MATHEMATICS AND ASTRONOMY: A JOINT LONG JOURNEY: Proceedings of the International Conference

The Joint Journey of Mathematics and Astronomy in the Ancient World
View Description Hide DescriptionIn this paper we shall show some ways in which mathematics intervened to allow ancient and medieval astronomers to solve problems that they had not been able to solve previously. We shall, as well, discuss some of the ways in which astronomical problems stimulated developments in ancient mathematics. We shall begin with mathematical astronomy in Babylon, during a period when astronomy was a purely numerical exercise.
We shall then look at some aspects of Greek astronomy, when theories of the motions of heavenly bodies based on the geometry of spheres and circles were first introduced. And we shall see how the Greeks also developed new numerical methods to use with their geometry to create a new astronomy.
Finally, we shall examine how medieval Islamic astronomers created a more powerful version of trigonometry to invent the science of astronomical timekeeping. And we shall also see how they exercised geometrical ingenuity to bring the theories of Ptolemy into closer correspondence with motions that might be physically possible.

A Strategic Plan for European Astronomy: The ASTRONET Infrastructure Roadmap
View Description Hide DescriptionEurope is at the forefront of a golden age of astronomical discovery. The ASTRONET Infrastructure Roadmap represents a community‐based, comprehensive plan that aims to maintain and strengthen Europe’s rôle over the next 10–20 years and also to capitalise on our endeavours for the greater good of the continent’s population. In this review, I first describe the background to ASTRONET. This is followed by an outline of the development of the Science Vision for astronomy, and then that of the Roadmap itself. Details are given of the working methods used and the conclusions reached, which in the context of the current meeting include not only those regarding future facilities, but also areas such are theory, computing and public engagement. Gaps and opportunities in our proposed provision are outlined before addressing the proposed next steps for the ASTRONET programme as a whole.

Periodic orbits and non integrability of autonomous Glydén‐like systems
View Description Hide DescriptionWe apply the averaging theory of first order for studying the periodic orbits for an autonomous variation of the Gyldén problem, i.e., a model of the planar two body problem where the gravitational constant undergoes small variations depending on the cartesian coordinate x. Of course this problem admits a Hamiltonian formulation. Two main results are shown.
First, we show that at any negative energy level the Hamiltonian system has at least two periodic orbits. These periodic orbits form in the whole phase space a continuous family of periodic orbits parameterized by the energy.
Second, using these two families of periodic orbits we can prove the non integrability of the Hamiltonian system in the sense of Liouville‐Arnold, independently of the class of differentiability of the second first integral.
Moreover the two tools that we use for proving our results, the first for studying periodic orbits and the second for studying the non Liouville‐Arnold integrability can be used for Hamiltonian systems with an arbitrary number of degrees of freedom.

Secular behavior of a pair of coplanar planets
View Description Hide DescriptionThis chapter is a review of the secular dynamics of two coplanar planets. The first section presents the development of the secular model. We use both analytical and semi‐numerical approaches, which employ an averaging of the short period gravitational interaction of the planets. We apply the secular analysis to the case of the two outer planets, c and d, of the ν Andromedae system. Assuming an edge‐on orbital configuration of the planets, we obtain the general structure of its secular phase space, and the boundaries of its secular stability.

Solar Eclipses for Science and for Public Education
View Description Hide DescriptionOn the rare occurrences of total solar eclipses, about 350 years apart at a given terrestrial location and 18 months apart in the world, people in the zone of totality are dazzled and impressed by the spectacular phenomena that become visible in the midst of atmospheric darkening. I describe a selection of scientific results about the solar chromosphere corona that are obtained at eclipses, from the historic discovery of helium to present‐day investigations of how the corona gets to be 2,000,000 K. I also describe how the attention to the eclipse, to astronomy, and to science in general that often accompanies eclipse day in not only the zone of totality but also to thousands of kilometers to either side, can lead to teaching opportunities not only for students participating in the scientific observing but also to all students and the general public.

Teaching and Learning Astronomy
View Description Hide DescriptionI review the teaching and learning of astronomy, in elementary and secondary school, colleges and universities, and for the public through astronomy outreach and communication. I describe International Year of Astronomy 2009, and some of the national and personal projects in which I am involved.

Predicting chaotic climates: from Earth to super‐Earths?
View Description Hide DescriptionThe prediction of atmospheric behaviour for the Earth has been a major arena for the application of complex mathematical ideas and techniques to geophysics and astronomy. Objective forecasting of the weather provided the first stimulus for the development of numerical methods to integrate the equations of fluid motion by L. F. Richardson in the early 20th century, leading on to their implementation in electronic computers in the 1940s and 1950s. Such an approach has now reached a highly sophisticated state with weather and climate models attempting to forecast weather and climate changes in immense detail. Such techniques have been applied to model the atmospheres of other planets in the Solar System since the 1960s, and are catching up rapidly in their sophistication with models used for the Earth. But with the expanding discoveries of planets around other stars, it is likely that alternative approaches may be needed that are more general but seek to quantify trends in the gross features of atmospheric circulation systems as a function of a small number of global parameters. By the study of simple analogues, either in the form of simplified numerical models or laboratory experiments, considerable insights may be gained as to the likely roles of planetary size, rotation, thermal stratification and other factors in determining the principal length scales, styles of global circulation and dominant waves and instability processes active in all planetary atmospheres. In this review, we explore aspects of these analogues and demonstrate the importance of a number of key dimensionless parameters, most notably thermal Rossby and Burger numbers and a measure of the dominant frictional or radiative timescale, in defining the type of circulation regime to be expected in a prototype planetary atmosphere subject to axisymmetric driving. These considerations help to place Mars, Venus, Titan and Earth into an appropriate context, and may also lay the foundations for predicting and understanding the climate and circulation regimes of (as yet undiscovered) Earth‐like extrasolar planets. However, as recent discoveries of ‘super‐Earth’ planets around some nearby stars are beginning to reveal, the parameter space determined from axisymmetrically‐forced prototype atmospheres may be incomplete and other factors, such as the possibility of tidally‐locked rotation and tidal forcing, may also need to be taken into account for some classes of extra‐solar planet.

Mathematical snapshots motivated by “dark matter”
View Description Hide DescriptionA standard difficulty in astronomy is the inability to observationally verify basic assumptions about the dynamics such as that the moment of inertia of a system can be set equal to a constant, or that where T and U are, respectively, the system’s kinetic energy and self‐potential. To address these kinds of problems, the approach developed here is to introduce long term consequences of the assumptions. To be useful, the consequences must be of the type that can be observationally checked. The choices of illustrating assumptions come from aspects of how mass is computed for dark matter considerations.

A Long Journey of Mathematics and Astronomy in Romania
View Description Hide DescriptionBucharest Astronomical Observatory celebrated recently its centenary. Its founders were all mathematicians or, better said, astronomers specialized in celestial mechanics. Their first doctoral theses were defended at Sorbonne, in the second half of the 19th century, under the guidance of the greatest specialists of the time. After they returned home, they continued what they had begun in Paris, namely celestial mechanics. The instruments they ordered and the first programmes of astronomical observations had an increasingly closer relation to mathematics, as they referred to astrometry and especially to stellar catalogues.
Naturally, there were also astrophysical concerns, timid ones in the beginning, and then ever larger, especially beginning with the International Geophysical Year.
The evolution of world astronomy, as well as that of Romania, seems to be following but one direction: astrophysics. The truth is that astrometry and celestial mechanics continue to lie at the basis of all astrophysical researches, actually in an entirely new and modern form. The astrometry schools recently organized, the new astrometry textbooks, as well as the IAU working groups dedicated to modern astrometry prove that the long journey of mathematics and astronomy is not over yet.

Scalar and Vector Spherical Functions: Solving the Problems of Astrometry and Stellar Kinematics
View Description Hide DescriptionThe paper presents a review of the methods and results obtained in astrometry and galactic astronomy with the help of spherical harmonics. As the starting point, the representation of systematic differences in positions and proper motions via orthogonal scalar functions is considered. Proposed in the second half of XX century this method became a standard tool in comparison of ground based and space astrometry catalogues. Two further developments of this approach made by the author is reported afterwards.
The first one is deriving the parameters of mutual orientation and rotation of frames. In contrast to widely used least square method the spherical function formalism provides not only the determination of orientation and spin but (a) takes into account only the harmonics which correspond to rotation, (b) tests them for pure rotation, (c) discovers the existence of quasi‐rotational terms which may smear rotation. Due to these properties the spherical harmonics yield realistic results even in the case when the observational data contain not only noise but other systematic terms that have nothing to do with rotation. The application of this method to the comparison of the catalogues FK5 and HIPPARCOS is given.
The second item is the vector spherical functions analysis of stellar kinematics. This approach allows detecting all systematic components in the stellar velocity field without being attached to a specific physical model. Comparison of the theoretical decomposition coefficients of the equations for a particular kinematical model with observational data reveals systematic components that are not described by this model. The formalism of vector spherical functions is well suited for analyzing the present and future (GAIA) catalogues containing three velocity vector components: the proper motions in both coordinates and the radial velocity. Application of this method to the proper motions of HIPPARCOS stars gives evidence that the proper motions under consideration contain systematic components that can not be interpreted in terms of the standard Ogorodnikov‐Milne model. The same result is also confirmed by an analysis of the radial velocities of these stars.

N‐parametric canonical perturbation method based on Lie transforms. Application to the analysis of perturbations on multiple stellar systems
View Description Hide DescriptionRecently, in order to analytically solve perturbation problems involving an arbitrary number of small parameters in the Hamiltonian formulation, a multiparametric theory based on Lie transforms theory was derived by the author [5]. It is a complete generalization of the Hori‐Deprit method for N parameters with N arbitrary.
This method has been used to solve the classical Gyldén‐Meščerskij problem—the relative motion of a binary system the components of which are losing mass over time—when the primary’s oblateness, as well as relativistic effects, are taken into account. In addition, speed and accuracy comparisons between this analytical method and a numerical one (implicit Runge‐Kutta) were also performed.

Heavenly Mathematics & Cultural Astronomy: A Course at the National University of Singapore
View Description Hide DescriptionI have introduced a general education course called Heavenly Mathematics and Cultural Astronomy [2] at the National University of Singapore. The goal of this course is to study astronomy in a cultural context with a tropical emphasis. Most astronomy books are written from a high northern latitude point of view, but Singapore is almost on the equator, so I aim to be “hemispherically‐correct”. Singapore is also a multi‐racial society, where public holidays are determined using the Gregorian, Chinese, Islamic and Indian calendars.
The course starts with an introduction to observational astronomy with an emphasis on the appearance of the Sun and the Moon from different parts of the world. I then give a fairly detailed description of the Gregorian, Chinese, Islamic and Indian calendars [1, 4, 5], and finish with a thorough discussion of the analemma, equation of time and navigation [3].
Being a mathematician, my approach is quite mathematical, but my emphasis is on geometrical reasoning. Formulas and computations may scare some students away, but they are surprisingly willing to struggle with complicated spatial visualization.

Linear stability in an extended ring system
View Description Hide DescriptionThe planar ring body problem consists of n bodies of equal mass m uniformly distributed around a central body of mass The bodies are rotating on its own plane about its center of mass with a constant angular velocity. Since Maxwell introduced the problem to understand the stability of Saturn’s rings, many authors have studied and extended the problem. In particular, we proved that if forces that are functions of the mutual distances are considered the n‐gon is a central configuration. Examples of this kind are the quasi‐homogeneous potentials.
In a previous work we analyzed the linear stability of a system where the potential of the central body is a Manev’s type potential. By introducing a perturbation parameter to the Newtonian potential associated with the central primary, we showed that unstable cases for the unperturbed problem, for may become stable for some values of the perturbation.
The purpose of this paper is to show that it is possible to increase the range of values of the mass parameter and the parameter in order to render a stable configuration. In order to get it, we introduce a second perturbation term (with parameter to the Newtonian potential of the bodies in the ring. We show some results for the problem with

Basic aerodynamical properties of meteoroid fragments in the terrestrial atmosphere
View Description Hide DescriptionThere is significant evidence that some fraction of meteoric bodies is destroyed in the atmosphere. The evolution of the fragment cloud depends on a large number of factors, among them: the meteoroidŠs altitude and velocity at the moment of breakup, fragments sizes and properties of a body material. The interaction of shock waves forming in front of the fragments may lead to both an increase and decrease of the midsection area of the fragment cloud [1], [2]. In this work, we consider the inter action of the fragments in a supersonic flow. The configuration properties of two spherical bodies of different radii are considered. Via numerical simulations, we calculate the pressure distribution in the flow around the two bodies for different relative positions. We construct the functions of the coefficients of transverse and drag forces from the angle between the central line of the two bodies and the flow direction for different distances between the two fragments. We find the conditions for the collimation effect, i.e., fragment involving into the wake of the leading (usually, the largest) fragment. We systematize the simulation results for drag and forces and infer the basic aerodynamic properties of the meteoroid fragments. Moreover, in the work dynamics of two spherical fragments in a supersonic stream is defined on the basis of numerical calculations. The aerodynamic properties are accurately expressed in a dynamic picture. Besides, it is obtained, that fragments connect and move as a unit due to aerodynamic interaction.

Mathematics and Astronomy: Inquire Based Scientific Education at School
View Description Hide DescriptionMathematics is the language of science however, in secondary and high school education students are not made aware of the strong implications behind this statement. This is partially caused because mathematical training and the modelling of nature are not taught together. Astronomy provides firm scientific grounds for this joint training; the mathematics needed is simple, the data can be acquired with simple instrumentation in any place on the planet and the physics is rich with a broad range of levels. In addition, astronomy and space exploration are extremely appealing to young (14–17 years old) students helping to motivate them to study science doing science, i.e. to introduce Inquiry Based Scientific Education (IBSE). Since 1997 a global consortium is being developed to introduce IBSE techniques in secondary/high school education on a global scale: the Global Hands‐On Universe association (www.globalhou.org) making use of the astronomical universe as a training lab. This contribution is a brief update on the current activities of the HOU consortium.
Relevant URLS: www.globalhou.org, www.euhou.net, www.houspain.com.

Kinematics in space with variable linear measurement standards
View Description Hide DescriptionProposed approach is based on the idea of linear measurement standards spontaneous shortening with time. Shown is the possibility to explain Hubble Law applying observational data and this approach for any distance being free of contradictions as an alternative to the dark energy hypothesis. This approach also permits to determine the inverse dependence of the value of cosmological constant on the square of radius of visible Universe.

Ancient Indian astro‐mathematical tradition: Evolution and linkages
View Description Hide DescriptionIndian astronomical tradition is characterized by antiquity, continuity and interaction with the outside world. From 6th century CE till the time of Kepler’s laws, Indian astronomers were probably the only ones in the world who could calculate eclipses with any degree of accuracy. In the 12th century, an astronomer in Central India, Padmanabha by name, predicted the lunar eclipse of 8 November 1128 and was rewarded by the king with a land grant (Mirashi 1933–34). The tradition was alive well into the 19th century. By means of shells arranged on the ground and using mathematical tables memorized “by means of certain artificial words and syllables”, a “Kalendar maker residing in Pondicherry” calculated the lunar eclipse of 31 May–1 June 1825, with an error of no more than +4 minutes for the beginning (Neugebauer 1983, p. 436). Even now, traditional astronomical almanacs in India, known as panchangas, used in India for ritual and religious purposes base their calculations on ancient texts. It is only in the case of eclipse that they borrow data from modern sources.
The beginnings of astronomy are related to the requirements of the ritual in early cultures. Ritual was seen as a means of securing divine approval and support for terrestrial actions. To be effective, it had to be elaborate and welltimed, so that a careful distinction could be made between auspicious and inauspicious times. Since planetary motions provided a natural means of time keeping and were seen as embodiment of divine signals, astronomy developed as an intellectual discipline (see Yano 2003). Similarly mathematics grew as an aid to designing sacrificial altars. The oldest geometry texts in India are the Sulvasutras which dealt with questions like the square root of two. Different scholars place the earlier of these texts anywhere between 800 BCE and 400 BCE. Astronomy texts are decidedly older. Subsequent developments in mathematics came about as an astronomical aid.

ITEMS Project: An online sequence for teaching mathematics and astronomy
View Description Hide DescriptionThis work describes an elearning sequence for teaching geometry and astronomy in lower secondary school created inside the ITEMS (Improving Teacher Education in Mathematics and Science) project. It is based on results from the astronomy education research about studentsŠ difficulties in understanding elementary astronomical observations and models. The sequence consists of a set of computer animations embedded in an elearning environment aimed at supporting students in learning about astronomy ideas that require the use of geometrical concepts and visual‐spatial reasoning.

Near field cosmological simulations: Is Dark Energy playing a role in our Local Neighbourhood?
View Description Hide DescriptionCould we find any manifestation of the dark energy in the local universe? Recently, this possibility has been argued by different authors using theoretical models and observational data [1] and simulations [2]. In order to check this important issue, we have run a set of different cosmological simulations with constrained intial conditions specially designed to represent the local mass distributions around us. Using this kind of simulations with different cosmological models, resolution and codes, we have proved that there is not such evidence of a dark energy manifestation in the local universe [3] [4].
These constrained simulations constitue an excellent laboratory to investigate how dark matter is distributed and structured in an environment pretty much the same as in the real universe [5].