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7 \newcommand{\vdag}{(v)^\dagger}
8 \newcommand{\myemail}{skywalker@galaxy.far.far.away}
9 \slugcomment{Not to appear in Nonlearned J., 45.}
10 \shorttitle{Djorgovski et al.}
11 \shortauthors{Collapsed Cores in Globular Clusters}
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71 Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AASTeX
78 \begin_inset Flex altaffilmark
81 \begin_layout Plain Layout
89 \begin_inset Flex altaffilmark
92 \begin_layout Plain Layout
101 \begin_layout Affiliation
102 Astronomy Department, University of California, Berkeley, CA 94720
109 \begin_inset Flex altaffilmark
112 \begin_layout Plain Layout
121 \begin_layout Affiliation
122 National Optical Astronomy Observatories, Tucson, AZ 85719
137 \begin_inset Flex altaffilmark
140 \begin_layout Plain Layout
149 \begin_layout Affiliation
150 Space Telescope Science Institute, Baltimore, MD 21218
153 \begin_layout Altaffilation
158 \begin_layout Plain Layout
165 Visiting Astronomer, Cerro Tololo Inter-American Observatory.
166 CTIO is operated by AURA, Inc.
167 \begin_inset space \space{}
170 under contract to the National Science Foundation.
173 \begin_layout Altaffilation
178 \begin_layout Plain Layout
185 Society of Fellows, Harvard University.
188 \begin_layout Altaffilation
193 \begin_layout Plain Layout
200 present address: Center for Astrophysics, 60 Garden Street, Cambridge, MA
204 \begin_layout Altaffilation
209 \begin_layout Plain Layout
216 Visiting Programmer, Space Telescope Science Institute
219 \begin_layout Altaffilation
224 \begin_layout Plain Layout
231 Patron, Alonso's Bar and Grill
234 \begin_layout Abstract
235 This is a preliminary report on surface photometry of the major fraction
236 of known globular clusters, to see which of them show the signs of a collapsed
238 We also explore some diversionary mathematics and recreational tables.
242 \begin_layout Keywords
243 clusters: globular, peanut---bosons: bozos
246 \begin_layout Section
250 \begin_layout Standard
251 A focal problem today in the dynamics of globular clusters is core collapse.
252 It has been predicted by theory for decades
256 \begin_layout Plain Layout
260 citep{hen61,lyn68,spi85}
265 , but observation has been less alert to the phenomenon.
266 For many years the central brightness peak in M15
270 \begin_layout Plain Layout
279 seemed a unique anomaly.
284 \begin_layout Plain Layout
293 suggested a central peak in NGC 6397, and a limited photographic survey
298 \begin_layout Plain Layout
302 citep[Paper I]{djo84}
307 found three more cases, including NGC 6624, whose sharp center had often
312 \begin_layout Plain Layout
325 \begin_layout Section
329 \begin_layout Standard
330 All our observations were short direct exposures with CCD's.
331 At Lick Observatory we used a TI 500
332 \begin_inset Formula $\times$
335 500 chip and a GEC 575
336 \begin_inset Formula $\times$
339 385, on the 1-m Nickel reflector.
340 The only filter available at Lick was red.
341 At CTIO we used a GEC 575
342 \begin_inset Formula $\times$
346 \begin_inset Formula $B,V,$
350 \begin_inset Formula $R$
353 filters, and an RCA 512
354 \begin_inset Formula $\times$
358 \begin_inset Formula $U,B,V,R,$
362 \begin_inset Formula $I$
365 filters, on the 1.5-m reflector.
366 In the CTIO observations we tried to concentrate on the shortest practicable
367 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
368 often kept us from observing in
369 \begin_inset Formula $U$
373 \begin_inset Formula $B$
377 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
378 sizes were around 3 arcmin.
381 \begin_layout Standard
382 The CCD images are unfortunately not always suitable, for very poor clusters
383 or for clusters with large cores.
384 Since the latter are easily studied by other means, we augmented our own
385 CCD profiles by collecting from the literature a number of star-count profiles
390 \begin_layout Plain Layout
394 citep{kin68,pet76,har84,ort85}
399 , as well as photoelectric profiles
403 \begin_layout Plain Layout
412 and electronographic profiles
416 \begin_layout Plain Layout
426 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
429 \begin_layout Section
433 \begin_layout Standard
434 It has been realized that helicity amplitudes provide a convenient means
439 \begin_layout Plain Layout
440 Footnotes can be inserted like this.
446 These amplitude-level techniques are particularly convenient for calculations
447 involving many Feynman diagrams, where the usual trace techniques for the
448 amplitude squared becomes unwieldy.
449 Our calculations use the helicity techniques developed by other authors
451 \begin_inset CommandInset citation
457 ; we briefly summarize below.
460 \begin_layout Subsection
464 \begin_layout Standard
465 \begin_inset CommandInset label
474 \begin_layout Standard
475 A tree-level amplitude in
476 \begin_inset Formula $e^{+}e^{-}$
479 collisions can be expressed in terms of fermion strings of the form
482 \bar{v}(p_{2},\sigma_{2})P_{-\tau}\hat{a}_{1}\hat{a}_{2}\cdots\hat{a}_{n}u(p_{1},\sigma_{1}),
488 \begin_inset Formula $p$
492 \begin_inset Formula $\sigma$
496 \begin_inset Formula $e^{\pm}$
499 four-momenta and helicities
500 \begin_inset Formula $(\sigma=\pm1)$
504 \begin_inset Formula $\hat{a}_{i}=a_{i}^{\mu}\gamma_{\nu}$
508 \begin_inset Formula $P_{\tau}=\frac{1}{2}(1+\tau\gamma_{5})$
511 is a chirality projection operator
512 \begin_inset Formula $(\tau=\pm1)$
517 \begin_inset Formula $a_{i}^{\mu}$
520 may be formed from particle four-momenta, gauge-boson polarization vectors
521 or fermion strings with an uncontracted Lorentz index associated with final-sta
525 \begin_layout NoteToEditor
526 Figures 1 and 2 should appear side-by-side in print
529 \begin_layout Standard
530 In the chiral representation the
531 \begin_inset Formula $\gamma$
534 matrices are expressed in terms of
535 \begin_inset Formula $2\times2$
539 \begin_inset Formula $\sigma$
542 and the unit matrix 1 as
545 \gamma^{\mu} & = & \left(\begin{array}{cc}
546 0 & \sigma_{+}^{\mu}\\
548 \end{array}\right),\gamma^{5}=\left(\begin{array}{cc}
551 \end{array}\right),\\
552 \sigma_{\pm}^{\mu} & = & ({\textbf{1}},\pm\sigma),
560 \hat{a}=\left(\begin{array}{cc}
563 \end{array}\right),(\hat{a})_{\pm}=a_{\mu}\sigma_{\pm}^{\mu},
568 The spinors are expressed in terms of two-component Weyl spinors as
571 u=\left(\begin{array}{c}
574 \end{array}\right),v={\textbf{(}}\vdag_{+}{\textbf{,}}\vdag_{-}{\textbf{)}}.
582 \begin_layout Standard
583 The Weyl spinors are given in terms of helicity eigenstates
584 \begin_inset Formula $\chi_{\lambda}(p)$
588 \begin_inset Formula $\lambda=\pm1$
594 \begin_layout MathLetters
597 u(p,\lambda)_{\pm} & = & (E\pm\lambda|{\textbf{p}}|)^{1/2}\chi_{\lambda}(p),\\
598 v(p,\lambda)_{\pm} & = & \pm\lambda(E\mp\lambda|{\textbf{p}}|)^{1/2}\chi_{-\lambda}(p)
606 \begin_layout Section
607 Floating material and so forth
610 \begin_layout Standard
611 Consider a task that computes profile parameters for a modified Lorentzian
615 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
623 d_{1}=\sqrt{\left(\begin{array}{c}
624 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
625 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}
633 d_{2}=\sqrt{\left(\begin{array}{c}
634 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
635 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}
643 x_{1}=(x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta
651 y_{1}=-(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta
659 \begin_layout Standard
661 \begin_inset Formula $x_{0}$
665 \begin_inset Formula $y_{0}$
668 is the star center, and
669 \begin_inset Formula $\Theta$
672 is the angle with the
673 \begin_inset Formula $x$
677 Results of this task are shown in table
682 \begin_inset CommandInset ref
689 It is not clear how these sorts of analyses may affect determination of
691 \begin_inset Formula $M_{\sun}$
694 , but the assumption is that the alternate results should be less than 90°
695 out of phase with previous values.
696 We have no observations of
700 \begin_layout Plain Layout
714 \begin_layout Plain Layout
723 of the electronically submitted abstracts for AAS meetings are error-free.
726 \begin_layout Acknowledgements
730 \begin_layout Standard
731 We are grateful to V.
736 Phillips for doing the math in section
741 \begin_inset CommandInset ref
748 More information on the AASTeX macros package are available at
749 \begin_inset Flex URL
752 \begin_layout Plain Layout
754 http://www.aas.org/publications/aastex
763 \begin_layout Plain Layout
767 anchor{ftp://www.aas.org/pubs/}{AAS ftp site}
775 \begin_layout Appendix
779 \begin_layout Section
783 \begin_layout Standard
784 Consider once again a task that computes profile parameters for a modified
785 Lorentzian of the form
788 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
796 \begin_layout MathLetters
799 d_{1}=\frac{3}{4}\sqrt{\left(\begin{array}{c}
800 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
801 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}
809 d_{2}=\case{3}{4}\sqrt{\left(\begin{array}{c}
810 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
811 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}
819 x_{1} & = & (x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta\\
820 y_{1} & = & -(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta
828 \begin_layout Standard
829 For completeness, here is one last equation.
841 \begin_layout References
842 \begin_inset CommandInset bibitem
844 label "Auri\\`ere(1982)"
854 \begin_layout Plain Layout
866 \begin_layout References
867 \begin_inset CommandInset bibitem
869 label "Canizares et al.(1978)"
877 A., Liller, W., and McClintock, J.
883 \begin_layout Plain Layout
895 \begin_layout References
896 \begin_inset CommandInset bibitem
898 label "Djorgovski and King(1984)"
903 Djorgovski, S., and King, I.
909 \begin_layout Plain Layout
921 \begin_layout References
922 \begin_inset CommandInset bibitem
924 label "Hagiwara and Zeppenfeld(1986)"
929 Hagiwara, K., and Zeppenfeld, D.
930 1986, Nucl.Phys., 274, 1
933 \begin_layout References
934 \begin_inset CommandInset bibitem
936 label "Harris and van den Bergh(1984)"
942 E., and van den Bergh, S.
947 \begin_layout Plain Layout
959 \begin_layout References
960 \begin_inset CommandInset bibitem
962 label "H\\`enon(1961)"
968 1961, Ann.d'Ap., 24, 369
971 \begin_layout References
972 \begin_inset CommandInset bibitem
985 \begin_layout Plain Layout
997 \begin_layout References
998 \begin_inset CommandInset bibitem
1007 1975, Dynamics of Stellar Systems, A.
1008 Hayli, Dordrecht: Reidel, 1975, 99
1011 \begin_layout References
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1013 LatexCommand bibitem
1014 label "King et al.(1968)"
1020 R., Hedemann, E., Hodge, S.
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1042 label "Kron et al.(1984)"
1049 V., and Wasserman, L.
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1070 label "Lynden-Bell and Wood(1968)"
1075 Lynden-Bell, D., and Wood, R.
1080 \begin_layout Plain Layout
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1122 label "Ortolani et al.(1985)"
1127 Ortolani, S., Rosino, L., and Sandage, A.
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1158 \begin_layout Plain Layout
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1172 LatexCommand bibitem
1173 label "Spitzer(1985)"
1179 1985, Dynamics of Star Clusters, J.
1181 Hut, Dordrecht: Reidel, 109
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2200 \begin_layout Plain Layout
2214 \begin_layout Table note
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2226 Sample footnote for table
2227 \begin_inset space ~
2231 \begin_inset CommandInset ref
2237 that was generated with the LaTeX table environment
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2252 Yet another sample footnote for table
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2266 \begin_layout Table note
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2278 Another sample footnote for table
2279 \begin_inset space ~
2283 \begin_inset CommandInset ref
2292 \begin_layout TableComments
2293 We can also attach a long-ish paragraph of explanatory material to a table.
2296 tablerefs to append a list of references.
2297 The following references were from a different table: I've patched them
2298 in here to show how they look, but don't take them too seriously---I certainly
2302 \begin_layout TableRefs
2303 (1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
2304 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
2305 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
2306 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
2307 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
2308 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
2309 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
2310 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
2311 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
2313 1991; (26) Olsen 1983.
2321 \begin_layout Standard
2325 \begin_layout Plain Layout
2327 % LyX can load deluxetable files verbatim.
2328 To see this in action, uncomment the following
2336 \begin_layout Standard
2340 \begin_layout Plain Layout
2348 \begin_inset Quotes eld
2355 \begin_layout Plain Layout
2363 \begin_inset Quotes erd
2370 \begin_layout Plain Layout
2372 ) and copy the file table.tex from the AASTeX 5.0 distribution into
2380 \begin_layout Standard
2384 \begin_layout Plain Layout
2386 % the same directory as this file.
2394 \begin_layout Standard
2398 \begin_layout Plain Layout
2406 \begin_inset CommandInset include