Special Issue on Astronomy and Symmetry

1. Introduction

Astronomy is undoubtedly one of the oldest branches of natural sciences. Our early ancestors were truly amazed by the heavenly bodies they could observe in the sky with their naked eye, and the phenomena these objects—stars, planets, the Sun, the Moon, comets, and occasionally transient events—produced. Celestial symmetry in its most evident form had to be apparent already for the very first humans who turned their attention towards the sky, because the brightest, extended objects, the Sun and the Moon, look circularly symmetric. Since then, astronomy has come a very long way. Currently, we know that the planets and minor bodies of the Solar System are orbiting around a massive central object, the Sun, which itself is just one of the countless stars in the Universe. We know how the stars are powered by nuclear fusion and that they are concentrating in gravitationally bound systems called galaxies. We have a fairly good knowledge of stellar evolution, the composition and structure of galaxies, including our own galaxy, the Milky Way. We could map the large-scale distribution of galaxies out to massive distances and have working cosmological models on how the Universe as a whole evolved in the past 14 billion years or so. Most of this knowledge is incorporated into modern human literacy. Astronomical discoveries establish new directions of research and trigger technological innovations and developments. Perhaps most importantly, astronomy still remains an amazing field of science in the eyes of the public.

The first telescopes were turned to the sky in the early 17th century. Since then, we have built giant optical telescopes on the ground, as well as placed advanced instruments on board Earth-orbiting satellites. We opened new windows to the Universe in other electromagnetic wavebands, first in the radio in the 1930s. Today’s multi-messenger astronomy involves detecting high-energy cosmic ray particles, neutrinos and gravitational waves originating from distant black hole and neutron star mergers. Humans sent automated probes to various bodies in the Solar System. Most recently, using radio interferometry, we could make direct images of shadows of supermassive black holes.

Observational and theoretical astronomy and astrophyics, as well as the celestial bodies and their phenomena, are diversified so much that symmetry can be found almost everywhere. Recently, Szabados [1] presented a collection of various degrees of symmetries on vastly different spatial scales in stellar, galactic and extragalactic astronomy. In that review, the non-exhaustive list of examples for spherical symmetry contains slowly rotating stars, globular clusters and massive clusters of galaxies. Symmetries or asymmetries can often be utilised to better understand and model astronomical objects and physical reasons behind astronomical phenomena.

In late 2019, our Special Issue on Astronomy and Symmetry (https://www.mdpi.com/journal/symmetry/special_issues/astronomy_symmetry, accessed on 2 August 2022) became open for submissions in the journal Symmetry, and we hoped to attract contributions from researchers in the fields of astronomy and astrophysics to an interdisciplinary journal which may be somewhat beyond their comfort zone. In the teaser, we wrote the following: “From meteoritic crystals to giant double-lobed radio galaxies, from light curves of transiting extrasolar planets to gravitationally-lensed images of distant quasars, from jets in young stellar objects to the morphology of planetary nebulae, from celestial mechanics to cosmology, symmetry is ubiquitous in the Universe, and therefore in astronomical research. This Special Issue of the interdisciplinary journal Symmetry aims to collect observational and theoretical contributions related to symmetry (or, actually, the lack of it) from various fields of astronomy, astrophysics, and closely-related disciplines”. According to a full-text search in the extensive publication database of the Smithsonian Astrophysical Observatory (SAO)—National Aeronautics and Space Administration (NASA) Astrophysics Data System (ADS) (https://ui.adsabs.harvard.edu/, accessed on 2 August 2022), the words symmetry, symmetric, asymmetry or asymmetric appear in more than 40,000 scientific papers published from 2020 up to the time of writing this editorial in 2022. Even though only seven of them found its way to this Special Issue, I hope that the readers will enjoy reading this collection, which covers a wide range of interesting topics, from celestial mechanics through stellar physics to extragalactic objects. To facilitate this, I briefly review each of the publications in the next section.

2. Contributions to the Special Issue

A fundamental problem in celestial mechanics is to describe the motion of n point-like bodies under the influence of mutual gravitational forces. The system of equations of motion is generally solvable only for $n=2$; therefore, special configurations are of great importance. In their study, Kovári & Érdi [2] deal with a so-called central configuration of four bodies where the forces on each body are directed toward the mass centre of the system. While central configurations promise analytic solutions, these are usually very complicated. A special case with $n=4$ is the axially symmetric ‘kite’ configuration where an axis of symmetry connects two bodies and the other two bodies of equal masses are located symmetrically with respect to this axis. What is even more special is that the authors in [2] apply the analytical solutions obtained for the above ‘kite’ configuration for the case where three of the four participating masses are equal. The numerical solutions give a complete description of this configuration by calculating the allowed ranges of the parameters describing the system.

The paper by Cseh [3] offers a novel view of an old problem. While studying the planetary motions in the Solar System some four centuries ago, Johannes Kepler tried to find a harmonic relationship between the distances of the known planets. Finally, he rejected his hypothesis, but the idea about the possibility of a regular sequence of planetary distances remained with us. The laws of gravity can successfully explain the motion of planets but tell us nothing about where exactly the planets have to be located. The author takes up a relation inspired by the hidden symmetry of the Kepler problem, which appears to be approximately valid in our Solar System, and tests it in a couple of selected multi-planet exoplanetary systems. Interestingly, those planetary distances seem to follow the same symmetry-inspired regularity. The conclusion in [3] is that although much research remains to be conducted in this topic, the exoplanetary systems that are being discovered provide us with a potentially very interesting possibility to reveal regularity in planet–star distances.

Returning to our own Solar System, the study of Hofmeister and Criss [4] deals with interplanetary gravitational interactions that are the strongest forces to perturb Keplerian planetary orbits around the Sun. To represent interplanetary interactions, which are transient, time-dependent and cyclical, the authors developed improved analytical and numerical models. Among their key findings, they mention that Mercury’s orbit is Keplerian, within the uncertainties of the available short-term observational data. Relativistic correction should be applied only after observational data are corrected for Earth’s non-Keplerian orbit, and precession models based on transient planetary interactions are considered.

Moving away from our immediate cosmic neighbourhood, Criss and Hofmeister [5] develop an elegant analytical inverse model, based on conjugate histograms of spin period and its reciprocal, angular velocity, to reveal how the rotation period of dwarf main-sequence stars changes with time. Obviously, ages are important inputs to astrophysical models describing stellar evolution. However, only our own Sun has an accurately determined age. A useful indicator of a star’s age is its axial spin rate. Generally speaking, young stars rotate fast while old ones rotate slowly. A fundamental question in gyrochronology is the relation between the spin rate and the stellar age. Fortunately, significant amounts of measurement data exist on large open clusters. Based on 15 clusters with at least 120 stellar light curve measurements, the authors find that exponential decay best describes the spin-down of stars smaller than 3 Solar masses, in contrast to earlier models. Moreover, the data indicate that, contrary to the generally assumed picture, the production of all stars in open clusters is not coeval. Notably, the approach of extracting a physical law from conjugate histograms could be useful for other applications as well [5].

The remaining three publications in the Special Issue deal with extragalactic objects and radio-emitting active galactic nuclei (AGNs). Krezinger et al. [6] report on high angular resolution radio interferometric observations of five low-luminosity AGNs in the redshift range $0.36<z<0.58$. These were pre-selected on the basis of kpc-scale double symmetric radio structure and flat or inverted continuum radio spectra in hopes of finding dual AGN cores among them. However, it turned out that the spectra of four sources were actually steep, contrary to earlier data published in the literature. Not surprisingly, these objects could not be confidently detected with the European Very Long Baseline Interferometry (VLBI) Network because their extended radio emission was resolved out on ∼1–10 milliarcsec angular scales. Nevertheless, one flat-spectrum quasar showed a compact, albeit single VLBI core at both observed frequencies (1.7 and 5 GHz).

Veres et al. [7] revisit the proposed counterpart of a $\gamma$-ray source detected with the Energetic Gamma Ray Experiment Telescope (EGRET) on board the Compton Gamma Ray Observatory satellite. Given the large positional error box (∼1${}^{\circ }$), it is not a trivial task to securely associate sources in the catalogue with extragalactic objects observed at other wavebands. It turns out that the overwhelming majority of extragalactic $\gamma$-ray sources are blazars, radio-loud AGNs producing a symmetric pair of relativistic plasma jets emanating from the vicinity of a supermassive black hole. One of the jets in blazars point nearly to our line of sight, rendering the radio emission Doppler-boosted. The authors analysed eight epochs of VLBI imaging data of the quasar J1826+3431. Its compact core and flux density variability support the blazar classification and strengthen the case for its association with the EGRET source.

Finally, I myself and collaborators [8] investigate and clarify the case of the radio-loud quasar CTD 135 (J2236+2828) in the light of recent observational data. This object has been proposed as a candidate compact symmetric object (CSO), based on its symmetric radio structure revealed by multi-frequency VLBI imaging. CSOs are young jetted AGNs for which its relativistic plasma jets are misaligned with respect to the line of sight, contrary to blazar jets. The peculiarity of CTD 135 as a CSO candidate was its detection in $\gamma$-rays, while $\gamma$-ray sources are usually blazars. A $\gamma$-ray CSO would be a rare and potentially very interesting object, but there is convincing evidence at multiple wavebands that CTD 135 is in fact a blazar rather than a CSO.