1 #LyX 1.3 created this file. For more info see http://www.lyx.org/
5 \newcommand{\vdag}{(v)^\dagger}
6 \newcommand{\myemail}{skywalker@galaxy.far.far.away}
7 \slugcomment{Not to appear in Nonlearned J., 45.}
8 \shorttitle{Djorgovski et al.}
9 \shortauthors{Collapsed Cores in Globular Clusters}
12 \inputencoding default
15 \paperfontsize default
22 \use_numerical_citations 0
23 \paperorientation portrait
26 \paragraph_separation indent
28 \quotes_language english
32 \paperpagestyle default
36 Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AASTeX
65 Astronomy Department, University of California, Berkeley, CA 94720
83 National Optical Astronomy Observatories, Tucson, AZ 85719
106 Space Telescope Science Institute, Baltimore, MD 21218
116 altaffiltext{1}{Visiting Astronomer, Cerro Tololo Inter-American Observatory.
117 CTIO is operated by AURA, Inc.
119 under contract to the National Science Foundation.}
123 altaffiltext{2}{Society of Fellows, Harvard University.}
127 altaffiltext{3}{present address: Center for Astrophysics, 60 Garden Street,
132 altaffiltext{4}{Visiting Programmer, Space Telescope Science Institute}
137 altaffiltext{5}{Patron, Alonso's Bar and Grill}
143 This is a preliminary report on surface photometry of the major fraction
144 of known globular clusters, to see which of them show the signs of a collapsed
146 We also explore some diversionary mathematics and recreational tables.
150 clusters: globular, peanut---bosons: bozos
156 A focal problem today in the dynamics of globular clusters is core collapse.
157 It has been predicted by theory for decades
164 citep{hen61,lyn68,spi85}
167 , but observation has been less alert to the phenomenon.
168 For many years the central brightness peak in M15
178 seemed a unique anomaly.
189 suggested a central peak in NGC 6397, and a limited photographic survey
197 citep[Paper I]{djo84}
200 found three more cases, including NGC 6624, whose sharp center had often
218 All our observations were short direct exposures with CCD's.
219 At Lick Observatory we used a TI 500
220 \begin_inset Formula \( \times \)
223 500 chip and a GEC 575
224 \begin_inset Formula \( \times \)
227 385, on the 1-m Nickel reflector.
228 The only filter available at Lick was red.
229 At CTIO we used a GEC 575
230 \begin_inset Formula \( \times \)
234 \begin_inset Formula \( B,V, \)
238 \begin_inset Formula \( R \)
241 filters, and an RCA 512
242 \begin_inset Formula \( \times \)
246 \begin_inset Formula \( U,B,V,R, \)
250 \begin_inset Formula \( I \)
253 filters, on the 1.5-m reflector.
254 In the CTIO observations we tried to concentrate on the shortest practicable
255 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
256 often kept us from observing in
257 \begin_inset Formula \( U \)
261 \begin_inset Formula \( B \)
265 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
266 sizes were around 3 arcmin.
269 The CCD images are unfortunately not always suitable, for very poor clusters
270 or for clusters with large cores.
271 Since the latter are easily studied by other means, we augmented our own
272 CCD profiles by collecting from the literature a number of star-count profiles
280 citep{kin68,pet76,har84,ort85}
283 , as well as photoelectric profiles
293 and electronographic profiles
304 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
310 It has been realized that helicity amplitudes provide a convenient means
317 Footnotes can be inserted like this.
321 These amplitude-level techniques are particularly convenient for calculations
322 involving many Feynman diagrams, where the usual trace techniques for the
323 amplitude squared becomes unwieldy.
324 Our calculations use the helicity techniques developed by other authors
326 \begin_inset LatexCommand \cite{hag86}
330 ; we briefly summarize below.
337 \begin_inset LatexCommand \label{bozomath}
344 A tree-level amplitude in
345 \begin_inset Formula \( e^{+}e^{-} \)
348 collisions can be expressed in terms of fermion strings of the form
349 \begin_inset Formula \begin{equation}
350 \bar{v}(p_{2},\sigma _{2})P_{-\tau }\hat{a}_{1}\hat{a}_{2}\cdots \hat{a}_{n}u(p_{1},\sigma _{1}),
356 \begin_inset Formula \( p \)
360 \begin_inset Formula \( \sigma \)
364 \begin_inset Formula \( e^{\pm } \)
367 four-momenta and helicities
368 \begin_inset Formula \( (\sigma =\pm 1) \)
372 \begin_inset Formula \( \hat{a}_{i}=a^{\mu }_{i}\gamma _{\nu } \)
376 \begin_inset Formula \( P_{\tau }=\frac{1}{2}(1+\tau \gamma _{5}) \)
379 is a chirality projection operator
380 \begin_inset Formula \( (\tau =\pm1 ) \)
385 \begin_inset Formula \( a^{\mu }_{i} \)
388 may be formed from particle four-momenta, gauge-boson polarization vectors
389 or fermion strings with an uncontracted Lorentz index associated with final-sta
393 Figures 1 and 2 should appear side-by-side in print
396 In the chiral representation the
397 \begin_inset Formula \( \gamma \)
400 matrices are expressed in terms of
401 \begin_inset Formula \( 2\times 2 \)
405 \begin_inset Formula \( \sigma \)
408 and the unit matrix 1 as
409 \begin_inset Formula \begin{eqnarray}
410 \gamma ^{\mu } & = & \left( \begin{array}{cc}
411 0 & \sigma ^{\mu }_{+}\\
412 \sigma ^{\mu }_{-} & 0
413 \end{array}\right) ,\gamma ^{5}=\left( \begin{array}{cc}
416 \end{array}\right) ,\nonumber \\
417 \sigma ^{\mu }_{\pm } & = & ({\textbf {1}},\pm \sigma ),\nonumber
423 \begin_inset Formula \begin{equation}
424 \hat{a}=\left( \begin{array}{cc}
427 \end{array}\right) ,(\hat{a})_{\pm }=a_{\mu }\sigma ^{\mu }_{\pm },
432 The spinors are expressed in terms of two-component Weyl spinors as
433 \begin_inset Formula \begin{equation}
434 u=\left( \begin{array}{c}
437 \end{array}\right) ,v={\textbf {(}}\vdag _{+}{\textbf {,}}\vdag _{-}{\textbf {)}}.
445 The Weyl spinors are given in terms of helicity eigenstates
446 \begin_inset Formula \( \chi _{\lambda }(p) \)
450 \begin_inset Formula \( \lambda =\pm1 \)
457 \begin_inset Formula \begin{eqnarray}
458 u(p,\lambda )_{\pm } & = & (E\pm \lambda |{\textbf {p}}|)^{1/2}\chi _{\lambda }(p),\\
459 v(p,\lambda )_{\pm } & = & \pm \lambda (E\mp \lambda |{\textbf {p}}|)^{1/2}\chi _{-\lambda }(p)
467 Floating material and so forth
470 Consider a task that computes profile parameters for a modified Lorentzian
472 \begin_inset Formula \begin{equation}
473 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
479 \begin_inset Formula \[
480 d_{1}=\sqrt{\left( \begin{array}{c}
481 \frac{x_{1}}{R_{maj}}
482 \end{array}\right) ^{2}+\left( \begin{array}{c}
483 \frac{y_{1}}{R_{min}}
484 \end{array}\right) ^{2}}\]
489 \begin_inset Formula \[
490 d_{2}=\sqrt{\left( \begin{array}{c}
491 \frac{x_{1}}{PR_{maj}}
492 \end{array}\right) ^{2}+\left( \begin{array}{c}
493 \case {y_{1}}{PR_{min}}
494 \end{array}\right) ^{2}}\]
499 \begin_inset Formula \[
500 x_{1}=(x-x_{0})\cos \Theta +(y-y_{0})\sin \Theta \]
505 \begin_inset Formula \[
506 y_{1}=-(x-x_{0})\sin \Theta +(y-y_{0})\cos \Theta \]
514 \begin_inset Formula \( x_{0} \)
518 \begin_inset Formula \( y_{0} \)
521 is the star center, and
522 \begin_inset Formula \( \Theta \)
525 is the angle with the
526 \begin_inset Formula \( x \)
530 Results of this task are shown in table\SpecialChar ~
532 \begin_inset LatexCommand \ref{tbl-2}
537 It is not clear how these sorts of analyses may affect determination of
539 \begin_inset Formula \( M_{\sun } \)
542 , but the assumption is that the alternate results should be less than 90
554 out of phase with previous values.
555 We have no observations of
576 of the electronically submitted abstracts for AAS meetings are error-free.
577 \layout Acknowledgements
581 We are grateful to V.
586 Phillips for doing the math in section\SpecialChar ~
588 \begin_inset LatexCommand \ref{bozomath}
593 More information on the AASTeX macros package are available at
595 \begin_inset LatexCommand \url{http://www.aas.org/publications/aastex}
607 anchor{ftp://www.aas.org/pubs/}{AAS ftp site}
618 Consider once again a task that computes profile parameters for a modified
619 Lorentzian of the form
620 \begin_inset Formula \begin{equation}
621 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
630 \begin_inset Formula \[
631 d_{1}=\frac{3}{4}\sqrt{\left( \begin{array}{c}
632 \frac{x_{1}}{R_{maj}}
633 \end{array}\right) ^{2}+\left( \begin{array}{c}
634 \frac{y_{1}}{R_{min}}
635 \end{array}\right) ^{2}}\]
640 \begin_inset Formula \begin{equation}
641 d_{2}=\case {3}{4}\sqrt{\left( \begin{array}{c}
642 \frac{x_{1}}{PR_{maj}}
643 \end{array}\right) ^{2}+\left( \begin{array}{c}
644 \case {y_{1}}{PR_{min}}
645 \end{array}\right) ^{2}}
651 \begin_inset Formula \begin{eqnarray}
652 x_{1} & = & (x-x_{0})\cos \Theta +(y-y_{0})\sin \Theta \\
653 y_{1} & = & -(x-x_{0})\sin \Theta +(y-y_{0})\cos \Theta
661 For completeness, here is one last equation.
663 \begin_inset Formula \begin{equation}
671 \bibitem [Auri\`ere(1982)]{aur82}
687 \bibitem [Canizares et al.(1978)]{can78}
692 A., Liller, W., and McClintock, J.
706 \bibitem [Djorgovski and King(1984)]{djo84}
708 Djorgovski, S., and King, I.
722 \bibitem [Hagiwara and Zeppenfeld(1986)]{hag86}
724 Hagiwara, K., and Zeppenfeld, D.
725 1986, Nucl.Phys., 274, 1
727 \bibitem [Harris and van den Bergh(1984)]{har84}
730 E., and van den Bergh, S.
743 \bibitem [H\`enon(1961)]{hen61}
747 1961, Ann.d'Ap., 24, 369
749 \bibitem [King(1966)]{kin66}
765 \bibitem [King(1975)]{kin75}
769 1975, Dynamics of Stellar Systems, A.
770 Hayli, Dordrecht: Reidel, 1975, 99
772 \bibitem [King et al.(1968)]{kin68}
775 R., Hedemann, E., Hodge, S.
790 \bibitem [Kron et al.(1984)]{kro84}
794 V., and Wasserman, L.
808 \bibitem [Lynden-Bell and Wood(1968)]{lyn68}
810 Lynden-Bell, D., and Wood, R.
823 \bibitem [Newell and O'Neil(1978)]{new78}
840 \bibitem [Ortolani et al.(1985)]{ort85}
842 Ortolani, S., Rosino, L., and Sandage, A.
855 \bibitem [Peterson(1976)]{pet76}
871 \bibitem [Spitzer(1985)]{spi85}
874 1985, Dynamics of Star Clusters, J.
876 Hut, Dordrecht: Reidel, 109
893 \begin_inset LatexCommand \label{fig1}
897 This is the first figure and it uses sgi9259.eps as its EPS figure file.
901 \begin_inset LatexCommand \label{fig2}
905 This is an example of a long figure caption that must be set as a paragraph.
906 The processor has to buffer the text of the caption, so it is good not
907 to be too wordy, but that would make for poor communication as well.
912 \begin_inset LatexCommand \label{fig3}
916 This figure has no associated EPS file, so the optional parameter is omitted.
933 \begin_inset Float table
938 \added_space_bottom medskip
939 Terribly relevant tabular information.
940 \begin_inset LatexCommand \label{tbl-2}
949 <lyxtabular version="3" rows="7" columns="13">
950 <features endhead="-1" endfirsthead="-1" endfoot="-1" endlastfoot="-1">
951 <column alignment="center" valignment="top" width="0pt">
952 <column alignment="right" valignment="top" width="0pt">
953 <column alignment="right" valignment="top" width="0pt">
954 <column alignment="right" valignment="top" width="0pt">
955 <column alignment="right" valignment="top" width="0pt">
956 <column alignment="right" valignment="top" width="0pt">
957 <column alignment="right" valignment="top" width="0pt">
958 <column alignment="right" valignment="top" width="0pt">
959 <column alignment="right" valignment="top" width="0pt">
960 <column alignment="right" valignment="top" width="0pt">
961 <column alignment="right" valignment="top" width="0pt">
962 <column alignment="right" valignment="top" width="0pt">
963 <column alignment="center" valignment="top" width="0pt">
965 <cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
973 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
981 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
987 \begin_inset Formula \( d_{x} \)
993 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
999 \begin_inset Formula \( d_{y} \)
1005 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1011 \begin_inset Formula \( n \)
1017 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1023 \begin_inset Formula \( \chi ^{2} \)
1029 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1035 \begin_inset Formula \( R_{maj} \)
1041 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1047 \begin_inset Formula \( R_{min} \)
1053 <cell multicolumn="1" alignment="center" valignment="top" usebox="none">
1059 \begin_inset Formula \( P \)
1075 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1081 \begin_inset Formula \( PR_{maj} \)
1087 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1093 \begin_inset Formula \( PR_{min} \)
1099 <cell multicolumn="1" alignment="center" valignment="top" usebox="none">
1105 \begin_inset Formula \( \Theta \)
1121 <cell alignment="center" valignment="top" usebox="none">
1131 <cell alignment="center" valignment="top" usebox="none">
1151 <cell alignment="right" valignment="top" usebox="none">
1159 <cell alignment="right" valignment="top" usebox="none">
1167 <cell alignment="right" valignment="top" usebox="none">
1175 <cell alignment="right" valignment="top" usebox="none">
1183 <cell alignment="right" valignment="top" usebox="none">
1191 <cell alignment="right" valignment="top" usebox="none">
1199 <cell alignment="right" valignment="top" usebox="none">
1207 <cell alignment="right" valignment="top" usebox="none">
1215 <cell alignment="right" valignment="top" usebox="none">
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1231 <cell alignment="right" valignment="top" usebox="none">
1239 <cell alignment="right" valignment="top" usebox="none">
1249 <cell alignment="center" valignment="top" usebox="none">
1257 <cell alignment="right" valignment="top" usebox="none">
1265 <cell alignment="right" valignment="top" usebox="none">
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1451 <cell alignment="right" valignment="top" usebox="none">
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1683 <cell alignment="right" valignment="top" usebox="none">
1693 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
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1812 tablenotetext{a}{Sample footnote for table~
1814 ref{tbl-2} that was generated with the
1822 tablenotetext{b}{Yet another sample footnote for table~
1828 tablenotetext{c}{Another sample footnote for table~
1834 \layout TableComments
1836 We can also attach a long-ish paragraph of explanatory material to a table.
1839 tablerefs to append a list of references.
1840 The following references were from a different table: I've patched them
1841 in here to show how they look, but don't take them too seriously---I certainly
1845 (1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
1846 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
1847 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
1848 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
1849 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
1850 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
1851 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
1852 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
1853 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
1855 1991; (26) Olsen 1983.
1865 % LyX can load deluxetable files verbatim.
1866 To see this in action, uncomment the following
1881 \begin_inset Quotes eld
1893 \begin_inset Quotes erd
1901 ) and copy the file table.tex from the AASTeX 5.0 distribution into
1912 % the same directory as this file.
1927 \begin_inset Include \input{table}