------------------------------------------------------------------------ From: Andrew King ark@star.le.ac.uk X-Sender: ark@theory1 To: gcnews@aoc.nrao.edu Subject: Galactic Centre Transients paper MIME-Version: 1.0 Content-ID: ------------- End Forwarded Message ------------- %astro-ph/0004277 %\documentstyle[epsf,referee]{mn} \documentstyle[epsf]{mn} \renewcommand{\baselinestretch}{1} \begin{document} \def\lsun{{\rm L_{\odot}}} \def\msun{{\rm M_{\odot}}} \def\rsun{{\rm R_{\odot}}} \title{The Population of Faint Transients in the Galactic Centre} \author{A.R. King$^1$} \institute{Astronomy Group, University of Leicester, Leicester LE1 7RH, U.K} \maketitle \begin{abstract} BeppoSAX has detected a population of faint transient X--ray sources in the Galactic Centre. I show that a simple irradiated--disc picture gives a consistent fit to the properties of this population, and that it probably consists of low--mass X--ray binaries which have evolved beyond their minimum orbital periods $\sim 80$~min. Since all post--minimum systems are transient, and neutron--star LMXBs are more common than black--hole LMXBs in the Galaxy, the majority of these systems should contain neutron stars, as observed. This picture predicts that the Galactic Centre transients should have orbital periods in the range $\sim 80 - 120$~minutes, and that most of them should repeat in the next few years. In this case, the total number of post--minimum transients in the Galaxy would be considerably smaller than the usual estimates of its total LMXB population. I discuss possible reasons for this. \end{abstract} {\bf Key Words:} accretion, accretion discs - instabilities - stars: X--rays: stars.\\ \section{Introduction} % Over the past 3~yr the Wide Field Cameras on BeppoSAX have detected 9 soft X--ray transients in a 40x40 degree field around the Galactic Centre (GC) (see Heise et al. 1998 for a review). These outbursts are rather faint ($L_X \la 3\times 10^{37}$~erg~s$^{-1}$) and have very short e--folding timescales $\tau \sim 2-6$~d. Of the 9 detected systems, 7 have shown Type~I X--ray bursts, indicating that the accretor is a neutron star. This statistic contrasts sharply with that for brighter transients in the Galaxy, which suggests that no more than one half contain neutron stars, while a large fraction are believed on the basis of dynamical mass measurements to contain black holes. The faintness and short decay timescales of the GC transient outbursts imply a remarkably low accreted mass \begin{equation} \Delta M \simeq 1.8\times 10^{-11}L_{37}\tau_4\msun, \label{m} \end{equation} where $L_{37}, \tau_4$ are $L_x, \tau$ in units of $10^{37}$~erg~s$^{-1}$ and 4~d respectively, and we have assumed that each gram of accreted matter releases $10^{20}$~erg. (This mass estimate would rise if a significant fraction of the accreted matter radiated at low efficiency, i.e. was advected. This is only possible for black--hole systems.) The limits quoted above show that the maximum value of $L_{37}\tau_4$ is about 4.5, allowing $\Delta M$ as large as $8\times 10^{-11}\msun$. Only two of the GC transients, SAX J1808.4-3658, and very recently SAX J1747.0-2853 (Markwardt et al. 2000) have been observed to recur. If all of the mass transferred from the companion is accreted efficiently by the compact primary, one can then estimate its mass transfer rate in SAX J1808.4-3658 as $\dot M_{\rm tr} \simeq 1\times 10^{-11}\ \msun\ {\rm yr}^{-1}$ (Chakrabarty \& Morgan, 1998). The non--recurrence of the other GC transients implies that their mass transfer rates must also be very low, i.e. \begin{equation} \dot M_{\rm tr} \la \Delta M/({\rm 3~yr}) \sim 6\times 10^{-12}L_{37}\tau_4\ \msun{\rm yr}^{-1}. \label{mdot} \end{equation} We shall see in Section 2 below that explicit modelling of the accretion disc stability in these systems leads to a similar limit. (Note that the estimate (\ref{mdot}) would no longer be an upper limit if, contrary to the assumption made above, a significant fraction of the transferred mass was lost from the system rather than accreted, cf Meyer \& Meyer--Hofmeister, 1994). There seem to be only two kinds of relatively abundant binary system which would have such low transfer rates: 1. wind--fed systems\\ 2. Roche--lobe--filling systems in which the donor star has an extremely low mass $M_2 < 0.1\msun$, with mass transfer driven by gravitational radiation. This gives a rate (cf King, Kolb \& Szuszkiewicz, 1997) \begin{equation} \dot M_{\rm GR} \simeq 1 \times 10^{-11}\: m_1^{2/3}\biggl({m_2\over 0.1}\biggr)^2\: \biggl({P\over 2\ {\rm hr}}\biggr)^{-8/3} \: \msun {\rm yr}^{-1} \, , \label{gr} \end{equation} where $m_1, m_2$ are the accretor and donor masses in $\msun$, and I have assumed $m_1 >> m_2$. Note that 1. need not automatically imply a high--mass donor, which would be in possible contradiction with the usual identification of soft X--ray transients as low--mass X--ray binaries (LMXBs). For the mass transfer rates (\ref{mdot}) are so low that even the comparatively weak wind of a low--mass main sequence star may be sufficient to power them. Alternatively, radio pulsar irradiation could excite a wind from a low--mass companion star. Orbital angular momentum losses may then shrink the binary to the point where the neutron star begins to capture this wind material and turns on as an X--ray source. The source will be transient because the accretion rate through the disc is too low to keep it fully ionized (see equation \ref{mtr} below). A possible example of a higher--mass wind--fed transient may be CI Cam (e.g. Marshall et al., 1998, Clark et al., 2000), which has been variously classified as a symbiotic system, or a B[e] binary. Any such system must be wide enough that the UV radiation from the high--mass donor is unable to keep the accretion disc ionized. Whatever the mass of the donor, accretion by wind capture will tend to produce an accretion disc of small radius: if much of the disc mass is accreted in an outburst, the outer radius of the disc will be determined by the low specific angular momentum of the captured wind material, naturally producing a small disc. This in turn would account for the low accreted mass $\Delta M$ and short decay times $\tau$ (see equations \ref{dm}, \ref{tth} below). One might test for the presence of high--mass companions of this type in the GC transients through optical/IR identifications. There are currently only two reported identifications, both of which are inconsistent with high--mass stars. In SAX J1808.4-3658 (Roche et al., 1998) the 2--hour orbital period rules out any such companion. In SAX J1810.8-2609, Greiner et al. (1999) find $R = 19$ fading to $R > 21.5$ for the optical/IR source. This variation means that this source is presumably dominated by the accretion disc, which is probably irradiated by the central X--rays. This is inconsistent with a high--mass companion, which would dominate the optical/infrared as it would have a similar effective temperature to the disc, but be much larger. These arguments fall short of proving that {\it no} GC transient is wind--fed, in either high--mass or low--mass versions. However the second possibility listed above seems inescapable. At the end of their lives as accreting binaries, LMXBs must reduce the donor mass $M_2$ to the point where this star begins to become degenerate (i.e. $M_2 \la 0.1\msun$). If the orbital evolution of LMXBs is similar to the standard picture assumed for cataclysmic variables (CVs), where the accretor is a white dwarf rather than a neutron star or black hole, the binary period will be close to $\sim 80$~minutes at this point (see e.g. King, 1988; Kolb \& Baraffe, 1999 for reviews). Subsequently this period begins to increase rather than decrease as orbital angular momentum is lost via gravitational radiation, as the donor expands in response to further mass loss. The mass transfer rate drops to values comparable with those of equation (\ref{mdot}). Because of this, the mass transfer timescale $M_2/\dot M_{\rm tr}$ becomes very long, eventually approaching a Hubble time. The system's evolution thus slows considerably; even a system `born' at the minimum period of $\sim 80$~minutes requires a time of order the age of the Galaxy to reach $P\simeq 2$~hr. Mass transfer continues, albeit at a very slow rate ($\sim (0.5 - 1)\times 10^{-11}\ \msun\ {\rm yr}^{-1}$). Hence there should exist a population of such extremely faint LMXB systems, directly analogous to the post--minimum--period population of CVs. While no post--minimum CV has been certainly identified, their LMXB analogues are potentially much easier to observe since they are all likely to be soft X--ray transients (King, Kolb \& Szuszkiewicz, 1997). As we shall see, their expected properties are very similar to those of the faint Galactic Centre transients. Note that there may be other more exotic ways of producing individual systems among the faint GC transients, as has for example been proposed for SAX J1808.4-3658 by Ergma \& Antipova (1999). However post--minimum systems offer the most likely way of producing them in significant numbers. The identification of the faint GC transients with post--minimum LMXBs has the desirable property of explaining why neutron stars are favoured among this population. Black--hole LMXBs are likely to be transient at all orbital periods (King, Kolb \& Szuszkiewicz, 1997), while neutron--star LMXBs with main--sequence donors (i.e. before they reach the minimum period) are likely to be persistent. Accordingly black--hole systems predominate among bright transients, since a bright neutron--star transient requires an unusual (nuclear--evolved) donor. However the calculations of King, Kolb \& Szuszkiewicz (1997) show that neutron--star LMXBs {\it will} become transient as they evolve beyond the the minimum period. Thus the ratio of neutron star to black hole systems must be much higher for post--minimum LMXBs than for bright transients. This naturally explains why the faint Galactic Centre transients seem mostly to contain neutron stars. In the next Section I show that the simple irradiated--disc model of soft X--ray transient (SXT) outbursts proposed by King \& Ritter (1998) predicts outburst masses $\Delta M$ and decay times $\tau$ in excellent agreement with the observational estimates given above, provided that the outer disc radius is small, i.e. $\sim 10^{10}$~cm. Moreover the maximum average mass transfer rate consistent with irradiation not suppressing the outbursts is close to the limit (\ref{mdot}). In Section 3 I discuss the two possible types of binary system considered above in the light of these disc properties. Section 4 is the Conclusion. \section{Accretion disc properties in the faint GC transients} Soft X--ray transient outbursts are thought to result from instabilities in LMXB accretion discs. Both the incidence and the nature of these outbursts are strongly affected by irradiation of the disc surfaces by the central X--rays. Irradiation can suppress outbursts in some LMXB discs by removing their hydrogen ionization zones (van Paradijs, 1996; King, Kolb \& Burderi, 1996; King, Kolb \& Szuszkiewicz, 1997), thus making them stable (persistent) at lower mass transfer rates than is true for the otherwise similar discs in CVs. The irradiation effect appears to be weaker if the accretor is a black hole rather than a neutron star, possibly because of the lack of a hard surface (King, Kolb \& Szuszkiewicz, 1997). The result is that neutron--star LMXBs with main--sequence companions tend to be persistent, while similar black--hole binaries are largely transient. If an LMXB disc goes into outburst, irradiation of the disc greatly prolongs the high state, and causes viscous rather than thermal effects to dominate the light--curve (King \& Ritter, 1998). This often produces an exponential decay, particularly if the whole disc is efficiently irradiated. Several predictions of this simple irradiated--disc picture are confirmed by observation (cf Shahbaz, Charles \& King, 1998). More detailed calculations with a full 1--D disc code (Dubus et al, 1999) give results very similar to those of King and Ritter (1998) so I shall use their simple analytic expressions in the following. As the observed decays are approximately exponential I assume that the entire disc of a faint transient (of radius $R$) is irradiated during an outburst. Most of this mass is accreted. Immediately before the outburst the surface density in the disc must have been close to the value \begin{equation} \Sigma_{\rm max} = 11.4R_{10}^{1.05}m_1^{-0.35}\alpha_c^{-0.86}\ {\rm g\ cm}^{-2} \end{equation} (Cannizzo, Shafter \& Wheeler, 1988) triggering the thermal instability through local viscous dissipation, where $R_{10} = R/10^{10}$~cm, with $R$ the radial disc coordinate, $m_1$ is the central accreting mass in $\msun$, and $\alpha_c$ is the cold--state viscosity parameter. Thus by integrating over $R$ we predict the mass accreted in the outburst as \begin{equation} \Delta M_{\rm pr} \simeq 1.5\times 10^{-11} m_1^{-0.35}\alpha_{0.05}^{-0.86}R_{10}^{3.05}\msun, \label{dm} \end{equation} where $\alpha_{0.05} = \alpha_c/0.05$. This relation can be compared with equation (8) of King \& Ritter (1998), who used a simpler form of $\Sigma_{\rm max}$. Equating this to the observational estimate (\ref{m}) we find a disc radius \begin{equation} R \simeq 1.2\times 10^{10} (L_{37}\tau_4)^{0.33}m_1^{0.11}\alpha_{0.05}^{0.28}\ {\rm cm}. \label{R} \end{equation} King \& Ritter (1998) predict that the e--folding time for the decay will be \begin{equation} \tau_{\rm pr} = {R^2\over 3\nu}, \end{equation} where $\nu = \alpha_hc_SH$ is the hot--state kinematic viscosity at the disc edge. Here $c_S, H$ are the local sound speed and scale height, and we have \begin{equation} H = {c_S\over \Omega} \label{H} \end{equation} with $\Omega = (GM_1/R^3)^{1/2}$ the Kepler frequency at disc radius $R$. Writing $T_4$ for the surface temperature at this point in units of $10^4$~K, we find the predicted timescale \begin{equation} \tau_{\rm pr} = 3.9\alpha_h^{-1}m_1^{0.5}R_{10}^{0.5}T_4^{-1}\ {\rm d}. \label{tth} \end{equation} The theoretical predictions (\ref{dm}, \ref{tth}) are in excellent agreement with the observational values $\Delta M$, $\tau$ and thus $L_X$ provided that the hot--state viscosity parameter takes a value $\alpha_h \sim T_4 \la 1$, as expected, and $R \sim 10^{10}$~cm. Evidently the distinctive feature of the discs in the faint GC transients is their very small size, which accounts for both their low peak luminosities and their rapid decays. We can now ask how low the mean mass transfer rate $\dot M_{\rm tr}$ must be if such a disc is to have ionization zones despite being so small. This requires that the surface temperature $T_{\rm irr}$ resulting from irradiation should be less than the ionization temperature $T_{\rm H} \sim 6500$~K at the disc edge. I take \begin{equation} T_{\rm irr}(R)^4 = {10^{20}\dot M_{\rm tr}\over 4\pi \sigma R^2} \biggl({H \over R}\biggr)^n \biggl[{{\rm d}\ln H\over {\rm d}\ln R} - 1\biggr], \label{eq3} \end{equation} (e.g. van Paradijs, 1996) where the factor in square brackets lies between 1/8 and 2/7 and the index $n = 1$ or 2 for a neutron star or black hole respectively (cf King, Kolb \& Szuszkiewicz, 1997). Requiring $T_{\rm irr} < T_{\rm H}$ and using the estimate (\ref{R}) gives \begin{equation} \dot M_{\rm tr} \la 1.3\times 10^{-11} (L_{37}\tau_4)^{0.66}m_1^{0.22}\alpha_{0.05}^{0.56}\ \msun {\rm yr}^{-1} \label{mtr} \end{equation} for a neutron star, and a limit about 10 times larger for a black hole. This is again in excellent agreement with the observational limit~(\ref{mdot}). \section{Binary models for the faint GC transients} The good agreement with simple irradiated--disc theory found above shows that the faint GC transient population must be binaries with two key properties (a) disc radii $R \sim 1.2\times 10^{10}$~cm, and (b) mean mass transfer rates $\dot M_{\rm tr} \sim (0.6 - 1)\times 10^{-11}\ \msun{\rm yr}^{-1}$. Both the binary types 1 (wind--fed) and 2 (low--mass donor) considered in the Introduction are able to reproduce these conditions. In the wind--fed case, the low specific angular momentum of matter captured from a wind will produce a small disc radius provided that most of the disc mass is accreted during an outburst. The low mass transfer rate is a natural consequence of inefficient wind capture. However, the specific values of $R$ and $\dot M_{\rm tr}$ depend in detail on the properties of the donor wind, and it is not obvious why these should cluster around the values producing (a) and (b) above. By contrast, the version of model 2 involving post--minimum LMXBs will naturally produce these values. If most of the disc mass is accreted in an outburst, as is implicit in the estimate (\ref{dm}), the disc radius $R$ will be close to the circularization radius \begin{equation} R_{\rm circ} \simeq (1+q)(0.7-0.227\log q)^4a, \label{circ} \end{equation} (e.g. Frank et al, 1992) where $q = M_2/M_1$ is the mass ratio and $a$ is the binary separation, i.e. \begin{equation} a = 3.53\times 10^{10}m_1^{1/3}P_{\rm hr}^{2/3}\ {\rm cm}. \label{a} \end{equation} With $M_1 = 1.4\msun, M_2 = 0.1\msun$ we find an expected disc radius \begin{equation} R \simeq R_{\rm circ} = 1.7\times 10^{10}\biggl({m_1\over 1.4}\biggr)^{1/3} \biggl({P\over 80\ {\rm min}}\biggr)^{2/3}\ {\rm cm}. \label{rcirc} \end{equation} Thus binary periods in the typical range 80~min -- 2~hr will produce discs of the right size (cf eqn. \ref{R}) for neutron--star masses $m_1 = 1.4$. Equating the two expressions (\ref{R}, \ref{rcirc}) gives the requirement \begin{equation} L_{37}\tau_4 = 5m_1^{2/3}\alpha_{0.05}^{-0.84} \biggl({P\over 2\ {\rm hr}}\biggr)^2, \label{Lt} \end{equation} which compares well with the observed range of this quantity given in the Introduction. The calculations of King, Kolb \& Szuszkiewicz (1997) show that such systems have transfer rates at or below the limit (b), which also ensure that the systems are indeed transient there (see their Fig. 3). For black--hole masses $m_1 \sim 7$ we see that the predicted disc radius $R$ tends to become rather larger than the estimate (\ref{R}), leading to more prolonged outbursts (transferred mass $\Delta M_{\rm pr} \sim 10^{-10}\msun$, decay timescale $\tau_{\rm pr} \sim 27$~d) than are typical of the faint GC transients (note that eqs. \ref{dm}, \ref{rcirc} imply that $\Delta M_{\rm pr} \propto m_1$, while eq. \ref{tth} implies $\tau_{\rm pr} \propto m_1$). Thus such systems would probably not be classified as faint transients. Moreover black--hole systems will have longer recurrence times $t_{\rm rep} \propto \Delta M/\dot M_{\rm tr} \propto m_1/m_1^{2/3} \sim m_1^{1/3}$ and thus lower discovery probability ($\dot M_{\rm tr} \propto m_1^{2/3}$ for gravitational radiation, cf eqn \ref{gr}). This already tends to suggest agreement with the observation that at least 7 out of 9 GC transients contain neutron stars. A still stronger reason comes from the fact that Fig. 3 of King, Kolb \& Szuszkiewicz (1997) shows that essentially {\it all} neutron--star LMXBs will be transient once they have evolved sufficiently far beyond the minimum period, and neutron--star systems are much more common than black--hole ones in general. Thus the ratio of neutron--star to black--hole systems among the faint GC transients should be \begin{eqnarray} \lefteqn{{N_{\rm NS\ GC transients}\over N_{\rm BH\ GC transients}} \simeq {N_{\rm NS\ post}\over N_{\rm BH\ post}}} \nonumber \\ \lefteqn{= {N_{\rm NS\ post}\over N_{\rm NS\ pre}}. {N_{\rm NS\ pre}\over N_{\rm BH\ pre}}. {N_{\rm BH\ pre}\over N_{\rm BH\ post}}}, \label{N} \end{eqnarray} where the $N$ are space densities and `pre' and `post' refer to the minimum period. Now Fig. 3 of King, Kolb \& Szuszkiewicz (1997) implies \begin{equation} {N_{\rm NS\ post}\over N_{\rm NS\ pre}} \ga {N_{\rm BH\ post}\over N_{\rm BH\ pre}}, \label{N2} \end{equation} i.e. the slow--down of neutron--star LMXB evolution after passing the minimum period is more dramatic than that of black--hole systems. Using (\ref{N2}) in (\ref{N}), the two outer factors combine to give a number $\ga 1$, so we get \begin{equation} {N_{\rm NS\ GC transients}\over N_{\rm BH\ GC transients}} \ga {N_{\rm NS\ pre}\over N_{\rm BH\ pre}} >> 1, \label{N4} \end{equation} where the last inequality expresses the fact that there are far more neutron--star then black--hole binaries. \section{Conclusions} I have shown that the simple irradiated--disc picture gives a consistent fit to the properties of the faint GC transients, and that this population probably consists of post--minimum LMXBs. Neutron star systems far outweigh black--hole ones here, mainly because {\it all} post--minimum systems are transient, and neutron--star LMXBs are simply more common than black--hole ones in the Galaxy (outbursts of the latter would also probably be too long to be classified among the faint transients). By contrast among brighter transients, which are generally pre--minimum systems, the greater incidence of transient behaviour among black--hole systems makes them prominent. If the identification as post--minimum LMXBs is correct, the faint transients should be binaries with periods in the range $80 - 120$~minutes, with extremely low--mass companions. An interesting point emerges from the fact that only two faint GC transients have yet been observed to repeat. From (\ref{mdot}) we see that if this state of affairs persists for a decade or so, the resulting upper limit on the mean mass transfer rate will become embarassingly low even for post--minimum LMXBs. On the other hand, a typical repetition time $t_{\rm rep}$ of order a few years would also severely limit the total number of such systems in the Galaxy: Heise et al. (1998) estimate from the sky and temporal coverage of the BeppoSAX WFC that there are about 18 faint transient outbursts in the Galaxy per year, leading to a total population $N_{\rm Gal} \sim 18(t_{\rm rep}/{\rm yr})$. This number ($50 \la N_{\rm Gal} \la 180$) is considerably smaller than the usual estimate of $\sim 1000$ LMXBs in the Galaxy, whereas one might expect it to be much larger, as the evolution of post--minimum systems is so slow. There are several possible reasons for this, of which two seem most likely. (a) If most LMXBs first reach contact at initial periods $P_i$ greater than a few hours (as indeed suggested by theoretical studies of LMXB formation, e.g. Kalogera \& Webbink, 1996; King \& Kolb 1997), their lifetimes before becoming post--minimum transients may be comparable with the age of the Galaxy. This lifetime is spent mostly near the minimum period (cf Kolb \& Baraffe, 1999) while according to King, Kolb \& Szuszkiewicz (1997), neutron--star LMXBs have to evolve somewhat beyond this period in order to become transient. A large fraction of the neutron--star LMXBs ever formed in the Galaxy may still not have reached this stage. (b) Isolated millisecond pulsars are thought to be neutron stars spun up in LMXBs, but which have evaporated their companions (see Bhattacharya \& van den Heuvel, 1991, for a review). This suggests that many neutron--star LMXBs may not even reach the theoretical minimum period, and thus do not become faint transients at all. It is sometimes hypothesized (cf Bhattacharya \& van den Heuvel, 1991) that most neutron--star LMXBs evaporate the companion star as the system attempts to cross the analogue of the CV period gap between 3~hr and 2~hr (see e.g. King, 1988 for a review of the latter). If so, this would limit post--minimum neutron--star systems to the rare examples first coming into contact at periods below the period gap, i.e. with $P_i \la 2$~hr. Of course there can be no such effect for black--hole LMXBs. At present we do not have any clear idea of the mean repetition time for the faint GC transients. The arguments above show that there are interesting consequences whatever this number turns out to be. Extensive X--ray monitoring of the Galactic Centre region clearly has much more to tell us about the stellar populations there. \section{Acknowledgment} I gratefully acknowledge the support of a PPARC Senior Fellowship. 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King]{A.R. King$^1$\\ $^1$ Astronomy Group, University of Leicester, Leicester LE1 7RH, U.K} \maketitle \begin{abstract} BeppoSAX has detected a population of faint transient X--ray sources in the Galactic Centre. I show that a simple irradiated--disc picture gives a consistent fit to the properties of this population, and that it probably consists of low--mass X--ray binaries which have evolved beyond their minimum orbital periods $\sim 80$~min. Since all post--minimum systems are transient, and neutron--star LMXBs are more common than black--hole LMXBs in the Galaxy, the majority of these systems should contain neutron stars, as observed. This picture predicts that the Galactic Centre transients should have orbital periods in the range $\sim 80 - 120$~minutes, and that most of them should repeat in the next few years. In this case, the total number of post--minimum transients in the Galaxy would be considerably smaller than the usual estimates of its total LMXB population. I discuss possible reasons for this. {\bf Key Words:} accretion, accretion discs - instabilities - stars: X--rays: stars.\\ \end{abstract} \section{Introduction} % Over the past 3~yr the Wide Field Cameras on BeppoSAX have detected 9 soft X--ray transients in a 40x40 degree field around the Galactic Centre (GC) (see Heise et al. 1998 for a review). These outbursts are rather faint ($L_X \la 3\times 10^{37}$~erg~s$^{-1}$) and have very short e--folding timescales $\tau \sim 2-6$~d. Of the 9 detected systems, 7 have shown Type~I X--ray bursts, indicating that the accretor is a neutron star. This statistic contrasts sharply with that for brighter transients in the Galaxy, which suggests that no more than one half contain neutron stars, while a large fraction are believed on the basis of dynamical mass measurements to contain black holes. The faintness and short decay timescales of the GC transient outbursts imply a remarkably low accreted mass \begin{equation} \Delta M \simeq 1.8\times 10^{-11}L_{37}\tau_4\msun, \label{m} \end{equation} where $L_{37}, \tau_4$ are $L_x, \tau$ in units of $10^{37}$~erg~s$^{-1}$ and 4~d respectively, and we have assumed that each gram of accreted matter releases $10^{20}$~erg. (This mass estimate would rise if a significant fraction of the accreted matter radiated at low efficiency, i.e. was advected. This is only possible for black--hole systems.) The limits quoted above show that the maximum value of $L_{37}\tau_4$ is about 4.5, allowing $\Delta M$ as large as $8\times 10^{-11}\msun$. Only two of the GC transients, SAX J1808.4-3658, and very recently SAX J1747.0-2853 (Markwardt et al. 2000) have been observed to recur. If all of the mass transferred from the companion is accreted efficiently by the compact primary, one can then estimate its mass transfer rate in SAX J1808.4-3658 as $\dot M_{\rm tr} \simeq 1\times 10^{-11}\ \msun\ {\rm yr}^{-1}$ (Chakrabarty \& Morgan, 1998). The non--recurrence of the other GC transients implies that their mass transfer rates must also be very low, i.e. \begin{equation} \dot M_{\rm tr} \la \Delta M/({\rm 3~yr}) \sim 6\times 10^{-12}L_{37}\tau_4\ \msun{\rm yr}^{-1}. \label{mdot} \end{equation} We shall see in Section 2 below that explicit modelling of the accretion disc stability in these systems leads to a similar limit. (Note that the estimate (\ref{mdot}) would no longer be an upper limit if, contrary to the assumption made above, a significant fraction of the transferred mass was lost from the system rather than accreted, cf Meyer \& Meyer--Hofmeister, 1994). There seem to be only two kinds of relatively abundant binary system which would have such low transfer rates: 1. wind--fed systems\\ 2. Roche--lobe--filling systems in which the donor star has an extremely low mass $M_2 < 0.1\msun$, with mass transfer driven by gravitational radiation. This gives a rate (cf King, Kolb \& Szuszkiewicz, 1997) \begin{equation} \dot M_{\rm GR} \simeq 1 \times 10^{-11}\: m_1^{2/3}\biggl({m_2\over 0.1}\biggr)^2\: \biggl({P\over 2\ {\rm hr}}\biggr)^{-8/3} \: \msun {\rm yr}^{-1} \, , \label{gr} \end{equation} where $m_1, m_2$ are the accretor and donor masses in $\msun$, and I have assumed $m_1 >> m_2$. Note that 1. need not automatically imply a high--mass donor, which would be in possible contradiction with the usual identification of soft X--ray transients as low--mass X--ray binaries (LMXBs). For the mass transfer rates (\ref{mdot}) are so low that even the comparatively weak wind of a low--mass main sequence star may be sufficient to power them. Alternatively, radio pulsar irradiation could excite a wind from a low--mass companion star. Orbital angular momentum losses may then shrink the binary to the point where the neutron star begins to capture this wind material and turns on as an X--ray source. The source will be transient because the accretion rate through the disc is too low to keep it fully ionized (see equation \ref{mtr} below). A possible example of a higher--mass wind--fed transient may be CI Cam (e.g. Marshall et al., 1998, Clark et al., 2000), which has been variously classified as a symbiotic system, or a B[e] binary. Any such system must be wide enough that the UV radiation from the high--mass donor is unable to keep the accretion disc ionized. Whatever the mass of the donor, accretion by wind capture will tend to produce an accretion disc of small radius: if much of the disc mass is accreted in an outburst, the outer radius of the disc will be determined by the low specific angular momentum of the captured wind material, naturally producing a small disc. This in turn would account for the low accreted mass $\Delta M$ and short decay times $\tau$ (see equations \ref{dm}, \ref{tth} below). One might test for the presence of high--mass companions of this type in the GC transients through optical/IR identifications. There are currently only two reported identifications, both of which are inconsistent with high--mass stars. In SAX J1808.4-3658 (Roche et al., 1998) the 2--hour orbital period rules out any such companion. In SAX J1810.8-2609, Greiner et al. (1999) find $R = 19$ fading to $R > 21.5$ for the optical/IR source. This variation means that this source is presumably dominated by the accretion disc, which is probably irradiated by the central X--rays. This is inconsistent with a high--mass companion, which would dominate the optical/infrared as it would have a similar effective temperature to the disc, but be much larger. These arguments fall short of proving that {\it no} GC transient is wind--fed, in either high--mass or low--mass versions. However the second possibility listed above seems inescapable. At the end of their lives as accreting binaries, LMXBs must reduce the donor mass $M_2$ to the point where this star begins to become degenerate (i.e. $M_2 \la 0.1\msun$). If the orbital evolution of LMXBs is similar to the standard picture assumed for cataclysmic variables (CVs), where the accretor is a white dwarf rather than a neutron star or black hole, the binary period will be close to $\sim 80$~minutes at this point (see e.g. King, 1988; Kolb \& Baraffe, 1999 for reviews). Subsequently this period begins to increase rather than decrease as orbital angular momentum is lost via gravitational radiation, as the donor expands in response to further mass loss. The mass transfer rate drops to values comparable with those of equation (\ref{mdot}). Because of this, the mass transfer timescale $M_2/\dot M_{\rm tr}$ becomes very long, eventually approaching a Hubble time. The system's evolution thus slows considerably; even a system `born' at the minimum period of $\sim 80$~minutes requires a time of order the age of the Galaxy to reach $P\simeq 2$~hr. Mass transfer continues, albeit at a very slow rate ($\sim (0.5 - 1)\times 10^{-11}\ \msun\ {\rm yr}^{-1}$). Hence there should exist a population of such extremely faint LMXB systems, directly analogous to the post--minimum--period population of CVs. While no post--minimum CV has been certainly identified, their LMXB analogues are potentially much easier to observe since they are all likely to be soft X--ray transients (King, Kolb \& Szuszkiewicz, 1997). As we shall see, their expected properties are very similar to those of the faint Galactic Centre transients. Note that there may be other more exotic ways of producing individual systems among the faint GC transients, as has for example been proposed for SAX J1808.4-3658 by Ergma \& Antipova (1999). However post--minimum systems offer the most likely way of producing them in significant numbers. The identification of the faint GC transients with post--minimum LMXBs has the desirable property of explaining why neutron stars are favoured among this population. Black--hole LMXBs are likely to be transient at all orbital periods (King, Kolb \& Szuszkiewicz, 1997), while neutron--star LMXBs with main--sequence donors (i.e. before they reach the minimum period) are likely to be persistent. Accordingly black--hole systems predominate among bright transients, since a bright neutron--star transient requires an unusual (nuclear--evolved) donor. However the calculations of King, Kolb \& Szuszkiewicz (1997) show that neutron--star LMXBs {\it will} become transient as they evolve beyond the the minimum period. Thus the ratio of neutron star to black hole systems must be much higher for post--minimum LMXBs than for bright transients. This naturally explains why the faint Galactic Centre transients seem mostly to contain neutron stars. In the next Section I show that the simple irradiated--disc model of soft X--ray transient (SXT) outbursts proposed by King \& Ritter (1998) predicts outburst masses $\Delta M$ and decay times $\tau$ in excellent agreement with the observational estimates given above, provided that the outer disc radius is small, i.e. $\sim 10^{10}$~cm. Moreover the maximum average mass transfer rate consistent with irradiation not suppressing the outbursts is close to the limit (\ref{mdot}). In Section 3 I discuss the two possible types of binary system considered above in the light of these disc properties. Section 4 is the Conclusion. \section{Accretion disc properties in the faint GC transients} Soft X--ray transient outbursts are thought to result from instabilities in LMXB accretion discs. Both the incidence and the nature of these outbursts are strongly affected by irradiation of the disc surfaces by the central X--rays. Irradiation can suppress outbursts in some LMXB discs by removing their hydrogen ionization zones (van Paradijs, 1996; King, Kolb \& Burderi, 1996; King, Kolb \& Szuszkiewicz, 1997), thus making them stable (persistent) at lower mass transfer rates than is true for the otherwise similar discs in CVs. The irradiation effect appears to be weaker if the accretor is a black hole rather than a neutron star, possibly because of the lack of a hard surface (King, Kolb \& Szuszkiewicz, 1997). The result is that neutron--star LMXBs with main--sequence companions tend to be persistent, while similar black--hole binaries are largely transient. If an LMXB disc goes into outburst, irradiation of the disc greatly prolongs the high state, and causes viscous rather than thermal effects to dominate the light--curve (King \& Ritter, 1998). This often produces an exponential decay, particularly if the whole disc is efficiently irradiated. Several predictions of this simple irradiated--disc picture are confirmed by observation (cf Shahbaz, Charles \& King, 1998). More detailed calculations with a full 1--D disc code (Dubus et al, 1999) give results very similar to those of King and Ritter (1998) so I shall use their simple analytic expressions in the following. As the observed decays are approximately exponential I assume that the entire disc of a faint transient (of radius $R$) is irradiated during an outburst. Most of this mass is accreted. Immediately before the outburst the surface density in the disc must have been close to the value \begin{equation} \Sigma_{\rm max} = 11.4R_{10}^{1.05}m_1^{-0.35}\alpha_c^{-0.86}\ {\rm g\ cm}^{-2} \end{equation} (Cannizzo, Shafter \& Wheeler, 1988) triggering the thermal instability through local viscous dissipation, where $R_{10} = R/10^{10}$~cm, with $R$ the radial disc coordinate, $m_1$ is the central accreting mass in $\msun$, and $\alpha_c$ is the cold--state viscosity parameter. Thus by integrating over $R$ we predict the mass accreted in the outburst as \begin{equation} \Delta M_{\rm pr} \simeq 1.5\times 10^{-11} m_1^{-0.35}\alpha_{0.05}^{-0.86}R_{10}^{3.05}\msun, \label{dm} \end{equation} where $\alpha_{0.05} = \alpha_c/0.05$. This relation can be compared with equation (8) of King \& Ritter (1998), who used a simpler form of $\Sigma_{\rm max}$. Equating this to the observational estimate (\ref{m}) we find a disc radius \begin{equation} R \simeq 1.2\times 10^{10} (L_{37}\tau_4)^{0.33}m_1^{0.11}\alpha_{0.05}^{0.28}\ {\rm cm}. \label{R} \end{equation} King \& Ritter (1998) predict that the e--folding time for the decay will be \begin{equation} \tau_{\rm pr} = {R^2\over 3\nu}, \end{equation} where $\nu = \alpha_hc_SH$ is the hot--state kinematic viscosity at the disc edge. Here $c_S, H$ are the local sound speed and scale height, and we have \begin{equation} H = {c_S\over \Omega} \label{H} \end{equation} with $\Omega = (GM_1/R^3)^{1/2}$ the Kepler frequency at disc radius $R$. Writing $T_4$ for the surface temperature at this point in units of $10^4$~K, we find the predicted timescale \begin{equation} \tau_{\rm pr} = 3.9\alpha_h^{-1}m_1^{0.5}R_{10}^{0.5}T_4^{-1}\ {\rm d}. \label{tth} \end{equation} The theoretical predictions (\ref{dm}, \ref{tth}) are in excellent agreement with the observational values $\Delta M$, $\tau$ and thus $L_X$ provided that the hot--state viscosity parameter takes a value $\alpha_h \sim T_4 \la 1$, as expected, and $R \sim 10^{10}$~cm. Evidently the distinctive feature of the discs in the faint GC transients is their very small size, which accounts for both their low peak luminosities and their rapid decays. We can now ask how low the mean mass transfer rate $\dot M_{\rm tr}$ must be if such a disc is to have ionization zones despite being so small. This requires that the surface temperature $T_{\rm irr}$ resulting from irradiation should be less than the ionization temperature $T_{\rm H} \sim 6500$~K at the disc edge. I take \begin{equation} T_{\rm irr}(R)^4 = {10^{20}\dot M_{\rm tr}\over 4\pi \sigma R^2} \biggl({H \over R}\biggr)^n \biggl[{{\rm d}\ln H\over {\rm d}\ln R} - 1\biggr], \label{eq3} \end{equation} (e.g. van Paradijs, 1996) where the factor in square brackets lies between 1/8 and 2/7 and the index $n = 1$ or 2 for a neutron star or black hole respectively (cf King, Kolb \& Szuszkiewicz, 1997). Requiring $T_{\rm irr} < T_{\rm H}$ and using the estimate (\ref{R}) gives \begin{equation} \dot M_{\rm tr} \la 1.3\times 10^{-11} (L_{37}\tau_4)^{0.66}m_1^{0.22}\alpha_{0.05}^{0.56}\ \msun {\rm yr}^{-1} \label{mtr} \end{equation} for a neutron star, and a limit about 10 times larger for a black hole. This is again in excellent agreement with the observational limit~(\ref{mdot}). \section{Binary models for the faint GC transients} The good agreement with simple irradiated--disc theory found above shows that the faint GC transient population must be binaries with two key properties (a) disc radii $R \sim 1.2\times 10^{10}$~cm, and (b) mean mass transfer rates $\dot M_{\rm tr} \sim (0.6 - 1)\times 10^{-11}\ \msun{\rm yr}^{-1}$. Both the binary types 1 (wind--fed) and 2 (low--mass donor) considered in the Introduction are able to reproduce these conditions. In the wind--fed case, the low specific angular momentum of matter captured from a wind will produce a small disc radius provided that most of the disc mass is accreted during an outburst. The low mass transfer rate is a natural consequence of inefficient wind capture. However, the specific values of $R$ and $\dot M_{\rm tr}$ depend in detail on the properties of the donor wind, and it is not obvious why these should cluster around the values producing (a) and (b) above. By contrast, the version of model 2 involving post--minimum LMXBs will naturally produce these values. If most of the disc mass is accreted in an outburst, as is implicit in the estimate (\ref{dm}), the disc radius $R$ will be close to the circularization radius \begin{equation} R_{\rm circ} \simeq (1+q)(0.7-0.227\log q)^4a, \label{circ} \end{equation} (e.g. Frank et al, 1992) where $q = M_2/M_1$ is the mass ratio and $a$ is the binary separation, i.e. \begin{equation} a = 3.53\times 10^{10}m_1^{1/3}P_{\rm hr}^{2/3}\ {\rm cm}. \label{a} \end{equation} With $M_1 = 1.4\msun, M_2 = 0.1\msun$ we find an expected disc radius \begin{equation} R \simeq R_{\rm circ} = 1.7\times 10^{10}\biggl({m_1\over 1.4}\biggr)^{1/3} \biggl({P\over 80\ {\rm min}}\biggr)^{2/3}\ {\rm cm}. \label{rcirc} \end{equation} Thus binary periods in the typical range 80~min -- 2~hr will produce discs of the right size (cf eqn. \ref{R}) for neutron--star masses $m_1 = 1.4$. Equating the two expressions (\ref{R}, \ref{rcirc}) gives the requirement \begin{equation} L_{37}\tau_4 = 5m_1^{2/3}\alpha_{0.05}^{-0.84} \biggl({P\over 2\ {\rm hr}}\biggr)^2, \label{Lt} \end{equation} which compares well with the observed range of this quantity given in the Introduction. The calculations of King, Kolb \& Szuszkiewicz (1997) show that such systems have transfer rates at or below the limit (b), which also ensure that the systems are indeed transient there (see their Fig. 3). For black--hole masses $m_1 \sim 7$ we see that the predicted disc radius $R$ tends to become rather larger than the estimate (\ref{R}), leading to more prolonged outbursts (transferred mass $\Delta M_{\rm pr} \sim 10^{-10}\msun$, decay timescale $\tau_{\rm pr} \sim 27$~d) than are typical of the faint GC transients (note that eqs. \ref{dm}, \ref{rcirc} imply that $\Delta M_{\rm pr} \propto m_1$, while eq. \ref{tth} implies $\tau_{\rm pr} \propto m_1$). Thus such systems would probably not be classified as faint transients. Moreover black--hole systems will have longer recurrence times $t_{\rm rep} \propto \Delta M/\dot M_{\rm tr} \propto m_1/m_1^{2/3} \sim m_1^{1/3}$ and thus lower discovery probability ($\dot M_{\rm tr} \propto m_1^{2/3}$ for gravitational radiation, cf eqn \ref{gr}). This already tends to suggest agreement with the observation that at least 7 out of 9 GC transients contain neutron stars. A still stronger reason comes from the fact that Fig. 3 of King, Kolb \& Szuszkiewicz (1997) shows that essentially {\it all} neutron--star LMXBs will be transient once they have evolved sufficiently far beyond the minimum period, and neutron--star systems are much more common than black--hole ones in general. Thus the ratio of neutron--star to black--hole systems among the faint GC transients should be \begin{eqnarray} \lefteqn{{N_{\rm NS\ GC transients}\over N_{\rm BH\ GC transients}} \simeq {N_{\rm NS\ post}\over N_{\rm BH\ post}}} \nonumber \\ \lefteqn{= {N_{\rm NS\ post}\over N_{\rm NS\ pre}}. {N_{\rm NS\ pre}\over N_{\rm BH\ pre}}. {N_{\rm BH\ pre}\over N_{\rm BH\ post}}}, \label{N} \end{eqnarray} where the $N$ are space densities and `pre' and `post' refer to the minimum period. Now Fig. 3 of King, Kolb \& Szuszkiewicz (1997) implies \begin{equation} {N_{\rm NS\ post}\over N_{\rm NS\ pre}} \ga {N_{\rm BH\ post}\over N_{\rm BH\ pre}}, \label{N2} \end{equation} i.e. the slow--down of neutron--star LMXB evolution after passing the minimum period is more dramatic than that of black--hole systems. Using (\ref{N2}) in (\ref{N}), the two outer factors combine to give a number $\ga 1$, so we get \begin{equation} {N_{\rm NS\ GC transients}\over N_{\rm BH\ GC transients}} \ga {N_{\rm NS\ pre}\over N_{\rm BH\ pre}} >> 1, \label{N4} \end{equation} where the last inequality expresses the fact that there are far more neutron--star then black--hole binaries. \section{Conclusions} I have shown that the simple irradiated--disc picture gives a consistent fit to the properties of the faint GC transients, and that this population probably consists of post--minimum LMXBs. Neutron star systems far outweigh black--hole ones here, mainly because {\it all} post--minimum systems are transient, and neutron--star LMXBs are simply more common than black--hole ones in the Galaxy (outbursts of the latter would also probably be too long to be classified among the faint transients). By contrast among brighter transients, which are generally pre--minimum systems, the greater incidence of transient behaviour among black--hole systems makes them prominent. If the identification as post--minimum LMXBs is correct, the faint transients should be binaries with periods in the range $80 - 120$~minutes, with extremely low--mass companions. An interesting point emerges from the fact that only two faint GC transients have yet been observed to repeat. From (\ref{mdot}) we see that if this state of affairs persists for a decade or so, the resulting upper limit on the mean mass transfer rate will become embarassingly low even for post--minimum LMXBs. On the other hand, a typical repetition time $t_{\rm rep}$ of order a few years would also severely limit the total number of such systems in the Galaxy: Heise et al. (1998) estimate from the sky and temporal coverage of the BeppoSAX WFC that there are about 18 faint transient outbursts in the Galaxy per year, leading to a total population $N_{\rm Gal} \sim 18(t_{\rm rep}/{\rm yr})$. This number ($50 \la N_{\rm Gal} \la 180$) is considerably smaller than the usual estimate of $\sim 1000$ LMXBs in the Galaxy, whereas one might expect it to be much larger, as the evolution of post--minimum systems is so slow. There are several possible reasons for this, of which two seem most likely. (a) If most LMXBs first reach contact at initial periods $P_i$ greater than a few hours (as indeed suggested by theoretical studies of LMXB formation, e.g. Kalogera \& Webbink, 1996; King \& Kolb 1997), their lifetimes before becoming post--minimum transients may be comparable with the age of the Galaxy. This lifetime is spent mostly near the minimum period (cf Kolb \& Baraffe, 1999) while according to King, Kolb \& Szuszkiewicz (1997), neutron--star LMXBs have to evolve somewhat beyond this period in order to become transient. A large fraction of the neutron--star LMXBs ever formed in the Galaxy may still not have reached this stage. (b) Isolated millisecond pulsars are thought to be neutron stars spun up in LMXBs, but which have evaporated their companions (see Bhattacharya \& van den Heuvel, 1991, for a review). This suggests that many neutron--star LMXBs may not even reach the theoretical minimum period, and thus do not become faint transients at all. It is sometimes hypothesized (cf Bhattacharya \& van den Heuvel, 1991) that most neutron--star LMXBs evaporate the companion star as the system attempts to cross the analogue of the CV period gap between 3~hr and 2~hr (see e.g. King, 1988 for a review of the latter). If so, this would limit post--minimum neutron--star systems to the rare examples first coming into contact at periods below the period gap, i.e. with $P_i \la 2$~hr. Of course there can be no such effect for black--hole LMXBs. At present we do not have any clear idea of the mean repetition time for the faint GC transients. The arguments above show that there are interesting consequences whatever this number turns out to be. Extensive X--ray monitoring of the Galactic Centre region clearly has much more to tell us about the stellar populations there. \section{Acknowledgment} I gratefully acknowledge the support of a PPARC Senior Fellowship. 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