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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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88 Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AAS\SpecialChar TeX
95 \begin_inset Flex altaffilmark
98 \begin_layout Plain Layout
106 \begin_inset Flex altaffilmark
109 \begin_layout Plain Layout
118 \begin_layout Affiliation
119 Astronomy Department, University of California, Berkeley, CA 94720
126 \begin_inset Flex altaffilmark
129 \begin_layout Plain Layout
138 \begin_layout Affiliation
139 National Optical Astronomy Observatories, Tucson, AZ 85719
154 \begin_inset Flex altaffilmark
157 \begin_layout Plain Layout
166 \begin_layout Affiliation
167 Space Telescope Science Institute, Baltimore, MD 21218
170 \begin_layout Altaffilation
171 \begin_inset Argument 1
174 \begin_layout Plain Layout
181 Visiting Astronomer, Cerro Tololo Inter-American Observatory.
182 CTIO is operated by AURA, Inc.
183 \begin_inset space \space{}
186 under contract to the National Science Foundation.
189 \begin_layout Altaffilation
190 \begin_inset Argument 1
193 \begin_layout Plain Layout
200 Society of Fellows, Harvard University.
203 \begin_layout Altaffilation
204 \begin_inset Argument 1
207 \begin_layout Plain Layout
214 present address: Center for Astrophysics, 60 Garden Street, Cambridge, MA
218 \begin_layout Altaffilation
219 \begin_inset Argument 1
222 \begin_layout Plain Layout
229 Visiting Programmer, Space Telescope Science Institute
232 \begin_layout Altaffilation
233 \begin_inset Argument 1
236 \begin_layout Plain Layout
243 Patron, Alonso's Bar and Grill
246 \begin_layout Abstract
247 This is a preliminary report on surface photometry of the major fraction
248 of known globular clusters, to see which of them show the signs of a collapsed
250 We also explore some diversionary mathematics and recreational tables.
254 \begin_layout Keywords
255 clusters: globular, peanut—bosons: bozos
258 \begin_layout Section
262 \begin_layout Standard
263 A focal problem today in the dynamics of globular clusters is core collapse.
264 It has been predicted by theory for decades
268 \begin_layout Plain Layout
272 citep{hen61,lyn68,spi85}
277 , but observation has been less alert to the phenomenon.
278 For many years the central brightness peak in M15
282 \begin_layout Plain Layout
291 seemed a unique anomaly.
296 \begin_layout Plain Layout
305 suggested a central peak in NGC 6397, and a limited photographic survey
310 \begin_layout Plain Layout
314 citep[Paper I]{djo84}
319 found three more cases, including NGC 6624, whose sharp center had often
324 \begin_layout Plain Layout
337 \begin_layout Section
341 \begin_layout Standard
342 All our observations were short direct exposures with CCD's.
343 At Lick Observatory we used a TI 500
344 \begin_inset Formula $\times$
347 500 chip and a GEC 575
348 \begin_inset Formula $\times$
351 385, on the 1-m Nickel reflector.
352 The only filter available at Lick was red.
353 At CTIO we used a GEC 575
354 \begin_inset Formula $\times$
358 \begin_inset Formula $B,V,$
362 \begin_inset Formula $R$
365 filters, and an RCA 512
366 \begin_inset Formula $\times$
370 \begin_inset Formula $U,B,V,R,$
374 \begin_inset Formula $I$
377 filters, on the 1.5-m reflector.
378 In the CTIO observations we tried to concentrate on the shortest practicable
379 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
380 often kept us from observing in
381 \begin_inset Formula $U$
385 \begin_inset Formula $B$
389 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
390 sizes were around 3 arcmin.
393 \begin_layout Standard
394 The CCD images are unfortunately not always suitable, for very poor clusters
395 or for clusters with large cores.
396 Since the latter are easily studied by other means, we augmented our own
397 CCD profiles by collecting from the literature a number of star-count profiles
402 \begin_layout Plain Layout
406 citep{kin68,pet76,har84,ort85}
411 , as well as photoelectric profiles
415 \begin_layout Plain Layout
424 and electronographic profiles
428 \begin_layout Plain Layout
438 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
441 \begin_layout Section
445 \begin_layout Standard
446 It has been realized that helicity amplitudes provide a convenient means
451 \begin_layout Plain Layout
452 Footnotes can be inserted like this.
458 These amplitude-level techniques are particularly convenient for calculations
459 involving many Feynman diagrams, where the usual trace techniques for the
460 amplitude squared becomes unwieldy.
461 Our calculations use the helicity techniques developed by other authors
463 \begin_inset CommandInset citation
470 ; we briefly summarize below.
473 \begin_layout Subsection
477 \begin_layout Standard
478 \begin_inset CommandInset label
487 \begin_layout Standard
488 A tree-level amplitude in
489 \begin_inset Formula $e^{+}e^{-}$
492 collisions can be expressed in terms of fermion strings of the form
495 \bar{v}(p_{2},\sigma_{2})P_{-\tau}\hat{a}_{1}\hat{a}_{2}\cdots\hat{a}_{n}u(p_{1},\sigma_{1}),
501 \begin_inset Formula $p$
505 \begin_inset Formula $\sigma$
509 \begin_inset Formula $e^{\pm}$
512 four-momenta and helicities
513 \begin_inset Formula $(\sigma=\pm1)$
517 \begin_inset Formula $\hat{a}_{i}=a_{i}^{\mu}\gamma_{\nu}$
521 \begin_inset Formula $P_{\tau}=\frac{1}{2}(1+\tau\gamma_{5})$
524 is a chirality projection operator
525 \begin_inset Formula $(\tau=\pm1)$
530 \begin_inset Formula $a_{i}^{\mu}$
533 may be formed from particle four-momenta, gauge-boson polarization vectors
534 or fermion strings with an uncontracted Lorentz index associated with final-sta
538 \begin_layout NoteToEditor
539 Figures 1 and 2 should appear side-by-side in print
542 \begin_layout Standard
543 In the chiral representation the
544 \begin_inset Formula $\gamma$
547 matrices are expressed in terms of
548 \begin_inset Formula $2\times2$
552 \begin_inset Formula $\sigma$
555 and the unit matrix 1 as
558 \gamma^{\mu} & = & \left(\begin{array}{cc}
559 0 & \sigma_{+}^{\mu}\\
561 \end{array}\right),\gamma^{5}=\left(\begin{array}{cc}
564 \end{array}\right),\\
565 \sigma_{\pm}^{\mu} & = & ({\textbf{1}},\pm\sigma),
573 \hat{a}=\left(\begin{array}{cc}
576 \end{array}\right),(\hat{a})_{\pm}=a_{\mu}\sigma_{\pm}^{\mu},
581 The spinors are expressed in terms of two-component Weyl spinors as
584 u=\left(\begin{array}{c}
587 \end{array}\right),v={\textbf{(}}\vdag_{+}{\textbf{,}}\vdag_{-}{\textbf{)}}.
595 \begin_layout Standard
596 The Weyl spinors are given in terms of helicity eigenstates
597 \begin_inset Formula $\chi_{\lambda}(p)$
601 \begin_inset Formula $\lambda=\pm1$
607 \begin_layout MathLetters
610 u(p,\lambda)_{\pm} & = & (E\pm\lambda|{\textbf{p}}|)^{1/2}\chi_{\lambda}(p),\\
611 v(p,\lambda)_{\pm} & = & \pm\lambda(E\mp\lambda|{\textbf{p}}|)^{1/2}\chi_{-\lambda}(p)
619 \begin_layout Section
620 Floating material and so forth
623 \begin_layout Standard
624 Consider a task that computes profile parameters for a modified Lorentzian
628 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
636 d_{1}=\sqrt{\left(\begin{array}{c}
637 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
638 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}
646 d_{2}=\sqrt{\left(\begin{array}{c}
647 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
648 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}
656 x_{1}=(x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta
664 y_{1}=-(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta
672 \begin_layout Standard
674 \begin_inset Formula $x_{0}$
678 \begin_inset Formula $y_{0}$
681 is the star center, and
682 \begin_inset Formula $\Theta$
685 is the angle with the
686 \begin_inset Formula $x$
690 Results of this task are shown in table
695 \begin_inset CommandInset ref
702 It is not clear how these sorts of analyses may affect determination of
704 \begin_inset Formula $M_{\sun}$
707 , but the assumption is that the alternate results should be less than 90°
708 out of phase with previous values.
709 We have no observations of
713 \begin_layout Plain Layout
727 \begin_layout Plain Layout
736 of the electronically submitted abstracts for AAS meetings are error-free.
739 \begin_layout Acknowledgements
743 \begin_layout Standard
744 We are grateful to V.
749 Phillips for doing the math in section
754 \begin_inset CommandInset ref
761 More information on the AAS\SpecialChar TeX
762 macros package are available at
763 \begin_inset Flex URL
766 \begin_layout Plain Layout
768 http://www.aas.org/publications/aastex
777 \begin_layout Plain Layout
786 ftp://www.aas.org/pubs/
790 \begin_layout Plain Layout
801 \begin_layout Plain Layout
811 \begin_layout Appendix
815 \begin_layout Section
819 \begin_layout Standard
820 Consider once again a task that computes profile parameters for a modified
821 Lorentzian of the form
824 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
832 \begin_layout MathLetters
835 d_{1}=\frac{3}{4}\sqrt{\left(\begin{array}{c}
836 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
837 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}
845 d_{2}=\case{3}{4}\sqrt{\left(\begin{array}{c}
846 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
847 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}
855 x_{1} & = & (x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta\\
856 y_{1} & = & -(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta
864 \begin_layout Standard
865 For completeness, here is one last equation.
877 \begin_layout References
878 \begin_inset CommandInset bibitem
880 label "Auri\\`ere(1982)"
891 \begin_layout Plain Layout
903 \begin_layout References
904 \begin_inset CommandInset bibitem
906 label "Canizares et al.(1978)"
915 A., Liller, W., and McClintock, J.
921 \begin_layout Plain Layout
933 \begin_layout References
934 \begin_inset CommandInset bibitem
936 label "Djorgovski and King(1984)"
942 Djorgovski, S., and King, I.
948 \begin_layout Plain Layout
960 \begin_layout References
961 \begin_inset CommandInset bibitem
963 label "Hagiwara and Zeppenfeld(1986)"
969 Hagiwara, K., and Zeppenfeld, D.
970 1986, Nucl.Phys., 274, 1
973 \begin_layout References
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976 label "Harris and van den Bergh(1984)"
983 E., and van den Bergh, S.
988 \begin_layout Plain Layout
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1010 1961, Ann.d'Ap., 24, 369
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1051 1975, Dynamics of Stellar Systems, A.
1052 Hayli, Dordrecht: Reidel, 1975, 99
1055 \begin_layout References
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1058 label "King et al.(1968)"
1065 R., Hedemann, E., Hodge, S.
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1087 label "Kron et al.(1984)"
1095 V., and Wasserman, L.
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1176 Ortolani, S., Rosino, L., and Sandage, A.
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1230 1985, Dynamics of Star Clusters, J.
1232 Hut, Dordrecht: Reidel, 109
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2301 Sample footnote for table
2302 \begin_inset space ~
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2312 that was generated with the \SpecialChar LaTeX
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2368 \begin_layout TableComments
2369 We can also attach a long-ish paragraph of explanatory material to a table.
2372 tablerefs to append a list of references.
2373 The following references were from a different table: I've patched them
2374 in here to show how they look, but don't take them too seriously—I certainly
2378 \begin_layout TableRefs
2379 (1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
2380 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
2381 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
2382 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
2383 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
2384 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
2385 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
2386 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
2387 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
2389 1991; (26) Olsen 1983.
2397 \begin_layout Standard
2401 \begin_layout Plain Layout
2403 % LyX can load deluxetable files verbatim.
2404 To see this in action, uncomment the following
2412 \begin_layout Standard
2416 \begin_layout Plain Layout
2424 \begin_inset Quotes eld
2431 \begin_layout Plain Layout
2439 \begin_inset Quotes erd
2446 \begin_layout Plain Layout
2448 ) and copy the file table.tex from the AASTeX 5.0 distribution into
2456 \begin_layout Standard
2460 \begin_layout Plain Layout
2462 % the same directory as this file.
2470 \begin_layout Standard
2474 \begin_layout Plain Layout
2482 \begin_inset CommandInset include