# Chained Flexible Capsule Endoscope: Unraveling the Conundrum of Size
Limitations and Functional Integration for Gastrointestinal Transitivity

Sishen Yuan1,†, Guang Li1,†, Baijia Liang1, Lailu Li1, Qingzhuo Zheng1,
Shuang Song2, Zhen Li3, Hongliang Ren1∗ The work was supported by Hong Kong
Research Grants Council (RGC) NSFC/RGC Joint Research Scheme N_CUHK420/22;
Collaborative Research Fund (CRF C4063-18G, C4026-21G), and General Research
Fund (GRF 14203323), Shenzhen-Hong Kong-Macau Technology Research Programme
(Type C) Grant 202108233000303 at The Chinese University of Hong Kong (CUHK).
(Corresponding author: Hongliang Ren. $\dagger$ indicates contributed
equally.) 1Sishen Yuan, Guang Li, Baijia Liang, Lailu Li, Qingzhuo Zheng, and
Hongliang Ren are with the Department of Electronic Engineering, The Chinese
University of Hong Kong, Hong Kong 2Shuang Song is with the Harbin Institute
of Technology, Shenzhen 518055, China 3Zhen Li is with the Department of
Gastroenterology, Laboratory of Translational Gastroenterology, and Robot
Engineering Laboratory for Precise Diagnosis and Therapyof GI Tumor, Qilu
Hospital of Shandong University, Cheeloo College of Medicine, Jinan, Shandong,
China

###### Abstract

Capsule endoscopes, predominantly serving diagnostic functions, provide lucid
internal imagery but are devoid of surgical or therapeutic capabilities.
Consequently, despite lesion detection, physicians frequently resort to
traditional endoscopic or open surgical procedures for treatment, resulting in
more complex, potentially risky interventions. To surmount these limitations,
this study introduces a chained flexible capsule endoscope (FCE) design
concept, specifically conceived to navigate the inherent volume constraints of
capsule endoscopes whilst augmenting their therapeutic functionalities. The
FCE’s distinctive flexibility originates from a conventional rotating joint
design and the incision pattern in the flexible material. In vitro experiments
validated the passive navigation ability of the FCE in rugged intestinal
tracts. Further, the FCE demonstrates consistent reptile-like peristalsis
under the influence of an external magnetic field, and possesses the
capability for film expansion and disintegration under high-frequency
electromagnetic stimulation. These findings illuminate a promising path toward
amplifying the therapeutic capacities of capsule endoscopes without
necessitating a size compromise.

##  I Introduction

Capsule endoscopy facilitates non-invasive inspection of the gastrointestinal
tract [1]. Since its introduction into clinical practice in 2001, it has
garnered widespread adoption globally. Capsule endoscopy exhibits notable
superiority in diagnosing gastrointestinal maladies, but often, diagnosis is
merely an initial step towards resolving the issue [2].

Upon identification of a lesion or anomaly, subsequent treatment or
intervention is typically necessitated, frequently requiring more intricate
surgical or therapeutic procedures. These encompass biological tissue
extraction, drug delivery, haemostasis, among other interventions. However,
the current suite of capsule endoscopes is predominantly deployed for
diagnostic, as opposed to therapeutic, objectives. They yield lucid internal
imagery but are bereft of capabilities to execute surgical or therapeutic
manoeuvres [3]. Consequently, despite lesion detection by the capsule
endoscope, physicians often resort to traditional endoscopic or open surgical
procedures for treatment. This necessitates a more complex and risky treatment
regimen, potentially prolonging the patient’s recovery. Therefore, there is an
unmet need to enhance the therapeutic capabilities of capsule endoscopy.

Figure 1: Design and functionality of the proposed flexible capsule endoscope.
(a) Illustration of the challenges faced by longer capsule endoscopes during
transit through the narrow, elongated lumen of the small intestine, where
their smooth passage is impeded. (b) Schematic representation of the ”rigid-
soft-rigid” configuration of our flexible capsule endoscope. The first ’rigid’
segment is modelled after existing commercial endoscopes, with the ’soft’
segment enabling passive deformation under intestinal compression stresses and
peristaltic forces, facilitating adaptation to the zigzag environment of the
small intestine. (c) The additional rigid section at the tail end of the
capsule is designed to induce a worm-like crawling propulsion motion via the
interaction of a pair of internal radial magnets under an external magnetic
field, and to integrate functional components. The figure also depicts the
expansion of the capsule endoscope in response to high-frequency
electromagnetic heating, leading to the expansion of an Ecoflex film. This
feature suggests potential applications for anchoring, drug delivery,
sampling, and scaffold deployment.

Numerous contemporary studies concentrate on functionalizing capsule
endoscopes, primarily for anchoring, biopsy, and drug delivery [4]. Anchoring
is generally achieved through electrothermal actuation of memory alloy wires
or wireless magnetic control. Implementing these mechanisms necessitates
either an enlargement of the capsule endoscope’s radial dimensions or an
increase in its axial rigidity scale [5, 6, 7]. Current dimensions do not
appear to account for the volume occupation of the endoscope’s actual vision
module. This paradox is similarly observed in biopsy-capable capsule robots
[8, 9, 10].

In response to these challenges, a collaborative strategy involving modular
capsule robots has been suggested for capsule robot volume expansion. During
the examination process, a capsule robot equipped with an image acquisition
function is initially used as the primary functional module. Subsequently,
according to the patient’s individual needs, an additional functional module
with corresponding capabilities is ingested. These multiple modules have the
potential to assemble in vivo and collectively execute medical tasks [11, 12,
13]. However, the in vivo assembly and control of these modules remain
formidable challenges. Additionally, their suitability for smooth intestinal
passage and potential perforation risks have not been fully explored.

This study presents a design concept for flexible capsule endoscopes (FCEs),
devised to reconcile the inherent volume constraints of capsule endoscopes
with the requirement for comprehensive functionalization. The flexibility of
the FCE stems from the traditional rotating joint design and the incision
pattern within the flexible material. We substantiated its feasibility for
passive navigation through a rugged intestinal tract via in vitro intestinal
passage experiments. Furthermore, we experimentally confirmed the consistent
reptile-like peristaltic motion of the FCE’s tail propulsion mechanism and the
entire body under the influence of an external magnetic field. Ultimately, we
demonstrate that the FCE can achieve film expansion and overall disintegration
under high-frequency electromagnetic stimulation.

##  II System Overview

The gastrointestinal (GI) tract region most rigorously navigated by capsule
endoscopes is the small intestine, notable for its slender form ($\sim$8 m)
[14], narrow cross-section, and thin intestinal wall. The passage capability
of currently available commercial endoscopes through this area is well-
documented, albeit with isolated incidences of hysteresis. Yet, an increase in
the length of capsule endoscopes evidently presents a challenge to their
smooth transit through this lumen, as illustrated in Figure 1a.

The FCE adopts a ”rigid-soft-rigid” configuration (Figure 1b). The rigid head
section, typically around 12 mm in diameter and 27 mm in length, aligns with
the dimensions of standard commercial endoscopes. The central soft section,
pivotal in extending the length of the capsule endoscope, facilitates passive
deformation under intestinal compressive stresses and peristaltic forces. This
design draws inspiration from the mechanics intrinsic to traditional rotary
joints and flexible surgical robots. Consequently, an additional rigid section
can be affixed at the tail end, a feature designed to enhance the diverse
functionalization of the FCE.

As the FCE navigates the small intestine, it frequently encounters narrowed
regions, enveloped by the inner intestinal wall. To facilitate the capsule
endoscope’s progress within the small intestine, we have engineered a pair of
radial magnets at the tail end (Fig. 1c). Under the influence of their
internal magnetic potential energy, they naturally adopt a closed state, but
are compelled to separate by the exertion of an external magnetic field. As a
result, a worm-like bionic reciprocal motion in the axial direction can be
realized. This offers propulsion to the head’s rigid part, bolstered by the
reactionary force of the intestinal wall.

Furthermore, weak acid and alkali solutions are integrated into the tail’s
functional components, segregated by hot-melt adhesive and coated round iron
sheet. Under high-frequency electromagnetic heating, the hot melt transitions
into a fluid state, instigating an acid-base reaction that generates carbon
dioxide. This elevates the internal pressure, culminating in the expansion of
the Ecoflex film. This innovative feature paves the way for future capsule
endoscopes, opening up possibilities for soft anchoring, drug delivery,
sampling, and stent deployment.

##  III FCE connector design and evaluation

Figure 2: Schematic of the rotary joint in a flexible capsule, constructed with a custom connection mechanism, 3D-printed malleable ring, and iron pins. (b) ”Passability” metric used for performance evaluation, where a tethered wire in a curved plastic hose replicates small intestine conditions. Results are summarized in Table 1.(c) Manual bending of the flexible capsule and measurement of extreme bending angle. Despite a stiffness of 70%, the softer, non-hollowed-out components do not cause appreciable deformation, with bending facilitated by the rotary joints and ring deformation. Component lengthening from 18mm to 24mm did not notably increase the bending angle. These findings may inform future design optimizations of flexible capsules. TABLE I: Examination results of material length and stiffness on the rotary joint performance in a flexible capsule robot. Length (mm) | Stiffness factor (%) | Passability |  | Extreme bending
---
angle ( °)
12 | 70 | Yes | 160
85 | No | 145
18 | 70 | Yes | 180
85 | No | 60
24 | 70 | Yes | 180
85 | No | 60

The configuration of the connection mechanism within an FCE is integral to its
performance. We have incorporated a classical rotary joint design (Fig. 2a) to
enhance passive bending flexibility. The joint comprises a bespoke connection
structure and a 3D-printed ring, fabricated from an elastic material, which
are interconnected via an iron hold. The soft connector is fabricated using a
Stratasys J826 3D printer (Stratasys, Eden Prairie, MN, USA) with Agilus30
Clear (Stratasys) serving as the print material. Subsequently, we adjusted the
length and stiffness factor of the material to empirically investigate the
direct influence of these physical quantities on the results. A succinct
encapsulation of the findings is presented in Table I. To evaluate whether
this design facilitates an increase in the rigid length of the capsule
endoscope, we introduced a metric termed ”passability”. A PVC clear plastic
hose was employed to fabricate a curved environment with a bending angle of
$\sim$300°, thereby replicating the extreme deformation angles of the small
bowel in vivo. A wire attached to the flexible capsule simulated the
peristaltic forces exerted by the small intestine (Fig. 2b). Furthermore,
manual bending of the flexible capsule was conducted (Fig. 2c), followed by
measurement of the ultimate bending angle. Observations indicated that even a
70% stiffness does not induce significant deformation as the softer components
are not hollowed out. Rather, the rotary joint’s rotation and the ring’s
deformation facilitated bending. Notably, extending the part from 18 to 24 mm
does not appreciably augment the ultimate bending angle. These insights
suggest that additional modifications to the primary section of the soft
component may be required to optimize the bending performance of FCE.

Figure 3: Analysis and Modelling of ’Soft’ Structure. (a) Shows parametric
representation of the structure. (b) Explores material property impact on
bending deformation. (c) Presents ANSYS simulation results aligning with
experimental data. (d) Indicates lower bending force with higher stiffness,
favouring VeroMagenta-V over VeroCyan-V as a connecting material. Figure 4:
Deformation Analysis of FCE. (a) Shows force analysis with stresses $q_{1}$,
$q_{2}$ and reaction forces $\textbf{F}{n1}$ and $\textbf{F}{n2}$. Deforming
areas $a_{1}^{{}^{\prime}}$ and $a_{2}^{{}^{\prime}}$ enable wider bending.
(b) Displays passive deformation via iron hold forces and reaction forces
$\textbf{F}{n1}^{{}^{\prime}}$ and $\textbf{F}{n2}^{{}^{\prime}}$ (c). (d)
Examines soft section deformation under section moments $\textbf{M}{i}$ and
$\textbf{M}{j}$. The moment-curvature relationship is derived from the Euler-
Bernoulli beam theory. Figure 5: (a) In vitro intestinal passage experiment to
assess the passive transport capability of the FCE. (b) Model’s interior,
constructed from pig small intestine, emulates a true intestinal wall. (c)
Depiction of one section of the model. (d) Model’s rigid, zigzag-patterned
support made of acrylic. (e) Force associated with the passive transit of the
FCE along the path, demonstrating an elevated tensile force requirement at
large corners and small curvatures, such as points B and D, yet confirming the
FCE’s effective adaptability to such conditions.

Incision tube structures, also referred to as torsos, comprise incision tubes
engineered to diminish rigidity and permit bending when externally actuated
via wires. Lateral incision tube structures, fabricated from superelastic
materials, are gaining prominence in the field of surgical robotics. Their
seamless integration with surgical instruments and provision of enhanced
maneuverability for surgeons is highly valued [15]. We were inspired by their
ability to attain large bending angles and tight bending radii for the design
of the primary structure of the soft segment, as depicted in Fig. 3. This
makes it particularly adept at maneuvering in narrow lumens.

Figure 3a illustrates the ”soft” structure’s parametric representation, as
summarized in Table. II. Figure 3b elucidates the impact of printed material
properties on the bending of a soft segment. One end of the segment is
secured, while the other is affixed to a force transducer using screws to
constrain displacement and engender the desired bending deformation. An
analogous simulation was performed utilizing [16], where Figure 3c presents
the resultant, converged displacement and stress distributions. These outcomes
exhibit a satisfactory congruence with the empirical data. Furthermore, the
experimental findings, as depicted in Fig. 3d, reveal that the required
bending force inversely correlates with the stiffness factor. This suggests
that VeroMagenta-V exhibits superior suitability as a connecting segment
compared to VeroCyan-V [17].

TABLE II: Parameters of the Prototype’s ’Soft’ Section (Unit: mm) L | f | w | r | e | c | d
---|---|---|---|---|---|---
43 | 2 | 9 | 3.5 | 9 | 1.5 | 2.5

Figure 4a details the force analysis of the FCE under passive deformation.
Here, $q_{1}$ and $q_{2}$ represent the stresses applied to the rigid section
of the intestinal wall. The points of contact, $a_{1}$ and $a_{2}$, between
the rigid connection structure and the 3D-printed soft component at the FCE’s
maximum bending, generate respective reaction forces $\textbf{F}_{n1}$ and
$\textbf{F}_{n2}$, thereby constraining further deformation. As
$a_{1}^{{}^{\prime}}$ ($a_{2}^{{}^{\prime}}$) on the soft section undergoes
deformation, it offers the FCE a broader bending angle across its body. The
passive deformation of the soft section is facilitated by the torque and
compression forces transmitted by the iron hold (Fig. 4b), along with the
reaction forces $\textbf{F}_{n1}^{{}^{\prime}}$ and
$\textbf{F}_{n2}^{{}^{\prime}}$, in the ultimate bending state (Fig. 4c). The
observed deformation within the soft section is characterized by the bending
of notched cells under the influence of section moments $\textbf{M}_{i}$ and
$\textbf{M}_{j}$ (Fig. 4d). Extending the principles of Euler-Bernoulli beam
theory to encompass constant-curvature beams, we establish a moment-curvature
relationship applicable to a generally curved beam, as expressed herein[18]:

$\displaystyle\small\frac{d\theta}{ds}=\frac{1}{R}-\frac{\textbf{M}_{i}+\textbf{M}_{j}}{EI}$
(1)

where $\frac{d\theta}{ds}$ is the curvature at $s$ and $s$ is the curvilinear
coordinate of the soft section. Parameters $R$, $E$, and $I$ denote the
initial radius of curvature, Young’s modulus, and the second moment of the
equivalent area, respectively.

To assess the passability of FCE, an in vitro intestinal passage experiment
was conducted (Fig. 5a). Here, FCE was maneuvered through an in vitro model
via a rope tether connected to a force transducer. The model’s interior was
constructed using pig small intestine to mimic a genuine intestinal wall (Fig.
5b), with a section’s appearance displayed in Fig. 5c. The model’s rigid
support, crafted from acrylic, features a zigzag design (Fig. 5d). Fig. 5e
illustrates the force involved in the passive movement of the FCE along the
path, highlighting an increase in the necessary tensile force for large
corners and small curvatures, such as points B and D. Nevertheless, passage is
feasible, affirming the FCE’s superior adaptability to such environments.

##  IV Integrated Functions

###  IV-A Active Locomotion

Figure 6: (a) Schematic of the experiment where the FCE’s head end is fixed
and external drive magnets rotate at 3Hz. (b-h) Sequence showing the combined
rotational and linear motion of the FCE tail, beneficial for forward
propulsion in confined environments.

We confirmed the FCE’s capability for passive intestinal delivery, and
subsequently conducted experimental verification of its ability to be remotely
actuated under the influence of an external magnetic drive system. Initially,
we verified that the internal radial magnet pair could convert the input
rotating magnetic field signal into axial displacement. Fig. 6a presents a
schematic, where the head end of the FCE is fixed, and the external drive
magnets are affixed to a torque motor rotating at 3Hz, with marker points
designated on the drive housing. Fig. 6 (b-h) depicts a motion sequence in
which the FCE tail can generate combined rotational and linear motion. This
feature may prove beneficial in confined environments, where the tail
facilitates forward propulsion by interacting with direct tissue contacts.

Building on this, we employed the in vitro intestinal model described in Fig.
5(d), boasting a diameter of approximately 20 mm, to authenticate the active
penetration capabilities of the FCEs within a tortuous, large cavity. The
relative positions of the in vitro gut model and the driving magnet were
manually adjusted, and the FCE exhibited a sensitive response (Fig. 7(a-f)).
When the FCE encountered a challenging zigzag narrow space (Fig. 7(g,h)), the
tail mechanism executed a propulsive function to aid the FCE’s onward
progression as the head’s forward travel faced constraints.

Prior experiments indicated a vigorous movement of FCEs that could prove
unsuitable for passage through the thin-walled small intestine, potentially
posing risks such as perforation. Drawing on methods from previous research,
we validated the approach in a porcine small intestine. Employing a wire, a
live knot was attached to a segment of the small intestine to simulate an
intestinal narrowing. The FCE navigated through the small intestine with
gentle motion and successfully traversed the live knot opening without
difficulty (Fig. 8).

Figure 7: (a-f) FCE’s response in the in vitro intestinal model, showcasing
its penetration capabilities. (g-h) FCE’s navigation in challenging zigzag
spaces, with the tail aiding forward progression when limited. Figure 8:
Validation of FCE’s movement in a porcine small intestine, addressing previous
concerns of potential perforation due to vigorous motion. A live knot,
simulating intestinal narrowing, was attached via a wire. The figure
illustrates the FCE’s gentle navigation and successful traversal through the
live knot opening.

###  IV-B Expansion and Separation

The FCE allows for an increase in capsule endoscope size, mitigating concerns
of retention in the gastrointestinal (GI) tract. To illustrate this FCE
advantage, we developed a thermal trigger mechanism to control the acid-base
reaction within the rigid structure of the FCE tail, encased in an ecoflex
film. The thermal triggering mechanism relies on a hot melt adhesive, which
melts at approximately 50°C [19]. Installed in a round iron plate as a valve
in its hot melt state, it is heated using a high-frequency electromagnetic
field. Upon transformation to a liquid state, the valve opens, and the
acid/base reaction generates carbon dioxide and water, enabling the film’s
expansion. This membrane expansion, a basic function, can be further utilized
for applications such as intestinal anchoring, biopsy of the intestinal wall
and drug administration in combination with microneedles, and potentially
stent deployment in narrow lumens (Fig. 9).

Figure 9: Demonstration of the FCE’s thermal trigger mechanism. The mechanism,
reliant on a hot melt adhesive, triggers an acid/base reaction upon heating,
leading to film expansion. This can be used for applications like intestinal
anchoring, wall biopsy, drug administration, and potential stent deployment.

In line with the aforementioned principle, we’ve substituted a portion of the
FCE’s connecting ring with a hot melt adhesive. The thermal effect of the iron
pin in the high-frequency electromagnetic field allows for the controlled
disconnection of the connecting ring. Consequently, the FCE can disassemble
and be naturally expelled from the body upon completion of its function (Fig.
10).

Figure 10: Depiction of FCE’s modified connecting ring with hot melt adhesive.
The thermal effect allows controlled ring disconnection, facilitating FCE
disassembly and natural expulsion post-function.

##  V Discussion and Conclusion

Our proposed flexible capsule endoscope incorporates a ”rigid-soft-rigid”
configuration. The first ’rigid’ component is modeled on commercially
available endoscopes, typically around 12 mm in diameter and 27 mm in length.
The ’soft’ component facilitates the lengthening of the capsule endoscope,
enabling it to adapt to the zigzag topography of the intestinal environment.
This segment allows the capsule endoscope to undergo passive deformation under
intestinal compression stresses and peristaltic forces, drawing on the
mechanical principles inherent in traditional rotating joints and flexible
surgical robots.

In this study, we present a narrowly focused investigation into the
engineering design alternatives and ex vivo feasibility of the FCE. Notably
absent, however, is a consideration of critical clinical parameters. Most
prominently, in Figure 5, the prototype exerts a force that, at its zenith,
approaches 2N—potentially sufficient to inflict mucosal injury or more
grievous perforations within the delicate milieu of the small intestine.
Furthermore, the current design’s intermediate soft segment, critical for
accommodating the undulating motions of the small intestinal tract, mandates
empirical validation through in vivo experimentation to confirm its compliance
and functional integrity. Our research indicates that magnetic origami
structures contribute to more ergonomic and intuitive human-computer
interfaces due to their flexible and adaptable nature [20, 21, 22]. Meanwhile,
the automated control of the FCE predicates upon real-time positional
feedback. The feasibility of such a system, while preliminarily substantiated
in prior investigations concerning magnetic localization techniques [23, 24,
25], remains an avenue for prospective development.

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Vol.0 (200x) No.0, 000–000

11institutetext: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing, 100012, P. R. China
11email<EMAIL_ADDRESS>22institutetext: Department of Physics &
Astronomy, The University of Sheffield, Hicks Building, Hounsfield Road,
Sheffield S3 7RH, UK
33institutetext: Graduate University, Chinese Academy of Sciences, Beijing,
100039, P. R. China
44institutetext: Department of Astronomy and Space Science, Chungnam National
University, Daejeon 305-764, Korea
55institutetext: Center for Space Astrophysics, Yonsei University, Seoul
120-749, Korea
66institutetext: California Institute of Technology, MC 405-47, 1200 E.
California Boulevard, Pasadena, CA 91125

# Old stellar population synthesis: New age and mass estimates for Mayall II =
G1

Jun Ma 11 Richard de Grijs 2211 Zhou Fan 1133 Soo-Chang Rey 44 Zhenyu Wu 11 Xu
Zhou 11 Jianghua Wu 11 Zhaoji Jiang 11 Jiansheng Chen 11 Kyungsook Lee 44 S.
T. Sohn 5566

(Received 2001 month day; accepted 2001 month day)

###### Abstract

Mayall II = G1 is one of the most luminous globular clusters (GCs) in M31.
Here, we determine its age and mass by comparing multicolor photometry with
theoretical stellar population synthesis models. Based on far- and near-
ultraviolet GALEX photometry, broad-band $UBVRI$, and infrared $JHK_{\rm s}$
2MASS data, we construct the most extensive spectral energy distribution of G1
to date, spanning the wavelength range from 1538 to 20,000Å. A quantitative
comparison with a variety of simple stellar population (SSP) models yields a
mean age that is consistent with G1 being among the oldest building blocks of
M31 and having formed within $\sim$1.7 Gyr after the Big Bang. Irrespective of
the SSP model or stellar initial mass function adopted, the resulting mass
estimates (of order $10^{7}M_{\odot}$) indicate that G1 is one of the most
massive GCs in the Local Group. However, we speculate that the cluster’s
exceptionally high mass suggests that it may not be a genuine GC. We also
derive that G1 may contain, on average, $(1.65\pm 0.63)\times 10^{2}L_{\odot}$
far-ultraviolet-bright, hot, extreme horizontal-branch stars, depending on the
SSP model adopted. On a generic level, we demonstrate that extensive multi-
passband photometry coupled with SSP analysis enables one to obtain age
estimates for old SSPs to a similar accuracy as from integrated spectroscopy
or resolved stellar photometry, provided that some of the free parameters can
be constrained independently.

###### keywords:

galaxies: individual (M31) – galaxies: star clusters – galaxies: stellar
content

##  1 Introduction

Globular clusters (GCs) are among the oldest bound stellar systems in the
Universe, and they thus provide a fossil record of the earliest stages of
galaxy formation and evolution. GCs are internally homogeneous in age and
metallicity, i.e. they are simple stellar systems composed of coeval stellar
populations. In addition, GCs are the oldest systems in our own and other
galaxies for which we can estimate reasonably reliable ages (and realistic
uncertainties); they can thus independently provide vitally important
information regarding the minimum age of the Universe. In a detailed study of
17 Galactic GCs, Chaboyer et al. (1998) used the improved Hipparcos parallaxes
having just become available at that time to determine updated distances, and
hence improved ages, of their GC sample. They concluded that the mean age of
the oldest GCs is $11.5\pm 1.3$ Gyr, although their age histogram (their fig.
2) shows a tail toward ages as old as $\sim 16$ Gyr. Gratton et al. (2003)
obtain improved ages (and distances) for three Galactic GCs, NGC 6397, NGC
6752, and 47 Tuc, and conclude that the age of the oldest GCs is $13.4\pm 0.8$
(random) $\pm 0.6$ (systematic) Gyr, in good agreement with the 3-year results
from the Wilkinson Microwave Anisotropy Probe (WMAP). This led them to suggest
that the oldest Galactic GCs formed within the first 1.7 Gyr after the Big
Bang, at the $1\sigma$ confidence level. We note that this is still fully
compatible with the 5-year WMAP results constraining the age of the Universe
to $13.73\pm 0.12$ Gyr (Hinshaw et al. 2008). While the ages of the oldest GCs
in the Galaxy are now reasonably well determined, this is certainly not the
case even for our nearest large neighbour, the Andromeda galaxy (M31).

The most direct method for determining the age of a star cluster is by means
of individual stellar photometry, since the main-sequence turn-off location is
mostly affected by age (see, e.g., Puzia et al. 2002b, and references
therein). However, this method has only been applied to Galactic GCs and to
GCs associated with the Milky Way’s satellites (e.g., Riich 2001), in which
individual stars can be resolved and their photometry determined to
satisfactory accuracy, to a few magnitudes fainter than the main-sequence
turn-offs. This is difficult, if not impossible, to achieve even for GCs as
close as those associated with M31 (but see Brown et al. 2004), at a distance
of $D=772\pm 44$ kpc (e.g., Ribas et al. 2005). Fortunately, starting from the
pioneering work of Tinsley (1968, 1972) and Searle et al. (1973), evolutionary
population synthesis modeling has become a powerful tool to address many
outstanding problems in astrophysics, from determining the ages of star
clusters to interpreting integrated spectrophotometric observations of
galaxies. Therefore, extragalactic GC ages can, in general, also be inferred
from composite colors and/or integrated spectroscopy.

The evolution of GCs is usually modeled by means of the simple stellar
population (SSP) approximation. An SSP is a single generation of coeval stars
formed from the same progenitor molecular cloud (thus implying a single
metallicity), and governed by a given stellar initial mass function (IMF). GCs
are ideal templates to test the compatibility between the population synthesis
models and reality. Barmby & Huchra (2000) compared the predicted SSP colors
of three stellar population synthesis models to the intrinsic broad-band
$UBVRIJHK$ colors of Galactic and M31 GCs, and concluded that the best-fitting
models match the cluster colors very well. Subsequently, many authors have
used SSP modeling to determine the parameters of cluster populations. For
instance, de Grijs et al. (2003a) determined the ages and masses of star
clusters in the fossil starburst region B of M82 by comparing the observed
cluster spectral energy distributions (SEDs) with both the Starburst99 SSP
models (Leitherer et al. 1999) and those developed by Bruzual & Charlot
(2000), based on Hubble Space Telescope (HST) observations from the blue
optical to the near-infrared (NIR) (see also de Grijs et al. 2003b, 2003c);
Bik et al. (2003) and Bastian et al. (2005) estimated ages, initial masses and
extinction values for M51 star cluster candidates by comparing the Starburst99
and the Frascati models (Romaniello 1998) for instantaneous star formation
with the observed SEDs based on HST/WFPC2 observations in six broad-band and
two narrow-band filters. Ma et al. (2006) and Fan et al. (2006) obtained age
estimates for M31 GCs by fitting theoretical stellar population synthesis
models (Bruzual & Charlot 2003, henceforth BC03) to their photometric
measurements in a large number of intermediate- and broad-band passbands from
the optical to the NIR. Based on the same method and models, Ma et al. (2007a)
constrained the age of the M31 GC S312, using multicolor photometry from the
near-ultraviolet (NUV) to the NIR, to $9.5^{+1.15}_{-0.99}$ Gyr. S312 is among
the first extragalactic GCs for which the age was estimated accurately using
main-sequence photometry, i.e., Brown et al. (2004) estimated the age of S312
at $10^{+2.5}_{-1}$ Gyr by means of a quantitative comparison with the
isochrones of VandenBerg et al. (2006). This was based on their analysis of
the cluster’s color-magnitude diagram (CMD) below the main-sequence turn-off
using extremely deep images obtained with the HST/Advanced Camera for Surveys
(ACS).

It is a common misconception that spectroscopic age estimates are always much
better than those based on broad-band photometry. Schweizer et al. (2004)
recently showed convincingly that spectroscopic age determinations are not
necessarily better or more accurate than photometrically obtained ages, at
least in the age range from $\sim 100-500$ Myr. Anders et al. (2004) published
a detailed theoretical investigation of the accuracy of retrieved star cluster
properties, including their ages, based on sophisticated fits of SSP models to
observed broad-band SEDs spanning varying wavelength ranges. They concluded
that if one has access to as large a wavelength range as possible, ideally
including both ultraviolet and NIR data points, the resulting age estimates
are reasonably accurate, even for ages as old as $\sim 10$ Gyr – particularly
if one or more of the other free parameters (e.g., metallicity or extinction)
can be constrained reliably and independently (see Anders et al. 2004, e.g.,
their fig. 14). We will use this promising approach as our basic premise in
this paper.

Of the Local Group members, M31 is particularly important as it provides a
direct comparison with our own Galaxy. In addition, it contains a large number
of GCs and GC candidates, including 496 genuine, 367 probable, and 301
possible GCs (Kim et al. 2007). The M31 GC system is among the extragalactic
GC systems studied most often (Harris 1991; Brodie & Strader 2006). As one of
the brightest M31 GCs, Mayall II = G1 has attracted much scientific interest
(see, e.g., Barmby et al. 2007; Ma et al. 2007b, and references therein).

In this paper, we first determine the age and mass of G1 by comparing
observational SEDs (§2) with population synthesis models (§3). We will use the
lessons learned from studies of broad-band photometric SED fits to minimize
the associated uncertainties. We discuss our results along the way, as
appropriate, and provide a summary in §4.

##  2 Ultraviolet, optical, and infrared observations of G1

###  2.1 Historical observations

G1 was first detected by Mayall & Eggen (1953) (their No. 2, and hence
referred to as Mayall II), who searched for nebulous objects associated with
M31 using a $6^{\circ}\times 6^{\circ}$ Palomar 48-inch Schmidt plate taken in
1948 and centered on M31. Subsequently, Sargent et al. (1977) rediscovered the
cluster (their No. 1, i.e., G1) based on their survey of 29 plates associated
with the general field of M31, which had been obtained at the $f/2.7$ prime
focus of the Kitt Peak National Observatory’s (KPNO) 4-m telescope. The
cluster is located in the halo of M31, at a projected distance of about 40 kpc
from the galaxy’s nucleus (see Meylan et al. 2001).

###  2.2 GALEX ultraviolet, optical broad-band, and 2MASS NIR photometry

Although the cluster is generally believed to be among the oldest GCs in M31,
to the best of our knowledge there is no CMD-based or spectroscopic age
estimate available in the literature to date. The lack of a CMD-based age
estimate is due to the challenges associated with probing the cluster’s CMD
down to below the main-sequence turn-off. The current deepest CMD of the
cluster (Meylan et al. 2001) does not reach these faint levels. Although both
integrated and spatially-resolved spectra of the cluster are available (e.g.,
Huchra et al. 1991; Gebhardt et al. 2005; Cohen 2006), they have thus far not
been used to determine an age for G1. This may be partially due to the limited
wavelength range covered by most of these spectra, and the difficulties one
faces when trying to constrain ages in the regime beyond $\sim 10$ Gyr (see
below).

To constrain the age of G1 accurately, with the smallest uncertainty allowed
by the observational data, we use as many photometric data points covering as
large a wavelength range as possible. Kaviraj et al. (2007) showed that the
combination of far (FUV) and near-ultraviolet photometry with optical
observations in the standard broad bands enables one to efficiently break the
age-metallicity degeneracy; Worthey (1994) showed that the age-metallicity
degeneracy associated with optical broad-band colors is $\Delta{\rm
age}/\Delta Z\sim 3/2$ (also see MacArthur et al. 2004). However, de Jong
(1996) showed that this degeneracy can be partially broken by adding NIR
photometry to the optical colors, which was recently supported by Wu et al.
(2005). Cardiel et al. (2003) found that inclusion of an infrared passband can
improve the predictive power of the stellar population diagnostics by $\sim$30
times compared to using optical photometry alone. Since NIR photometry is less
sensitive to interstellar extinction than the classical optical passbands,
Kissler-Patig et al. (2002) and Puzia et al. (2002a) also suggested that it
provides useful complementary information that can help to disentangle the
age-metallicity degeneracy (also see Galleti et al. 2004).

The M31 field was observed as part of the Nearby Galaxies Survey (NGS) by the
Galaxy Evolution Explorer (GALEX) in two ultraviolet bands (see for details
from Rey et al 2005, 2007). Rey et al. (2007) published photometric data for
485 and 273 M31 GCs in the GALEX NUV and FUV bands, respectively. G1 was
detected in these two ultraviolet bands. The GALEX photometric system is
calibrated to match the spectrophotometric AB system.

van den Bergh (1969) determined photo-electric photometry for 45 M31 GCs,
including G1, in the $UBV$ bands. Using CCD imaging from the KPNO 0.9m
telescope, Reed, Harris & Harris (1994) published integrated $BVR$ magnitudes
and color indices for 41 GCs and GC candidates, including G1, in the outer
halo of M31. We compared the photometry of G1 in the $B$ and $V$ bands between
these two studies; the results match closely. In this paper, we adopt the CCD
$BVR$ photometry of Reed, Harris & Harris (1994), and the photographic
$U$-band photometry of van den Bergh (1969), with a photometric uncertainty of
0.08 mag as suggested by Galleti et al. (2004). Based on HST images, Barmby &
Huchra (2001) detected and published photometry for 114 GC candidates
associated with M31, including 32 new objects. Their $V$-band photometry is in
good agreement with that of van den Bergh (1969) and Reed, Harris & Harris
(1994), although they do not provide their photometric uncertainties. However,
Barmby & Huchra (2001) compared their HST photometry with the ground-based
measurements compiled by Barmby et al. (2000), and found that the median
offset in $I$ is $0.06\pm 0.04$ mag. Therefore, we adopt $0.06$ mag as the
photometric uncertainty in the $I$ band. Using the Two Micron All Sky Survey
(2MASS) database, Galleti et al. (2004) identified 693 known and candidate GCs
in M31, and listed their 2MASS $JHK_{\rm s}$ magnitudes. Galleti et al. (2004)
transformed all 2MASS magnitudes to the CIT photometric system (Elias et al.
1982, 1983) using the color transformations in Carpenter (2001). However, we
need the original 2MASS $JHK_{\rm s}$ magnitudes to compare our observational
SEDs with the SSP models, so we reversed this transformation using the same
procedures. Since Galleti et al. (2004) do not provide the photometric
uncertainties in $JHK_{\rm s}$, we obtained these by comparing the magnitudes
with fig. 2 of Carpenter et al. (2001), where the observed photometric rms
uncertainties in the time series are shown as a function of magnitude, for
stars brighter than the observational completeness limits. In fact, the
photometric uncertainties adopted do not affect our results significantly, as
we showed in Fan et al. (2006) (see their section 4.3 for details). The full
set of ultraviolet, optical broad-band, and 2MASS NIR photometry of G1 is
listed in Table 1\. The $UBVRI$ and 2MASS magnitudes are given in the Vega
system Schneider et al. (1977). For convenience, we converted all
observational magnitudes to the AB system, following the procedures
recommended in BC03.

Table 1: Ultraviolet, optical broad-band, and NIR 2MASS photometry of G1. Filter | Magnitude (uncertainty) | Reference
---|---|---
FUV | 18.972 (0.031) | Rey et al. (2007)
NUV | 18.014 (0.012) |
$U$ | 14.85 (0.08) | van den Bergh (1969)
$B$ | 14.584 (0.013) | Reed, Harris & Harris (1994)
$V$ | 13.750 (0.007) |
$R$ | 13.191 (0.010) |
$I$ | 12.684 (0.060) | Barmby & Huchra (2001)
$J$ | 11.858 (0.054) | Galleti et al. (2004)
$H$ | 11.127 (0.054) |
$K_{\rm s}$ | 11.016 (0.054) |

###  2.3 Reddening and metallicity

To obtain the intrinsic SED of G1, the photometric data must first be
dereddened. Since G1 is located in the halo of M31, i.e., far away from the
galaxy’s disk, it is (for all practical purposes) only affected by Galactic
foreground extinction. In fact, some authors have demonstrated that G1 is
affected by a negligible amount of reddening. Meylan et al. (2001) used
HST/WFPC2 observations in the F555W and F814W filters, and applied Sarajedini
(1994)’s method to simultaneously determine the cluster’s reddening and
metallicity; they obtained a reddening of $E(V-I)=0.05\pm 0.02$ mag toward G1,
which is less than the Galactic foreground extinction. van den Bergh (1969)
studied the reddening in the halo of M31 by comparing the colors of clusters
with the same line-strength index in the Galaxy and in M31, and obtained a
mean reddening of $E(B-V)=0.08\pm 0.02$ mag for the clusters in the halo of
M31. Barmby et al. (2000) determined the reddening for each individual cluster
using correlations between optical and infrared colors and metallicity, and by
defining various ‘reddening-free’ parameters based on their large database of
multi-color photometry. For G1, Barmby et al. (2000, also P. Barmby, priv.
comm.) obtained $E(B-V)=0.09\pm 0.02$ mag. In this paper, we adopt the
reddening value from Barmby et al. (2000). The values for the extinction
coefficient, $R_{\lambda}$, were obtained by interpolating the interstellar
extinction curve of Cardelli et al. (1989).

Cluster SEDs are determined by the combination of their ages and
metallicities, which is often referred to as the age-metallicity degeneracy.
Therefore, the age of a cluster can only be constrained accurately if the
metallicity is known with confidence, from independent determinations. There
exist two metallicity determinations for G1. Huchra et al. (1991) derived
metallicities for 150 M31 GCs, including G1, using the strengths of six
absorption features in the clusters’ integrated spectra. The resulting
metallicity of G1 is $\rm[Fe/H]=-1.08\pm 0.09$. Meylan et al. (2001) used
HST/WFPC2 photometry to construct deep CMDs for G1, combined with the shape of
the red-giant branch as calibrated by Sarajedini et al. (2000), to derive the
mean metallicity of G1 on the scale of Zinn & West (1984), $\rm[Fe/H]=-0.95\pm
0.09$. In this paper, we adopt $\rm[Fe/H]=-1.08\pm 0.09$ for G1.

##  3 The stellar population of G1

###  3.1 Stellar populations and synthetic photometry

In this section, we compare the SED of G1 with theoretical stellar population
synthesis models. We start by using the BC03 SSP models, which have been
upgraded from the earlier Bruzual & Charlot (1993, 1996) versions, and now
provide the evolution of the spectra and photometric properties for a wider
range of stellar metallicities. BC03 provide 26 SSP models (both of high and
low spectral resolution) using the Padova-1994 evolutionary tracks, half of
which were computed based on the Salpeter (1955) IMF with lower and upper-mass
cut-offs of $m_{\rm L}=0.1~{}M_{\odot}$ and $m_{\rm U}=100~{}M_{\odot}$,
respectively. The other thirteen were computed using the Chabrier (2003) IMF
with the same mass cut-offs. In addition, BC03 provide 26 SSP models using the
Padova-2000 evolutionary tracks. In this paper, we will use all of these SSP
models to determine the most appropriate age and mass for G1. These SSP models
contain 221 spectra describing the spectral evolution of SSPs from $1.0\times
10^{5}$ yr to 20 Gyr. The evolving spectra include the contribution of the
stellar component at wavelengths from 91Å to $160\mu$m.

Since our observational data are integrated luminosities through a given set
of filters, we convolved the theoretical SSP SEDs of BC03 with the FUV and
NUV, broad-band $UBVRI$, and 2MASS $JHK_{\rm s}$ filter response curves to
obtain synthetic ultraviolet, optical, and NIR photometry for comparison. The
synthetic magnitude in the AB magnitude system for the $i{\rm th}$ filter can
be computed as

$m_{\lambda_{i}}=-2.5\log\frac{\int_{\lambda}F_{\lambda}\varphi_{i}(\lambda){\rm
d}\lambda}{\int_{\lambda}\varphi_{i}(\lambda){\rm d}\lambda}-48.60,$ (1)

where $F_{\lambda}$ is the theoretical SED and $\varphi_{i}$ the response
curve of the $i{\rm th}$ filter. $F_{\lambda}$ varies as a function of age and
metallicity.

###  3.2 Fit results

We use a $\chi^{2}$ minimization approach to examine which SSP models are most
compatible with the observed SEDs, following

$\chi^{2}=\sum_{i=1}^{10}{\frac{[m_{\lambda_{i}}^{\rm
intr}-m_{\lambda_{i}}^{\rm mod}(t)]^{2}}{\sigma_{i}^{2}}},$ (2)

where $m_{\lambda_{i}}^{\rm mod}(t)$ is the integrated magnitude in the $i{\rm
th}$ filter of a theoretical SSP at age $t$, $m_{\lambda_{i}}^{\rm intr}$
represents the intrinsic integrated magnitude in the same filter, and

$\sigma_{i}^{2}=\sigma_{{\rm obs},i}^{2}+\sigma_{{\rm mod},i}^{2}.$ (3)

Here, $\sigma_{{\rm obs},i}^{2}$ is the observational uncertainty, and
$\sigma_{{\rm mod},i}^{2}$ is the uncertainty associated with the model
itself, for the $i{\rm th}$ filter. Charlot et al. (1996) estimated the
uncertainty associated with the term $\sigma_{{\rm mod},i}^{2}$ by comparing
the colors obtained from different stellar evolutionary tracks and spectral
libraries. Following Wu et al. (2005), we adopt $\sigma_{{\rm
mod},i}^{2}=0.05$.

The BC03 SSP models based on the Padova-1994 evolutionary tracks include six
initial metallicities, $Z=0.0001,0.0004,0.004,0.008,0.02\,(Z_{\odot})$, and
0.05, corresponding to ${\rm[Fe/H]}=-2.25,-1.65,-0.64,-0.33,+0.09$, and
$+0.56$. However, the BC03 SSP models based on the Padova-2000 evolutionary
tracks include six partially different initial metallicities, $Z=0.0004$,
0.001, 0.004, 0.008, 0.019 $(Z_{\odot})$, and 0.03, i.e.,
${\rm[Fe/H]}=-1.65,-1.25,-0.64,-0.33,+0.07$, and $+0.29$. Spectra for other
metallicities can, in principle, be obtained by interpolation of the
appropriate spectra for any of these metallicities, although this is not
necessarily advisable or straightforward (Frayn & Gilmore 2002). Instead, we
adopt the most appropriate model metallicity for the analysis performed in
this paper. Since we have good estimates of the metallicity and reddening
values of G1 (see §2.3), the cluster age is therefore the sole parameter to be
estimated (for a given IMF and extinction law, both of which we assume to be
universal).

None of the SSP models fit the photometric data point in the GALEX FUV band as
well as the other nine data points. (We checked that the image of G1 in the
FUV band is not affected by instrumental problems.) Given that G1 contains an
old stellar population, the most likely physical explanation for this FUV
excess compared to the ‘standard’ BC03 SSP models is the presence of a
significant number of FUV-bright, hot, extreme horizontal-branch (EHB) stars
giving rise to the well-known ‘ultraviolet upturn’ below $\lambda\simeq 2000$Å
(see, e.g., the review of O’Connell 1999, and references therein; see also
Landsman et al. 1998; Sohn et al. 2006). (Alternative species, such as AGB-
manqué stars or blue stragglers are expected to be fewer in number in any
‘normal’ stellar population.)

Since ‘standard’ SSP models do not contain EHB populations, we are forced to
deselect the photometric FUV data point when applying our fitting routines. In
Fig. 1, we show the intrinsic SED of G1 and the integrated SEDs of the best-
fitting models. The dereddened data are shown as the symbols with error bars
(vertical errors for photometric uncertainties and horizontal error bars for
the approximate wavelength coverage of each filter); open circles represent
the calculated magnitudes of the model SED for each filter, obtained by
convolving the theoretical SSP SEDs with the appropriate filter response
curves. The best reduced-$\chi^{2}$ values and ages are listed in Table 2. The
mass of G1, also listed in Table 2, can be estimated by comparing the measured
luminosity in the $V$ band with the theoretical mass-to-light ($M/L$) ratios.
These $M/L$ ratios are a function of the cluster age and metallicity. The
mass-to-light ratios of G1, calculated based on the metallicity adopted and
the age obtained in this paper, are listed in Table 2 for the BC03 SSP models.
Based on its present luminosity, $V=13.750\pm 0.007$ mag, and extinction,
$E(B-V)=0.09$ mag, the cluster’s visual magnitude corrected for the extinction
is $V_{0}=13.471\pm 0.007$ mag, assuming a Cardelli et al. (1989) Galactic
reddening law with $A_{V}=0.279$ mag.

(We note that the NUV data point is also marginally affected by the onset of
the UV upturn, which causes a slight mismatch between the observations and the
best-fitting theoretical SSP models.)

Figure 1: Best-fitting integrated theoretical BC03 SEDs compared to the
intrinsic SED of G1. The photometric measurements are shown as the symbols
with error bars (vertical for uncertainties and horizontal for the approximate
wavelength coverage of each filter). Open circles represent the calculated
magnitudes of the model SED for each filter. We did not use the FUV
photometric data point for the fits (see text).

###  3.3 Age and mass

In the previous section we determined the best-fitting age and mass of G1
based on different theoretical SSP models. From Table 2 we conclude that,
within the errors, the ages obtained from the different BC03 models are
internally consistent. The mean age of G1 is $18.23\pm{1.76}$ Gyr. This is in
excellent agreement with the only other available (rough) age estimate for the
cluster by Meylan et al. (2001), who estimated its age to be $\sim$15 Gyr.
However, we note that the age of G1 obtained in this paper is older than the
current-best estimate of the age of the Universe, of order 13.7 Gyr, as
discussed in §3.1. We will discuss this problem in §4.

Table 2: Age and mass estimates of G1 based on the BC03 models. Evolutionary Track | IMF | Age | $\chi^{2}/\rm{degree~{}~{}of~{}~{}freedom}$ | $M/L_{V}$ | Mass
---|---|---|---|---|---
|  | (Gyr) |  | $(M/L_{V})_{\odot}$ | $(10^{7}M_{\odot})$
Padova 1994 | Salpeter (1955) | $19.68\pm{0.75}$ | 3.04 | 5.10 | $1.06\pm{0.07}$
Padova 1994 | Chabrier (2003) | $19.79\pm{0.50}$ | 2.69 | 3.15 | $0.65\pm{0.04}$
Padova 2000 | Salpeter (1955) | $15.44\pm{0.78}$ | 5.36 | 4.14 | $0.86\pm{0.05}$
Padova 2000 | Chabrier (2003) | $18.01\pm{2.00}$ | 5.27 | 2.79 | $0.58\pm{0.04}$

We conclude that the various mass estimates listed in Table 2 place G1 firmly
at the top of the cluster mass function in the Local Group. Meylan et al.
(2001) presented three estimates of the total mass of G1, (i) a King-model
mass (King 1966) of $1.5\times 10^{7}M_{\odot}$, (ii) a virial mass of
$0.75\times 10^{7}~{}M_{\odot}$; and (iii) a mass based on a King-Michie model
(as defined by Gunn & Griffi 1979) fitted simultaneously to the surface
brightness profile and the central velocity dispersion value, estimated
between $1.4\times 10^{7}M_{\odot}$ and $1.7\times 10^{7}M_{\odot}$. Our
results are in reasonable agreement with Meylan et al. (2001), although we are
aware that the King and King-Michie mass estimates of Meylan et al. (2001) are
up to a factor of two greater than our photometric mass estimates. This is not
too surprising in view of the model assumptions made. Cohen (2006) recently
obtained an optical velocity dispersion for the cluster using the Keck/HIRES
spectrograph, and derived an aperture-corrected line-of-sight velocity
dispersion, $\sigma_{\rm los}=25.5\pm 1.5$ km s-1 (where we have averaged the
values she obtained for the two reddest wavelength ranges analyzed; see also
Djorgovski et al. 2002). We recently redetermined a projected half-light
radius for G1 of $r_{\rm h}=6.5\pm 0.3$ pc (Ma et al. 2007b). Thus, based on
these most recent results, the dynamical (virial) mass of G1 is $M_{\rm
vir}=(7.37\pm 2.15)\times 10^{6}M_{\odot}$. This is in excellent agreement
with the photometric mass estimates obtained in this paper. In turn, this
strongly supports the notion that G1 must have had a close-to-‘normal’ stellar
IMF, in order for it to have survived dissolution due to internal two-body
relaxation until the present time (see also Ma et al. 2006). In particular,
this is driven by the observation that if the IMF is too shallow, i.e., if a
cluster is significantly depleted in low-mass stars compared to (for instance)
the solar neighborhood, it will disperse within a few orbital periods around
its host galaxy’s center, and most likely within about a Gyr of its formation
(e.g., Gnedin & Ostriker 1997; Goodwin 1997; Smith & Gallagher 2001; Mengel et
al. 2002; Rose, Kouwenhoven & de Grijs, in prep.).

From the recent work of Ma et al. (2006), the intrinsically most luminous M31
GC, 037-B327, has been suggested to be the most massive GC in the Local Group,
with a total mass of $\sim(3.0\pm 0.5)\times 10^{7}M_{\odot}$, also determined
photometrically and somewhat depending on the SSP models used, the metallicity
and age adopted, and the IMF representation. However, Cohen (2006) pointed out
that the photometric mass of this cluster had likely been overestimated due to
an incorrect extinction correction. Nevertheless, she also confirmed the
nature of 037-B327 as one of the most massive GCs in the Local Group, with a
dynamical mass similar to that of G1. It is intriguing that these two most
massive GC in M31 both are significantly more massive than the most massive
Galactic GC, $\omega$ Cen [$\sim(2.9-5.1)\times 10^{6}M_{\odot}$; Meylan
(2002)].

In fact, the high mass of these clusters raises additional, intriguing
questions regarding the nature of these objects in general, and of G1 in
particular (see also Federici et al. 2007; Ma et al. 2007b, and references
therein). It has been speculated that these objects may be nucleated dwarf
galaxies instead of genuine GCs. In support of this notion, we point out that
Gieles et al. (2006) suggest that there may be a physical upper limit to the
mass of a star cluster that is not merely the result of size-of-sample
effects. This maximum mass depends to some extent on the galactic environment;
for their example galaxies, M51 and the Antennae system, they find a physical
upper limit to the stellar mass of $\sim(10^{5}-10^{6})M_{\odot}$. Our values
derived for both the photometric and the virial mass of G1 are well above
these suggested upper mass limits. This may, therefore, provide an additional
(although circumstantial) proverbial nail in the coffin of G1 as a normal GC.

###  3.4 Luminosity of the hot, extreme horizontal-branch stars

As discussed in §3.3, none of the BC03 SSP models fit the photometric data
point in the GALEX FUV band as well as the other nine data points. EHB stars
may be responsible for this excess in FUV band. In fact, Rich et al. (1996)
found some bluer horizontal-branch stars extending to $(V-I)=0.0$ mag, based
on their HST/WFPC2 observations. In this section, we will calculate the
luminosity of the EHB stars. We assume that the excess in the GALEX FUV band
is solely due to these EHB stars. The magnitude differences between the four
SSP models employed in this paper and the photometric data points are 0.63,
1.00, 0.56, and 0.94 mag, respectively, corresponding $(2.44,1.07,2.11,\mbox{
and }1.00)\times 10^{2}L_{\odot}$, respectively, with a mean number of EHB
stars in G1 of $(1.65\pm 0.63)\times 10^{2}L_{\odot}$.

###  3.5 Comparison between G1 and S312

Brown et al. (2004) showed that a 10 Gyr old population in M31 has a main-
sequence turnoff at about $m_{\rm F814W}=28.8$ mag. Their deep observations
needed exposures of 39.1 hours in F606W and 45.5 hours F814W, spanning 120
orbits of HST/ACS imaging observations (Brown et al. 2003, 2004). In the
future, such deep HST observations will only be obtained for a very small
number of fields (e.g., Rich et al. 2005). As discussed in §1, Ma et al.
(2007a) constrained the age of the M31 GC S312 by comparing multicolor
photometry and theoretical stellar population synthesis models. It is
encouraging that the age obtained by Ma et al. (2007a) is in good agreement
with the previous determination based on main-sequence photometry (Brown et
al. 2004), i.e., $9.5^{+1.15}_{-0.99}$ Gyr versus $10^{+2.5}_{-1}$ Gyr. S312
is one of the few extragalactic GCs for which the age can be determined from
main-sequence photometry. By comparing the ages of S312 and G1 determined
using the same method, we can conclude that S312 is younger than the majority
of the Galactic GCs at the same metallicity, and G1 is as old as the oldest
Galactic GCs. In fact, if we try to fit the intrinsic SEDs of G1 by the
theoretical BC03 SSP SEDs at an age of 10 Gyr, the resulting fit is very poor
indeed, particularly in the ultraviolet. Therefore, we conclude that the
method used in this paper, by which the ages of both S312 and G1 have been
determined, can be used to determine the ages of old stellar populations to
satisfactory precision, and in particular to distinguish between young and old
populations.

##  4 Summary and Conclusions

In this paper, we first determined the age and mass of the M31 GC G1, as well
as the realistic uncertainties associated with these estimates, by comparing
its multicolor photometry with theoretical stellar population synthesis
models. Our multicolor photometric data were obtained from GALEX FUV and NUV,
broad-band optical $UBVRI$, and 2MASS $JHK_{\rm s}$ observations, which form
an SED covering the wavelength range from 2267 to 20,000Å. Our results confirm
that G1 is one of the oldest and most massive GCs in the Local Group – that
is, if it is indeed a genuine GC given that its mass is well in excess of the
physical maximum mass predicted by the models of Gieles et al. (2006).

The age and mass obtained in this paper are somewhat dependent on the SSP
model adopted. It is evident that the age of $18.23\pm{1.76}$ Gyr for G1 based
on the BC03 models is greater than the currently accepted age of the Universe.
However, we must keep in mind that the BC03 SSP models were calculated for
ages up to 20 Gyr. In fact, ages derived for objects such as GCs and galaxies
in excess of that of the Universe only mean that these objects are among the
oldest objects in the Universe.

In the context of the BC03 models and their associated age range up to 20 Gyr,
our derived age for G1 is consistent with the suggestion by the WMAP team that
the oldest GCs may have formed within the first 1.7 Gyr after the Big Bang
(see §1).

The integrated FUV flux depends mainly on the fractional number of horizontal-
branch (HB) stars with temperatures hotter than $T_{e}\sim 10,000$ K, with a
modest dependence on their temperature distribution (see Rey et al. 2007 and
references therein). Older GCs produce more of these hot HB stars and they are
thus more likely to produce stronger FUV fluxes at a given metallicity (see
Rey et al. 2007 and references therein) Lee et al. (2003) showed that the
addition of FUV photometry to optical data can discriminate cleanly among
young ($<$1 Gyr), intermediate-age (3–5 Gyr), and old ($>$ 14 Gyr) GCs. Young
and very old GCs exhibit a significant FUV-to-optical spectral continuum
slope, but intermediate-age clusters are relatively faint in the FUV (see fig.
2 of Lee et al. 2003). Figure 6 of Rey et al. (2007) implies that the age of
G1 may be similar to that of the oldest GCs.

Overall, we therefore conclude that G1 is indeed among the oldest and most
massive building blocks of M31, and provides a key limitation to the age of
the Universe, although we caution that our results also provide circumstantial
support to the suggestion that the cluster may not be a genuine GC.

## Acknowledgments

We are indebted to the referee for thoughtful comments and insightful
suggestions that improved this paper greatly. This work has been supported by
the Chinese National Natural Science Foundation through Grant Nos. 10873016,
10803007, 10473012, 10573020, 10633020, 10673012, and 10603006; and by
National Basic Research Program of China (973 Program) No. 2007CB815403. RdG
acknowledges partial financial support from the Royal Society in the form of a
UK-China International Joint Project. SCR acknowledges partial support from
KOSEF through the Astrophysical Research Center for the Structure and
Evolution of the Cosmos (ARCSEC). This paper makes use of data from the Two
Micron All Sky Survey, which is a joint project of the University of
Massachusetts and the Infrared Processing and Analysis Center, funded by NASA
and the National Science Foundation. This paper is also partially based on
archival observations with the NASA/ESA Hubble Space Telescope, obtained at
the Space Telescope Science Institute (STScI), which is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS 5-26555. This research has made use of NASA’s Astrophysics Data
System Abstract Service. This reasearch is partially based on archival data
from the NASA GALEX mission developed in cooperation with the Centre National
d’Etudes Spatiales of France and the Korean Ministry of Science and
Technology.

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