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 \font_typewriter default
18 \font_default_family default
24 \paperfontsize default
32 \paperorientation portrait
35 \paragraph_separation indent
37 \quotes_language english
40 \paperpagestyle default
41 \tracking_changes false
49 Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AASTeX
60 \begin_layout Standard
73 \begin_layout Standard
84 \begin_layout Affiliation
86 Astronomy Department, University of California, Berkeley, CA 94720
97 \begin_layout Standard
108 \begin_layout Affiliation
110 National Optical Astronomy Observatories, Tucson, AZ 85719
130 \begin_layout Standard
141 \begin_layout Affiliation
143 Space Telescope Science Institute, Baltimore, MD 21218
146 \begin_layout Standard
152 \begin_layout Standard
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 Standard
165 \begin_layout Standard
168 altaffiltext{2}{Society of Fellows, Harvard University.}
171 \begin_layout Standard
175 \begin_layout Standard
178 altaffiltext{3}{present address: Center for Astrophysics, 60 Garden Street,
182 \begin_layout Standard
186 \begin_layout Standard
189 altaffiltext{4}{Visiting Programmer, Space Telescope Science Institute}
193 \begin_layout Standard
197 \begin_layout Standard
200 altaffiltext{5}{Patron, Alonso's Bar and Grill}
208 \begin_layout Abstract
210 This is a preliminary report on surface photometry of the major fraction
211 of known globular clusters, to see which of them show the signs of a collapsed
213 We also explore some diversionary mathematics and recreational tables.
217 \begin_layout Keywords
219 clusters: globular, peanut---bosons: bozos
222 \begin_layout Section
227 \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 Standard
237 citep{hen61,lyn68,spi85}
242 , but observation has been less alert to the phenomenon.
243 For many years the central brightness peak in M15
247 \begin_layout Standard
255 seemed a unique anomaly.
260 \begin_layout Standard
268 suggested a central peak in NGC 6397, and a limited photographic survey
273 \begin_layout Standard
276 citep[Paper I]{djo84}
281 found three more cases, including NGC 6624, whose sharp center had often
286 \begin_layout Standard
298 \begin_layout Section
303 \begin_layout Standard
305 All our observations were short direct exposures with CCD's.
306 At Lick Observatory we used a TI 500
307 \begin_inset Formula \( \times \)
310 500 chip and a GEC 575
311 \begin_inset Formula \( \times \)
314 385, on the 1-m Nickel reflector.
315 The only filter available at Lick was red.
316 At CTIO we used a GEC 575
317 \begin_inset Formula \( \times \)
321 \begin_inset Formula \( B,V, \)
325 \begin_inset Formula \( R \)
328 filters, and an RCA 512
329 \begin_inset Formula \( \times \)
333 \begin_inset Formula \( U,B,V,R, \)
337 \begin_inset Formula \( I \)
340 filters, on the 1.5-m reflector.
341 In the CTIO observations we tried to concentrate on the shortest practicable
342 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
343 often kept us from observing in
344 \begin_inset Formula \( U \)
348 \begin_inset Formula \( B \)
352 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
353 sizes were around 3 arcmin.
356 \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 Standard
369 citep{kin68,pet76,har84,ort85}
374 , as well as photoelectric profiles
378 \begin_layout Standard
386 and electronographic profiles
390 \begin_layout Standard
399 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
402 \begin_layout Section
407 \begin_layout Standard
409 It has been realized that helicity amplitudes provide a convenient means
414 \begin_layout Standard
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 LatexCommand cite
431 ; we briefly summarize below.
434 \begin_layout Subsection
439 \begin_layout Standard
442 \begin_inset LatexCommand label
449 \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}),
463 \begin_inset Formula \( p \)
467 \begin_inset Formula \( \sigma \)
471 \begin_inset Formula \( e^{\pm } \)
474 four-momenta and helicities
475 \begin_inset Formula \( (\sigma =\pm 1) \)
479 \begin_inset Formula \( \hat{a}_{i}=a^{\mu }_{i}\gamma _{\nu } \)
483 \begin_inset Formula \( P_{\tau }=\frac{1}{2}(1+\tau \gamma _{5}) \)
486 is a chirality projection operator
487 \begin_inset Formula \( (\tau =\pm1 ) \)
492 \begin_inset Formula \( a^{\mu }_{i} \)
495 may be formed from particle four-momenta, gauge-boson polarization vectors
496 or fermion strings with an uncontracted Lorentz index associated with final-sta
500 \begin_layout NoteToEditor
502 Figures 1 and 2 should appear side-by-side in print
505 \begin_layout Standard
507 In the chiral representation the
508 \begin_inset Formula \( \gamma \)
511 matrices are expressed in terms of
512 \begin_inset Formula \( 2\times 2 \)
516 \begin_inset Formula \( \sigma \)
519 and the unit matrix 1 as
520 \begin_inset Formula \begin{eqnarray}
521 \gamma ^{\mu } & = & \left( \begin{array}{cc}
522 0 & \sigma ^{\mu }_{+}\\
523 \sigma ^{\mu }_{-} & 0
524 \end{array}\right) ,\gamma ^{5}=\left( \begin{array}{cc}
527 \end{array}\right) ,\nonumber \\
528 \sigma ^{\mu }_{\pm } & = & ({\textbf {1}},\pm \sigma ),\nonumber
534 \begin_inset Formula \begin{equation}
535 \hat{a}=\left( \begin{array}{cc}
538 \end{array}\right) ,(\hat{a})_{\pm }=a_{\mu }\sigma ^{\mu }_{\pm },
543 The spinors are expressed in terms of two-component Weyl spinors as
544 \begin_inset Formula \begin{equation}
545 u=\left( \begin{array}{c}
548 \end{array}\right) ,v={\textbf {(}}\vdag _{+}{\textbf {,}}\vdag _{-}{\textbf {)}}.
556 \begin_layout Standard
558 The Weyl spinors are given in terms of helicity eigenstates
559 \begin_inset Formula \( \chi _{\lambda }(p) \)
563 \begin_inset Formula \( \lambda =\pm1 \)
569 \begin_layout MathLetters
572 \begin_inset Formula \begin{eqnarray}
573 u(p,\lambda )_{\pm } & = & (E\pm \lambda |{\textbf {p}}|)^{1/2}\chi _{\lambda }(p),\\
574 v(p,\lambda )_{\pm } & = & \pm \lambda (E\mp \lambda |{\textbf {p}}|)^{1/2}\chi _{-\lambda }(p)
582 \begin_layout Section
584 Floating material and so forth
587 \begin_layout Standard
589 Consider a task that computes profile parameters for a modified Lorentzian
591 \begin_inset Formula \begin{equation}
592 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
598 \begin_inset Formula \[
599 d_{1}=\sqrt{\left( \begin{array}{c}
600 \frac{x_{1}}{R_{maj}}
601 \end{array}\right) ^{2}+\left( \begin{array}{c}
602 \frac{y_{1}}{R_{min}}
603 \end{array}\right) ^{2}}\]
608 \begin_inset Formula \[
609 d_{2}=\sqrt{\left( \begin{array}{c}
610 \frac{x_{1}}{PR_{maj}}
611 \end{array}\right) ^{2}+\left( \begin{array}{c}
612 \case {y_{1}}{PR_{min}}
613 \end{array}\right) ^{2}}\]
618 \begin_inset Formula \[
619 x_{1}=(x-x_{0})\cos \Theta +(y-y_{0})\sin \Theta \]
624 \begin_inset Formula \[
625 y_{1}=-(x-x_{0})\sin \Theta +(y-y_{0})\cos \Theta \]
632 \begin_layout Standard
635 \begin_inset Formula \( x_{0} \)
639 \begin_inset Formula \( y_{0} \)
642 is the star center, and
643 \begin_inset Formula \( \Theta \)
646 is the angle with the
647 \begin_inset Formula \( x \)
651 Results of this task are shown in table\InsetSpace ~
653 \begin_inset LatexCommand ref
658 It is not clear how these sorts of analyses may affect determination of
660 \begin_inset Formula \( M_{\sun } \)
663 , but the assumption is that the alternate results should be less than 90
667 \begin_layout Standard
677 out of phase with previous values.
678 We have no observations of
682 \begin_layout Standard
695 \begin_layout Standard
703 of the electronically submitted abstracts for AAS meetings are error-free.
706 \begin_layout Acknowledgements
710 \begin_layout Standard
712 We are grateful to V.
717 Phillips for doing the math in section\InsetSpace ~
719 \begin_inset LatexCommand ref
724 More information on the AASTeX macros package are available at
726 \begin_inset LatexCommand url
727 target "http://www.aas.org/publications/aastex"
735 \begin_layout Standard
738 anchor{ftp://www.aas.org/pubs/}{AAS ftp site}
746 \begin_layout Appendix
750 \begin_layout Section
755 \begin_layout Standard
757 Consider once again a task that computes profile parameters for a modified
758 Lorentzian of the form
759 \begin_inset Formula \begin{equation}
760 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
768 \begin_layout MathLetters
771 \begin_inset Formula \[
772 d_{1}=\frac{3}{4}\sqrt{\left( \begin{array}{c}
773 \frac{x_{1}}{R_{maj}}
774 \end{array}\right) ^{2}+\left( \begin{array}{c}
775 \frac{y_{1}}{R_{min}}
776 \end{array}\right) ^{2}}\]
781 \begin_inset Formula \begin{equation}
782 d_{2}=\case {3}{4}\sqrt{\left( \begin{array}{c}
783 \frac{x_{1}}{PR_{maj}}
784 \end{array}\right) ^{2}+\left( \begin{array}{c}
785 \case {y_{1}}{PR_{min}}
786 \end{array}\right) ^{2}}
792 \begin_inset Formula \begin{eqnarray}
793 x_{1} & = & (x-x_{0})\cos \Theta +(y-y_{0})\sin \Theta \\
794 y_{1} & = & -(x-x_{0})\sin \Theta +(y-y_{0})\cos \Theta
802 \begin_layout Standard
804 For completeness, here is one last equation.
806 \begin_inset Formula \begin{equation}
815 \begin_layout References
816 \begin_inset LatexCommand bibitem
817 label "Auri\`ere(1982)"
828 \begin_layout Standard
839 \begin_layout References
840 \begin_inset LatexCommand bibitem
841 label "Canizares et al.(1978)"
849 A., Liller, W., and McClintock, J.
855 \begin_layout Standard
866 \begin_layout References
867 \begin_inset LatexCommand bibitem
868 label "Djorgovski and King(1984)"
873 Djorgovski, S., and King, I.
879 \begin_layout Standard
890 \begin_layout References
891 \begin_inset LatexCommand bibitem
892 label "Hagiwara and Zeppenfeld(1986)"
897 Hagiwara, K., and Zeppenfeld, D.
898 1986, Nucl.Phys., 274, 1
901 \begin_layout References
902 \begin_inset LatexCommand bibitem
903 label "Harris and van den Bergh(1984)"
909 E., and van den Bergh, S.
914 \begin_layout Standard
925 \begin_layout References
926 \begin_inset LatexCommand bibitem
927 label "H\`enon(1961)"
934 1961, Ann.d'Ap., 24, 369
937 \begin_layout References
938 \begin_inset LatexCommand bibitem
950 \begin_layout Standard
961 \begin_layout References
962 \begin_inset LatexCommand bibitem
970 1975, Dynamics of Stellar Systems, A.
971 Hayli, Dordrecht: Reidel, 1975, 99
974 \begin_layout References
975 \begin_inset LatexCommand bibitem
976 label "King et al.(1968)"
982 R., Hedemann, E., Hodge, S.
989 \begin_layout Standard
1000 \begin_layout References
1001 \begin_inset LatexCommand bibitem
1002 label "Kron et al.(1984)"
1009 V., and Wasserman, L.
1015 \begin_layout Standard
1026 \begin_layout References
1027 \begin_inset LatexCommand bibitem
1028 label "Lynden-Bell and Wood(1968)"
1033 Lynden-Bell, D., and Wood, R.
1038 \begin_layout Standard
1049 \begin_layout References
1050 \begin_inset LatexCommand bibitem
1051 label "Newell and O'Neil(1978)"
1063 \begin_layout Standard
1074 \begin_layout References
1075 \begin_inset LatexCommand bibitem
1076 label "Ortolani et al.(1985)"
1081 Ortolani, S., Rosino, L., and Sandage, A.
1086 \begin_layout Standard
1097 \begin_layout References
1098 \begin_inset LatexCommand bibitem
1099 label "Peterson(1976)"
1110 \begin_layout Standard
1121 \begin_layout References
1122 \begin_inset LatexCommand bibitem
1123 label "Spitzer(1985)"
1129 1985, Dynamics of Star Clusters, J.
1131 Hut, Dordrecht: Reidel, 109
1134 \begin_layout Standard
1140 \begin_layout Standard
1151 \begin_layout FigCaption
1154 \begin_inset LatexCommand label
1158 This is the first figure and it uses sgi9259.eps as its EPS figure file.
1161 \begin_layout FigCaption
1164 \begin_inset LatexCommand label
1168 This is an example of a long figure caption that must be set as a paragraph.
1169 The processor has to buffer the text of the caption, so it is good not
1170 to be too wordy, but that would make for poor communication as well.
1174 \begin_layout FigCaption
1177 \begin_inset LatexCommand label
1181 This figure has no associated EPS file, so the optional parameter is omitted.
1185 \begin_layout Standard
1191 \begin_layout Standard
1202 \begin_layout Standard
1204 \begin_inset Float table
1210 \begin_inset Caption
1214 Terribly relevant tabular information.
1215 \begin_inset LatexCommand label
1227 \begin_layout Standard
1230 \begin_inset VSpace medskip
1237 \begin_layout Standard
1240 \begin_inset Tabular
1241 <lyxtabular version="3" rows="7" columns="13">
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1243 <column alignment="center" valignment="top" width="0pt">
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1260 \begin_layout Standard
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1270 \begin_layout Standard
1277 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1280 \begin_layout Standard
1283 \begin_inset Formula \( d_{x} \)
1291 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1294 \begin_layout Standard
1297 \begin_inset Formula \( d_{y} \)
1305 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1308 \begin_layout Standard
1311 \begin_inset Formula \( n \)
1319 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1322 \begin_layout Standard
1325 \begin_inset Formula \( \chi ^{2} \)
1333 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1336 \begin_layout Standard
1339 \begin_inset Formula \( R_{maj} \)
1347 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1350 \begin_layout Standard
1353 \begin_inset Formula \( R_{min} \)
1361 <cell multicolumn="1" alignment="center" valignment="top" usebox="none">
1364 \begin_layout Standard
1367 \begin_inset Formula \( P \)
1374 \begin_layout Standard
1387 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1390 \begin_layout Standard
1393 \begin_inset Formula \( PR_{maj} \)
1401 <cell alignment="right" valignment="top" topline="true" bottomline="true" usebox="none">
1404 \begin_layout Standard
1407 \begin_inset Formula \( PR_{min} \)
1415 <cell multicolumn="1" alignment="center" valignment="top" usebox="none">
1418 \begin_layout Standard
1421 \begin_inset Formula \( \Theta \)
1428 \begin_layout Standard
1441 <cell alignment="center" valignment="top" usebox="none">
1444 \begin_layout Standard
1453 <cell alignment="center" valignment="top" usebox="none">
1456 \begin_layout Standard
1462 \begin_layout Standard
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1480 \begin_layout Standard
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1500 \begin_layout Standard
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2298 tablenotetext{a}{Sample footnote for table~
2300 ref{tbl-2} that was generated with the
2307 \begin_layout Standard
2311 \begin_layout Standard
2314 tablenotetext{b}{Yet another sample footnote for table~
2319 \begin_layout Standard
2323 \begin_layout Standard
2326 tablenotetext{c}{Another sample footnote for table~
2336 \begin_layout TableComments
2338 We can also attach a long-ish paragraph of explanatory material to a table.
2341 tablerefs to append a list of references.
2342 The following references were from a different table: I've patched them
2343 in here to show how they look, but don't take them too seriously---I certainly
2347 \begin_layout TableRefs
2349 (1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
2350 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
2351 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
2352 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
2353 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
2354 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
2355 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
2356 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
2357 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
2359 1991; (26) Olsen 1983.
2366 \begin_layout Standard
2372 \begin_layout Standard
2373 % LyX can load deluxetable files verbatim.
2374 To see this in action, uncomment the following
2382 \begin_layout Standard
2388 \begin_layout Standard
2395 \begin_inset Quotes eld
2402 \begin_layout Standard
2409 \begin_inset Quotes erd
2416 \begin_layout Standard
2417 ) and copy the file table.tex from the AASTeX 5.0 distribution into
2425 \begin_layout Standard
2431 \begin_layout Standard
2432 % the same directory as this file.
2440 \begin_layout Standard
2446 \begin_layout Standard
2453 \begin_inset Include \input{table}