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\shorttitle{Djorgovski et al.}
\shortauthors{Collapsed Cores in Globular Clusters}
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Sample document for contributions to journals of the American Astronomical
 Society (AAS).
 Please consult the 
\begin_inset CommandInset href
LatexCommand href
name "AASTeX user guide"
target "http://journals.aas.org/authors/aastex/aasguide.html"
literal "false"

\end_inset

 for details on how to use AASTeX.
 
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\noindent
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An up-to-date sample LaTeX article using AASTeX is available under 
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status open

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http://journals.aas.org/authors/aastex/aastex.html#_download
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.
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\begin_inset Note Note
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\noindent
\align left
The AASTeX document class uses a different name since version 6.0 (2016/02/16).
 Depending on the version installed on your system, set the document class
 to American Astronomical Society (AASTeX v.
 6) or American Astronomical Society (AASTeX v.
 5).
 
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\noindent
\align left
New features (cf.
 
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http://journals.aas.org/authors/aastex/revisionhistory.html
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) are only partially supported by LyX (see 
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https://www.lyx.org/trac/ticket/10027
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).
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\end_layout

\begin_layout Title
Collapsed Cores in Globular Clusters,  Gauge-Boson Couplings, and AAS\SpecialChar TeX
 Examples
\end_layout

\begin_layout Author
S.
 Djorgovski
\begin_inset Flex altaffilmark
status open

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1,2,3
\end_layout

\end_inset

 and Ivan R.
 King
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1
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\end_layout

\begin_layout Affiliation
Astronomy Department, University of California, Berkeley, CA 94720
\end_layout

\begin_layout Author
C.
 D.
 Biemesderfer
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4,5
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\end_layout

\begin_layout Affiliation
National Optical Astronomy Observatories, Tucson, AZ 85719
\end_layout

\begin_layout Email
aastex-help@aas.org
\end_layout

\begin_layout And

\end_layout

\begin_layout Author
R.
 J.
 Hanisch
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5
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\begin_layout Affiliation
Space Telescope Science Institute, Baltimore, MD 21218
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1
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Visiting Astronomer, Cerro Tololo Inter-American Observatory.
 CTIO is operated by AURA, Inc.
\begin_inset space \space{}
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under contract to the National Science Foundation.
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2
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Society of Fellows, Harvard University.
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3
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present address: Center for Astrophysics, 60 Garden Street, Cambridge, MA
 02138
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4
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Visiting Programmer, Space Telescope Science Institute
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5
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Patron, Alonso's Bar and Grill
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\begin_layout Abstract
This is a preliminary report on surface photometry of the major fraction
 of known globular clusters, to see which of them show the signs of a collapsed
 core.
 We also explore some diversionary mathematics and recreational tables.
 
\end_layout

\begin_layout Keywords
clusters: globular, peanut—bosons: bozos
\end_layout

\begin_layout Section
Introduction
\end_layout

\begin_layout Standard
A focal problem today in the dynamics of globular clusters is core collapse.
 It has been predicted by theory for decades 
\begin_inset ERT
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\backslash
citep{hen61,lyn68,spi85}
\end_layout

\end_inset

, but observation has been less alert to the phenomenon.
 For many years the central brightness peak in M15 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
citep{kin75,new78}
\end_layout

\end_inset

 seemed a unique anomaly.
 Then 
\begin_inset ERT
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\backslash
citet{aur82}
\end_layout

\end_inset

 suggested a central peak in NGC 6397, and a limited photographic survey
 of ours 
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\backslash
citep[Paper I]{djo84}
\end_layout

\end_inset

 found three more cases, including NGC 6624, whose sharp center had often
 been remarked on 
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\backslash
citep{can78}
\end_layout

\end_inset

.
 
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\begin_layout Section
Observations
\end_layout

\begin_layout Standard
All our observations were short direct exposures with CCD's.
 At Lick Observatory we used a TI 500
\begin_inset Formula $\times$
\end_inset

500 chip and a GEC 575
\begin_inset Formula $\times$
\end_inset

385, on the 1-m Nickel reflector.
 The only filter available at Lick was red.
 At CTIO we used a GEC 575
\begin_inset Formula $\times$
\end_inset

385, with 
\begin_inset Formula $B,V,$
\end_inset

 and 
\begin_inset Formula $R$
\end_inset

 filters, and an RCA 512
\begin_inset Formula $\times$
\end_inset

320, with 
\begin_inset Formula $U,B,V,R,$
\end_inset

 and 
\begin_inset Formula $I$
\end_inset

 filters, on the 1.5-m reflector.
 In the CTIO observations we tried to concentrate on the shortest practicable
 wavelengths; but faintness, reddening, and poor short-wavelength sensitivity
 often kept us from observing in 
\begin_inset Formula $U$
\end_inset

 or even in 
\begin_inset Formula $B$
\end_inset

.
 All four cameras had scales of the order of 0.4 arcsec/pixel, and our field
 sizes were around 3 arcmin.
\end_layout

\begin_layout Standard
The CCD images are unfortunately not always suitable, for very poor clusters
 or for clusters with large cores.
 Since the latter are easily studied by other means, we augmented our own
 CCD profiles by collecting from the literature a number of star-count profiles
 
\begin_inset ERT
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\backslash
citep{kin68,pet76,har84,ort85}
\end_layout

\end_inset

, as well as photoelectric profiles 
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\backslash
citep{kin66,kin75}
\end_layout

\end_inset

 and electronographic profiles 
\begin_inset ERT
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\backslash
citep{kro84}
\end_layout

\end_inset

.
 In a few cases we judged normality by eye estimates on one of the Sky Surveys.
\end_layout

\begin_layout Section
Helicity Amplitudes
\end_layout

\begin_layout Standard
It has been realized that helicity amplitudes provide a convenient means
 for Feynman diagram
\begin_inset Foot
status collapsed

\begin_layout Plain Layout
Footnotes can be inserted like this.
\end_layout

\end_inset

 evaluations.
 These amplitude-level techniques are particularly convenient for calculations
 involving many Feynman diagrams, where the usual trace techniques for the
 amplitude squared becomes unwieldy.
 Our calculations use the helicity techniques developed by other authors
 
\begin_inset CommandInset citation
LatexCommand cite
key "hag86"
literal "true"

\end_inset

; we briefly summarize below.
\end_layout

\begin_layout Subsection
Formalism
\end_layout

\begin_layout Standard
\begin_inset CommandInset label
LatexCommand label
name "bozomath"

\end_inset


\end_layout

\begin_layout Standard
A tree-level amplitude in 
\begin_inset Formula $e^{+}e^{-}$
\end_inset

 collisions can be expressed in terms of fermion strings of the form 
\begin_inset Formula 
\begin{equation}
\bar{v}(p_{2},\sigma_{2})P_{-\tau}\hat{a}_{1}\hat{a}_{2}\cdots\hat{a}_{n}u(p_{1},\sigma_{1}),
\end{equation}

\end_inset

 where 
\begin_inset Formula $p$
\end_inset

 and 
\begin_inset Formula $\sigma$
\end_inset

 label the initial 
\begin_inset Formula $e^{\pm}$
\end_inset

 four-momenta and helicities 
\begin_inset Formula $(\sigma=\pm1)$
\end_inset

, 
\begin_inset Formula $\hat{a}_{i}=a_{i}^{\mu}\gamma_{\nu}$
\end_inset

 and 
\begin_inset Formula $P_{\tau}=\frac{1}{2}(1+\tau\gamma_{5})$
\end_inset

 is a chirality projection operator 
\begin_inset Formula $(\tau=\pm1)$
\end_inset

.
 The 
\begin_inset Formula $a_{i}^{\mu}$
\end_inset

 may be formed from particle four-momenta, gauge-boson polarization vectors
 or fermion strings with an uncontracted Lorentz index associated with final-sta
te fermions.
\end_layout

\begin_layout NoteToEditor
Figures 1 and 2 should appear side-by-side in print
\end_layout

\begin_layout Standard
In the chiral representation the 
\begin_inset Formula $\gamma$
\end_inset

 matrices are expressed in terms of 
\begin_inset Formula $2\times2$
\end_inset

 Pauli matrices 
\begin_inset Formula $\sigma$
\end_inset

 and the unit matrix 1 as 
\begin_inset Formula 
\begin{eqnarray*}
\gamma^{\mu} & = & \left(\begin{array}{cc}
0 & \sigma_{+}^{\mu}\\
\sigma_{-}^{\mu} & 0
\end{array}\right),\gamma^{5}=\left(\begin{array}{cc}
-1 & 0\\
0 & 1
\end{array}\right),\\
\sigma_{\pm}^{\mu} & = & ({\textbf{1}},\pm\sigma),
\end{eqnarray*}

\end_inset

 giving 
\begin_inset Formula 
\begin{equation}
\hat{a}=\left(\begin{array}{cc}
0 & (\hat{a})_{+}\\
(\hat{a})_{-} & 0
\end{array}\right),(\hat{a})_{\pm}=a_{\mu}\sigma_{\pm}^{\mu},
\end{equation}

\end_inset

 The spinors are expressed in terms of two-component Weyl spinors as 
\begin_inset Formula 
\begin{equation}
u=\left(\begin{array}{c}
(u)_{-}\\
(u)_{+}
\end{array}\right),v={\textbf{(}}\vdag_{+}{\textbf{,}}\vdag_{-}{\textbf{)}}.
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
The Weyl spinors are given in terms of helicity eigenstates 
\begin_inset Formula $\chi_{\lambda}(p)$
\end_inset

 with 
\begin_inset Formula $\lambda=\pm1$
\end_inset

 by 
\end_layout

\begin_layout MathLetters
\begin_inset Formula 
\begin{eqnarray}
u(p,\lambda)_{\pm} & = & (E\pm\lambda|{\textbf{p}}|)^{1/2}\chi_{\lambda}(p),\\
v(p,\lambda)_{\pm} & = & \pm\lambda(E\mp\lambda|{\textbf{p}}|)^{1/2}\chi_{-\lambda}(p)
\end{eqnarray}

\end_inset


\end_layout

\begin_layout Section
Floating material and so forth
\end_layout

\begin_layout Standard
Consider a task that computes profile parameters for a modified Lorentzian
 of the form 
\begin_inset Formula 
\begin{equation}
I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
\end{equation}

\end_inset

 where 
\begin_inset Formula 
\[
d_{1}=\sqrt{\left(\begin{array}{c}
\frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
\frac{y_{1}}{R_{min}}\end{array}\right)^{2}}
\]

\end_inset


\begin_inset Formula 
\[
d_{2}=\sqrt{\left(\begin{array}{c}
\frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
\case{y_{1}}{PR_{min}}\end{array}\right)^{2}}
\]

\end_inset


\begin_inset Formula 
\[
x_{1}=(x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta
\]

\end_inset


\begin_inset Formula 
\[
y_{1}=-(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta
\]

\end_inset


\end_layout

\begin_layout Standard
In these expressions 
\begin_inset Formula $x_{0}$
\end_inset

,
\begin_inset Formula $y_{0}$
\end_inset

 is the star center, and 
\begin_inset Formula $\Theta$
\end_inset

 is the angle with the 
\begin_inset Formula $x$
\end_inset

 axis.
 Results of this task are shown in table
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "tbl-2"

\end_inset

.
 It is not clear how these sorts of analyses may affect determination of
 
\begin_inset Formula $M_{\text{\sun}}$
\end_inset

, but the assumption is that the alternate results should be less than 90°
 out of phase with previous values.
 We have no observations of 
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
ion{Ca}{2}
\end_layout

\end_inset

.
 Roughly 
\begin_inset Formula $\nicefrac{4}{5}$
\end_inset

 of the electronically submitted abstracts for AAS meetings are error-free.
 
\end_layout

\begin_layout Acknowledgements

\end_layout

\begin_layout Standard
We are grateful to V.
 Barger, T.
 Han, and R.
 J.
 N.
 Phillips for doing the math in section
\begin_inset space ~
\end_inset


\begin_inset CommandInset ref
LatexCommand ref
reference "bozomath"

\end_inset

.
 More information on the AAS\SpecialChar TeX
 macros package are available at 
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\begin_layout Plain Layout

http://www.aas.org/publications/aastex
\end_layout

\end_inset

 or the 
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anchor{
\end_layout

\end_inset

ftp://www.aas.org/pubs/
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}{
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AAS ftp site
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}
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\end_inset

.
 
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\begin_layout Software
IRAF, AIPS, Astropy, ...
 
\end_layout

\begin_layout Appendix

\end_layout

\begin_layout Section
Appendicial material
\end_layout

\begin_layout Standard
Consider once again a task that computes profile parameters for a modified
 Lorentzian of the form 
\begin_inset Formula 
\begin{equation}
I=\frac{1}{1+d_{1}^{P(1+d_{2})}}
\end{equation}

\end_inset

 where 
\end_layout

\begin_layout MathLetters
\begin_inset Formula 
\[
d_{1}=\frac{3}{4}\sqrt{\left(\begin{array}{c}
\frac{x_{1}}{R_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
\frac{y_{1}}{R_{min}}\end{array}\right)^{2}}
\]

\end_inset


\begin_inset Formula 
\begin{equation}
d_{2}=\case{3}{4}\sqrt{\left(\begin{array}{c}
\frac{x_{1}}{PR_{maj}}\end{array}\right)^{2}+\left(\begin{array}{c}
\case{y_{1}}{PR_{min}}\end{array}\right)^{2}}
\end{equation}

\end_inset


\begin_inset Formula 
\begin{eqnarray}
x_{1} & = & (x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta\\
y_{1} & = & -(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta
\end{eqnarray}

\end_inset


\end_layout

\begin_layout Standard
For completeness, here is one last equation.
 
\begin_inset Formula 
\begin{equation}
e=mc^{2}
\end{equation}

\end_inset


\end_layout

\begin_layout References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "Auri\\`ere(1982)"
key "aur82"
literal "true"

\end_inset

Aurière, M.
 1982, 
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aap
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, 109, 301 
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\begin_layout References
\begin_inset CommandInset bibitem
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label "Canizares et al.(1978)"
key "can78"
literal "true"

\end_inset

Canizares, C.
 R., Grindlay, J.
 E., Hiltner, W.
 A., Liller, W., and McClintock, J.
 E.
 1978, 
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, 224, 39 
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\begin_layout References
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label "Djorgovski and King(1984)"
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Djorgovski, S., and King, I.
 R.
 1984, 
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, 277, L49 
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label "Hagiwara and Zeppenfeld(1986)"
key "hag86"
literal "true"

\end_inset

Hagiwara, K., and Zeppenfeld, D.
 1986, Nucl.Phys., 274, 1 
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\begin_inset CommandInset bibitem
LatexCommand bibitem
label "Harris and van den Bergh(1984)"
key "har84"
literal "true"

\end_inset

Harris, W.
 E., and van den Bergh, S.
 1984, 
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, 89, 1816 
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\begin_layout References
\begin_inset CommandInset bibitem
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label "H\\`enon(1961)"
key "hen61"
literal "true"

\end_inset

Hénon, M.
 1961, Ann.d'Ap., 24, 369 
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\begin_layout References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "King(1966)"
key "kin66"
literal "true"

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King, I.
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 1966, 
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, 71, 276 
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\begin_layout References
\begin_inset CommandInset bibitem
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label "King(1975)"
key "kin75"
literal "true"

\end_inset

King, I.
 R.
 1975, Dynamics of Stellar Systems, A.
 Hayli, Dordrecht: Reidel, 1975, 99 
\end_layout

\begin_layout References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "King et al.(1968)"
key "kin68"
literal "true"

\end_inset

King, I.
 R., Hedemann, E., Hodge, S.
 M., and White, R.
 E.
 1968, 
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, 73, 456 
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\begin_inset CommandInset bibitem
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label "Kron et al.(1984)"
key "kro84"
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Kron, G.
 E., Hewitt, A.
 V., and Wasserman, L.
 H.
 1984, 
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\begin_layout References
\begin_inset CommandInset bibitem
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label "Lynden-Bell and Wood(1968)"
key "lyn68"
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Lynden-Bell, D., and Wood, R.
 1968, 
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\begin_layout References
\begin_inset CommandInset bibitem
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label "Newell and O'Neil(1978)"
key "new78"
literal "true"

\end_inset

Newell, E.
 B., and O'Neil, E.
 J.
 1978, 
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, 37, 27 
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label "Ortolani et al.(1985)"
key "ort85"
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Ortolani, S., Rosino, L., and Sandage, A.
 1985, 
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Peterson, C.
 J.
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\begin_layout References
\begin_inset CommandInset bibitem
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label "Spitzer(1985)"
key "spi85"
literal "true"

\end_inset

Spitzer, L.
 1985, Dynamics of Star Clusters, J.
 Goodman and P.
 Hut, Dordrecht: Reidel, 109 
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 to be too wordy, but that would make for poor communication as well.
 
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 that was generated with the \SpecialChar LaTeX
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\begin_layout TableComments
We can also attach a long-ish paragraph of explanatory material to a table.
 Use 
\backslash
tablerefs to append a list of references.
 The following references were from a different table: I've patched them
 in here to show how they look, but don't take them too seriously—I certainly
 have not.
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(1) Barbuy, Spite, & Spite 1985; (2) Bond 1980; (3) Carbon et al.
 1987; (4) Hobbs & Duncan 1987; (5) Gilroy et al.
 1988: (6) Gratton & Ortolani 1986; (7) Gratton & Sneden 1987; (8) Gratton
 & Sneden (1988); (9) Gratton & Sneden 1991; (10) Kraft et al.
 1982; (11) LCL, or Laird, 1990; (12) Leep & Wallerstein 1981; (13) Luck
 & Bond 1981; (14) Luck & Bond 1985; (15) Magain 1987; (16) Magain 1989;
 (17) Peterson 1981; (18) Peterson, Kurucz, & Carney 1990; (19) RMB; (20)
 Schuster & Nissen 1988; (21) Schuster & Nissen 1989b; (22) Spite et al.
 1984; (23) Spite & Spite 1986; (24) Hobbs & Thorburn 1991; (25) Hobbs et
 al.
 1991; (26) Olsen 1983.
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LyX can load deluxetable files verbatim.
 To see this in action, dissolve the following Note and copy the file table.tex
 from the AASTeX 5.0 distribution into the same directory as this file.
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