Celestial Sphere Research Articles

Stellar Aberration from Earth and from a Satellite

In his endeavor to find a concrete evidence in favor of the Copernican picture of the solar system, the English astronomer James Bradley made a series of astronomical observations during the period (1725-1728) aiming to detect a stellar parallax. His findings, which manifested indeed an annual apparent cyclic motion of a star, were however at conflict with what is expected in a parallax. To his surprise, the result of every measurement obtained corresponded to what he expected to get in a measurement done three months earlier. Bradley realized that he was witnessing a new physical effect, and he presented an explanation that conceived light as a corpuscular stream travelling at finite velocity. Despite that Bradley’s explanation of the stellar aberration effect was inadequate, the equation which he derived to quantify the aberration angle, predicted a better estimation of light velocity, and the aberration phenomenon itself was a concrete support of heliocentrism. Stellar aberration as well as some other optical experiments, whose explanations posed challenges to the existing physical theories in the late nineteenth century paved the way for the emergence of the special theory of relativity. In the current work we employ the theory of universal space and time to show that a given direction in a frame of reference is tilted when observed in a moving frame by an angle that depends on the direction itself and the velocity of the moving frame. The latter fact is utilized to explain stellar aberration, determine the deviation of a star’s vision direction from its true one, and deduce its apparent position at any instant as a function of its latitude and time. The novel concept of aberration correction vector is employed to derive the apparent elliptic path of an observed celestial object at any time. The concept of graded inertial frames is introduced and utilized to deal with aberration when observed from a satellite in a similar way to its treatment when observed from Earth. The transformation matrix between a geocentric frame and a satellite’s non-rotating frame is derived and used to transform temporary Earthly vision directions to the satellite’s frame. Furthermore, the transformed vectors are adopted as transient fixed directions relative to which the vision directions of a star from the satellite are specified throughout one revolution. Satellites connective matrices are constructed to make geometric information regarding the celestial sphere in one frame immediately usable by observers on Earth and in all other satellites.

Paul’s Three Paths To Salvation

Paul’s Three Paths To Salvation

Conformal wave expansions for flat space amplitudes

The extended BMS algebra contains a conformal subgroup that acts on the celestial sphere as SO(1, 3). It is of interest to perform mode expansions of free fields in Minkowski spacetime that realize this symmetry in a simple way. In the present work we perform such a mode expansion for massive scalar fields using the unitary principal series representations of SO(1, 3) with a view to developing a holographic approach to gravity in asymptotically flat spacetime. These mode expansions are also of use in studying holography in three-dimensional de Sitter spacetime.

(2, 2) Scattering and the celestial torus

Analytic continuation from Minkowski space to (2, 2) split signature spacetime has proven to be a powerful tool for the study of scattering amplitudes. Here we show that, under this continuation, null infinity becomes the product of a null interval with a celestial torus (replacing the celestial sphere) and has only one connected component. Spacelike and timelike infinity are time-periodic quotients of AdS3. These three components of infinity combine to an S3 represented as a toric fibration over the interval. Privileged scattering states of scalars organize into SL(2, ℝ)L×SL(2, ℝ)R conformal primary wave functions and their descendants with real integral or half-integral conformal weights, giving the normally continuous scattering problem a discrete character.

A mestizo cosmographer in the New Kingdom of Granada: astronomy and chronology in Sánchez de Cozar Guanientá’s Tratado (c.1696)

ABSTRACT This article interprets a recently recovered manuscript, Tratado de astronomía y la reformaçión del tiempo, composed by Antonio Sánchez in New Granada c.1696, in the context of the Spanish and Renaissance cosmographies. Sánchez’s Tratado proposes a spherical astronomy, in which celestial bodies – including comets — move in orbs containing pyramidal knots that explain the changing speed observed in the motion of planets. From this astronomy and following the peninsular style of repertorios, Sánchez derives two major conclusions: the corrected length of the solar year and a revised birth date of Jesus. Taking as center of reference Vélez, where Sánchez was based, these claims led to conclusions in domains ranging from calendric astronomy to eschatology, including the incorporation of the indigenous peoples into salvation narrative and a demonstration of the arrangement of the celestial orbs at the Last Judgment. Sánchez’s Tratado constitutes an expansion of the Spanish mathematical cosmography that sheds light on the production of knowledge in the Spanish-American world and, at the same time, provides elements to reassess our understanding of the global circulation of Renaissance and early modern ideas.

Functional Principal Component Analysis of Radio–Optical Reference Frame Tie

Abstract The Gaia optical reference frame is intrinsically undefined with respect to global orientation and spin, so it needs to be anchored in the radio-based International Celestial Reference Frame (ICRF) to provide a referenced and quasi-inertial celestial coordinate system. The link between the two fundamental frames is realized through two samples of distant extragalactic sources, mostly active galactic nuclei and quasars, but only the smaller sample of radio-loud ICRF sources with optical counterparts is available to determine the mutual orientation. The robustness of this link can be mathematically formulated in the framework of functional principal component analysis using a set of vector spherical harmonics to represent the differences in celestial positions of the common objects. The weakest eigenvectors are computed, which describe the greatest deficiency of the link. The deficient or poorly determined terms are specific vector fields on the sphere that carry the largest errors of absolute astrometry using Gaia in reference to the ICRF. This analysis provides guidelines for the future development of the ICRF maximizing the accuracy of the link over the entire celestial sphere. A measure of the robustness of a least-squares solution, which can be applied to any linear model fitting problem, is introduced to help discriminate between reference frame tie models of different degrees.

Dark Energy Survey Year 3 results: Curved-sky weak lensing mass map reconstruction

ABSTRACT We present reconstructed convergence maps, mass maps, from the Dark Energy Survey (DES) third year (Y3) weak gravitational lensing data set. The mass maps are weighted projections of the density field (primarily dark matter) in the foreground of the observed galaxies. We use four reconstruction methods, each is a maximum a posteriori estimate with a different model for the prior probability of the map: Kaiser–Squires, null B-mode prior, Gaussian prior, and a sparsity prior. All methods are implemented on the celestial sphere to accommodate the large sky coverage of the DES Y3 data. We compare the methods using realistic ΛCDM simulations with mock data that are closely matched to the DES Y3 data. We quantify the performance of the methods at the map level and then apply the reconstruction methods to the DES Y3 data, performing tests for systematic error effects. The maps are compared with optical foreground cosmic-web structures and are used to evaluate the lensing signal from cosmic-void profiles. The recovered dark matter map covers the largest sky fraction of any galaxy weak lensing map to date.

Roden Crater: Home Edition

Roden Crater, Arizona, engages with the phenomenology of light and the celestial sphere. Because such phenomena elude capture by photography, first-hand experience is necessary. However, the project will not be open to the public for years, as artist James Turrell wants to first achieve an overall sense of completion. As a geographer and landscape architect, my fascination with phenomenology and in particular with Roden Crater grew into a series of experiments at a domestic scale, to explore the potential for generating parallel experiences. This paper documents the practice of making 1:1 scale experiments that are based on Roden Crater, and the ways in which phenomena can be amplified and experienced. The experiments transformed my sensation of the sky and the perception of being on a moving planet, acts of defamiliarization and strangemaking that intensify our seeing of place

Non-abelian infrared divergences on the celestial sphere

We consider the infrared factorisation of non-abelian multi-particle scattering amplitudes, and we study the form of the universal colour operator responsible for infrared divergences, when expressed in terms of coordinates on the ‘celestial sphere’ intersecting the future light-cone at asymptotic distances. We find that colour-dipole contributions to the infrared operator, to all orders in perturbation theory, have a remarkably simple expression in these coordinates, with scale and coupling dependence factorised from kinematics and colour. Generalising earlier suggestions in the abelian theory, we then show that the infrared operator can be computed as a correlator of vertex operators in a conformal field theory of Lie-algebra-valued free bosons on the celestial sphere. We verify by means of the OPE that the theory correctly predicts the all-order structure of collinear limits, and the tree-level factorisation of soft real radiation.

Light-ray operators, detectors and gravitational event shapes

Light-ray operators naturally arise from integrating Einstein equations at null infinity along the light-cone time. We associate light-ray operators to physical detectors on the celestial sphere and we provide explicit expressions in perturbation theory for their hard modes using the steepest descent technique. We then study their algebra in generic 4-dimensional QFTs of massless particles with integer spin, comparing with complexified Cordova-Shao algebra. For the case of gravity, the Bondi news squared term provides an extension of the ANEC operator at infinity to a shear-inclusive ANEC, which as a quantum operator gives the energy of all quanta of radiation in a particular direction on the sky. We finally provide a direct connection of the action of the shear-inclusive ANEC with detector event shapes and we study infrared-safe gravitational wave event shapes produced in the scattering of massive compact objects, computing the energy flux at infinity in the classical limit at leading order in the soft expansion.

An analytic solution of full-sky spherical geometry for satellite relative motions

Herein, an exact and efficient analytic solution for an unperturbed satellite relative motion was developed using a direct geometrical approach. The derivation of the relative motion geometrically interpreted the projected Keplerian orbits of the satellites on a sphere (Earth and celestial spheres) using the solutions of full-sky spherical triangles. The results were basic and computationally faster than the vector and plane geometry solutions owing to the advantages of the full-sky spherical geometry. Accordingly, the validity of the proposed solution was evaluated by comparing it with other analytic relative motion theories in terms of modeling accuracy and efficiency. The modeling accuracy showed an equivalent performance with Vadali’s nonlinear unit sphere approach, which is essentially equal to the Yan–Alfriend nonlinear theory. Moreover, the efficiency was demonstrated by the lowest computational cost compared with other models. In conclusion, the proposed modeling approach illustrates a compact and efficient closed-form solution for satellite relative motion dynamics.

Gauge and gravity amplitudes on the celestial sphere

The analytic structures of scattering amplitudes in gauge theory and gravity are examined on the celestial sphere. The celestial amplitudes in the two theories - computed by employing a regulated Mellin transform - can be compared at low multiplicity. It is established by direct computation that up to five external particles, the double copy relations of Kawai, Lewellen and Tye continue to hold identically, modulo certain multiplicative factors which are explicitly determined. Supersymmetric representations of the amplitudes are utilized throughout, manifesting the double copy structure between $\mathcal{N}=4$ super Yang-Mills and $\mathcal{N}=8$ supergravity on the celestial sphere.

Soft gravity by squaring soft QED on the celestial sphere

We recast the soft $S$-matrices on the celestial sphere as correlation functions of certain $2$-dimensional models of topological defects. In pointing out the double copy structure between the soft photon and soft graviton cases, we arrive at a putative classical double copy between the corresponding topological models and a rederivation of gauge invariance and the equivalence principle as Ward identities of the $2$-dimensional theories.

Circular Truncated Conical Mirror with Cavity Shape and Inner Reflection for Concentrating Diffused Light to Collector Mouth from a Half Celestial Sphere

In this study, we designed a circular truncated conical mirror to concentrate diffused light to the collector mouth from a half celestial sphere and evaluated its performance. The conical mirror has a circular truncated cone-shaped cavity, which reflects light on the cavity’s inner surface. The top of the truncated cone shape is the incident mouth, the bottom is the collector mouth, and the sides are the mirror surfaces. The incident mouth size is smaller than the collector mouth size. All light incident on the incident mouth is either reflected or not reflected on the mirror surface to reach the collector mouth. Light incident at a shallow angle to the central axis of the conical mirror is not reflected on the mirror surface, while light incident at a deep angle is reflected on the mirror surface. The conical mirror can concentrate all diffused light incident at the incident mouth from a half celestial sphere of 360◦ in the horizontal direction and 180◦ in the vertical direction to the collector mouth. A coordinate system is established for the conical mirror, and a design method is proposed. Using our design method, the conical mirror is manufactured from the determined design values. A plane mirror, concave mirror, compound parabolic mirror, and Fresnel lens, comprising the conical mirror, are used as the measured objects to compare the manufactured conical mirror performance. An experimental setup is constructed to evaluate the performance of the measured objects. Using the constructed experimental setup, diffused light is irradiated to the measured objects as scattered light. The illuminance of the light concentrated in the collector mouth of the measured objects is measured for the irradiated diffused light. The measured illuminance is then used to compare and evaluate the light-concentrating efficiency of the measured objects.

Synthetic RGB photometry of bright stars: definition of the standard photometric system and UCM library of spectrophotometric spectra

ABSTRACT Although the use of RGB photometry has exploded in the last decades due to the advent of high-quality and inexpensive digital cameras equipped with Bayer-like colour filter systems, there is surprisingly no catalogue of bright stars that can be used for calibration purposes. Since due to their excessive brightness, accurate enough spectrophotometric measurements of bright stars typically cannot be performed with modern large telescopes, we have employed historical 13-colour medium-narrow-band photometric data, gathered with quite reliable photomultipliers, to fit the spectrum of 1346 bright stars using stellar atmosphere models. This not only constitutes a useful compilation of bright spectrophotometric standards well spread in the celestial sphere, the UCM library of spectrophotometric spectra, but allows the generation of a catalogue of reference RGB magnitudes, with typical random uncertainties ∼0.01 mag. For that purpose, we have defined a new set of spectral sensitivity curves, computed as the median of 28 sets of empirical sensitivity curves from the literature, that can be used to establish a standard RGB photometric system. Conversions between RGB magnitudes computed with any of these sets of empirical RGB curves and those determined with the new standard photometric system are provided. Even though particular RGB measurements from single cameras are not expected to provide extremely accurate photometric data, the repeatability and multiplicity of observations will allow access to a large amount of exploitable data in many astronomical fields, such as the detailed monitoring of light pollution and its impact on the night sky brightness, or the study of meteors, Solar system bodies, variable stars, and transient objects. In addition, the RGB magnitudes presented here make the sky an accessible and free laboratory for the calibration of the cameras themselves.

A comparison theorem for cosmological lightcones

Let (M, g) denote a cosmological spacetime describing the evolution of a universe which is isotropic and homogeneous on large scales, but highly inhomogeneous on smaller scales. We consider two past lightcones, the first, {{mathcal {C}}_{L}^{-}}(p, g), is associated with the physical observer pin ,M who describes the actual physical spacetime geometry of (M, g) at the length scale L, whereas the second, {mathcal {C}_{L}^{-}}(p, hat{g}), is associated with an idealized version of the observer p who, notwithstanding the presence of local inhomogeneities at the given scale L, wish to model (M, g) with a member (M, hat{g}) of the family of Friedmann–Lemaitre–Robertson–Walker spacetimes. In such a framework, we discuss a number of mathematical results that allows a rigorous comparison between the two lightcones {mathcal {C}_{L}^{-}}(p, g) and {mathcal {C}_{L}^{-}}(p, hat{g}). In particular, we introduce a scale-dependent (L) lightcone-comparison functional, defined by a harmonic type energy, associated with a natural map between the physical {mathcal {C}_{L}^{-}}(p, g) and the FLRW reference lightcone {mathcal {C}_{L}^{-}}(p, hat{g}). This functional has a number of remarkable properties, in particular it vanishes iff, at the given length-scale, the corresponding lightcone surface sections (the celestial spheres) are isometric. We discuss in detail its variational analysis and prove the existence of a minimum that characterizes a natural scale-dependent distance functional between the two lightcones. We also indicate how it is possible to extend our results to the case when caustics develop on the physical past lightcone {mathcal {C}_{L}^{-}}(p, g). Finally, by exploiting causal diamond theory, we show how the distance functional is related (to leading order in the scale L) to spacetime scalar curvature in the causal past of the two lightcones, and briefly illustrate a number of its possible applications.

Potential asteroid discoveries by the ESA Gaia mission

Context. Since July 2014, the Gaia mission of the European Space Agency has been surveying the entire sky down to magnitude 20.7 in the visible. In addition to the millions of daily observations of stars, thousands of Solar System objects (SSOs) are observed. By comparing their positions, as measured by Gaia, to those of known objects, a daily processing pipeline filters known objects from potential discoveries. However, owing to Gaia’s specific observing mode, which follows a predetermined scanning law designed for stars as “fixed” objects on the celestial sphere, potential newly discovered moving objects are characterized by very few observations, which are acquired over a limited time. Furthermore, these objects cannot be specifically targeted by Gaia itself after their first detection. This aspect was recognized early on in the design of the Gaia data processing. Aims. A daily processing pipeline dedicated to these candidate discoveries was set up to release calls for observations to a network of ground-based telescopes. Their aim is to acquire follow-up astrometry and to characterize these objects. Methods. From the astrometry measured by Gaia, preliminary orbital solutions are determined, allowing us to predict the position of these potentially newly discovered objects in the sky while accounting for the large parallax between Gaia and the Earth (separated by 0.01 au). A specific task within the Gaia Data Processing and Analysis Consortium has been responsible for the distribution of requests for follow-up observations of potential Gaia SSO discoveries. Since late 2016, these calls for observations (nicknamed “alerts”) have been published via a Web interface with a quasi-daily frequency, together with observing guides, which is freely available to anyone worldwide. Results. Between November 2016 and the end of the first year of the extended mission (July 2020), over 1700 alerts were published, leading to the successful recovery of more than 200 objects. Among them, six have a provisional designation assigned with the Gaia observations; the others were previously known objects with poorly characterized orbits, precluding identification at the time of Gaia observations. There is a clear trend for objects with a high inclination to be unidentified, revealing a clear bias in the current census of SSOs against high-inclination populations.

Lightlike geodesics and gravitational lensing in the spacetime of an accelerating black hole

The C-metric is a solution to Einstein’s vacuum field equation that describes an accelerating black hole. In this paper we discuss the propagation of light rays and the resulting lensing features in this metric. We first solve the lightlike geodesic equation using elliptic integrals and Jacobi elliptic functions. Then we fix a static observer in the region of outer communication of the C-metric and introduce an orthonormal tetrad to parameterise the directions of the light rays ending at the position of the observer using latitude-longitude coordinates on the observer’s celestial sphere. In this parameterisation we rederive the angular radius of the shadow, we formulate a lens equation, and we derive the redshift and the travel time of light rays. We discuss the relevance of our theoretical results for detecting accelerating black holes described by the C-metric and for distinguishing them from non-accelerating black holes.

Theology and Science in Copernicus’ Universe

The publication of Copernicus’ On the Revolutions of the Heavenly Spheres (since now on referred to as Revolutions) marked the beginning of the Scientific Revolution. Christian doctrine played a key role for the emergence of the scientific turning point, that brought about the transition from a qualitative to a quantitative approach to natural phenomena. Although the Polish scientist was not a philosopher in the ordinary sense of the term, he shared with many other protagonists of modern science the idea of the universe as mathematical harmony created by God. In this sense, modern scientific thought completed the development that took place since the Scholastic Age. In the historical period between the XII and the XVII century, indeed, Christianity proved a fundamental factor for a considerable growth in natural knowledge.

ʾABŪ SAʿĪD AL-SIǦZĪ AND THE “STRUCTURE OF THE ORBS,” THE EARLIEST KNOWN WORK ON HAYʾA

AbstractʾAbū Saʿīd al-Siǧzī was a prominent fourth/tenth century astronomer and mathematician who was one of the first contributors to the new genre of ʿilm al-hayʾa (science of the configuration). However, little is known about the initial steps taken in the formation of the discipline, or its independence from other astronomical writings and practices. In this paper, I will discuss new findings about Siǧzī’s life to determine details of his biography and a more precise time period for his scientific activities. I then describe, for the first time, a composition in theoretical astronomy from the fourth/tenth century, the “Structure of the orbs” (Tarkīb al-aflāk) by Siǧzī to show its place in the formation of the discipline of ʿilm al-hayʾa, comparing Siǧzī’s book to the earlier work of al-Farġānī on the size of the earth and the number of celestial spheres. I conclude that Siǧzī’s Tarkīb al-aflāk is the earliest known example of a book that contains only topics found in later hayʾa and may be the first appearance of the ninth celestial orb that became standard in the later genre.