Collapsed Cores in Globular Clusters, Gauge-Boson Couplings, and AASTEX Macro Sample

S. Djorgovski^{1,2,3} and Ivan R. King^{1} Astronomy Department, University of California, Berkeley, CA 94720

C. D. Biemesderfer^{4,5} National Optical Astronomy Observatories, Tucson, AZ 85719

and

R. J. Hanisch^{5} Space Telescope Science Institute, Baltimore, MD 21218

Received ; accepted

Not to appear in Nonlearned J., 45.

1Visiting Astronomer, Cerro Tololo Inter-American Observatory. CTIO is operated by AURA, Inc. under contract to the National Science Foundation.

2Society of Fellows, Harvard University. 3present address: Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 4Visiting Programmer, Space Telescope Science Institute 5Patron, Alonso’s Bar and Grill

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 show off the results of some recreational mathematics, and give pause to consider the dangers of the too fertile mind.

Subject headings: clusters: globular, peanut — bosons: bozos

1. Introduction

A focal problem today in the dynamics of globular clusters is core collapse. It has been predicted by theory for decades (Hènon 1961, Lynden-Bell and Wood 1968, Spitzer 1985), but observation has been less alert to the phenomenon. For many years the central brightness peak in M15 (King 1975, Newell and O’Neil 1978) seemed a unique anomaly. Then Aurière (1982) suggested a central peak in NGC 6397, and a limited photographic survey of ours (Djorgovski and King 1984, Paper I) found three more cases, including NGC 6624, whose sharp center had often been remarked on (e.g., Canizares et al. 1978).

2. Observations

All our observations were short direct exposures with CCD’s. At Lick Observatory we used a TI 500×500 chip and a GEC 575×385, on the 1-m Nickel reflector. The only filter available at Lick was red. At CTIO we used a GEC 575×385, with B, V, and R filters, and an RCA 512×320, with U, B, V, R, and 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 or even in 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 (King et al., 1968, Peterson 1976, Harris and van den Bergh 1984, Ortolani et al., 1985), as well as photoelectric profiles (King 1966) and electronographic profiles (Kron et al., 1984). In a few cases we judged normality by eye estimates on one of the Sky Surveys.

EDITOR: PLACE TABLE 2 HERE.

EDITOR: PLACE FIGURE 1 HERE.

3. Helicity Amplitudes

It has been realized that helicity amplitudes provide a convenient means for Feynman diagram^{6} 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 (Hagiwara and Zeppenfeld 1986); we briefly summarize below.

EDITOR: PLACE FIGURE 3 HERE.

3.1. Formalism

A tree-level amplitude in e^{+}e^{−} collisions can be expressed in terms of fermion strings of the form

v̄(p_{2}, σ_{2})P_{−τ} a̸_{1} a̸_{2} ⋯ a̸_{n}u(p_{1}, σ_{1}) ,  (1)

where p and σ label the initial e^{±} four-momenta and helicities (σ = ±1), a̸_{i} = a_{i}^{μ}γ_{ν} and P_{τ} = \frac{1}{2}(1 + τγ_{5}) is a chirality projection operator (τ = ±1). The 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.

6Footnotes can be inserted like this.

In the chiral ^{E1} representation the γ matrices are expressed in terms of 2 × 2 Pauli matrices σ and the unit matrix 1 as

γ^{μ} = ( \begin{matrix}0 & σ_{+}^{μ} \\ σ_{−}^{μ} & 0 \end{matrix} ) , γ^{5} = ( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) ,

(2a)

σ_{±}^{μ} = (1, ±σ) ,

giving

a̸ = ( \begin{matrix}0 & (a̸)_{+} \\ (a̸)_{−} & 0 \end{matrix} ) , (a̸)_{±} = a_{μ}σ_{±}^{μ} ,  (3)

The spinors are expressed in terms of two-component Weyl spinors as

u = ( \begin{matrix}(u)_{−} \\ (u)_{+} \end{matrix} ) , v = ((v)_{+}^{†}, (v)_{−}^{†}) .  (3A)

The Weyl spinors are given in terms of helicity eigenstates χ_{λ}(p) with λ = ±1 by

u(p, λ)_{±} = (E ± λ|p|)^{1/2}χ_{λ}(p) ,

(4)

v(p, λ)_{±} = ±λ(E ∓ λ|p|)^{1/2}χ_{−λ}(p)

4. Floating material and so forth

Consider a task that computes profile parameters for a modified Lorentzian of the form

I = \frac{1}{1 + d_{1}^{P (1+d_{2})}}  (5)

E1NOTE TO EDITOR: This is a note to the copy editor that is inserted by the author using the \notetoeditor command. where

d_{1} = \sqrt{( _{\frac{x_{1}}{R_{maj}}} )^{2} + ( \frac{y_{1}}{R_{min}} )^{2}}
d_{2} = \sqrt{( _{\frac{x_{1}}{P R_{maj}}} )^{2} + ( \frac{y_{1}}{P R_{min}} )^{2}}
x_{1} = (x − x_{0}) cos Θ + (y − y_{0}) sin Θ
y_{1} = −(x − x_{0}) sin Θ + (y − y_{0}) cos Θ

In these expressions x_{0},y_{0} is the star center, and Θ is the angle with the x axis. Results of this task are shown in table 1. It is not clear how these sorts of analyses may affect determination of M⊙ and 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 Ca II. Roughly 4/_{5} of the electronically submitted abstracts for AAS meetings are error-free.

EDITOR: PLACE TABLE 1 HERE.

EDITOR: PLACE TABLE 1A HERE.

EDITOR: PLACE FIGURE 2 HERE.

We are grateful to V. Barger, T. Han, and R. J. N. Phillips for doing the math in section 3.1.

A. Floating material and so forth

Consider a task that computes profile parameters for a modified Lorentzian of the form

I = \frac{1}{1 + d_{1}^{P (1+d_{2})}}  (A1)

where

d_{1} = \frac{3}{4}\sqrt{( _{\frac{x_{1}}{R_{maj}}} )^{2} + ( \frac{y_{1}}{R_{min}} )^{2}}

d_{2} = \frac{3}{4} \sqrt{( _{\frac{x_{1}}{P R_{maj}}} )^{2} + ( \frac{y_{1}}{P R_{min}} )^{2}}  (A2a)
x_{1} = (x − x_{0}) cos Θ + (y − y_{0}) sin Θ  (A2b)
y_{1} = −(x − x_{0}) sin Θ + (y − y_{0}) cos Θ  (A2c)

For completeness, here is one last equation.

e = mc^{2}  (A3)

Table 1. Terribly relevant tabular information.

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

aSample footnote for table 1 that was generated with the deluxetable environment bAnother sample footnote for table 1 cFootnote with no call out dAnother footnote with no call out eA further additional footnote with no call out

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

Table 1A: More terribly relevant tabular information. Notice that it is possible, but not necessarily desirable, to have more than one table on a page where each can have associated independent notes. We extend the caption with further pointless drivel to see the effects of lengthy text on caption formatting.

tSample footnote for table 1A that was generated with the L^{A}TEX table environment vYet another sample footnote for table 1A uAnother sample footnote for table 1A

Note. — We can also attach a long-ish paragraph of explanatory material to a table. This would be done for journals where long captions are not permitted (usually because the caption is regarded as the table’s title). A different command would be used if the paragraph contained a list of references for the table.

REFERENCES

Aurière, M. 1982, A&A, 109, 301 Canizares, C. R., Grindlay, J. E., Hiltner, W. A., Liller, W., and McClintock, J. E. 1978,

ApJ, 224, 39 Djorgovski, S., and King, I. R. 1984, ApJ, 277, L49 Hagiwara, K., and Zeppenfeld, D. 1986, Nucl.Phys., 274, 1 Harris, W. E., and van den Bergh, S. 1984, AJ, 89, 1816 Hénon, M. 1961, Ann.d’Ap., 24, 369 King, I. R. 1966, AJ, 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, AJ, 73, 456 Kron, G. E., Hewitt, A. V., and Wasserman, L. H. 1984, PASP, 96, 198 Lynden-Bell, D., and Wood, R. 1968, MNRAS, 138, 495 Newell, E. B., and O’Neil, E. J. 1978, ApJS, 37, 27 Ortolani, S., Rosino, L., and Sandage, A. 1985, AJ, 90, 473 Peterson, C. J. 1976, AJ, 81, 617 Spitzer, L. 1985, Dynamics of Star Clusters, J. Goodman and P. Hut, Dordrecht: Reidel,

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This manuscript was prepared with the AAS L^{A}TEX macros v4.0.

Fig. 1.— This is the first figure and it uses sgi9259.eps as its EPS figure file. Fig. 2.— This figure has no associated EPS file, so the optional parameter is omitted. Fig. 3.— 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.