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Page 202 of 851

  • Introduction

  • Bruce MacEvoy
  • Illustrated by Wil Tirion
  • Book:
    The Cambridge Double Star Atlas
    Published online:
    05 May 2016
    Print publication:
    10 December 2015, pp 1-20

  • Summary

    This new edition of the Cambridge Double Star Atlas is designed to improve its utility for amateur astronomers of all skill levels.

    For the first time in a publication of this type, the focus is squarely on double stars as physical systems, so far as these can be identified with existing data. Using the procedures described in Appendix A, the target list of double stars has been increased to 2,500 systems by adding 1,100 “high probability” physical double and multiple stars and deleting more than 850 systems beyond the reach of amateur telescopes or lacking any evidence of a physical connection. Wil Tirion has completely relabeled the Atlas charts to reflect these changes, and left in place the previous edition's double star icons as a basis for comparison. This new edition provides a selection based on evidence rather than traditional opinion, so that the twenty-first century astronomer can explore with fresh eyes the astonishing actual variety in double stars.

    Continuing the emphasis on physical systems, this Atlas explains the origin and dynamic properties of double stars and the role they have played in our understanding of star formation and stellar evolution. The elements of binary orbits, stellar spectral types, and methods of detecting and cataloging double stars, are explained to enrich the observer's understanding of double star astronomy. There is also practical guidance for the visual astronomer – information on optics, equipment preparation, useful accessories, viewing techniques and opportunities for amateur research. The references suggest both print and online double star resources. Finally, over 330 systems in the target list are marked with a star (★) in the left margin. These indicate “showpiece” systems of intrinsic beauty or charm, “challenge” pairs of close separation or large brightness contrast, and several systems that have been important in the history of astronomy. From most observing locations, at least three dozen of these targets will be in view at any time of night on any evening of the year.

    Jim Mullaney's choice of nineteenth century double star catalog labels has been retained as a tribute both to his original Atlas concept and to the bygone astral explorers who discovered over 90% of the systems in the target list (see Appendix D).

  • Appendix A - The target list

  • Bruce MacEvoy
  • Illustrated by Wil Tirion
  • Book:
    The Cambridge Double Star Atlas
    Published online:
    05 May 2016
    Print publication:
    10 December 2015, pp 85-164

  • Summary

    Two criteria were used to select the following 2,500 “high probability” double stars from the 110,950 unique systems in the January, 2015 Washington Double Star Catalog.

    The first is visual limits to magnitude and resolution. With few exceptions, targets are only included down to a combined visual magnitude of 7.75 and secondary stars down to magnitude 13.5; separations are not less than the Abbe resolution limit of Ro = 0.5″ feasible with a 250 mm or larger objective. Over 90% of the pairs can be resolved with a 150 mm aperture.

    The second is evidence of a physical bond. This includes a projected separation less than 5,000 AU, a divergence between proper motion vectors less than 30% of the larger proper motion, a note in WDS that the system is physical based on CPM and/or parallax data, or an orbital solution (weighted by the quality rating assigned in the 6th Catalog of Orbits of Visual Binary Stars). Negative indicators included a projected separation greater than 50,000 AU, a CPM deviation greater than 80%, a WDS flag indicating a “bogus binary” or optical pair, or a linear solution (a straight line path rather than curved orbital motion). Every record in WDS was evaluated against these criteria and the highest scoring systems and components (within the visual limits) were selected. These systems were further vetted using sky survey images at SIMBAD, observing notes and other references. Although this procedure cannot ensure every target is a physical system, it does reveal more accurately the true variety in physical double and multiple stars outside the traditional and ambiguous criterion of “looking like” a double star.

    To conserve space, data are listed on a single line. This includes the Bayer, Flamsteed or Gould (southern hemisphere) designation, the catalog ID (as it appears in the charts); the component letter codes; the celestial coordinates for epoch J2000; magnitudes, position angle (θ) and separation (ρ) for the listed components; distance from the Sun (in parsecs); spectral types (with alternate or dual types, most special codes and giant subtypes omitted for simplicity), and the Henry Draper (HD) and Smithsonian (SAO) catalog numbers. Distances estimated by spectroscopic parallax are flagged with an asterisk (*).

  • 14 - Observational constraints

  • from IV - Expansion models
  • William D. Heacox, University of Hawaii, Hilo
  • Book:
    The Expanding Universe
    Published online:
    28 May 2018
    Print publication:
    26 November 2015, pp 164-174

  • Summary

    Solutions to the Friedmann Equations constitute a four-parameter family of expansion functions a (t); it is the cosmologists’ job to determine which of them is correct, or at least plausible. One way to do so is to measure the Universe's mass/energy densities so as to determine the inputs to the Friedmann Equations, but this approach is inherently inaccurate due to difficulties in determining such densities observationally (Chapter 12) and to the possibility of forms of energy density that leave no direct trace (e.g., dark energy). The alternative is to infer such densities from evidence left by expansion in the form of spatial and kinematic structures in the observable Universe. Most of these hinge upon observational estimates of distances to galaxies.

    Primary expansion diagnostics

    Distance estimates on cosmological scales are sufficiently difficult as to have generated a small industry among cosmologists over the past several decades. The overall outline of such methods is hierarchical: distances on one scale are calibrated with those on a previous, smaller scale. Thus:

  • • a fundamental distance is the Astronomical Unit (AU), conveniently measured by bouncing radar beams off the surface of the Sun;

  • • distances to relatively nearby stars are then measured in terms of trigonometric parallaxes based on the AU (shifts in apparent star position as the Earth revolves about the Sun);

  • • distances to other, more distant stars are estimated from stellar properties (e.g., spectral class luminosities) calibrated from nearby stellar parallaxes.

    • All this is sufficient to determine distances within local portions of our Galaxy. To extend this procedure to cosmological distances we require standards that can be seen at such distances and that have either luminosities or other characteristics of known values. Bright objects of known or presumed luminosities are particularly helpful in that they can be used to infer luminosity distances via Equations (9.63) or (9.66); such creatures are known prosaically as ‘standard candles’. Historically, the most useful standard candles have been Cepheid variable stars, whose luminosity can be inferred from their readily observable pulsation periods.

  • 4 - General covariance

  • from II - General Relativity
  • William D. Heacox, University of Hawaii, Hilo
  • Book:
    The Expanding Universe
    Published online:
    28 May 2018
    Print publication:
    26 November 2015, pp 27-37

  • Summary

    A major accomplishment of Einstein's Theory of Special Relativity (SR) was the demonstration that the laws of physics took the same form for all inertial observers. In more relevant language we would say that the laws of physics were independent of the reference frame and coordinate system chosen for their expression – but only for inertial reference frames or, what is equivalent, for coordinate systems related by Lorentz transforms. This covariance of inertial coordinate systems allowed Einstein to write the laws of mechanics and of electrodynamics in ways that revealed new aspects (e.g., E = mc2) and extended their validity to reference frames moving at high velocity.

    It was thus a principal purpose of General Relativity (GR) to extend this Principle of General Covariance – that the equations of physics were invariant to change of coordinate systems – to all reference frames, including accelerating ones. Needed for this purpose are tools of mathematical physics that preserve equalities under all changes of coordinate systems and thus – as Einstein would put it – free physics from the tyranny of coordinates. The required tools were providentially at hand in the form of the Absolute Differential Calculus, a coordinate-independent version of calculus being developed by (mostly Italian) mathematicians. Among the tools in that development were mathematical quantities associated with geometry that all transformed in the same manner under changes in coordinate systems, so that equalities in any coordinate system led to equalities in all. These were tensors, different forms for which have since been employed in many areas of advanced physics.

    The idea of tensors is simple enough: if tensors A and B are equal in coordinate system S, and transform to A′ and B′ in system S′, then it follows that A′ = B′ there also. The trick is to find tensors to represent physical quantities of interest; fortunately for GR, in which geometrically related objects play a central role, tensor representations exist for most applications.

    The remainder of this chapter is devoted to an explanation of the most basic properties of tensors and of their manipulations in the service of the differential geometry employed in GR. Tensor analysis is not really difficult, but it is different from the mathematics to which students have normally been exposed and so requires one's attention in order to understand.

  • Preface

  • William D. Heacox, University of Hawaii, Hilo
  • Book:
    The Expanding Universe
    Published online:
    28 May 2018
    Print publication:
    26 November 2015, pp xi-xiii

  • Summary

    The pace of change in cosmology has accelerated remarkably in the years bracketing the turn of the twenty-first century, so that many of the classical texts are becoming dated. The purpose of this text is to provide a coherent description of current theory underlying modern cosmology, at a level appropriate for advanced undergraduate students. To do so, the book is loosely organized around two pedagogical principles.

    First, while the development of physical cosmology is heavily mathematical, the book emphasizes physical concepts over mathematical results wherever possible. The mathematics of General Relativity and of relativistic cosmology are beautiful, elegant, and seductive. It is a real temptation to develop theoretical cosmology as a purely mathematical structure, much as can be done with classical thermodynamics. But to do so is to lose sight of the deeper meaning of cosmology and to leave the student unprepared for the changes in the field that are almost certainly coming. In Einstein's inimitable phrasing, “Mathematics is all very well, but Nature leads us by the nose.” The book endeavors to lead the student gently, if not always easily, toward a useful understanding of the physical underpinnings of modern cosmology.

    Cosmology is an inherently uncertain science, because of both the remoteness (spatial and temporal) of its subjects and the incompleteness of its observational foundations. It is thus not surprising that recent technological advances in observational astronomy have produced something of a revolution in cosmological theory, from inflation to dark energy to new theories of galaxy origins. But interpretations of cosmological observations are typically based on conceptual models and (in some cases) underlying physics of uncertain validity, so wherever possible the book derives and interprets its results in a manner conducive to re-interpretation when new observations and/or physics so permit. The book is also at some pains to point out the uncertainties of cosmology theory arising from incomplete observational evidence and adoption of specific physical models. Modern cosmology is truly an intellectual wonder, but is likely to experience considerable revision as new observations and physics come to bear upon it. This book will, hopefully, prepare students for such changes.

  • 3 - Relativistic cosmology

  • from I - Conceptual foundations
  • William D. Heacox, University of Hawaii, Hilo
  • Book:
    The Expanding Universe
    Published online:
    28 May 2018
    Print publication:
    26 November 2015, pp 17-24

  • Summary

    Cosmology, as the term is currently used, is the study of the structure, contents, and evolution of the Universe on the largest scales. Relativistic cosmological models characterizing the evolution of the Universe on such scales are quantitative forms for the metric tensor components as derived from the Einstein Field Equations. In many cases these are superficially similar to the Newtonian forms of Chapter 1, but they differ from Newtonian models in conceptual interpretations and in several key details. In particular, relativistic cosmological models incorporate all forms of gravitating energy and matter, not just ordinary matter alone; and the character of the models reflects the curvature of space-time in place of Newtonian total energy.

    Cosmological coordinates

    The field equations are so complex that their practical solution requires the adoption of a coordinate system fully expressing the symmetries of the application. In cosmology, those symmetries usually arise from the Cosmological Principle: on sufficiently large scales the Universe is homogeneous and isotropic, the same everywhere and in all directions (at any given time). The system of galaxies in an expanding, spatially uniform Universe may then be thought of as one in which the galaxies are all falling upward in a uniform gravitational field. This suggests adoption of an appropriately symmetric coordinate system falling with the galaxies, which (by the Equivalence Principle) effectively defines an inertial reference frame, inside which the galaxies are not moving with respect to each other and the physics of SR are valid. Such a coordinate system – which may usefully be visualized as an expanding grid that carries galaxies with it as it expands – is commonly called a co-moving coordinate system. Adoption of such a system further suggests the validity of a universal time system, one that applies to all galaxies and that greatly simplifies the physics of universal expansion.

    The choice of coordinate system leads to expressions for the metric tensor components in terms of the coordinates. In such a co-moving system the coordinates of a grid point (i.e., galaxy) do not change as a consequence of the expansion of the Universe; the only time-varying element is the Expansion Function a(t), which, as in Newtonian cosmology, is non-dimensional and normalized so that a(t0)= 1.

Page 202 of 851