1 #LyX 1.4.0cvs 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
33 \paperpagestyle default
34 \tracking_changes false
42 Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AASTeX
53 \begin_layout Standard
66 \begin_layout Standard
77 \begin_layout Affiliation
79 Astronomy Department, University of California, Berkeley, CA 94720
90 \begin_layout Standard
101 \begin_layout Affiliation
103 National Optical Astronomy Observatories, Tucson, AZ 85719
123 \begin_layout Standard
134 \begin_layout Affiliation
136 Space Telescope Science Institute, Baltimore, MD 21218
139 \begin_layout Standard
145 \begin_layout Standard
148 altaffiltext{1}{Visiting Astronomer, Cerro Tololo Inter-American Observatory.
149 CTIO is operated by AURA, Inc.
151 under contract to the National Science Foundation.}
154 \begin_layout Standard
158 \begin_layout Standard
161 altaffiltext{2}{Society of Fellows, Harvard University.}
164 \begin_layout Standard
168 \begin_layout Standard
171 altaffiltext{3}{present address: Center for Astrophysics, 60 Garden Street,
175 \begin_layout Standard
179 \begin_layout Standard
182 altaffiltext{4}{Visiting Programmer, Space Telescope Science Institute}
186 \begin_layout Standard
190 \begin_layout Standard
193 altaffiltext{5}{Patron, Alonso's Bar and Grill}
201 \begin_layout Abstract
203 This is a preliminary report on surface photometry of the major fraction
204 of known globular clusters, to see which of them show the signs of a collapsed
206 We also explore some diversionary mathematics and recreational tables.
210 \begin_layout Keywords
212 clusters: globular, peanut---bosons: bozos
215 \begin_layout Section
220 \begin_layout Standard
222 A focal problem today in the dynamics of globular clusters is core collapse.
223 It has been predicted by theory for decades
227 \begin_layout Standard
230 citep{hen61,lyn68,spi85}
235 , but observation has been less alert to the phenomenon.
236 For many years the central brightness peak in M15
240 \begin_layout Standard
248 seemed a unique anomaly.
253 \begin_layout Standard
261 suggested a central peak in NGC 6397, and a limited photographic survey
266 \begin_layout Standard
269 citep[Paper I]{djo84}
274 found three more cases, including NGC 6624, whose sharp center had often
279 \begin_layout Standard
291 \begin_layout Section
296 \begin_layout Standard
298 All our observations were short direct exposures with CCD's.
299 At Lick Observatory we used a TI 500
300 \begin_inset Formula \( \times \)
303 500 chip and a GEC 575
304 \begin_inset Formula \( \times \)
307 385, on the 1-m Nickel reflector.
308 The only filter available at Lick was red.
309 At CTIO we used a GEC 575
310 \begin_inset Formula \( \times \)
314 \begin_inset Formula \( B,V, \)
318 \begin_inset Formula \( R \)
321 filters, and an RCA 512
322 \begin_inset Formula \( \times \)
326 \begin_inset Formula \( U,B,V,R, \)
330 \begin_inset Formula \( I \)
333 filters, on the 1.5-m reflector.
334 In the CTIO observations we tried to concentrate on the shortest practicable
335 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
336 often kept us from observing in
337 \begin_inset Formula \( U \)
341 \begin_inset Formula \( B \)
345 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
346 sizes were around 3 arcmin.
349 \begin_layout Standard
351 The CCD images are unfortunately not always suitable, for very poor clusters
352 or for clusters with large cores.
353 Since the latter are easily studied by other means, we augmented our own
354 CCD profiles by collecting from the literature a number of star-count profiles
359 \begin_layout Standard
362 citep{kin68,pet76,har84,ort85}
367 , as well as photoelectric profiles
371 \begin_layout Standard
379 and electronographic profiles
383 \begin_layout Standard
392 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
395 \begin_layout Section
400 \begin_layout Standard
402 It has been realized that helicity amplitudes provide a convenient means
407 \begin_layout Standard
409 Footnotes can be inserted like this.
415 These amplitude-level techniques are particularly convenient for calculations
416 involving many Feynman diagrams, where the usual trace techniques for the
417 amplitude squared becomes unwieldy.
418 Our calculations use the helicity techniques developed by other authors
420 \begin_inset LatexCommand \cite{hag86}
424 ; we briefly summarize below.
427 \begin_layout Subsection
432 \begin_layout Standard
435 \begin_inset LatexCommand \label{bozomath}
442 \begin_layout Standard
444 A tree-level amplitude in
445 \begin_inset Formula \( e^{+}e^{-} \)
448 collisions can be expressed in terms of fermion strings of the form
449 \begin_inset Formula \begin{equation}
450 \bar{v}(p_{2},\sigma _{2})P_{-\tau }\hat{a}_{1}\hat{a}_{2}\cdots \hat{a}_{n}u(p_{1},\sigma _{1}),
456 \begin_inset Formula \( p \)
460 \begin_inset Formula \( \sigma \)
464 \begin_inset Formula \( e^{\pm } \)
467 four-momenta and helicities
468 \begin_inset Formula \( (\sigma =\pm 1) \)
472 \begin_inset Formula \( \hat{a}_{i}=a^{\mu }_{i}\gamma _{\nu } \)
476 \begin_inset Formula \( P_{\tau }=\frac{1}{2}(1+\tau \gamma _{5}) \)
479 is a chirality projection operator
480 \begin_inset Formula \( (\tau =\pm1 ) \)
485 \begin_inset Formula \( a^{\mu }_{i} \)
488 may be formed from particle four-momenta, gauge-boson polarization vectors
489 or fermion strings with an uncontracted Lorentz index associated with final-sta
493 \begin_layout NoteToEditor
495 Figures 1 and 2 should appear side-by-side in print
498 \begin_layout Standard
500 In the chiral representation the
501 \begin_inset Formula \( \gamma \)
504 matrices are expressed in terms of
505 \begin_inset Formula \( 2\times 2 \)
509 \begin_inset Formula \( \sigma \)
512 and the unit matrix 1 as
513 \begin_inset Formula \begin{eqnarray}
514 \gamma ^{\mu } & = & \left( \begin{array}{cc}
515 0 & \sigma ^{\mu }_{+}\\
516 \sigma ^{\mu }_{-} & 0
517 \end{array}\right) ,\gamma ^{5}=\left( \begin{array}{cc}
520 \end{array}\right) ,\nonumber \\
521 \sigma ^{\mu }_{\pm } & = & ({\textbf {1}},\pm \sigma ),\nonumber
527 \begin_inset Formula \begin{equation}
528 \hat{a}=\left( \begin{array}{cc}
531 \end{array}\right) ,(\hat{a})_{\pm }=a_{\mu }\sigma ^{\mu }_{\pm },
536 The spinors are expressed in terms of two-component Weyl spinors as
537 \begin_inset Formula \begin{equation}
538 u=\left( \begin{array}{c}
541 \end{array}\right) ,v={\textbf {(}}\vdag _{+}{\textbf {,}}\vdag _{-}{\textbf {)}}.
549 \begin_layout Standard
551 The Weyl spinors are given in terms of helicity eigenstates
552 \begin_inset Formula \( \chi _{\lambda }(p) \)
556 \begin_inset Formula \( \lambda =\pm1 \)
562 \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)
575 \begin_layout Section
577 Floating material and so forth
580 \begin_layout Standard
582 Consider a task that computes profile parameters for a modified Lorentzian
584 \begin_inset Formula \begin{equation}
585 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
591 \begin_inset Formula \[
592 d_{1}=\sqrt{\left( \begin{array}{c}
593 \frac{x_{1}}{R_{maj}}
594 \end{array}\right) ^{2}+\left( \begin{array}{c}
595 \frac{y_{1}}{R_{min}}
596 \end{array}\right) ^{2}}\]
601 \begin_inset Formula \[
602 d_{2}=\sqrt{\left( \begin{array}{c}
603 \frac{x_{1}}{PR_{maj}}
604 \end{array}\right) ^{2}+\left( \begin{array}{c}
605 \case {y_{1}}{PR_{min}}
606 \end{array}\right) ^{2}}\]
611 \begin_inset Formula \[
612 x_{1}=(x-x_{0})\cos \Theta +(y-y_{0})\sin \Theta \]
617 \begin_inset Formula \[
618 y_{1}=-(x-x_{0})\sin \Theta +(y-y_{0})\cos \Theta \]
625 \begin_layout Standard
628 \begin_inset Formula \( x_{0} \)
632 \begin_inset Formula \( y_{0} \)
635 is the star center, and
636 \begin_inset Formula \( \Theta \)
639 is the angle with the
640 \begin_inset Formula \( x \)
644 Results of this task are shown in table\InsetSpace ~
646 \begin_inset LatexCommand \ref{tbl-2}
651 It is not clear how these sorts of analyses may affect determination of
653 \begin_inset Formula \( M_{\sun } \)
656 , but the assumption is that the alternate results should be less than 90
660 \begin_layout Standard
670 out of phase with previous values.
671 We have no observations of
675 \begin_layout Standard
688 \begin_layout Standard
696 of the electronically submitted abstracts for AAS meetings are error-free.
699 \begin_layout Acknowledgements
703 \begin_layout Standard
705 We are grateful to V.
710 Phillips for doing the math in section\InsetSpace ~
712 \begin_inset LatexCommand \ref{bozomath}
717 More information on the AASTeX macros package are available at
719 \begin_inset LatexCommand \url{http://www.aas.org/publications/aastex}
728 \begin_layout Standard
731 anchor{ftp://www.aas.org/pubs/}{AAS ftp site}
739 \begin_layout Appendix
743 \begin_layout Section
748 \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})}}
761 \begin_layout MathLetters
764 \begin_inset Formula \[
765 d_{1}=\frac{3}{4}\sqrt{\left( \begin{array}{c}
766 \frac{x_{1}}{R_{maj}}
767 \end{array}\right) ^{2}+\left( \begin{array}{c}
768 \frac{y_{1}}{R_{min}}
769 \end{array}\right) ^{2}}\]
774 \begin_inset Formula \begin{equation}
775 d_{2}=\case {3}{4}\sqrt{\left( \begin{array}{c}
776 \frac{x_{1}}{PR_{maj}}
777 \end{array}\right) ^{2}+\left( \begin{array}{c}
778 \case {y_{1}}{PR_{min}}
779 \end{array}\right) ^{2}}
785 \begin_inset Formula \begin{eqnarray}
786 x_{1} & = & (x-x_{0})\cos \Theta +(y-y_{0})\sin \Theta \\
787 y_{1} & = & -(x-x_{0})\sin \Theta +(y-y_{0})\cos \Theta
795 \begin_layout Standard
797 For completeness, here is one last equation.
799 \begin_inset Formula \begin{equation}
808 \begin_layout References
809 \bibitem [Auri\`ere(1982)]{aur82}
817 \begin_layout Standard
828 \begin_layout References
829 \bibitem [Canizares et al.(1978)]{can78}
834 A., Liller, W., and McClintock, J.
840 \begin_layout Standard
851 \begin_layout References
852 \bibitem [Djorgovski and King(1984)]{djo84}
854 Djorgovski, S., and King, I.
860 \begin_layout Standard
871 \begin_layout References
872 \bibitem [Hagiwara and Zeppenfeld(1986)]{hag86}
874 Hagiwara, K., and Zeppenfeld, D.
875 1986, Nucl.Phys., 274, 1
878 \begin_layout References
879 \bibitem [Harris and van den Bergh(1984)]{har84}
882 E., and van den Bergh, S.
887 \begin_layout Standard
898 \begin_layout References
899 \bibitem [H\`enon(1961)]{hen61}
903 1961, Ann.d'Ap., 24, 369
906 \begin_layout References
907 \bibitem [King(1966)]{kin66}
915 \begin_layout Standard
926 \begin_layout References
927 \bibitem [King(1975)]{kin75}
931 1975, Dynamics of Stellar Systems, A.
932 Hayli, Dordrecht: Reidel, 1975, 99
935 \begin_layout References
936 \bibitem [King et al.(1968)]{kin68}
939 R., Hedemann, E., Hodge, S.
946 \begin_layout Standard
957 \begin_layout References
958 \bibitem [Kron et al.(1984)]{kro84}
962 V., and Wasserman, L.
968 \begin_layout Standard
979 \begin_layout References
980 \bibitem [Lynden-Bell and Wood(1968)]{lyn68}
982 Lynden-Bell, D., and Wood, R.
987 \begin_layout Standard
998 \begin_layout References
999 \bibitem [Newell and O'Neil(1978)]{new78}
1008 \begin_layout Standard
1019 \begin_layout References
1020 \bibitem [Ortolani et al.(1985)]{ort85}
1022 Ortolani, S., Rosino, L., and Sandage, A.
1027 \begin_layout Standard
1038 \begin_layout References
1039 \bibitem [Peterson(1976)]{pet76}
1047 \begin_layout Standard
1058 \begin_layout References
1059 \bibitem [Spitzer(1985)]{spi85}
1062 1985, Dynamics of Star Clusters, J.
1064 Hut, Dordrecht: Reidel, 109
1067 \begin_layout Standard
1073 \begin_layout Standard
1084 \begin_layout FigCaption
1087 \begin_inset LatexCommand \label{fig1}
1091 This is the first figure and it uses sgi9259.eps as its EPS figure file.
1094 \begin_layout FigCaption
1097 \begin_inset LatexCommand \label{fig2}
1101 This is an example of a long figure caption that must be set as a paragraph.
1102 The processor has to buffer the text of the caption, so it is good not
1103 to be too wordy, but that would make for poor communication as well.
1107 \begin_layout FigCaption
1110 \begin_inset LatexCommand \label{fig3}
1114 This figure has no associated EPS file, so the optional parameter is omitted.
1118 \begin_layout Standard
1124 \begin_layout Standard
1135 \begin_layout Standard
1137 \begin_inset Float table
1142 \begin_layout Caption
1144 Terribly relevant tabular information.
1145 \begin_inset LatexCommand \label{tbl-2}
1152 \begin_layout Standard
1155 \begin_inset VSpace medskip
1162 \begin_layout Standard
1165 \begin_inset Tabular
1166 <lyxtabular version="3" rows="7" columns="13">
1167 <features endhead="-1" endfirsthead="-1" endfoot="-1" endlastfoot="-1">
1168 <column alignment="center" valignment="top" width="0pt">
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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="right" valignment="top" width="0pt">
1180 <column alignment="center" valignment="top" width="0pt">
1182 <cell alignment="center" valignment="top" topline="true" bottomline="true" usebox="none">
1185 \begin_layout Standard
1192 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1195 \begin_layout Standard
1202 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1205 \begin_layout Standard
1208 \begin_inset Formula \( d_{x} \)
1216 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1219 \begin_layout Standard
1222 \begin_inset Formula \( d_{y} \)
1230 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1233 \begin_layout Standard
1236 \begin_inset Formula \( n \)
1244 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1247 \begin_layout Standard
1250 \begin_inset Formula \( \chi ^{2} \)
1258 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1261 \begin_layout Standard
1264 \begin_inset Formula \( R_{maj} \)
1272 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1275 \begin_layout Standard
1278 \begin_inset Formula \( R_{min} \)
1286 <cell multicolumn="1" alignment="center" valignment="top" usebox="none">
1289 \begin_layout Standard
1292 \begin_inset Formula \( P \)
1299 \begin_layout Standard
1312 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1315 \begin_layout Standard
1318 \begin_inset Formula \( PR_{maj} \)
1326 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1329 \begin_layout Standard
1332 \begin_inset Formula \( PR_{min} \)
1340 <cell multicolumn="1" alignment="center" valignment="top" usebox="none">
1343 \begin_layout Standard
1346 \begin_inset Formula \( \Theta \)
1353 \begin_layout Standard
1366 <cell alignment="center" valignment="top" usebox="none">
1369 \begin_layout Standard
1378 <cell alignment="center" valignment="top" usebox="none">
1381 \begin_layout Standard
1387 \begin_layout Standard
1402 <cell alignment="right" valignment="top" usebox="none">
1405 \begin_layout Standard
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1415 \begin_layout Standard
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1425 \begin_layout Standard
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2076 <cell alignment="center" valignment="top" bottomline="true" usebox="none">
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2199 \begin_layout Standard
2214 \begin_layout Standard
2220 \begin_layout Standard
2223 tablenotetext{a}{Sample footnote for table~
2225 ref{tbl-2} that was generated with the
2232 \begin_layout Standard
2236 \begin_layout Standard
2239 tablenotetext{b}{Yet another sample footnote for table~
2244 \begin_layout Standard
2248 \begin_layout Standard
2251 tablenotetext{c}{Another sample footnote for table~
2261 \begin_layout TableComments
2263 We can also attach a long-ish paragraph of explanatory material to a table.
2266 tablerefs to append a list of references.
2267 The following references were from a different table: I've patched them
2268 in here to show how they look, but don't take them too seriously---I certainly
2272 \begin_layout TableRefs
2274 (1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
2275 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
2276 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
2277 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
2278 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
2279 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
2280 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
2281 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
2282 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
2284 1991; (26) Olsen 1983.
2291 \begin_layout Standard
2297 \begin_layout Standard
2298 % LyX can load deluxetable files verbatim.
2299 To see this in action, uncomment the following
2307 \begin_layout Standard
2313 \begin_layout Standard
2320 \begin_inset Quotes eld
2327 \begin_layout Standard
2334 \begin_inset Quotes erd
2341 \begin_layout Standard
2342 ) and copy the file table.tex from the AASTeX 5.0 distribution into
2350 \begin_layout Standard
2356 \begin_layout Standard
2357 % the same directory as this file.
2365 \begin_layout Standard
2371 \begin_layout Standard
2378 \begin_inset Include \input{table}