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11 \shorttitle{Djorgovski et al.}
12 \shortauthors{Collapsed Cores in Globular Clusters}
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92 \begin_layout Standard
93 \begin_inset Note Note
96 \begin_layout Plain Layout
99 Sample document for contributions to journals of the American Astronomical
102 \begin_inset CommandInset href
104 name "AASTeX user guide"
105 target "http://journals.aas.org/authors/aastex/aasguide.html"
110 for details on how to use AASTeX.
114 \begin_layout Plain Layout
117 An up-to-date sample LaTeX article using AASTeX is available under
118 \begin_inset Flex URL
121 \begin_layout Plain Layout
123 http://journals.aas.org/authors/aastex/aastex.html#_download
134 \begin_inset Note Note
137 \begin_layout Plain Layout
140 The AASTeX document class uses a different name since version 6.0 (2016/02/16).
141 Depending on the version installed on your system, set the document class
142 to American Astronomical Society (AASTeX v.
143 6) or American Astronomical Society (AASTeX v.
148 \begin_layout Plain Layout
153 \begin_inset Flex URL
156 \begin_layout Plain Layout
158 http://journals.aas.org/authors/aastex/revisionhistory.html
163 ) are only partially supported by LyX (see
164 \begin_inset Flex URL
167 \begin_layout Plain Layout
169 https://www.lyx.org/trac/ticket/10027
183 Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AAS\SpecialChar TeX
189 Djorgovski and Ivan R.
193 \begin_layout Affiliation
194 Astronomy Department, University of California, Berkeley, CA 94720
197 \begin_layout Altaffilation
199 \begin_inset Newline newline
202 Cerro Tololo Inter-American Observatory.
203 \begin_inset Newline newline
206 CTIO is operated by AURA Inc.
207 \begin_inset space \space{}
210 under contract to the National Science Foundation.
213 \begin_layout Altaffilation
214 Society of Fellows, Harvard University.
217 \begin_layout Altaffilation
218 present address: Center for Astrophysics
219 \begin_inset Newline newline
222 60 Garden Street, Cambridge, MA 02138
231 \begin_layout Affiliation
232 National Optical Astronomy Observatories, Tucson, AZ 85719
235 \begin_layout Altaffilation
236 Visiting Programmer, Space Telescope Science Institute
239 \begin_layout Altaffilation
240 Patron, Alonso's Bar and Grill
253 \begin_layout Affiliation
254 Space Telescope Science Institute, Baltimore, MD 21218
257 \begin_layout Altaffilation
258 Patron, Alonso's Bar and Grill
261 \begin_layout Abstract
262 This is a preliminary report on surface photometry of the major fraction
263 of known globular clusters, to see which of them show the signs of a collapsed
265 We also explore some diversionary mathematics and recreational tables.
269 \begin_layout Keywords
270 clusters: globular, peanut—bosons: bozos
273 \begin_layout Section
277 \begin_layout Standard
278 A focal problem today in the dynamics of globular clusters is core collapse.
279 It has been predicted by theory for decades
280 \begin_inset CommandInset citation
282 key "hen61,lyn68,spi85"
287 , but observation has been less alert to the phenomenon.
288 For many years the central brightness peak in M15
289 \begin_inset CommandInset citation
296 seemed a unique anomaly.
298 \begin_inset CommandInset citation
305 suggested a central peak in NGC 6397, and a limited photographic survey
307 \begin_inset CommandInset citation
315 found three more cases, including NGC 6624, whose sharp center had often
317 \begin_inset CommandInset citation
328 \begin_layout Section
332 \begin_layout Standard
333 All our observations were short direct exposures with CCD's.
334 At Lick Observatory we used a TI 500
335 \begin_inset Formula $\times$
338 500 chip and a GEC 575
339 \begin_inset Formula $\times$
342 385, on the 1-m Nickel reflector.
343 The only filter available at Lick was red.
344 At CTIO we used a GEC 575
345 \begin_inset Formula $\times$
349 \begin_inset Formula $B,V,$
353 \begin_inset Formula $R$
356 filters, and an RCA 512
357 \begin_inset Formula $\times$
361 \begin_inset Formula $U,B,V,R,$
365 \begin_inset Formula $I$
368 filters, on the 1.5-m reflector.
369 In the CTIO observations we tried to concentrate on the shortest practicable
370 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
371 often kept us from observing in
372 \begin_inset Formula $U$
376 \begin_inset Formula $B$
380 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
381 sizes were around 3 arcmin.
384 \begin_layout Standard
385 The CCD images are unfortunately not always suitable, for very poor clusters
386 or for clusters with large cores.
387 Since the latter are easily studied by other means, we augmented our own
388 CCD profiles by collecting from the literature a number of star-count profiles
390 \begin_inset CommandInset citation
392 key "kin68,pet76,har84,ort85"
397 , as well as photoelectric profiles
398 \begin_inset CommandInset citation
405 and electronographic profiles
406 \begin_inset CommandInset citation
414 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
417 \begin_layout Section
421 \begin_layout Standard
422 It has been realized that helicity amplitudes provide a convenient means
427 \begin_layout Plain Layout
428 Footnotes can be inserted like this.
434 These amplitude-level techniques are particularly convenient for calculations
435 involving many Feynman diagrams, where the usual trace techniques for the
436 amplitude squared becomes unwieldy.
437 Our calculations use the helicity techniques developed by other authors
439 \begin_inset CommandInset citation
446 ; we briefly summarize below.
449 \begin_layout Subsection
453 \begin_layout Standard
454 \begin_inset CommandInset label
463 \begin_layout Standard
464 A tree-level amplitude in
465 \begin_inset Formula $e^{+}e^{-}$
468 collisions can be expressed in terms of fermion strings of the form
471 \bar{v}(p_{2},\sigma_{2})P_{-\tau}\hat{a}_{1}\hat{a}_{2}\cdots\hat{a}_{n}u(p_{1},\sigma_{1}),
477 \begin_inset Formula $p$
481 \begin_inset Formula $\sigma$
485 \begin_inset Formula $e^{\pm}$
488 four-momenta and helicities
489 \begin_inset Formula $(\sigma=\pm1)$
493 \begin_inset Formula $\hat{a}_{i}=a_{i}^{\mu}\gamma_{\nu}$
497 \begin_inset Formula $P_{\tau}=\frac{1}{2}(1+\tau\gamma_{5})$
500 is a chirality projection operator
501 \begin_inset Formula $(\tau=\pm1)$
506 \begin_inset Formula $a_{i}^{\mu}$
509 may be formed from particle four-momenta, gauge-boson polarization vectors
510 or fermion strings with an uncontracted Lorentz index associated with final-sta
514 \begin_layout NoteToEditor
515 Figures 1 and 2 should appear side-by-side in print
518 \begin_layout Standard
519 In the chiral representation the
520 \begin_inset Formula $\gamma$
523 matrices are expressed in terms of
524 \begin_inset Formula $2\times2$
528 \begin_inset Formula $\sigma$
531 and the unit matrix 1 as
534 \gamma^{\mu} & = & \left(\begin{array}{cc}
535 0 & \sigma_{+}^{\mu}\\
537 \end{array}\right),\gamma^{5}=\left(\begin{array}{cc}
540 \end{array}\right),\\
541 \sigma_{\pm}^{\mu} & = & ({\textbf{1}},\pm\sigma),
549 \hat{a}=\left(\begin{array}{cc}
552 \end{array}\right),(\hat{a})_{\pm}=a_{\mu}\sigma_{\pm}^{\mu},
557 The spinors are expressed in terms of two-component Weyl spinors as
560 u=\left(\begin{array}{c}
563 \end{array}\right),v={\textbf{(}}\vdag_{+}{\textbf{,}}\vdag_{-}{\textbf{)}}.
571 \begin_layout Standard
572 The Weyl spinors are given in terms of helicity eigenstates
573 \begin_inset Formula $\chi_{\lambda}(p)$
577 \begin_inset Formula $\lambda=\pm1$
583 \begin_layout MathLetters
586 u(p,\lambda)_{\pm} & = & (E\pm\lambda|{\textbf{p}}|)^{1/2}\chi_{\lambda}(p),\\
587 v(p,\lambda)_{\pm} & = & \pm\lambda(E\mp\lambda|{\textbf{p}}|)^{1/2}\chi_{-\lambda}(p)
595 \begin_layout Section
596 Floating material and so forth
599 \begin_layout Standard
600 Consider a task that computes profile parameters for a modified Lorentzian
604 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
612 d_{1}=\sqrt{\left(\begin{array}{c}
613 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
614 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}
622 d_{2}=\sqrt{\left(\begin{array}{c}
623 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
624 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}
632 x_{1}=(x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta
640 y_{1}=-(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta
648 \begin_layout Standard
650 \begin_inset Formula $x_{0}$
654 \begin_inset Formula $y_{0}$
657 is the star center, and
658 \begin_inset Formula $\Theta$
661 is the angle with the
662 \begin_inset Formula $x$
666 Results of this task are shown in table
671 \begin_inset CommandInset ref
678 It is not clear how these sorts of analyses may affect determination of
680 \begin_inset Formula $M_{\text{\sun}}$
683 , but the assumption is that the alternate results should be less than 90°
684 out of phase with previous values.
685 We have no observations of
688 \begin_layout Standard
692 \begin_layout Plain Layout
708 \begin_inset Formula $\nicefrac{4}{5}$
711 of the electronically submitted abstracts for AAS meetings are error-free.
715 \begin_layout Acknowledgements
716 We are grateful to V.
721 Phillips for doing the math in section
726 \begin_inset CommandInset ref
733 More information on the AAS\SpecialChar TeX
734 macros package are available at
735 \begin_inset Flex URL
738 \begin_layout Plain Layout
740 http://www.aas.org/publications/aastex
746 \begin_inset CommandInset href
749 target "ftp://www.aas.org/pubs/AAS ftp site"
757 \begin_layout Software
758 IRAF, AIPS, Astropy, ...
761 \begin_layout Appendix
765 \begin_layout Section
769 \begin_layout Standard
770 Consider once again a task that computes profile parameters for a modified
771 Lorentzian of the form
774 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
782 \begin_layout MathLetters
785 d_{1}=\frac{3}{4}\sqrt{\left(\begin{array}{c}
786 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
787 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}
795 d_{2}=\case{3}{4}\sqrt{\left(\begin{array}{c}
796 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
797 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}
805 x_{1} & = & (x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta\\
806 y_{1} & = & -(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta
814 \begin_layout Standard
815 For completeness, here is one last equation.
827 \begin_layout References
828 \begin_inset CommandInset bibitem
830 label "Auri\\`ere(1982)"
841 \begin_layout Plain Layout
853 \begin_layout References
854 \begin_inset CommandInset bibitem
856 label "Canizares et al.(1978)"
865 A., Liller, W., and McClintock, J.
871 \begin_layout Plain Layout
883 \begin_layout References
884 \begin_inset CommandInset bibitem
886 label "Djorgovski and King(1984)"
892 Djorgovski, S., and King, I.
897 \begin_layout Standard
901 \begin_layout Plain Layout
918 \begin_layout References
919 \begin_inset CommandInset bibitem
921 label "Hagiwara and Zeppenfeld(1986)"
927 Hagiwara, K., and Zeppenfeld, D.
928 1986, Nucl.Phys., 274, 1
931 \begin_layout References
932 \begin_inset CommandInset bibitem
934 label "Harris and van den Bergh(1984)"
941 E., and van den Bergh, S.
946 \begin_layout Plain Layout
958 \begin_layout References
959 \begin_inset CommandInset bibitem
961 label "H\\`enon(1961)"
968 1961, Ann.d'Ap., 24, 369
971 \begin_layout References
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986 \begin_layout Plain Layout
998 \begin_layout References
999 \begin_inset CommandInset bibitem
1000 LatexCommand bibitem
1009 1975, Dynamics of Stellar Systems, A.
1010 Hayli, Dordrecht: Reidel, 1975, 99
1013 \begin_layout References
1014 \begin_inset CommandInset bibitem
1015 LatexCommand bibitem
1016 label "King et al.(1968)"
1023 R., Hedemann, E., Hodge, S.
1030 \begin_layout Plain Layout
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1044 LatexCommand bibitem
1045 label "Kron et al.(1984)"
1053 V., and Wasserman, L.
1059 \begin_layout Plain Layout
1071 \begin_layout References
1072 \begin_inset CommandInset bibitem
1073 LatexCommand bibitem
1074 label "Lynden-Bell and Wood(1968)"
1080 Lynden-Bell, D., and Wood, R.
1082 \begin_inset Preview
1084 \begin_layout Standard
1088 \begin_layout Plain Layout
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1108 label "Newell and O'Neil(1978)"
1121 \begin_layout Plain Layout
1133 \begin_layout References
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1135 LatexCommand bibitem
1136 label "Ortolani et al.(1985)"
1142 Ortolani, S., Rosino, L., and Sandage, A.
1147 \begin_layout Plain Layout
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1161 LatexCommand bibitem
1162 label "Peterson(1976)"
1174 \begin_layout Plain Layout
1186 \begin_layout References
1187 \begin_inset CommandInset bibitem
1188 LatexCommand bibitem
1189 label "Spitzer(1985)"
1196 1985, Dynamics of Star Clusters, J.
1198 Hut, Dordrecht: Reidel, 109
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2204 <cell alignment="right" valignment="top" usebox="none">
2207 \begin_layout Plain Layout
2213 <cell alignment="right" valignment="top" usebox="none">
2216 \begin_layout Plain Layout
2222 <cell alignment="right" valignment="top" usebox="none">
2225 \begin_layout Plain Layout
2231 <cell alignment="right" valignment="top" usebox="none">
2234 \begin_layout Plain Layout
2240 <cell alignment="right" valignment="top" usebox="none">
2243 \begin_layout Plain Layout
2257 \begin_layout Table note
2262 \begin_layout Plain Layout
2269 Sample footnote for table
2270 \begin_inset space ~
2274 \begin_inset CommandInset ref
2280 that was generated with the \SpecialChar LaTeX
2284 \begin_layout Table note
2289 \begin_layout Plain Layout
2296 Yet another sample footnote for table
2297 \begin_inset space ~
2301 \begin_inset CommandInset ref
2310 \begin_layout Table note
2315 \begin_layout Plain Layout
2322 Another sample footnote for table
2323 \begin_inset space ~
2327 \begin_inset CommandInset ref
2336 \begin_layout TableComments
2337 We can also attach a long-ish paragraph of explanatory material to a table.
2340 tablerefs to append a list of references.
2341 The following references were from a different table: I've patched them
2342 in here to show how they look, but don't take them too seriously—I certainly
2346 \begin_layout TableRefs
2347 (1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
2348 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
2349 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
2350 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
2351 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
2352 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
2353 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
2354 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
2355 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
2357 1991; (26) Olsen 1983.
2365 \begin_layout Standard
2366 \begin_inset Note Note
2369 \begin_layout Plain Layout
2370 LyX can load deluxetable files verbatim.
2371 To see this in action, dissolve the following Note and copy the file table.tex
2372 from the AASTeX 5.0 distribution into the same directory as this file.
2380 \begin_layout Standard
2381 \begin_inset Note Note
2384 \begin_layout Plain Layout
2385 \begin_inset CommandInset include