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About 1772 Bode announced a law which gives approximately the distances from the sun of all the planets then known. This law seems to be merely a curious coincidence, having no natural basic or physical foundation. Briefly stated it is as follows:“Write a series of 4’s; under each one after the first write a series as follows:3,6,12,24,48,etc.; the sums, when divided by 10, will give approximately the relative distances of the successive planets in terms of the distance of the earth as the astronomical unit.” The algebiaic expression is 4+3*2^(n-2).This expression, however, does not give the first term of the series, which is 4 instead of 5.5。
as it would be from the formula for n=1. The law, while fairly accurate for the planets known at that time, and although it helped to suspect the existence and distance of Neptune，and suggested a missing planet between Mars and Jupiter which later led to the discovery of the asteroids, yet failed to give the distance of Neptune with even approximate exactness, differing by about 29%. It has long been recognized that a more satisfactory empirical law is desirable. Attempts have been made to devise such a law but without satisfactory results.
In thinking about this matter, I have devised a law which seems to be more general in its application and to have, perhaps, a real physical significance. It may be stated as follows:“The radii of orbits of any natural bodies moving around a central mass are proportional to the squares of the integral numbers 1, 2, 3, 4 etc., except for the effects of the mutual attractions of the bodies themselves.”Stated in this general way, the law, if it has real physical significance, should apply to the planets of the solar system, to the satellites of any planet, and to the orbits of electrons in the atom. The deviation of any particular body should be accurately accounted for by the attractions of its neighbors.
The law may be stated algebraically as follows: R=Kn^2,where R is the radius of the orbit, n is an integer and K is a constant for any one system of bodies, its value probably depending on the mass of the central body of the system. When K has once been determined for any system, the distances of all bodies in that system follow at once from their ordinal numbers. However, it is not necessary to assume that all possible orbits are actually occupied. Thus, in the solar system, if K is taken as 4000000 miles and Mercury is given the number 3, Venus No.4, Earth No.5, Mars No.6, Jupiter No.7, Saturn No.15, Uranus No.21 and Neptune No.26, the distances may be calculated with an accuracy considerably greater than that given by Bode’s law. The missing numbers may correspond to the orbits of small undiscovered planets or to unoccupied orbits. The accompanying tabulation shows how the law compares with the actual distances of the planets and with Bode’s law. It is noticed that the percent of error, in most cases, is less than that of Bode’s law and that the new law does not break down in the case of Neptune, as Bode’s law does. COMPARISON OF PLANETARY DISTANCES COMPUTED BY THE LAW R=Kn^2, WITH THE OBSERVED DISTANCES AND WITH THOSE GIVEN BY BODE’S LAW. (All distances in millions of miles. K=4000000 miles.)
The largest percent of difference occurs in the cases of Venus and the Earth, the distance of the former being about 3.2 million miles greater and that of the latter about 7million miles less than the distances suggested by the law. Jupiter, on account of its great mass and the rather large distances separating it from the other planets, would be only slightly perturbed and should conform quite closely to the law. This, we see, it does, the difference being only 0.14 of one percent. The fact that Uranus and Neptune are both somewhat farther out than the law may require suggests the existence of a still more distant planet [1], probably of enormous mass. If we assign to this planet the ordinal number 31 in the system, its distance would be 3844 million miles. Astronomers have searched for such a planet, but it has never been discovered. Since the distances of the planets are connected with their periods by Kepler’s Law which states that the squares of the periods are proportional to the cubes of the mean distances, it should be possible to comput the period and therefore the orbital velocity of any planet from its order number in the system. The formula would be V=U/n. Thus, if the unit velocity be taken as U=90 miles per second, the orbital velocity of any planet would be 90 mi. per sec. divided by its order number. The following table shows how well the velocities computed in this way agree with the observed velocities.
This remarkable agreement seems to indicate that the law has a real physical significance. Bode’s law can not give any information as to orbital velocities. I have endeavored to apply this law to the satellites of Jupiter, Saturn, and Uranus, and apparently it applies with some degree of success although on account of the small distances separating the satellites from each other they would naturally produce large perturbations of each other’s orbits, and the law could not be expected to agree with the observed distances with the same degree of accuracy with which it applies to the planets themselves.
The radii of the stable orbits of the electron in the Bohr model of the hydrogen atom follow this law exactly, as there are in this case no perturbations due to other electrons.
I hope, in a later paper, to give more of the details and tests of my theory, as well as methods of calculating the value of K for any system.
Baker University, Baldwin City, Kansas.
原载《Popular astronomy》，1927,18:327-329
[1]. There are two possible orbits outside that of Neptune, one of which(most likely No.31) must be occupied by a real planet, possibly both of them. They would be suggested as follows: No.31.
Distance about 3844 million miles.
Velocity about 2.9 miles per sec.
Diameter about 5 times that of the earth.
Density about two (?) times that of water.
Eccentricity about. 08 or. 09.
No.36.
Distance about 5184 million miles.
Velocity about 2.5 miles per second.
Diameter about 6.8 times that of earth.
Density about 2.4 times that of water.
Eccentricity about. 21.
一条关于行星距离和轨道速度的新规律 大约在1722年波德发表了一条关于太阳与所有当时已知行星之间大致距离的规律，这条规律看上去只是一个不同寻常的巧合，没有自然基础或者是物理上的依据。简要的说这条规律是这样的：首先写出一系列4，再依次写出：3，6，12，24，48，等等，求和，然后再除以10，将连续的给出行星与太阳之间的距离，前提是把地球到太阳的距离作为1个天文单位。其代数表达式是：4+3*2^(n-2)。然而这个公式并没有给出通项中的第一项，而是以4代替了原本n=1时的5.5。这条规律与当时已知行星的位置符合的相当准确。虽然它促进了海王星存在的猜想及其它的距离，还有它提出了在火星和木星之间有一个消失的行星，导致之后发现了小行星带，但是它却没有给出相对准确的海王星的距离，误差达29%。它长时间被公认为是一条令人满意的经验公式。人们努力尝试设计出一条像这样好的规律，但是没有得到令人满意的结果。 有鉴于此，我设计了一条在它的适用范围内看上去更普遍的规律，也许，这是一条具备物理意义的规律。它可以这样表述：任何围绕中心天体运转的自然物体的轨道半径是与整数1,2,3,4等等的平方成正比的。除了天体之间的相互吸引的影响之外。这样一般的陈述，如果它有物理意义，将应用于太阳系内的行星、任何行星的卫星、原子的电子轨道。对于任何值得注意的天体的偏离必须准确地考虑相邻天体之间的吸引。
这条规律的代数表达式可以写成：R=Kn^2，R是轨道半径，n是整数，K是任何一个天体系统的常数。它的数值很可能取决于系统中中心天体的质量。在某系统中，当K一旦被确定了，那么这个系统所有的天体的距离对应着它的序数也被确定。然而，假定所有可能的轨道实际上是被占用的是不必要的。于是，在太阳系中，如果K取4000000英里，水星对应的是序数3号，金星是4号，地球是5号，火星是6号，木星是11号，土星是15号，天王星是21号，海王星是26号。计算出的距离比波德定则更好，更准确。消失的序数行星轨道符合那些小而未被发现的行星或者未被占用的轨道。附表展示了新规律行星的距离与实际距离和波德定则三者的比较。请注意到它的误差比例，大多数情况下，新规律的误差比波德定则更小。新规律在海王星的问题上并没有失效，但波德定则却失效了。 通过新规律 R=Kn^2计算出的行星距离，与观测得到的距离，波德定则计算的距离相比较。 （所有的距离单位是百万英里，K=4000000英里）
在新规律中，最大误差出现在金星和地球上，计算出的金星距离比观测到的距离少了3.2百万英里，而地球的距离又多了7百万英里。木星，由于它的巨大质量和相当大的距离，与其它行星区分开来。未来只要略微的扰动将完全紧密的符合规律。在这，我们可以看出，它做到了，误差只有0.14%。事实上，规律暗示，离天王星、海王星更远的地方可能存在一颗行星。它可能有巨大的质量，如果我们指定31号行星，它的距离将是3844百万英里。天文学家寻找过这颗行星，但是它从未被发现过。
开普勒定律表明：行星的距离与周期是有关联的，表述为：周期的平方与距离的立方是成正比的。计算出系统中任何序号行星的周期和轨道速度是有可能的，它的公式是：V=U/n，如果，单位速度U=90英里每秒，任何行星的轨道速度将是90英里每秒除以它的序号数，下面的表格展示了计算出的数据与观测数据符合的多么好。
这不同寻常的完美吻合像是暗示这条规律有实在的物理意义。在轨道速度方面，波德定则给不出任何信息。 我努力的将这条规律应用到木星、土星、天王星的卫星上，显然它取得了一定的成功，尽管在区别于行星的短距离的卫星上，它们将自然的产生大的摄动，不像它运用到行星中那样，观测到的距离没有像预期的那样准确的符合规律。 在波尔氢原子模型中，当没有由其它电子引起的摄动的时候，稳定的电子轨道半径也严格的遵循这条规律。 我希望，在这之后，能有文章为我的理论做出更多的解释和检验，还有计算任何系统的K值的方法。 拜克大学，鲍德温市，堪萨斯州
原载《大众天文学》，1927,18:327-329 [1].海王星外可能存在两条轨道，一条（很可能是31号）很可能被一颗行星所占据。也许两条都被占据了。它们是下面两条。 31号
距离大约3884百万英里。
速度大约2.9英里每秒。
直径大约为地球的5倍。
密度大约是水的2（？）倍。
离心率大约是0.08或0.09。
36号
距离大约5184百万英里。
速度大约2.5英里每秒。
直径大约为地球的6.8倍。
密度大约是水的2.4倍。
离心率大约是0.21。
备注： 这篇文章是束先生20岁时发表的，原载于《Popular astronomy》，而后收录于《束星北学术论文选集》。这是束先生发表的第一篇论文，也是唯一一篇与天文有关的论文。翻译工作由鄙人完成，由于鄙人能力有限，有些语句翻译不准还望各位指正。当时冥王星尚未被发现，有兴趣的朋友可以把冥王星的数据代入新规律中检验一番。如何从新规律 R=Kn^2推导到V=U/n，之后我做了说明。另外，我制作了一些图形，以便更加直观的展现出三者之间的比较。最后，对于我来说能够宣传束先生的理论，已是莫大的荣幸。
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