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11 \shorttitle{Djorgovski et al.}
12 \shortauthors{Collapsed Cores in Globular Clusters}
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95 \begin_layout Standard
96 \begin_inset Note Note
99 \begin_layout Plain Layout
102 Sample document for contributions to journals of the American Astronomical
105 \begin_inset CommandInset href
107 name "AASTeX user guide"
108 target "http://journals.aas.org/authors/aastex/aasguide.html"
113 for details on how to use AASTeX.
117 \begin_layout Plain Layout
120 An up-to-date sample LaTeX article using AASTeX is available under
121 \begin_inset Flex URL
124 \begin_layout Plain Layout
126 http://journals.aas.org/authors/aastex/aastex.html#_download
137 \begin_inset Note Note
140 \begin_layout Plain Layout
143 The AASTeX document class uses a different name since version 6.0 (2016/02/16).
144 Depending on the version installed on your system, set the document class
145 to American Astronomical Society (AASTeX v.
146 6) or American Astronomical Society (AASTeX v.
151 \begin_layout Plain Layout
156 \begin_inset Flex URL
159 \begin_layout Plain Layout
161 http://journals.aas.org/authors/aastex/revisionhistory.html
166 ) are only partially supported by LyX (see
167 \begin_inset Flex URL
170 \begin_layout Plain Layout
172 https://www.lyx.org/trac/ticket/10027
186 Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AAS\SpecialChar TeX
192 Djorgovski and Ivan R.
196 \begin_layout Affiliation
197 Astronomy Department, University of California, Berkeley, CA 94720
200 \begin_layout Altaffilation
202 \begin_inset Newline newline
205 Cerro Tololo Inter-American Observatory.
206 \begin_inset Newline newline
209 CTIO is operated by AURA Inc.
210 \begin_inset space \space{}
213 under contract to the National Science Foundation.
216 \begin_layout Altaffilation
217 Society of Fellows, Harvard University.
220 \begin_layout Altaffilation
221 present address: Center for Astrophysics
222 \begin_inset Newline newline
225 60 Garden Street, Cambridge, MA 02138
234 \begin_layout Affiliation
235 National Optical Astronomy Observatories, Tucson, AZ 85719
238 \begin_layout Altaffilation
239 Visiting Programmer, Space Telescope Science Institute
242 \begin_layout Altaffilation
243 Patron, Alonso's Bar and Grill
256 \begin_layout Affiliation
257 Space Telescope Science Institute, Baltimore, MD 21218
260 \begin_layout Altaffilation
261 Patron, Alonso's Bar and Grill
264 \begin_layout Abstract
265 This is a preliminary report on surface photometry of the major fraction
266 of known globular clusters, to see which of them show the signs of a collapsed
268 We also explore some diversionary mathematics and recreational tables.
272 \begin_layout Keywords
273 clusters: globular, peanut—bosons: bozos
276 \begin_layout Section
280 \begin_layout Standard
281 A focal problem today in the dynamics of globular clusters is core collapse.
282 It has been predicted by theory for decades
283 \begin_inset CommandInset citation
285 key "hen61,lyn68,spi85"
290 , but observation has been less alert to the phenomenon.
291 For many years the central brightness peak in M15
292 \begin_inset CommandInset citation
299 seemed a unique anomaly.
301 \begin_inset CommandInset citation
308 suggested a central peak in NGC 6397, and a limited photographic survey
310 \begin_inset CommandInset citation
318 found three more cases, including NGC 6624, whose sharp center had often
320 \begin_inset CommandInset citation
331 \begin_layout Section
335 \begin_layout Standard
336 All our observations were short direct exposures with CCD's.
337 At Lick Observatory we used a TI 500
338 \begin_inset Formula $\times$
341 500 chip and a GEC 575
342 \begin_inset Formula $\times$
345 385, on the 1-m Nickel reflector.
346 The only filter available at Lick was red.
347 At CTIO we used a GEC 575
348 \begin_inset Formula $\times$
352 \begin_inset Formula $B,V,$
356 \begin_inset Formula $R$
359 filters, and an RCA 512
360 \begin_inset Formula $\times$
364 \begin_inset Formula $U,B,V,R,$
368 \begin_inset Formula $I$
371 filters, on the 1.5-m reflector.
372 In the CTIO observations we tried to concentrate on the shortest practicable
373 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
374 often kept us from observing in
375 \begin_inset Formula $U$
379 \begin_inset Formula $B$
383 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
384 sizes were around 3 arcmin.
387 \begin_layout Standard
388 The CCD images are unfortunately not always suitable, for very poor clusters
389 or for clusters with large cores.
390 Since the latter are easily studied by other means, we augmented our own
391 CCD profiles by collecting from the literature a number of star-count profiles
393 \begin_inset CommandInset citation
395 key "kin68,pet76,har84,ort85"
400 , as well as photoelectric profiles
401 \begin_inset CommandInset citation
408 and electronographic profiles
409 \begin_inset CommandInset citation
417 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
420 \begin_layout Section
424 \begin_layout Standard
425 It has been realized that helicity amplitudes provide a convenient means
430 \begin_layout Plain Layout
431 Footnotes can be inserted like this.
437 These amplitude-level techniques are particularly convenient for calculations
438 involving many Feynman diagrams, where the usual trace techniques for the
439 amplitude squared becomes unwieldy.
440 Our calculations use the helicity techniques developed by other authors
442 \begin_inset CommandInset citation
449 ; we briefly summarize below.
452 \begin_layout Subsection
456 \begin_layout Standard
457 \begin_inset CommandInset label
466 \begin_layout Standard
467 A tree-level amplitude in
468 \begin_inset Formula $e^{+}e^{-}$
471 collisions can be expressed in terms of fermion strings of the form
474 \bar{v}(p_{2},\sigma_{2})P_{-\tau}\hat{a}_{1}\hat{a}_{2}\cdots\hat{a}_{n}u(p_{1},\sigma_{1}),
480 \begin_inset Formula $p$
484 \begin_inset Formula $\sigma$
488 \begin_inset Formula $e^{\pm}$
491 four-momenta and helicities
492 \begin_inset Formula $(\sigma=\pm1)$
496 \begin_inset Formula $\hat{a}_{i}=a_{i}^{\mu}\gamma_{\nu}$
500 \begin_inset Formula $P_{\tau}=\frac{1}{2}(1+\tau\gamma_{5})$
503 is a chirality projection operator
504 \begin_inset Formula $(\tau=\pm1)$
509 \begin_inset Formula $a_{i}^{\mu}$
512 may be formed from particle four-momenta, gauge-boson polarization vectors
513 or fermion strings with an uncontracted Lorentz index associated with final-sta
517 \begin_layout NoteToEditor
518 Figures 1 and 2 should appear side-by-side in print
521 \begin_layout Standard
522 In the chiral representation the
523 \begin_inset Formula $\gamma$
526 matrices are expressed in terms of
527 \begin_inset Formula $2\times2$
531 \begin_inset Formula $\sigma$
534 and the unit matrix 1 as
537 \gamma^{\mu} & = & \left(\begin{array}{cc}
538 0 & \sigma_{+}^{\mu}\\
540 \end{array}\right),\gamma^{5}=\left(\begin{array}{cc}
543 \end{array}\right),\\
544 \sigma_{\pm}^{\mu} & = & (\textbf{1},\pm\sigma),
552 \hat{a}=\left(\begin{array}{cc}
555 \end{array}\right),(\hat{a})_{\pm}=a_{\mu}\sigma_{\pm}^{\mu},
560 The spinors are expressed in terms of two-component Weyl spinors as
563 u=\left(\begin{array}{c}
566 \end{array}\right),v=\textbf{(}v_{+}^{\dagger}\textbf{,}v_{-}^{\dagger}\textbf{)}.
574 \begin_layout Standard
575 The Weyl spinors are given in terms of helicity eigenstates
576 \begin_inset Formula $\chi_{\lambda}(p)$
580 \begin_inset Formula $\lambda=\pm1$
586 \begin_layout MathLetters
589 u(p,\lambda)_{\pm} & = & (E\pm\lambda|\textbf{p}|)^{1/2}\chi_{\lambda}(p),\\
590 v(p,\lambda)_{\pm} & = & \pm\lambda(E\mp\lambda|\textbf{p}|)^{1/2}\chi_{-\lambda}(p)
598 \begin_layout Section
599 Floating material and so forth
602 \begin_layout Standard
603 Consider a task that computes profile parameters for a modified Lorentzian
607 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
615 d_{1}=\sqrt{\left(\begin{array}{c}
616 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
617 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}
625 d_{2}=\sqrt{\left(\begin{array}{c}
626 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
627 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}
635 x_{1}=(x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta
643 y_{1}=-(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta
651 \begin_layout Standard
653 \begin_inset Formula $x_{0}$
657 \begin_inset Formula $y_{0}$
660 is the star center, and
661 \begin_inset Formula $\Theta$
664 is the angle with the
665 \begin_inset Formula $x$
669 Results of this task are shown in table
674 \begin_inset CommandInset ref
681 It is not clear how these sorts of analyses may affect determination of
683 \begin_inset Formula $M_{\text{\sun}}$
686 , but the assumption is that the alternate results should be less than 90°
687 out of phase with previous values.
688 We have no observations of
691 \begin_layout Standard
695 \begin_layout Plain Layout
711 \begin_inset Formula $\nicefrac{4}{5}$
714 of the electronically submitted abstracts for AAS meetings are error-free.
718 \begin_layout Acknowledgements
719 We are grateful to V.
724 Phillips for doing the math in section
729 \begin_inset CommandInset ref
736 More information on the AAS\SpecialChar TeX
737 macros package are available at
738 \begin_inset Flex URL
741 \begin_layout Plain Layout
743 http://www.aas.org/publications/aastex
749 \begin_inset CommandInset href
752 target "ftp://www.aas.org/pubs/AAS ftp site"
760 \begin_layout Software
761 IRAF, AIPS, Astropy, ...
764 \begin_layout Appendix
768 \begin_layout Section
772 \begin_layout Standard
773 Consider once again a task that computes profile parameters for a modified
774 Lorentzian of the form
777 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
785 \begin_layout MathLetters
788 d_{1}=\frac{3}{4}\sqrt{\left(\begin{array}{c}
789 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
790 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}
798 d_{2}=\case{3}{4}\sqrt{\left(\begin{array}{c}
799 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
800 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}
808 x_{1} & = & (x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta\\
809 y_{1} & = & -(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta
817 \begin_layout Standard
818 For completeness, here is one last equation.
830 \begin_layout Bibliography
831 \begin_inset CommandInset bibitem
833 label "Auri\\`ere(1982)"
844 \begin_layout Plain Layout
856 \begin_layout Bibliography
857 \begin_inset CommandInset bibitem
859 label "Canizares et al.(1978)"
868 A., Liller, W., and McClintock, J.
874 \begin_layout Plain Layout
886 \begin_layout Bibliography
887 \begin_inset CommandInset bibitem
889 label "Djorgovski and King(1984)"
895 Djorgovski, S., and King, I.
900 \begin_layout Standard
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921 \begin_layout Bibliography
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924 label "Hagiwara and Zeppenfeld(1986)"
930 Hagiwara, K., and Zeppenfeld, D.
931 1986, Nucl.Phys., 274, 1
934 \begin_layout Bibliography
935 \begin_inset CommandInset bibitem
937 label "Harris and van den Bergh(1984)"
944 E., and van den Bergh, S.
949 \begin_layout Plain Layout
961 \begin_layout Bibliography
962 \begin_inset CommandInset bibitem
964 label "H\\`enon(1961)"
971 1961, Ann.d'Ap., 24, 369
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989 \begin_layout Plain Layout
1001 \begin_layout Bibliography
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1003 LatexCommand bibitem
1012 1975, Dynamics of Stellar Systems, A.
1013 Hayli, Dordrecht: Reidel, 1975, 99
1016 \begin_layout Bibliography
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1018 LatexCommand bibitem
1019 label "King et al.(1968)"
1026 R., Hedemann, E., Hodge, S.
1033 \begin_layout Plain Layout
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1056 V., and Wasserman, L.
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1076 LatexCommand bibitem
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1083 Lynden-Bell, D., and Wood, R.
1085 \begin_inset Preview
1087 \begin_layout Standard
1091 \begin_layout Plain Layout
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1124 \begin_layout Plain Layout
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1137 \begin_inset CommandInset bibitem
1138 LatexCommand bibitem
1139 label "Ortolani et al.(1985)"
1145 Ortolani, S., Rosino, L., and Sandage, A.
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1164 LatexCommand bibitem
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1177 \begin_layout Plain Layout
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1190 \begin_inset CommandInset bibitem
1191 LatexCommand bibitem
1192 label "Spitzer(1985)"
1199 1985, Dynamics of Star Clusters, J.
1201 Hut, Dordrecht: Reidel, 109
1204 \begin_layout Standard
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2198 <cell alignment="right" valignment="top" usebox="none">
2201 \begin_layout Plain Layout
2207 <cell alignment="right" valignment="top" usebox="none">
2210 \begin_layout Plain Layout
2216 <cell alignment="right" valignment="top" usebox="none">
2219 \begin_layout Plain Layout
2225 <cell alignment="right" valignment="top" usebox="none">
2228 \begin_layout Plain Layout
2234 <cell alignment="right" valignment="top" usebox="none">
2237 \begin_layout Plain Layout
2243 <cell alignment="right" valignment="top" usebox="none">
2246 \begin_layout Plain Layout
2260 \begin_layout Table note
2265 \begin_layout Plain Layout
2272 Sample footnote for table
2273 \begin_inset space ~
2277 \begin_inset CommandInset ref
2283 that was generated with the \SpecialChar LaTeX
2287 \begin_layout Table note
2292 \begin_layout Plain Layout
2299 Yet another sample footnote for table
2300 \begin_inset space ~
2304 \begin_inset CommandInset ref
2313 \begin_layout Table note
2318 \begin_layout Plain Layout
2325 Another sample footnote for table
2326 \begin_inset space ~
2330 \begin_inset CommandInset ref
2339 \begin_layout TableComments
2340 We can also attach a long-ish paragraph of explanatory material to a table.
2343 tablerefs to append a list of references.
2344 The following references were from a different table: I've patched them
2345 in here to show how they look, but don't take them too seriously—I certainly
2349 \begin_layout TableRefs
2350 (1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
2351 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
2352 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
2353 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
2354 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
2355 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
2356 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
2357 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
2358 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
2360 1991; (26) Olsen 1983.
2368 \begin_layout Standard
2369 \begin_inset Note Note
2372 \begin_layout Plain Layout
2373 LyX can load deluxetable files verbatim.
2374 To see this in action, dissolve the following Note and copy the file table.tex
2375 from the AASTeX 5.0 distribution into the same directory as this file.
2383 \begin_layout Standard
2384 \begin_inset Note Note
2387 \begin_layout Plain Layout
2388 \begin_inset CommandInset include