1 #LyX 1.5.0svn 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
17 \paperfontsize default
24 \paperorientation portrait
27 \paragraph_separation indent
29 \quotes_language english
32 \paperpagestyle default
33 \tracking_changes false
41 Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AASTeX
52 \begin_layout Standard
65 \begin_layout Standard
76 \begin_layout Affiliation
78 Astronomy Department, University of California, Berkeley, CA 94720
89 \begin_layout Standard
100 \begin_layout Affiliation
102 National Optical Astronomy Observatories, Tucson, AZ 85719
122 \begin_layout Standard
133 \begin_layout Affiliation
135 Space Telescope Science Institute, Baltimore, MD 21218
138 \begin_layout Standard
144 \begin_layout Standard
147 altaffiltext{1}{Visiting Astronomer, Cerro Tololo Inter-American Observatory.
148 CTIO is operated by AURA, Inc.
150 under contract to the National Science Foundation.}
153 \begin_layout Standard
157 \begin_layout Standard
160 altaffiltext{2}{Society of Fellows, Harvard University.}
163 \begin_layout Standard
167 \begin_layout Standard
170 altaffiltext{3}{present address: Center for Astrophysics, 60 Garden Street,
174 \begin_layout Standard
178 \begin_layout Standard
181 altaffiltext{4}{Visiting Programmer, Space Telescope Science Institute}
185 \begin_layout Standard
189 \begin_layout Standard
192 altaffiltext{5}{Patron, Alonso's Bar and Grill}
200 \begin_layout Abstract
202 This is a preliminary report on surface photometry of the major fraction
203 of known globular clusters, to see which of them show the signs of a collapsed
205 We also explore some diversionary mathematics and recreational tables.
209 \begin_layout Keywords
211 clusters: globular, peanut---bosons: bozos
214 \begin_layout Section
219 \begin_layout Standard
221 A focal problem today in the dynamics of globular clusters is core collapse.
222 It has been predicted by theory for decades
226 \begin_layout Standard
229 citep{hen61,lyn68,spi85}
234 , but observation has been less alert to the phenomenon.
235 For many years the central brightness peak in M15
239 \begin_layout Standard
247 seemed a unique anomaly.
252 \begin_layout Standard
260 suggested a central peak in NGC 6397, and a limited photographic survey
265 \begin_layout Standard
268 citep[Paper I]{djo84}
273 found three more cases, including NGC 6624, whose sharp center had often
278 \begin_layout Standard
290 \begin_layout Section
295 \begin_layout Standard
297 All our observations were short direct exposures with CCD's.
298 At Lick Observatory we used a TI 500
299 \begin_inset Formula \( \times \)
302 500 chip and a GEC 575
303 \begin_inset Formula \( \times \)
306 385, on the 1-m Nickel reflector.
307 The only filter available at Lick was red.
308 At CTIO we used a GEC 575
309 \begin_inset Formula \( \times \)
313 \begin_inset Formula \( B,V, \)
317 \begin_inset Formula \( R \)
320 filters, and an RCA 512
321 \begin_inset Formula \( \times \)
325 \begin_inset Formula \( U,B,V,R, \)
329 \begin_inset Formula \( I \)
332 filters, on the 1.5-m reflector.
333 In the CTIO observations we tried to concentrate on the shortest practicable
334 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
335 often kept us from observing in
336 \begin_inset Formula \( U \)
340 \begin_inset Formula \( B \)
344 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
345 sizes were around 3 arcmin.
348 \begin_layout Standard
350 The CCD images are unfortunately not always suitable, for very poor clusters
351 or for clusters with large cores.
352 Since the latter are easily studied by other means, we augmented our own
353 CCD profiles by collecting from the literature a number of star-count profiles
358 \begin_layout Standard
361 citep{kin68,pet76,har84,ort85}
366 , as well as photoelectric profiles
370 \begin_layout Standard
378 and electronographic profiles
382 \begin_layout Standard
391 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
394 \begin_layout Section
399 \begin_layout Standard
401 It has been realized that helicity amplitudes provide a convenient means
406 \begin_layout Standard
408 Footnotes can be inserted like this.
414 These amplitude-level techniques are particularly convenient for calculations
415 involving many Feynman diagrams, where the usual trace techniques for the
416 amplitude squared becomes unwieldy.
417 Our calculations use the helicity techniques developed by other authors
419 \begin_inset LatexCommand \cite{hag86}
423 ; we briefly summarize below.
426 \begin_layout Subsection
431 \begin_layout Standard
434 \begin_inset LatexCommand \label{bozomath}
441 \begin_layout Standard
443 A tree-level amplitude in
444 \begin_inset Formula \( e^{+}e^{-} \)
447 collisions can be expressed in terms of fermion strings of the form
448 \begin_inset Formula \begin{equation}
449 \bar{v}(p_{2},\sigma _{2})P_{-\tau }\hat{a}_{1}\hat{a}_{2}\cdots \hat{a}_{n}u(p_{1},\sigma _{1}),
455 \begin_inset Formula \( p \)
459 \begin_inset Formula \( \sigma \)
463 \begin_inset Formula \( e^{\pm } \)
466 four-momenta and helicities
467 \begin_inset Formula \( (\sigma =\pm 1) \)
471 \begin_inset Formula \( \hat{a}_{i}=a^{\mu }_{i}\gamma _{\nu } \)
475 \begin_inset Formula \( P_{\tau }=\frac{1}{2}(1+\tau \gamma _{5}) \)
478 is a chirality projection operator
479 \begin_inset Formula \( (\tau =\pm1 ) \)
484 \begin_inset Formula \( a^{\mu }_{i} \)
487 may be formed from particle four-momenta, gauge-boson polarization vectors
488 or fermion strings with an uncontracted Lorentz index associated with final-sta
492 \begin_layout NoteToEditor
494 Figures 1 and 2 should appear side-by-side in print
497 \begin_layout Standard
499 In the chiral representation the
500 \begin_inset Formula \( \gamma \)
503 matrices are expressed in terms of
504 \begin_inset Formula \( 2\times 2 \)
508 \begin_inset Formula \( \sigma \)
511 and the unit matrix 1 as
512 \begin_inset Formula \begin{eqnarray}
513 \gamma ^{\mu } & = & \left( \begin{array}{cc}
514 0 & \sigma ^{\mu }_{+}\\
515 \sigma ^{\mu }_{-} & 0
516 \end{array}\right) ,\gamma ^{5}=\left( \begin{array}{cc}
519 \end{array}\right) ,\nonumber \\
520 \sigma ^{\mu }_{\pm } & = & ({\textbf {1}},\pm \sigma ),\nonumber
526 \begin_inset Formula \begin{equation}
527 \hat{a}=\left( \begin{array}{cc}
530 \end{array}\right) ,(\hat{a})_{\pm }=a_{\mu }\sigma ^{\mu }_{\pm },
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}
540 \end{array}\right) ,v={\textbf {(}}\vdag _{+}{\textbf {,}}\vdag _{-}{\textbf {)}}.
548 \begin_layout Standard
550 The Weyl spinors are given in terms of helicity eigenstates
551 \begin_inset Formula \( \chi _{\lambda }(p) \)
555 \begin_inset Formula \( \lambda =\pm1 \)
561 \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)
574 \begin_layout Section
576 Floating material and so forth
579 \begin_layout Standard
581 Consider a task that computes profile parameters for a modified Lorentzian
583 \begin_inset Formula \begin{equation}
584 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
590 \begin_inset Formula \[
591 d_{1}=\sqrt{\left( \begin{array}{c}
592 \frac{x_{1}}{R_{maj}}
593 \end{array}\right) ^{2}+\left( \begin{array}{c}
594 \frac{y_{1}}{R_{min}}
595 \end{array}\right) ^{2}}\]
600 \begin_inset Formula \[
601 d_{2}=\sqrt{\left( \begin{array}{c}
602 \frac{x_{1}}{PR_{maj}}
603 \end{array}\right) ^{2}+\left( \begin{array}{c}
604 \case {y_{1}}{PR_{min}}
605 \end{array}\right) ^{2}}\]
610 \begin_inset Formula \[
611 x_{1}=(x-x_{0})\cos \Theta +(y-y_{0})\sin \Theta \]
616 \begin_inset Formula \[
617 y_{1}=-(x-x_{0})\sin \Theta +(y-y_{0})\cos \Theta \]
624 \begin_layout Standard
627 \begin_inset Formula \( x_{0} \)
631 \begin_inset Formula \( y_{0} \)
634 is the star center, and
635 \begin_inset Formula \( \Theta \)
638 is the angle with the
639 \begin_inset Formula \( x \)
643 Results of this task are shown in table\InsetSpace ~
645 \begin_inset LatexCommand \ref{tbl-2}
650 It is not clear how these sorts of analyses may affect determination of
652 \begin_inset Formula \( M_{\sun } \)
655 , but the assumption is that the alternate results should be less than 90
659 \begin_layout Standard
669 out of phase with previous values.
670 We have no observations of
674 \begin_layout Standard
687 \begin_layout Standard
695 of the electronically submitted abstracts for AAS meetings are error-free.
698 \begin_layout Acknowledgements
702 \begin_layout Standard
704 We are grateful to V.
709 Phillips for doing the math in section\InsetSpace ~
711 \begin_inset LatexCommand \ref{bozomath}
716 More information on the AASTeX macros package are available at
718 \begin_inset LatexCommand \url{http://www.aas.org/publications/aastex}
727 \begin_layout Standard
730 anchor{ftp://www.aas.org/pubs/}{AAS ftp site}
738 \begin_layout Appendix
742 \begin_layout Section
747 \begin_layout Standard
749 Consider once again a task that computes profile parameters for a modified
750 Lorentzian of the form
751 \begin_inset Formula \begin{equation}
752 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
760 \begin_layout MathLetters
763 \begin_inset Formula \[
764 d_{1}=\frac{3}{4}\sqrt{\left( \begin{array}{c}
765 \frac{x_{1}}{R_{maj}}
766 \end{array}\right) ^{2}+\left( \begin{array}{c}
767 \frac{y_{1}}{R_{min}}
768 \end{array}\right) ^{2}}\]
773 \begin_inset Formula \begin{equation}
774 d_{2}=\case {3}{4}\sqrt{\left( \begin{array}{c}
775 \frac{x_{1}}{PR_{maj}}
776 \end{array}\right) ^{2}+\left( \begin{array}{c}
777 \case {y_{1}}{PR_{min}}
778 \end{array}\right) ^{2}}
784 \begin_inset Formula \begin{eqnarray}
785 x_{1} & = & (x-x_{0})\cos \Theta +(y-y_{0})\sin \Theta \\
786 y_{1} & = & -(x-x_{0})\sin \Theta +(y-y_{0})\cos \Theta
794 \begin_layout Standard
796 For completeness, here is one last equation.
798 \begin_inset Formula \begin{equation}
807 \begin_layout References
808 \bibitem [Auri\`ere(1982)]{aur82}
816 \begin_layout Standard
827 \begin_layout References
828 \bibitem [Canizares et al.(1978)]{can78}
833 A., Liller, W., and McClintock, J.
839 \begin_layout Standard
850 \begin_layout References
851 \bibitem [Djorgovski and King(1984)]{djo84}
853 Djorgovski, S., and King, I.
859 \begin_layout Standard
870 \begin_layout References
871 \bibitem [Hagiwara and Zeppenfeld(1986)]{hag86}
873 Hagiwara, K., and Zeppenfeld, D.
874 1986, Nucl.Phys., 274, 1
877 \begin_layout References
878 \bibitem [Harris and van den Bergh(1984)]{har84}
881 E., and van den Bergh, S.
886 \begin_layout Standard
897 \begin_layout References
898 \bibitem [H\`enon(1961)]{hen61}
902 1961, Ann.d'Ap., 24, 369
905 \begin_layout References
906 \bibitem [King(1966)]{kin66}
914 \begin_layout Standard
925 \begin_layout References
926 \bibitem [King(1975)]{kin75}
930 1975, Dynamics of Stellar Systems, A.
931 Hayli, Dordrecht: Reidel, 1975, 99
934 \begin_layout References
935 \bibitem [King et al.(1968)]{kin68}
938 R., Hedemann, E., Hodge, S.
945 \begin_layout Standard
956 \begin_layout References
957 \bibitem [Kron et al.(1984)]{kro84}
961 V., and Wasserman, L.
967 \begin_layout Standard
978 \begin_layout References
979 \bibitem [Lynden-Bell and Wood(1968)]{lyn68}
981 Lynden-Bell, D., and Wood, R.
986 \begin_layout Standard
997 \begin_layout References
998 \bibitem [Newell and O'Neil(1978)]{new78}
1007 \begin_layout Standard
1018 \begin_layout References
1019 \bibitem [Ortolani et al.(1985)]{ort85}
1021 Ortolani, S., Rosino, L., and Sandage, A.
1026 \begin_layout Standard
1037 \begin_layout References
1038 \bibitem [Peterson(1976)]{pet76}
1046 \begin_layout Standard
1057 \begin_layout References
1058 \bibitem [Spitzer(1985)]{spi85}
1061 1985, Dynamics of Star Clusters, J.
1063 Hut, Dordrecht: Reidel, 109
1066 \begin_layout Standard
1072 \begin_layout Standard
1083 \begin_layout FigCaption
1086 \begin_inset LatexCommand \label{fig1}
1090 This is the first figure and it uses sgi9259.eps as its EPS figure file.
1093 \begin_layout FigCaption
1096 \begin_inset LatexCommand \label{fig2}
1100 This is an example of a long figure caption that must be set as a paragraph.
1101 The processor has to buffer the text of the caption, so it is good not
1102 to be too wordy, but that would make for poor communication as well.
1106 \begin_layout FigCaption
1109 \begin_inset LatexCommand \label{fig3}
1113 This figure has no associated EPS file, so the optional parameter is omitted.
1117 \begin_layout Standard
1123 \begin_layout Standard
1134 \begin_layout Standard
1136 \begin_inset Float table
1141 \begin_layout Caption
1143 Terribly relevant tabular information.
1144 \begin_inset LatexCommand \label{tbl-2}
1151 \begin_layout Standard
1154 \begin_inset VSpace medskip
1161 \begin_layout Standard
1164 \begin_inset Tabular
1165 <lyxtabular version="3" rows="7" columns="13">
1166 <features endhead="-1" endfirsthead="-1" endfoot="-1" endlastfoot="-1">
1167 <column alignment="center" valignment="top" width="0pt">
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1171 <column alignment="right" valignment="top" width="0pt">
1172 <column alignment="right" valignment="top" width="0pt">
1173 <column alignment="right" valignment="top" width="0pt">
1174 <column alignment="right" valignment="top" width="0pt">
1175 <column alignment="right" valignment="top" width="0pt">
1176 <column alignment="right" valignment="top" width="0pt">
1177 <column alignment="right" valignment="top" width="0pt">
1178 <column alignment="right" valignment="top" width="0pt">
1179 <column alignment="center" valignment="top" width="0pt">
1181 <cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
1184 \begin_layout Standard
1191 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1194 \begin_layout Standard
1201 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1204 \begin_layout Standard
1207 \begin_inset Formula \( d_{x} \)
1215 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1218 \begin_layout Standard
1221 \begin_inset Formula \( d_{y} \)
1229 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1232 \begin_layout Standard
1235 \begin_inset Formula \( n \)
1243 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1246 \begin_layout Standard
1249 \begin_inset Formula \( \chi ^{2} \)
1257 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1260 \begin_layout Standard
1263 \begin_inset Formula \( R_{maj} \)
1271 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1274 \begin_layout Standard
1277 \begin_inset Formula \( R_{min} \)
1285 <cell multicolumn="1" alignment="center" valignment="top" usebox="none">
1288 \begin_layout Standard
1291 \begin_inset Formula \( P \)
1298 \begin_layout Standard
1311 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1314 \begin_layout Standard
1317 \begin_inset Formula \( PR_{maj} \)
1325 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1328 \begin_layout Standard
1331 \begin_inset Formula \( PR_{min} \)
1339 <cell multicolumn="1" alignment="center" valignment="top" usebox="none">
1342 \begin_layout Standard
1345 \begin_inset Formula \( \Theta \)
1352 \begin_layout Standard
1365 <cell alignment="center" valignment="top" usebox="none">
1368 \begin_layout Standard
1377 <cell alignment="center" valignment="top" usebox="none">
1380 \begin_layout Standard
1386 \begin_layout Standard
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1404 \begin_layout Standard
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1414 \begin_layout Standard
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1424 \begin_layout Standard
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2213 \begin_layout Standard
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2222 tablenotetext{a}{Sample footnote for table~
2224 ref{tbl-2} that was generated with the
2231 \begin_layout Standard
2235 \begin_layout Standard
2238 tablenotetext{b}{Yet another sample footnote for table~
2243 \begin_layout Standard
2247 \begin_layout Standard
2250 tablenotetext{c}{Another sample footnote for table~
2260 \begin_layout TableComments
2262 We can also attach a long-ish paragraph of explanatory material to a table.
2265 tablerefs to append a list of references.
2266 The following references were from a different table: I've patched them
2267 in here to show how they look, but don't take them too seriously---I certainly
2271 \begin_layout TableRefs
2273 (1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
2274 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
2275 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
2276 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
2277 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
2278 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
2279 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
2280 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
2281 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
2283 1991; (26) Olsen 1983.
2290 \begin_layout Standard
2296 \begin_layout Standard
2297 % LyX can load deluxetable files verbatim.
2298 To see this in action, uncomment the following
2306 \begin_layout Standard
2312 \begin_layout Standard
2319 \begin_inset Quotes eld
2326 \begin_layout Standard
2333 \begin_inset Quotes erd
2340 \begin_layout Standard
2341 ) and copy the file table.tex from the AASTeX 5.0 distribution into
2349 \begin_layout Standard
2355 \begin_layout Standard
2356 % the same directory as this file.
2364 \begin_layout Standard
2370 \begin_layout Standard
2377 \begin_inset Include \input{table}