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}
13 \use_default_options false
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53 Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AASTeX
63 \begin_layout Plain Layout
77 \begin_layout Plain Layout
89 \begin_layout Affiliation
90 Astronomy Department, University of California, Berkeley, CA 94720
100 \begin_layout Plain Layout
112 \begin_layout Affiliation
113 National Optical Astronomy Observatories, Tucson, AZ 85719
131 \begin_layout Plain Layout
143 \begin_layout Affiliation
144 Space Telescope Science Institute, Baltimore, MD 21218
147 \begin_layout Standard
151 \begin_layout Plain Layout
155 altaffiltext{1}{Visiting Astronomer, Cerro Tololo Inter-American Observatory.
156 CTIO is operated by AURA, Inc.
158 under contract to the National Science Foundation.}
161 \begin_layout Plain Layout
165 \begin_layout Plain Layout
169 altaffiltext{2}{Society of Fellows, Harvard University.}
172 \begin_layout Plain Layout
176 \begin_layout Plain Layout
180 altaffiltext{3}{present address: Center for Astrophysics, 60 Garden Street,
184 \begin_layout Plain Layout
188 \begin_layout Plain Layout
192 altaffiltext{4}{Visiting Programmer, Space Telescope Science Institute}
196 \begin_layout Plain Layout
200 \begin_layout Plain Layout
204 altaffiltext{5}{Patron, Alonso's Bar and Grill}
212 \begin_layout Abstract
213 This is a preliminary report on surface photometry of the major fraction
214 of known globular clusters, to see which of them show the signs of a collapsed
216 We also explore some diversionary mathematics and recreational tables.
220 \begin_layout Keywords
221 clusters: globular, peanut---bosons: bozos
224 \begin_layout Section
228 \begin_layout Standard
229 A focal problem today in the dynamics of globular clusters is core collapse.
230 It has been predicted by theory for decades
234 \begin_layout Plain Layout
238 citep{hen61,lyn68,spi85}
243 , but observation has been less alert to the phenomenon.
244 For many years the central brightness peak in M15
248 \begin_layout Plain Layout
257 seemed a unique anomaly.
262 \begin_layout Plain Layout
271 suggested a central peak in NGC 6397, and a limited photographic survey
276 \begin_layout Plain Layout
280 citep[Paper I]{djo84}
285 found three more cases, including NGC 6624, whose sharp center had often
290 \begin_layout Plain Layout
303 \begin_layout Section
307 \begin_layout Standard
308 All our observations were short direct exposures with CCD's.
309 At Lick Observatory we used a TI 500
310 \begin_inset Formula $\times$
313 500 chip and a GEC 575
314 \begin_inset Formula $\times$
317 385, on the 1-m Nickel reflector.
318 The only filter available at Lick was red.
319 At CTIO we used a GEC 575
320 \begin_inset Formula $\times$
324 \begin_inset Formula $B,V,$
328 \begin_inset Formula $R$
331 filters, and an RCA 512
332 \begin_inset Formula $\times$
336 \begin_inset Formula $U,B,V,R,$
340 \begin_inset Formula $I$
343 filters, on the 1.5-m reflector.
344 In the CTIO observations we tried to concentrate on the shortest practicable
345 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
346 often kept us from observing in
347 \begin_inset Formula $U$
351 \begin_inset Formula $B$
355 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
356 sizes were around 3 arcmin.
359 \begin_layout Standard
360 The CCD images are unfortunately not always suitable, for very poor clusters
361 or for clusters with large cores.
362 Since the latter are easily studied by other means, we augmented our own
363 CCD profiles by collecting from the literature a number of star-count profiles
368 \begin_layout Plain Layout
372 citep{kin68,pet76,har84,ort85}
377 , as well as photoelectric profiles
381 \begin_layout Plain Layout
390 and electronographic profiles
394 \begin_layout Plain Layout
404 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
407 \begin_layout Section
411 \begin_layout Standard
412 It has been realized that helicity amplitudes provide a convenient means
417 \begin_layout Plain Layout
418 Footnotes can be inserted like this.
424 These amplitude-level techniques are particularly convenient for calculations
425 involving many Feynman diagrams, where the usual trace techniques for the
426 amplitude squared becomes unwieldy.
427 Our calculations use the helicity techniques developed by other authors
429 \begin_inset CommandInset citation
435 ; we briefly summarize below.
438 \begin_layout Subsection
442 \begin_layout Standard
443 \begin_inset CommandInset label
452 \begin_layout Standard
453 A tree-level amplitude in
454 \begin_inset Formula $e^{+}e^{-}$
457 collisions can be expressed in terms of fermion strings of the form
458 \begin_inset Formula \begin{equation}
459 \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}
464 \begin_inset Formula $p$
468 \begin_inset Formula $\sigma$
472 \begin_inset Formula $e^{\pm}$
475 four-momenta and helicities
476 \begin_inset Formula $(\sigma=\pm1)$
480 \begin_inset Formula $\hat{a}_{i}=a_{i}^{\mu}\gamma_{\nu}$
484 \begin_inset Formula $P_{\tau}=\frac{1}{2}(1+\tau\gamma_{5})$
487 is a chirality projection operator
488 \begin_inset Formula $(\tau=\pm1)$
493 \begin_inset Formula $a_{i}^{\mu}$
496 may be formed from particle four-momenta, gauge-boson polarization vectors
497 or fermion strings with an uncontracted Lorentz index associated with final-sta
501 \begin_layout NoteToEditor
502 Figures 1 and 2 should appear side-by-side in print
505 \begin_layout Standard
506 In the chiral representation the
507 \begin_inset Formula $\gamma$
510 matrices are expressed in terms of
511 \begin_inset Formula $2\times2$
515 \begin_inset Formula $\sigma$
518 and the unit matrix 1 as
519 \begin_inset Formula \begin{eqnarray*}
520 \gamma^{\mu} & = & \left(\begin{array}{cc}
521 0 & \sigma_{+}^{\mu}\\
522 \sigma_{-}^{\mu} & 0\end{array}\right),\gamma^{5}=\left(\begin{array}{cc}
524 0 & 1\end{array}\right),\\
525 \sigma_{\pm}^{\mu} & = & ({\textbf{1}},\pm\sigma),\end{eqnarray*}
530 \begin_inset Formula \begin{equation}
531 \hat{a}=\left(\begin{array}{cc}
533 (\hat{a})_{-} & 0\end{array}\right),(\hat{a})_{\pm}=a_{\mu}\sigma_{\pm}^{\mu},\end{equation}
537 The spinors are expressed in terms of two-component Weyl spinors as
538 \begin_inset Formula \begin{equation}
539 u=\left(\begin{array}{c}
541 (u)_{+}\end{array}\right),v={\textbf{(}}\vdag_{+}{\textbf{,}}\vdag_{-}{\textbf{)}}.\end{equation}
548 \begin_layout Standard
549 The Weyl spinors are given in terms of helicity eigenstates
550 \begin_inset Formula $\chi_{\lambda}(p)$
554 \begin_inset Formula $\lambda=\pm1$
560 \begin_layout MathLetters
561 \begin_inset Formula \begin{eqnarray}
562 u(p,\lambda)_{\pm} & = & (E\pm\lambda|{\textbf{p}}|)^{1/2}\chi_{\lambda}(p),\\
563 v(p,\lambda)_{\pm} & = & \pm\lambda(E\mp\lambda|{\textbf{p}}|)^{1/2}\chi_{-\lambda}(p)\end{eqnarray}
570 \begin_layout Section
571 Floating material and so forth
574 \begin_layout Standard
575 Consider a task that computes profile parameters for a modified Lorentzian
577 \begin_inset Formula \begin{equation}
578 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}\end{equation}
583 \begin_inset Formula \[
584 d_{1}=\sqrt{\left(\begin{array}{c}
585 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
586 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}\]
591 \begin_inset Formula \[
592 d_{2}=\sqrt{\left(\begin{array}{c}
593 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
594 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}\]
599 \begin_inset Formula \[
600 x_{1}=(x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta\]
605 \begin_inset Formula \[
606 y_{1}=-(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta\]
613 \begin_layout Standard
615 \begin_inset Formula $x_{0}$
619 \begin_inset Formula $y_{0}$
622 is the star center, and
623 \begin_inset Formula $\Theta$
626 is the angle with the
627 \begin_inset Formula $x$
631 Results of this task are shown in table
636 \begin_inset CommandInset ref
643 It is not clear how these sorts of analyses may affect determination of
645 \begin_inset Formula $M_{\sun}$
648 , but the assumption is that the alternate results should be less than 90
652 \begin_layout Plain Layout
663 out of phase with previous values.
664 We have no observations of
668 \begin_layout Plain Layout
682 \begin_layout Plain Layout
691 of the electronically submitted abstracts for AAS meetings are error-free.
694 \begin_layout Acknowledgements
698 \begin_layout Standard
699 We are grateful to V.
704 Phillips for doing the math in section
709 \begin_inset CommandInset ref
716 More information on the AASTeX macros package are available at
717 \begin_inset Flex URL
720 \begin_layout Plain Layout
722 http://www.aas.org/publications/aastex
731 \begin_layout Plain Layout
735 anchor{ftp://www.aas.org/pubs/}{AAS ftp site}
743 \begin_layout Appendix
747 \begin_layout Section
751 \begin_layout Standard
752 Consider once again a task that computes profile parameters for a modified
753 Lorentzian of the form
754 \begin_inset Formula \begin{equation}
755 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}\end{equation}
762 \begin_layout MathLetters
763 \begin_inset Formula \[
764 d_{1}=\frac{3}{4}\sqrt{\left(\begin{array}{c}
765 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
766 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}\]
771 \begin_inset Formula \begin{equation}
772 d_{2}=\case{3}{4}\sqrt{\left(\begin{array}{c}
773 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
774 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}\end{equation}
779 \begin_inset Formula \begin{eqnarray}
780 x_{1} & = & (x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta\\
781 y_{1} & = & -(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta\end{eqnarray}
788 \begin_layout Standard
789 For completeness, here is one last equation.
791 \begin_inset Formula \begin{equation}
792 e=mc^{2}\end{equation}
799 \begin_layout References
800 \begin_inset CommandInset bibitem
802 label "Auri\\`ere(1982)"
812 \begin_layout Plain Layout
824 \begin_layout References
825 \begin_inset CommandInset bibitem
827 label "Canizares et al.(1978)"
835 A., Liller, W., and McClintock, J.
841 \begin_layout Plain Layout
853 \begin_layout References
854 \begin_inset CommandInset bibitem
856 label "Djorgovski and King(1984)"
861 Djorgovski, S., and King, I.
867 \begin_layout Plain Layout
879 \begin_layout References
880 \begin_inset CommandInset bibitem
882 label "Hagiwara and Zeppenfeld(1986)"
887 Hagiwara, K., and Zeppenfeld, D.
888 1986, Nucl.Phys., 274, 1
891 \begin_layout References
892 \begin_inset CommandInset bibitem
894 label "Harris and van den Bergh(1984)"
900 E., and van den Bergh, S.
905 \begin_layout Plain Layout
917 \begin_layout References
918 \begin_inset CommandInset bibitem
920 label "H\\`enon(1961)"
926 1961, Ann.d'Ap., 24, 369
929 \begin_layout References
930 \begin_inset CommandInset bibitem
943 \begin_layout Plain Layout
955 \begin_layout References
956 \begin_inset CommandInset bibitem
965 1975, Dynamics of Stellar Systems, A.
966 Hayli, Dordrecht: Reidel, 1975, 99
969 \begin_layout References
970 \begin_inset CommandInset bibitem
972 label "King et al.(1968)"
978 R., Hedemann, E., Hodge, S.
985 \begin_layout Plain Layout
997 \begin_layout References
998 \begin_inset CommandInset bibitem
1000 label "Kron et al.(1984)"
1007 V., and Wasserman, L.
1013 \begin_layout Plain Layout
1025 \begin_layout References
1026 \begin_inset CommandInset bibitem
1027 LatexCommand bibitem
1028 label "Lynden-Bell and Wood(1968)"
1033 Lynden-Bell, D., and Wood, R.
1038 \begin_layout Plain Layout
1050 \begin_layout References
1051 \begin_inset CommandInset bibitem
1052 LatexCommand bibitem
1053 label "Newell and O'Neil(1978)"
1065 \begin_layout Plain Layout
1077 \begin_layout References
1078 \begin_inset CommandInset bibitem
1079 LatexCommand bibitem
1080 label "Ortolani et al.(1985)"
1085 Ortolani, S., Rosino, L., and Sandage, A.
1090 \begin_layout Plain Layout
1102 \begin_layout References
1103 \begin_inset CommandInset bibitem
1104 LatexCommand bibitem
1105 label "Peterson(1976)"
1116 \begin_layout Plain Layout
1128 \begin_layout References
1129 \begin_inset CommandInset bibitem
1130 LatexCommand bibitem
1131 label "Spitzer(1985)"
1137 1985, Dynamics of Star Clusters, J.
1139 Hut, Dordrecht: Reidel, 109
1142 \begin_layout Standard
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1175 This is an example of a long figure caption that must be set as a paragraph.
1176 The processor has to buffer the text of the caption, so it is good not
1177 to be too wordy, but that would make for poor communication as well.
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1192 \begin_layout Standard
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1208 \begin_layout Standard
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1218 Terribly relevant tabular information.
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1365 \begin_layout Plain Layout
1379 <cell alignment="right" valignment="top" usebox="none">
1382 \begin_layout Plain Layout
1384 \begin_inset Formula $PR_{maj}$
1392 <cell alignment="right" valignment="top" usebox="none">
1395 \begin_layout Plain Layout
1397 \begin_inset Formula $PR_{min}$
1405 <cell multicolumn="1" alignment="center" valignment="top" usebox="none">
1408 \begin_layout Plain Layout
1409 \begin_inset Formula $\Theta$
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2209 tablenotetext{a}{Sample footnote for table~
2211 ref{tbl-2} that was generated with the
2218 \begin_layout Plain Layout
2222 \begin_layout Plain Layout
2226 tablenotetext{b}{Yet another sample footnote for table~
2231 \begin_layout Plain Layout
2235 \begin_layout Plain Layout
2239 tablenotetext{c}{Another sample footnote for table~
2249 \begin_layout TableComments
2250 We can also attach a long-ish paragraph of explanatory material to a table.
2253 tablerefs to append a list of references.
2254 The following references were from a different table: I've patched them
2255 in here to show how they look, but don't take them too seriously---I certainly
2259 \begin_layout TableRefs
2260 (1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
2261 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
2262 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
2263 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
2264 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
2265 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
2266 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
2267 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
2268 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
2270 1991; (26) Olsen 1983.
2278 \begin_layout Standard
2282 \begin_layout Plain Layout
2284 % LyX can load deluxetable files verbatim.
2285 To see this in action, uncomment the following
2293 \begin_layout Standard
2297 \begin_layout Plain Layout
2305 \begin_inset Quotes eld
2312 \begin_layout Plain Layout
2320 \begin_inset Quotes erd
2327 \begin_layout Plain Layout
2329 ) and copy the file table.tex from the AASTeX 5.0 distribution into
2337 \begin_layout Standard
2341 \begin_layout Plain Layout
2343 % the same directory as this file.
2351 \begin_layout Standard
2355 \begin_layout Plain Layout
2363 \begin_inset CommandInset include