We shall show that the fundamental attributes of a planet – mass, diameter, distance from the Sun, duration of a day, duration of an orbit are all statistically correlated. We shall also see the correlation is maximum when Pluto is discarded from the calculations, thereby providing a statistical justification of the demotion of Pluto as a planet.
Note that from these fundamental attributes, the derived attributes such as escape velocity, density, orbit velocity can be calculated. So in this analysis our scope is limited to the abive five fundamental attributes.
1. We define SMR of a set of data A as the ratio of the sample standard deviation of the data to the maximum value of the data in A and denote it by SMR(A).
2. We define the follow sets of fundamental planetary attributes. It is to be noted that Pluto has been demoted as a planet so we are left with only eight planets namely Mercury, Venus, Mars, Earth, Jupiter, Saturn, Uranus and Neptune.
M = Set of masses of the planets
D = Set of the diameters of planets
R = Set of the distance for the planets form the sum
T = Set of the Orbital period of the planets
X = Set of the rotation period of the planets
3. All data in this analysis is collected form NASA website. Link http://nssdc.gsfc.nasa.gov/planetary/factsheet/
SMR(M) = 0.331
SMR(D) = 0.370
SMR(R) = 0.368
SMR(T) = 0.360
SMR(X) = 0.331
The above values of SMR have a mean value of 0.359 and very small standard deviation of 0.011. Hence statistically we can say that each of these value is close to the arithmetic mean of the SMRs.
Inference: The SMR of the fundamental attributes of a planet are roughly equal.
Even if we include Pluto in the calculations, the mean value if 0.352 with a standard deviation of 0.020.
Let us calculate the mean value and standard deviation of the SMR when one planet goes missing. Because of Observation 2, we can include Pluto in our analysis without caring for its classification.
Without Pluto : Mean = 0.359, SD = 0.011
Without Neptune: Mean = 0.358, SD = 0.021
Without Uranus : Mean = 0.373, SD = 0.021
Without Saturn : Mean = 0.361, SD = 0.025
Without Jupiter : Mean = 0.359, SD = 0.022
Without Mars : Mean = 0.366, SD = 0.016
Without Earth : Mean = 0.366, SD = 0.017
Without Venus : Mean = 0.366, SD = 0.018
Without Mercury : Mean = 0.365, SD = 0.015
Inference 3.1: It is interesting to note that that for each of the five fundamental quantities, the mean value of SMR when one planet goes missing is roughly equal. This shows that there is a statistical correlation between the fundamental attributes of the planets.
Inference 3.2: The standard deviation is least when Pluto is removed, showing that Pluto fits least with the other eight planets. This gives a statistical justification of the demotion of Pluto as a planet.
Inference 3.3: The mean and standard deviation is greater for larger planet and lesser for smaller planets. A possible inference of this could be that larger planets play a bigger role in the stability of the solar system.
A physical interpretation of the above observations could be that even if one of the planets of the solar system mysteriously vanishes without altering the other planets then, on a large scale, the rest of the planets will continue to behave roughly the same as they had before the incident. This is somewhat like a clever inbuilt damage control mechanism. Quantitatively we can say that:
1. In the Solar System, the large and small values of the fundamental planetary attributes are almost equally distributed in such a way that SMR of any fundamental attribute the planets are roughly equal.
2. The Solar System is tries to conserve its SMR of its fundamental physical attributes.
Is this a coincidence?
Mathematically this is not a wonder. We can have infinitely many set of data showing similar properties. But the question here to be asked is
1. Why should the planets be one among these infinite sets? There are infinitely many sets of data which do not show this correlation.
2. The more important question is weather the planets are trying to following some mathematical regularities due to which they produced favorable data for the correlation.
For a set of positive real numbers SMR can be between 0 and 0.5; for example, the SMR for the distance of the ten nearest stars from the earth is 0.212 with a very high standard deviation of 2.237. So there is no particular reason why SMR of planetary data should all be roughly equal to 0.36. It may still be a coincidence that the data turned out to be favorable. However the chance of such coincidences decrease rapidly when we see that despite removing any one planet, SMR remained roughly at 0.36 with a very low standard deviation.
Finally, history has it that the Titus-Bode rule failed for Neptune and Pluto. But our observation holds holds not only for the eight major planet and Pluto but also for the hypothetical scenario when one planet mysteriously disappears. If a coincidence matches not just present data but even hypothetical scenarios, then it is no longer a coincidence.
Consider a finite set of positive reals having cardinality n.