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}
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53 Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AASTeX
60 \begin_inset Flex altaffilmark
63 \begin_layout Plain Layout
71 \begin_inset Flex altaffilmark
74 \begin_layout Plain Layout
83 \begin_layout Affiliation
84 Astronomy Department, University of California, Berkeley, CA 94720
91 \begin_inset Flex altaffilmark
94 \begin_layout Plain Layout
103 \begin_layout Affiliation
104 National Optical Astronomy Observatories, Tucson, AZ 85719
119 \begin_inset Flex altaffilmark
122 \begin_layout Plain Layout
131 \begin_layout Affiliation
132 Space Telescope Science Institute, Baltimore, MD 21218
135 \begin_layout Altaffilation
140 \begin_layout Plain Layout
147 Visiting Astronomer, Cerro Tololo Inter-American Observatory.
148 CTIO is operated by AURA, Inc.
149 \begin_inset space \space{}
152 under contract to the National Science Foundation.
155 \begin_layout Altaffilation
160 \begin_layout Plain Layout
167 Society of Fellows, Harvard University.
170 \begin_layout Altaffilation
175 \begin_layout Plain Layout
182 present address: Center for Astrophysics, 60 Garden Street, Cambridge, MA
186 \begin_layout Altaffilation
191 \begin_layout Plain Layout
198 Visiting Programmer, Space Telescope Science Institute
201 \begin_layout Altaffilation
206 \begin_layout Plain Layout
213 Patron, Alonso's Bar and Grill
216 \begin_layout Abstract
217 This is a preliminary report on surface photometry of the major fraction
218 of known globular clusters, to see which of them show the signs of a collapsed
220 We also explore some diversionary mathematics and recreational tables.
224 \begin_layout Keywords
225 clusters: globular, peanut---bosons: bozos
228 \begin_layout Section
232 \begin_layout Standard
233 A focal problem today in the dynamics of globular clusters is core collapse.
234 It has been predicted by theory for decades
238 \begin_layout Plain Layout
242 citep{hen61,lyn68,spi85}
247 , but observation has been less alert to the phenomenon.
248 For many years the central brightness peak in M15
252 \begin_layout Plain Layout
261 seemed a unique anomaly.
266 \begin_layout Plain Layout
275 suggested a central peak in NGC 6397, and a limited photographic survey
280 \begin_layout Plain Layout
284 citep[Paper I]{djo84}
289 found three more cases, including NGC 6624, whose sharp center had often
294 \begin_layout Plain Layout
307 \begin_layout Section
311 \begin_layout Standard
312 All our observations were short direct exposures with CCD's.
313 At Lick Observatory we used a TI 500
314 \begin_inset Formula $\times$
317 500 chip and a GEC 575
318 \begin_inset Formula $\times$
321 385, on the 1-m Nickel reflector.
322 The only filter available at Lick was red.
323 At CTIO we used a GEC 575
324 \begin_inset Formula $\times$
328 \begin_inset Formula $B,V,$
332 \begin_inset Formula $R$
335 filters, and an RCA 512
336 \begin_inset Formula $\times$
340 \begin_inset Formula $U,B,V,R,$
344 \begin_inset Formula $I$
347 filters, on the 1.5-m reflector.
348 In the CTIO observations we tried to concentrate on the shortest practicable
349 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
350 often kept us from observing in
351 \begin_inset Formula $U$
355 \begin_inset Formula $B$
359 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
360 sizes were around 3 arcmin.
363 \begin_layout Standard
364 The CCD images are unfortunately not always suitable, for very poor clusters
365 or for clusters with large cores.
366 Since the latter are easily studied by other means, we augmented our own
367 CCD profiles by collecting from the literature a number of star-count profiles
372 \begin_layout Plain Layout
376 citep{kin68,pet76,har84,ort85}
381 , as well as photoelectric profiles
385 \begin_layout Plain Layout
394 and electronographic profiles
398 \begin_layout Plain Layout
408 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
411 \begin_layout Section
415 \begin_layout Standard
416 It has been realized that helicity amplitudes provide a convenient means
421 \begin_layout Plain Layout
422 Footnotes can be inserted like this.
428 These amplitude-level techniques are particularly convenient for calculations
429 involving many Feynman diagrams, where the usual trace techniques for the
430 amplitude squared becomes unwieldy.
431 Our calculations use the helicity techniques developed by other authors
433 \begin_inset CommandInset citation
439 ; we briefly summarize below.
442 \begin_layout Subsection
446 \begin_layout Standard
447 \begin_inset CommandInset label
456 \begin_layout Standard
457 A tree-level amplitude in
458 \begin_inset Formula $e^{+}e^{-}$
461 collisions can be expressed in terms of fermion strings of the form
462 \begin_inset Formula \begin{equation}
463 \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}
468 \begin_inset Formula $p$
472 \begin_inset Formula $\sigma$
476 \begin_inset Formula $e^{\pm}$
479 four-momenta and helicities
480 \begin_inset Formula $(\sigma=\pm1)$
484 \begin_inset Formula $\hat{a}_{i}=a_{i}^{\mu}\gamma_{\nu}$
488 \begin_inset Formula $P_{\tau}=\frac{1}{2}(1+\tau\gamma_{5})$
491 is a chirality projection operator
492 \begin_inset Formula $(\tau=\pm1)$
497 \begin_inset Formula $a_{i}^{\mu}$
500 may be formed from particle four-momenta, gauge-boson polarization vectors
501 or fermion strings with an uncontracted Lorentz index associated with final-sta
505 \begin_layout NoteToEditor
506 Figures 1 and 2 should appear side-by-side in print
509 \begin_layout Standard
510 In the chiral representation the
511 \begin_inset Formula $\gamma$
514 matrices are expressed in terms of
515 \begin_inset Formula $2\times2$
519 \begin_inset Formula $\sigma$
522 and the unit matrix 1 as
523 \begin_inset Formula \begin{eqnarray*}
524 \gamma^{\mu} & = & \left(\begin{array}{cc}
525 0 & \sigma_{+}^{\mu}\\
526 \sigma_{-}^{\mu} & 0\end{array}\right),\gamma^{5}=\left(\begin{array}{cc}
528 0 & 1\end{array}\right),\\
529 \sigma_{\pm}^{\mu} & = & ({\textbf{1}},\pm\sigma),\end{eqnarray*}
534 \begin_inset Formula \begin{equation}
535 \hat{a}=\left(\begin{array}{cc}
537 (\hat{a})_{-} & 0\end{array}\right),(\hat{a})_{\pm}=a_{\mu}\sigma_{\pm}^{\mu},\end{equation}
541 The spinors are expressed in terms of two-component Weyl spinors as
542 \begin_inset Formula \begin{equation}
543 u=\left(\begin{array}{c}
545 (u)_{+}\end{array}\right),v={\textbf{(}}\vdag_{+}{\textbf{,}}\vdag_{-}{\textbf{)}}.\end{equation}
552 \begin_layout Standard
553 The Weyl spinors are given in terms of helicity eigenstates
554 \begin_inset Formula $\chi_{\lambda}(p)$
558 \begin_inset Formula $\lambda=\pm1$
564 \begin_layout MathLetters
565 \begin_inset Formula \begin{eqnarray}
566 u(p,\lambda)_{\pm} & = & (E\pm\lambda|{\textbf{p}}|)^{1/2}\chi_{\lambda}(p),\\
567 v(p,\lambda)_{\pm} & = & \pm\lambda(E\mp\lambda|{\textbf{p}}|)^{1/2}\chi_{-\lambda}(p)\end{eqnarray}
574 \begin_layout Section
575 Floating material and so forth
578 \begin_layout Standard
579 Consider a task that computes profile parameters for a modified Lorentzian
581 \begin_inset Formula \begin{equation}
582 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}\end{equation}
587 \begin_inset Formula \[
588 d_{1}=\sqrt{\left(\begin{array}{c}
589 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
590 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}\]
595 \begin_inset Formula \[
596 d_{2}=\sqrt{\left(\begin{array}{c}
597 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
598 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}\]
603 \begin_inset Formula \[
604 x_{1}=(x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta\]
609 \begin_inset Formula \[
610 y_{1}=-(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta\]
617 \begin_layout Standard
619 \begin_inset Formula $x_{0}$
623 \begin_inset Formula $y_{0}$
626 is the star center, and
627 \begin_inset Formula $\Theta$
630 is the angle with the
631 \begin_inset Formula $x$
635 Results of this task are shown in table
640 \begin_inset CommandInset ref
647 It is not clear how these sorts of analyses may affect determination of
649 \begin_inset Formula $M_{\sun}$
652 , but the assumption is that the alternate results should be less than 90°
653 out of phase with previous values.
654 We have no observations of
658 \begin_layout Plain Layout
672 \begin_layout Plain Layout
681 of the electronically submitted abstracts for AAS meetings are error-free.
684 \begin_layout Acknowledgements
688 \begin_layout Standard
689 We are grateful to V.
694 Phillips for doing the math in section
699 \begin_inset CommandInset ref
706 More information on the AASTeX macros package are available at
707 \begin_inset Flex URL
710 \begin_layout Plain Layout
712 http://www.aas.org/publications/aastex
721 \begin_layout Plain Layout
725 anchor{ftp://www.aas.org/pubs/}{AAS ftp site}
733 \begin_layout Appendix
737 \begin_layout Section
741 \begin_layout Standard
742 Consider once again a task that computes profile parameters for a modified
743 Lorentzian of the form
744 \begin_inset Formula \begin{equation}
745 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}\end{equation}
752 \begin_layout MathLetters
753 \begin_inset Formula \[
754 d_{1}=\frac{3}{4}\sqrt{\left(\begin{array}{c}
755 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
756 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}\]
761 \begin_inset Formula \begin{equation}
762 d_{2}=\case{3}{4}\sqrt{\left(\begin{array}{c}
763 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
764 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}\end{equation}
769 \begin_inset Formula \begin{eqnarray}
770 x_{1} & = & (x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta\\
771 y_{1} & = & -(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta\end{eqnarray}
778 \begin_layout Standard
779 For completeness, here is one last equation.
781 \begin_inset Formula \begin{equation}
782 e=mc^{2}\end{equation}
789 \begin_layout References
790 \begin_inset CommandInset bibitem
792 label "Auri\\`ere(1982)"
802 \begin_layout Plain Layout
814 \begin_layout References
815 \begin_inset CommandInset bibitem
817 label "Canizares et al.(1978)"
825 A., Liller, W., and McClintock, J.
831 \begin_layout Plain Layout
843 \begin_layout References
844 \begin_inset CommandInset bibitem
846 label "Djorgovski and King(1984)"
851 Djorgovski, S., and King, I.
857 \begin_layout Plain Layout
869 \begin_layout References
870 \begin_inset CommandInset bibitem
872 label "Hagiwara and Zeppenfeld(1986)"
877 Hagiwara, K., and Zeppenfeld, D.
878 1986, Nucl.Phys., 274, 1
881 \begin_layout References
882 \begin_inset CommandInset bibitem
884 label "Harris and van den Bergh(1984)"
890 E., and van den Bergh, S.
895 \begin_layout Plain Layout
907 \begin_layout References
908 \begin_inset CommandInset bibitem
910 label "H\\`enon(1961)"
916 1961, Ann.d'Ap., 24, 369
919 \begin_layout References
920 \begin_inset CommandInset bibitem
933 \begin_layout Plain Layout
945 \begin_layout References
946 \begin_inset CommandInset bibitem
955 1975, Dynamics of Stellar Systems, A.
956 Hayli, Dordrecht: Reidel, 1975, 99
959 \begin_layout References
960 \begin_inset CommandInset bibitem
962 label "King et al.(1968)"
968 R., Hedemann, E., Hodge, S.
975 \begin_layout Plain Layout
987 \begin_layout References
988 \begin_inset CommandInset bibitem
990 label "Kron et al.(1984)"
997 V., and Wasserman, L.
1003 \begin_layout Plain Layout
1015 \begin_layout References
1016 \begin_inset CommandInset bibitem
1017 LatexCommand bibitem
1018 label "Lynden-Bell and Wood(1968)"
1023 Lynden-Bell, D., and Wood, R.
1028 \begin_layout Plain Layout
1040 \begin_layout References
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1042 LatexCommand bibitem
1043 label "Newell and O'Neil(1978)"
1055 \begin_layout Plain Layout
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1069 LatexCommand bibitem
1070 label "Ortolani et al.(1985)"
1075 Ortolani, S., Rosino, L., and Sandage, A.
1080 \begin_layout Plain Layout
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1094 LatexCommand bibitem
1095 label "Peterson(1976)"
1106 \begin_layout Plain Layout
1118 \begin_layout References
1119 \begin_inset CommandInset bibitem
1120 LatexCommand bibitem
1121 label "Spitzer(1985)"
1127 1985, Dynamics of Star Clusters, J.
1129 Hut, Dordrecht: Reidel, 109
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1157 The processor has to buffer the text of the caption, so it is good not
1158 to be too wordy, but that would make for poor communication as well.
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1367 \begin_inset Formula $PR_{min}$
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2148 \begin_layout Plain Layout
2162 \begin_layout Table note
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2174 Sample footnote for table
2175 \begin_inset space ~
2179 \begin_inset CommandInset ref
2185 that was generated with the LaTeX table environment
2188 \begin_layout Table note
2193 \begin_layout Plain Layout
2200 Yet another sample footnote for table
2201 \begin_inset space ~
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2214 \begin_layout Table note
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2226 Another sample footnote for table
2227 \begin_inset space ~
2231 \begin_inset CommandInset ref
2240 \begin_layout TableComments
2241 We can also attach a long-ish paragraph of explanatory material to a table.
2244 tablerefs to append a list of references.
2245 The following references were from a different table: I've patched them
2246 in here to show how they look, but don't take them too seriously---I certainly
2250 \begin_layout TableRefs
2251 (1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
2252 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
2253 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
2254 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
2255 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
2256 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
2257 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
2258 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
2259 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
2261 1991; (26) Olsen 1983.
2269 \begin_layout Standard
2273 \begin_layout Plain Layout
2275 % LyX can load deluxetable files verbatim.
2276 To see this in action, uncomment the following
2284 \begin_layout Standard
2288 \begin_layout Plain Layout
2296 \begin_inset Quotes eld
2303 \begin_layout Plain Layout
2311 \begin_inset Quotes erd
2318 \begin_layout Plain Layout
2320 ) and copy the file table.tex from the AASTeX 5.0 distribution into
2328 \begin_layout Standard
2332 \begin_layout Plain Layout
2334 % the same directory as this file.
2342 \begin_layout Standard
2346 \begin_layout Plain Layout
2354 \begin_inset CommandInset include