Predictions of planets for 586 exosystems and a method for predicting planets in multi-planetary exosystems Valeri Beck1* 1Alfa Carbon® Beck, Berlin, Germany ABSTRACT We have applied a model of planetary system formation in the field of a standing sound wave (the ‘SSW-Model’) to the prediction of planets in 586 multi-planetary exosystems (384 bi- planetary, 127 tri-planetary and 75 exosystems with four or more confirmed planets). We have verified our predictions using transit-like events from the NASA Threshold-Crossing Event (TCE) catalogue, finding that more than 80% of these events are included in our list of exoplanet predictions. We describe a method for predicting the periods and semi-major axes of additional planets for exosystems with two or more confirmed planets. Our method can significantly facilitate targeted searches for planets in exosystems, enabling the detection of many new exoplanets, including those located in the habitable zones. KEY WORDS: planets and satellites: detection–planets and satellites: formation–planets and satellites: general–planets and satellites: rings–protoplanetary discs 1. INTRODUCTION In this paper, based on our model for the formation of the Solar System (Beck & Beck 2016; hereafter BB16), we present predictions for exoplanets in multi-planetary exosystems. This model explains very well many facts about, and observations of, our Solar System and exoplanetary systems, which current theories of the origins of planetary systems fail to explain. In our model, planets in the system are arranged in proportion to r4, where r is the distance from the centre of the system. When we applied this distribution to the ‘Trappist-1’ exoplanetary system (Fig. 1), we were astonished by how well the planets in this system are arranged in a straight line on a graph of ∜(an) vs. n, where an is the semi-major axis of the planet with ordinal number n. * E-Mail: [email protected] 1 0.5 TRAPPIST-1 h TRAPPIST-1 g 0.45 TRAPPIST-1 f 0.4 TRAPPIST-1 e TRAPPIST-1 d Fourth root (a) root Fourth 0.35 TRAPPIST-1 c TRAPPIST-1 b 0.3 0 1 2 3 4 5 6 7 Figure 1. Distribution of planets in the ‘Trappist-1’ exoplanetary system We have also applied this correlation to other exoplanetary systems with 6–8 confirmed planets from a NASA Exoplanet Archive1. We obtain similar distributions of the planets in straight lines for all of the systems, with one or several gaps between the known planets (Fig. 2). HD 10180 HD 219134 HD 34445 Kepler-11 Kepler-20 Kepler-80 KOI-351 TRAPPIST-1 Fourth root (a) root Fourth 0 1 2 3 4 5 6 7 8 9 10 11 12 13 Figure 2. Distribution of planets in the exoplanetary system with six or more planets In Figs 1 and 2, it is clear that there are definite patterns in the distances of the planets from the centre of the system and that, in many exosystems, one can hypothesise the existence of undiscovered planets between the known ones. 1 https://exoplanetarchive.ipac.caltech.edu 2 For our Solar System, the so-called Titius–Bode Law (TB relation) has been known for more than 250 years; it describes in one simple mathematical formula the average distances of the planets from the Sun. This relation attracted great attention after the discovery of Uranus and the minor planet Ceres, providing the first instance of successful predictions of previously unknown planets. Boivard and Lineweaver (2013) used a generalised TB relation to predict 141 additional planets in 68 multi-planetary exosystems. Using these predictions, Huang and Bakos (2014) discovered five planetary candidates within the limits of the predicted periods, but they concluded that such a small number of discovered candidates is insufficient to confirm the usefulness of the TB relation to predict new exoplanets. Although the TB relation itself cannot be accepted as being generally applicable to exoplanetary systems, the idea of using the observed patterns in exosystems to predict new planets may be very productive, if there is a ‘correct’ theory or model for the origin of planetary systems. The TB relation does not have any physical foundation; our model, on the contrary, describes the formation of planetary systems from a purely physical standpoint, and although it still lacks a complete mathematical description, it may nevertheless be useful for the practical prediction of new exoplanets in multi-planetary systems. In Section 3, we present our predictions for 586 exosystems, and Section 4 verifies our predictions for a number of systems from the Kepler mission data. Furthermore, all five planetary candidates discovered by Huang and Bakos are present in our predictions. 2. PHYSICAL GROUNDING OF THE MODEL Many researchers (e.g. Loomis et al. 2017; Debes et al. 2013; Williams and Lucas 2011; Avenhaus et al., 2018; Andrews et al. 2016; Churcher et al., 2011; Matthews et al. 2015; Isella et al. 2018; Clarke et al 2018) are working to understand and explain the observed patterns in the distributions of planets in the numerous recently discovered multi-planetary systems. However, they have experienced only minor success, since there is at present no physically sound theory that takes into account most of the known facts. In contrast, our model is a theoretical construction that takes into account, from a purely physical point of view, most of the observational data and currently known facts about the formation and evolution of the Solar System and exoplanetary systems. It is generally accepted that planetary systems are formed as a result of the collapse of a rotating cloud of gas and dust. In the process of contracting, the rotation of the cloud accelerates, and it forms a protoplanetary disk, within which the planets form. But what is a protoplanetary ‘disk’? More than 99% of the initial protoplanetary cloud consists of gas (e.g. Ansdell M. et al. 2016). Due to centrifugal force, the rotating cloud takes the form of an ellipsoid of revolution, but it is prevented from collapsing into a thin, flat disk by gas pressure. Fig. 3 shows some images of protoplanetary disks obtained by the ALMA radio array1. Surprisingly, most appear to be oriented perpendicular to, or with only a slight deviation from 90° relative to, the line of sight from an observer on Earth. If these objects were actually flat disks, then it would appear that the Solar System occupies some favoured position in our galaxy toward which the emerging planetary systems ‘look’. Of course, this cannot be the case. Our model of the formation of a planetary system in the field of a standing sound wave (the SSW-Model) perfectly explains this contradiction. Each ALMA image is in reality the projection of a three-dimensional object onto a plane perpendicular to our line of sight; if such an object is an ellipsoid of revolution (a protoplanetary gas–dust cloud), we will obtain images exactly like those in Fig. 3, i.e. circles or ellipses, regardless of the place from 1 The Atacama Large Millimeter/submillimeter Array 3 which we observe the object. Ring-like structures in these images are dust spheres (ellipsoids of revolution) located inside one another that concentrate in the antinodes of the standing wave. We see them as rings because a line of sight near the edge of each sphere crosses a much greater thickness of dust than does one near the centre (see BB16 for more detail). In more evolved gas and dust ellipsoids, the dust has already concentrated largely toward the plane of rotation. Note that ALMA observes mainly dust, or in some cases gases such as CO (e.g. Huang J. et al. (2018), but not hydrogen or helium, which make up the bulk of the protoplanetary cloud. Based on the ALMA images, we conclude that the maximum degree of compression of the gas ellipsoid does not exceed 1:2; to call such a cloud a disk is at the very least incorrect. Figure 3. ALMA’s high-resolution images of nearby protoplanetary disks, which are the results of the Disk Substructures at High Angular Resolution Project (DSHARP). Credit: ALMA (ESO/NAOJ/NRAO), S. Andrews et al.; N. Lira The dust in the original protoplanetary cloud is distributed more or less evenly, and its behaviour is completely determined by the gravitational influence of the much larger mass of gas. The low concentration of dust in the cloud in turn causes a very low rate of adhesion of the dust particles among themselves, so it is problematic to conclude that planets may form in a short time (according to some estimates, 1 million years or even 100 thousand years, e.g. 4 Fedele et al. 2018). Our model assumes a perhaps more realistic timescale of 2–10 million years. In a collapsing gas cloud, as in any gaseous medium, sound waves can propagate. In BB16, we show that gas pressure oscillations occur spontaneously in such a cloud and establish a standing acoustic wave in which the energy losses are compensated for by periodic explosive processes in the central protostar. These trigger the oscillations that produce the standing wave. The sound pressure, directed from the centre to the periphery, stabilises the gas–dust cloud and slows down the matter falling onto the protostar. A reverse process is also possible, in which the substance of the protostar is ejected into the surrounding space, becoming distributed throughout the cloud during the periodic explosions. In this quasi- stationary state, the rotational speeds of the matter in different parts of the protoplanetary cloud are redistributed in such a way that rotation occurs at Keplerian velocities. Such a ‘sounding’ state of the protoplanetary system can last for millions of years, until the conditions that support the standing wave change.