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#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
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\begin_preamble
\newcommand{\vdag}{(v)^\dagger}
\newcommand{\myemail}{skywalker@galaxy.far.far.away}
\slugcomment{Not to appear in Nonlearned J., 45.}
\shorttitle{Djorgovski et al.}
\shortauthors{Collapsed Cores in Globular Clusters}
\end_preamble
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\begin_body
\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
\begin_layout Plain Layout
1,2,3
\end_layout
\end_inset
and Ivan R.
King
\begin_inset Flex altaffilmark
status collapsed
\begin_layout Plain Layout
1
\end_layout
\end_inset
\end_layout
\begin_layout Affiliation
Astronomy Department, University of California, Berkeley, CA 94720
\end_layout
\begin_layout Author
C.
D.
Biemesderfer
\begin_inset Flex altaffilmark
status collapsed
\begin_layout Plain Layout
4,5
\end_layout
\end_inset
\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
\begin_inset Flex altaffilmark
status collapsed
\begin_layout Plain Layout
5
\end_layout
\end_inset
\end_layout
\begin_layout Affiliation
Space Telescope Science Institute, Baltimore, MD 21218
\end_layout
\begin_layout Altaffilation
\begin_inset Argument 1
status open
\begin_layout Plain Layout
1
\end_layout
\end_inset
Visiting Astronomer, Cerro Tololo Inter-American Observatory.
CTIO is operated by AURA, Inc.
\begin_inset space \space{}
\end_inset
under contract to the National Science Foundation.
\end_layout
\begin_layout Altaffilation
\begin_inset Argument 1
status open
\begin_layout Plain Layout
2
\end_layout
\end_inset
Society of Fellows, Harvard University.
\end_layout
\begin_layout Altaffilation
\begin_inset Argument 1
status open
\begin_layout Plain Layout
3
\end_layout
\end_inset
present address: Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138
\end_layout
\begin_layout Altaffilation
\begin_inset Argument 1
status open
\begin_layout Plain Layout
4
\end_layout
\end_inset
Visiting Programmer, Space Telescope Science Institute
\end_layout
\begin_layout Altaffilation
\begin_inset Argument 1
status open
\begin_layout Plain Layout
5
\end_layout
\end_inset
Patron, Alonso's Bar and Grill
\end_layout
\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
status collapsed
\begin_layout Plain Layout
\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
status collapsed
\begin_layout Plain Layout
\backslash
citet{aur82}
\end_layout
\end_inset
suggested a central peak in NGC 6397, and a limited photographic survey
of ours
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
citep[Paper I]{djo84}
\end_layout
\end_inset
found three more cases, including NGC 6624, whose sharp center had often
been remarked on
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
citep{can78}
\end_layout
\end_inset
.
\end_layout
\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
status collapsed
\begin_layout Plain Layout
\backslash
citep{kin68,pet76,har84,ort85}
\end_layout
\end_inset
, as well as photoelectric profiles
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
citep{kin66,kin75}
\end_layout
\end_inset
and electronographic profiles
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\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 $M_{\sun}$
\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
status collapsed
\begin_layout Plain Layout
\backslash
ion{Ca}{2}
\end_layout
\end_inset
.
Roughly
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
slantfrac{4}{5}
\end_layout
\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
\begin_inset Flex URL
status collapsed
\begin_layout Plain Layout
http://www.aas.org/publications/aastex
\end_layout
\end_inset
or the
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
anchor{
\end_layout
\end_inset
ftp://www.aas.org/pubs/
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}{
\end_layout
\end_inset
AAS ftp site
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
.
\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,
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
aap
\end_layout
\end_inset
, 109, 301
\end_layout
\begin_layout References
\begin_inset CommandInset bibitem
LatexCommand bibitem
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,
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
apj
\end_layout
\end_inset
, 224, 39
\end_layout
\begin_layout References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "Djorgovski and King(1984)"
key "djo84"
literal "true"
\end_inset
Djorgovski, S., and King, I.
R.
1984,
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
apjl
\end_layout
\end_inset
, 277, L49
\end_layout
\begin_layout References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "Hagiwara and Zeppenfeld(1986)"
key "hag86"
literal "true"
\end_inset
Hagiwara, K., and Zeppenfeld, D.
1986, Nucl.Phys., 274, 1
\end_layout
\begin_layout References
\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,
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
aj
\end_layout
\end_inset
, 89, 1816
\end_layout
\begin_layout References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "H\\`enon(1961)"
key "hen61"
literal "true"
\end_inset
Hénon, M.
1961, Ann.d'Ap., 24, 369
\end_layout
\begin_layout References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "King(1966)"
key "kin66"
literal "true"
\end_inset
King, I.
R.
1966,
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
aj
\end_layout
\end_inset
, 71, 276
\end_layout
\begin_layout References
\begin_inset CommandInset bibitem
LatexCommand bibitem
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,
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
aj
\end_layout
\end_inset
, 73, 456
\end_layout
\begin_layout References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "Kron et al.(1984)"
key "kro84"
literal "true"
\end_inset
Kron, G.
E., Hewitt, A.
V., and Wasserman, L.
H.
1984,
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
pasp
\end_layout
\end_inset
, 96, 198
\end_layout
\begin_layout References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "Lynden-Bell and Wood(1968)"
key "lyn68"
literal "true"
\end_inset
Lynden-Bell, D., and Wood, R.
1968,
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
mnras
\end_layout
\end_inset
, 138, 495
\end_layout
\begin_layout References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "Newell and O'Neil(1978)"
key "new78"
literal "true"
\end_inset
Newell, E.
B., and O'Neil, E.
J.
1978,
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
apjs
\end_layout
\end_inset
, 37, 27
\end_layout
\begin_layout References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "Ortolani et al.(1985)"
key "ort85"
literal "true"
\end_inset
Ortolani, S., Rosino, L., and Sandage, A.
1985,
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
aj
\end_layout
\end_inset
, 90, 473
\end_layout
\begin_layout References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "Peterson(1976)"
key "pet76"
literal "true"
\end_inset
Peterson, C.
J.
1976,
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
aj
\end_layout
\end_inset
, 81, 617
\end_layout
\begin_layout References
\begin_inset CommandInset bibitem
LatexCommand bibitem
label "Spitzer(1985)"
key "spi85"
literal "true"
\end_inset
Spitzer, L.
1985, Dynamics of Star Clusters, J.
Goodman and P.
Hut, Dordrecht: Reidel, 109
\end_layout
\begin_layout Standard
\begin_inset Newpage clearpage
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Caption FigCaption
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig1"
\end_inset
This is the first figure and it uses sgi9259.eps as its EPS figure file.
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Caption FigCaption
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig2"
\end_inset
This is an example of a long figure caption that must be set as a paragraph.
The processor has to buffer the text of the caption, so it is good not
to be too wordy, but that would make for poor communication as well.
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Caption FigCaption
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "fig3"
\end_inset
This figure has no associated EPS file, so the optional parameter is omitted.
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Newpage clearpage
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Float table
wide false
sideways false
status open
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Terribly relevant tabular information.
\begin_inset CommandInset label
LatexCommand label
name "tbl-2"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset VSpace medskip
\end_inset
\end_layout
\begin_layout Plain Layout
\align center
\begin_inset Tabular
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Height
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a
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b
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Ref.
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1
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33472.5
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-0.1
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0.4
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53
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27.4
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2.065
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1.940
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3.900
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68.3
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116.2
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-27.639
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27802.4
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60
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3.7
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1.628
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1.510
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2.156
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6.8
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7.5
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-26.764
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1.551
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2.159
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-40.272
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41
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54.8
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2.282
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2.156
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4.313
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2.343
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\end_inset
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a
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}{
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Sample footnote for table
\begin_inset space ~
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\begin_inset CommandInset ref
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reference "tbl-2"
\end_inset
that was generated with the \SpecialChar LaTeX
table environment
\end_layout
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b
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\end_inset
\end_layout
\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.
\end_layout
\begin_layout TableRefs
(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, uncomment the following
\end_layout
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% line (delete the
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\begin_inset Quotes eld
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%
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) and copy the file table.tex from the AASTeX 5.0 distribution into
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% the same directory as this file.
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%
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\begin_inset CommandInset include
LatexCommand input
filename "table"
\end_inset
\end_layout
\end_body
\end_document
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