1 #This file was created by <mike> Sun Jul 25 23:59:27 1999
2 #LyX 1.0 (C) 1995-1999 Matthias Ettrich and the LyX Team
6 \newcommand{\vdag}{(v)^\dagger}
7 \newcommand{\myemail}{skywalker@galaxy.far.far.away}
8 \slugcomment{Not to appear in Nonlearned J., 45.}
9 \shorttitle{Djorgovski et al.}
10 \shortauthors{Collapsed Cores in Globular Clusters}
13 \inputencoding default
16 \paperfontsize default
22 \paperorientation portrait
25 \paragraph_separation indent
27 \quotes_language english
31 \paperpagestyle default
35 Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AASTeX
57 Astronomy Department, University of California, Berkeley, CA 94720
69 National Optical Astronomy Observatories, Tucson, AZ 85719
86 Space Telescope Science Institute, Baltimore, MD 21218
93 altaffiltext{1}{Visiting Astronomer, Cerro Tololo Inter-American Observatory.
94 CTIO is operated by AURA, Inc.
96 under contract to the National Science Foundation.}
103 altaffiltext{2}{Society of Fellows, Harvard University.}
110 altaffiltext{3}{present address: Center for Astrophysics, 60 Garden Street,
118 altaffiltext{4}{Visiting Programmer, Space Telescope Science Institute}
126 altaffiltext{5}{Patron, Alonso's Bar and Grill}
129 This is a preliminary report on surface photometry of the major fraction
130 of known globular clusters, to see which of them show the signs of a collapsed
132 We also explore some diversionary mathematics and recreational tables.
136 clusters: globular, peanut---bosons: bozos
142 A focal problem today in the dynamics of globular clusters is core collapse.
143 It has been predicted by theory for decades
147 citep{hen61,lyn68,spi85}
149 , but observation has been less alert to the phenomenon.
150 For many years the central brightness peak in M15
156 seemed a unique anomaly.
163 suggested a central peak in NGC 6397, and a limited photographic survey
168 citep[Paper I]{djo84}
170 found three more cases, including NGC 6624, whose sharp center had often
184 All our observations were short direct exposures with CCD's.
185 At Lick Observatory we used a TI 500
186 \begin_inset Formula \( \times \)
189 500 chip and a GEC 575
190 \begin_inset Formula \( \times \)
193 385, on the 1-m Nickel reflector.
194 The only filter available at Lick was red.
195 At CTIO we used a GEC 575
196 \begin_inset Formula \( \times \)
200 \begin_inset Formula \( B,V, \)
204 \begin_inset Formula \( R \)
207 filters, and an RCA 512
208 \begin_inset Formula \( \times \)
212 \begin_inset Formula \( U,B,V,R, \)
216 \begin_inset Formula \( I \)
219 filters, on the 1.5-m reflector.
220 In the CTIO observations we tried to concentrate on the shortest practicable
221 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
222 often kept us from observing in
223 \begin_inset Formula \( U \)
227 \begin_inset Formula \( B \)
231 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
232 sizes were around 3 arcmin.
235 The CCD images are unfortunately not always suitable, for very poor clusters
236 or for clusters with large cores.
237 Since the latter are easily studied by other means, we augmented our own
238 CCD profiles by collecting from the literature a number of star-count profiles
243 citep{kin68,pet76,har84,ort85}
245 , as well as photoelectric profiles
251 and electronographic profiles
258 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
264 It has been realized that helicity amplitudes provide a convenient means
266 \begin_float footnote
269 Footnotes can be inserted like this.
272 These amplitude-level techniques are particularly convenient for calculations
273 involving many Feynman diagrams, where the usual trace techniques for the
274 amplitude squared becomes unwieldy.
275 Our calculations use the helicity techniques developed by other authors
277 \begin_inset LatexCommand \cite{hag86}
281 ; we briefly summarize below.
288 \begin_inset LatexCommand \label{bozomath}
295 A tree-level amplitude in
296 \begin_inset Formula \( e^{+}e^{-} \)
299 collisions can be expressed in terms of fermion strings of the form
302 \bar{v}(p_{2},\sigma _{2})P_{-\tau }\hat{a}_{1}\hat{a}_{2}\cdots \hat{a}_{n}u(p_{1},\sigma _{1}),
308 \begin_inset Formula \( p \)
312 \begin_inset Formula \( \sigma \)
316 \begin_inset Formula \( e^{\pm } \)
319 four-momenta and helicities
320 \begin_inset Formula \( (\sigma =\pm 1) \)
324 \begin_inset Formula \( \hat{a}_{i}=a^{\mu }_{i}\gamma _{\nu } \)
328 \begin_inset Formula \( P_{\tau }=\frac{1}{2}(1+\tau \gamma _{5}) \)
331 is a chirality projection operator
332 \begin_inset Formula \( (\tau =\pm1 ) \)
337 \begin_inset Formula \( a^{\mu }_{i} \)
340 may be formed from particle four-momenta, gauge-boson polarization vectors
341 or fermion strings with an uncontracted Lorentz index associated with final-sta
345 Figures 1 and 2 should appear side-by-side in print
348 In the chiral representation the
349 \begin_inset Formula \( \gamma \)
352 matrices are expressed in terms of
353 \begin_inset Formula \( 2\times 2 \)
357 \begin_inset Formula \( \sigma \)
360 and the unit matrix 1 as
363 \gamma ^{\mu } & = & \left( \begin{array}{cc}
364 0 & \sigma ^{\mu }_{+}\\
365 \sigma ^{\mu }_{-} & 0
366 \end{array}\right) ,\gamma ^{5}=\left( \begin{array}{cc}
369 \end{array}\right) ,\nonumber \\
370 \sigma ^{\mu }_{\pm } & = & ({\textbf {1}},\pm \sigma ),\nonumber
378 \hat{a}=\left( \begin{array}{cc}
381 \end{array}\right) ,(\hat{a})_{\pm }=a_{\mu }\sigma ^{\mu }_{\pm },
386 The spinors are expressed in terms of two-component Weyl spinors as
389 u=\left( \begin{array}{c}
392 \end{array}\right) ,v={\textbf {(}}\vdag _{+}{\textbf {,}}\vdag _{-}{\textbf {)}}.
400 The Weyl spinors are given in terms of helicity eigenstates
401 \begin_inset Formula \( \chi _{\lambda }(p) \)
405 \begin_inset Formula \( \lambda =\pm1 \)
414 u(p,\lambda )_{\pm } & = & (E\pm \lambda |{\textbf {p}}|)^{1/2}\chi _{\lambda }(p),\\
415 v(p,\lambda )_{\pm } & = & \pm \lambda (E\mp \lambda |{\textbf {p}}|)^{1/2}\chi _{-\lambda }(p)
423 Floating material and so forth
426 Consider a task that computes profile parameters for a modified Lorentzian
430 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
438 d_{1}=\sqrt{\left( \begin{array}{c}
439 \frac{x_{1}}{R_{maj}}
440 \end{array}\right) ^{2}+\left( \begin{array}{c}
441 \frac{y_{1}}{R_{min}}
442 \end{array}\right) ^{2}}\]
449 d_{2}=\sqrt{\left( \begin{array}{c}
450 \frac{x_{1}}{PR_{maj}}
451 \end{array}\right) ^{2}+\left( \begin{array}{c}
452 \case {y_{1}}{PR_{min}}
453 \end{array}\right) ^{2}}\]
460 x_{1}=(x-x_{0})\cos \Theta +(y-y_{0})\sin \Theta \]
467 y_{1}=-(x-x_{0})\sin \Theta +(y-y_{0})\cos \Theta \]
475 \begin_inset Formula \( x_{0} \)
479 \begin_inset Formula \( y_{0} \)
482 is the star center, and
483 \begin_inset Formula \( \Theta \)
486 is the angle with the
487 \begin_inset Formula \( x \)
491 Results of this task are shown in table
494 \begin_inset LatexCommand \ref{tbl-2}
499 It is not clear how these sorts of analyses may affect determination of
501 \begin_inset Formula \( M_{\sun } \)
504 , but the assumption is that the alternate results should be less than 90
512 out of phase with previous values.
513 We have no observations of
526 of the electronically submitted abstracts for AAS meetings are error-free.
527 \layout Acknowledgements
531 We are grateful to V.
536 Phillips for doing the math in section
539 \begin_inset LatexCommand \ref{bozomath}
544 More information on the AASTeX macros package are available at
547 \begin_inset LatexCommand \url{http://www.aas.org/publications/aastex}
557 anchor{ftp://www.aas.org/pubs/}{AAS ftp site}
567 Consider once again a task that computes profile parameters for a modified
568 Lorentzian of the form
571 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
582 d_{1}=\frac{3}{4}\sqrt{\left( \begin{array}{c}
583 \frac{x_{1}}{R_{maj}}
584 \end{array}\right) ^{2}+\left( \begin{array}{c}
585 \frac{y_{1}}{R_{min}}
586 \end{array}\right) ^{2}}\]
593 d_{2}=\case {3}{4}\sqrt{\left( \begin{array}{c}
594 \frac{x_{1}}{PR_{maj}}
595 \end{array}\right) ^{2}+\left( \begin{array}{c}
596 \case {y_{1}}{PR_{min}}
597 \end{array}\right) ^{2}}
605 x_{1} & = & (x-x_{0})\cos \Theta +(y-y_{0})\sin \Theta \\
606 y_{1} & = & -(x-x_{0})\sin \Theta +(y-y_{0})\cos \Theta
614 For completeness, here is one last equation.
625 \bibitem [Auri\`ere(1982)]{aur82}
637 \bibitem [Canizares et al.(1978)]{can78}
642 A., Liller, W., and McClintock, J.
652 \bibitem [Djorgovski and King(1984)]{djo84}
654 Djorgovski, S., and King, I.
664 \bibitem [Hagiwara and Zeppenfeld(1986)]{hag86}
666 Hagiwara, K., and Zeppenfeld, D.
667 1986, Nucl.Phys., 274, 1
669 \bibitem [Harris and van den Bergh(1984)]{har84}
672 E., and van den Bergh, S.
681 \bibitem [H\`enon(1961)]{hen61}
685 1961, Ann.d'Ap., 24, 369
687 \bibitem [King(1966)]{kin66}
699 \bibitem [King(1975)]{kin75}
703 1975, Dynamics of Stellar Systems, A.
704 Hayli, Dordrecht: Reidel, 1975, 99
706 \bibitem [King et al.(1968)]{kin68}
709 R., Hedemann, E., Hodge, S.
720 \bibitem [Kron et al.(1984)]{kro84}
724 V., and Wasserman, L.
734 \bibitem [Lynden-Bell and Wood(1968)]{lyn68}
736 Lynden-Bell, D., and Wood, R.
745 \bibitem [Newell and O'Neil(1978)]{new78}
758 \bibitem [Ortolani et al.(1985)]{ort85}
760 Ortolani, S., Rosino, L., and Sandage, A.
769 \bibitem [Peterson(1976)]{pet76}
781 \bibitem [Spitzer(1985)]{spi85}
784 1985, Dynamics of Star Clusters, J.
786 Hut, Dordrecht: Reidel, 109
797 \begin_inset LatexCommand \label{fig1}
801 This is the first figure and it uses sgi9259.eps as its EPS figure file.
805 \begin_inset LatexCommand \label{fig2}
809 This is an example of a long figure caption that must be set as a paragraph.
810 The processor has to buffer the text of the caption, so it is good not
811 to be too wordy, but that would make for poor communication as well.
816 \begin_inset LatexCommand \label{fig3}
820 This figure has no associated EPS file, so the optional parameter is omitted.
833 \added_space_bottom medskip
834 Terribly relevant tabular information.
835 \begin_inset LatexCommand \label{tbl-2}
961 \begin_inset Formula \( d_{x} \)
967 \begin_inset Formula \( d_{y} \)
973 \begin_inset Formula \( n \)
979 \begin_inset Formula \( \chi ^{2} \)
985 \begin_inset Formula \( R_{maj} \)
991 \begin_inset Formula \( R_{min} \)
997 \begin_inset Formula \( P \)
1009 \begin_inset Formula \( PR_{maj} \)
1015 \begin_inset Formula \( PR_{min} \)
1021 \begin_inset Formula \( \Theta \)
1215 tablenotetext{a}{Sample footnote for table~
1217 ref{tbl-2} that was generated with the
1228 tablenotetext{b}{Yet another sample footnote for table~
1237 tablenotetext{c}{Another sample footnote for table~
1240 \layout TableComments
1242 We can also attach a long-ish paragraph of explanatory material to a table.
1245 tablerefs to append a list of references.
1246 The following references were from a different table: I've patched them
1247 in here to show how they look, but don't take them too seriously---I certainly
1251 (1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
1252 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
1253 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
1254 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
1255 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
1256 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
1257 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
1258 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
1259 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
1261 1991; (26) Olsen 1983.
1267 % LyX can load deluxetable files verbatim.
1268 To see this in action, uncomment the following
1276 \begin_inset Quotes eld
1284 \begin_inset Quotes erd
1289 ) and copy the file table.tex from the AASTeX 5.0 distribution into
1294 % the same directory as this file.
1300 \begin_inset Include \input{table}