S. Djorgovski and Ivan R. King Astronomy Department, University of California, Berkeley, CA 94720 Visiting Astronomer Cerro Tololo Inter-American Observatory.CTIO is operated by AURA Inc. under contract to the National Science Foundation. Society of Fellows, Harvard University. present address: Center for Astrophysics60 Garden Street, Cambridge, MA 02138 C. D. Biemesderfer National Optical Astronomy Observatories, Tucson, AZ 85719 Visiting Programmer, Space Telescope Science Institute Patron, Alonso's Bar and Grill aastex-help@aas.org R. J. Hanisch Space Telescope Science Institute, Baltimore, MD 21218 Patron, Alonso's Bar and Grill clusters: globular, peanut—bosons: bozos 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.

A focal problem today in the dynamics of globular clusters is core collapse. It has been predicted by theory for decades , , , but observation has been less alert to the phenomenon. For many years the central brightness peak in M15 , seemed a unique anomaly. Then suggested a central peak in NGC 6397, and a limited photographic survey of ours found three more cases, including NGC 6624, whose sharp center had often been remarked on .

All our observations were short direct exposures with CCD's. At Lick Observatory we used a TI 500 \times × 500 chip and a GEC 575 \times × 385, on the 1-m Nickel reflector. The only filter available at Lick was red. At CTIO we used a GEC 575 \times × 385, with B,V, B,V, and R R filters, and an RCA 512 \times × 320, with U,B,V,R, U,B,V,R, and I I 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 U U or even in B B . All four cameras had scales of the order of 0.4 arcsec/pixel, and our field sizes were around 3 arcmin. 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 , , , , as well as photoelectric profiles , and electronographic profiles . In a few cases we judged normality by eye estimates on one of the Sky Surveys.

It has been realized that helicity amplitudes provide a convenient means for Feynman diagram Footnotes can be inserted like this. 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 ; we briefly summarize below.

A tree-level amplitude in e^{+}e^{-} e + e - collisions can be expressed in terms of fermion strings of the form \bar{v}(p_{2},\sigma_{2})P_{-\tau}\hat{a}_{1}\hat{a}_{2}\cdots\hat{a}_{n}u(p_{1},\sigma_{1}), v ¯ ( p 2 , σ 2 ) P -τ a ˆ 1 a ˆ 2 ⋯ a ˆ n u( p 1 , σ 1 ) , where p p and \sigma σ label the initial e^{\pm} e ± four-momenta and helicities (\sigma=\pm1) ( σ=±1 ) , \hat{a}_{i}=a_{i}^{\mu}\gamma_{\nu} a ˆ i = a i μ γ ν and P_{\tau}=\frac{1}{2}(1+\tau\gamma_{5}) P τ = 1 2 ( 1+τ γ 5 ) is a chirality projection operator (\tau=\pm1) ( τ=±1 ) . The a_{i}^{\mu} a i μ may be formed from particle four-momenta, gauge-boson polarization vectors or fermion strings with an uncontracted Lorentz index associated with final-state fermions. Figures 1 and 2 should appear side-by-side in print In the chiral representation the \gamma γ matrices are expressed in terms of 2\times2 2×2 Pauli matrices \sigma σ and the unit matrix 1 as \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), γ μ = ( 0 σ + μ σ - μ 0 ), γ 5 =( -1 0 0 1 ), σ ± μ = ( 1 ,±σ ) , giving \hat{a}=\left(\begin{array}{cc} 0 & (\hat{a})_{+}\\ (\hat{a})_{-} & 0 \end{array}\right),(\hat{a})_{\pm}=a_{\mu}\sigma_{\pm}^{\mu}, a ˆ =( 0 ( a ˆ ) + ( a ˆ ) - 0 ),( a ˆ ) ± = a μ σ ± μ , The spinors are expressed in terms of two-component Weyl spinors as u=\left(\begin{array}{c} (u)_{-}\\ (u)_{+} \end{array}\right),v={\textbf{(}}\vdag_{+}{\textbf{,}}\vdag_{-}{\textbf{)}}. MathML export failed. Please report this as a bug. The Weyl spinors are given in terms of helicity eigenstates \chi_{\lambda}(p) χ λ ( p ) with \lambda=\pm1 λ=±1 by 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) u(p,λ ) ± = (E±λ| p | ) 1/2 χ λ ( p ) , v(p,λ ) ± = ±λ(E∓λ| p | ) 1/2 χ -λ ( p )

Consider a task that computes profile parameters for a modified Lorentzian of the form I=\frac{1}{1+d_{1}^{P(1+d_{2})}} I= 1 1+ d 1 P( 1+ d 2 ) where 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}} d 1 = ( x 1 R maj ) 2 + ( y 1 R min ) 2 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}} MathML export failed. Please report this as a bug. x_{1}=(x-x_{0})\cos\Theta+(y-y_{0})\sin\Theta x 1 =( x- x 0 ) cosΘ+( y- y 0 ) sinΘ y_{1}=-(x-x_{0})\sin\Theta+(y-y_{0})\cos\Theta y 1 =-( x- x 0 ) sinΘ+( y- y 0 ) cosΘ In these expressions x_{0} x 0 , y_{0} y 0 is the star center, and \Theta Θ is the angle with the x x axis. Results of this task are shown in table . It is not clear how these sorts of analyses may affect determination of M_{\text{\sun}} M ☼ , but the assumption is that the alternate results should be less than 90° out of phase with previous values. We have no observations of . Roughly \nicefrac{4}{5} 4 5 of the electronically submitted abstracts for AAS meetings are error-free. We are grateful to V. Barger, T. Han, and R. J. N. Phillips for doing the math in section . More information on the AASTeX macros package are available at http://www.aas.org/publications/aastex or the AAS ftp site. IRAF, AIPS, Astropy, ...

Aurière, M. 1982, , 109, 301 Canizares, C. R., Grindlay, J. E., Hiltner, W. A., Liller, W., and McClintock, J. E. 1978, , 224, 39 Djorgovski, S., and King, I. R. 1984, , 277, L49 Hagiwara, K., and Zeppenfeld, D. 1986, Nucl.Phys., 274, 1 Harris, W. E., and van den Bergh, S. 1984, , 89, 1816 Hénon, M. 1961, Ann.d'Ap., 24, 369 King, I. R. 1966, , 71, 276 King, I. R. 1975, Dynamics of Stellar Systems, A. Hayli, Dordrecht: Reidel, 1975, 99 King, I. R., Hedemann, E., Hodge, S. M., and White, R. E. 1968, , 73, 456 Kron, G. E., Hewitt, A. V., and Wasserman, L. H. 1984, , 96, 198 Lynden-Bell, D., and Wood, R. 1968, , 138, 495 Newell, E. B., and O'Neil, E. J. 1978, , 37, 27 Ortolani, S., Rosino, L., and Sandage, A. 1985, , 90, 473 Peterson, C. J. 1976, , 81, 617 Spitzer, L. 1985, Dynamics of Star Clusters, J. Goodman and P. Hut, Dordrecht: Reidel, 109 a Sample footnote for table  that was generated with the LaTeX table environmentb Yet another sample footnote for table c Another sample footnote for table We can also attach a long-ish paragraph of explanatory material to a table. Use \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.(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.

Terribly relevant tabular information.

Star	Height	d_{x} d x	d_{y} d y	n n	\chi^{2} χ 2	R_{maj} R maj	R_{min} R min	P P a	PR_{maj} P R maj	PR_{min} P R min	\Theta Θ b	Ref.
1	33472.5	-0.1	0.4	53	27.4	2.065	1.940	3.900	68.3	116.2	-27.639	1,2
2	27802.4	-0.3	-0.2	60	3.7	1.628	1.510	2.156	6.8	7.5	-26.764	3
3	29210.6	0.9	0.3	60	3.4	1.622	1.551	2.159	6.7	7.3	-40.272	4
4	32733.8	-1.2c	-0.5	41	54.8	2.282	2.156	4.313	117.4	78.2	-35.847	5,6
5	9607.4	-0.4	-0.4	60	1.4	1.669c	1.574	2.343	8.0	8.9	-33.417	7
6	31638.6	1.6	0.1	39	315.2	3.433	3.075	7.488	92.1	25.3	-12.052	8