1 #LyX 2.1 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}
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80 Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AASTeX
87 \begin_inset Flex altaffilmark
90 \begin_layout Plain Layout
98 \begin_inset Flex altaffilmark
101 \begin_layout Plain Layout
110 \begin_layout Affiliation
111 Astronomy Department, University of California, Berkeley, CA 94720
118 \begin_inset Flex altaffilmark
121 \begin_layout Plain Layout
130 \begin_layout Affiliation
131 National Optical Astronomy Observatories, Tucson, AZ 85719
146 \begin_inset Flex altaffilmark
149 \begin_layout Plain Layout
158 \begin_layout Affiliation
159 Space Telescope Science Institute, Baltimore, MD 21218
162 \begin_layout Altaffilation
163 \begin_inset Argument 1
166 \begin_layout Plain Layout
172 Visiting Astronomer, Cerro Tololo Inter-American Observatory.
173 CTIO is operated by AURA, Inc.
174 \begin_inset space \space{}
177 under contract to the National Science Foundation.
180 \begin_layout Altaffilation
181 \begin_inset Argument 1
184 \begin_layout Plain Layout
190 Society of Fellows, Harvard University.
193 \begin_layout Altaffilation
194 \begin_inset Argument 1
197 \begin_layout Plain Layout
203 present address: Center for Astrophysics, 60 Garden Street, Cambridge, MA
207 \begin_layout Altaffilation
208 \begin_inset Argument 1
211 \begin_layout Plain Layout
217 Visiting Programmer, Space Telescope Science Institute
220 \begin_layout Altaffilation
221 \begin_inset Argument 1
224 \begin_layout Plain Layout
230 Patron, Alonso's Bar and Grill
233 \begin_layout Abstract
234 This is a preliminary report on surface photometry of the major fraction
235 of known globular clusters, to see which of them show the signs of a collapsed
237 We also explore some diversionary mathematics and recreational tables.
241 \begin_layout Keywords
242 clusters: globular, peanut---bosons: bozos
245 \begin_layout Section
249 \begin_layout Standard
250 A focal problem today in the dynamics of globular clusters is core collapse.
251 It has been predicted by theory for decades
255 \begin_layout Plain Layout
259 citep{hen61,lyn68,spi85}
264 , but observation has been less alert to the phenomenon.
265 For many years the central brightness peak in M15
269 \begin_layout Plain Layout
278 seemed a unique anomaly.
283 \begin_layout Plain Layout
292 suggested a central peak in NGC 6397, and a limited photographic survey
297 \begin_layout Plain Layout
301 citep[Paper I]{djo84}
306 found three more cases, including NGC 6624, whose sharp center had often
311 \begin_layout Plain Layout
324 \begin_layout Section
328 \begin_layout Standard
329 All our observations were short direct exposures with CCD's.
330 At Lick Observatory we used a TI 500
331 \begin_inset Formula $\times$
334 500 chip and a GEC 575
335 \begin_inset Formula $\times$
338 385, on the 1-m Nickel reflector.
339 The only filter available at Lick was red.
340 At CTIO we used a GEC 575
341 \begin_inset Formula $\times$
345 \begin_inset Formula $B,V,$
349 \begin_inset Formula $R$
352 filters, and an RCA 512
353 \begin_inset Formula $\times$
357 \begin_inset Formula $U,B,V,R,$
361 \begin_inset Formula $I$
364 filters, on the 1.5-m reflector.
365 In the CTIO observations we tried to concentrate on the shortest practicable
366 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
367 often kept us from observing in
368 \begin_inset Formula $U$
372 \begin_inset Formula $B$
376 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
377 sizes were around 3 arcmin.
380 \begin_layout Standard
381 The CCD images are unfortunately not always suitable, for very poor clusters
382 or for clusters with large cores.
383 Since the latter are easily studied by other means, we augmented our own
384 CCD profiles by collecting from the literature a number of star-count profiles
389 \begin_layout Plain Layout
393 citep{kin68,pet76,har84,ort85}
398 , as well as photoelectric profiles
402 \begin_layout Plain Layout
411 and electronographic profiles
415 \begin_layout Plain Layout
425 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
428 \begin_layout Section
432 \begin_layout Standard
433 It has been realized that helicity amplitudes provide a convenient means
438 \begin_layout Plain Layout
439 Footnotes can be inserted like this.
445 These amplitude-level techniques are particularly convenient for calculations
446 involving many Feynman diagrams, where the usual trace techniques for the
447 amplitude squared becomes unwieldy.
448 Our calculations use the helicity techniques developed by other authors
450 \begin_inset CommandInset citation
456 ; we briefly summarize below.
459 \begin_layout Subsection
463 \begin_layout Standard
464 \begin_inset CommandInset label
473 \begin_layout Standard
474 A tree-level amplitude in
475 \begin_inset Formula $e^{+}e^{-}$
478 collisions can be expressed in terms of fermion strings of the form
481 \bar{v}(p_{2},\sigma_{2})P_{-\tau}\hat{a}_{1}\hat{a}_{2}\cdots\hat{a}_{n}u(p_{1},\sigma_{1}),
487 \begin_inset Formula $p$
491 \begin_inset Formula $\sigma$
495 \begin_inset Formula $e^{\pm}$
498 four-momenta and helicities
499 \begin_inset Formula $(\sigma=\pm1)$
503 \begin_inset Formula $\hat{a}_{i}=a_{i}^{\mu}\gamma_{\nu}$
507 \begin_inset Formula $P_{\tau}=\frac{1}{2}(1+\tau\gamma_{5})$
510 is a chirality projection operator
511 \begin_inset Formula $(\tau=\pm1)$
516 \begin_inset Formula $a_{i}^{\mu}$
519 may be formed from particle four-momenta, gauge-boson polarization vectors
520 or fermion strings with an uncontracted Lorentz index associated with final-sta
524 \begin_layout NoteToEditor
525 Figures 1 and 2 should appear side-by-side in print
528 \begin_layout Standard
529 In the chiral representation the
530 \begin_inset Formula $\gamma$
533 matrices are expressed in terms of
534 \begin_inset Formula $2\times2$
538 \begin_inset Formula $\sigma$
541 and the unit matrix 1 as
544 \gamma^{\mu} & = & \left(\begin{array}{cc}
545 0 & \sigma_{+}^{\mu}\\
547 \end{array}\right),\gamma^{5}=\left(\begin{array}{cc}
550 \end{array}\right),\\
551 \sigma_{\pm}^{\mu} & = & ({\textbf{1}},\pm\sigma),
559 \hat{a}=\left(\begin{array}{cc}
562 \end{array}\right),(\hat{a})_{\pm}=a_{\mu}\sigma_{\pm}^{\mu},
567 The spinors are expressed in terms of two-component Weyl spinors as
570 u=\left(\begin{array}{c}
573 \end{array}\right),v={\textbf{(}}\vdag_{+}{\textbf{,}}\vdag_{-}{\textbf{)}}.
581 \begin_layout Standard
582 The Weyl spinors are given in terms of helicity eigenstates
583 \begin_inset Formula $\chi_{\lambda}(p)$
587 \begin_inset Formula $\lambda=\pm1$
593 \begin_layout MathLetters
596 u(p,\lambda)_{\pm} & = & (E\pm\lambda|{\textbf{p}}|)^{1/2}\chi_{\lambda}(p),\\
597 v(p,\lambda)_{\pm} & = & \pm\lambda(E\mp\lambda|{\textbf{p}}|)^{1/2}\chi_{-\lambda}(p)
605 \begin_layout Section
606 Floating material and so forth
609 \begin_layout Standard
610 Consider a task that computes profile parameters for a modified Lorentzian
614 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
622 d_{1}=\sqrt{\left(\begin{array}{c}
623 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
624 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}
632 d_{2}=\sqrt{\left(\begin{array}{c}
633 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
634 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}
642 x_{1}=(x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta
650 y_{1}=-(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta
658 \begin_layout Standard
660 \begin_inset Formula $x_{0}$
664 \begin_inset Formula $y_{0}$
667 is the star center, and
668 \begin_inset Formula $\Theta$
671 is the angle with the
672 \begin_inset Formula $x$
676 Results of this task are shown in table
681 \begin_inset CommandInset ref
688 It is not clear how these sorts of analyses may affect determination of
690 \begin_inset Formula $M_{\sun}$
693 , but the assumption is that the alternate results should be less than 90°
694 out of phase with previous values.
695 We have no observations of
699 \begin_layout Plain Layout
713 \begin_layout Plain Layout
722 of the electronically submitted abstracts for AAS meetings are error-free.
725 \begin_layout Acknowledgements
729 \begin_layout Standard
730 We are grateful to V.
735 Phillips for doing the math in section
740 \begin_inset CommandInset ref
747 More information on the AASTeX macros package are available at
748 \begin_inset Flex URL
751 \begin_layout Plain Layout
753 http://www.aas.org/publications/aastex
762 \begin_layout Plain Layout
771 ftp://www.aas.org/pubs/
775 \begin_layout Plain Layout
786 \begin_layout Plain Layout
796 \begin_layout Appendix
800 \begin_layout Section
804 \begin_layout Standard
805 Consider once again a task that computes profile parameters for a modified
806 Lorentzian of the form
809 I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
817 \begin_layout MathLetters
820 d_{1}=\frac{3}{4}\sqrt{\left(\begin{array}{c}
821 \frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
822 \frac{y_{1}}{R_{min}}\end{array}\right)^{2}}
830 d_{2}=\case{3}{4}\sqrt{\left(\begin{array}{c}
831 \frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
832 \case{y_{1}}{PR_{min}}\end{array}\right)^{2}}
840 x_{1} & = & (x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta\\
841 y_{1} & = & -(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta
849 \begin_layout Standard
850 For completeness, here is one last equation.
862 \begin_layout References
863 \begin_inset CommandInset bibitem
865 label "Auri\\`ere(1982)"
875 \begin_layout Plain Layout
887 \begin_layout References
888 \begin_inset CommandInset bibitem
890 label "Canizares et al.(1978)"
898 A., Liller, W., and McClintock, J.
904 \begin_layout Plain Layout
916 \begin_layout References
917 \begin_inset CommandInset bibitem
919 label "Djorgovski and King(1984)"
924 Djorgovski, S., and King, I.
930 \begin_layout Plain Layout
942 \begin_layout References
943 \begin_inset CommandInset bibitem
945 label "Hagiwara and Zeppenfeld(1986)"
950 Hagiwara, K., and Zeppenfeld, D.
951 1986, Nucl.Phys., 274, 1
954 \begin_layout References
955 \begin_inset CommandInset bibitem
957 label "Harris and van den Bergh(1984)"
963 E., and van den Bergh, S.
968 \begin_layout Plain Layout
980 \begin_layout References
981 \begin_inset CommandInset bibitem
983 label "H\\`enon(1961)"
989 1961, Ann.d'Ap., 24, 369
992 \begin_layout References
993 \begin_inset CommandInset bibitem
1006 \begin_layout Plain Layout
1018 \begin_layout References
1019 \begin_inset CommandInset bibitem
1020 LatexCommand bibitem
1028 1975, Dynamics of Stellar Systems, A.
1029 Hayli, Dordrecht: Reidel, 1975, 99
1032 \begin_layout References
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1034 LatexCommand bibitem
1035 label "King et al.(1968)"
1041 R., Hedemann, E., Hodge, S.
1048 \begin_layout Plain Layout
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1063 label "Kron et al.(1984)"
1070 V., and Wasserman, L.
1076 \begin_layout Plain Layout
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1091 label "Lynden-Bell and Wood(1968)"
1096 Lynden-Bell, D., and Wood, R.
1101 \begin_layout Plain Layout
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1116 label "Newell and O'Neil(1978)"
1128 \begin_layout Plain Layout
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1143 label "Ortolani et al.(1985)"
1148 Ortolani, S., Rosino, L., and Sandage, A.
1153 \begin_layout Plain Layout
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1168 label "Peterson(1976)"
1179 \begin_layout Plain Layout
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1193 LatexCommand bibitem
1194 label "Spitzer(1985)"
1200 1985, Dynamics of Star Clusters, J.
1202 Hut, Dordrecht: Reidel, 109
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1287 Terribly relevant tabular information.
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2259 \begin_layout Table note
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2271 Sample footnote for table
2272 \begin_inset space ~
2276 \begin_inset CommandInset ref
2282 that was generated with the LaTeX table environment
2285 \begin_layout Table note
2290 \begin_layout Plain Layout
2297 Yet another sample footnote for table
2298 \begin_inset space ~
2302 \begin_inset CommandInset ref
2311 \begin_layout Table note
2316 \begin_layout Plain Layout
2323 Another sample footnote for table
2324 \begin_inset space ~
2328 \begin_inset CommandInset ref
2337 \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
2348 (1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
2349 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
2350 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
2351 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
2352 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
2353 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
2354 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
2355 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
2356 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
2358 1991; (26) Olsen 1983.
2366 \begin_layout Standard
2370 \begin_layout Plain Layout
2372 % LyX can load deluxetable files verbatim.
2373 To see this in action, uncomment the following
2381 \begin_layout Standard
2385 \begin_layout Plain Layout
2393 \begin_inset Quotes eld
2400 \begin_layout Plain Layout
2408 \begin_inset Quotes erd
2415 \begin_layout Plain Layout
2417 ) and copy the file table.tex from the AASTeX 5.0 distribution into
2425 \begin_layout Standard
2429 \begin_layout Plain Layout
2431 % the same directory as this file.
2439 \begin_layout Standard
2443 \begin_layout Plain Layout
2451 \begin_inset CommandInset include