From: "Robert Dawson" Newsgroups: sci.math.symbolic Subject: Re: Amazingly Enigmatic Gravity System Date: 13 May 1998 18:04:02 GMT Aleksandr Timofeev wrote > 1. Can you working out probability of this physical phenomenon? > The problem of calculation of probability of existence of this physical > phenomenon is not univalent. This problem has some approaches depending on > the physical interpretation of an offered experimental data. Integer > Planet Notations Mass | Ratio Experemental number > of mass value | value commensur- > value | ability > | > Jupiter MJU or 1 317.735 |(MJU+MSA)/(MUR+MNE)= 12.995971 ~ 13 > Saturn MSA or 2 95.147 | MJU/(MUR+MNE) = 10.001011 ~ 10 > Neptune MNE or 3 17.23 | MSA/(MUR+MNE) = 2.994869 ~ 3 > Uranus MUR or 4 14.54 | (MJU+MSA)/MNE = 23.9630 ~ 24 > Earth MTE or 5 1.000 | MUR/(MTE+MVE) = 8.011019 ~ 8 > Venus MVE or 6 0.815 | (MNE+MUR)/MVE = 38.9816 ~ 39 > Mars MMA or 7 0.108 | (MTE+MVE)/MME = 33.0000 ~ 33 > Mercury MME or 8 0.055 | MVE/(MMA+MME) = 5.0000 ~ 5 I cannot give an exact probability, of course, but I can say that it is not as improbable as it might appear. Firstly, not all of these are independent; the first row follows as a necessary concomittant of the two below it! Secondly, the exact values in the last two rows arise from the limited-precision values for the masses of the four inner planets; the last couple zeros are not significant, as a ratio of two numbers of that precision exhibits some granularity. If n/55 is going to be 33.00 to two places, it has to be 33.0000000..... and not 33.001. Thirdly, any of 8 planets and 28 pairs of planets can appear in numerator or denominator. Assuming no planet appears twice, that gives us 56 values of type p/p 168 values of type (p+p)/p 168 values of type p/(p+p) 420 values of type (p+p)/(p+p) = 752 possible values. In fact, as there is no reason to omit Pluto, or the Sun, or to exclude using the mass of the Earth-Moon system as a whole [no other satellite has a significant proportion of its primary's mass], there are three other possible numbers that presumably were examined but omitted. This gives a total of 3080 possible combinations. Half of these will have values larger than unity (as they appear in reciprocal pairs). The Poisson distribution tells us that among 1500 independent random real numbers, randomized by a smooth and well-spread-out distribution, the expected number that are within 1 part in 1000 of being integers is about 3; the probability that there will be none is less than 5%. We expect about 30 to be within 1 part in 100 of unity, and about 120 to be as close as 23.96 is. Even within the original 8 numbers, the expected numers of close hits is not small. In fact, Aleksandr Timofeev has found many fewer than that. This may be partially explained by the lack of algebraic independence; but I would expect, if the less significant digits of the planetary masses really *are* random, there should be more than a few more such "coincidences" waiting to be discovered, mostly in the form (p+p)/(p+p). -Robert Dawson