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

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

                         aastex-help@aas.org

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 explore some diversionary mathematics and recreational tables.

Subject headings: globular clusters: general — globular clusters: individual(NGC 6397, NGC 6624, NGC 7078, Terzan 8

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 & 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 & 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 & King 1984, Paper I) found three more cases, NGC 6624, NGC 7078, and Terzan 8), whose sharp center had often been remarked on (Canizares et al. 1978).

2. Observations

All our observations were short direct exposures with CCD’s. We also have a random Chandra data set ADS/Sa.ASCA#X/86008020 and a neat HST FOS spectrum that readers can access via the links in the electronic edition. Unfortunately this has nothing whatsoever to do with this research. 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 & van den Bergh 1984; Ortolani et al. 1985), as well as photoelectric profiles (King 1966, 1975) and electronographic profiles (Kron et al. 1984). In a few cases we judged normality by eye estimates on one of the Sky Surveys.

3. Helicity Amplitudes

It has been realized that helicity amplitudes provide a convenient means for Feynman diagram^{1} 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 & Zeppenfeld 1986); we briefly summarize below.

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} â_{1}â_{2} ⋯ â_{n}u(p_{1}, σ_{1}),  (1)

where p and σ label the initial e^{±} four-momenta and helicities (σ = ±1), â_{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.

In the chiral ^{E1} representation the γ matrices are expressed in terms of 2 × 2 Pauli

1Footnotes can be inserted like this.

E1NOTE TO EDITOR: Figures 1 and 2 should appear side-by-side in print 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} ) ,
σ_{±}^{μ} = (1, ±σ),

giving

â = ( \begin{matrix}0 & (â)_{+} \\ (â)_{−} & 0 \end{matrix} ) , (â)_{±} = a_{μ}σ_{±}^{μ},  (2)

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

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

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),  (4a)
v(p, λ)_{±} = ±λ(E ∓ λ|p|)^{1/2}χ_{−λ}(p)  (4b)

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)

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_{⊙}, 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 \frac{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 3.1. More information on the AASTeX macros package is available at http://www.aas.org/publications/aastex. For technical support, please write to aastex-help@aas.org.

Facilities: Nickel, HST (STIS), CXO (ASIS).

A. Appendix material

Consider once again 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)

REFERENCES

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

ApJ, 224, 39 Djorgovski, S., & King, I. R. 1984, ApJ, 277, L49 Hagiwara, K., & Zeppenfeld, D. 1986, Nucl.Phys., 274, 1 Harris, W. E., & van den Bergh, S. 1984, AJ, 89, 1816 Hénon, M. 1961, Ann.d’Ap., 24, 369 Heiles, C. & Troland, T. H., 2003, ApJS, preprint doi:10.1086/381753 Kim, W.-T., Ostriker, E., & Stone, J. M., 2003, ApJ, 599, 1157 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., & White, R. E. 1968, AJ, 73, 456 Kron, G. E., Hewitt, A. V., & Wasserman, L. H. 1984, PASP, 96, 198 Lynden-Bell, D., & Wood, R. 1968, MNRAS, 138, 495 Newell, E. B., & O’Neil, E. J. 1978, ApJS, 37, 27 Ortolani, S., Rosino, L., & Sandage, A. 1985, AJ, 90, 473 Peterson, C. J. 1976, AJ, 81, 617 Rudnick, G. et al., 2003, ApJ, 599, 847 Spitzer, L. 1985, Dynamics of Star Clusters, J. Goodman & P. Hut, Dordrecht: Reidel, 109 Treu, T. et al., 2003, ApJ, 591, 53

This manuscript was prepared with the AAS L^{A}TEX macros v5.2.

Fig. 1.— Derived spectra for 3C138 (see Heiles & Troland 2003). Plots for all sources are available in the electronic edition of The Astrophysical Journal.

Fig. 2.— A panel taken from Figure 2 of Rudnick et al. (2003). See the electronic edition of the Journal for a color version of this figure.

Fig. 3.— Animation still frame taken from Kim, Ostricker, & Stone (2003). This figure is also available as an mpeg animation in the electronic edition of the Astrophysical Journal.

Table 1. Sample table taken from Treu et al. (2003)

POS chip ID  X  Y  RA  DEC  IAU± δ IAU IAP1± δ IAP1 IAP2 ± δ IAP2 star E Comment
0  2  1 1370.99 57.35 6.651120 17.131149 21.344±0.006  2 4.385±0.016  23.528±0.013 0.0  9 -
0  2  2 1476.62  8.03 6.651480 17.129572 21.641±0.005  2 3.141±0.007  22.007±0.004 0.0  9 -
0  2  3 1079.62 28.92 6.652430 17.135000 23.953±0.030  2 4.890±0.023  24.240±0.023 0.0  - -
0  2  4  114.58 21.22 6.655560 17.148020 23.801±0.025  2 5.039±0.026  24.112±0.021 0.0  - -
0  2  5  46.78 19.46 6.655800 17.148932 23.012±0.012  2 3.924±0.012  23.282±0.011 0.0  - -
0  2  6 1441.84 16.16 6.651480 17.130072 24.393±0.045  2 6.099±0.062  25.119±0.049 0.0  - -
0  2  7  205.43  3.96 6.655520 17.146742 24.424±0.032  2 5.028±0.025  24.597±0.027 0.0  - -
0  2  8 1321.63  9.76 6.651950 17.131672 22.189±0.011  2 4.743±0.021  23.298±0.011 0.0  4 edge

Note. — Table 1 is published in its entirety in the electronic edition of the Astrophysical Journal. A portion is shown here for guidance regarding its form and content.

aSample footnote for table 1 that was generated with the deluxetable environment bAnother sample footnote for table 1

Table 2: More 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^{c} -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^{c} 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 2 that was generated with the L^{A}TEX table environment bYet another sample footnote for table 2 cAnother sample footnote for table 2

Note. — We can also attach a long-ish paragraph of explanatory material to a table.

Table 3. Literature Data for Program Stars

Star  V b−y m_{1} c_{1} ref T_{eff} log g v_{turb} [Fe/H] ref
HD 97 9.7 0.51 0.15 0.35 2 ⋯ ⋯ ⋯ −1.50 2
5015 ⋯  ⋯ −1.50 10
HD 2665 7.7 0.54 0.09 0.34 2 ⋯ ⋯ ⋯ −2.30 2
5000 2.50 2.4 −1.99 5
5120 3.00 2.0 −1.69 7
4980 ⋯  ⋯ −2.05 10
HD 4306 9.0 0.52 0.05 0.35 20, 2 ⋯ ⋯ ⋯ −2.70 2
5000 1.75 2.0 −2.70 13
5000 1.50 1.8 −2.65 14
4950 2.10 2.0 −2.92 8
5000 2.25 2.0 −2.83 18
⋯ ⋯ ⋯ −2.80 21
4930 ⋯  ⋯ −2.45 10
HD 5426 9.6 0.50 0.08 0.34 2  ⋯  ⋯  ⋯ −2.30 2
HD 6755 7.7 0.49 0.12 0.28 20, 2 ⋯  ⋯  ⋯ −1.70 2
5200 2.50 2.4 −1.56 5
5260 3.00 2.7 −1.67 7
⋯ ⋯ ⋯ −1.58 21
5200 ⋯  ⋯ −1.80 10
4600 ⋯  ⋯ −2.75 10
HD 94028 8.2 0.34 0.08 0.25 20 5795 4.00 ⋯ −1.70 22

Table 3—Continued

Star  V b−y m_{1} c_{1}  ref T_{eff} log g v_{turb} [Fe/H] ref
5860 ⋯  ⋯  −1.70 4
5910 3.80 ⋯  −1.76 15
5800 ⋯  ⋯  −1.67 17
5902 ⋯  ⋯  −1.50 11
5900 ⋯  ⋯  −1.57 3
⋯  ⋯  ⋯  −1.32 21
HD 97916 9.2 0.29 0.10 0.41 20 6125 4.00 ⋯  −1.10 22
6160 ⋯ ⋯ −1.39 3
6240 3.70 ⋯  −1.28 15
5950 ⋯  ⋯  −1.50 17
6204 ⋯  ⋯  −1.36 11

This is a cut-in head

+26^{∘}2606 9.7 0.34 0.05 0.28 20,11 5980 ⋯  ⋯ < −2.20 19
5950 ⋯  ⋯  −2.89 24
+26^{∘}3578 9.4 0.31 0.05 0.37 20,11 5830 ⋯  ⋯  −2.60 4
5800 ⋯  ⋯  −2.62 17
6177 ⋯  ⋯  −2.51 11
6000 3.25 ⋯  −2.20 22
6140 3.50 ⋯  −2.57 15

Table 3—Continued

Star  V b−y m_{1} c_{1}  ref T_{eff} log g v_{turb} [Fe/H] ref
+30^{∘}2611 9.2 0.82 0.33 0.55 2  ⋯  ⋯  ⋯ −1.70 2
4400 1.80 ⋯ −1.70 12
4400 0.90 1.7 −1.20 14
4260 ⋯  ⋯ −1.55 10
+37^{∘}1458 8.9 0.44 0.07 0.22 20,11 5296 ⋯  ⋯ −2.39 11
5420 ⋯  ⋯ −2.43 3
+58^{∘}1218 10.0 0.51 0.03 0.36 2 ⋯ ⋯ ⋯ −2.80 2
5000 1.10 2.2 −2.71 14
5000 2.20 1.8 −2.46 5
4980 ⋯  ⋯ −2.55 10
+72^{∘}0094 10.2 0.31 0.09 0.26 12 6160 ⋯  ⋯ −1.80 19

I’m a side head:

G5–36  10.8 0.40 0.07 0.28 20  ⋯  ⋯  ⋯ −1.19 21
G18–54 10.7 0.37 0.08 0.28 20  ⋯  ⋯  ⋯ −1.34 21
G20–08  9.9 0.36 0.05 0.25 20,11 5849 ⋯  ⋯ −2.59 11
⋯  ⋯  ⋯ −2.03 21
G20–15 10.6 0.45 0.03 0.27 20,11 5657 ⋯  ⋯ −2.00 11
6020 ⋯  ⋯ −1.56 3
⋯  ⋯  ⋯ −1.58 21
G21–22 10.7 0.38 0.07 0.27 20,11 ⋯  ⋯  ⋯ −1.23 21
G24–03 10.5 0.36 0.06 0.27 20,11 5866 ⋯  ⋯ −1.78 11

Table 3—Continued

Star  V b−y m_{1} c_{1} ref T_{eff} log g v_{turb} [Fe/H] ref
⋯  ⋯  ⋯ −1.70 21
G30–52  8.6 0.50 0.25 0.27 11 4757 ⋯  ⋯ −2.12 11
4880 ⋯  ⋯ −2.14 3
G33–09  10.6 0.41 0.10 0.28 20 5575 ⋯  ⋯ −1.48 11
G66–22  10.5 0.46 0.16 0.28 11 5060 ⋯  ⋯ −1.77 3
⋯  ⋯  ⋯ −1.04 21
G90–03  10.4 0.37 0.04 0.29 20 ⋯  ⋯  ⋯ −2.01 21
LP 608–62^{a} 10.5 0.30 0.07 0.35 11 6250 ⋯  ⋯ −2.70 4

aStar LP 608–62 is also known as BD+1^{∘}2341p. We will make this footnote extra long so that it extends over two lines.

References. — (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.