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11 \shorttitle{Djorgovski et al.}
12 \shortauthors{Collapsed Cores in Globular Clusters}
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86 \begin_layout Standard
87 \begin_inset Note Note
90 \begin_layout Plain Layout
93 Sample document for contributions to journals of the American Astronomical
96 \begin_inset CommandInset href
98 name "AASTeX user guide"
99 target "http://journals.aas.org/authors/aastex/aasguide.html"
104 for details on how to use AASTeX.
108 \begin_layout Plain Layout
111 An up-to-date sample LaTeX article using AASTeX is available under
112 \begin_inset Flex URL
115 \begin_layout Plain Layout
117 http://journals.aas.org/authors/aastex/aastex.html#_download
128 \begin_inset Note Note
131 \begin_layout Plain Layout
134 The AASTeX document class uses a different name since version 6.0 (2016/02/16).
135 Depending on the version installed on your system, set the document class
136 to American Astronomical Society (AASTeX v.
137 6) or American Astronomical Society (AASTeX v.
142 \begin_layout Plain Layout
147 \begin_inset Flex URL
150 \begin_layout Plain Layout
152 http://journals.aas.org/authors/aastex/revisionhistory.html
157 ) are only partially supported by LyX (see
158 \begin_inset Flex URL
161 \begin_layout Plain Layout
163 https://www.lyx.org/trac/ticket/10027
177 Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AAS\SpecialChar TeX
183 Djorgovski and Ivan R.
187 \begin_layout Affiliation
188 Astronomy Department, University of California, Berkeley, CA 94720
191 \begin_layout Altaffilation
193 \begin_inset Newline newline
196 Cerro Tololo Inter-American Observatory.
197 \begin_inset Newline newline
200 CTIO is operated by AURA Inc.
201 \begin_inset space \space{}
204 under contract to the National Science Foundation.
207 \begin_layout Altaffilation
208 Society of Fellows, Harvard University.
211 \begin_layout Altaffilation
212 present address: Center for Astrophysics
213 \begin_inset Newline newline
216 60 Garden Street, Cambridge, MA 02138
225 \begin_layout Affiliation
226 National Optical Astronomy Observatories, Tucson, AZ 85719
229 \begin_layout Altaffilation
230 Visiting Programmer, Space Telescope Science Institute
233 \begin_layout Altaffilation
234 Patron, Alonso's Bar and Grill
247 \begin_layout Affiliation
248 Space Telescope Science Institute, Baltimore, MD 21218
251 \begin_layout Altaffilation
252 Patron, Alonso's Bar and Grill
255 \begin_layout Abstract
256 This is a preliminary report on surface photometry of the major fraction
257 of known globular clusters, to see which of them show the signs of a collapsed
259 We also explore some diversionary mathematics and recreational tables.
263 \begin_layout Keywords
264 clusters: globular, peanut—bosons: bozos
267 \begin_layout Section
271 \begin_layout Standard
272 A focal problem today in the dynamics of globular clusters is core collapse.
273 It has been predicted by theory for decades
274 \begin_inset CommandInset citation
276 key "hen61,lyn68,spi85"
281 , but observation has been less alert to the phenomenon.
282 For many years the central brightness peak in M15
283 \begin_inset CommandInset citation
290 seemed a unique anomaly.
292 \begin_inset CommandInset citation
299 suggested a central peak in NGC 6397, and a limited photographic survey
301 \begin_inset CommandInset citation
309 found three more cases, including NGC 6624, whose sharp center had often
311 \begin_inset CommandInset citation
322 \begin_layout Section
326 \begin_layout Standard
327 All our observations were short direct exposures with CCD's.
328 At Lick Observatory we used a TI 500
329 \begin_inset Formula $\times$
332 500 chip and a GEC 575
333 \begin_inset Formula $\times$
336 385, on the 1-m Nickel reflector.
337 The only filter available at Lick was red.
338 At CTIO we used a GEC 575
339 \begin_inset Formula $\times$
343 \begin_inset Formula $B,V,$
347 \begin_inset Formula $R$
350 filters, and an RCA 512
351 \begin_inset Formula $\times$
355 \begin_inset Formula $U,B,V,R,$
359 \begin_inset Formula $I$
362 filters, on the 1.5-m reflector.
363 In the CTIO observations we tried to concentrate on the shortest practicable
364 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
365 often kept us from observing in
366 \begin_inset Formula $U$
370 \begin_inset Formula $B$
374 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
375 sizes were around 3 arcmin.
378 \begin_layout Standard
379 The CCD images are unfortunately not always suitable, for very poor clusters
380 or for clusters with large cores.
381 Since the latter are easily studied by other means, we augmented our own
382 CCD profiles by collecting from the literature a number of star-count profiles
384 \begin_inset CommandInset citation
386 key "kin68,pet76,har84,ort85"
391 , as well as photoelectric profiles
392 \begin_inset CommandInset citation
399 and electronographic profiles
400 \begin_inset CommandInset citation
408 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
411 \begin_layout Section
415 \begin_layout Standard
416 It has been realized that helicity amplitudes provide a convenient means
421 \begin_layout Plain Layout
422 Footnotes can be inserted like this.
428 These amplitude-level techniques are particularly convenient for calculations
429 involving many Feynman diagrams, where the usual trace techniques for the
430 amplitude squared becomes unwieldy.
431 Our calculations use the helicity techniques developed by other authors
433 \begin_inset CommandInset citation
440 ; we briefly summarize below.
443 \begin_layout Subsection
447 \begin_layout Standard
448 \begin_inset CommandInset label
457 \begin_layout Standard
458 A tree-level amplitude in
459 \begin_inset Formula $e^{+}e^{-}$
462 collisions can be expressed in terms of fermion strings of the form
465 \bar{v}(p_{2},\sigma_{2})P_{-\tau}\hat{a}_{1}\hat{a}_{2}\cdots\hat{a}_{n}u(p_{1},\sigma_{1}),
471 \begin_inset Formula $p$
475 \begin_inset Formula $\sigma$
479 \begin_inset Formula $e^{\pm}$
482 four-momenta and helicities
483 \begin_inset Formula $(\sigma=\pm1)$
487 \begin_inset Formula $\hat{a}_{i}=a_{i}^{\mu}\gamma_{\nu}$
491 \begin_inset Formula $P_{\tau}=\frac{1}{2}(1+\tau\gamma_{5})$
494 is a chirality projection operator
495 \begin_inset Formula $(\tau=\pm1)$
500 \begin_inset Formula $a_{i}^{\mu}$
503 may be formed from particle four-momenta, gauge-boson polarization vectors
504 or fermion strings with an uncontracted Lorentz index associated with final-sta
508 \begin_layout NoteToEditor
509 Figures 1 and 2 should appear side-by-side in print
512 \begin_layout Standard
513 In the chiral representation the
514 \begin_inset Formula $\gamma$
517 matrices are expressed in terms of
518 \begin_inset Formula $2\times2$
522 \begin_inset Formula $\sigma$
525 and the unit matrix 1 as
528 \gamma^{\mu} & = & \left(\begin{array}{cc}
529 0 & \sigma_{+}^{\mu}\\
531 \end{array}\right),\gamma^{5}=\left(\begin{array}{cc}
534 \end{array}\right),\\
535 \sigma_{\pm}^{\mu} & = & ({\textbf{1}},\pm\sigma),
543 \hat{a}=\left(\begin{array}{cc}
546 \end{array}\right),(\hat{a})_{\pm}=a_{\mu}\sigma_{\pm}^{\mu},
551 The spinors are expressed in terms of two-component Weyl spinors as
554 u=\left(\begin{array}{c}
557 \end{array}\right),v={\textbf{(}}\vdag_{+}{\textbf{,}}\vdag_{-}{\textbf{)}}.
565 \begin_layout Standard
566 The Weyl spinors are given in terms of helicity eigenstates
567 \begin_inset Formula $\chi_{\lambda}(p)$
571 \begin_inset Formula $\lambda=\pm1$
577 \begin_layout MathLetters
580 u(p,\lambda)_{\pm} & = & (E\pm\lambda|{\textbf{p}}|)^{1/2}\chi_{\lambda}(p),\\
581 v(p,\lambda)_{\pm} & = & \pm\lambda(E\mp\lambda|{\textbf{p}}|)^{1/2}\chi_{-\lambda}(p)
589 \begin_layout Section
590 Floating material and so forth
593 \begin_layout Standard
594 Consider a task that computes profile parameters for a modified Lorentzian
598 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
606 d_{1}=\sqrt{\left(\begin{array}{c}
607 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
608 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}
616 d_{2}=\sqrt{\left(\begin{array}{c}
617 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
618 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}
626 x_{1}=(x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta
634 y_{1}=-(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta
642 \begin_layout Standard
644 \begin_inset Formula $x_{0}$
648 \begin_inset Formula $y_{0}$
651 is the star center, and
652 \begin_inset Formula $\Theta$
655 is the angle with the
656 \begin_inset Formula $x$
660 Results of this task are shown in table
665 \begin_inset CommandInset ref
672 It is not clear how these sorts of analyses may affect determination of
674 \begin_inset Formula $M_{\text{\sun}}$
677 , but the assumption is that the alternate results should be less than 90°
678 out of phase with previous values.
679 We have no observations of
682 \begin_layout Standard
686 \begin_layout Plain Layout
702 \begin_inset Formula $\nicefrac{4}{5}$
705 of the electronically submitted abstracts for AAS meetings are error-free.
709 \begin_layout Acknowledgements
710 We are grateful to V.
715 Phillips for doing the math in section
720 \begin_inset CommandInset ref
727 More information on the AAS\SpecialChar TeX
728 macros package are available at
729 \begin_inset Flex URL
732 \begin_layout Plain Layout
734 http://www.aas.org/publications/aastex
740 \begin_inset CommandInset href
743 target "ftp://www.aas.org/pubs/AAS ftp site"
751 \begin_layout Software
752 IRAF, AIPS, Astropy, ...
755 \begin_layout Appendix
759 \begin_layout Section
763 \begin_layout Standard
764 Consider once again a task that computes profile parameters for a modified
765 Lorentzian of the form
768 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
776 \begin_layout MathLetters
779 d_{1}=\frac{3}{4}\sqrt{\left(\begin{array}{c}
780 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
781 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}
789 d_{2}=\case{3}{4}\sqrt{\left(\begin{array}{c}
790 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
791 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}
799 x_{1} & = & (x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta\\
800 y_{1} & = & -(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta
808 \begin_layout Standard
809 For completeness, here is one last equation.
821 \begin_layout References
822 \begin_inset CommandInset bibitem
824 label "Auri\\`ere(1982)"
835 \begin_layout Plain Layout
847 \begin_layout References
848 \begin_inset CommandInset bibitem
850 label "Canizares et al.(1978)"
859 A., Liller, W., and McClintock, J.
865 \begin_layout Plain Layout
877 \begin_layout References
878 \begin_inset CommandInset bibitem
880 label "Djorgovski and King(1984)"
886 Djorgovski, S., and King, I.
891 \begin_layout Standard
895 \begin_layout Plain Layout
912 \begin_layout References
913 \begin_inset CommandInset bibitem
915 label "Hagiwara and Zeppenfeld(1986)"
921 Hagiwara, K., and Zeppenfeld, D.
922 1986, Nucl.Phys., 274, 1
925 \begin_layout References
926 \begin_inset CommandInset bibitem
928 label "Harris and van den Bergh(1984)"
935 E., and van den Bergh, S.
940 \begin_layout Plain Layout
952 \begin_layout References
953 \begin_inset CommandInset bibitem
955 label "H\\`enon(1961)"
962 1961, Ann.d'Ap., 24, 369
965 \begin_layout References
966 \begin_inset CommandInset bibitem
980 \begin_layout Plain Layout
992 \begin_layout References
993 \begin_inset CommandInset bibitem
1003 1975, Dynamics of Stellar Systems, A.
1004 Hayli, Dordrecht: Reidel, 1975, 99
1007 \begin_layout References
1008 \begin_inset CommandInset bibitem
1009 LatexCommand bibitem
1010 label "King et al.(1968)"
1017 R., Hedemann, E., Hodge, S.
1024 \begin_layout Plain Layout
1036 \begin_layout References
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1038 LatexCommand bibitem
1039 label "Kron et al.(1984)"
1047 V., and Wasserman, L.
1053 \begin_layout Plain Layout
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1067 LatexCommand bibitem
1068 label "Lynden-Bell and Wood(1968)"
1074 Lynden-Bell, D., and Wood, R.
1076 \begin_inset Preview
1078 \begin_layout Standard
1082 \begin_layout Plain Layout
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1115 \begin_layout Plain Layout
1127 \begin_layout References
1128 \begin_inset CommandInset bibitem
1129 LatexCommand bibitem
1130 label "Ortolani et al.(1985)"
1136 Ortolani, S., Rosino, L., and Sandage, A.
1141 \begin_layout Plain Layout
1153 \begin_layout References
1154 \begin_inset CommandInset bibitem
1155 LatexCommand bibitem
1156 label "Peterson(1976)"
1168 \begin_layout Plain Layout
1180 \begin_layout References
1181 \begin_inset CommandInset bibitem
1182 LatexCommand bibitem
1183 label "Spitzer(1985)"
1190 1985, Dynamics of Star Clusters, J.
1192 Hut, Dordrecht: Reidel, 109
1195 \begin_layout Standard
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2214 <cell alignment="right" valignment="top" usebox="none">
2217 \begin_layout Plain Layout
2223 <cell alignment="right" valignment="top" usebox="none">
2226 \begin_layout Plain Layout
2232 <cell alignment="right" valignment="top" usebox="none">
2235 \begin_layout Plain Layout
2249 \begin_layout Table note
2254 \begin_layout Plain Layout
2261 Sample footnote for table
2262 \begin_inset space ~
2266 \begin_inset CommandInset ref
2272 that was generated with the \SpecialChar LaTeX
2276 \begin_layout Table note
2281 \begin_layout Plain Layout
2288 Yet another sample footnote for table
2289 \begin_inset space ~
2293 \begin_inset CommandInset ref
2302 \begin_layout Table note
2307 \begin_layout Plain Layout
2314 Another sample footnote for table
2315 \begin_inset space ~
2319 \begin_inset CommandInset ref
2328 \begin_layout TableComments
2329 We can also attach a long-ish paragraph of explanatory material to a table.
2332 tablerefs to append a list of references.
2333 The following references were from a different table: I've patched them
2334 in here to show how they look, but don't take them too seriously—I certainly
2338 \begin_layout TableRefs
2339 (1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
2340 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
2341 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
2342 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
2343 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
2344 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
2345 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
2346 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
2347 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
2349 1991; (26) Olsen 1983.
2357 \begin_layout Standard
2358 \begin_inset Note Note
2361 \begin_layout Plain Layout
2362 LyX can load deluxetable files verbatim.
2363 To see this in action, dissolve the following Note and copy the file table.tex
2364 from the AASTeX 5.0 distribution into the same directory as this file.
2372 \begin_layout Standard
2373 \begin_inset Note Note
2376 \begin_layout Plain Layout
2377 \begin_inset CommandInset include