1 #LyX 1.6.0 created this file. For more info see http://www.lyx.org/
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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52 Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AASTeX
59 \begin_inset Flex altaffilmark
62 \begin_layout Plain Layout
70 \begin_inset Flex altaffilmark
73 \begin_layout Plain Layout
82 \begin_layout Affiliation
83 Astronomy Department, University of California, Berkeley, CA 94720
90 \begin_inset Flex altaffilmark
93 \begin_layout Plain Layout
102 \begin_layout Affiliation
103 National Optical Astronomy Observatories, Tucson, AZ 85719
118 \begin_inset Flex altaffilmark
121 \begin_layout Plain Layout
130 \begin_layout Affiliation
131 Space Telescope Science Institute, Baltimore, MD 21218
134 \begin_layout Altaffilation
139 \begin_layout Plain Layout
146 Visiting Astronomer, Cerro Tololo Inter-American Observatory.
147 CTIO is operated by AURA, Inc.
148 \begin_inset space \space{}
151 under contract to the National Science Foundation.
154 \begin_layout Altaffilation
159 \begin_layout Plain Layout
166 Society of Fellows, Harvard University.
169 \begin_layout Altaffilation
174 \begin_layout Plain Layout
181 present address: Center for Astrophysics, 60 Garden Street, Cambridge, MA
185 \begin_layout Altaffilation
190 \begin_layout Plain Layout
197 Visiting Programmer, Space Telescope Science Institute
200 \begin_layout Altaffilation
205 \begin_layout Plain Layout
212 Patron, Alonso's Bar and Grill
215 \begin_layout Abstract
216 This is a preliminary report on surface photometry of the major fraction
217 of known globular clusters, to see which of them show the signs of a collapsed
219 We also explore some diversionary mathematics and recreational tables.
223 \begin_layout Keywords
224 clusters: globular, peanut---bosons: bozos
227 \begin_layout Section
231 \begin_layout Standard
232 A focal problem today in the dynamics of globular clusters is core collapse.
233 It has been predicted by theory for decades
237 \begin_layout Plain Layout
241 citep{hen61,lyn68,spi85}
246 , but observation has been less alert to the phenomenon.
247 For many years the central brightness peak in M15
251 \begin_layout Plain Layout
260 seemed a unique anomaly.
265 \begin_layout Plain Layout
274 suggested a central peak in NGC 6397, and a limited photographic survey
279 \begin_layout Plain Layout
283 citep[Paper I]{djo84}
288 found three more cases, including NGC 6624, whose sharp center had often
293 \begin_layout Plain Layout
306 \begin_layout Section
310 \begin_layout Standard
311 All our observations were short direct exposures with CCD's.
312 At Lick Observatory we used a TI 500
313 \begin_inset Formula $\times$
316 500 chip and a GEC 575
317 \begin_inset Formula $\times$
320 385, on the 1-m Nickel reflector.
321 The only filter available at Lick was red.
322 At CTIO we used a GEC 575
323 \begin_inset Formula $\times$
327 \begin_inset Formula $B,V,$
331 \begin_inset Formula $R$
334 filters, and an RCA 512
335 \begin_inset Formula $\times$
339 \begin_inset Formula $U,B,V,R,$
343 \begin_inset Formula $I$
346 filters, on the 1.5-m reflector.
347 In the CTIO observations we tried to concentrate on the shortest practicable
348 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
349 often kept us from observing in
350 \begin_inset Formula $U$
354 \begin_inset Formula $B$
358 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
359 sizes were around 3 arcmin.
362 \begin_layout Standard
363 The CCD images are unfortunately not always suitable, for very poor clusters
364 or for clusters with large cores.
365 Since the latter are easily studied by other means, we augmented our own
366 CCD profiles by collecting from the literature a number of star-count profiles
371 \begin_layout Plain Layout
375 citep{kin68,pet76,har84,ort85}
380 , as well as photoelectric profiles
384 \begin_layout Plain Layout
393 and electronographic profiles
397 \begin_layout Plain Layout
407 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
410 \begin_layout Section
414 \begin_layout Standard
415 It has been realized that helicity amplitudes provide a convenient means
420 \begin_layout Plain Layout
421 Footnotes can be inserted like this.
427 These amplitude-level techniques are particularly convenient for calculations
428 involving many Feynman diagrams, where the usual trace techniques for the
429 amplitude squared becomes unwieldy.
430 Our calculations use the helicity techniques developed by other authors
432 \begin_inset CommandInset citation
438 ; we briefly summarize below.
441 \begin_layout Subsection
445 \begin_layout Standard
446 \begin_inset CommandInset label
455 \begin_layout Standard
456 A tree-level amplitude in
457 \begin_inset Formula $e^{+}e^{-}$
460 collisions can be expressed in terms of fermion strings of the form
461 \begin_inset Formula \begin{equation}
462 \bar{v}(p_{2},\sigma_{2})P_{-\tau}\hat{a}_{1}\hat{a}_{2}\cdots\hat{a}_{n}u(p_{1},\sigma_{1}),\end{equation}
467 \begin_inset Formula $p$
471 \begin_inset Formula $\sigma$
475 \begin_inset Formula $e^{\pm}$
478 four-momenta and helicities
479 \begin_inset Formula $(\sigma=\pm1)$
483 \begin_inset Formula $\hat{a}_{i}=a_{i}^{\mu}\gamma_{\nu}$
487 \begin_inset Formula $P_{\tau}=\frac{1}{2}(1+\tau\gamma_{5})$
490 is a chirality projection operator
491 \begin_inset Formula $(\tau=\pm1)$
496 \begin_inset Formula $a_{i}^{\mu}$
499 may be formed from particle four-momenta, gauge-boson polarization vectors
500 or fermion strings with an uncontracted Lorentz index associated with final-sta
504 \begin_layout NoteToEditor
505 Figures 1 and 2 should appear side-by-side in print
508 \begin_layout Standard
509 In the chiral representation the
510 \begin_inset Formula $\gamma$
513 matrices are expressed in terms of
514 \begin_inset Formula $2\times2$
518 \begin_inset Formula $\sigma$
521 and the unit matrix 1 as
522 \begin_inset Formula \begin{eqnarray*}
523 \gamma^{\mu} & = & \left(\begin{array}{cc}
524 0 & \sigma_{+}^{\mu}\\
525 \sigma_{-}^{\mu} & 0\end{array}\right),\gamma^{5}=\left(\begin{array}{cc}
527 0 & 1\end{array}\right),\\
528 \sigma_{\pm}^{\mu} & = & ({\textbf{1}},\pm\sigma),\end{eqnarray*}
533 \begin_inset Formula \begin{equation}
534 \hat{a}=\left(\begin{array}{cc}
536 (\hat{a})_{-} & 0\end{array}\right),(\hat{a})_{\pm}=a_{\mu}\sigma_{\pm}^{\mu},\end{equation}
540 The spinors are expressed in terms of two-component Weyl spinors as
541 \begin_inset Formula \begin{equation}
542 u=\left(\begin{array}{c}
544 (u)_{+}\end{array}\right),v={\textbf{(}}\vdag_{+}{\textbf{,}}\vdag_{-}{\textbf{)}}.\end{equation}
551 \begin_layout Standard
552 The Weyl spinors are given in terms of helicity eigenstates
553 \begin_inset Formula $\chi_{\lambda}(p)$
557 \begin_inset Formula $\lambda=\pm1$
563 \begin_layout MathLetters
564 \begin_inset Formula \begin{eqnarray}
565 u(p,\lambda)_{\pm} & = & (E\pm\lambda|{\textbf{p}}|)^{1/2}\chi_{\lambda}(p),\\
566 v(p,\lambda)_{\pm} & = & \pm\lambda(E\mp\lambda|{\textbf{p}}|)^{1/2}\chi_{-\lambda}(p)\end{eqnarray}
573 \begin_layout Section
574 Floating material and so forth
577 \begin_layout Standard
578 Consider a task that computes profile parameters for a modified Lorentzian
580 \begin_inset Formula \begin{equation}
581 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}\end{equation}
586 \begin_inset Formula \[
587 d_{1}=\sqrt{\left(\begin{array}{c}
588 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
589 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}\]
594 \begin_inset Formula \[
595 d_{2}=\sqrt{\left(\begin{array}{c}
596 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
597 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}\]
602 \begin_inset Formula \[
603 x_{1}=(x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta\]
608 \begin_inset Formula \[
609 y_{1}=-(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta\]
616 \begin_layout Standard
618 \begin_inset Formula $x_{0}$
622 \begin_inset Formula $y_{0}$
625 is the star center, and
626 \begin_inset Formula $\Theta$
629 is the angle with the
630 \begin_inset Formula $x$
634 Results of this task are shown in table
639 \begin_inset CommandInset ref
646 It is not clear how these sorts of analyses may affect determination of
648 \begin_inset Formula $M_{\sun}$
651 , but the assumption is that the alternate results should be less than 90°
652 out of phase with previous values.
653 We have no observations of
657 \begin_layout Plain Layout
671 \begin_layout Plain Layout
680 of the electronically submitted abstracts for AAS meetings are error-free.
683 \begin_layout Acknowledgements
687 \begin_layout Standard
688 We are grateful to V.
693 Phillips for doing the math in section
698 \begin_inset CommandInset ref
705 More information on the AASTeX macros package are available at
706 \begin_inset Flex URL
709 \begin_layout Plain Layout
711 http://www.aas.org/publications/aastex
720 \begin_layout Plain Layout
724 anchor{ftp://www.aas.org/pubs/}{AAS ftp site}
732 \begin_layout Appendix
736 \begin_layout Section
740 \begin_layout Standard
741 Consider once again a task that computes profile parameters for a modified
742 Lorentzian of the form
743 \begin_inset Formula \begin{equation}
744 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}\end{equation}
751 \begin_layout MathLetters
752 \begin_inset Formula \[
753 d_{1}=\frac{3}{4}\sqrt{\left(\begin{array}{c}
754 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
755 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}\]
760 \begin_inset Formula \begin{equation}
761 d_{2}=\case{3}{4}\sqrt{\left(\begin{array}{c}
762 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
763 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}\end{equation}
768 \begin_inset Formula \begin{eqnarray}
769 x_{1} & = & (x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta\\
770 y_{1} & = & -(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta\end{eqnarray}
777 \begin_layout Standard
778 For completeness, here is one last equation.
780 \begin_inset Formula \begin{equation}
781 e=mc^{2}\end{equation}
788 \begin_layout References
789 \begin_inset CommandInset bibitem
791 label "Auri\\`ere(1982)"
801 \begin_layout Plain Layout
813 \begin_layout References
814 \begin_inset CommandInset bibitem
816 label "Canizares et al.(1978)"
824 A., Liller, W., and McClintock, J.
830 \begin_layout Plain Layout
842 \begin_layout References
843 \begin_inset CommandInset bibitem
845 label "Djorgovski and King(1984)"
850 Djorgovski, S., and King, I.
856 \begin_layout Plain Layout
868 \begin_layout References
869 \begin_inset CommandInset bibitem
871 label "Hagiwara and Zeppenfeld(1986)"
876 Hagiwara, K., and Zeppenfeld, D.
877 1986, Nucl.Phys., 274, 1
880 \begin_layout References
881 \begin_inset CommandInset bibitem
883 label "Harris and van den Bergh(1984)"
889 E., and van den Bergh, S.
894 \begin_layout Plain Layout
906 \begin_layout References
907 \begin_inset CommandInset bibitem
909 label "H\\`enon(1961)"
915 1961, Ann.d'Ap., 24, 369
918 \begin_layout References
919 \begin_inset CommandInset bibitem
932 \begin_layout Plain Layout
944 \begin_layout References
945 \begin_inset CommandInset bibitem
954 1975, Dynamics of Stellar Systems, A.
955 Hayli, Dordrecht: Reidel, 1975, 99
958 \begin_layout References
959 \begin_inset CommandInset bibitem
961 label "King et al.(1968)"
967 R., Hedemann, E., Hodge, S.
974 \begin_layout Plain Layout
986 \begin_layout References
987 \begin_inset CommandInset bibitem
989 label "Kron et al.(1984)"
996 V., and Wasserman, L.
1002 \begin_layout Plain Layout
1014 \begin_layout References
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1016 LatexCommand bibitem
1017 label "Lynden-Bell and Wood(1968)"
1022 Lynden-Bell, D., and Wood, R.
1027 \begin_layout Plain Layout
1039 \begin_layout References
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1041 LatexCommand bibitem
1042 label "Newell and O'Neil(1978)"
1054 \begin_layout Plain Layout
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1068 LatexCommand bibitem
1069 label "Ortolani et al.(1985)"
1074 Ortolani, S., Rosino, L., and Sandage, A.
1079 \begin_layout Plain Layout
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1093 LatexCommand bibitem
1094 label "Peterson(1976)"
1105 \begin_layout Plain Layout
1117 \begin_layout References
1118 \begin_inset CommandInset bibitem
1119 LatexCommand bibitem
1120 label "Spitzer(1985)"
1126 1985, Dynamics of Star Clusters, J.
1128 Hut, Dordrecht: Reidel, 109
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1156 The processor has to buffer the text of the caption, so it is good not
1157 to be too wordy, but that would make for poor communication as well.
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1179 \begin_layout Standard
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2147 \begin_layout Plain Layout
2161 \begin_layout Table note
2166 \begin_layout Plain Layout
2173 Sample footnote for table
2174 \begin_inset space ~
2178 \begin_inset CommandInset ref
2184 that was generated with the LaTeX table environment
2187 \begin_layout Table note
2192 \begin_layout Plain Layout
2199 Yet another sample footnote for table
2200 \begin_inset space ~
2204 \begin_inset CommandInset ref
2213 \begin_layout Table note
2218 \begin_layout Plain Layout
2225 Another sample footnote for table
2226 \begin_inset space ~
2230 \begin_inset CommandInset ref
2239 \begin_layout TableComments
2240 We can also attach a long-ish paragraph of explanatory material to a table.
2243 tablerefs to append a list of references.
2244 The following references were from a different table: I've patched them
2245 in here to show how they look, but don't take them too seriously---I certainly
2249 \begin_layout TableRefs
2250 (1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
2251 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
2252 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
2253 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
2254 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
2255 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
2256 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
2257 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
2258 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
2260 1991; (26) Olsen 1983.
2268 \begin_layout Standard
2272 \begin_layout Plain Layout
2274 % LyX can load deluxetable files verbatim.
2275 To see this in action, uncomment the following
2283 \begin_layout Standard
2287 \begin_layout Plain Layout
2295 \begin_inset Quotes eld
2302 \begin_layout Plain Layout
2310 \begin_inset Quotes erd
2317 \begin_layout Plain Layout
2319 ) and copy the file table.tex from the AASTeX 5.0 distribution into
2327 \begin_layout Standard
2331 \begin_layout Plain Layout
2333 % the same directory as this file.
2341 \begin_layout Standard
2345 \begin_layout Plain Layout
2353 \begin_inset CommandInset include