#LyX 1.5.0svn created this file. For more info see http://www.lyx.org/ \lyxformat 276 \begin_document \begin_header \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 \font_roman default \font_sans default \font_typewriter default \font_default_family default \font_sc false \font_osf false \font_sf_scale 100 \font_tt_scale 100 \graphics default \paperfontsize default \spacing single \papersize default \use_geometry false \use_amsmath 0 \use_esint 0 \cite_engine basic \use_bibtopic false \paperorientation portrait \secnumdepth 3 \tocdepth 3 \paragraph_separation indent \defskip medskip \quotes_language english \papercolumns 1 \papersides 1 \paperpagestyle default \tracking_changes false \output_changes false \end_header \begin_body \begin_layout Title Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AASTeX Examples \end_layout \begin_layout Author S. Djorgovski \begin_inset ERT status collapsed \begin_layout Standard \backslash altaffilmark{1,2,3} \end_layout \end_inset and Ivan R. King \begin_inset ERT status collapsed \begin_layout Standard \backslash altaffilmark{1} \end_layout \end_inset \end_layout \begin_layout Affiliation Astronomy Department, University of California, Berkeley, CA 94720 \end_layout \begin_layout Author C. D. Biemesderfer \begin_inset ERT status collapsed \begin_layout Standard \backslash altaffilmark{4,5} \end_layout \end_inset \end_layout \begin_layout Affiliation National Optical Astronomy Observatories, Tucson, AZ 85719 \end_layout \begin_layout Email aastex-help@aas.org \end_layout \begin_layout And \end_layout \begin_layout Author R. J. Hanisch \begin_inset ERT status collapsed \begin_layout Standard \backslash altaffilmark{5} \end_layout \end_inset \end_layout \begin_layout Affiliation Space Telescope Science Institute, Baltimore, MD 21218 \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_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.} \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash altaffiltext{2}{Society of Fellows, Harvard University.} \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash altaffiltext{3}{present address: Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138} \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash altaffiltext{4}{Visiting Programmer, Space Telescope Science Institute} \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash altaffiltext{5}{Patron, Alonso's Bar and Grill} \end_layout \end_inset \end_layout \begin_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. \end_layout \begin_layout Keywords clusters: globular, peanut---bosons: bozos \end_layout \begin_layout Section Introduction \end_layout \begin_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 \begin_layout Standard \backslash citep{hen61,lyn68,spi85} \end_layout \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 \begin_layout Standard \backslash citep{kin75,new78} \end_layout \end_inset seemed a unique anomaly. Then \begin_inset ERT status collapsed \begin_layout Standard \backslash citet{aur82} \end_layout \end_inset suggested a central peak in NGC 6397, and a limited photographic survey of ours \begin_inset ERT status collapsed \begin_layout Standard \backslash citep[Paper I]{djo84} \end_layout \end_inset found three more cases, including NGC 6624, whose sharp center had often been remarked on \begin_inset ERT status collapsed \begin_layout Standard \backslash citep{can78} \end_layout \end_inset . \end_layout \begin_layout Section Observations \end_layout \begin_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. \end_layout \begin_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 \begin_layout Standard \backslash citep{kin68,pet76,har84,ort85} \end_layout \end_inset , as well as photoelectric profiles \begin_inset ERT status collapsed \begin_layout Standard \backslash citep{kin66,kin75} \end_layout \end_inset and electronographic profiles \begin_inset ERT status collapsed \begin_layout Standard \backslash citep{kro84} \end_layout \end_inset . In a few cases we judged normality by eye estimates on one of the Sky Surveys. \end_layout \begin_layout Section Helicity Amplitudes \end_layout \begin_layout Standard It has been realized that helicity amplitudes provide a convenient means for Feynman diagram \begin_inset Foot status collapsed \begin_layout Standard Footnotes can be inserted like this. \end_layout \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 key "hag86" \end_inset ; we briefly summarize below. \end_layout \begin_layout Subsection Formalism \end_layout \begin_layout Standard \begin_inset LatexCommand label name "bozomath" \end_inset \end_layout \begin_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. \end_layout \begin_layout NoteToEditor Figures 1 and 2 should appear side-by-side in print \end_layout \begin_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 \end_layout \begin_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 \end_layout \begin_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 \end_layout \begin_layout Section Floating material and so forth \end_layout \begin_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 \end_layout \begin_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\InsetSpace ~ \begin_inset LatexCommand ref reference "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 \begin_layout Standard \backslash degr \backslash \end_layout \end_inset out of phase with previous values. We have no observations of \begin_inset ERT status collapsed \begin_layout Standard \backslash ion{Ca}{2} \end_layout \end_inset . Roughly \begin_inset ERT status collapsed \begin_layout Standard \backslash slantfrac{4}{5} \end_layout \end_inset of the electronically submitted abstracts for AAS meetings are error-free. \end_layout \begin_layout Acknowledgements \end_layout \begin_layout Standard We are grateful to V. Barger, T. Han, and R. J. N. Phillips for doing the math in section\InsetSpace ~ \begin_inset LatexCommand ref reference "bozomath" \end_inset . More information on the AASTeX macros package are available at \begin_inset LatexCommand url target "http://www.aas.org/publications/aastex" \end_inset or the \begin_inset ERT status collapsed \begin_layout Standard \backslash anchor{ftp://www.aas.org/pubs/}{AAS ftp site} \end_layout \end_inset . \end_layout \begin_layout Appendix \end_layout \begin_layout Section Appendicial material \end_layout \begin_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 \end_layout \begin_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 \end_layout \begin_layout Standard For completeness, here is one last equation. \begin_inset Formula \begin{equation} e=mc^{2} \end{equation} \end_inset \end_layout \begin_layout References \begin_inset LatexCommand bibitem label "Auri\`ere(1982)" key "aur82" \end_inset Auriè re, M. 1982, \begin_inset ERT status collapsed \begin_layout Standard \backslash aap \end_layout \end_inset , 109, 301 \end_layout \begin_layout References \begin_inset LatexCommand bibitem label "Canizares et al.(1978)" key "can78" \end_inset Canizares, C. R., Grindlay, J. E., Hiltner, W. A., Liller, W., and McClintock, J. E. 1978, \begin_inset ERT status collapsed \begin_layout Standard \backslash apj \end_layout \end_inset , 224, 39 \end_layout \begin_layout References \begin_inset LatexCommand bibitem label "Djorgovski and King(1984)" key "djo84" \end_inset Djorgovski, S., and King, I. R. 1984, \begin_inset ERT status collapsed \begin_layout Standard \backslash apjl \end_layout \end_inset , 277, L49 \end_layout \begin_layout References \begin_inset LatexCommand bibitem label "Hagiwara and Zeppenfeld(1986)" key "hag86" \end_inset Hagiwara, K., and Zeppenfeld, D. 1986, Nucl.Phys., 274, 1 \end_layout \begin_layout References \begin_inset LatexCommand bibitem label "Harris and van den Bergh(1984)" key "har84" \end_inset Harris, W. E., and van den Bergh, S. 1984, \begin_inset ERT status collapsed \begin_layout Standard \backslash aj \end_layout \end_inset , 89, 1816 \end_layout \begin_layout References \begin_inset LatexCommand bibitem label "H\`enon(1961)" key "hen61" \end_inset Hé non, M. 1961, Ann.d'Ap., 24, 369 \end_layout \begin_layout References \begin_inset LatexCommand bibitem label "King(1966)" key "kin66" \end_inset King, I. R. 1966, \begin_inset ERT status collapsed \begin_layout Standard \backslash aj \end_layout \end_inset , 71, 276 \end_layout \begin_layout References \begin_inset LatexCommand bibitem label "King(1975)" key "kin75" \end_inset King, I. R. 1975, Dynamics of Stellar Systems, A. Hayli, Dordrecht: Reidel, 1975, 99 \end_layout \begin_layout References \begin_inset LatexCommand bibitem label "King et al.(1968)" key "kin68" \end_inset King, I. R., Hedemann, E., Hodge, S. M., and White, R. E. 1968, \begin_inset ERT status collapsed \begin_layout Standard \backslash aj \end_layout \end_inset , 73, 456 \end_layout \begin_layout References \begin_inset LatexCommand bibitem label "Kron et al.(1984)" key "kro84" \end_inset Kron, G. E., Hewitt, A. V., and Wasserman, L. H. 1984, \begin_inset ERT status collapsed \begin_layout Standard \backslash pasp \end_layout \end_inset , 96, 198 \end_layout \begin_layout References \begin_inset LatexCommand bibitem label "Lynden-Bell and Wood(1968)" key "lyn68" \end_inset Lynden-Bell, D., and Wood, R. 1968, \begin_inset ERT status collapsed \begin_layout Standard \backslash mnras \end_layout \end_inset , 138, 495 \end_layout \begin_layout References \begin_inset LatexCommand bibitem label "Newell and O'Neil(1978)" key "new78" \end_inset Newell, E. B., and O'Neil, E. J. 1978, \begin_inset ERT status collapsed \begin_layout Standard \backslash apjs \end_layout \end_inset , 37, 27 \end_layout \begin_layout References \begin_inset LatexCommand bibitem label "Ortolani et al.(1985)" key "ort85" \end_inset Ortolani, S., Rosino, L., and Sandage, A. 1985, \begin_inset ERT status collapsed \begin_layout Standard \backslash aj \end_layout \end_inset , 90, 473 \end_layout \begin_layout References \begin_inset LatexCommand bibitem label "Peterson(1976)" key "pet76" \end_inset Peterson, C. J. 1976, \begin_inset ERT status collapsed \begin_layout Standard \backslash aj \end_layout \end_inset , 81, 617 \end_layout \begin_layout References \begin_inset LatexCommand bibitem label "Spitzer(1985)" key "spi85" \end_inset Spitzer, L. 1985, Dynamics of Star Clusters, J. Goodman and P. 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\end_layout \end_inset \begin_inset Text \begin_layout Standard 8 \end_layout \end_inset \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard \backslash tablenotetext{a}{Sample footnote for table~ \backslash ref{tbl-2} that was generated with the \backslash LaTeX \backslash table environment} \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash tablenotetext{b}{Yet another sample footnote for table~ \backslash ref{tbl-2}} \end_layout \begin_layout Standard \end_layout \begin_layout Standard \backslash tablenotetext{c}{Another sample footnote for table~ \backslash ref{tbl-2}} \end_layout \end_inset \end_layout \begin_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. \end_layout \begin_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_layout \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard % LyX can load deluxetable files verbatim. To see this in action, uncomment the following \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard % line (delete the \end_layout \end_inset \begin_inset Quotes eld \end_inset \begin_inset ERT status collapsed \begin_layout Standard % \end_layout \end_inset \begin_inset Quotes erd \end_inset \begin_inset ERT status collapsed \begin_layout Standard ) and copy the file table.tex from the AASTeX 5.0 distribution into \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard % the same directory as this file. \end_layout \end_inset \end_layout \begin_layout Standard \begin_inset ERT status collapsed \begin_layout Standard % \end_layout \end_inset \begin_inset Include \input{table} \end_inset \end_layout \end_body \end_document