#  Comment on “Nonideal Fields Solve the Injection Problem in Relativistic
Reconnection”

Fan Guo  Los Alamos National Laboratory, Los Alamos, NM 87545, USA
<EMAIL_ADDRESS>Xiaocan Li  Dartmouth College, Hanover, NH 03755, USA  Omar
French  Department of Physics, 390 UCB, University of Colorado, Boulder, CO
80309, USA  William Daughton  Los Alamos National Laboratory, Los Alamos, NM
87545, USA  William Matthaeus  Department of Physics and Astronomy, University
of Delaware, Newark, DE 19716, USA  Qile Zhang  Los Alamos National
Laboratory, Los Alamos, NM 87545, USA  Yi-Hsin Liu  Dartmouth College,
Hanover, NH 03755, USA  Patrick Kilian  Space Science Institute, 4765 Walnut
St, Suite B, Boulder, CO 80301  Grant Johnson  Princeton Plasma Physics
Laboratory 100 Stellarator Rd, Princeton, NJ 08540  Hui Li  Los Alamos
National Laboratory, Los Alamos, NM 87545, USA

In a recent Letter, Sironi [1] (S22) reported the correlation between
particles accelerated into high energy and their crossings of regions with
electric field larger than magnetic field ($E>B$ regions) in kinetic
simulations of relativistic magnetic reconnection [3, 4, 2, 5, 6, 7]. They
claim that electric fields in $E>B$ regions (for a vanishing guide field)
dominate in accelerating particles to the injection energy $\gamma_{\rm
inj}\sim\sigma$ (magnetization). They suggest that the diffusion regions host
particles for a sufficient time for efficient injection. S22 presented test-
particle simulations showing that if particle energies are reset to low
energies in $E>B$ regions, efficient injection is suppressed. This issue has
strong implications for modeling large-scale reconnection system, and thus
important to resolve.

This Comment re-examines these claims by analyzing a simulation resembling the
reference case in S22 using VPIC [8]. We find nearly no particle stayed in
$E>B$ regions long enough to achieve injection by the reconnecting electric
field. The acceleration in $E>B$ regions only contributes a small fraction to
the injection energy ($\sim 10\%\gamma_{\rm inj}$ on average) [6]. The
energization before any $E>B$ crossings has a comparable contribution,
indicating $E>B$ regions are not unique in pre-accelerating particles. A new
test-particle simulation shows that zero-outing electric fields in $E>B$
regions does not strongly influence the injection. We suggest that the
procedure used in S22 to exclude $E>B$ acceleration partly removes
acceleration outside $E>B$ regions, leading to a false conclusion that
injection by $E>B$ regions is a necessary prerequisite.

Figure 1: a) Distribution function for the time duration of $E>B$ particles
within $E>B$ regions. The estimated time limits for achieving injection are
labeled; b) Distribution of energy gain during injection (for $E>B$ particles)
for $E>B$ regions (blue), before $E>B$ crossing (red), and outside $E>B$
regions after the first crossing (black); c) Spectra for self-consistent
particles (blue), test-particles that do not see electric fields in $E>B$
regions (red) and test-particles with a resetting energy approach (green;
resembling S22).

The initial magnetic field
$\mathbf{B}=B_{0}\tanh(z/\lambda)\mathbf{e_{x}}+B_{0}\operatorname{sech}(z/\lambda)\mathbf{e_{y}}$.
$B_{0}$ is the reconnecting-field magnitude and $\lambda(=6$ skin depth
$d_{e}$) is the half-layer-thickness. $\sigma=50$ and temperature
$kT=0.36m_{e}c^{2}$. The box size is $L_{x}\times L_{z}=1600d_{e}\times
1200d_{e}$ and the simulation lasts $2.5L_{x}/c$, with a small perturbation
added to trigger reconnection. Each $d_{e}$ is resolved by $4$ cells with
$100$ positron-electron-pairs per cell. Boundary conditions are periodic in
the $x$-direction and conducting (reflecting) in the $z$-direction for fields
(particles). We trace $1.28$ million particles uniformly and record the
electromagnetic fields they experience at every time-step [9]. The
reconnection dynamics and nonthermal energy spectra in the magnetically-
dominated regime have been well documented [3, 4, 2, 5, 6, 7, 10].

During injection $\gamma\rightarrow\sigma$($\sigma/4$), $77.5\%$($51.0\%$) of
the injected tracers have $E>B$ crossings (“$E>B$ particles”). S22 found a
stronger correlation, since they label all particles that ever crossed $E>B$
regions during the entire simulation, rather than just during injection [11].
Clearly, there is a significant fraction of particles injected without the
need to cross the $E>B$ regions [12]. Nevertheless, it is still interesting to
explore if $E>B$ regions are important for particles that crossed those
regions before achieving injection.

During injection, $E>B$ particles can have multiple $E>B$ crossings. Our
analysis includes all the duration that particles are in $E>B$ regions. The
time tells the limit of acceleration in the regions
$\Delta\gamma_{E>B}\lesssim\int qrB_{0}cdt/(m_{e}c^{2})$, where reconnection
rate $r\sim 0.1$. For $\sigma=50$, $\omega_{\rm pe}t_{\rm inj}\gtrsim 50$ is
needed for $\gamma_{\rm inj}=\sigma$ ($\omega_{\rm pe}t_{\rm inj}\gtrsim 12.5$
for $\gamma_{\rm inj}=\sigma/4$). Fig. 1a shows the time distribution of $E>B$
particles stayed in $E>B$ regions during injection. The mean time that
particles stay in those regions is $\omega_{\rm pe}\bar{t}=4.2(1.8)$ for
$\gamma_{\rm inj}=\sigma$($\sigma/4$) and nearly no $E>B$ particles have time
to reach $\gamma_{\rm inj}$. Fig. 1b shows the distribution of particle energy
gain (during injection) in $E>B$ regions (blue), before $E>B$ crossings (red),
and outside $E>B$ regions after the first $E>B$ crossing (black).
Consistently, the acceleration in $E>B$ regions is too little for direct
injections, with $\Delta\bar{\gamma}_{E>B}=4.8(1.6)$ for $\gamma_{\rm
inj}=\sigma(\sigma/4)$. Interestingly, we find comparable acceleration before
particles encounter any $E>B$, giving $\Delta\bar{\gamma}_{b,E>B}=5.6(2.5)$
for $\gamma_{\rm inj}=\sigma(\sigma/4)$. This suggests that $E>B$ acceleration
is not unique in pre-accelerating particles. Note that this result is
consistent with Fig. 3 in S22, but unfortunately overlooked in their
interpretation. Fig. 1b also shows that most acceleration during injection
occurs outside $E>B$ regions. We evolve a test-particle component in the
simulation that does not “see” the electric field in $E>B$ regions (so no
acceleration during each crossing), and find $84\%$ ($94\%$) for $\gamma_{\rm
inj}=\sigma$($\sigma/4$) compare to self-consistent particles are still
injected. There is no significant difference between energy spectra of the
test-particles and self-consistent particles (Fig. 1c). In contrast, when
particle energies are reset to an energy of $10kT$ during $E>B$ crossings
(resembling S22), particle injection is suppressed. Obviously, this difference
is because resetting particle energy removes the acceleration before and
between $E>B$ crossings.

Our analysis demonstrated that the apparent correlation between particle
injection and $E>B$ crossings do not have direct physical relation. Most
acceleration for $E>B$ particles is not achieved by $E>B$ regions. We have
reached the same conclusion for different $\sigma$ and domain sizes, which
will be presented elsewhere.

Acknowledgment.— We are thankful for discussions with Lorenzo Sironi, as well
as discussions with Joel Dahlin, Jim Drake, Colby Haggerty, Dmitri Uzdensky,
and Greg Werner. We acknowledge the support from Los Alamos National
Laboratory through the LDRD program, DOE office of science, and NASA programs
through the Astrophysical Theory Program. The work by X.L. and Y. L. is funded
by the National Science Foundation grant PHY-1902867 through the NSF/DOE
Partnership in Basic Plasma Science and Engineering and NASA 80NSSC21K2048.
The simulations used resources provided by the Los Alamos National Laboratory
Institutional Computing Program, the National Energy Research Scientific
Computing Center (NERSC) and the Texas Advanced Computing Center (TACC).

## References

  * Sironi [2022] Sironi, L. 2022, Phys. Rev. Lett., 128, 145102. doi:10.1103/PhysRevLett.128.145102
  * Guo et al. [2015] Guo, F., Liu, Y.-H., Daughton, W., et al. 2015, ApJ, 806, 167. doi:10.1088/0004-637X/806/2/167
  * Sironi & Spitkovsky [2014] Sironi, L. & Spitkovsky, A. 2014, ApJ, 783, L21. doi:10.1088/2041-8205/783/1/L21
  * Guo et al. [2014] Guo, F., Li, H., Daughton, W., et al. 2014, Phys. Rev. Lett., 113, 155005. doi:10.1103/PhysRevLett.113.155005
  * Werner et al. [2016] Werner, G. R., Uzdensky, D. A., Cerutti, B., et al. 2016, ApJ, 816, L8. doi:10.3847/2041-8205/816/1/L8
  * Guo et al. [2019] Guo, F., Li, X., Daughton, W., et al. 2019, ApJ, 879, L23. doi:10.3847/2041-8213/ab2a15
  * Guo et al. [2020] Guo, F., Liu, Y.-H., Li, X., et al. 2020, Physics of Plasmas, 27, 080501. doi:10.1063/5.0012094
  * Bowers et al. [2008] Bowers, K. J., Albright, B. J., Yin, L., et al. 2008, Physics of Plasmas, 15, 055703. doi:10.1063/1.2840133
  * Guo et al. [2021] Guo, F., Li, X., Daughton, W., et al. 2021, ApJ, 919, 111. doi:10.3847/1538-4357/ac0918
  * Sironi et al. [2016] Sironi, L., Giannios, D., & Petropoulou, M. 2016, MNRAS, 462, 48. doi:10.1093/mnras/stw1620
  * Sironi [2022] Sironi, L. 2022, private communication
  * French [2022] French, O. et al. 2022, in preparation
  * Liu et al. [2015] Liu, Y.-H., Guo, F., Daughton, W., et al. 2015, Phys. Rev. Lett., 114, 095002. doi:10.1103/PhysRevLett.114.095002
  * Liu et al. [2017] Liu, Y.-H., Hesse, M., Guo, F., et al. 2017, Phys. Rev. Lett., 118, 085101. doi:10.1103/PhysRevLett.118.085101
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††thanks: * Corresponding author

# Does The Monge Theorem Apply To Some Non-Euclidean Geometries ?

Temel Ermiş  Department of Mathematics and Computer Sciences,
Eskisehir Osmangazi University, 26040 Eskisehir, Turkey<EMAIL_ADDRESS>Özcan Gelişgen  Department of Mathematics and Computer Sciences,
Eskisehir Osmangazi University, 26040 Eskisehir, Turkey<EMAIL_ADDRESS>
(Date: January 1, 2004)

###### Abstract.

In geometry, Monge’s theorem states that for any three non-overlapping circles
of distinct radii in the two dimensional analytical plane equipped with the
Euclidean metric, none of which is completely inside one of the others, the
intersection points of each of the three pairs of external tangent lines are
collinear. So, it is clearly observed that Monge’s theorem is an application
of Desargues’ theorem. Our main motivation in this study is to show whether
Monge theorem is still valid even if the plane is equipped with the metrics
alpha and $L_{p}$.

###### Key words and phrases:

The Monge Theorem, Alpha Plane Geometry, $L_{p}$ Plane Geometry

###### 1991 Mathematics Subject Classification:

Primary 51K05; Secondary 51N20; 51F99

##  1\. Introduction

Euclidean distance is the most common use of distance. In most cases when
people said about distance, they will refer to Euclidean distance. The
Euclidean distance between two points is defined as the length of the segment
between two points. Although it is the most popular distance function, it is
not practical when we measure the distance which we actually move in the real
world ( we live on a spherical Earth rather than on a Euclidean $3-$space ! ).
We must think of the distance as though a car would drive in the urban
geography where physical obstacles have to be avoided. So, one had to travel
through horizontal and vertical streets to get from one location to another.
To compensate disadvantage of the Euclidean distance, the taxicab geometry was
first introduced by K. Menger and has developed by E. F. Krause using the
taxicab metric $d_{T}$ of which paths composed of the line segments parallel
to coordinate axes (see Figure 1) ([5], [10]). Its three-dimensional version
has been introduced in [1]. Later, researchers have wondered whether there are
alternative distance functions of which paths are different from path of
Euclidean metric. For example, G. Chen [2] developed Chinese checker distance
$d_{CC}$ in $\mathbb{R}^{2}\ $of which paths are similar to the movement made
by Chinese checker (see Figure 1). Another example, S. Tian gave a family of
metrics in [15], alpha distance $d_{\alpha}$ for
$\alpha\in\left[0,\pi/4\right]$, which includes the taxicab and Chinese
checker metrics as special cases. Then, Kaya et al. have given the most
general form of $d_{\alpha}$ on $n-$dimensional analytical space for
$\alpha\in\left[0,\pi/2\right)$. When we examine the common features of the
metrics $d_{M}$, $d_{T}$, $d_{CC}\ $and$\ d_{\alpha}$, we see that these
metrics whose paths are parallel to at least one of the coordinate axes. So,
it is a logical question ” Are there metric or metrics of which paths are not
parallel to the coordinate axes” ( Yes, there are !). The references [3] and
[11] can be reviewed for the answer to this question. Meanwhile, the alpha
metric is defined as follows;

######  Definition 1.1.

Let $A=\left(x_{a},y_{a}\right)$ and $B=\left(x_{b},y_{b}\right)$ be two any
points in $\mathbb{R}^{2}$ such that
$\Delta_{AB}{\small=}\max\left\\{\left|x_{a}-x_{b}\right|,\left|y_{a}-y_{b}\right|\right\\}$
and
$\delta_{AB}{\small=}\min\left\\{\left|x_{a}{\small-}x_{b}\right|,\left|y_{a}{\small-}y_{b}\right|\right\\}$.
Then, the Euclidean metric can be given by

$d_{E}\left(A,B\right)=\left(\Delta_{AB}^{2}+\delta_{AB}^{2}\right)^{1/2}\text{.}$

Also, the metrics $d_{M},\ d_{T}$, $d_{CC}\ $and$\ d_{\alpha}$ are defined as
following Figure 1.

Figure 1. The paths of distances $d_{M},\ d_{T}$, $d_{CC}\ $and$\ d_{\alpha}$.

$\alpha-$(alpha) plane geometry, which includes the taxicab and Chinese
checker geometry, is a Minkowski geometry. Minkowski geometry is a non
Euclidean geometry in a finite number of dimensions that is different from
elliptic and hyperbolic geometry (from Minkowskian geometry of space-time).
Here the linear structure is same as the Euclidean one but distance is not
uniform in all directions (see for details [4], [6], [8], [9], [12], [14]).
That is, $\alpha-$plane is almost the same as Euclidean plane since the points
are the same, the lines are the same, and the angles are measured in the same
way. Instead of the usual circle in Euclidean plane geometry, unit ball is a
certain symmetric closed (see Figure 2). Since the $\alpha-$plane geometry has
a different distance function, it seems interesting to study the
$\alpha-$analog of the topics that include the concepts of distance in the
Euclidean geometry. One of the famous theorems that includes the concept of
distance is the Monge Theorem [7], [16]. Monge’s theorem says that for any
three nonintersecting circles in a plane, none of which is equal radius, the
intersection points of each of the three pairs of external tangent lines are
collinear. For any two circles in a plane, an external tangent is a line that
is tangent to both circles but does not pass between them. There are two such
external tangent lines for any two circles. Each such pair has a unique
intersection point in the plane. In the case of two of the circles being of
equal size, the two external tangent lines are parallel. If the two external
tangents are considered to intersect at the point at infinity, then the other
two intersection points must be on a line passing through the same point at
infinity, so the line between them takes the same angle as the external
tangent.

We will show that Monge theorem is still valid even if the plane is equipped
with the metrics $\alpha$ and $L_{p}\ $in the next section. First, let us
recall the definition of the well-known $L_{p}-$metric.

######  Definition 1.2.

Let $X=\left(x_{1},\ldots,x_{n}\right)$ and
$Y=\left(y_{1},\ldots,y_{n}\right)$ be two vectors in the $n-$dimensional real
vector space $\mathbb{R}^{n}$. The $l_{p}-$metric $d_{l_{p}}$, $1\leq
p\leq\infty$, is a norm metric on $\mathbb{R}^{n}$ ( or on $\mathbb{C}^{n}$),
is defined by

$\left\|X-Y\right\|_{p}\text{,}$

where the $l_{p}-$norm $\left\|.\right\|_{p}$ is defined by

$\left\|X\right\|_{p}=\left(\underset{i=1}{\overset{n}{\sum}}\left|x_{i}\right|^{p}\right)^{\frac{1}{p}}\text{.}$

If $p\rightarrow\infty$, we obtain
$\left\|X\right\|_{\infty}=\underset{p\rightarrow\infty}{\lim}\left(\underset{i=1}{\overset{n}{\sum}}\left|x_{i}\right|^{p}\right)^{\frac{1}{p}}=\underset{1\leq
i\leq n}{\max}\left|x_{i}\right|$. Also, the the pair
$\left(\mathbb{R}^{n},d_{l_{p}}\right)$ is called $l_{p}-$(metric) space.

##  2\. The Unit Circles in $\mathbb{R}^{2}$ and Monge’ s Theorem

As is known to all, the circle is simply the set of points that are an equal
distance from a certain central point. That is, let $M$ be given point in the
plane, and $r$ be a positive real number. The set of points
$\left\\{X\in\mathbb{R}^{2}:d\left(M,X\right)=r\right\\}$ is called a circle,
the point $M$ is called center of the circle, and $r$ is called the length of
the radius or simply radius of the circle. But there’s a big assumption in
definition of the circle: ” distance ”. While we usually use the Euclidean
distance $d_{E}$, there are other valid notions of distance we could have used
instead, which we have mentioned above.

Figure 2. The unit circles in the $\alpha$ and $l_{p}-$plane.

It is easy to see that if $\alpha\in\left(0,\pi/2\right)$, then unit
$\alpha-$circle is an octagon with corner points$\ A_{1}=\left(1,0\right)$,
$A_{2}=\left(1/k,1/k\right)$,$A_{3}{\small=}\left(0,1\right)$,
$A_{4}{\small=}\left(-1/k,1/k\right)$, $A_{5}{\small=}\left(-1,0\right)$,
$A_{6}{\small=}\left(-1/k,-1/k\right)$, $A_{7}=\left(0,-1\right)$ and
$A_{8}=\left(1/k,-1/k\right)$, where$\ k=1+\sec\alpha-\tan\alpha$. Note that
$A_{2}$ and $A_{6}$ are on the line $y=x$; $A_{4}$ and $A_{8}$ are on the line
$y=-x$. When $\alpha\rightarrow\pi/2$ and $\alpha=0$, the $\alpha-$circle is
the circle with respect to the metrics $d_{M}$ and $d_{T}$, respectively (see
Figure 2). Also, $\alpha-$circle is the circle with respect to the metrics
$d_{CC}$ such that $\alpha=\pi/4$. Similarly, when $p\rightarrow\infty$,$\
p=1$ and $p=2$, the circle in $l_{p}-$plane is the circle with respect to the
metrics $d_{M}$,$\ d_{T}$ and $d_{E}$, respectively (see Figure 2). Now, when
we take the circles $\alpha$ and $l_{p}$ instead of Euclidean circles, we will
investigate whether the Monge theorem is still valid.

Figure 3.

Consider Figure 3. Let $C_{i}$ be the $\alpha-$circles (or $l_{p}-$circles )
with center $M_{i}=\left(x_{i},y_{i}\right)$ and radius $r_{i}$ for $i=1,2,3$.
Then, the coordinates of points on the $\alpha-$circles (or $l_{p}-$circles
)$\ C_{1}$, $C_{2}$ and $C_{3}$ can be easily given with respect to the
coordinates of circles’s centers and radii. Because we know the coordinates of
points $A,F,D$ and $G$ on the circles, we can find the equations of the
tangent lines $l_{AF}$ and $l_{DG}$. Consequently, coordinates of the point
$P_{12}$ of intersection of the tangent lines $l_{AF}$ and $l_{DG}\ $ be able
to calculated as
$P_{12}=\left(\dfrac{r_{1}x_{2}-r_{2}x_{1}}{r_{1}-r_{2}},\dfrac{r_{1}y_{2}-r_{2}y_{1}}{r_{1}-r_{2}}\right)$.
Also, the line $l_{M_{1}M_{2}}$ through the center points $M_{1}$ and $M_{2}$
passes through the point $P_{12}$. Similarly, after calculating the equations
of tangent lines $l_{BI}$ and $l_{EJ}$ for the $\alpha-$circles (or
$l_{p}-$circles ) $C_{1}\ $and $C_{3}$, their intersection point of lines
$l_{BI}$ and $l_{EJ}$ is found as
$P_{13}=\left(\dfrac{r_{1}x_{3}-r_{3}x_{1}}{r_{1}-r_{3}},\dfrac{r_{1}y_{3}-r_{3}y_{1}}{r_{1}-r_{3}}\right)$.
Finally, coordinates of the point $P_{23}\ $be able to calculated as
$P_{23}=\left(\dfrac{r_{2}x_{3}-r_{3}x_{2}}{r_{2}-r_{3}},\dfrac{r_{2}y_{3}-r_{3}y_{2}}{r_{2}-r_{3}}\right)$.
By simplifying long and boring calculations, we get the equation of the line
passing through points $P_{12}$ and $P_{13}$ as

$\displaystyle y$ $\displaystyle=$
$\displaystyle\dfrac{y_{1}\left(r_{2}-r_{3}\right)-y_{2}\left(r_{1}-r_{3}\right)+y_{3}\left(r_{1}-r_{2}\right)}{x_{1}\left(r_{2}-r_{3}\right)-x_{2}\left(r_{1}-r_{3}\right)+x_{3}\left(r_{1}-r_{2}\right)}x$
$\displaystyle+\dfrac{r_{1}\left(x_{2}y_{3}-x_{3}y_{2}\right)-r_{2}\left(x_{1}y_{3}-x_{3}y_{1}\right)+r_{3}\left(x_{1}y_{2}-x_{2}y_{1}\right)}{x_{1}\left(r_{2}-r_{3}\right)-x_{2}\left(r_{1}-r_{3}\right)+x_{3}\left(r_{1}-r_{2}\right)}\text{.}$

Similarly, we get the equation of the line passing through points $P_{13}$ and
$P_{23}$ in the same way as the line above. Consequently, the points$\
P_{12}$, $P_{13}$ and $P_{23}\ $are collinear since the these points determine
one line called the axis of similitude (or Monge Line). Thus, using the
notations in [13], we can give the following main theorem.

######  Theorem 2.1.

Let $C_{i}$ and $C_{j}$ be two $\alpha-$circles (or $l_{p}-$circles ) with
centers $M_{i}=\left(x_{i},y_{i}\right)$ and $M_{j}=\left(x_{i},y_{i}\right)$
of radii $r_{i}$ and $r_{j}$ for $i,j\in\left\\{1,2,3\right\\}$, respectively.
Then, the common tangents of the circles $C_{i}$ and $C_{j}$ intersect at the
point

$P_{ij}=\left(\dfrac{r_{i}x_{j}-r_{j}x_{i}}{r_{i}-r_{j}},\dfrac{r_{i}y_{j}-r_{j}y_{i}}{r_{i}-r_{j}}\right)\text{.}$

Also, the points $P_{12}$,$\ P_{13}$ and $P_{23}\ $of intersection of the
three pairs of tangent lines lie on a line. This line is called Monge line
whose equation

$\left|\begin{array}[]{ccc}x_{1}&x_{2}&x_{3}\\\ r_{1}&r_{2}&r_{3}\\\
1&1&1\end{array}\right|y=\left|\begin{array}[]{ccc}y_{1}&y_{2}&y_{3}\\\
r_{1}&r_{2}&r_{3}\\\
1&1&1\end{array}\right|x-\left|\begin{array}[]{ccc}x_{1}&x_{2}&x_{3}\\\
y_{1}&y_{2}&y_{3}\\\ r_{1}&r_{2}&r_{3}\end{array}\right|\text{.}$

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# Age and mass constraints for a young massive cluster in M31 based on
spectral-energy-distribution fitting

Jun Ma,11affiliation: National Astronomical Observatories, Chinese Academy of
Sciences, Beijing 100012, P. R. China;
<EMAIL_ADDRESS>22affiliation: Key Laboratory of Optical Astronomy, National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R.
China Song Wang,11affiliation: National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100012, P. R. China;
<EMAIL_ADDRESS>33affiliation: Graduate University, Chinese Academy of
Sciences, Beijing 100039, P. R. China Zhenyu Wu,11affiliation: National
Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, P. R.
China;
<EMAIL_ADDRESS>Zhou Fan,11affiliation: National Astronomical Observatories,
Chinese Academy of Sciences, Beijing 100012, P. R. China;
<EMAIL_ADDRESS>Yanbin Yang,11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China;
<EMAIL_ADDRESS>Tianmeng Zhang,11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China;
<EMAIL_ADDRESS>Jianghua Wu,11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China;
<EMAIL_ADDRESS>Xu Zhou,11affiliation: National Astronomical Observatories,
Chinese Academy of Sciences, Beijing 100012, P. R. China;
<EMAIL_ADDRESS>Zhaoji Jiang,11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China;
<EMAIL_ADDRESS>and Jiansheng Chen11affiliation: National Astronomical
Observatories, Chinese Academy of Sciences, Beijing 100012, P. R. China;
<EMAIL_ADDRESS>
###### Abstract

VDB0-B195D is a massive, blue star cluster in M31. It was observed as part of
the Beijing-Arizona-Taiwan-Connecticut (BATC) Multicolor Sky Survey using 15
intermediate-band filters covering a wavelength range of 3000–10,000 Å. Based
on aperture photometry, we obtain its spectral-energy distribution (SED) as
defined by the 15 BATC filters. We apply previously established relations
between the BATC intermediate-band and the Johnson-Cousins $UBVRI$ broad-band
systems to convert our BATC photometry to the standard system. A detailed
comparison shows that our newly derived $VRI$ magnitudes are fully consistent
with previous results, while our new $B$ magnitude agrees to within $2\sigma$.
In addition, we determine the cluster’s age and mass by comparing its SED
(from 3000 to 20,000Å, comprising photometric data in the 15 BATC intermediate
bands, optical broad-band $BVRI$, and 2MASS near-infrared $JHK_{\rm s}$ data)
with theoretical stellar population synthesis models, resulting in age and
mass determinations of $60.0\pm 8.0$ Myr and $(1.1-1.6)\times
10^{5}M_{\odot}$, respectively. This age and mass confirms previous
suggestions that VDB0-B195D is a young massive cluster in M31.

###### Subject headings:

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

††slugcomment: AJ, in press

##  1\. Introduction

Young massive star clusters (YMCs) are among the main objects resulting from
violent star-forming episodes triggered by galaxy collisions, mergers, and
close encounters (see de Grijs & Parmentier, 2007, and references therein).
They are also referred to as ‘young populous clusters,’ a term first coined by
Hodge (1961), who used it to describe 23 clusters containing bright, blue
stars in the Large Magellanic Cloud. In Hodge (1961), the ‘young’ aspect is
demonstrated by the fact that all clusters have main sequences that extend to
absolute magnitudes brighter than $M_{V}=0$, while ‘populous’ describes their
richness (stellar membership). However, YMCs are also observed in quiescent
galaxies (Larsen & Richtler, 1999) and in the disks of isolated spirals,
although higher cluster-formation efficiencies are associated with
environments exhibiting high star-formation rates (see Larsen, 2004; Cao & Wu,
2007, and references therein). It has become clear that, in many ways, YMCs
resemble young versions of the old globular clusters (GCs) associated with all
large galaxies (see Larsen et al., 2004, and references therein). YMCs are
seemingly absent in the Milky Way; possibly the best example of a Galactic YMC
is Westerlund 1, a heavily reddened cluster with an age and mass of 4–5 Myr
(Crowther et al., 2006) and $M_{\rm cl}\sim 10^{5}~{}M_{\odot}$ (Clark et al.,
2005), respectively.

Since the pioneering work of Tinsley (1968, 1972) and Searle (1973),
evolutionary population synthesis modeling has become a powerful tool to
interpret integrated spectrophotometric observations of galaxies and their
components, such as star clusters (e.g., Anders et al., 2004). The evolution
of star clusters is usually modeled by means of the simple stellar population
(SSP) approximation. An SSP is defined as 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).

Age and metallicity are two basic star cluster parameters. The most direct
method to determine a cluster’s age is by employing main-sequence photometry,
since the absolute magnitude of the main-sequence turnoff is predominantly
affected by age (see Puzia et al., 2002, and references therein). However,
until recently (cf. Perina et al., 2009), this method was only applied to the
star clusters in the Milky Way and its satellites (e.g., Rich et al., 2001),
although Brown et al. (2004) estimated the age of an M31 GC using extremely
deep images observed with the Hubble Space Telescope (HST)’s Advanced Camera
for Surveys. Generally, the ages of extragalactic star clusters are determined
by comparing their observed spectral-energy distributions (SEDs) and/or
spectroscopy with the predictions of SSP models (Williams & Hodge, 2001a, b;
de Grijs et al., 2003a, b, c; Bik et al., 2003; Jiang et al., 2003; Beasley et
al., 2004; Puzia et al., 2005; Ma et al., 2006; Fan et al., 2006; Ma et al.,
2007, 2009; Caldwell et al., 2009; Wang et al., 2010). Nevertheless, SSP
models assume that cluster IMFs are fully populated, i.e., that clusters
contain infinite numbers of stars with a continuous distribution of stellar
masses, and that all evolutionary stages are well sampled. Real clusters,
however, contain a finite number of stars. Therefore, a disagreement between
the observed cluster colors and theoretical colors derived from SSP models may
become apparent (see Piskunov et al., 2009; Popescu & Hanson, 2010, and
references therein). Other limitations inherent to SSP models arise from our
poor understanding of some advanced stellar evolutionary stages, such as the
supergiant and the asymptotic-giant-branch (AGB) phases (see Bruzual &
Charlot, 2003, and references therein).

Located at a distance of $785\pm 25$ kpc, corresponding to a distance modulus
of $(m-M)_{0}=24.47\pm 0.07$ mag (McConnachie et al., 2005), M31 is the
nearest and largest spiral galaxy in the Local Group of galaxies. It has been
the subject of many GC studies and surveys, dating back to the early study of
Hubble (1932). Based on previous publications (Hubble, 1932; Seyfert & Nassau,
1945; Hiltner, 1958; Mayall & Eggen, 1953; Kron & Mayall, 1960), Vetes̆nik
(1962) compiled the first large M31 GC catalog, containing $UBV$ photometric
data of approximately 300 GC candidates. Over the past decades, several major
catalogs of M31 GCs and GC candidates have been published, including major
efforts by the Bologna group (Battistini et al., 1980, 1987, 1993), Barmby et
al. (2000), Galleti et al. (2004, 2005, 2006, 2007), Kim et al. (2007),
Caldwell et al. (2009), and Peacock et al. (2010). Following on from the first
extensive spectroscopic survey of M31 GCs by van den Bergh (1969), a
significant number of authors (e.g., Huchra et al., 1982, 1991; Dubath &
Grillmair, 1997; Federici et al., 1993; Jablonka et al., 1998; Barmby et al.,
2000; Perrett et al., 2002; Galleti et al., 2006; Lee et al., 2008, and
references therein) have studied their spatial, kinematic, and chemical
(metallicity) properties.

M31 is known to host a large number of young star clusters (e.g., Fusi Pecci
et al., 2005; Caldwell et al., 2009; Wang et al., 2010, and references
therein). Fusi Pecci et al. (2005) presented a comprehensive study of 67 very
blue star clusters, which they referred to as ‘blue luminous compact clusters’
(BLCCs). Since they are quite bright ($-6.5\leq M_{V}\leq-10.0$ mag) and very
young ($<2$ Gyr), BLCCs may be equivalent to YMCs (see for details Perina et
al., 2009, 2010). To ascertain their properties, Perina et al. (2009, 2010)
performed an imaging survey of 20 BLCCs in the disk of M31 using the HST’s
Wide Field and Planetary Camera-2 (WFPC2). They obtained the reddening values,
ages, and metallicities of their sample clusters by comparing the observed
color-magnitude diagrams (CMDs) and luminosity functions with theoretical
models.

VDB0-B195D was first detected by van den Bergh (1969). Its color is extremely
blue (e.g., $U-B=-0.48$ mag; van den Bergh, 1969) and it is very bright in
blue bands (e.g., $U=14.66$ mag; van den Bergh, 1969). As a consequence, van
den Bergh (1969) asserted that VDB0-B195D is the brightest open cluster in
M31. He determined an integrated stellar spectral type equivalent to A0, which
implies that the cluster contains massive stars. In addition, VDB0-B195D is
particularly extended and most previous photometric studies did not include
the full extent of the object’s light distribution (see for details Perina et
al., 2009). We will provide an overview of previous studies that included the
cluster in §2.1. It was observed as part of the galaxy calibration program of
the Beijing-Arizona-Taiwan-Connecticut (BATC) Multicolor Sky Survey (e.g., Fan
et al., 1996; Zheng et al., 1999) in 15 intermediate-band filters. Combined
with photometry in optical broad-band $BVRI$ and near-infrared $JHK_{\rm s}$
filters from the Two Micron All Sky Survey (2MASS) taken from Perina et al.
(2009), we obtained the SED of VDB0-B195D in 22 filters, covering the
wavelengh range from 3000 to 20,000 Å.

In this paper, we describe the details of the observations and our approach to
the data reduction in §2. In §3, we determine the age and mass of VDB0-B195D
by comparing observational SEDs with population synthesis models. We discuss
the implications of our results and provide a summary in §4.

##  2\. Optical and near-infrared observations of the YMC VDB0-B195D

###  2.1. Historical overview

VDB0-B195D was first given the designation ‘0’ (i.e., VDB0), the brightest
open cluster in M31, in the catalog of van den Bergh (1969). Battistini et al.
(1987) identified VDB0-B195D independently and called it B195D. In Battistini
et al. (1987), B195D was given a low level of confidence (class D) of being a
genuine cluster (classes A and B were assigned very high and high levels of
confidence, respectively). It was only recently independently confirmed to be
a single object. Caldwell et al. (2009) presented a new catalog containing 670
likely star clusters, stars, possible stars, and galaxies in the field of M31,
all with updated high-quality coordinates accurate to $0.2^{\prime\prime}$,
based on images from either the Local Group Galaxies Survey (LGGS) (Massey,
2006) or the Digitized Sky Survey (DSS). They use the designation VDB0-B195D,
associated with $\rm\alpha_{0}=00^{\rm h}40^{\rm m}29^{\rm s}.43$ and
$\rm\delta_{0}=+40^{\circ}36^{\prime}14^{\prime\prime}.8$ (J2000.0), which are
the coordinates we adopt in this paper. Independently, Perina et al. (2009)
studied the properties of VDB0-B195D in detail based on their HST/WFPC2
imaging survey of young massive GCs in M31. They initially selected VDB0-B195D
as two YMCs in M31, but their WFPC2 images showed unequivocally that these two
sample objects are, in fact, the same cluster. In addition, the HST images
clearly confirmed that VDB0-B195D is a real cluster. However, it is difficult
to establish whether it is more similar to ordinary open clusters, similar to
those in the disk of the Milky Way, than to YMCs that may evolve to become
disk GCs (see for details Perina et al., 2009). Spectral observations of
VDB0-B195D were obtained by van den Bergh (1969)—yielding classification
spectra and the object’s radial velocity—and Perrett et al. (2002), who used
them for determination of its radial velocity and metallicity.

###  2.2. Archival images of the BATC Multicolor Sky Survey

Observations of the YMC VDB0-B195D were obtained with the BATC 60/90cm Schmidt
telescope located at the XingLong station of the National Astronomical
Observatory of China (NAOC). This telescope is equipped with 15 intermediate-
band filters covering the optical wavelength range from 3000 to 10,000 Å. The
filter system was specifically designed to avoid contamination by the
brightest and most variable night-sky emission lines. Descriptions of the BATC
photometric system can be found in Fan et al. (1996). Before February 2006, a
Ford Aerospace 2k$\times$2k thick CCD camera was installed, with a pixel size
of 15 $\mu$m and a field of view of $58^{\prime}\times 58^{\prime}$, yielding
a resolution of $1.7^{\prime\prime}$ pixel-1. Since February 2006, a new E2V
4k$\times$4k thinned CCD with a pixel size of 12 $\mu$m has been in operation,
featuring a resolution of $1.3^{\prime\prime}$ pixel-1. The blue quantum
efficiency of the new, thinned CCD is 92.2% at 4000 Å, which is much higher
than for the old, thick device (see for details Fan et al., 2009). A field
including VDB0-B195D in the $a$–$c$ filters was observed with the thinned CCD,
and in $d$–$p$ bands with the thick CCD. Fig. 1 shows a finding chart of
VDB0-B195D in the BATC $g$ band (centered at 5795 Å), obtained with the NAOC
60/90cm Schmidt telescope. We adopt an aperture with a radius of
$15^{\prime\prime}$ (shown in Fig. 1) for the integrated photometry discussed
in this paper.

Figure 1.— Image of VDB0-B195D in the BATC $g$ band, obtained with the NAOC
60/90cm Schmidt telescope. VDB0-B195D is circled using an aperture with a
radius of $15^{\prime\prime}$. The field of view of the image is
$11^{\prime}\times 11^{\prime}$.

The BATC survey team obtained 61 images of VDB0-B195D in 15 BATC filters
between January 2004 and November 2006. Fan et al. (2009) performed the data
reduction of these images, which formed part of their M31-7 field. Table 1
contains the observation log, including the BATC filter names, the central
wavelength and bandwidth of each filter, the number of images observed through
each filter, and the total observing time per filter. Multiple images through
the same filter were combined to improve image quality (i.e., increase the
signal-to-noise ratio and remove spurious signal).

###  2.3. Intermediate-band photometry of VDB0-B195D

We determined the intermediate-band magnitudes of VDB0-B195D on the combined
images using a standard aperture photometry approach, i.e., the phot routine
in daophot (Stetson, 1987). Calibration of the magnitude zero level in the
BATC photometric system is similar to that of the spectrophotometric AB
magnitude system. For flux calibration, the Oke-Gunn (Oke & Gunn, 1983)
primary flux standard stars HD 19445, HD 84937, BD +26∘2606, and BD +17∘4708
were observed during photometric nights (Yan et al., 2000). VDB0-B195D is
located in the M31-7 field of Fan et al. (2009). The absolute flux of the
M31-7 field was calibrated based on secondary standard transformations using
the M31-1 field, which was calibrated, in turn, by the four Oke-Gunn primary
flux standard stars by Jiang et al. (2003). Since VDB0-B195D is an extended
object, an appropriate aperture size must be adopted to determine its total
luminosity. The (radial) photometric asymptotic growth curve, in all BATC
bands, flattens out at a radius of $\sim 15^{\prime\prime}$. Inspection
ensured that this aperture is adequate for photometry, i.e., VDB0-B195D does
not show any obvious signal beyond this radius. In addition, this aperture is
nearly the same as that adopted by Perina et al. (2009) to determine the
cluster’s photometry in the $BVRI$ bands, based on the M31 imaging survey of
Massey (2006) (see §2.4 below). Therefore, we use an aperture with $r\approx
15^{\prime\prime}$ for integrated photometry, i.e., $r=9$ pixels for the
2k$\times$2k thick CCD camera, and $r=12$ pixels for the 4k$\times$4k thinned
CCD camera. VDB0-B195D is projected onto the disk of M31, where the background
is bright and fluctuates, potentially as a function of distance from the
cluster center. To avoid contamination from background fluctuations, we
adopted annuli for background subtraction spanning between 10 and 15 pixels
for the 2k$\times$2k thick CCD camera, and from 13 to 20 pixels for the
4$\times$4k thinned CCD camera, both corresponding to $\sim 17$–26′′. While
these annuli are spatially as close as possible to the region dominated by
cluster light (so that any differences in background flux are minimized), they
are wide enough to average out any expected background fluctuations. The
calibrated photometry of VDB0-B195D in 15 filters is summarized in column (6)
of Table 1, in conjunction with the $1\sigma$ magnitude uncertainties, which
include uncertainties from the calibration errors of both the M31-1 field
standard stars (see for details Fan et al., 2009; Jiang et al., 2003) and ‘the
secondary standard stars’ in common between the M31-1 and M31-7 fields used
for calculation of the mean magnitude offsets between the standard and
instrumental magnitudes (see for details Fan et al., 2009), as well as those
resulting from our daophot application.

###  2.4. Optical broad-band and near-infrared 2MASS photometry of VDB0-B195D

Four independent sets of photometric data exist for VDB0-B195D. van den Bergh
(1969) obtained $UBV$ photometry using observations of the 200-inch Hale
telescope, Battistini et al. (1987) performed $UBVR$ photometry based on
photographic plates observed with the 152 cm Ritchey-Chrétien $f$/8 telescope
of the University of Bologna in Loiano, King & Lupton (1991) obtained $UBV$
photometry for VDB0-B195D using observations with the University of Hawaii’s
2.2 m telescope on Mauna Kea using the $f$/10 secondary and coronene-coated
$584\times 416$ GEC CCD, and Sharov et al. (1995) performed $UBV$ photometry
based on photo-electric observations with the 2.6 m Shain telescope of the
Crimean Astrophysical Observatory. In addition, in the Revised Bologna
Catalogue (RBC) of M31 GCs published by Galleti et al. (2004), the photometric
data of VDB0-B195D in optical bands are based on Battistini et al. (1987) and
Sharov et al. (1995), and transformed to the reference system of Barmby et al.
(2000) by applying offsets derived from objects in common between the relevant
catalog and the data set of Barmby et al. (2000). In the RBC, VDB0-B195D was
regarded as two objects. We list these photometric data in Table 2 for
comparison. Note that, in the latest RBC incarnation (version 3.5, updated on
27 March 2008), VDB0-B195D is included as a single object.

Galleti et al. (2004) also determined 2MASS $JHK_{\rm s}$ photometric
magnitudes for VDB0-B195D (transformed to the CIT photometric system; Elias et
al., 1982, 1983), which we have included in Table 3. In addition, Perina et
al. (2009) realized that VDB0-B195D is a particularly extended object and that
it is possible that the photometry of Sharov et al. (1995) (compiled in the
RBC) was obtained with apertures that were not large enough to include all of
its flux. Therefore, they redetermined its photometric values in the $BVRI$
bands based on the M31 imaging survey of Massey (2006) using an aperture with
$r=14.4^{\prime\prime}$, which are also listed in Table 3.

From a comparison of the values in Tables 2 and 3, it is clear that the
magnitudes of van den Bergh (1969) are brighter, while the results of the
three other references are consistent. The magnitudes determined by Perina et
al. (2009) are much brighter, however, because of their careful inclusion of
all of the cluster’s flux. To compare our photometric results with previously
published values, we transformed the magnitudes of VDB0-B195D in the BATC
intermediate bands to broad-band $UBVRI$-equivalent photometry based on the
relationships obtained by Zhou et al. (2003). These are also listed in Table
3, and the uncertainties include those originating from the transformation
based on the relationships of Zhou et al. (2003) and their calibration errors
(column 5 of their Table 3). In Fig. 2, we show the result of the comparison.
In general, the other photometric data are fainter than ours and those of
Perina et al. (2009). Fig. 2 and Table 3 show that our new $VRI$ magnitudes
agree with the results of Perina et al. (2009), and that the $B$ magnitude
obtained in this paper is 0.32 mag brighter than that of Perina et al. (2009).
Considering the photometric errors of both Perina et al. (2009) and our
current study, these two $B$-band photometric results are consistent within
$2\sigma$. In addition, we should keep in mind that, although the $VRI$
magnitudes obtained in this paper are consistent with the results of Perina et
al. (2009) within $1\sigma$, the disagreement in $B$ magnitudes at this level
is understandable. This is caused by the fact that the original photometry in
the present paper was obtained in the proprietary BATC filters and transformed
to the $UBVRI$ system using transformation equations. Zhou et al. (2003)
determined these conversions based on the broad-band $UBVRI$ magnitudes of 48
stars from Landolt (1983, 1992) and Galadí-Enríquez et al. (2000) in the
Landolt SA95 field, and their photometric data in the 15 BATC intermediate-
band filters. In addition, the central wavelengths and bandwidths of the BATC
and $UBVRI$ systems differ. In fact, a similar significant disagreement of
$B$-band photometric data for some M31 GCs was reported by Wang et al. (2010),
citing similar arguments.

Figure 2.— Comparison of photometric data from different sources with new
determinations in this paper for VDB0-B195D. The data points shown as black
dots are from Perina et al. (2009).

##  3\. Stellar population of VDB0-B195D

###  3.1. Stellar populations and synthetic photometry

To determine the age and mass of VDB0-B195D, we compared its SED with
theoretical stellar population synthesis models. The SED consists of
photometric data in the 15 BATC intermediate bands obtained in this paper and
optical broad-band $BVRI$ and 2MASS near-infrared $JHK_{s}$ data from Perina
et al. (2009), listed in Table 3. We used the galev SSP models (e.g., Kurth et
al., 1999; Schulz et al., 2002; Anders & Fritze-v. Alvensleben, 2003) for our
comparisons. The galev SSPs are based on the Padova stellar isochrones, with
the most recent versions using the updated Bertelli et al. (1994) isochrones
(which include the thermally pulsing asymptotic giant-branch phase), and a
Salpeter (1955) stellar IMF with lower- and upper-mass limits of 0.10 and
between 50 and 70 $M_{\odot}$, respectively, depending on metallicity. The
full set of models spans the wavelength range from 91Å to 160 $\mu$m. These
models cover ages from $4\times 10^{6}$ to $1.6\times 10^{10}$ yr, with an age
resolution of 4 Myr for ages up to 2.35 Gyr, and 20 Myr for greater ages. The
galev SSP models include five initial metallicities,
$Z=0.0004,0.004,0.008,0.02$ (solar metallicity), and 0.05.

Since our observational data consist of integrated luminosities through the
set of BATC filters, we convolved the galev SSP SEDs with the BATC
intermediate-, optical broad-band $BVRI$, and 2MASS filter-response curves to
obtain synthetic optical and near-infrared photometry for comparison. The
synthetic $i^{\rm th}$ filter magnitude can be computed as

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

where $F_{\nu}$ is the theoretical SED and $\varphi_{i}$ the response curve of
the $i^{\rm th}$ filter of the BATC, $BVRI$, and 2MASS photometric systems.
Here, $F_{\nu}$ varies with age and metallicity. Since the observed magnitudes
in the $BVRI$ and 2MASS photometric systems are given in the Vega system, we
transformed them to the AB system for our fits.

###  3.2. Reddening and metallicity of VDB0-B195D

To obtain the intrinsic SED of VDB0-B195D, its photometry must be dereddened.
To date, only Perina et al. (2009) obtained reddening values for VDB0-B195D.
They compared the observed CMD with theoretical isochrones and determined
$E(B-V)=0.20\pm 0.03$ mag. Caldwell et al. (2009) were unable to derive the
cluster’s reddening value because of the presence of a foreground field star,
so they adopted $E(B-V)=0.28\pm 0.17$ mag (external rms error), equivalent to
the mean reddening of the young clusters in M31. In this paper, we therefore
adopt the reddening value from Perina et al. (2009).

In addition, cluster SEDs are affected by age and metallicity effects.
Therefore, we can only accurately constrain a cluster’s age if the metallicity
is known. Perina et al. (2009) found that the CMD of VDB0-B195D, based on
their HST/WFPC2 observations, is best reproduced by the solar-metallicity
models of Girardi et al. (2002). We therefore adopt solar metallicity for
VDB0-B195D.

###  3.3. The ‘lowest-luminosity-limit’ test

The lowest-luminosity limit (LLL; Cerviño & Luridiana, 2004) implies that it
is meaningless to compare a cluster with population synthesis models to obtain
its age and mass if its integrated luminosity is lower than the luminosity of
the most luminous star included in the model for the relevant age. The LLL
method states that clusters fainter than this limit cannot be analyzed using
standard procedures such as $\chi^{2}$ minimization of the observed values
with respect to the mean SSP models (see also Barker et al., 2008). Below the
LLL, cluster ages and masses cannot be obtained self-consistently. To take
into account the effects on the integrated luminosities of statistically
sampling the stellar IMF (e.g., Cerviño et al., 2000, 2002; Cerviño &
Luridiana, 2004), we used the theoretical Padova isochrones at
http://stev.oapd.inaf.it/cmd (CMD2.2). This interactive Web interface provides
isochrones for a number of photometric systems, including optical broad-band,
2MASS, and the BATC data used here. We obtained the solar-metallicity
($Z=0.019$) isochrones of Marigo et al. (2008), as recommended by CMD2.2,
based on the (Salpeter, 1955) IMF so as to match the IMF selection for our age
and mass determinations of VDB0-B195D in §3.4 (see §3.1 for details).

Figure 3 shows the LLL values as a function of age for the different filters
used in this paper. These luminosities are obtained by identifying the most
luminous star on each isochrone for the relevant passband. The gray area shows
the cluster’s absolute luminosity, assuming a distance modulus of
$(m-M)_{0}=24.47$ mag (785 kpc) for M31 (McConnachie et al., 2005). The upper
luminosity limit has been corrected for extinction, based on a reddening value
of $E(B-V)=0.20$ mag. The interstellar extinction curve, $A_{\lambda}$, is
taken from Cardelli et al. (1989), $R_{V}=A_{V}/E(B-V)=3.1$.

We see that VDB0-B195D does not lie below the LLL in any of the passbands used
here. This means that, in general, VDB0-B195D can host the most luminous star
that would be present theoretically for the given age of the cluster.

Figure 3.— Lowest-luminosity limit for the filters used in this paper. The
curves indicate the luminosities of the most luminous star on each isochrone
for the relevant passband. The light-gray area shows the absolute magnitudes
of VDB0-B195D based on a reddening value of $E(B-V)=0.20$ mag (Perina et al.,
2009). We used a distance modulus of $(m-M)_{0}=24.47$ mag (785 kpc) for M31
(McConnachie et al., 2005) to calculate the absolute magnitudes.

###  3.4. Fit results

In the previous section, the LLL test proves that the luminosity of VDB0-195D
is higher than the luminosity of its brightest star expected for a given
cluster age, i.e., that using SSP models is not completely meaningless. In
addition, the bright absolute magnitude of VDB0-195D allows us to consider a
possibility that the cluster is massive enough and IMF sampling effects should
not strongly impact the fitting results. So we will determine the cluster’s
age and mass estimates based on direct comparisons with SSP mean values in
this section. However, we should keep in mind that this approach is a
compromise. In fact, the fitting results (Fig. 4 and Table 5) show probable
problem even for relative massive clusters.

We use a $\chi^{2}$ minimization test to determine which galev SSP models are
most compatible with the observed SEDs,

$\chi^{2}=\sum_{i=1}^{22}{\frac{[m_{\nu_{i}}^{\rm intr}-m_{\nu_{i}}^{\rm
mod}(t)]^{2}}{\sigma_{i}^{2}}},$ (2)

where $m_{\nu_{i}}^{\rm mod}(t)$ is the integrated magnitude in the $i^{\rm
th}$ filter of a theoretical SSP at age $t$ (for solar metallicity),
$m_{\nu_{i}}^{\rm intr}$ is the intrinsic, integrated magnitude, and
$\sigma_{i}$ is the magnitude uncertainty, defined as

$\sigma_{i}^{2}=\sigma_{{\rm obs},i}^{2}+\sigma_{{\rm
mod},i}^{2}+(R_{\lambda_{i}}*\sigma_{\rm red})^{2}+\sigma_{{\rm md},i}^{2}.$
(3)

Here, $\sigma_{{\rm obs},i}$ is the observational uncertainty from column (6)
of Table 1 and column (2) of Table 3, $\sigma_{{\rm mod},i}$ is the
uncertainty associated with the model itself, $\sigma_{\rm red}$ is the
uncertainty in the reddening value, and
$R_{\lambda_{i}}=A_{\lambda_{i}}/E(B-V)$, where $A_{\lambda_{i}}$ is taken
from Cardelli et al. (1989), $R_{V}=A_{V}/E(B-V)=3.1$, and $\sigma_{{\rm
md},i}$ is the uncertainty in the distance modulus, for the $i^{\rm th}$
filter. Charlot et al. (1996) estimated the uncertainty associated with the
term $\sigma_{{\rm mod},i}$ by comparing the colors obtained from different
stellar evolutionary tracks and spectral libraries. Following Ma et al. (2007,
2009), we adopt $\sigma_{{\rm mod},i}=0.05$ mag.

Perina et al. (2009) pointed out that VDB0-B195D is a particularly extended
object and that the photometric measurements of van den Bergh (1969),
Battistini et al. (1987), King & Lupton (1991), and Sharov et al. (1995) did
not include all of its flux. Therefore, we adopt the photometry of Perina et
al. (2009) to fit the observed SED with theoretical SSPs for our age
determination. The fit yielding the minimum $\chi^{2}$ value ($\chi^{2}({\rm
min})$) was adopted as the best fit and we adopted the corresponding age
value, $60.0\pm 8.0$ Myr. In addition, our best-fitting age estimate of
$60.0\pm 10.0$ Myr results from using the (redder) $k$–$p$ and $IJHK_{\rm s}$
photometry; using only the blue part of the cluster’s SED ($B,a$–$e$, where
any effects caused by stochasticity may be smaller) yields an age of $72.0\pm
34.0$ Myr. The uncertainty was estimated using confidence limits. If
$\chi^{2}/{\nu}<\chi^{2}({\rm min})/{\nu}+1$, the resulting age is within the
68.3% probability range; here, ${\nu}=21$ is the number of free parameters,
i.e., the number of observational data points minus the number of parameters
used in the theoretical model. Therefore, the accepted age range is derived
from those fits that have $\chi^{2}({\rm
min})/{\nu}<\chi^{2}/{\nu}<\chi^{2}({\rm min})/{\nu}+1$. The best
reduced-$\chi^{2}$—defined as $\chi_{\nu}^{2}({\rm min})=\chi^{2}({\rm
min})/{\nu}$—and age are listed in Table 4. The best fit to the SED of
VDB0-B195D is shown in Fig. 4, where we display the intrinsic cluster SED
(symbols with error bars), as well as the integrated SED (open circles) and
spectrum of the best-fitting model. From Fig. 4, we note that the
observational data in the $b$, $d$, $o$, and $p$ BATC filters and in the
$K_{\rm s}$ band do not match the best-fitting model very well (the difference
is approximately 0.3 mag). Photometric uncertainties in these filters may
cause some differences, although this might not be the main reason for the
discrepancy. As we know, observational star clusters’ SEDs are affected by
age, metallicity and reddening. If the reddening value and metallicity adopted
in this paper are not problematic, discrepancy between our observations and
the best-fitting model may reflect the difficulty in achieving an appropriate
(but formal) fit of an SED of a single, real cluster by SSP models. However,
as we will see below, the reddening value adopted in this paper may be bigger
than the actual reddening of VDB0-B195D. In addition, the differences between
the photometric data and the model in Fig. 4 show a somewhat systematic
behavior with wavelength: in bluer passbands the cluster seems to be more
luminous than predicted by the model, while in redder passbands it is fainter
than the corresponding model predictions. A blue excess and red deficiency in
the observed SED with respect to the model predictions may indicate a shortage
of red giants (RGs), which can occur when the cluster is either younger or
less massive (or both) than the corresponding best-fitting model suggests. In
other words, IMF discreteness may play a role: due to a relatively longer
main-sequence (MS) phase and shorter RG phase, a random young cluster is
typically bluer than predicted by SSP models. At the same time, we find that
the reddening value adopted affects the fitting result greatly. In fact, the
best fit to the SED of VDB0-B195D improves a great deal when adopting a
smaller reddening value such as $E(B-V)=0.1$: $\chi_{\nu}^{2}({\rm
min})=0.73$; the resulting age ($64.0\pm 8.0$ Myr) is nearly the same as one
($60.0\pm 8.0$ Myr) obtained with $E(B-V)=0.2$.

We next determined the mass of VDB0-B195D. The galev models include absolute
magnitudes (in the Vega system) in 77 filters for SSPs of
$10^{6}~{}M_{\odot}$, including 66 filters of the HST, Johnson $UBVRI$ (see
for details Landolt, 1983), Cousins $RI$ (see for details Landolt, 1983), and
$JHK$ (Bessell & Brett, 1988) systems. The difference between the intrinsic
absolute magnitudes and those given by the model provides a direct measurement
of the cluster mass, in units of $10^{6}~{}M_{\odot}$. However, we should keep
in mind that this is only correct for cluster masses above
$10^{6}~{}M_{\odot}$. We estimated the mass of VDB0-B195D using magnitudes in
all of the $BVRI$ and $JHK_{\rm s}$ bands. Therefore, we transformed the 2MASS
$JHK_{\rm s}$ magnitudes to the photometric system of Bessell & Brett (1988)
using the equations given by Carpenter (2001). The resulting mass
determinations for VDB0-B195D are listed in Table 5 with their $1\sigma$
uncertainties including contributions from uncertainties in extinction and
distance modulus. From Table 5, we see that the mass of VDB0-B195D obtained
based on the magnitudes in different filters is very different. (The highest
mass obtained, based on the $B$-band magnitude, is $~{}0.5\times
10^{5}~{}M_{\odot}$ more massive than that obtained using the $K_{s}$
magnitude.) In addition, the mass estimates differ systematically with
filters. Provided that VDB0-B195D is massive enough to be fitted by SSP
models, a systematic trend of masses based on different passbands may indicate
a problem with reddening value adopted for the cluster. If the actual
reddening is smaller than the adopted value, the actual luminosity would be
overestimated. This effect is small in redder filters but strong in bluer
filters. As discussed in age estimation, a smaller reddening value can improve
the fitting result greatly. In fact, a smaller reddening value can reduce the
mass discrepancies based on the magnitudes in different filters. When we
adopted $E(B-V)=0.1$, the mass of VDB0-B195D obtained based on the magnitudes
in different filters is the same within 1$\sigma$. We list these estimates in
Table 7. From Table 5, we know that the mass of VDB0-B195D obtained in paper
is between $(1.1-1.6)\times 10^{5}~{}M_{\odot}$ when the reddening value is
adopted to be $(B-V)=0.2$.

Figure 4.— Best-fitting, integrated theoretical galev SEDs compared to the
intrinsic SED of VDB0-B195D. The photometric measurements are shown as 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 used a distance modulus of
$(m-M)_{0}=24.47$ mag (785 kpc) for M31 (McConnachie et al., 2005) to
calculate the absolute magnitudes.

##  4\. Summary and discussion

VDB0-B195D was previously shown to be a massive cluster based on HST/WFPC2
observations. Its color is extremely blue and it is very bright, particularly
in blue bands. In addition, VDB0-B195D is an extended object, and most
previous photometric measurements did not include its full flux distribution
(see Perina et al., 2009, for details).

In this paper, we obtained the cluster’s SED in the 15 BATC intermediate-band
filters. We subsequently determined its age and mass by comparing our
multicolor photometry with theoretical stellar population synthesis models.
Our multicolor photometric data consist of 15 intermediate-band filters
obtained in this paper, and broad-band $BVRI$ and 2MASS $JHK_{\rm s}$ from
Perina et al. (2009), covering a wavelength range from 3000 to 20,000 Å. Our
results show that VDB0-B195D is a genuine YMC in M31.

To understand the real nature of the BLCCs, Perina et al. (2009, 2010)
performed an HST imaging survey of 20 BLCCs in M31’s disk. As a test case,
Perina et al. (2009) presented details of the data-reduction pipeline that
will be applied to all survey data and describe its application to VDB0-B195D.
They estimated the object’s age, by comparison of the observed CMD with
theoretical isochrones from Girardi et al. (2002), at $\simeq 25$ Myr. In
addition, they constrained realistic upper and lower limits to the cluster’s
age, independent of the adopted metallicity, within the relatively narrow
range from 12 to 63 Myr. Using Maraston’s SSP models of solar metallicity
(Maraston, 1998, 2005), Salpeter (1955) and Kroupa (2001) IMFs, and
photometric values in the $V$ and 2MASS $J$, $H$, and $K_{\rm s}$ bands,
Perina et al. (2009) concluded that the mass of VDB0-B195D is $>2.4\times
10^{4}~{}M_{\odot}$, with their best estimates in the range $\simeq(4-9)\times
10^{4}~{}M_{\odot}$.

Caldwell et al. (2009) presented an updated catalog of 1300 objects in M31,
including spectroscopic and imaging surveys, based on images from either the
LGGS or the DSS and spectra taken with the Hectospec fiber positioner and
spectrograph on the 6.5 m MMT. They derived ages and reddening values for 140
young clusters by comparing their observed spectra with model spectra from the
Starburst99 SSP suite (Leitherer et al., 1999). The results show that these
clusters are less than 2 Gyr old, while most have ages between $10^{8}$ and
$10^{9}$ yr (the age of VDB0-B195D they derive is $\log{\rm age/yr}=7.6$). In
addition, Caldwell et al. (2009) also estimated the masses of these young
clusters using $V$-band photometry and model mass-to-light ratios (Leitherer
et al., 1999) corresponding to the derived spectroscopic ages. This resulted
in masses ranging from $2.5\times 10^{2}$ to $1.5\times 10^{5}~{}M_{\odot}$.
The mass of VDB0-B195D obtained by Caldwell et al. (2009) is $\log M_{\rm
cl}/M_{\odot}=5.1$ (no uncertainty quoted).

We compare the various age and mass estimates of VDB0-B195D in Table 6\. Our
newly obtained age is older than the estimates of both Perina et al. (2009)
and Caldwell et al. (2009), while the mass obtained in this paper is higher
than the estimate of Perina et al. (2009) and consistent with the
determination of Caldwell et al. (2009). However, our results are in agreement
with those of both Perina et al. (2009) and Caldwell et al. (2009) within
$3\sigma$. The age and mass obtained in this paper confirms that VDB0-B195D is
genuinely a YMC in M31.

As we know, SSP models describe a very special case of a continuous
distribution of stellar mass (or light) along isochrones. This is well
approximated by clusters with masses larger than $10^{6}~{}M_{\odot}$. Also,
for cluster masses of about $10^{5}~{}M_{\odot}$, SSP models can probably
still be applied since a systematic difference between SSP models and
observations should, on average, be smaller than 0.05 mag for clusters older
than 10 Myr (see Fig. 3 in Piskunov et al. 2009). However, from the results of
this paper, we may conclude that, probably, a formal fitting of SSP models to
observed SEDs cannot be used without caution even for relatively massive (or
apparently massive) clusters, and it is highly doubtful that this approach can
be applied in a routine work providing accurate cluster parameters. The
relative accuracy of 10% for age and 20% found for the mass of VDB0-B195D
seems to be rather formal and not very confident. In addition, observational
star clusters’ SEDs are affected by reddening, an effect that is also
difficult to separate from the combined effects of age and metallicity
(Calzetti 1997; Vazdekis et al. 1997; Origlia et al. 1999). Only the
metallicity and reddening are derived accurately (and, ideally,
independently), these degeneracies are largely (if not entirely) reduced, and
ages can then also be estimated accurately based on a comparison of multicolor
photometry spanning a significant wavelength range (de Grijs et al. 2003b;
Anders et al. 2004) with theoretical stellar population synthesis models. It
is true that the discrepancy between our observations and the best-fitting
model is great, and the mass of VDB0-B195D obtained based on the magnitudes in
different filters is very different. However, when we adopt a smaller
reddening value, the results improve greatly. So, we conclude that the actual
reddening value of VDB0-B195D may be smaller than $E(B-V)=0.2$.

We are indebted to the anonymous referee for very carefully reading our
manuscript, and for many thoughtful comments and insightful suggestions that
improved this paper significantly. We are grateful to Dr. Richard de Grijs for
the help in terms of scientific input and proofreading. This work was
supported by the Chinese National Natural Science Foundation (grants 10873016,
10633020, 10603006, and 10803007) and by the National Basic Research Program
of China (973 Program), No. 2007CB815403.

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Table 1BATC photometry of the M31 YMC VDB0-B195D. Filter | Central wavelength | Bandwidth | Number of images | Exposure time | Magnitude
---|---|---|---|---|---
| (Å) | (Å) |  | (hours) |
$a$ | 3360 | 222 | 6 | 2.0 | $15.51\pm 0.14$
$b$ | 3890 | 187 | 6 | 2.0 | $14.73\pm 0.11$
$c$ | 4210 | 185 | 3 | 0.8 | $14.70\pm 0.07$
$d$ | 4550 | 222 | 3 | 1.0 | $14.49\pm 0.10$
$e$ | 4920 | 225 | 3 | 1.0 | $14.67\pm 0.05$
$f$ | 5270 | 211 | 3 | 1.0 | $14.65\pm 0.05$
$g$ | 5795 | 176 | 3 | 1.0 | $14.59\pm 0.04$
$h$ | 6075 | 190 | 3 | 1.0 | $14.56\pm 0.02$
$i$ | 6660 | 312 | 3 | 1.0 | $14.50\pm 0.02$
$j$ | 7050 | 121 | 5 | 1.7 | $14.51\pm 0.06$
$k$ | 7490 | 125 | 3 | 1.0 | $14.47\pm 0.05$
$m$ | 8020 | 179 | 3 | 1.0 | $14.43\pm 0.07$
$n$ | 8480 | 152 | 6 | 2.0 | $14.49\pm 0.05$
$o$ | 9190 | 194 | 6 | 2.0 | $14.49\pm 0.05$
$p$ | 9745 | 188 | 6 | 2.0 | $14.50\pm 0.05$
Table 2Comparison of broad-band photometry of VDB0-B195D.

Filter | ${\rm Mag}^{a}$ | ${\rm Mag}^{b}$ | ${\rm Mag}^{c}$ | ${\rm Mag}^{d}$ | ${\rm Mag}^{e}$ | ${\rm Mag}^{f}$
---|---|---|---|---|---|---
$U$ | 14.66 | 15.11 | $15.12\pm 0.012$ | $14.97\pm 0.01$ | 15.110 | 15.140
$B$ | 15.14 | 15.39 | $15.49\pm 0.010$ | $15.31\pm 0.01$ | 15.410 | 15.510
$V$ | 14.94 | 15.19 | $15.32\pm 0.013$ | $15.06\pm 0.01$ | 15.190 | 15.280
$R$ |  | 15.27 |  |  | 14.920 |

avan den Bergh (1969); bBattistini et al. (1987); cKing & Lupton (1991),
uncertainties are the median uncertainties in the mean for all sample cluster
measurements; dSharov et al. (1995); ePhotometry from Galleti et al. (2004),
based on Battistini et al. (1987); fPhotometry from Galleti et al. (2004),
based on Sharov et al. (1995).

Table 3Recently determined photometry for VDB0-B195D.

Filter | ${\rm Mag}^{a}$ | ${\rm Mag}^{b}$ | ${\rm Mag}^{c}$
---|---|---|---
$U$ |  |  | $14.37\pm 0.22$
$B$ | $14.94\pm 0.09$ |  | $14.62\pm 0.13$
$V$ | $14.67\pm 0.05$ |  | $14.67\pm 0.05$
$R$ | $14.45\pm 0.11$ |  | $14.60\pm 0.06$
$I$ | $14.01\pm 0.11$ |  | $14.19\pm 0.10$
$J$ | $13.26\pm 0.07$ | $13.78\pm 0.03$ |
$H$ | $12.76\pm 0.12$ | $13.15\pm 0.04$ |
$K_{s}$ | $12.77\pm 0.15$ | $12.96\pm 0.03$ |

aPerina et al. (2009); bGalleti et al. (2004); cThis paper.

Table 4Age estimate of VDB0-B195D based on the the galev models. Age | log (Age) | $\chi_{\nu}^{2}({\rm min})$
---|---|---
(Myr) | [yr] | (per degree of freedom)
$60.0\pm 8.0$ | $7.78\pm 0.05$ | 2.2
Table 5Mass estimates (and uncertainties) of VDB0-B195D based on the galev models. $B$ | $V$ | $R$ | $I$ | $J$ | $H$ | $K_{\rm s}$
---|---|---|---|---|---|---
|  |  | Mass $(10^{5}~{}M_{\odot})$ |  |  |
$1.6\pm 0.18$ | $1.6\pm 0.13$ | $1.4\pm 0.17$ | $1.4\pm 0.16$ | $1.4\pm 0.13$ | $1.3\pm 0.16$ | $1.1\pm 0.17$
Table 6Comparison of age and mass estimates of VDB0-B195D.

Age1 | Age2 | log (Age)3 | log (Age)2 | Mass1 | Mass2 | log (Mass)3 | log (Mass)2
---|---|---|---|---|---|---|---
(Myr) | (Myr) | [yr] | [yr] | ($10^{4}~{}M_{\odot}$) | ($10^{5}~{}M_{\odot}$) | [$M_{\odot}$] | [$M_{\odot}$]
25 | $60.0\pm 8.0$ | 7.6 | $7.78\pm 0.05$ | $4-9$ | $1.1-1.6$ | $5.1$ | $5.0-5.2$

1Perina et al. (2009); 2This paper; 3Caldwell et al. (2009).

Table 7Mass estimates (and uncertainties) of VDB0-B195D based on the galev models with $E(B-V)=0.1$. $B$ | $V$ | $R$ | $I$ | $J$ | $H$ | $K_{\rm s}$
---|---|---|---|---|---|---
|  |  | Mass $(10^{5}~{}M_{\odot})$ |  |  |
$1.1\pm 0.12$ | $1.2\pm 0.10$ | $1.1\pm 0.14$ | $1.2\pm 0.14$ | $1.3\pm 0.12$ | $1.2\pm 0.16$ | $1.1\pm 0.17$