1 #LyX 2.4 created this file. For more info see https://www.lyx.org/
5 \save_transient_properties true
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9 \newcommand{\vdag}{(v)^\dagger}
10 \newcommand{\myemail}{skywalker@galaxy.far.far.away}
11 %\slugcomment{Not to appear in Nonlearned J., 45.}
12 \shorttitle{Djorgovski et al.}
13 \shortauthors{Collapsed Cores in Globular Clusters}
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56 \cite_engine_type authoryear
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89 \begin_layout Standard
90 \begin_inset Note Note
93 \begin_layout Plain Layout
96 Sample document for contributions to journals of the American Astronomical
99 \begin_inset CommandInset href
101 name "AASTeX user guide"
102 target "http://journals.aas.org/authors/aastex/aasguide.html"
107 for details on how to use AASTeX.
111 \begin_layout Plain Layout
114 An up-to-date sample LaTeX article using AASTeX is available under
115 \begin_inset Flex URL
118 \begin_layout Plain Layout
120 http://journals.aas.org/authors/aastex/aastex.html#_download
131 \begin_inset Note Note
134 \begin_layout Plain Layout
137 The AASTeX document class uses a different name since version 6.0 (2016/02/16).
138 Depending on the version installed on your system, set the document class
139 to American Astronomical Society (AASTeX v.
140 6) or American Astronomical Society (AASTeX v.
145 \begin_layout Plain Layout
150 \begin_inset Flex URL
153 \begin_layout Plain Layout
155 http://journals.aas.org/authors/aastex/revisionhistory.html
160 ) are only partially supported by LyX (see
161 \begin_inset Flex URL
164 \begin_layout Plain Layout
166 https://www.lyx.org/trac/ticket/10027
180 Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AAS\SpecialChar TeX
186 Djorgovski and Ivan R.
190 \begin_layout Affiliation
191 Astronomy Department, University of California, Berkeley, CA 94720
194 \begin_layout Altaffilation
196 \begin_inset Newline newline
199 Cerro Tololo Inter-American Observatory.
200 \begin_inset Newline newline
203 CTIO is operated by AURA Inc.
204 \begin_inset space \space{}
207 under contract to the National Science Foundation.
210 \begin_layout Altaffilation
211 Society of Fellows, Harvard University.
214 \begin_layout Altaffilation
215 present address: Center for Astrophysics
216 \begin_inset Newline newline
219 60 Garden Street, Cambridge, MA 02138
228 \begin_layout Affiliation
229 National Optical Astronomy Observatories, Tucson, AZ 85719
232 \begin_layout Altaffilation
233 Visiting Programmer, Space Telescope Science Institute
236 \begin_layout Altaffilation
237 Patron, Alonso's Bar and Grill
250 \begin_layout Affiliation
251 Space Telescope Science Institute, Baltimore, MD 21218
254 \begin_layout Altaffilation
255 Patron, Alonso's Bar and Grill
258 \begin_layout Abstract
259 This is a preliminary report on surface photometry of the major fraction
260 of known globular clusters, to see which of them show the signs of a collapsed
262 We also explore some diversionary mathematics and recreational tables.
266 \begin_layout Keywords
267 clusters: globular, peanut—bosons: bozos
270 \begin_layout Section
274 \begin_layout Standard
275 A focal problem today in the dynamics of globular clusters is core collapse.
276 It has been predicted by theory for decades
277 \begin_inset CommandInset citation
279 key "hen61,lyn68,spi85"
284 , but observation has been less alert to the phenomenon.
285 For many years the central brightness peak in M15
286 \begin_inset CommandInset citation
293 seemed a unique anomaly.
295 \begin_inset CommandInset citation
302 suggested a central peak in NGC 6397, and a limited photographic survey
307 \begin_layout Plain Layout
311 citep[Paper I]{djo84}
316 found three more cases, including NGC 6624, whose sharp center had often
318 \begin_inset CommandInset citation
329 \begin_layout Section
333 \begin_layout Standard
334 All our observations were short direct exposures with CCD's.
335 At Lick Observatory we used a TI 500
336 \begin_inset Formula $\times$
339 500 chip and a GEC 575
340 \begin_inset Formula $\times$
343 385, on the 1-m Nickel reflector.
344 The only filter available at Lick was red.
345 At CTIO we used a GEC 575
346 \begin_inset Formula $\times$
350 \begin_inset Formula $B,V,$
354 \begin_inset Formula $R$
357 filters, and an RCA 512
358 \begin_inset Formula $\times$
362 \begin_inset Formula $U,B,V,R,$
366 \begin_inset Formula $I$
369 filters, on the 1.5-m reflector.
370 In the CTIO observations we tried to concentrate on the shortest practicable
371 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
372 often kept us from observing in
373 \begin_inset Formula $U$
377 \begin_inset Formula $B$
381 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
382 sizes were around 3 arcmin.
385 \begin_layout Standard
386 The CCD images are unfortunately not always suitable, for very poor clusters
387 or for clusters with large cores.
388 Since the latter are easily studied by other means, we augmented our own
389 CCD profiles by collecting from the literature a number of star-count profiles
391 \begin_inset CommandInset citation
393 key "kin68,pet76,har84,ort85"
398 , as well as photoelectric profiles
399 \begin_inset CommandInset citation
406 and electronographic profiles
407 \begin_inset CommandInset citation
415 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
418 \begin_layout Section
422 \begin_layout Standard
423 It has been realized that helicity amplitudes provide a convenient means
428 \begin_layout Plain Layout
429 Footnotes can be inserted like this.
435 These amplitude-level techniques are particularly convenient for calculations
436 involving many Feynman diagrams, where the usual trace techniques for the
437 amplitude squared becomes unwieldy.
438 Our calculations use the helicity techniques developed by other authors
440 \begin_inset CommandInset citation
447 ; we briefly summarize below.
450 \begin_layout Subsection
454 \begin_layout Standard
455 \begin_inset CommandInset label
464 \begin_layout Standard
465 A tree-level amplitude in
466 \begin_inset Formula $e^{+}e^{-}$
469 collisions can be expressed in terms of fermion strings of the form
472 \bar{v}(p_{2},\sigma_{2})P_{-\tau}\hat{a}_{1}\hat{a}_{2}\cdots\hat{a}_{n}u(p_{1},\sigma_{1}),
478 \begin_inset Formula $p$
482 \begin_inset Formula $\sigma$
486 \begin_inset Formula $e^{\pm}$
489 four-momenta and helicities
490 \begin_inset Formula $(\sigma=\pm1)$
494 \begin_inset Formula $\hat{a}_{i}=a_{i}^{\mu}\gamma_{\nu}$
498 \begin_inset Formula $P_{\tau}=\frac{1}{2}(1+\tau\gamma_{5})$
501 is a chirality projection operator
502 \begin_inset Formula $(\tau=\pm1)$
507 \begin_inset Formula $a_{i}^{\mu}$
510 may be formed from particle four-momenta, gauge-boson polarization vectors
511 or fermion strings with an uncontracted Lorentz index associated with final-sta
515 \begin_layout NoteToEditor
516 Figures 1 and 2 should appear side-by-side in print
519 \begin_layout Standard
520 In the chiral representation the
521 \begin_inset Formula $\gamma$
524 matrices are expressed in terms of
525 \begin_inset Formula $2\times2$
529 \begin_inset Formula $\sigma$
532 and the unit matrix 1 as
535 \gamma^{\mu} & = & \left(\begin{array}{cc}
536 0 & \sigma_{+}^{\mu}\\
538 \end{array}\right),\gamma^{5}=\left(\begin{array}{cc}
541 \end{array}\right),\\
542 \sigma_{\pm}^{\mu} & = & ({\textbf{1}},\pm\sigma),
550 \hat{a}=\left(\begin{array}{cc}
553 \end{array}\right),(\hat{a})_{\pm}=a_{\mu}\sigma_{\pm}^{\mu},
558 The spinors are expressed in terms of two-component Weyl spinors as
561 u=\left(\begin{array}{c}
564 \end{array}\right),v={\textbf{(}}\vdag_{+}{\textbf{,}}\vdag_{-}{\textbf{)}}.
572 \begin_layout Standard
573 The Weyl spinors are given in terms of helicity eigenstates
574 \begin_inset Formula $\chi_{\lambda}(p)$
578 \begin_inset Formula $\lambda=\pm1$
584 \begin_layout MathLetters
587 u(p,\lambda)_{\pm} & = & (E\pm\lambda|{\textbf{p}}|)^{1/2}\chi_{\lambda}(p),\\
588 v(p,\lambda)_{\pm} & = & \pm\lambda(E\mp\lambda|{\textbf{p}}|)^{1/2}\chi_{-\lambda}(p)
596 \begin_layout Section
597 Floating material and so forth
600 \begin_layout Standard
601 Consider a task that computes profile parameters for a modified Lorentzian
605 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
613 d_{1}=\sqrt{\left(\begin{array}{c}
614 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
615 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}
623 d_{2}=\sqrt{\left(\begin{array}{c}
624 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
625 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}
633 x_{1}=(x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta
641 y_{1}=-(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta
649 \begin_layout Standard
651 \begin_inset Formula $x_{0}$
655 \begin_inset Formula $y_{0}$
658 is the star center, and
659 \begin_inset Formula $\Theta$
662 is the angle with the
663 \begin_inset Formula $x$
667 Results of this task are shown in table
672 \begin_inset CommandInset ref
679 It is not clear how these sorts of analyses may affect determination of
681 \begin_inset Formula $M_{\text{\sun}}$
684 , but the assumption is that the alternate results should be less than 90°
685 out of phase with previous values.
686 We have no observations of
690 \begin_layout Plain Layout
701 \begin_inset Formula $\nicefrac{4}{5}$
704 of the electronically submitted abstracts for AAS meetings are error-free.
708 \begin_layout Acknowledgements
709 We are grateful to V.
714 Phillips for doing the math in section
719 \begin_inset CommandInset ref
726 More information on the AAS\SpecialChar TeX
727 macros package are available at
728 \begin_inset Flex URL
731 \begin_layout Plain Layout
733 http://www.aas.org/publications/aastex
742 \begin_layout Plain Layout
751 ftp://www.aas.org/pubs/
755 \begin_layout Plain Layout
766 \begin_layout Plain Layout
777 \begin_layout Software
778 IRAF, AIPS, Astropy, ...
782 \begin_layout Appendix
786 \begin_layout Section
790 \begin_layout Standard
791 Consider once again a task that computes profile parameters for a modified
792 Lorentzian of the form
795 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
803 \begin_layout MathLetters
806 d_{1}=\frac{3}{4}\sqrt{\left(\begin{array}{c}
807 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
808 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}
816 d_{2}=\case{3}{4}\sqrt{\left(\begin{array}{c}
817 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
818 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}
826 x_{1} & = & (x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta\\
827 y_{1} & = & -(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta
835 \begin_layout Standard
836 For completeness, here is one last equation.
848 \begin_layout References
849 \begin_inset CommandInset bibitem
851 label "Auri\\`ere(1982)"
862 \begin_layout Plain Layout
874 \begin_layout References
875 \begin_inset CommandInset bibitem
877 label "Canizares et al.(1978)"
886 A., Liller, W., and McClintock, J.
892 \begin_layout Plain Layout
904 \begin_layout References
905 \begin_inset CommandInset bibitem
907 label "Djorgovski and King(1984)"
913 Djorgovski, S., and King, I.
919 \begin_layout Plain Layout
931 \begin_layout References
932 \begin_inset CommandInset bibitem
934 label "Hagiwara and Zeppenfeld(1986)"
940 Hagiwara, K., and Zeppenfeld, D.
941 1986, Nucl.Phys., 274, 1
944 \begin_layout References
945 \begin_inset CommandInset bibitem
947 label "Harris and van den Bergh(1984)"
954 E., and van den Bergh, S.
959 \begin_layout Plain Layout
971 \begin_layout References
972 \begin_inset CommandInset bibitem
974 label "H\\`enon(1961)"
981 1961, Ann.d'Ap., 24, 369
984 \begin_layout References
985 \begin_inset CommandInset bibitem
999 \begin_layout Plain Layout
1011 \begin_layout References
1012 \begin_inset CommandInset bibitem
1013 LatexCommand bibitem
1022 1975, Dynamics of Stellar Systems, A.
1023 Hayli, Dordrecht: Reidel, 1975, 99
1026 \begin_layout References
1027 \begin_inset CommandInset bibitem
1028 LatexCommand bibitem
1029 label "King et al.(1968)"
1036 R., Hedemann, E., Hodge, S.
1043 \begin_layout Plain Layout
1055 \begin_layout References
1056 \begin_inset CommandInset bibitem
1057 LatexCommand bibitem
1058 label "Kron et al.(1984)"
1066 V., and Wasserman, L.
1072 \begin_layout Plain Layout
1084 \begin_layout References
1085 \begin_inset CommandInset bibitem
1086 LatexCommand bibitem
1087 label "Lynden-Bell and Wood(1968)"
1093 Lynden-Bell, D., and Wood, R.
1098 \begin_layout Plain Layout
1110 \begin_layout References
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1112 LatexCommand bibitem
1113 label "Newell and O'Neil(1978)"
1126 \begin_layout Plain Layout
1138 \begin_layout References
1139 \begin_inset CommandInset bibitem
1140 LatexCommand bibitem
1141 label "Ortolani et al.(1985)"
1147 Ortolani, S., Rosino, L., and Sandage, A.
1152 \begin_layout Plain Layout
1164 \begin_layout References
1165 \begin_inset CommandInset bibitem
1166 LatexCommand bibitem
1167 label "Peterson(1976)"
1179 \begin_layout Plain Layout
1191 \begin_layout References
1192 \begin_inset CommandInset bibitem
1193 LatexCommand bibitem
1194 label "Spitzer(1985)"
1201 1985, Dynamics of Star Clusters, J.
1203 Hut, Dordrecht: Reidel, 109
1206 \begin_layout Standard
1207 \begin_inset Newpage clearpage
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1243 to be too wordy, but that would make for poor communication as well.
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2218 <cell alignment="right" valignment="top" usebox="none">
2221 \begin_layout Plain Layout
2227 <cell alignment="right" valignment="top" usebox="none">
2230 \begin_layout Plain Layout
2236 <cell alignment="right" valignment="top" usebox="none">
2239 \begin_layout Plain Layout
2245 <cell alignment="right" valignment="top" usebox="none">
2248 \begin_layout Plain Layout
2262 \begin_layout Table note
2267 \begin_layout Plain Layout
2274 Sample footnote for table
2275 \begin_inset space ~
2279 \begin_inset CommandInset ref
2285 that was generated with the \SpecialChar LaTeX
2289 \begin_layout Table note
2294 \begin_layout Plain Layout
2301 Yet another sample footnote for table
2302 \begin_inset space ~
2306 \begin_inset CommandInset ref
2315 \begin_layout Table note
2320 \begin_layout Plain Layout
2327 Another sample footnote for table
2328 \begin_inset space ~
2332 \begin_inset CommandInset ref
2341 \begin_layout TableComments
2342 We can also attach a long-ish paragraph of explanatory material to a table.
2345 tablerefs to append a list of references.
2346 The following references were from a different table: I've patched them
2347 in here to show how they look, but don't take them too seriously—I certainly
2351 \begin_layout TableRefs
2352 (1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
2353 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
2354 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
2355 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
2356 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
2357 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
2358 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
2359 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
2360 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
2362 1991; (26) Olsen 1983.
2370 \begin_layout Standard
2371 \begin_inset Note Note
2374 \begin_layout Plain Layout
2375 LyX can load deluxetable files verbatim.
2376 To see this in action, dissolve the following Note and copy the file table.tex
2377 from the AASTeX 5.0 distribution into the same directory as this file.
2385 \begin_layout Standard
2386 \begin_inset Note Note
2389 \begin_layout Plain Layout
2390 \begin_inset CommandInset include