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1289 lines
38 KiB
XML
1289 lines
38 KiB
XML
<?xml version="1.0" encoding="UTF-8"?>
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<!-- This DocBook file was created by LyX 2.4.0dev
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See http://www.lyx.org/ for more information -->
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<article xml:lang="en_US" xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:xi="http://www.w3.org/2001/XInclude" version="5.2">
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<info>
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<title>Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AASTeX Examples</title>
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<author>
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<personname>S. Djorgovski and Ivan R. King</personname>
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<affiliation>
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<orgname>Astronomy Department, University of California, Berkeley, CA 94720</orgname>
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</affiliation>
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<affiliation role='alternate'>
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<orgname>Visiting Astronomer Cerro Tololo Inter-American Observatory.CTIO is operated by AURA Inc. under contract to the National Science Foundation.</orgname>
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</affiliation>
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<affiliation role='alternate'>
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<orgname>Society of Fellows, Harvard University.</orgname>
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</affiliation>
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<affiliation role='alternate'>
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<orgname>present address: Center for Astrophysics60 Garden Street, Cambridge, MA 02138</orgname>
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</affiliation>
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</author>
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<author>
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<personname>C. D. Biemesderfer</personname>
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<affiliation>
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<orgname>National Optical Astronomy Observatories, Tucson, AZ 85719</orgname>
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</affiliation>
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<affiliation role='alternate'>
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<orgname>Visiting Programmer, Space Telescope Science Institute</orgname>
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</affiliation>
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<affiliation role='alternate'>
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<orgname>Patron, Alonso's Bar and Grill</orgname>
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</affiliation>
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<email>aastex-help@aas.org</email>
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</author>
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<author>
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<personname>R. J. Hanisch</personname>
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<affiliation>
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<orgname>Space Telescope Science Institute, Baltimore, MD 21218</orgname>
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</affiliation>
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<affiliation role='alternate'>
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<orgname>Patron, Alonso's Bar and Grill</orgname>
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</affiliation>
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</author>
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<keywordset>
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<keyword>clusters: globular, peanut—bosons: bozos</keyword>
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</keywordset>
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<abstract>
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<para>This is a preliminary report on surface photometry of the major fraction of known globular clusters, to see which of them show the signs of a collapsed core. We also explore some diversionary mathematics and recreational tables. </para>
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</abstract>
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</info>
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<section>
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<title>Introduction</title>
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<para>A focal problem today in the dynamics of globular clusters is core collapse. It has been predicted by theory for decades <biblioref endterm="hen61" />, <biblioref endterm="lyn68" />, <biblioref endterm="spi85" />, but observation has been less alert to the phenomenon. For many years the central brightness peak in M15 <biblioref endterm="kin75" />, <biblioref endterm="new78" /> seemed a unique anomaly. Then <biblioref endterm="aur82" /> suggested a central peak in NGC 6397, and a limited photographic survey of ours <biblioref endterm="djo84" /> found three more cases, including NGC 6624, whose sharp center had often been remarked on <biblioref endterm="can78" />. </para>
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</section>
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<section>
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<title>Observations</title>
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<para>All our observations were short direct exposures with CCD's. At Lick Observatory we used a TI 500<inlineequation>
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<alt role='tex'>\times</alt>
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<m:math>
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<m:mrow><m:mo>×</m:mo>
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</m:mrow>
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</m:math>
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</inlineequation>500 chip and a GEC 575<inlineequation>
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<alt role='tex'>\times</alt>
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<m:math>
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<m:mrow><m:mo>×</m:mo>
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</m:mrow>
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</m:math>
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</inlineequation>385, on the 1-m Nickel reflector. The only filter available at Lick was red. At CTIO we used a GEC 575<inlineequation>
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<alt role='tex'>\times</alt>
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<m:math>
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<m:mrow><m:mo>×</m:mo>
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</m:mrow>
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</m:math>
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</inlineequation>385, with <inlineequation>
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<alt role='tex'>B,V,</alt>
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<m:math>
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<m:mrow>
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<m:mrow><m:mi>B</m:mi><m:mo>,</m:mo><m:mi>V</m:mi><m:mo>,</m:mo>
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</m:mrow>
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</m:mrow>
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</m:math>
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</inlineequation> and <inlineequation>
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<alt role='tex'>R</alt>
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<m:math>
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<m:mrow><m:mi>R</m:mi>
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</m:mrow>
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</m:math>
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</inlineequation> filters, and an RCA 512<inlineequation>
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<alt role='tex'>\times</alt>
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<m:math>
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<m:mrow><m:mo>×</m:mo>
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</m:mrow>
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</m:math>
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</inlineequation>320, with <inlineequation>
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<alt role='tex'>U,B,V,R,</alt>
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<m:math>
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<m:mrow>
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<m:mrow><m:mi>U</m:mi><m:mo>,</m:mo><m:mi>B</m:mi><m:mo>,</m:mo><m:mi>V</m:mi><m:mo>,</m:mo><m:mi>R</m:mi><m:mo>,</m:mo>
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</m:mrow>
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</m:mrow>
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</m:math>
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</inlineequation> and <inlineequation>
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<alt role='tex'>I</alt>
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<m:math>
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<m:mrow><m:mi>I</m:mi>
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</m:mrow>
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</m:math>
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</inlineequation> filters, on the 1.5-m reflector. In the CTIO observations we tried to concentrate on the shortest practicable wavelengths; but faintness, reddening, and poor short-wavelength sensitivity often kept us from observing in <inlineequation>
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<alt role='tex'>U</alt>
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<m:math>
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<m:mrow><m:mi>U</m:mi>
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</m:mrow>
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</m:math>
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</inlineequation> or even in <inlineequation>
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<alt role='tex'>B</alt>
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<m:math>
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<m:mrow><m:mi>B</m:mi>
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</m:mrow>
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</m:math>
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</inlineequation>. All four cameras had scales of the order of 0.4 arcsec/pixel, and our field sizes were around 3 arcmin.</para>
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<para>The CCD images are unfortunately not always suitable, for very poor clusters or for clusters with large cores. Since the latter are easily studied by other means, we augmented our own CCD profiles by collecting from the literature a number of star-count profiles <biblioref endterm="kin68" />, <biblioref endterm="pet76" />, <biblioref endterm="har84" />, <biblioref endterm="ort85" />, as well as photoelectric profiles <biblioref endterm="kin66" />, <biblioref endterm="kin75" /> and electronographic profiles <biblioref endterm="kro84" />. In a few cases we judged normality by eye estimates on one of the Sky Surveys.</para>
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</section>
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<section>
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<title>Helicity Amplitudes</title>
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<para>It has been realized that helicity amplitudes provide a convenient means for Feynman diagram<footnote>
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<para>Footnotes can be inserted like this.</para>
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</footnote> evaluations. These amplitude-level techniques are particularly convenient for calculations involving many Feynman diagrams, where the usual trace techniques for the amplitude squared becomes unwieldy. Our calculations use the helicity techniques developed by other authors <biblioref endterm="hag86" />; we briefly summarize below.</para>
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<section>
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<title>Formalism</title>
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<para><anchor xml:id="bozomath" /></para>
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<para>A tree-level amplitude in <inlineequation>
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<alt role='tex'>e^{+}e^{-}</alt>
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<m:math>
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<m:mrow>
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<m:mrow>
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<m:msup>
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<m:mrow><m:mi>e</m:mi>
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</m:mrow>
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<m:mrow><m:mo>+</m:mo>
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</m:mrow>
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</m:msup>
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<m:msup>
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<m:mrow><m:mi>e</m:mi>
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</m:mrow>
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<m:mrow><m:mo>-</m:mo>
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</m:mrow>
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</m:msup>
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</m:mrow>
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</m:mrow>
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</m:math>
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</inlineequation> collisions can be expressed in terms of fermion strings of the form
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<informalequation>
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<alt role='tex'>\bar{v}(p_{2},\sigma_{2})P_{-\tau}\hat{a}_{1}\hat{a}_{2}\cdots\hat{a}_{n}u(p_{1},\sigma_{1}),</alt>
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<m:math>
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<m:mrow>
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<m:mrow>
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<m:mover>
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<m:mrow><m:mi>v</m:mi>
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</m:mrow><m:mo stretchy="true">¯</m:mo>
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</m:mover><m:mo form='prefix' fence='true' stretchy='true' symmetric='true'><m:mrow>(</m:mrow></m:mo>
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<m:mrow>
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<m:msub>
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<m:mrow><m:mi>p</m:mi>
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</m:mrow>
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<m:mrow><m:mn>2</m:mn>
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</m:mrow>
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</m:msub><m:mo>,</m:mo>
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<m:msub>
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<m:mrow><m:mi>σ</m:mi>
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</m:mrow>
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<m:mrow><m:mn>2</m:mn>
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</m:mrow>
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</m:msub>
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</m:mrow>
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<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'><m:mrow>)</m:mrow></m:mo>
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<m:msub>
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<m:mrow><m:mi>P</m:mi>
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</m:mrow>
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<m:mrow>
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<m:mrow><m:mo>-</m:mo><m:mi>τ</m:mi>
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</m:mrow>
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</m:mrow>
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</m:msub>
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<m:msub>
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<m:mrow>
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<m:mover>
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<m:mrow><m:mi>a</m:mi>
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</m:mrow><m:mo stretchy="true">ˆ</m:mo>
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</m:mover>
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</m:mrow>
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<m:mrow><m:mn>1</m:mn>
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</m:mrow>
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</m:msub>
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<m:msub>
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<m:mrow>
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<m:mover>
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<m:mrow><m:mi>a</m:mi>
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</m:mrow><m:mo stretchy="true">ˆ</m:mo>
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</m:mover>
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</m:mrow>
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<m:mrow><m:mn>2</m:mn>
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</m:mrow>
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</m:msub>
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<m:mi>⋯
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</m:mi>
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<m:msub>
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<m:mrow>
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<m:mover>
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<m:mrow><m:mi>a</m:mi>
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</m:mrow><m:mo stretchy="true">ˆ</m:mo>
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</m:mover>
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</m:mrow>
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<m:mrow><m:mi>n</m:mi>
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</m:mrow>
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</m:msub><m:mi>u</m:mi><m:mo form='prefix' fence='true' stretchy='true' symmetric='true'><m:mrow>(</m:mrow></m:mo>
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<m:mrow>
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<m:msub>
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<m:mrow><m:mi>p</m:mi>
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</m:mrow>
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<m:mrow><m:mn>1</m:mn>
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</m:mrow>
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</m:msub><m:mo>,</m:mo>
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<m:msub>
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<m:mrow><m:mi>σ</m:mi>
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</m:mrow>
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<m:mrow><m:mn>1</m:mn>
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</m:mrow>
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</m:msub>
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</m:mrow>
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<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'><m:mrow>)</m:mrow></m:mo>
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<m:mo>,</m:mo>
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</m:mrow>
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</m:mrow>
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</m:math>
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</informalequation>
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where <inlineequation>
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<alt role='tex'>p</alt>
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<m:math>
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<m:mrow><m:mi>p</m:mi>
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</m:mrow>
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</m:math>
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</inlineequation> and <inlineequation>
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<alt role='tex'>\sigma</alt>
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<m:math>
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<m:mrow><m:mi>σ</m:mi>
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</m:mrow>
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</m:math>
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</inlineequation> label the initial <inlineequation>
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<alt role='tex'>e^{\pm}</alt>
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<m:math>
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<m:mrow>
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<m:msup>
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<m:mrow><m:mi>e</m:mi>
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</m:mrow>
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<m:mrow><m:mo>±</m:mo>
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</m:mrow>
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</m:msup>
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</m:mrow>
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</m:math>
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</inlineequation> four-momenta and helicities <inlineequation>
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<alt role='tex'>(\sigma=\pm1)</alt>
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<m:math>
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<m:mrow><m:mo form='prefix' fence='true' stretchy='true' symmetric='true'><m:mrow>(</m:mrow></m:mo>
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<m:mrow><m:mi>σ</m:mi><m:mo>=</m:mo><m:mo>±</m:mo><m:mn>1</m:mn>
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</m:mrow>
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<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'><m:mrow>)</m:mrow></m:mo>
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</m:mrow>
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</m:math>
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</inlineequation>, <inlineequation>
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<alt role='tex'>\hat{a}_{i}=a_{i}^{\mu}\gamma_{\nu}</alt>
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<m:math>
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<m:mrow>
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<m:mrow>
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<m:msub>
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<m:mrow>
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<m:mover>
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<m:mrow><m:mi>a</m:mi>
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</m:mrow><m:mo stretchy="true">ˆ</m:mo>
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</m:mover>
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</m:mrow>
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<m:mrow><m:mi>i</m:mi>
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</m:mrow>
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</m:msub><m:mo>=</m:mo>
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<m:msubsup>
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<m:mrow><m:mi>a</m:mi>
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</m:mrow>
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<m:mrow><m:mi>i</m:mi>
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</m:mrow>
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<m:mrow><m:mi>μ</m:mi>
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</m:mrow>
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</m:msubsup>
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<m:msub>
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<m:mrow><m:mi>γ</m:mi>
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</m:mrow>
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<m:mrow><m:mi>ν</m:mi>
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</m:mrow>
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</m:msub>
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</m:mrow>
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</m:mrow>
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</m:math>
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</inlineequation> and <inlineequation>
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<alt role='tex'>P_{\tau}=\frac{1}{2}(1+\tau\gamma_{5})</alt>
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<m:math>
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<m:mrow>
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<m:mrow>
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<m:msub>
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<m:mrow><m:mi>P</m:mi>
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</m:mrow>
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<m:mrow><m:mi>τ</m:mi>
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</m:mrow>
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</m:msub><m:mo>=</m:mo>
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<m:mfrac>
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<m:mrow><m:mn>1</m:mn>
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</m:mrow>
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<m:mrow><m:mn>2</m:mn>
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</m:mrow>
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</m:mfrac><m:mo form='prefix' fence='true' stretchy='true' symmetric='true'><m:mrow>(</m:mrow></m:mo>
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<m:mrow><m:mn>1</m:mn><m:mo>+</m:mo><m:mi>τ</m:mi>
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<m:msub>
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<m:mrow><m:mi>γ</m:mi>
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</m:mrow>
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<m:mrow><m:mn>5</m:mn>
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</m:mrow>
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</m:msub>
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</m:mrow>
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<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'><m:mrow>)</m:mrow></m:mo>
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</m:mrow>
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</m:mrow>
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</m:math>
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</inlineequation> is a chirality projection operator <inlineequation>
|
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<alt role='tex'>(\tau=\pm1)</alt>
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<m:math>
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<m:mrow><m:mo form='prefix' fence='true' stretchy='true' symmetric='true'><m:mrow>(</m:mrow></m:mo>
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<m:mrow><m:mi>τ</m:mi><m:mo>=</m:mo><m:mo>±</m:mo><m:mn>1</m:mn>
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</m:mrow>
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<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'><m:mrow>)</m:mrow></m:mo>
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</m:mrow>
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</m:math>
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</inlineequation>. The <inlineequation>
|
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<alt role='tex'>a_{i}^{\mu}</alt>
|
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<m:math>
|
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<m:mrow>
|
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<m:msubsup>
|
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<m:mrow><m:mi>a</m:mi>
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</m:mrow>
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<m:mrow><m:mi>i</m:mi>
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</m:mrow>
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<m:mrow><m:mi>μ</m:mi>
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</m:mrow>
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</m:msubsup>
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</m:mrow>
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</m:math>
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</inlineequation> may be formed from particle four-momenta, gauge-boson polarization vectors or fermion strings with an uncontracted Lorentz index associated with final-state fermions.</para>
|
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<remark role='to-editor'>Figures 1 and 2 should appear side-by-side in print</remark>
|
|
<para>In the chiral representation the <inlineequation>
|
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<alt role='tex'>\gamma</alt>
|
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<m:math>
|
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|
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<m:mrow><m:mi>γ</m:mi>
|
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</m:mrow>
|
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</m:math>
|
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</inlineequation> matrices are expressed in terms of <inlineequation>
|
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<alt role='tex'>2\times2</alt>
|
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<m:math>
|
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<m:mrow>
|
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<m:mrow><m:mn>2</m:mn><m:mo>×</m:mo><m:mn>2</m:mn>
|
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</m:mrow>
|
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</m:mrow>
|
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</m:math>
|
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</inlineequation> Pauli matrices <inlineequation>
|
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<alt role='tex'>\sigma</alt>
|
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<m:math>
|
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|
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<m:mrow><m:mi>σ</m:mi>
|
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</m:mrow>
|
|
</m:math>
|
|
</inlineequation> and the unit matrix 1 as
|
|
<informalequation>
|
|
<alt role='tex'>\gamma^{\mu} & = & \left(\begin{array}{cc}
|
|
0 & \sigma_{+}^{\mu}\\
|
|
\sigma_{-}^{\mu} & 0
|
|
\end{array}\right),\gamma^{5}=\left(\begin{array}{cc}
|
|
-1 & 0\\
|
|
0 & 1
|
|
\end{array}\right),\\
|
|
\sigma_{\pm}^{\mu} & = & ({\textbf{1}},\pm\sigma),
|
|
</alt>
|
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<m:math>
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|
|
<m:mtable>
|
|
<m:mtr>
|
|
<m:mtd>
|
|
<m:msup>
|
|
<m:mrow><m:mi>γ</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mi>μ</m:mi>
|
|
</m:mrow>
|
|
</m:msup>
|
|
</m:mtd>
|
|
<m:mtd><m:mo>=</m:mo>
|
|
</m:mtd>
|
|
<m:mtd>
|
|
<m:mrow><m:mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='thinmathspace'>(</m:mo>
|
|
<m:mtable>
|
|
<m:mtr>
|
|
<m:mtd><m:mn>0</m:mn>
|
|
</m:mtd>
|
|
<m:mtd>
|
|
<m:msubsup>
|
|
<m:mrow><m:mi>σ</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mo>+</m:mo>
|
|
</m:mrow>
|
|
<m:mrow><m:mi>μ</m:mi>
|
|
</m:mrow>
|
|
</m:msubsup>
|
|
</m:mtd>
|
|
</m:mtr>
|
|
<m:mtr>
|
|
<m:mtd>
|
|
<m:msubsup>
|
|
<m:mrow><m:mi>σ</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mo>-</m:mo>
|
|
</m:mrow>
|
|
<m:mrow><m:mi>μ</m:mi>
|
|
</m:mrow>
|
|
</m:msubsup>
|
|
</m:mtd>
|
|
<m:mtd><m:mn>0</m:mn>
|
|
</m:mtd>
|
|
</m:mtr>
|
|
</m:mtable><m:mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='thinmathspace'>)</m:mo><m:mo>,</m:mo>
|
|
<m:msup>
|
|
<m:mrow><m:mi>γ</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mn>5</m:mn>
|
|
</m:mrow>
|
|
</m:msup><m:mo>=</m:mo><m:mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='thinmathspace'>(</m:mo>
|
|
<m:mtable>
|
|
<m:mtr>
|
|
<m:mtd>
|
|
<m:mrow><m:mo>-</m:mo><m:mn>1</m:mn>
|
|
</m:mrow>
|
|
</m:mtd>
|
|
<m:mtd><m:mn>0</m:mn>
|
|
</m:mtd>
|
|
</m:mtr>
|
|
<m:mtr>
|
|
<m:mtd><m:mn>0</m:mn>
|
|
</m:mtd>
|
|
<m:mtd><m:mn>1</m:mn>
|
|
</m:mtd>
|
|
</m:mtr>
|
|
</m:mtable><m:mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='thinmathspace'>)</m:mo><m:mo>,</m:mo>
|
|
</m:mrow>
|
|
</m:mtd>
|
|
</m:mtr>
|
|
<m:mtr>
|
|
<m:mtd>
|
|
<m:msubsup>
|
|
<m:mrow><m:mi>σ</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mo>±</m:mo>
|
|
</m:mrow>
|
|
<m:mrow><m:mi>μ</m:mi>
|
|
</m:mrow>
|
|
</m:msubsup>
|
|
</m:mtd>
|
|
<m:mtd><m:mo>=</m:mo>
|
|
</m:mtd>
|
|
<m:mtd>
|
|
<m:mrow><m:mo form='prefix' fence='true' stretchy='true' symmetric='true'><m:mrow>(</m:mrow></m:mo>
|
|
|
|
<m:mrow>
|
|
<m:mrow>
|
|
<m:mstyle mathvariant='bold'><m:mn>1</m:mn>
|
|
</m:mstyle>
|
|
</m:mrow><m:mo>,</m:mo><m:mo>±</m:mo><m:mi>σ</m:mi>
|
|
</m:mrow>
|
|
<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'><m:mrow>)</m:mrow></m:mo>
|
|
<m:mo>,</m:mo>
|
|
</m:mrow>
|
|
</m:mtd>
|
|
</m:mtr>
|
|
</m:mtable>
|
|
</m:math>
|
|
</informalequation>
|
|
giving
|
|
<informalequation>
|
|
<alt role='tex'>\hat{a}=\left(\begin{array}{cc}
|
|
0 & (\hat{a})_{+}\\
|
|
(\hat{a})_{-} & 0
|
|
\end{array}\right),(\hat{a})_{\pm}=a_{\mu}\sigma_{\pm}^{\mu},</alt>
|
|
<m:math>
|
|
|
|
<m:mrow>
|
|
<m:mrow>
|
|
<m:mover>
|
|
<m:mrow><m:mi>a</m:mi>
|
|
</m:mrow><m:mo stretchy="true">ˆ</m:mo>
|
|
</m:mover><m:mo>=</m:mo><m:mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='thinmathspace'>(</m:mo>
|
|
<m:mtable>
|
|
<m:mtr>
|
|
<m:mtd><m:mn>0</m:mn>
|
|
</m:mtd>
|
|
<m:mtd>
|
|
<m:mrow><m:mo>(</m:mo>
|
|
<m:mover>
|
|
<m:mrow><m:mi>a</m:mi>
|
|
</m:mrow><m:mo stretchy="true">ˆ</m:mo>
|
|
</m:mover>
|
|
<m:msub>
|
|
<m:mrow><m:mo>)</m:mo>
|
|
</m:mrow>
|
|
<m:mrow><m:mo>+</m:mo>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
</m:mtd>
|
|
</m:mtr>
|
|
<m:mtr>
|
|
<m:mtd>
|
|
<m:mrow><m:mo>(</m:mo>
|
|
<m:mover>
|
|
<m:mrow><m:mi>a</m:mi>
|
|
</m:mrow><m:mo stretchy="true">ˆ</m:mo>
|
|
</m:mover>
|
|
<m:msub>
|
|
<m:mrow><m:mo>)</m:mo>
|
|
</m:mrow>
|
|
<m:mrow><m:mo>-</m:mo>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
</m:mtd>
|
|
<m:mtd><m:mn>0</m:mn>
|
|
</m:mtd>
|
|
</m:mtr>
|
|
</m:mtable><m:mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='thinmathspace'>)</m:mo><m:mo>,</m:mo><m:mo>(</m:mo>
|
|
<m:mover>
|
|
<m:mrow><m:mi>a</m:mi>
|
|
</m:mrow><m:mo stretchy="true">ˆ</m:mo>
|
|
</m:mover>
|
|
<m:msub>
|
|
<m:mrow><m:mo>)</m:mo>
|
|
</m:mrow>
|
|
<m:mrow><m:mo>±</m:mo>
|
|
</m:mrow>
|
|
</m:msub><m:mo>=</m:mo>
|
|
<m:msub>
|
|
<m:mrow><m:mi>a</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mi>μ</m:mi>
|
|
</m:mrow>
|
|
</m:msub>
|
|
<m:msubsup>
|
|
<m:mrow><m:mi>σ</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mo>±</m:mo>
|
|
</m:mrow>
|
|
<m:mrow><m:mi>μ</m:mi>
|
|
</m:mrow>
|
|
</m:msubsup><m:mo>,</m:mo>
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:math>
|
|
</informalequation>
|
|
The spinors are expressed in terms of two-component Weyl spinors as
|
|
<informalequation>
|
|
<alt role='tex'>u=\left(\begin{array}{c}
|
|
(u)_{-}\\
|
|
(u)_{+}
|
|
\end{array}\right),v={\textbf{(}}\vdag_{+}{\textbf{,}}\vdag_{-}{\textbf{)}}.</alt>
|
|
<mathphrase>MathML export failed. Please report this as a bug.</mathphrase>
|
|
</informalequation>
|
|
</para>
|
|
<para>The Weyl spinors are given in terms of helicity eigenstates <inlineequation>
|
|
<alt role='tex'>\chi_{\lambda}(p)</alt>
|
|
<m:math>
|
|
|
|
<m:mrow>
|
|
<m:mrow>
|
|
<m:msub>
|
|
<m:mrow><m:mi>χ</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mi>λ</m:mi>
|
|
</m:mrow>
|
|
</m:msub><m:mo form='prefix' fence='true' stretchy='true' symmetric='true'><m:mrow>(</m:mrow></m:mo>
|
|
<m:mi>p</m:mi>
|
|
<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'><m:mrow>)</m:mrow></m:mo>
|
|
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation> with <inlineequation>
|
|
<alt role='tex'>\lambda=\pm1</alt>
|
|
<m:math>
|
|
|
|
<m:mrow>
|
|
<m:mrow><m:mi>λ</m:mi><m:mo>=</m:mo><m:mo>±</m:mo><m:mn>1</m:mn>
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation> by </para>
|
|
<informalequation>
|
|
<alt role='tex'>u(p,\lambda)_{\pm} & = & (E\pm\lambda|{\textbf{p}}|)^{1/2}\chi_{\lambda}(p),\\
|
|
v(p,\lambda)_{\pm} & = & \pm\lambda(E\mp\lambda|{\textbf{p}}|)^{1/2}\chi_{-\lambda}(p)
|
|
</alt>
|
|
<m:math>
|
|
|
|
<m:mtable>
|
|
<m:mtr>
|
|
<m:mtd>
|
|
<m:mrow><m:mi>u</m:mi><m:mo>(</m:mo><m:mi>p</m:mi><m:mo>,</m:mo><m:mi>λ</m:mi>
|
|
<m:msub>
|
|
<m:mrow><m:mo>)</m:mo>
|
|
</m:mrow>
|
|
<m:mrow><m:mo>±</m:mo>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
</m:mtd>
|
|
<m:mtd><m:mo>=</m:mo>
|
|
</m:mtd>
|
|
<m:mtd>
|
|
<m:mrow><m:mo>(</m:mo><m:mi>E</m:mi><m:mo>±</m:mo><m:mi>λ</m:mi><m:mo>|</m:mo>
|
|
<m:mrow>
|
|
<m:mstyle mathvariant='bold'><m:mi>p</m:mi>
|
|
</m:mstyle>
|
|
</m:mrow><m:mo>|</m:mo>
|
|
<m:msup>
|
|
<m:mrow><m:mo>)</m:mo>
|
|
</m:mrow>
|
|
<m:mrow>
|
|
<m:mrow><m:mn>1</m:mn><m:mo>/</m:mo><m:mn>2</m:mn>
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:msup>
|
|
<m:msub>
|
|
<m:mrow><m:mi>χ</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mi>λ</m:mi>
|
|
</m:mrow>
|
|
</m:msub><m:mo form='prefix' fence='true' stretchy='true' symmetric='true'><m:mrow>(</m:mrow></m:mo>
|
|
<m:mi>p</m:mi>
|
|
<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'><m:mrow>)</m:mrow></m:mo>
|
|
<m:mo>,</m:mo>
|
|
</m:mrow>
|
|
</m:mtd>
|
|
</m:mtr>
|
|
<m:mtr>
|
|
<m:mtd>
|
|
<m:mrow><m:mi>v</m:mi><m:mo>(</m:mo><m:mi>p</m:mi><m:mo>,</m:mo><m:mi>λ</m:mi>
|
|
<m:msub>
|
|
<m:mrow><m:mo>)</m:mo>
|
|
</m:mrow>
|
|
<m:mrow><m:mo>±</m:mo>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
</m:mtd>
|
|
<m:mtd><m:mo>=</m:mo>
|
|
</m:mtd>
|
|
<m:mtd>
|
|
<m:mrow><m:mo>±</m:mo><m:mi>λ</m:mi><m:mo>(</m:mo><m:mi>E</m:mi><m:mo>∓</m:mo><m:mi>λ</m:mi><m:mo>|</m:mo>
|
|
<m:mrow>
|
|
<m:mstyle mathvariant='bold'><m:mi>p</m:mi>
|
|
</m:mstyle>
|
|
</m:mrow><m:mo>|</m:mo>
|
|
<m:msup>
|
|
<m:mrow><m:mo>)</m:mo>
|
|
</m:mrow>
|
|
<m:mrow>
|
|
<m:mrow><m:mn>1</m:mn><m:mo>/</m:mo><m:mn>2</m:mn>
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:msup>
|
|
<m:msub>
|
|
<m:mrow><m:mi>χ</m:mi>
|
|
</m:mrow>
|
|
<m:mrow>
|
|
<m:mrow><m:mo>-</m:mo><m:mi>λ</m:mi>
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:msub><m:mo form='prefix' fence='true' stretchy='true' symmetric='true'><m:mrow>(</m:mrow></m:mo>
|
|
<m:mi>p</m:mi>
|
|
<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'><m:mrow>)</m:mrow></m:mo>
|
|
|
|
</m:mrow>
|
|
</m:mtd>
|
|
</m:mtr>
|
|
</m:mtable>
|
|
</m:math>
|
|
</informalequation>
|
|
</section>
|
|
</section>
|
|
<section>
|
|
<title>Floating material and so forth</title>
|
|
<para>Consider a task that computes profile parameters for a modified Lorentzian of the form
|
|
<informalequation>
|
|
<alt role='tex'>I=\frac{1}{1+d_{1}^{P(1+d_{2})}}</alt>
|
|
<m:math>
|
|
|
|
<m:mrow>
|
|
<m:mrow><m:mi>I</m:mi><m:mo>=</m:mo>
|
|
<m:mfrac>
|
|
<m:mrow><m:mn>1</m:mn>
|
|
</m:mrow>
|
|
<m:mrow>
|
|
<m:mrow><m:mn>1</m:mn><m:mo>+</m:mo>
|
|
<m:msubsup>
|
|
<m:mrow><m:mi>d</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mn>1</m:mn>
|
|
</m:mrow>
|
|
<m:mrow>
|
|
<m:mrow><m:mi>P</m:mi><m:mo form='prefix' fence='true' stretchy='true' symmetric='true'><m:mrow>(</m:mrow></m:mo>
|
|
|
|
<m:mrow><m:mn>1</m:mn><m:mo>+</m:mo>
|
|
<m:msub>
|
|
<m:mrow><m:mi>d</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mn>2</m:mn>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'><m:mrow>)</m:mrow></m:mo>
|
|
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:msubsup>
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:mfrac>
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:math>
|
|
</informalequation>
|
|
where
|
|
<informalequation>
|
|
<alt role='tex'>d_{1}=\sqrt{\left(\begin{array}{c}
|
|
\frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
|
|
\frac{y_{1}}{R_{min}}\end{array}\right)^{2}}</alt>
|
|
<m:math>
|
|
|
|
<m:mrow>
|
|
<m:mrow>
|
|
<m:msub>
|
|
<m:mrow><m:mi>d</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mn>1</m:mn>
|
|
</m:mrow>
|
|
</m:msub><m:mo>=</m:mo>
|
|
<m:msqrt>
|
|
<m:mrow>
|
|
<m:msup>
|
|
<m:mrow><m:mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='thinmathspace'>(</m:mo>
|
|
<m:mtable>
|
|
<m:mtr>
|
|
<m:mtd>
|
|
<m:mfrac>
|
|
<m:mrow>
|
|
<m:msub>
|
|
<m:mrow><m:mi>x</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mn>1</m:mn>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
<m:mrow>
|
|
<m:msub>
|
|
<m:mrow><m:mi>R</m:mi>
|
|
</m:mrow>
|
|
<m:mrow>
|
|
<m:mrow><m:mi>m</m:mi><m:mi>a</m:mi><m:mi>j</m:mi>
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
</m:mfrac>
|
|
</m:mtd>
|
|
</m:mtr>
|
|
</m:mtable><m:mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='thinmathspace'>)</m:mo>
|
|
</m:mrow>
|
|
<m:mrow><m:mn>2</m:mn>
|
|
</m:mrow>
|
|
</m:msup><m:mo>+</m:mo>
|
|
<m:msup>
|
|
<m:mrow><m:mo form='prefix' fence='true' stretchy='true' symmetric='true' lspace='thinmathspace'>(</m:mo>
|
|
<m:mtable>
|
|
<m:mtr>
|
|
<m:mtd>
|
|
<m:mfrac>
|
|
<m:mrow>
|
|
<m:msub>
|
|
<m:mrow><m:mi>y</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mn>1</m:mn>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
<m:mrow>
|
|
<m:msub>
|
|
<m:mrow><m:mi>R</m:mi>
|
|
</m:mrow>
|
|
<m:mrow>
|
|
<m:mrow><m:mi>m</m:mi><m:mi>i</m:mi><m:mi>n</m:mi>
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
</m:mfrac>
|
|
</m:mtd>
|
|
</m:mtr>
|
|
</m:mtable><m:mo form='postfix' fence='true' stretchy='true' symmetric='true' lspace='thinmathspace'>)</m:mo>
|
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</m:mrow>
|
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<m:mrow><m:mn>2</m:mn>
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</m:mrow>
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</m:msup>
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</m:mrow>
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</m:msqrt>
|
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</m:mrow>
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</m:mrow>
|
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</m:math>
|
|
</informalequation>
|
|
<informalequation>
|
|
<alt role='tex'>d_{2}=\sqrt{\left(\begin{array}{c}
|
|
\frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
|
|
\case{y_{1}}{PR_{min}}\end{array}\right)^{2}}</alt>
|
|
<mathphrase>MathML export failed. Please report this as a bug.</mathphrase>
|
|
</informalequation>
|
|
<informalequation>
|
|
<alt role='tex'>x_{1}=(x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta</alt>
|
|
<m:math>
|
|
|
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<m:mrow>
|
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<m:mrow>
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<m:msub>
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<m:mrow><m:mi>x</m:mi>
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</m:mrow>
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<m:mrow><m:mn>1</m:mn>
|
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</m:mrow>
|
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</m:msub><m:mo>=</m:mo><m:mo form='prefix' fence='true' stretchy='true' symmetric='true'><m:mrow>(</m:mrow></m:mo>
|
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<m:mrow><m:mi>x</m:mi><m:mo>-</m:mo>
|
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<m:msub>
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<m:mrow><m:mi>x</m:mi>
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</m:mrow>
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<m:mrow><m:mn>0</m:mn>
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</m:mrow>
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</m:msub>
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</m:mrow>
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<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'><m:mrow>)</m:mrow></m:mo>
|
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<m:mo>cos</m:mo><m:mo>Θ</m:mo><m:mo>+</m:mo><m:mo form='prefix' fence='true' stretchy='true' symmetric='true'><m:mrow>(</m:mrow></m:mo>
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<m:mrow><m:mi>y</m:mi><m:mo>-</m:mo>
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<m:msub>
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<m:mrow><m:mi>y</m:mi>
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</m:mrow>
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<m:mrow><m:mn>0</m:mn>
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</m:mrow>
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</m:msub>
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</m:mrow>
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<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'><m:mrow>)</m:mrow></m:mo>
|
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<m:mo>sin</m:mo><m:mo>Θ</m:mo>
|
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</m:mrow>
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</m:mrow>
|
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</m:math>
|
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</informalequation>
|
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<informalequation>
|
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<alt role='tex'>y_{1}=-(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta</alt>
|
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<m:math>
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<m:mrow>
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<m:mrow>
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<m:msub>
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<m:mrow><m:mi>y</m:mi>
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</m:mrow>
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<m:mrow><m:mn>1</m:mn>
|
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</m:mrow>
|
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</m:msub><m:mo>=</m:mo><m:mo>-</m:mo><m:mo form='prefix' fence='true' stretchy='true' symmetric='true'><m:mrow>(</m:mrow></m:mo>
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<m:mrow><m:mi>x</m:mi><m:mo>-</m:mo>
|
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<m:msub>
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<m:mrow><m:mi>x</m:mi>
|
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</m:mrow>
|
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<m:mrow><m:mn>0</m:mn>
|
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</m:mrow>
|
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</m:msub>
|
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</m:mrow>
|
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<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'><m:mrow>)</m:mrow></m:mo>
|
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<m:mo>sin</m:mo><m:mo>Θ</m:mo><m:mo>+</m:mo><m:mo form='prefix' fence='true' stretchy='true' symmetric='true'><m:mrow>(</m:mrow></m:mo>
|
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<m:mrow><m:mi>y</m:mi><m:mo>-</m:mo>
|
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<m:msub>
|
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<m:mrow><m:mi>y</m:mi>
|
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</m:mrow>
|
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<m:mrow><m:mn>0</m:mn>
|
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</m:mrow>
|
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</m:msub>
|
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</m:mrow>
|
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<m:mo form='postfix' fence='true' stretchy='true' symmetric='true'><m:mrow>)</m:mrow></m:mo>
|
|
<m:mo>cos</m:mo><m:mo>Θ</m:mo>
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:math>
|
|
</informalequation>
|
|
</para>
|
|
<para>In these expressions <inlineequation>
|
|
<alt role='tex'>x_{0}</alt>
|
|
<m:math>
|
|
|
|
<m:mrow>
|
|
<m:msub>
|
|
<m:mrow><m:mi>x</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mn>0</m:mn>
|
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</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation>,<inlineequation>
|
|
<alt role='tex'>y_{0}</alt>
|
|
<m:math>
|
|
|
|
<m:mrow>
|
|
<m:msub>
|
|
<m:mrow><m:mi>y</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mn>0</m:mn>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation> is the star center, and <inlineequation>
|
|
<alt role='tex'>\Theta</alt>
|
|
<m:math>
|
|
|
|
<m:mrow><m:mo>Θ</m:mo>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation> is the angle with the <inlineequation>
|
|
<alt role='tex'>x</alt>
|
|
<m:math>
|
|
|
|
<m:mrow><m:mi>x</m:mi>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation> axis. Results of this task are shown in table <xref linkend="tbl-2" />. It is not clear how these sorts of analyses may affect determination of <inlineequation>
|
|
<alt role='tex'>M_{\text{\sun}}</alt>
|
|
<m:math>
|
|
|
|
<m:mrow>
|
|
<m:msub>
|
|
<m:mrow><m:mi>M</m:mi>
|
|
</m:mrow>
|
|
<m:mrow>
|
|
<m:mstyle mathvariant='normal'><m:mo>☼</m:mo>
|
|
</m:mstyle>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation>, but the assumption is that the alternate results should be less than 90° out of phase with previous values. We have no observations of <!-- \ion{Ca}{2} -->
|
|
. Roughly <inlineequation>
|
|
<alt role='tex'>\nicefrac{4}{5}</alt>
|
|
<m:math>
|
|
|
|
<m:mrow>
|
|
<m:mfrac bevelled='true'>
|
|
<m:mrow><m:mn>4</m:mn>
|
|
</m:mrow>
|
|
<m:mrow><m:mn>5</m:mn>
|
|
</m:mrow>
|
|
</m:mfrac>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation> of the electronically submitted abstracts for AAS meetings are error-free. </para>
|
|
<acknowledgements>
|
|
<para>We are grateful to V. Barger, T. Han, and R. J. N. Phillips for doing the math in section <xref linkend="bozomath" />. More information on the AASTeX macros package are available at <link xlink:href="http://www.aas.org/publications/aastex">http://www.aas.org/publications/aastex</link> or the <link xlink:href="ftp://www.aas.org/pubs/AAS ftp site">AAS ftp site</link>.</para>
|
|
</acknowledgements>
|
|
<remark role='software'>IRAF, AIPS, Astropy, ...</remark>
|
|
</section>
|
|
<bibliography>
|
|
<bibliomixed xml:id='aur82'>Aurière, M. 1982, <!-- \aap -->
|
|
, 109, 301 </bibliomixed>
|
|
<bibliomixed xml:id='can78'>Canizares, C. R., Grindlay, J. E., Hiltner, W. A., Liller, W., and McClintock, J. E. 1978, <!-- \apj -->
|
|
, 224, 39 </bibliomixed>
|
|
<bibliomixed xml:id='djo84'>Djorgovski, S., and King, I. R. 1984, <!-- \apjl -->
|
|
, 277, L49 </bibliomixed>
|
|
<bibliomixed xml:id='hag86'>Hagiwara, K., and Zeppenfeld, D. 1986, Nucl.Phys., 274, 1 </bibliomixed>
|
|
<bibliomixed xml:id='har84'>Harris, W. E., and van den Bergh, S. 1984, <!-- \aj -->
|
|
, 89, 1816 </bibliomixed>
|
|
<bibliomixed xml:id='hen61'>Hénon, M. 1961, Ann.d'Ap., 24, 369 </bibliomixed>
|
|
<bibliomixed xml:id='kin66'>King, I. R. 1966, <!-- \aj -->
|
|
, 71, 276 </bibliomixed>
|
|
<bibliomixed xml:id='kin75'>King, I. R. 1975, Dynamics of Stellar Systems, A. Hayli, Dordrecht: Reidel, 1975, 99 </bibliomixed>
|
|
<bibliomixed xml:id='kin68'>King, I. R., Hedemann, E., Hodge, S. M., and White, R. E. 1968, <!-- \aj -->
|
|
, 73, 456 </bibliomixed>
|
|
<bibliomixed xml:id='kro84'>Kron, G. E., Hewitt, A. V., and Wasserman, L. H. 1984, <!-- \pasp -->
|
|
, 96, 198 </bibliomixed>
|
|
<bibliomixed xml:id='lyn68'>Lynden-Bell, D., and Wood, R. 1968, <!-- \mnras -->
|
|
, 138, 495 </bibliomixed>
|
|
<bibliomixed xml:id='new78'>Newell, E. B., and O'Neil, E. J. 1978, <!-- \apjs -->
|
|
, 37, 27 </bibliomixed>
|
|
<bibliomixed xml:id='ort85'>Ortolani, S., Rosino, L., and Sandage, A. 1985, <!-- \aj -->
|
|
, 90, 473 </bibliomixed>
|
|
<bibliomixed xml:id='pet76'>Peterson, C. J. 1976, <!-- \aj -->
|
|
, 81, 617 </bibliomixed>
|
|
<bibliomixed xml:id='spi85'>Spitzer, L. 1985, Dynamics of Star Clusters, J. Goodman and P. Hut, Dordrecht: Reidel, 109 </bibliomixed>
|
|
</bibliography>
|
|
<table xml:id="tbl-2">
|
|
<caption>Terribly relevant tabular information.</caption>
|
|
<tbody>
|
|
<tr>
|
|
<td align='center' valign='top'>Star </td>
|
|
<td align='right' valign='top'> Height </td>
|
|
<td align='right' valign='top'> <inlineequation>
|
|
<alt role='tex'>d_{x}</alt>
|
|
<m:math>
|
|
|
|
<m:mrow>
|
|
<m:msub>
|
|
<m:mrow><m:mi>d</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mi>x</m:mi>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation></td>
|
|
<td align='right' valign='top'> <inlineequation>
|
|
<alt role='tex'>d_{y}</alt>
|
|
<m:math>
|
|
|
|
<m:mrow>
|
|
<m:msub>
|
|
<m:mrow><m:mi>d</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mi>y</m:mi>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation></td>
|
|
<td align='right' valign='top'> <inlineequation>
|
|
<alt role='tex'>n</alt>
|
|
<m:math>
|
|
|
|
<m:mrow><m:mi>n</m:mi>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation></td>
|
|
<td align='right' valign='top'> <inlineequation>
|
|
<alt role='tex'>\chi^{2}</alt>
|
|
<m:math>
|
|
|
|
<m:mrow>
|
|
<m:msup>
|
|
<m:mrow><m:mi>χ</m:mi>
|
|
</m:mrow>
|
|
<m:mrow><m:mn>2</m:mn>
|
|
</m:mrow>
|
|
</m:msup>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation></td>
|
|
<td align='right' valign='top'> <inlineequation>
|
|
<alt role='tex'>R_{maj}</alt>
|
|
<m:math>
|
|
|
|
<m:mrow>
|
|
<m:msub>
|
|
<m:mrow><m:mi>R</m:mi>
|
|
</m:mrow>
|
|
<m:mrow>
|
|
<m:mrow><m:mi>m</m:mi><m:mi>a</m:mi><m:mi>j</m:mi>
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation></td>
|
|
<td align='right' valign='top'> <inlineequation>
|
|
<alt role='tex'>R_{min}</alt>
|
|
<m:math>
|
|
|
|
<m:mrow>
|
|
<m:msub>
|
|
<m:mrow><m:mi>R</m:mi>
|
|
</m:mrow>
|
|
<m:mrow>
|
|
<m:mrow><m:mi>m</m:mi><m:mi>i</m:mi><m:mi>n</m:mi>
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation></td>
|
|
<td align='center' valign='top' colspan='1'><inlineequation>
|
|
<alt role='tex'>P</alt>
|
|
<m:math>
|
|
|
|
<m:mrow><m:mi>P</m:mi>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation><remark role='tablenotemark'>a</remark></td>
|
|
<td align='right' valign='top'> <inlineequation>
|
|
<alt role='tex'>PR_{maj}</alt>
|
|
<m:math>
|
|
|
|
<m:mrow>
|
|
<m:mrow><m:mi>P</m:mi>
|
|
<m:msub>
|
|
<m:mrow><m:mi>R</m:mi>
|
|
</m:mrow>
|
|
<m:mrow>
|
|
<m:mrow><m:mi>m</m:mi><m:mi>a</m:mi><m:mi>j</m:mi>
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation></td>
|
|
<td align='right' valign='top'> <inlineequation>
|
|
<alt role='tex'>PR_{min}</alt>
|
|
<m:math>
|
|
|
|
<m:mrow>
|
|
<m:mrow><m:mi>P</m:mi>
|
|
<m:msub>
|
|
<m:mrow><m:mi>R</m:mi>
|
|
</m:mrow>
|
|
<m:mrow>
|
|
<m:mrow><m:mi>m</m:mi><m:mi>i</m:mi><m:mi>n</m:mi>
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:msub>
|
|
</m:mrow>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation></td>
|
|
<td align='center' valign='top' colspan='1'><inlineequation>
|
|
<alt role='tex'>\Theta</alt>
|
|
<m:math>
|
|
|
|
<m:mrow><m:mo>Θ</m:mo>
|
|
</m:mrow>
|
|
</m:math>
|
|
</inlineequation><remark role='tablenotemark'>b</remark></td>
|
|
<td align='center' valign='top'>Ref.</td>
|
|
</tr>
|
|
<tr>
|
|
<td align='center' valign='top'><!-- \tableline\tableline -->
|
|
1 </td>
|
|
<td align='right' valign='top'>33472.5 </td>
|
|
<td align='right' valign='top'>-0.1 </td>
|
|
<td align='right' valign='top'>0.4 </td>
|
|
<td align='right' valign='top'>53 </td>
|
|
<td align='right' valign='top'>27.4 </td>
|
|
<td align='right' valign='top'>2.065 </td>
|
|
<td align='right' valign='top'>1.940 </td>
|
|
<td align='right' valign='top'>3.900 </td>
|
|
<td align='right' valign='top'>68.3 </td>
|
|
<td align='right' valign='top'>116.2 </td>
|
|
<td align='right' valign='top'>-27.639</td>
|
|
<td align='center' valign='top'>1,2</td>
|
|
</tr>
|
|
<tr>
|
|
<td align='center' valign='top'> 2 </td>
|
|
<td align='right' valign='top'>27802.4 </td>
|
|
<td align='right' valign='top'>-0.3 </td>
|
|
<td align='right' valign='top'>-0.2 </td>
|
|
<td align='right' valign='top'>60 </td>
|
|
<td align='right' valign='top'>3.7 </td>
|
|
<td align='right' valign='top'>1.628 </td>
|
|
<td align='right' valign='top'>1.510 </td>
|
|
<td align='right' valign='top'>2.156 </td>
|
|
<td align='right' valign='top'>6.8 </td>
|
|
<td align='right' valign='top'>7.5 </td>
|
|
<td align='right' valign='top'>-26.764</td>
|
|
<td align='center' valign='top'>3</td>
|
|
</tr>
|
|
<tr>
|
|
<td align='center' valign='top'> 3 </td>
|
|
<td align='right' valign='top'>29210.6 </td>
|
|
<td align='right' valign='top'>0.9 </td>
|
|
<td align='right' valign='top'>0.3 </td>
|
|
<td align='right' valign='top'>60 </td>
|
|
<td align='right' valign='top'>3.4 </td>
|
|
<td align='right' valign='top'>1.622 </td>
|
|
<td align='right' valign='top'>1.551 </td>
|
|
<td align='right' valign='top'>2.159 </td>
|
|
<td align='right' valign='top'>6.7 </td>
|
|
<td align='right' valign='top'>7.3 </td>
|
|
<td align='right' valign='top'>-40.272</td>
|
|
<td align='center' valign='top'>4</td>
|
|
</tr>
|
|
<tr>
|
|
<td align='center' valign='top'> 4 </td>
|
|
<td align='right' valign='top'>32733.8 </td>
|
|
<td align='right' valign='top'>-1.2<remark role='tablenotemark'>c</remark></td>
|
|
<td align='right' valign='top'>-0.5 </td>
|
|
<td align='right' valign='top'>41 </td>
|
|
<td align='right' valign='top'>54.8 </td>
|
|
<td align='right' valign='top'>2.282 </td>
|
|
<td align='right' valign='top'>2.156 </td>
|
|
<td align='right' valign='top'>4.313 </td>
|
|
<td align='right' valign='top'>117.4 </td>
|
|
<td align='right' valign='top'>78.2 </td>
|
|
<td align='right' valign='top'>-35.847</td>
|
|
<td align='center' valign='top'>5,6</td>
|
|
</tr>
|
|
<tr>
|
|
<td align='center' valign='top'> 5 </td>
|
|
<td align='right' valign='top'> 9607.4 </td>
|
|
<td align='right' valign='top'>-0.4 </td>
|
|
<td align='right' valign='top'>-0.4 </td>
|
|
<td align='right' valign='top'>60 </td>
|
|
<td align='right' valign='top'>1.4 </td>
|
|
<td align='right' valign='top'>1.669<remark role='tablenotemark'>c</remark></td>
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<td align='right' valign='top'>1.574 </td>
|
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<td align='right' valign='top'>2.343 </td>
|
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<td align='right' valign='top'>8.0 </td>
|
|
<td align='right' valign='top'>8.9 </td>
|
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<td align='right' valign='top'>-33.417</td>
|
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<td align='center' valign='top'>7</td>
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</tr>
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<tr>
|
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<td align='center' valign='top'> 6 </td>
|
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<td align='right' valign='top'>31638.6 </td>
|
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<td align='right' valign='top'>1.6 </td>
|
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<td align='right' valign='top'>0.1 </td>
|
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<td align='right' valign='top'>39 </td>
|
|
<td align='right' valign='top'>315.2 </td>
|
|
<td align='right' valign='top'> 3.433 </td>
|
|
<td align='right' valign='top'>3.075 </td>
|
|
<td align='right' valign='top'>7.488 </td>
|
|
<td align='right' valign='top'>92.1 </td>
|
|
<td align='right' valign='top'>25.3 </td>
|
|
<td align='right' valign='top'>-12.052 </td>
|
|
<td align='center' valign='top'>8</td>
|
|
</tr>
|
|
</tbody>
|
|
|
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<remark role='tablenote'>a<!-- }{ -->
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Sample footnote for table <xref linkend="tbl-2" /> that was generated with the LaTeX table environment</remark>
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<remark role='tablenote'>b<!-- }{ -->
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Yet another sample footnote for table <xref linkend="tbl-2" /></remark>
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<remark role='tablenote'>c<!-- }{ -->
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Another sample footnote for table <xref linkend="tbl-2" /></remark>
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<remark role='tablecomments'>We can also attach a long-ish paragraph of explanatory material to a table. Use \tablerefs to append a list of references. The following references were from a different table: I've patched them in here to show how they look, but don't take them too seriously—I certainly have not.</remark>
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<remark role='tablerefs'>(1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al. 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al. 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al. 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989; (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20) Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al. 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et al. 1991; (26) Olsen 1983.</remark>
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</table>
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</article> |