CSIRO PUBLISHING www.publish.csiro.au/journals/pasa Publications of the Astronomical Society of Australia, 2011, 28, 177–201 Quantization of Planetary Systems and its Dependency on Stellar Rotation Jean-Paul A. Zoghbi American University of Beirut, Bliss St., Beirut, Lebanon. Corresponding author. Email: [email protected] Received 2009 September 24, accepted 2011 February 28 Abstract: With the discovery of now more than 500 exoplanets, we present a statistical analysis of the planetary orbital periods and their relationship to the rotation periods of their parent stars. We test whether the structural variables of planetary orbits, i.e. planetary angular momentum and orbital period, are ‘quantized’ in integer or half-integer multiples of the parent star’s rotation period. The Solar System is first shown to exhibit quantized planetary orbits that correlate with the Sun’s rotation period. The analysis is then expanded over 443 exoplanets to statistically validate this quantization and its association with stellar rotation. The results imply that the exoplanetary orbital periods are highly correlated with the parent star’s rotation periods and follow a discrete half-integer relationship with orbital ranks n ¼ 0.5, 1.0, 1.5, 2.0, 2.5, etc. The probability of obtaining these results by pure chance is p , 0.024. We discuss various mechanisms that could justify this planetary quantization, such as the hybrid gravitational instability models of planet formation, along with possible physical mechanisms such as the inner disc’s magnetospheric truncation, tidal dissipation, and resonance trapping. In conclusion, we statistically demonstrate that a quantized orbital structure should emerge from the formation processes of planetary systems and that this orbital quantization is highly dependent on the parent star’s rotation period. Keywords: planetary systems: formation — star: rotation — solar system: formation 1 Introduction important role in the final orbital configuration. In all of The discovery of now more than 500 exoplanets has these mechanisms, the stellar rotation period is a critical provided the opportunity to study the various properties of parameter. The main motivation in this paper is therefore planetary systems and has considerably advanced our (i) to statistically search for any apparent quantum-like understanding of planetary formation processes. One features in the orbital structure of exoplanetary systems; long-suspected property of planetary systems has been the (ii) to determine whether the quantization parameters are quantum-like feature that resembles the mathematical related to any specific physical system property (the regularity of the empirical Titius–Bode (TB) law in the stellar rotation rate is examined in this paper); and (iii) to Solar System. Various research papers have suggested shed some light on the nature of the possible physical that such ‘quantized’ features and empirical relationships processes that might lead to this apparent quantization. might be possible in extra-solar multi-planetary systems, We will argue on dynamical terms that a quasi-quantum such as Nottale (1996; 1997b), Nottale et al. (1997a; model might emerge naturally from the formation pro- 2004), Rubcic & Rubcic (1998; 1999), Poveda & Lara cesses that determine the final configuration of a plane- (2008), and Chang (2010), just to mention a few. In case tary system. they truly exist, one main question that needs to be The plan of this paper is as follows: Section 2 describes answered is what physical processes might cause these the basic methodology and simple quantum-like model. ‘quantization’ features to develop. The gravitational In Section 3, the model is applied to the Solar System. instability model of planet formation has been success- In Section 4, the analysis is expanded over a sample of fully used in the past to explain ‘discrete’ power law 443 exoplanets, for which we could obtain stellar distributions in planetary spacing (Griv & Gedalin 2005). rotation periods. In Section 5, a statistical analysis of Similarly, hybrid models of planetary formation (e.g. the results is presented demonstrating that the specific Durisen et al. 2005), are characterised by concentric dense angular momenta of all planetary orbits generally follow gas rings that are produced by resonances and discrete half-integer multiples of the specific angular momentum spiral modes which, in theory, can be correlated with at the parent star’s corotation radius. Section 6 briefly orbital ‘quantization’ features. Also similarly, tidal dis- proposes various physical mechanisms that may justify sipation and angular momentum transfer, along with the obtained results. Prospects and conclusions are drawn mean-motion resonances and resonance trapping, play an in Section 7. Ó Astronomical Society of Australia 2011 10.1071/AS09062 1323-3580/11/03177 Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 26 Sep 2021 at 03:04:14, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1071/AS09062 178 J.-P. A. Zoghbi 2 Methodology For nearly circular orbits, Newton’s force-balance equa- We will be testing for the quantization of planetary tion of motion gives angular momentum, i.e. we will test whether planetary GM angular momenta have discrete values in multiples of a r ¼ ; and ð6Þ n v 2 ‘ground-state’ system-specific parameter. Within an order n of magnitude estimate, the corotation orbit represents ¼ ¼ð Þ1=2 ð Þ an approximate inferior limit to the position of planetary Jn vnrn GMrn 7 orbits. This is confirmed by various physical mechanisms 2 that are discussed in Section 6, such as spin-orbit coupling Combining equations (1), (5), and (7), we obtain an n law th and tidal dissipation, as well as disc-locking and magnetic for the quantized semi-major axis rn of the n orbit, in O braking which create a barrier and inferior limit to plan- terms of the corotation radius r0, spin rotation rate s of etary migration. On that basis, we postulate that the the central star, and orbital rank n by corotation orbit represents the ground-state orbit of 1 ¼ =3 planetary systems and assign to it the orbital rank n 1. ¼ 2 ¼ 2 GM ð Þ rn n r0 n 2 8 However, this does not negate the possibility of having Os physical objects orbiting inside the corotation orbit. Nevertheless, the corotation orbit (at n ¼ 1) is particularly From Kepler’s third law and Equation (8), the quantized th chosen because of its importance as a base reference to the orbital period Pn of the n planetary orbit is also given orbital parameters of the entire planetary system, and in in terms of the corotation orbital period P0 and orbital particular, their relationship to the parent star’s rotation rank n by period. The corotation radius r is defined in terms of the star’s 1=3 0 ¼ 3 ¼ 3 ; ¼ Pn ð Þ rotation rate O by Pn n P0 n Prot or n 9 s Prot 1= GM 3 r ¼ ; ð1Þ Where Pn is the planet’s orbital period and P0 is the 0 O 2 s corotation period which is by definition equal to Prot, the star’s rotation period. where G is the gravitational constant and M and O are the s 3 Solar System Application and Results mass and rotation rate of the parent star respectively. Similarly, the mean motion orbital velocity v0 and 3.1 The Solar System Orbital Ranks specific angular momentum J0 (per unit mass) at the The quantum-like model described in Section 2 is first corotation radius are given by applied to the planets of the Solar System in order to discern any discrete pattern in their orbital ranks n.We 1=2 GM 1=3 will calculate the orbital ranks n using the Sun’s rotation v0 ¼ ¼ ðÞGMOs ; ð2Þ r0 period PSun taken from Allan’s Astrophysical Quantities (Cox 1999) as 25.38 days and the solar rotation rate 1= O ¼  À6 À1 G2M 2 3 Sun 2.8 10 rd s . The Sun’s corotation specific J ¼ v r ¼ ð3Þ 0 0 0 O angular momentum J0 is calculated from Equation (3) and s will represent the base quantization parameter for all possible planetary orbits in the Solar System. The plan- If our planetary quantization hypothesis is valid, then the etary orbital ranks are first calculated using the Sun’s specific angular momentum Jn of any planetary orbit n present rotation period (25.38 days) and presented in would follow discrete and quantized multiples of the Table 1. However, since the Sun’s rotation rate has specific angular momentum J0 at the corotation orbit ¼ already decayed with age through angular momentum n 1. loss, the orbital ranks are also calculated using the Sun’s rotation rate at the early stage of planet formation. The 1=3 G2M2 orbital parameters of the Solar System are assumed to Jn ¼ nJ0 ¼ n ð4Þ Os have settled into a long-term stable configuration at around 650 Myr or so. Using data on solar-type stars in the In other words, the ratio of the specific orbital angular Hyades (age ,650 Myr), we selected star VB-15 which th momentum Jn of a planet in the n planetary orbit to the has a B–V index similar to the Sun’s to estimate the Sun’s specific angular momentum J0 at the ‘ground-state’ rotation period of 8 days (Radick et al. 1987) at the corotation orbit ought to increase incrementally by a planets’ formation age, i.e. the time when the solar system discrete value. planets’ orbits have stabilised and the proposed quanti- zation has ‘frozen in’. The planets’ orbital ranks n are J n ¼ n ð5Þ calculated from Equation (5) for each planetary orbit J0 using both the Sun’s present and earlier formation rotation Downloaded from https://www.cambridge.org/core.