. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT "translates" into our SEH by allowing an arbitrary probability distribution for each factor. This is both astrobiologically realistic and useful for any further investigations. (3) An application of our SEH then follows. The (average) distance between any two nearby habitable planets in the Galaxy may be shown to be inversely proportional to the cubic root of N_(Hab)- Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density function, apparently previously unknown and dubbed "Maccone distribution" by Paul Davies in 2008. (4) Data Enrichment Principle. It should be noticed that ANY positive number of random variables in the SEH is compatible with the CLT. So, our generalization allows for many more factors to be added in the future as long as more refined scientific knowledge about each factor will be known to the scientists. This capability to make room for more future factors in the SEH we call the "Data Enrichment Principle", and we regard it as the key to more profound future results in the fields of Astrobiology and SETI. (5) A practical example is then given of how our SEH works numerically. We work out in detail the case where each of the ten random variables is uniformly distributed around its own mean value as given by Dole back in 1964 and has an assumed standard deviation of 10%. The conclusion is that the average number of habitable planets in the Galaxy should be around 100 million ± 200 million, and the average distance in between any couple of nearby habitable planets should be about 88 light years ± 40 light years. (6) Finally, we match our SEH results against the results of the Statistical Drake Equation that we introduced in our 2008 IAC presentation. As expected, the number of currently communicating ET civilizations in the Galaxy turns out to be much smaller than the number of habitable planets (about 10,000 against 100 million, i.e. one ET civilization out of 10,000 habitable planets). And the average distance between any two nearby habitable planets turns out to be much smaller than the average distance between any two neighboring ET civilizations: 88 light years vs. 2000 light years, respectively. This means an ET average distance about 20 times higher than the average distance between any couple of adjacent habitable planets.

机译：可居住行星的统计数据可能基于斯蒂芬·H·多尔（Stephen H. Dole）在其《人类可居住的行星》（1964）一书中首次提出的十个（甚至更多）天体生物学要求。在本文中，我们首先提供原始的统计归纳法，而现在则过于简单化了Dole方程。换句话说，现在将十个正数的乘积变成十个正数随机变量的乘积。我们称其为SEH，是“可居住者统计方程式”的首字母缩写。然后推导SEH的数学结构。该证明基于统计的中心极限定理（CLT）。用宽松的术语来说，CLT指出，任意数量的独立随机变量的总和接近高斯（即正态）随机变量，每个变量可以任意分布。这被称为CLT的Lyapunov形式，或CLT的Lindeberg形式，这取决于各种概率分布的第三阶矩所假定的数学约束。总而言之，我们表明（1）新的随机变量N_（Hab）产生银河系中可居住的行星（即可居住的行星）的数量遵循对数正态分布。通过构造，该对数正态分布的平均值是统计Dole方程式给出的宜居行星总数。但是，现在我们还可以导出这个新的对数正态N_（Hab）随机变量的标准偏差，众数，中位数和所有时刻。（2）十个（或更多）天体生物学因素现在是正随机变量。每个随机变量的概率分布可以是任意的。所谓的李雅普诺夫（Lyapunov）或林德伯格（Lindeberg）形式的CLT（都没有假定因子分布相同）允许这样做。换句话说，CLT通过允许每个因素的任意概率分布来“转换”为我们的SEH。这在天文生物学上是现实的，对于任何进一步的研究都是有用的。（3）然后是我们的SEH的应用程序。星系中任何两个邻近的宜居行星之间的（平均）距离可能与N_（Hab）的立方根成反比。然后，在我们的方法中，该距离成为一个新的随机变量。我们推导了相关的概率密度函数，该函数以前是未知的，并在2008年被Paul Davies称为“ Maccone分布”。（4）数据丰富原理。应当注意，SEH中任何正数的随机变量都与CLT兼容。因此，只要科学家能够了解关于每个因素的更完善的科学知识，我们的概括就可以在将来添加更多因素。这种为SEH中的更多未来因素腾出空间的能力称为“数据丰富原则”，我们将其视为在天体生物学和SETI领域获得更深刻的未来成果的关键。（5）然后给出一个实际的例子，说明我们的SEH如何进行数字运算。我们详细研究了以下情况：十个随机变量中的每一个均按照Dole在1964年给出的平均值围绕其自己的平均值均匀分布，并且假设标准偏差为10％。结论是，银河系中的宜居行星的平均数量应为1亿±2亿个，附近任何两个宜居行星之间的平均距离应为88光年±40光年。（6）最后，我们将SEH结果与我们在2008 IAC演示中引入的统计Drake方程的结果进行匹配。正如预期的那样，银河系中目前正在交流的ET文明的数量远小于可居住的行星的数量（约10,000对1亿，即10,000个可居住的行星中的一个ET文明）。而且，附近任何两个宜居行星之间的平均距离远小于任何两个相邻的ET文明之间的平均距离：分别为88光年与2000光年。这意味着ET的平均距离比任何两个相邻的可居住行星之间的平均距离高大约20倍。

SETI and SEH (Statistical Equation for Habitables)

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