#LyX 1.3 created this file. For more info see http://www.lyx.org/ \lyxformat 221 \textclass aastex \begin_preamble \newcommand{\vdag}{(v)^\dagger} \newcommand{\myemail}{skywalker@galaxy.far.far.away} \slugcomment{Not to appear in Nonlearned J., 45.} \shorttitle{Djorgovski et al.} \shortauthors{Collapsed Cores in Globular Clusters} \end_preamble \language english \inputencoding default \fontscheme default \graphics default \paperfontsize default \spacing single \papersize Default \paperpackage a4 \use_geometry 0 \use_amsmath 0 \use_natbib 0 \use_numerical_citations 0 \paperorientation portrait \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \quotes_times 2 \papercolumns 1 \papersides 1 \paperpagestyle default \layout Title Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AASTeX Examples \layout Author S. Djorgovski \begin_inset ERT status Collapsed \layout Standard \backslash altaffilmark{1,2,3} \end_inset and Ivan R. King \begin_inset ERT status Collapsed \layout Standard \backslash altaffilmark{1} \end_inset \layout Affiliation Astronomy Department, University of California, Berkeley, CA 94720 \layout Author C. D. Biemesderfer \begin_inset ERT status Collapsed \layout Standard \backslash altaffilmark{4,5} \end_inset \layout Affiliation National Optical Astronomy Observatories, Tucson, AZ 85719 \layout Email aastex-help@aas.org \layout And \layout Author R. J. Hanisch \begin_inset ERT status Collapsed \layout Standard \backslash altaffilmark{5} \end_inset \layout Affiliation Space Telescope Science Institute, Baltimore, MD 21218 \layout Standard \begin_inset ERT status Collapsed \layout Standard \backslash altaffiltext{1}{Visiting Astronomer, Cerro Tololo Inter-American Observatory. CTIO is operated by AURA, Inc. \backslash under contract to the National Science Foundation.} \layout Standard \backslash altaffiltext{2}{Society of Fellows, Harvard University.} \layout Standard \backslash altaffiltext{3}{present address: Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138} \layout Standard \backslash altaffiltext{4}{Visiting Programmer, Space Telescope Science Institute} \layout Standard \backslash altaffiltext{5}{Patron, Alonso's Bar and Grill} \end_inset \layout Abstract 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. \layout Keywords clusters: globular, peanut---bosons: bozos \layout Section Introduction \layout Standard A focal problem today in the dynamics of globular clusters is core collapse. It has been predicted by theory for decades \begin_inset ERT status Collapsed \layout Standard \backslash citep{hen61,lyn68,spi85} \end_inset , but observation has been less alert to the phenomenon. For many years the central brightness peak in M15 \begin_inset ERT status Collapsed \layout Standard \backslash citep{kin75,new78} \end_inset seemed a unique anomaly. Then \begin_inset ERT status Collapsed \layout Standard \backslash citet{aur82} \end_inset suggested a central peak in NGC 6397, and a limited photographic survey of ours \begin_inset ERT status Collapsed \layout Standard \backslash citep[Paper I]{djo84} \end_inset found three more cases, including NGC 6624, whose sharp center had often been remarked on \begin_inset ERT status Collapsed \layout Standard \backslash citep{can78} \end_inset . \layout Section Observations \layout Standard All our observations were short direct exposures with CCD's. At Lick Observatory we used a TI 500 \begin_inset Formula \( \times \) \end_inset 500 chip and a GEC 575 \begin_inset Formula \( \times \) \end_inset 385, on the 1-m Nickel reflector. The only filter available at Lick was red. At CTIO we used a GEC 575 \begin_inset Formula \( \times \) \end_inset 385, with \begin_inset Formula \( B,V, \) \end_inset and \begin_inset Formula \( R \) \end_inset filters, and an RCA 512 \begin_inset Formula \( \times \) \end_inset 320, with \begin_inset Formula \( U,B,V,R, \) \end_inset and \begin_inset Formula \( I \) \end_inset 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 \begin_inset Formula \( U \) \end_inset or even in \begin_inset Formula \( B \) \end_inset . All four cameras had scales of the order of 0.4 arcsec/pixel, and our field sizes were around 3 arcmin. \layout Standard 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 \begin_inset ERT status Collapsed \layout Standard \backslash citep{kin68,pet76,har84,ort85} \end_inset , as well as photoelectric profiles \begin_inset ERT status Collapsed \layout Standard \backslash citep{kin66,kin75} \end_inset and electronographic profiles \begin_inset ERT status Collapsed \layout Standard \backslash citep{kro84} \end_inset . In a few cases we judged normality by eye estimates on one of the Sky Surveys. \layout Section Helicity Amplitudes \layout Standard It has been realized that helicity amplitudes provide a convenient means for Feynman diagram \begin_inset Foot collapsed true \layout Standard Footnotes can be inserted like this. \end_inset 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 \begin_inset LatexCommand \cite{hag86} \end_inset ; we briefly summarize below. \layout Subsection Formalism \layout Standard \begin_inset LatexCommand \label{bozomath} \end_inset \layout Standard A tree-level amplitude in \begin_inset Formula \( e^{+}e^{-} \) \end_inset collisions can be expressed in terms of fermion strings of the form \begin_inset Formula \begin{equation} \bar{v}(p_{2},\sigma _{2})P_{-\tau }\hat{a}_{1}\hat{a}_{2}\cdots \hat{a}_{n}u(p_{1},\sigma _{1}), \end{equation} \end_inset where \begin_inset Formula \( p \) \end_inset and \begin_inset Formula \( \sigma \) \end_inset label the initial \begin_inset Formula \( e^{\pm } \) \end_inset four-momenta and helicities \begin_inset Formula \( (\sigma =\pm 1) \) \end_inset , \begin_inset Formula \( \hat{a}_{i}=a^{\mu }_{i}\gamma _{\nu } \) \end_inset and \begin_inset Formula \( P_{\tau }=\frac{1}{2}(1+\tau \gamma _{5}) \) \end_inset is a chirality projection operator \begin_inset Formula \( (\tau =\pm1 ) \) \end_inset . The \begin_inset Formula \( a^{\mu }_{i} \) \end_inset may be formed from particle four-momenta, gauge-boson polarization vectors or fermion strings with an uncontracted Lorentz index associated with final-sta te fermions. \layout NoteToEditor Figures 1 and 2 should appear side-by-side in print \layout Standard In the chiral representation the \begin_inset Formula \( \gamma \) \end_inset matrices are expressed in terms of \begin_inset Formula \( 2\times 2 \) \end_inset Pauli matrices \begin_inset Formula \( \sigma \) \end_inset and the unit matrix 1 as \begin_inset Formula \begin{eqnarray} \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) ,\nonumber \\ \sigma ^{\mu }_{\pm } & = & ({\textbf {1}},\pm \sigma ),\nonumber \end{eqnarray} \end_inset giving \begin_inset Formula \begin{equation} \hat{a}=\left( \begin{array}{cc} 0 & (\hat{a})_{+}\\ (\hat{a})_{-} & 0 \end{array}\right) ,(\hat{a})_{\pm }=a_{\mu }\sigma ^{\mu }_{\pm }, \end{equation} \end_inset The spinors are expressed in terms of two-component Weyl spinors as \begin_inset Formula \begin{equation} u=\left( \begin{array}{c} (u)_{-}\\ (u)_{+} \end{array}\right) ,v={\textbf {(}}\vdag _{+}{\textbf {,}}\vdag _{-}{\textbf {)}}. \end{equation} \end_inset \layout Standard The Weyl spinors are given in terms of helicity eigenstates \begin_inset Formula \( \chi _{\lambda }(p) \) \end_inset with \begin_inset Formula \( \lambda =\pm1 \) \end_inset by \layout MathLetters \begin_inset Formula \begin{eqnarray} 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) \end{eqnarray} \end_inset \layout Section Floating material and so forth \layout Standard Consider a task that computes profile parameters for a modified Lorentzian of the form \begin_inset Formula \begin{equation} I=\frac{1}{1+d_{1}^{P(1+d_{2})}} \end{equation} \end_inset where \begin_inset Formula \[ d_{1}=\sqrt{\left( \begin{array}{c} \frac{x_{1}}{R_{maj}} \end{array}\right) ^{2}+\left( \begin{array}{c} \frac{y_{1}}{R_{min}} \end{array}\right) ^{2}}\] \end_inset \begin_inset Formula \[ d_{2}=\sqrt{\left( \begin{array}{c} \frac{x_{1}}{PR_{maj}} \end{array}\right) ^{2}+\left( \begin{array}{c} \case {y_{1}}{PR_{min}} \end{array}\right) ^{2}}\] \end_inset \begin_inset Formula \[ x_{1}=(x-x_{0})\cos \Theta +(y-y_{0})\sin \Theta \] \end_inset \begin_inset Formula \[ y_{1}=-(x-x_{0})\sin \Theta +(y-y_{0})\cos \Theta \] \end_inset \layout Standard In these expressions \begin_inset Formula \( x_{0} \) \end_inset , \begin_inset Formula \( y_{0} \) \end_inset is the star center, and \begin_inset Formula \( \Theta \) \end_inset is the angle with the \begin_inset Formula \( x \) \end_inset axis. Results of this task are shown in table\SpecialChar ~ \begin_inset LatexCommand \ref{tbl-2} \end_inset . It is not clear how these sorts of analyses may affect determination of \begin_inset Formula \( M_{\sun } \) \end_inset , but the assumption is that the alternate results should be less than 90 \begin_inset ERT status Collapsed \layout Standard \backslash degr \backslash \end_inset out of phase with previous values. We have no observations of \begin_inset ERT status Collapsed \layout Standard \backslash ion{Ca}{2} \end_inset . Roughly \begin_inset ERT status Collapsed \layout Standard \backslash slantfrac{4}{5} \end_inset of the electronically submitted abstracts for AAS meetings are error-free. \layout Acknowledgements \layout Standard We are grateful to V. Barger, T. Han, and R. J. N. Phillips for doing the math in section\SpecialChar ~ \begin_inset LatexCommand \ref{bozomath} \end_inset . More information on the AASTeX macros package are available at \begin_inset LatexCommand \url{http://www.aas.org/publications/aastex} \end_inset or the \begin_inset ERT status Collapsed \layout Standard \backslash anchor{ftp://www.aas.org/pubs/}{AAS ftp site} \end_inset . \layout Appendix \layout Section Appendicial material \layout Standard Consider once again a task that computes profile parameters for a modified Lorentzian of the form \begin_inset Formula \begin{equation} I=\frac{1}{1+d_{1}^{P(1+d_{2})}} \end{equation} \end_inset where \layout MathLetters \begin_inset Formula \[ d_{1}=\frac{3}{4}\sqrt{\left( \begin{array}{c} \frac{x_{1}}{R_{maj}} \end{array}\right) ^{2}+\left( \begin{array}{c} \frac{y_{1}}{R_{min}} \end{array}\right) ^{2}}\] \end_inset \begin_inset Formula \begin{equation} d_{2}=\case {3}{4}\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}} \end{equation} \end_inset \begin_inset Formula \begin{eqnarray} x_{1} & = & (x-x_{0})\cos \Theta +(y-y_{0})\sin \Theta \\ y_{1} & = & -(x-x_{0})\sin \Theta +(y-y_{0})\cos \Theta \end{eqnarray} \end_inset \layout Standard For completeness, here is one last equation. \begin_inset Formula \begin{equation} e=mc^{2} \end{equation} \end_inset \layout References \bibitem [Auri\`ere(1982)]{aur82} Auri\i \`{e} re, M. 1982, \begin_inset ERT status Collapsed \layout Standard \backslash aap \end_inset , 109, 301 \layout References \bibitem [Canizares et al.(1978)]{can78} Canizares, C. R., Grindlay, J. E., Hiltner, W. A., Liller, W., and McClintock, J. E. 1978, \begin_inset ERT status Collapsed \layout Standard \backslash apj \end_inset , 224, 39 \layout References \bibitem [Djorgovski and King(1984)]{djo84} Djorgovski, S., and King, I. R. 1984, \begin_inset ERT status Collapsed \layout Standard \backslash apjl \end_inset , 277, L49 \layout References \bibitem [Hagiwara and Zeppenfeld(1986)]{hag86} Hagiwara, K., and Zeppenfeld, D. 1986, Nucl.Phys., 274, 1 \layout References \bibitem [Harris and van den Bergh(1984)]{har84} Harris, W. E., and van den Bergh, S. 1984, \begin_inset ERT status Collapsed \layout Standard \backslash aj \end_inset , 89, 1816 \layout References \bibitem [H\`enon(1961)]{hen61} H\i \'{e} non, M. 1961, Ann.d'Ap., 24, 369 \layout References \bibitem [King(1966)]{kin66} King, I. R. 1966, \begin_inset ERT status Collapsed \layout Standard \backslash aj \end_inset , 71, 276 \layout References \bibitem [King(1975)]{kin75} King, I. R. 1975, Dynamics of Stellar Systems, A. Hayli, Dordrecht: Reidel, 1975, 99 \layout References \bibitem [King et al.(1968)]{kin68} King, I. R., Hedemann, E., Hodge, S. M., and White, R. E. 1968, \begin_inset ERT status Collapsed \layout Standard \backslash aj \end_inset , 73, 456 \layout References \bibitem [Kron et al.(1984)]{kro84} Kron, G. E., Hewitt, A. V., and Wasserman, L. H. 1984, \begin_inset ERT status Collapsed \layout Standard \backslash pasp \end_inset , 96, 198 \layout References \bibitem [Lynden-Bell and Wood(1968)]{lyn68} Lynden-Bell, D., and Wood, R. 1968, \begin_inset ERT status Collapsed \layout Standard \backslash mnras \end_inset , 138, 495 \layout References \bibitem [Newell and O'Neil(1978)]{new78} Newell, E. B., and O'Neil, E. J. 1978, \begin_inset ERT status Collapsed \layout Standard \backslash apjs \end_inset , 37, 27 \layout References \bibitem [Ortolani et al.(1985)]{ort85} Ortolani, S., Rosino, L., and Sandage, A. 1985, \begin_inset ERT status Collapsed \layout Standard \backslash aj \end_inset , 90, 473 \layout References \bibitem [Peterson(1976)]{pet76} Peterson, C. J. 1976, \begin_inset ERT status Collapsed \layout Standard \backslash aj \end_inset , 81, 617 \layout References \bibitem [Spitzer(1985)]{spi85} Spitzer, L. 1985, Dynamics of Star Clusters, J. Goodman and P. Hut, Dordrecht: Reidel, 109 \layout Standard \begin_inset ERT status Collapsed \layout Standard \backslash clearpage \end_inset \layout FigCaption \begin_inset LatexCommand \label{fig1} \end_inset This is the first figure and it uses sgi9259.eps as its EPS figure file. \layout FigCaption \begin_inset LatexCommand \label{fig2} \end_inset This is an example of a long figure caption that must be set as a paragraph. The processor has to buffer the text of the caption, so it is good not to be too wordy, but that would make for poor communication as well. \layout FigCaption \begin_inset LatexCommand \label{fig3} \end_inset This figure has no associated EPS file, so the optional parameter is omitted. \layout Standard \begin_inset ERT status Collapsed \layout Standard \backslash clearpage \end_inset \layout Standard \begin_inset Float table wide false collapsed false \layout Caption \added_space_bottom medskip Terribly relevant tabular information. \begin_inset LatexCommand \label{tbl-2} \end_inset \layout Standard \begin_inset Tabular \begin_inset Text \layout Standard Star \end_inset \begin_inset Text \layout Standard Height \end_inset \begin_inset Text \layout Standard \begin_inset Formula \( d_{x} \) \end_inset \end_inset \begin_inset Text \layout Standard \begin_inset Formula \( d_{y} \) \end_inset \end_inset \begin_inset Text \layout Standard \begin_inset Formula \( n \) \end_inset \end_inset 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\layout Standard 25.3 \end_inset \begin_inset Text \layout Standard -12.052 \end_inset \begin_inset Text \layout Standard 8 \end_inset \end_inset \layout Standard \begin_inset ERT status Collapsed \layout Standard \backslash tablenotetext{a}{Sample footnote for table~ \backslash ref{tbl-2} that was generated with the \backslash LaTeX \backslash table environment} \layout Standard \backslash tablenotetext{b}{Yet another sample footnote for table~ \backslash ref{tbl-2}} \layout Standard \backslash tablenotetext{c}{Another sample footnote for table~ \backslash ref{tbl-2}} \end_inset \layout TableComments We can also attach a long-ish paragraph of explanatory material to a table. Use \backslash 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. \layout 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. \end_inset \layout Standard \begin_inset ERT status Collapsed \layout Standard % LyX can load deluxetable files verbatim. To see this in action, uncomment the following \end_inset \layout Standard \begin_inset ERT status Collapsed \layout Standard % line (delete the \end_inset \begin_inset Quotes eld \end_inset \begin_inset ERT status Collapsed \layout Standard % \end_inset \begin_inset Quotes erd \end_inset \begin_inset ERT status Collapsed \layout Standard ) and copy the file table.tex from the AASTeX 5.0 distribution into \end_inset \layout Standard \begin_inset ERT status Collapsed \layout Standard % the same directory as this file. \end_inset \layout Standard \begin_inset ERT status Collapsed \layout Standard % \end_inset \begin_inset Include \input{table} \end_inset \the_end