1 #LyX 1.6.0svn 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}
14 \inputencoding default
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39 \quotes_language english
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51 Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AASTeX
61 \begin_layout Plain Layout
75 \begin_layout Plain Layout
87 \begin_layout Affiliation
88 Astronomy Department, University of California, Berkeley, CA 94720
98 \begin_layout Plain Layout
110 \begin_layout Affiliation
111 National Optical Astronomy Observatories, Tucson, AZ 85719
129 \begin_layout Plain Layout
141 \begin_layout Affiliation
142 Space Telescope Science Institute, Baltimore, MD 21218
145 \begin_layout Standard
149 \begin_layout Plain Layout
153 altaffiltext{1}{Visiting Astronomer, Cerro Tololo Inter-American Observatory.
154 CTIO is operated by AURA, Inc.
156 under contract to the National Science Foundation.}
159 \begin_layout Plain Layout
163 \begin_layout Plain Layout
167 altaffiltext{2}{Society of Fellows, Harvard University.}
170 \begin_layout Plain Layout
174 \begin_layout Plain Layout
178 altaffiltext{3}{present address: Center for Astrophysics, 60 Garden Street,
182 \begin_layout Plain Layout
186 \begin_layout Plain Layout
190 altaffiltext{4}{Visiting Programmer, Space Telescope Science Institute}
194 \begin_layout Plain Layout
198 \begin_layout Plain Layout
202 altaffiltext{5}{Patron, Alonso's Bar and Grill}
210 \begin_layout Abstract
211 This is a preliminary report on surface photometry of the major fraction
212 of known globular clusters, to see which of them show the signs of a collapsed
214 We also explore some diversionary mathematics and recreational tables.
218 \begin_layout Keywords
219 clusters: globular, peanut---bosons: bozos
222 \begin_layout Section
226 \begin_layout Standard
227 A focal problem today in the dynamics of globular clusters is core collapse.
228 It has been predicted by theory for decades
232 \begin_layout Plain Layout
236 citep{hen61,lyn68,spi85}
241 , but observation has been less alert to the phenomenon.
242 For many years the central brightness peak in M15
246 \begin_layout Plain Layout
255 seemed a unique anomaly.
260 \begin_layout Plain Layout
269 suggested a central peak in NGC 6397, and a limited photographic survey
274 \begin_layout Plain Layout
278 citep[Paper I]{djo84}
283 found three more cases, including NGC 6624, whose sharp center had often
288 \begin_layout Plain Layout
301 \begin_layout Section
305 \begin_layout Standard
306 All our observations were short direct exposures with CCD's.
307 At Lick Observatory we used a TI 500
308 \begin_inset Formula $\times$
311 500 chip and a GEC 575
312 \begin_inset Formula $\times$
315 385, on the 1-m Nickel reflector.
316 The only filter available at Lick was red.
317 At CTIO we used a GEC 575
318 \begin_inset Formula $\times$
322 \begin_inset Formula $B,V,$
326 \begin_inset Formula $R$
329 filters, and an RCA 512
330 \begin_inset Formula $\times$
334 \begin_inset Formula $U,B,V,R,$
338 \begin_inset Formula $I$
341 filters, on the 1.5-m reflector.
342 In the CTIO observations we tried to concentrate on the shortest practicable
343 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
344 often kept us from observing in
345 \begin_inset Formula $U$
349 \begin_inset Formula $B$
353 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
354 sizes were around 3 arcmin.
357 \begin_layout Standard
358 The CCD images are unfortunately not always suitable, for very poor clusters
359 or for clusters with large cores.
360 Since the latter are easily studied by other means, we augmented our own
361 CCD profiles by collecting from the literature a number of star-count profiles
366 \begin_layout Plain Layout
370 citep{kin68,pet76,har84,ort85}
375 , as well as photoelectric profiles
379 \begin_layout Plain Layout
388 and electronographic profiles
392 \begin_layout Plain Layout
402 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
405 \begin_layout Section
409 \begin_layout Standard
410 It has been realized that helicity amplitudes provide a convenient means
415 \begin_layout Plain Layout
416 Footnotes can be inserted like this.
422 These amplitude-level techniques are particularly convenient for calculations
423 involving many Feynman diagrams, where the usual trace techniques for the
424 amplitude squared becomes unwieldy.
425 Our calculations use the helicity techniques developed by other authors
427 \begin_inset CommandInset citation
433 ; we briefly summarize below.
436 \begin_layout Subsection
440 \begin_layout Standard
441 \begin_inset CommandInset label
450 \begin_layout Standard
451 A tree-level amplitude in
452 \begin_inset Formula $e^{+}e^{-}$
455 collisions can be expressed in terms of fermion strings of the form
456 \begin_inset Formula \begin{equation}
457 \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}
462 \begin_inset Formula $p$
466 \begin_inset Formula $\sigma$
470 \begin_inset Formula $e^{\pm}$
473 four-momenta and helicities
474 \begin_inset Formula $(\sigma=\pm1)$
478 \begin_inset Formula $\hat{a}_{i}=a_{i}^{\mu}\gamma_{\nu}$
482 \begin_inset Formula $P_{\tau}=\frac{1}{2}(1+\tau\gamma_{5})$
485 is a chirality projection operator
486 \begin_inset Formula $(\tau=\pm1)$
491 \begin_inset Formula $a_{i}^{\mu}$
494 may be formed from particle four-momenta, gauge-boson polarization vectors
495 or fermion strings with an uncontracted Lorentz index associated with final-sta
499 \begin_layout NoteToEditor
500 Figures 1 and 2 should appear side-by-side in print
503 \begin_layout Standard
504 In the chiral representation the
505 \begin_inset Formula $\gamma$
508 matrices are expressed in terms of
509 \begin_inset Formula $2\times2$
513 \begin_inset Formula $\sigma$
516 and the unit matrix 1 as
517 \begin_inset Formula \begin{eqnarray*}
518 \gamma^{\mu} & = & \left(\begin{array}{cc}
519 0 & \sigma_{+}^{\mu}\\
520 \sigma_{-}^{\mu} & 0\end{array}\right),\gamma^{5}=\left(\begin{array}{cc}
522 0 & 1\end{array}\right),\\
523 \sigma_{\pm}^{\mu} & = & ({\textbf{1}},\pm\sigma),\end{eqnarray*}
528 \begin_inset Formula \begin{equation}
529 \hat{a}=\left(\begin{array}{cc}
531 (\hat{a})_{-} & 0\end{array}\right),(\hat{a})_{\pm}=a_{\mu}\sigma_{\pm}^{\mu},\end{equation}
535 The spinors are expressed in terms of two-component Weyl spinors as
536 \begin_inset Formula \begin{equation}
537 u=\left(\begin{array}{c}
539 (u)_{+}\end{array}\right),v={\textbf{(}}\vdag_{+}{\textbf{,}}\vdag_{-}{\textbf{)}}.\end{equation}
546 \begin_layout Standard
547 The Weyl spinors are given in terms of helicity eigenstates
548 \begin_inset Formula $\chi_{\lambda}(p)$
552 \begin_inset Formula $\lambda=\pm1$
558 \begin_layout MathLetters
559 \begin_inset Formula \begin{eqnarray}
560 u(p,\lambda)_{\pm} & = & (E\pm\lambda|{\textbf{p}}|)^{1/2}\chi_{\lambda}(p),\\
561 v(p,\lambda)_{\pm} & = & \pm\lambda(E\mp\lambda|{\textbf{p}}|)^{1/2}\chi_{-\lambda}(p)\end{eqnarray}
568 \begin_layout Section
569 Floating material and so forth
572 \begin_layout Standard
573 Consider a task that computes profile parameters for a modified Lorentzian
575 \begin_inset Formula \begin{equation}
576 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}\end{equation}
581 \begin_inset Formula \[
582 d_{1}=\sqrt{\left(\begin{array}{c}
583 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
584 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}\]
589 \begin_inset Formula \[
590 d_{2}=\sqrt{\left(\begin{array}{c}
591 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
592 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}\]
597 \begin_inset Formula \[
598 x_{1}=(x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta\]
603 \begin_inset Formula \[
604 y_{1}=-(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta\]
611 \begin_layout Standard
613 \begin_inset Formula $x_{0}$
617 \begin_inset Formula $y_{0}$
620 is the star center, and
621 \begin_inset Formula $\Theta$
624 is the angle with the
625 \begin_inset Formula $x$
629 Results of this task are shown in table
634 \begin_inset CommandInset ref
641 It is not clear how these sorts of analyses may affect determination of
643 \begin_inset Formula $M_{\sun}$
646 , but the assumption is that the alternate results should be less than 90
650 \begin_layout Plain Layout
661 out of phase with previous values.
662 We have no observations of
666 \begin_layout Plain Layout
680 \begin_layout Plain Layout
689 of the electronically submitted abstracts for AAS meetings are error-free.
692 \begin_layout Acknowledgements
696 \begin_layout Standard
697 We are grateful to V.
702 Phillips for doing the math in section
707 \begin_inset CommandInset ref
714 More information on the AASTeX macros package are available at
715 \begin_inset Flex URL
718 \begin_layout Plain Layout
720 http://www.aas.org/publications/aastex
729 \begin_layout Plain Layout
733 anchor{ftp://www.aas.org/pubs/}{AAS ftp site}
741 \begin_layout Appendix
745 \begin_layout Section
749 \begin_layout Standard
750 Consider once again a task that computes profile parameters for a modified
751 Lorentzian of the form
752 \begin_inset Formula \begin{equation}
753 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}\end{equation}
760 \begin_layout MathLetters
761 \begin_inset Formula \[
762 d_{1}=\frac{3}{4}\sqrt{\left(\begin{array}{c}
763 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
764 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}\]
769 \begin_inset Formula \begin{equation}
770 d_{2}=\case{3}{4}\sqrt{\left(\begin{array}{c}
771 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
772 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}\end{equation}
777 \begin_inset Formula \begin{eqnarray}
778 x_{1} & = & (x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta\\
779 y_{1} & = & -(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta\end{eqnarray}
786 \begin_layout Standard
787 For completeness, here is one last equation.
789 \begin_inset Formula \begin{equation}
790 e=mc^{2}\end{equation}
797 \begin_layout References
798 \begin_inset CommandInset bibitem
800 label "Auri\\`ere(1982)"
810 \begin_layout Plain Layout
822 \begin_layout References
823 \begin_inset CommandInset bibitem
825 label "Canizares et al.(1978)"
833 A., Liller, W., and McClintock, J.
839 \begin_layout Plain Layout
851 \begin_layout References
852 \begin_inset CommandInset bibitem
854 label "Djorgovski and King(1984)"
859 Djorgovski, S., and King, I.
865 \begin_layout Plain Layout
877 \begin_layout References
878 \begin_inset CommandInset bibitem
880 label "Hagiwara and Zeppenfeld(1986)"
885 Hagiwara, K., and Zeppenfeld, D.
886 1986, Nucl.Phys., 274, 1
889 \begin_layout References
890 \begin_inset CommandInset bibitem
892 label "Harris and van den Bergh(1984)"
898 E., and van den Bergh, S.
903 \begin_layout Plain Layout
915 \begin_layout References
916 \begin_inset CommandInset bibitem
918 label "H\\`enon(1961)"
924 1961, Ann.d'Ap., 24, 369
927 \begin_layout References
928 \begin_inset CommandInset bibitem
941 \begin_layout Plain Layout
953 \begin_layout References
954 \begin_inset CommandInset bibitem
963 1975, Dynamics of Stellar Systems, A.
964 Hayli, Dordrecht: Reidel, 1975, 99
967 \begin_layout References
968 \begin_inset CommandInset bibitem
970 label "King et al.(1968)"
976 R., Hedemann, E., Hodge, S.
983 \begin_layout Plain Layout
995 \begin_layout References
996 \begin_inset CommandInset bibitem
998 label "Kron et al.(1984)"
1005 V., and Wasserman, L.
1011 \begin_layout Plain Layout
1023 \begin_layout References
1024 \begin_inset CommandInset bibitem
1025 LatexCommand bibitem
1026 label "Lynden-Bell and Wood(1968)"
1031 Lynden-Bell, D., and Wood, R.
1036 \begin_layout Plain Layout
1048 \begin_layout References
1049 \begin_inset CommandInset bibitem
1050 LatexCommand bibitem
1051 label "Newell and O'Neil(1978)"
1063 \begin_layout Plain Layout
1075 \begin_layout References
1076 \begin_inset CommandInset bibitem
1077 LatexCommand bibitem
1078 label "Ortolani et al.(1985)"
1083 Ortolani, S., Rosino, L., and Sandage, A.
1088 \begin_layout Plain Layout
1100 \begin_layout References
1101 \begin_inset CommandInset bibitem
1102 LatexCommand bibitem
1103 label "Peterson(1976)"
1114 \begin_layout Plain Layout
1126 \begin_layout References
1127 \begin_inset CommandInset bibitem
1128 LatexCommand bibitem
1129 label "Spitzer(1985)"
1135 1985, Dynamics of Star Clusters, J.
1137 Hut, Dordrecht: Reidel, 109
1140 \begin_layout Standard
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1174 The processor has to buffer the text of the caption, so it is good not
1175 to be too wordy, but that would make for poor communication as well.
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1190 \begin_layout Standard
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1206 \begin_layout Standard
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1216 Terribly relevant tabular information.
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1363 \begin_layout Plain Layout
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1380 \begin_layout Plain Layout
1382 \begin_inset Formula $PR_{maj}$
1390 <cell alignment="right" valignment="top" usebox="none">
1393 \begin_layout Plain Layout
1395 \begin_inset Formula $PR_{min}$
1403 <cell multicolumn="1" alignment="center" valignment="top" usebox="none">
1406 \begin_layout Plain Layout
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2207 tablenotetext{a}{Sample footnote for table~
2209 ref{tbl-2} that was generated with the
2216 \begin_layout Plain Layout
2220 \begin_layout Plain Layout
2224 tablenotetext{b}{Yet another sample footnote for table~
2229 \begin_layout Plain Layout
2233 \begin_layout Plain Layout
2237 tablenotetext{c}{Another sample footnote for table~
2247 \begin_layout TableComments
2248 We can also attach a long-ish paragraph of explanatory material to a table.
2251 tablerefs to append a list of references.
2252 The following references were from a different table: I've patched them
2253 in here to show how they look, but don't take them too seriously---I certainly
2257 \begin_layout TableRefs
2258 (1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
2259 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
2260 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
2261 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
2262 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
2263 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
2264 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
2265 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
2266 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
2268 1991; (26) Olsen 1983.
2276 \begin_layout Standard
2280 \begin_layout Plain Layout
2282 % LyX can load deluxetable files verbatim.
2283 To see this in action, uncomment the following
2291 \begin_layout Standard
2295 \begin_layout Plain Layout
2303 \begin_inset Quotes eld
2310 \begin_layout Plain Layout
2318 \begin_inset Quotes erd
2325 \begin_layout Plain Layout
2327 ) and copy the file table.tex from the AASTeX 5.0 distribution into
2335 \begin_layout Standard
2339 \begin_layout Plain Layout
2341 % the same directory as this file.
2349 \begin_layout Standard
2353 \begin_layout Plain Layout
2361 \begin_inset CommandInset include