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From Wikipedia:

No solid theoretical explanation underlies the Titius–Bode law – but it is possible that, given a combination of orbital resonance and shortage of degrees of freedom, any stable planetary system has a high probability of satisfying a Titius–Bode-type relationship. Since it may be a mathematical coincidence rather than a "law of nature," it is sometimes referred to as a rule instead of "law."[18] On the one hand, astrophysicist Alan Boss states that it is just a coincidence, and the planetary science journal Icarus no longer accepts papers attempting to provide improved versions of the "law."

I'm rather baffled by the way the theoretical explanation is presented. How the question is not yet settled. Don't we have enough planets in our Galaxy to rule out this is a mathematical coincidence, or actually to assure the law? I've see this related question here, but I feel like it does not so much address my issue here. My question is rather not if Titius–Bode works or not, but rather why do we still have this questions open in the 21st century.

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    $\begingroup$ There's a lot of bias in exoplanet detection, particularly due to size, so I suspect that's one of the main reasons. You'd need a reliable data set of solar-systems, where you are sure you've got all the planets to test it. We don't have that. $\endgroup$
    – user438383
    May 31, 2022 at 14:12
  • $\begingroup$ @user438383, oh I see. I probably overestimated the number of identified exoplanets then. It just seem every 6 month I see news of a new "like Earth" exoplanet.; So either being Earth-Like is not so rare or it is the same one :). $\endgroup$
    – d_e
    May 31, 2022 at 14:45
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    $\begingroup$ Finding other stellar planetary systems with an identical or otherwise regular distance law would not help you at all to understand it. You need a physical theory that correctly reproduces the law. $\endgroup$
    – Thomas
    May 31, 2022 at 17:38
  • $\begingroup$ @Thomas, maybe but it seems to me more philosophy of science. Generally, I personally, don't think we need to find physical explanation for observed phenomena that can expressed mathematically -- even if the "law" is only provisional. $\endgroup$
    – d_e
    May 31, 2022 at 19:48
  • $\begingroup$ @d_e: To the extent that the observations are truly reliable, that's fair enough, but we still want a physical explanation, because it might enable us to make further deductions or observations and broaden our understanding of the theory as a whole. $\endgroup$
    – Kevin
    Jun 1, 2022 at 4:22

2 Answers 2

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How it is possible is illustrated by the counter question - how would you define whether it is a coincidence or not? How accurately do the planets in a system have to follow the TB relation to decide that they are in fact following it? It is an ill-posed question and depending on how you pose it you could get different answers.

In order to address the problem using exoplanetary systems you have to have a big sample of stars that have multiple ($3+$) planets. There are nowhere near as many of these as there are stars where 1-2 planets have been detected. That is because the methods of planet detection - particularly transiting planets - are most sensitive to close-in planets and may easily miss other planets in the system that are not quite in the same plane as the transiting planet(s). A further complication is that stars where multiple exoplanets are seen may not be like our own, because their ecliptic planes are much flatter than that of the Solar System.

There are about 230 systems of $3+$ detected planets that can be looked at presently (Mousavi-Sadr 2021). Those authors conclude that about half these systems follow a logarithmically spaced period distribution better than our own Solar System does (and half don't). But note that these period distributions have two free parameters for each system, they are not fixed at the values appropriate for our Solar System.

Some authors have advocated searching for new planets using the predictions of TB-like relationships deduced from planets already detected (e.g., Bovaird et al. 2015). But where these predictions have been tested, the success rate is very low and it is unclear whether that is because the predictions are bad or just because there are plenty of other explanations to do with the potential size of the new planet or its orbital inclination that would lead to them not being detected.

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There's a discussion in Murray & Dermott's Solar System Dynamics, Section 1.5 'The Titius-Bode "Law"'. They generated orbits at random and tried to fit a Titius-Bode style curve, and were generally successful. Conclusion: "This leads us to suggest that the "law" as applied to other systems, including the planets themselves, is also without significance".

why do we still have this questions open in the 21st century?

I imagine it is because it is a nice, simple formula. It differs from Kepler's method of predicting planetary distances (inscribing regular polyhedra) in one way: it isn't as easy to prove that it is wrong.

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