\section{Introduction}}

   In conventional thermodynamics, phase transition is an important subject -- it explains how a substance undergoes a change from one phase to another. For black holes, the investigation of phase transition was started long ago by Davis \cite{Davies:1978mf} and was taken up later in a more rigorous way by Hawking and Page \cite{Hawking:1982dh}. Over the years, many different ways have been proposed to characterize and explore the different phases of a black hole. Recently two approaches attracted a lot of attention. In one approach, one mainly looks for the divergence of specific heat and inverse of isothermal compressibility \cite{Banerjee1}--\cite{Lala:2012jp}. Here black hole mass is treated as internal energy and cosmological constant ($\Lambda$), if it appears, is taken as a constant number. In the other approach, which is confined to AdS black holes, mass is treated as enthalpy and $\Lambda$ is regarded as a pressure term \cite{Dolan:2010ha,Dolan:2011xt}.

   An interesting feature of the second approach is that the thermodynamical variables of the black hole satisfy a van der Waal like equation of state and the critical exponents are identical to those of standard van der Waals system. This was first shown for charged $AdS$ black hole \cite{Kubiznak:2012wp} and then for higher dimensional charged and rotating black hole \cite{Gunasekaran:2012dq}. Later a similar study was done for charged topological black hole \cite{Zhao:2013oza}, quantum corrected black hole \cite{Ma:2014vxa}, dilaton black hole \cite{Dehghani:2014caa,xo}, Gauss-Bonnet black hole \cite{Cai:2013qga,Zou:2014mha} and also for Lovelock gravity \cite{Dolan:2014vba}-\cite{Xu:2014tja}, quasitopological gravity \cite{Mann} and conformal gravity \cite{Xu:2014kwa}. This extended phase space approach is also found to be quite successful in equilibrium state space geometry \cite{Hendi:2015hoa,Hendi:2016njy}.

   A remarkable feature of these studies \cite{Kubiznak:2012wp}-\cite{xo} and similar other studies \cite{Zou:2013owa}-\cite{Azreg-Ainou:2014twa} (for review, see \cite{Altamirano:2014tva,Kubiznak:2016qmn}) is that the values of the critical exponents are independent of the metric if one restricts oneself to any one of the two approaches. Thus there is a clear signature of universality with the universality class being characterized by the approach one choses to use. It is only too natural to ask: what makes them universal?

    Very recently we provided an method of exploring the critical exponents in the context of the first approach with a minimal amount of information about the metric  \cite{Mandal:2016anc}. Here black hole satisfies usual first law of thermodynamics, its mass is interpreted as internal energy and $\Lambda$ is treated as a pure constant having no significant role to play. We demonstrated how it is possible to extract the critical exponents with just a single assumption -- a phase transition point exists. It turned out that, this assumption makes the whole analysis very easy and transparent. In this paper our goal is the same but this time our study is addressed to the second approach. Here, we want to treat black hole mass as enthalpy so that $\Lambda$ can be treated as an intensive variable (i.e., a pressure-like term).

    Reassuringly, using this assumption, we have been successful in calculating the critical exponents elegantly. This satisfactorily complements our previous work \cite{Mandal:2016anc}.

  Let us now mention the things we achieved here. Present analysis clearly shows that if there is a van der Waal type phase transition for a set of black holes, the values of critical exponents must be same for that set and these values can be calculated without any further assumption about the spacetime. Although people intuitively expects this result, it has not been shown explicitly earlier. As already mentioned, before the calculation of critical exponents researchers give significant effort to find a spacetime which exhibits liquid-vapour type of phase transition. Our work shows that once this type of spacetime is found, calculation of critical exponents for that spacetime is not really necessary. In this sense, present paper nicely complement the existing literature.

   Before going into the main analysis, let us mention the essential inputs that have been used to achieve our goal. Since the general structure of the Smarr formula and the first law of thermodynamics are very much universal, they are considered to be valid in all black holes. In addition to this we also assume that there is a van der Waal like critical point about which two phases of black holes exist. These simple ingredients turn out to be adequate for our purpose.

   {\it Notations}: $C_V$ and $K_T$ denote the specific heat at constant volume ($V$) and isothermal compressibility at constant temperature ($T$), respectively. $P$ stands for the pressure of the system. The critical values of pressure, temperature and volume are specified by $P_c$, $T_c$ and $V_c$. We mark order parameter, the difference between the volumes of two phases, by $\eta$.

\vskip 2mm
\noindent
{\section{Critical Phenomena: a unified picture}}
  The critical exponents ($\alpha, \beta, \gamma, \delta$) for a van der Waals thermodynamic system are defined as \cite{book}
\begin{eqnarray}
&&C_v \sim |t|^{-\alpha}~;
\nonumber
\\
&& \eta \sim v_l-v_s\sim |t|^{\beta}~;
\nonumber
\\
&& K_T = -\frac{1}{v}\Big(\frac{\partial v}{\partial P}\Big)_T \sim |t|^{-\gamma}~;
\nonumber
\\
&&P-P_c \sim |v-v_c|^{\delta}~;
\label{exponent}
\end{eqnarray}
where $t=T/T_c-1$. Here $v$ stands for specific volume which for a gaseous system is defined as volume per molecules of the system \cite{book}. Usually, every quantity is calculated in terms of the specific volume. $l$ and $s$ in the subscript refer to the two phases of the system; e.g. in the case of normal thermodynamics, $l$ and $s$ may be vapour and liquid phases.
The critical point is a point of inflection, denoted by two conditions $(\partial P/\partial v)_T = 0 = (\partial^2P/\partial v^2)_T$ where $P$ is expressed as a function of temperature and specific volume.
For the AdS black hole case, it has been observed that the specific volume is a function of $r_+$, the location of event horizon (e.g. see \cite{Kubiznak:2012wp} for AdS-RN black hole and for other cases, refs. can be availed from \cite{Altamirano:2014tva,Kubiznak:2016qmn}). On the other hand, as we shall see later, thermodynamic volume $V$, in general, is also a function of $r_+$ only {\footnote{In general, the thermodynamic volume ($V$) is a function of $r_+$, angular momentum and charges of the black hole \cite{Cvetic:2010jb}. We shall explain later that the whole analysis for Van der Waals like phase transition happens for fixed values of angular momentum and charges and so, in principle, we can take $V$ as a function of $r_+$ only.}}.  Therefore it is obvious that $P$ can also be expressed as function of $T$ and $V$. Now to find the critical exponent $\alpha$ one needs to find the entropy. This is done by taking the derivative of Helmholtz free energy (expressed in terms of $T$ and $r_+$), with respect to $T$, keeping $r_+$ constant. On the other hand, for other exponents, $P=P(T,v)$ has to be expanded around the critical point. One can verify that the expansion for $P (T,v)$ will have the same form as that for $P(T,V)$ where the variable $(v/v_c)-1$ plays the same role as $V/V_c - 1$. Hence, to find the critical exponents, we can work with $V$ instead of $v$ as well. So in our analysis, at the critical point, we impose the conditions  as
\begin{equation}
\frac{\partial P}{\partial V}=0 = \frac{\partial^2P}{\partial V^2}~.
\end{equation}
Here in taking the derivatives, $T$ is kept constant and $P$ is expressed as a function of $V$ and $T$. The same was also mentioned and adopted in the original work \cite{Kubiznak:2012wp} (see footnote $5$ of this paper).
We shall use these conditions together with the basic definitions to find the values of the critical exponents for a AdS black hole.

  To start with, let us consider the two general results which are universally established: one is the general form of Smarr forumla and the other one is the first law of thermodynamics.
  The Smarr formula for a $D$-dimensional AdS black hole, in general, can be taken as
\begin{equation}
M=f_1(D)TS-f_2(D)PV + f_3(D)XY~,
\label{Smarr}
\end{equation}
where $M$, $T$ and $S$ are mass, temperature and entropy of the horizon while $P$ and $V$ are thermodynamic pressure and volume, respectively. The values of the unknown functions $f_1, f_2, f_3$ depend on the spacetime dimensions only. For example, in $D=4$, one has $f_1=2$, $f_2=2$ and $f_3=1$. We shall see that the explicit forms of them are not required for our main purpose. In the above equation, $X$ stands for electric potential $\Phi$ or angular velocity $\Omega$ or both whereas $Y$ refers to charge $Q$ or angular momentum $J$ or both for a black hole. The corresponding first law of thermodynamics is given by
\begin{equation}
dM = TdS+XdY+VdP~.
\label{law1}
\end{equation}
Note that, in the above expression $M$ is not the energy and the present form is not identical to the standard form of first law\footnote{Discussion on ``non--standard'' first law of thermodynamics and Smarr formula due to $\Lambda$ can be found in  \cite{Dolan:2010ha}, \cite{Dolan:2011xt}, \cite{Caldarelli:1999xj}-\cite{Dolan:2011jm}.}. But this can be expressed in the standard form:
\begin{equation}
dE=TdS+XdY-PdV
\end{equation}
if one identifies the energy of the black hole as
\begin{equation}
E=M-PV~.
\label{energy}
\end{equation}
Thus $M$ is the enthalpy of black hole. Using (\ref{Smarr}), the Helmohtz free energy ($F$) of the system is found to be
\begin{equation}
F=E-TS=(f_1-1)TS - (f_2+1)PV+f_3XY~.
\label{Hfree}
\end{equation}

   Now for a black hole, if one keeps $Y$ constant then mass ($M$) has to be a function of pressure ($P$) and location of the horizon ($r_+$); i.e. $M=M(r_+,P)$. The reason is as follows. The location of the horizon is defined by the vanishing of the metric coefficient and in general this coefficient is a function of $M$, $Y$ and $P$ (as $P=-\Lambda/8\pi$) for a AdS black hole. Therefore, $r_+=r_+(M,Y,P)$ and hence for a fixed value of $Y$, $M$ is a function of $r_+$ and $P$. On the other hand, we have  $V=V(r_+)$ and in general $X=X(r_+)$. Here it must be pointed out that, in general $V$ depends on not only $r_+$ but also on the angular momentum or the existing charges of the black hole; i.e. on $Y$ (For example, see \cite{Cvetic:2010jb}). But since the analysis is done for fixed value of $Y$, without any loss of generality, we can take $V$ as function of $r_+$ only. Of course, if there is any deviation from it, this has to be treated separately. The first law of thermodynamics (\ref{law1}) shows that $S$ can be expressed as a function of the black hole parameters; i.e. $S = S(M,Y,P)$. So by the above argument we have $S=S(r_+,P)$. Therefore the Smarr formula (\ref{Smarr}) shows that for a fixed value of $Y$, one finds the solution for pressure from the equation
   \begin{equation}
    P = \frac{f_1TS(r_+,P)}{f_2V(r_+)}-\frac{M(r_+,P)}{f_2V(r_+)}+\frac{f_3X(r_+)Y}{f_2V(r_+)}~,
    \label{P}
   \end{equation}
which has to be function of both temperature and horizon radius; i.e. $P=P(T,r_+)$. Hence the free energy (\ref{Hfree}) as well as $S$ are in general  functions of both $T$ and $r_+$. Since entropy can be calculated from the relation $S=-(\partial F/\partial T)_V$ ($\equiv$ $-(\partial F/\partial T)_{r_+}$ as $V=V(r_+)$), the evaluated value can be, in general, function of both temperature and horizon radius.  Now we find from (\ref{Hfree})
\begin{eqnarray}
-S=\Big(\frac{\partial F}{\partial T}\Big)_{r_+} &=& (f_1-1)S + (f_1-1) T \Big(\frac{\partial S}{\partial T}\Big)_{r_+}
\nonumber
\\
&-&(f_2+1)V(r_+)\Big(\frac{\partial P}{\partial T}\Big)_{r_+}~.
\label{PF}
\end{eqnarray}
This gives,
\begin{eqnarray}
0 &=& f_1S + (f_1-1) T \Big(\frac{\partial S}{\partial T}\Big)_{r_+}
\nonumber
\\
&-&(f_2+1)V(r_+)\Big(\frac{\partial P}{\partial T}\Big)_{r_+}~.
\label{PF1}
\end{eqnarray}

For a large class of black holes, it is found that pressure is a linear function of temperature, i.e.
\begin{eqnarray}
P=p_0(r_+)+Tp_1(r_+)~.
\label{PTR}
\end{eqnarray}
This, we want to take as an input for the AdS black holes because it has been observed that those which exhibit van der Waals like behaviour, have the above feature when $P$ is expressed as a function of $T$ and $r_+$. The same also happens if one looks at the van der Waals equation of state (e.g. see \cite{book,Kubiznak:2012wp}). Another way to see it is as follows. In later discussion we shall see that near the critical point, $P$ to the leading order, is a linear function of $t=(T/T_c)-1$. Since all the critical exponents are determined by the analysis very near to the critical point, there is no loss of generality to take an expression like (\ref{PTR}).
This, when substituted in (\ref{PF1}), gives
\begin{eqnarray}
f_1S+ (f_1-1) T \Big(\frac{\partial S}{\partial T}\Big)_{r_+}-f_1a(r_+)=0
\end{eqnarray}
where $a(r_+)=\frac{1}{f_1}(f_2+1)V(r_+)p_1(r_+)$. The solution of above first order differential equation is
\begin{eqnarray}
S=a(r_+)+C(r_+)T^{-\frac{f_1}{f_1-1}}
\end{eqnarray}
where $C(r_+)$ is an integration constant. Since $f_1/(f_1-1)$ is always positive for $D\geq 4$, the last term of the R.H.S. diverges as $T\rightarrow 0$. So to be consistent with the third law of thermodynamics, $C(r_+)$ must be zero. This leads to $(\partial S/\partial T)_V =(\partial S/\partial T)_{r_+}= 0$;
i.e. the specific heat at constant volume $C_V = T(\partial S/\partial T)_{V} = 0$ and hence by (\ref{exponent}) one finds the value of the critical exponent $\alpha=0$.

  Next we shall find the other critical exponents. Remember that the pressure here is in general a function of both $T$ and $r_+$. Now since thermodynamic volume $V$ is a function of $r_+$ only, we take $P$ as function of $T$ and $V$ for our purpose. Let us expand $P(T,V)$ around the critical values $T_c$ and $V_c$:
  \begin{eqnarray}
  P = &&P_c + \Big[\Big(\frac{\partial P}{\partial T}\Big)_{V}\Big]_c (T-T_c)
  \nonumber
  \\
  &&+ \frac{1}{2!} \Big[\Big(\frac{\partial^2 P}{\partial^2 T}\Big)_{V}\Big]_c (T-T_c)^2
  \nonumber
  \\
  &&+ \Big[\Big(\frac{\partial^2P}{\partial T\partial V}\Big)\Big]_c(T-T_c)(V-V_c)
  \nonumber
  \\
  &&+\frac{1}{3!}\Big[\Big(\frac{\partial^3P}{\partial V^3}\Big)_T\Big]_c (V-V_c)^3+\dots~.
  \end{eqnarray}
In the above expression, we have used the fact that at the critical point $(\partial P/\partial V)_c=0=(\partial^2P/\partial V^2)_c$. Defining two new variables $t=T/T_c-1$ and $\omega = V/V_c-1$, and above expression is written as,\begin{equation}
P=P_c+Rt+Bt\omega+D\omega^3+Kt^2;
\label{Pexpansion}
\end{equation}
where $R, B, D,K$, etc. are constants calculated from the derivatives at the critical point. Here we have ignored the other higher order terms since they are very small.

   Now for the usual van der Waal's system, below critical point there is a portion between the vapor and liquid phases where both of them coexists which is either unstable or meta-stable and therefore experimentally this exact form of the isotherm is not observed. Hence the ends of the vapor and liquid phases are usually connected by a straight line, parallel to the volume axis, to match with the obtained isotherm for a substance by direct experiment. Then one obtains two portions of van der Waal's isotherm and the straight line is drawn in a such a way that the area of both of them are equal so that one has $\oint VdP = 0$. This is known as Maxwell's area law. Since we are treating the AdS black holes similar to van der Waal's system, this law can be used here also \cite{Wei:2014qwa,Belhaj:2014eha}. In addition, by keeping the analogy between the volumes of vapor and liquid phases, it will be denoted that $\omega_l$ and $\omega_s$ correspond to the large and small volumes of black hole in two different phases. In between these two we have mixture of both of them.
   Since we are interested in the isotherms, to use Maxwell's area law we find $dP = (Bt+3D\omega^2)d\omega$ for constant $t$, which yields
\begin{equation}
\int_{\omega_l}^{\omega_s}\omega(Bt+3D\omega^2)d\omega + \int_{\omega_l}^{\omega_s}(Bt+3D\omega^2)d\omega =0~.
\label{omega}
\end{equation}
Next note that for the usual system, the end point of vapor and the staring point of liquid have same pressure. Similarly, here also the pressure does not change, i.e. $P_l=P_s$, and then (\ref{Pexpansion}) implies
\begin{equation}
Bt(\omega_l-\omega_s) + D(\omega_l^3-\omega_s^3)=0~.
\label{first}
\end{equation}
So the second integral of (\ref{omega}) vanishes. Therefore it reduces to
\begin{equation}
Bt(\omega_l^2-\omega_s^2)+\frac{3D}{2}(\omega_l^4-\omega_s^4) = 0~.
\label{second}
\end{equation}
One can now easily find the non-trivial solutions of the above two equations. These are $\omega_l =(-Bt/D)^{1/2}$ and $\omega_s=-(-Bt/D)^{1/2}$. Therefore we find
\begin{equation}
\eta \sim V_l-V_s = (\omega_l-\omega_s)V_c\sim |t|^{1/2}~,
\end{equation}
which yields $\beta=1/2$.

     To find $\gamma$ we need to calculate $K_T$, given in (\ref{exponent}). So we first find $(\partial P/\partial V)_T$ from (\ref{Pexpansion}). Upto first (leading) order it is given by
     \begin{equation}
     \Big(\frac{\partial P}{\partial V}\Big)_T \simeq \frac{B}{V_c}t~,
     \end{equation}
where one needed to use $\partial \omega/\partial V = 1/V_c$. Therefore the value of $K_T$ near the critical point is
\begin{equation}
K_T \simeq \frac{1}{Bt}\sim t^{-1}~.
\end{equation}
This implies $\gamma=1$. Next for $T=T_c$, the expression (\ref{Pexpansion}) for pressure yields
\begin{equation}
P-P_c\sim \omega^3\sim (V-V_c)^3~;
\end{equation}
i.e. the value of the critical exponent is $\delta=3$.
It must be pointed out that the derived critical exponents satisfy the following scaling laws:
\begin{eqnarray}
&&\alpha+2\beta+\gamma=2;
\nonumber
\\
&&\gamma=\beta(\delta-1)~.
\label{scaling}
\end{eqnarray}
As is well known, these scaling laws are universal in nature. Their thermodynamic analysis can be found in \cite{Stanley1}.

    Note that in our analysis we consider the thermodynamic volume as function of horizon radius only. In general, this is not always true. For example when black hole has some hair, $V$ is not just the volume inside the black hole but also the integral of the scalar potential and hence it depends on both $r_+$ and the charges corresponding to the hairs (for instance, see \cite{Cvetic:2010jb,Caceres:2015vsa}). Also for some black hole solutions, it may be possible that the entropy is not only a function of $r_+$ and $T$ but also function of other parameters of the spacetime.  There is a hairy case \cite{Hennigar:2015wxa} in which this happens though volume is function of $r_+$ alone. Then it might come in mind that the present general approach incorporates only those cases which are consistent with the {\it no hair theorem}. In this regard, remember that in our present analysis, $Y$ is kept constant; i.e. we are taking the canonical ensemble. Then if there is any hairy charge that should not be varied also and hence $V$ will be a function of $r_+$ alone. In this situation, such solutions behaves as Van der-Waal's system. Of course for an analysis in the grand canonical ensemble, one needs to be careful and must deal them in a separate way.
There is another example \cite{Sadeghi:2016dvc} where $S=S(r_+,T)$ corresponding to a solution with logarithmic correction to entropy. This can be dealt in this present approach.

\vskip 2mm
\noindent
{\section{Conclusions}}
Using the tools of thermodynamics, phase transition of completely dissimilar systems like chemical, magnetic, hydrodynamic etc. has been studied thoroughly. Black hole phase transition is a relatively new observation which deserves careful study. Though the first order phase transition from non-extremal to extremal black hole is known for some time, there are various other types of phase transition. A new type of phase transition has been found recently where cosmological constant ($\Lambda$) is treated as dynamical variable (instead of a constant) equivalent to pressure of hydrodynamic system \cite{Kubiznak:2012wp}. In this interpretation, phase transition of different black holes has been shown to be quite analogous to van der Waal's system. Interestingly critical exponents found from different metrics are same. This naturally deserves some explanation.

It is well known that critical exponents of quite different systems can be same. But for black holes, this point is not well appreciated. In our previous work \cite{Mandal:2016anc} we showed that, starting from few very general assumptions about the black hole spacetime, critical exponents can be calculated. In that work we did not treat $\Lambda$ as a dynamical variable. Present paper is a continuation of our previous work. Here we take a different interpretation of $\Lambda$. In this work we showed that, the single assumption of existence of liquid-vapour type phase transition for a spacetime determines the values of critical exponents. No other assumption regarding the variables or dimensions of spacetime is necessary.

Till now values of critical exponents are obtained first by finding a suitable metric which exhibits this type of phase transition and then by explicit calculation using the definitions of exponents. Though expected, it remained unexplained why these values are same for different $AdS$ black holes. In this paper, we give a completely satisfactory answer to this question. Thus our present work nicely complements the existing scientific works in this field.

It may be mentioned that, the analysis shows that the values of the critical exponents are very much universal in nature and also they are independent of spacetime dimensions. In literature, different critical exponents have been found very recently \cite{Mann}, usually known as ``nonstandard'' critical exponents, for black hole solutions in higher curvature gravity theory. These are not due to the analysis around the standard critical point; rather around {\it isolated} one. In that respect our analysis is different from them. Of course it would be very much worthwhile to look at such non-standard phase transition.

\vskip 4mm
{\section*{Acknowledgments}}
We thank to the anonymous referees for pointing out some important issues which helped to improve the earlier version. We also thank Juan F. Pedraza for making several interesting and valuable comments on the first version of our paper.
The research of one of the authors (BRM) is supported by a START-UP RESEARCH GRANT (No. SG/PHY/P/BRM/01) from Indian Institute of Technology
Guwahati, India.


\section{Introduction}

Low Surface Brightness (LSB) galaxies are the most unevolved class of galaxies
in our nearby Universe (Impey \& Bothun \cite{ImpeyBothun1997}). They are
optically dim with diffuse stellar disks (Auld et al. \cite{auld.etal.2006}),
massive HI gas disks (O'Neil et al. \cite{oneil.etal.2004};
Matthews, van Driel, Monnier-Ragaigne \cite{matthews.etal.2001}) but have low star formation rates
compared to regular spiral galaxies (McGaugh \cite{McGaugh.1994}). They are halo dominated galaxies
(de Blok \& McGaugh \cite{deblok.etal.1996}; Kuzio de Naray, McGaugh \& de Blok \cite{KuziodeNaray.etal.2008};
Coccato et al. \cite{Coccato.etal.2008}); this may account for the weak spiral arms and small bar perturbations
observed in these galaxies (Mihos, de Blok \& McGaugh \cite{Mihos.etal.1997};
Mayer \& Wadsley \cite{Mayer.Wadsley.2004}). Although the most commonly observed LSB galaxies are
the dwarf LSB galaxies (Sabatini et al. \cite{Sabatini.etal.2003}), a significant fraction of
LSB galaxies are
large spirals having prominent bulges (Beijersbergen, de Blok \& van der Hulst \cite{Beijersbergen.etal.1999}).
These giant LSB (GLSB) galaxies have extended LSB disks that are poor in star formation
and dust (Rahman et al. \cite{Rahman.etal.2007}; Hinz et al. \cite{Hinz.etal.2007}).
The bulge dominated GLSB galaxies often show
AGN activity (Schombert \cite{Schombert.1998}; Das et al. \cite{Das.etal.2009}).

Even though the optical properties of LSB galaxies have been investigated in great depth, not much
is known about their molecular gas content. This is important as a knowledge of the cold gas distribution in
LSB galaxies will help us understand star formation processes in these galaxies. Surveys of LSB galaxies
show that they have fairly massive HI disks that may be more than twice the size of the optical disk
(de Blok et al. \cite{deblok.etal.1996}; Pickering et al. \cite{pickering.etal.1997};
Das et al. \cite{Das.etal.2007}). In this paper we examine the molecular gas distribution in a GLSB galaxy
and see how it relates to the overall star formation in its' disk. Molecular gas has been detected in only
a handful of LSB galaxies (O'Neil, Hofner \& Schinnerer \cite{oneil.etal.2000};
Matthews \& Gao \cite{matthews.gao.2001}; O'Neil, Schinnerer \& Hofner \cite{oneil.etal.2003};
Matthews et al. \cite{matthews.etal.2005}; Das et al. \cite{Das.etal.2006}). In most cases the galaxies
were large spirals with extended optically dim disks. The low detection rate of
molecular emission from LSB galaxies is probably due to several factors related to the poor star formation
rate in these galaxies (e.g.~de Blok \& van der Hulst \cite{deblok.vanderHulst.1998}); factors such as
the lower dust content, lower metallicity and the lower surface denity of cold, neutral gas in these galaxies.
All of these properties lead to a slower rate of gas cooling and molecule formation.  For the few galaxies
where molecular gas has been detected, not much is known about the gas extent and distribution. Such
information is important if we want to understand star formation and disk evolution in LSB galaxies.

To investigate the molecular gas and star formation in GLSB galaxies we studied the CO distribution in
a galaxy where molecular gas has been detected, F568-6 or Malin~2 as it is widely known
(Das et al. \cite{Das.etal.2006}). It is a nearly face-on GLSB galaxy at a distance of 201~Mpc.
It has a prominent bulge and a very extended LSB disk. Its parameters are summarised in Table~1.
There are several localized star forming regions distributed over its inner disk.
Its metallicity is one third of the solar metallicity in value which is
relatively high for an LSB galaxy (McGaugh \cite{McGaugh.1994}).
The CO observations of Malin~2 were conducted using the HERA instrument mounted on the
30~m IRAM telescope. Our
main aim was to examine the molecular gas distribution; determine its extent,
total gas mass and surface density.

\section{Observations}

During March 2007 we observed the CO(2--1) line in Malin~2 with the HERA beam array
(Schuster et al. \cite{Schuster.etal.2004}) on the IRAM 30m
telescope at a fequency of 220.372~GHz. We specifically used this array as it has a
wide field of view and good sensitivity. HERA is made of 9 receivers
in a $3\times3$ array spaced by $24^{\prime\prime}$ on the sky. The backend used was the
Wideband Line Multiple Autocorrelator (WILMA). The total bandwidth was 930~Hz; it was made up of
465 channels of 2MHz each. The typical system temperatures $T_{sys}$ were in the range 200-250 K for 8
receivers; one receiver had a systematically higher $T_{sys}$ in the range 350-450K. The
mean FWHM of each of the nine beams is $11.7^{\prime\prime}$. As our main goal was to detect
the CO emission line from the
disk of Malin~2, we kept the array fixed on the sky in the standard  pointed mode with the central
beam pointed to the galactic nucleus; the field was tracked to get deep integrations at fixed
positions in the disk. This enabled us to achieve good sensitivity
but was at the cost of undersampling the source.  We obtained deep
observations at nine positions of the galaxy with a total on source integration time of
11 hours (Figure~1). The observations were frequency switched. The intrinsic velocity resolution
was 2.7~km~s$^{-1}$; the data was then smoothed using a Hanning squared function.

We reduced the data using the CLASS software of the GILDAS package \footnote{\url{http://www.iram.fr/IRAMFR/GILDAS}}
by fitting a first order baseline to all spectra within a window going
from -400 to +400\,km\,s$^{-1}$ about the galaxys' systemic velocity of 13830~km~s$^{-1}$. This
window was the same for all nine spectra. The noise
level is not the same for all receivers; the lowest is 0.7mK
and the highest is 1.6 mK in the 10.9 km/s channels. The noise is also not uniform across the band.
Hence we computed the baseline and the noise in the -400 +400 window using emission free channels only.
All the details are given in Table~2. The conversion factor from K to Jy is 8.6~Jy~K$^{-1}$.
To convert from the antenna temperature
scale $T_A^*$ to the main beam temperature $T_{mb}$ we multiplied $T_A^*$ by the factor
$F_{eff}/B_{eff}~=~1.67$ where $F_{eff}$ is the forward beam efficiency and $B_{eff}$ is the
main beam efficiency.

   \begin{table}
      \caption[]{Galaxy Parameters - Malin~2}
         \label{table1}
     $$
         \begin{array}{p{0.35\linewidth}ll}
            \hline
            \noalign{\smallskip}
            Parameter      &  Value & Reference \\
            \noalign{\smallskip}
            \hline
            \noalign{\smallskip}
            Other Names  & F568-6  & NED \\
            Distance & 201~Mpc & NED \\
            Heliocentric Velocity & 13830~km~s^{-1} & NED \\
            Position (epoch 2000) & 10^{h}39^{m}52^{s}.5, +20^{\circ}50^{\prime}49^{\prime\prime} & 2MASS \\
            Size ($D_{25}^{\prime}$) & 1.67 & (a) \\
            Position angle & 70^{\circ} & NED \\
            Inclination   & 38^{\circ} & (a) \\
            HI Linewidth ($W_{20,corr}$) & 637~km~s^{-1} & (b) (c) \\
            HI Mass & 4.2\times10^{10}~M_{\odot} & (b) \\
            Dynamical Mass & 2.5\times10^{12}~M_{\odot} & (b) \\
             (R~$<107$~Kpc) &                           &     \\
            Disk Central Brightness & 22.1~mag~arcsec^{-2} & (b) (d) \\
            Disk Scale Length & 18.8~Kpc & (b) \\
            Total R Band Magnitude & -23.6~mag & (b) \\
            \noalign{\smallskip}
            \hline
         \end{array}
     $$
\begin{list}{}{}
\item[$^{\mathrm{a}}$] Matthews et al. (2001)
\item[$^{\mathrm{b}}$] Pickering et al. (1997)
\item[$^{\mathrm{c}}$] Corrected for inclination
\item[$^{\mathrm{d}}$] Not corrected for extinction

\end{list}
   \end{table}

\section{Results}

We have detected CO(2-1) emission from several positions in the disk as well
as from the center of Malin~2. In
the following paragraphs we present our results and discuss their implications.\\

\noindent
{\bf 1.~CO(2-1) Detections~:~}Figure~2 shows the CO(2-1) emission spectra observed from
nine locations across Malin~2. The offsets from the galaxy center
are indicated in each box. CO(2-1) emission was detected from four out of nine positions
at line intensities above $3\sigma$. The line at (-24, 24) is a a hint of emission rather
than a sure detection. At (0, 0) the line is broad but is a $3\sigma$ detection.
We estimated the line parameters (flux, width and central velocity)
by fitting a gaussian to each spectrum. We determined the noise level for each of the nine
positions. The results are listed in Table~2 and the best fit gaussians are overlaid on the
spectra in Figure~2.

\noindent
{\bf 2.~Comparison with previous CO observations~:~}
Molecular gas has been detected earlier in this galaxy by Das et al.
(2006) with the IRAM 30m telescope and the BIMA array. However, due to a
correlator setup problem the BIMA map was incomplete, it was covering a
velocity range corresponding to CO(1--0) line emission from the east side
of the galaxy only. In addition, the 30m telescope was pointed toward two
directions only at 7$''$ and $\sim$35$''$ east of the nucleus (see
Fig.\,1). Although the source was observed simultaneously at 115\,GHz and
230\,GHz with the 30m telescope, only CO(1--0) emission was detected;
the CO(2--1) line was not detected from either positions.
Our present  observations cover the CO emission over a wider
velocity and spatial extent than these previous observations and
hence give a better idea of the distribution of molecular gas in the inner
disk of Malin 2. These
observations are also the first detections of the CO(2--1) line in Malin 2.
Our detection lies below the detection limit of the older observations of
Das et al. (2006). Their CO(2--1) emission spectrum had a noise (rms)
of 2.6~mK whereas our line detections have a peak of 2  to 3~mK (Figure~2).

Near the galaxy center, where there is overlap with the previous CO
detection,  the CO line
velocities are in the same direction and are similar in shape and in width.
Close to the center of the galaxy the CO(1--0) line as detected by
Das et al. (2006) is broad ($\sim$200\,km\,s$^{-1}$) and asymmetric with the
red side being slightly more prominent than the blue side.
The CO(2-1) line detected at the center of the galaxy by our new HERA
observations is $243\pm76~km~s^{-1}$ wide and at a centroid position
offset by $77\pm34~km~s^{-1}$ relative to the systemic velocity given in Table~1.
Investigating the cause of this offset would require interferometric observations.
As the beams and pointing directions are different (23$''$ for the CO(1--0) line and
11$''$ for the CO(2--1) line) the fluxes cannot be directly compared. But if the molecular
gas were uniformly distributed we would expect a line ratio in temperature scale close in the
range of values found by Braine \& Combes (1992) with the same telescope
i.e., 0.4 to 1.2. We checked this for the HERA central beam whose
pointing direction is the closest to the central pointing of Das et al (2006). A rough comparison
of the brightness in $mK~km~s^{-1}$ gives
$I_{\rm CO(2-1)}/I_{\rm CO(1-0)}$~=~0.43/1.01~=~0.43
(where the CO(2--1) line intensity from the HERA central pointing
is 0.43 and the CO(1--0) intensity from IRAM is 1.01). This value is at the lower end of the line ratio range
for most galaxies but given the uncertainties (e.g.~gas distribution), the value shows that the CO(2--1)
line intensity we measure is consistent with the previous CO(1--0) detection.

\noindent
{\bf 3.~Comparison with HI observations~:~}
High sensitivity HI VLA observations of Malin 2 were presented by Pickering et al. (1997). These reveal the
distribution of the HI gas and its kinematics at a spatial resolution of $\sim$19$''$. The HI disk
is lop-sided, the HI emission being significantly more prominent and extending farther out on
the western side of the nucleus than on the eastern side. The central region is essentialy devoid
of HI gas. Although we have only a partial view of the molecular disk due to our under-sampled data
its properties have similitude with  HI.
Based on the HI velocity field, the centroid velocity of the HI line profiles at the
positions (24,0), (-24,0) and (-24,24), respectively +90, -150 and -110 km\,s$^{-1}$ relative to the
systemic velocity, are in excellent agreement to those for the molecular gas (Tab. 2, col {\it e}).
Hence the HI and molecular gas appear to share a common velocity assymmetry when considering the
positions (24,0) and (-24,0).

The higher spatial and spectral resolutions in CO allow to better constrain
the velocity dispersion of the gaseous component. The CO profile with the
narrowest line width is at (-24,0). Given the spectral response, this leads to
a deconvolved velocity dispersion of 13 km\,s$^{-1}$. It is the best constraint
that we can get for the velocity dispersion of the gas $\sigma_{gas}$ in the disk of
Malin 2 given the facts that {\it a)} it is obtained in a direction close to
the major axis where beam smearing effects due to the rotation of the disk are minimum
and {\it b)} we do not observe the blue-shifted wing or
secondary velocity component that Pickering et al. discovered and had to blank
to determine and analyze the rotation curve.  These authors interpreted this
secondary component as high velocity clouds, gas infalling into
the disk southwest of the nucleus. It is best seen $\sim$20'' southwest of the nucleus
where it is blue-shifted by as much as $\sim$100\,km\,s$^{-1}$. Given the correspondance
with HII regions in that direction and the fact that there is a relatively moderate HI column
density there they suggested that this infall induces star formation
(Wyder et al. \cite{Wyder.etal.2009}).
We do not detect CO emission at (-24,-24) which is only $\sim$6'' further away in the southwest.
It could be that the disk component in that region is depeleted not only in HI but also
in molecular gas. CO observations spatially fully sampled would be required to confirm
this hypothesis.

In Table 2, the maximum CO(2--1) line velocities are reached in the offsets (24,0) and (-24,0) which is
consistent with the
position angle being close to horizontal $\sim70^{\circ}$ (2MASS images; Jarrett et al. \cite{Jarrett.etal.2003}).
We note a velocity asymmetry between the emission at (24,0) and that at (-24,0), which could be due to infalling
gas or a lopsided potential. We can estimate the rotational velocity from these lines.
Assuming an inclination of 38$^{\circ}$ (Matthews et al. \cite{matthews.etal.2001})
it is of the order ($\sim 150/\sin(i)\,km\,s^{-1}$ which implies a dynamical mass within
the 24$''$ (i.e. 23\,kpc) radius of $\sim3\times10^{11}~M_{\odot}$.

The broad line at the center has a width close to twice the velocities at
(24,0) and (-24,0) offsets. This could be due to the rotation curve turning point
being inside the central beam, i.e., inside a radius of 6$''$ (6\,kpc).
This central region which is essentially devoid of HI is where there is the strongest
emission in CO; higher angular resolution CO observations are required to determine the
rotation curve in that region. Would the curve continue to steadily decrease up to
the nucleus that would point to a model where the gas in the central bulge region
is kinematically hot, e.g. because of gas infall to feed the AGN.

\noindent
{\bf 4.~Molecular Gas Mass:~}
We derived the molecular gas surface densities for each line detection and the results are summarised in Table~2.
The column densitites are derived from the line fluxes using
$N(H_{2})~=~X~(1/0.9)~\int~T_{mb}[CO(2-1)]~dv$; where I(2-1)/I(1-0)~=~0.9
(Braine \& Combes \cite{Braine.Combes.1992}) and the $CO$ to $H_{2}$
conversion factor is given by $X=2\times 10^{20}~cm^{-2}(K~km~s^{-1})$
for cool virialized molecular clouds (Dickman et al. \cite{Dickman.etal.1986}). The
value of $X$ is somewhat uncertain for LSB disks, given their low metallicities and our poor
knowledge of the ISM conditions in these galaxies. However, for a first estimate we
used this value to determine the surface densities of molecular gas $\Sigma_{mol}$
in Malin~2 (Col. f Table~2). Assuming the eight directions observed in the disk form a
distribution of surface brightness values which is representative for the disk of
Malin~2 up to a radius of $40^{\prime\prime }$, the mean CO(2--1) brightness
is $\sim0.15~K~km~s^{-1}$.
The central region has a beam averaged surface density that is 3 times higher than
the mean brightness of the disk. The lower limit for $M_{mol}$ is $\sim~4.9\times10^{8}M_{\odot}~$ and is
given by the sum of the four detections in column~{\it f} of Table~2 multiplied by the beam solid angle.
To determine the upper limit for $M_{mol}$, we used the noise
(see Table~2, column~{\it a}) to derive upper limits for the molecular gas surface densities for the regions
where there were no detections including (-24,24) where there is a hint of emission. The upper limit to the
total mass of molecular gas within $R~<~40^{\prime\prime}$
is thus $M_{mol}\sim~8.3\times10^{8}~M_{\odot}$.
If the molecular disk does not extend beyond $40^{\prime\prime}$, then the molecular gas mass
corresponds to $\sim1.2 - 2$\% of the HI gas mass in the galaxy. This is similar to that observed in normal spiral
galaxies that have molecular gas masses typically a few percent of the total HI mass.

\noindent
{\bf 5.~Star Formation threshold~:~}
The molecular gas disk in Malin~2 is extended and relatively massive with a significant amount of
molecular gas but its star formation activity is low compared to normal spirals
(Wyder et al. \cite{Wyder.etal.2009}). For galactic disks star formation appears to be controled by the
onset of gravitational instabilities (Kennicutt \cite{Kennicutt.1989}). A simple single-fluid
Toomre disk stability model predicts threshold densities in agreement with the observations.
This critical density is given by
$\Sigma_{crit}~=~\alpha\kappa\sigma_{gas}/3.36~G$
where $\kappa$ is the epicyclic frequency and $\sigma_{gas}$ the velocity dispersion of the gas.
From a sample of spiral galaxies Kennicutt found $\alpha$ $\simeq$ 0.69.
In his study, to evaluate  $\alpha$,
Kennicutt assumed that all the galaxies in his sample have approximately the same velocity dispersion,
$\sim6 km~s^{-1}$, and that it is independent of the radial distance within a galaxy. This is a good
approximation given what is known for the galaxies of his sample. Hence the prediction is that,
to have disk galaxies such as the LSBs immune to star formation, the gas surface density should
be below this critical density.

Pickering et al. (\cite{pickering.etal.1997}) determined rotation curves for 4 GLSBs. These allow
them to determine the epicyclic
frequency necessary to obtain the radial profiles of $\Sigma_{crit}$. They show that, by assuming that
$\alpha = 0.69$ is valid for these LSB galaxies and that $\sigma_{gas}=10 km~s^{-1}$, the critical
density is, in some regions, close or lower than the observed HI surface density, in spite of the fact
that there is no conspicuious star formation activity there. This leads them to postulate
that $\sigma_{gas}$ must be larger than the assumed value.

In the specific case of Malin 2 Pickering et al. find that the
HI surface density never exceeds the critical density except at the location where HI is
the brightest (this direction is located a half way between the positions (-24,24) and (-24,0)).
When considering the 9 directions listed in table 2 only (0,0) coincides with an H$\alpha$
source. Using the present CO observations we can examine two questions:
{\it 1)} does the gas surface density remain below the critical density
when the contribution from the molecular gas component is taken into account
as did Kennicutt in his analysis? {\it 2)} is $\sigma_{gas} \le 10 km~s^{-1}$ used to
determine  $\Sigma_{crit}$ adapted in the case of giant LSBs such as Malin 2?
Column {\it g} in Table 2 gives the HI surface brightnesses. Comparing
these with the molecular surface densities (col. {\it f}), the molecular gas surface
densities appear to amount to typically 1/2 to 1/3 the HI surface densities,
with the exception for the position (-24,0) where HI largely dominates. Column {\it h} gives
the total gas surface density and in col. {\it i} this quantity relative to the critical
density. By using $\alpha = 0.69$, for internal consistency, the values in col. {\it h} and {\it i}
are determined using the same hypothesis as those adopted by Kennicutt for the conversion
factor X and the term Z which accounts for the elements heavier than H.
For  $\sigma_{gas}$ we adopted the value of $13~km~s^{-1}$
that we measured with good S/N at (-24,0) (sect 3.3). Better constrained than the $10~km~s^{-1}$
adopted by Pickering et al., this value is still conservative given the fact that
it is the lowest velocity dispersion (col. {\it d}) amongst the 5 directions where CO
is detected. Formally, all the values in col.  {\it i} being smaller than unity it could be
concluded that the disk of Malin 2 is sufficiently stable to prevent star formation. However,
the directions (-24,0) and (-24,24) have surface densities close to the threshold preventing
us to consider this as firm solid results. Also
the observed kinematics in western and north-western parts of Malin 2 appears complex because
of high velocity clouds (HVC). It could be that the star formation activity seen in Malin 2
is more related to the interaction of these HVCs with the disk than to growth of density
fluctuations originating from instabilities in the self-graviting rotating disk. A fully sampled
image of the Malin 2 and high signal-to-noise to disentangle different velocity components
and measure the line widthes are necessary to better understand the properties of Malin 2.

   \begin{figure}
   \centering
   \includegraphics[width=8cm,angle=-90]{F5686_pointing.3.ps}
      \caption{The figure shows the R band image of the galaxy Malin~2 with
the HERA footprint overlaid as nine black circles. The red circles correspond to
the previous IRAM observations at 115GHz (Das et al. 2006). Each circle has a
diameter equal to the FWHM of the corresponding beam.}
         \label{Fig1}
   \end{figure}
   \begin{figure*}
   \centering
   \includegraphics[width=8cm,angle=-90]{allF568.ps}
   \caption{The panel shows the CO(2-1) spectra obtained with the nine HERA receivers.
The offsets in arcsecond from the galaxy center are marked on the
top left hand corner of each figure. CO(2-1) emission was detected from five
locations. The gaussian fits for these detections are overlaid on the spectra.
The x axes are the velocity offsets from the galaxy systemic velocity ($v_{\rm hel}=13,830~km~s^{-1}$)
and the y axes are the measured antenna temperatures ($T_A^*$) in $mK$.}
    \label{Fig1}%
    \end{figure*}
   \begin{table*}
      \caption[]{CO(2-1) fluxes, kinematical parameters, molecular, HI and total gas surface densities, }
\vspace{-3mm}
         \label{table1}
     $$
         \begin{array}{ccccrrrrcccc}
            \hline
            \noalign{\smallskip}
\multicolumn{2}{c}{Offset}
&  rms~noise
& {\Delta~I_{CO}}
& \multicolumn{2}{c}{\Delta~V_{FWHM}}
& \multicolumn{2}{c}{<V>}
& \Sigma_{\rm mol}
& \Sigma_{\rm HI}
&  \multicolumn{1}{c}{\Sigma_{\rm gas}}
&{\frac{\Sigma_{\rm gas}}{\Sigma_{\rm crit}}}\\
\multicolumn{2}{c}{(arcsec)}
& (mK)
& (K~km~s^{-1})
& \multicolumn{2}{c} {(km~s^{-1})}
& \multicolumn{2}{c}{(km~s^{-1})}
& (M_{\odot}pc^{-2})
& (M_{\odot}pc^{-2})
& (M_{\odot}pc^{-2})
& \\
\multicolumn{2}{c}{a}
&b
&c
&\multicolumn{2}{c}{d}
&\multicolumn{2}{c}{e}
&f
&g
&h
&i\\
            \noalign{\smallskip}
            \hline
            \noalign{\smallskip}
+24 & +24  &  0.7    & <~0.12     &\multicolumn{2}{c}{0.39} & \multicolumn{2}{c}{....} & <~0.4 & 2.1 & <~3.0 & ... \\
+24 & +00  &  0.9    & 0.37\pm0.07 &  82~\pm&20  & 91~\pm&~~7  & 1.2\pm0.2                     & 2.5 &     4.8 & 0.7 \\
+24 & -24  &  0.8    & <~0.13     & \multicolumn{2}{c}{0.45}  & \multicolumn{2}{c}{....} & <~0.5 & 1.8 & <~2.8 & ... \\
+00 & +24  &  0.9    & 0.28\pm0.05 &  46~\pm&10  & 14~\pm&~~4  & 0.9\pm0.2                     & 2.7 &     4.6 & 0.7 \\
+00 & +00  &  0.8    & 0.45\pm0.11 & 243~\pm&76 & 77~\pm&34    & 1.5\pm0.4                     & 0.9 &     3.4 &     \\
+00 & -24  &  0.9    & <~0.15      & \multicolumn{2}{c}{0.50} & \multicolumn{2}{c}{....} & <~0.5 & 2.2 & <~3.3 & ... \\
-24 & +24  &  1.6    & \sim0.35\pm0.13 &  79~\pm&41 & -102~\pm&14  & \sim1.2\pm0.4               & 2.2 &  \sim4.4 & \sim0.9 \\
-24 & +00  &  0.9    & 0.20\pm0.04 &  33~\pm&~7  & -142~\pm&~~4& 0.7\pm0.1                     & 4.0 &     5.7 & 0.9 \\
-24 & -24  &  0.7    & <~0.12      & \multicolumn{2}{c}{0.39} & \multicolumn{2}{c}{....} & <~0.4 & 1.8 & <~2.7 & ... \\
            \noalign{\smallskip}
            \hline
         \end{array}
     $$
\begin{list}{}{}
\item[$^{\mathrm{a}}$] $\Delta\alpha$ and $\Delta\delta$ offsets relative to the position of the nucleus $\alpha,\delta$ (J2000): $10^{h}39^{m}52^{s}.5, +20^{\circ}50^{\prime}49^{\prime\prime}$
\item[$^{\mathrm{b}}$] at 10.9 $km~s^{-1}$ velocity resolution.
\item[$^{\mathrm{c}}$] $\Delta I_{\rm{CO(1-0)}}=\int T_{\rm MB} dv$ with $T_{\rm MB}=1.67 T_A^*$. For the non-detections,
we take $T_A^*$=noise and assume $\Delta V=100~km~s^{-1}$.
\item[$^{\mathrm{d}}$] fitted line widths, not deconvolved by the spectral response.
\item[$^{\mathrm{e}}$] fitted, relative to the systemic velocity 13830$~km~s^{-1}$.
\item[$^{\mathrm{f}}$] $\Sigma_{\rm mol}~=~2m_{p}N(H_{2})$. Values for X=2.0 10$^{20}~cm^{-2}~K~km~s^{-1}$ (ref. sect. 3.4).
\item[$^{\mathrm{g}}$] from Fig. 3 of Pickering et al. (\cite{pickering.etal.1997}).
\item[$^{\mathrm{h}}$] $\Sigma_{\rm gas} = (1+Z)~(\Sigma_{\rm HI}+\frac{X}{2.0 10^{20}}\Sigma_{\rm mol})~cos(i).$
                       with Z=0.45, X=2.8 10$^{20}~cm^{-2}~K~km~s^{-1}$ and i=38$^{\circ}$.
                       Values in brackets assume $\Sigma_{\rm mol} \ll \Sigma_{\rm HI}$
\item[$^{\mathrm{i}}$] using $\Sigma_{\rm crit}$ from Fig. 10 of Pickering et al. (\cite{pickering.etal.1997}),
                       re-scaled for $\sigma_{gas}=13~km~s^{-1}$
\end{list}
   \end{table*}

\section{Conclusions}

   \begin{enumerate}
      \item We report the first detection of CO(2-1) line emission from the disk of a giant
LSB galaxy. Our observations reveal the presence of extended molecular gas in the disk of the LSB galaxy
Malin~2. The radial extent of the molecular disk is at least 34~kpc and its mean line brightness
$\sim 0.15 K~km~s^{-1}$ (on the $T_{mb}$ temperature scale). With an angular
resolution of $11.7^{\prime\prime}$ this is $\sim 2/3$ the beam-averaged integrated line intensity
in the central region.
      \item When we compared the molecular and HI gas masses of Malin~2, we found that the
molecular gas fraction is $\sim$1.2~-~2\% of the  HI gas mass.
At radii $\sim~$15 kpc in the directions where CO is detected, the molecular gas
fraction is typically 30 to 50\% of the HI gas surface densities.
       \item We estimated the total surface density of the neutral gas and found that it is always
below the critical threshold density for gravitational instability.
Hence the disk of Malin 2 is overall stable against large scale star formation.
   \end{enumerate}

\begin{acknowledgements}
We are grateful to the IRAM staff at Pico Veleta for excellent support at the
telescope. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain).
The authors would also like to thank Alice Quillen for providing the R-band images of Malin~2.
This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated
by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the
National Aeronautics and Space Administration.
\end{acknowledgements}