A large population of neutron star low-mass X-ray binaries with long outburst recurrence time ?

Low-mass X-ray binaries (LMXBs) with neutron stars show quite different features which depend on the rate of mass transfer from the donor star. With a high transfer rate the Z sources are in a persistent soft spectral state, with a moderate rate the transient Atoll sources have outburst cycles like the black hole X-ray binaries. The observations document very long outburst recurrence times for quite a number of sources. We follow with our computations the evolution of the accretion disc until the onset of the ionization instability. For sources with a low mass transfer rate the accumulation of matter in the disc is essentially reduced due to the continuous evaporation of matter from the disc to the coronal flow. Different mass transfer rates result in nearly the same amount of matter accumulated for the outburst which means the outburst properties are similar for sources with short and sources with long outburst cycles, contrary to some expectations. Then of systems with long recurrence time less sources will be detected and the total population of LMXBs could be larger than it appears. This would relieve the apparent problem that the observed number of LMXBs as progenitors of millisecond pulsars (MSP) is too small compared to the number of MSP. Concerning the few quasi-persistent sources with year-long soft states we argue that these states are not outbursts, but quasi-stationary hot states as in Z sources.


INTRODUCTION
In low-mass X-ray binaries matter is transferred from a low-mass star onto a black hole or a neutron star.The mass overflow can be caused by the evolution of the donor star or loss of angular momentum due to either gravitational radiation or magnetic braking in binaries with short orbital periods.In most cases matter flows via an accretion disc toward the compact object.
The vertical structure of an accretion disc in close binary stars is determined by the amount of matter accumulated in the outer disc and can be either a hot or a cool state.For a certain range both states are possible and the ionization instability can trigger the transition from an initially cool state into a hot state, the outburst.This instability was first understood for accretion onto a white dwarf in cataclysmic variables (Meyer & Meyer-Hofmeister 1981), and later recognized as essential for the evolution of X-ray binary stars (Meyer-Hofmeister & Ritter 1993).In low-mass X-ray binaries (LMXB) the matter is accreted onto a black hole or a neutron star.The mass transfer rates depend on the mass and the evolutionary state of the secondary star.A detailed description of the computation of the vertical structure, the disk evolution and the instability behaviour of black hole and neutron star LMXBs is given in the review of Lasota (2001) for a wide range of parameters.A recent ⋆ emm@MPA-Garching.MPG.DE review of the evolution of neutron star LMXBs leading to the formation of millisecond pulsars is given by D' Antona & Tailo (2022).
An important process for disc evolution, especially in discs around a black hole or a neutron star, is the evaporation of matter from the disc to the corona (Meyer & Meyer-Hofmeister 1994;Liu et al. 1999;Meyer et al. 2000a,b).If the rate of mass transfer from the secondary star is low the evaporation of matter from the inner disc can lead to an extended truncation of the inner disc appearing as a hard spectral state.This leads to outburst recurrence times of decades of years.
There are differences between black hole LMXBs and neutron star LMXBs.A possible magnetic field in neutron star LMXBs, if strong enough, can cause a truncation of the inner disc within the Alfven radius (e.g.Frank et al. 2002) and influence the outburst behaviour.The irradiation from the surface of the neutron star is essential and its effect was early discussed by van Paradijs (1996); King et al. (1996King et al. ( , 1997) ) and Burderi et al. (1998); Different luminosities are observed in LMXBs.For black hole LMXBs the observations document large outbursts with luminosities of 10 37 − 10 39 erg s −1 after long quiescence with low luminosities of 10 30 − 10 33 erg s −1 (Yan & Yu 2015;Tetarenko et al. 2016).In neutron star binaries the outburst peak luminosities are about a factor of 10 lower than those of black hole sources and quiescent levels are typically near 10 32 − 10 33 erg s −1 (Tomsick et al. 2004).
The neutron star LMXBs show different phenomena according to the mass transfer from the secondary star.With the highest accretion rates Z sources are persistently in a soft spectral state.With also high rates the bright atoll sources remain almost permanently in the bright soft state (Muñoz-Darias et al. 2014).Transient atoll sources have lower accretion rates and are observed to change between soft and hard state in outburst cycles.Muñoz-Darias et al. (2014) studied in their investigation of hysteresis the spectral states of 50 neutron star binaries monitored by the Rossi X-ray Timing Explorer (RXTE), including also sources with very long recurrence time or long lasting outbursts.Recently Maccarone et al. (2022) discussed the observation of transient neutron star LMXBs and pointed out that about half of the known sources have recurrence times in excess of a decade (for outbursts at the sensitivity of MAXI) and argued that a much larger number of sources would not be discovered.With his investigation he clarified that long recurrence times are not only observed for black hole LMXBs.This was suggested by Hailey et al. (2018), but already in contradiction to an example of a binary evolution computed by Dubus et al. (2001).This gives rise to a study of the accretion disc evolution in neutron star LMXBs to understand the long recurrence times and the outbursts of these sources.
The aim of our computations is to study the disc evolution in binaries with long recurrence times.We describe in Sect.2 the computational procedure which is used to determine the accumulation of matter in the disc.In Sect.3 we show the results for disc evolution and reveal that during the quiescent phases evaporation of matter from the disc to the coronal flow plays an important role.The consequences of evaporation for the recurrence times and outburst strength and duration are discussed in Sect. 4. In Sect.5 we compare these results with observations for binaries with long recurrence times.In contradiction to perhaps expected powerful outbursts after a long lasting accumulation of matter the observed outburst luminosities are comparable to those in systems with short outbursts cycles.This affects the discovery rate of LMXBs and is of importance for the total number of LMXBs.In the context of the recyling scenario (D'Antona & Tailo 2022) the population of LMXBs as the progenitors of MSPs should be comparable with the population of MSPs.But too few LMXBs are observed.In Sect.6 we try to explain the nature of very long outbursts of quasi-persistent X-ray transients (Wijnands et al. 2003), which are probably not outburst cycles, but sources in a hot quasi-stationary state of the disc.Conclusions follow in Sect.7.

MODELLING THE ACCRETION DISC EVOLUTION
The structure of accretion discs in neutron star LMXBs shows similar features as that of discs around black holes and also in cataclysmic variables around white dwarfs.A standard modelling is adequate including the gravity of the compact primary star and additional features as the effect of a neutron star magnetic field and the radiation from the neutron star surface.Taking into account the influence of irradiation on the transient nature the ionization instability is expected as the cause for the outburst behaviour for all these binaries (van Paradijs 1996).
In quiescence the structure of accretion discs around a neutron star or a black hole primary for most binaries consists of a geometrically thin disc together with a hot corona, an inner ADAF (advectiondominated accretion flow, developed by Narayan, for a review see Narayan et al. (1998)).As studied originally for discs in cataclysmic variables and applied to black hole LMXBs matter can be evaporated from the cool thin disc to a coronal flow in a "siphon-flow" process (Meyer & Meyer-Hofmeister 1994).The evaporation then leads to the truncation of the thin disc, which depends on the mass flow rate in the thin disc.If the accretion rate is low the innermost part is completely evaporated into an ADAF and the spectrum is hard.Otherwise, the disc reaches inward to the last stable orbit and the spectrum is soft.
Our main computations concern the accumulation of matter in quiescence.We take the computer code as composed for studying the long recurrence times of outbursts in black hole LMXBs, especially the disc evolution of A0620-00 (Meyer-Hofmeister & Meyer 1999).We use for the cool state a viscosity parameter 0.05, which led for A0620-00 to a good agreement of the total matter accumulated and the recurrence time with observations (for a discussion of very low values see Sect.5).For the viscosity in the hot state we use the value 0.2.The physics of the interaction of disc and corona, which causes the evaporation of matter from the disc to the corona, was included as described in Liu et al. (1997).The critical values of surface density Σ A (r) and Σ B (r) for an unstable accretion disc structure are taken from the relations derived by Ludwig et al. (1994).These relations were used for the computation of the evolution of the disc in cataclysmic variables (Liu et al. 1997) and black hole binaries (Meyer-Hofmeister & Meyer 1999).
with M 1 the primary star mass, r the disc radius, α h and α c the vis- cosity values for the hot and the cool state respectively.During the disc evolution the inner radius is determined by the evaporation process within short time, independent of the initially assumed value.The outer boundary of the disc is given by the tidal radius or for small mass ratios by the 3:1 resonance between Kepler binary period and Kepler rotation in the disc, which ever is smaller (Frank et al. 2002;Mennickent et al. 2016).The resonance transfers any surplus of angular momentum outflow in the disc to the orbit (Whitehurst 1988;Lubow 1991).The extension of the disc depends on the mass ratio.For the neutron star mass we take 1.4M ⊙ .We assumed the masses of the secondary stars based on the investigation of orbital period changes during evolution by Piro & Bildsten (2002) (for short periods only auxiliary).
The accumulation of matter in the disc during a long lasting quiescence essentially depends on the evaporation process: the corona is fed by matter from the disc underneath, so that an equilibrium establishes between the cool accretion stream and the hot flow.We use the results derived by Liu et al. (1995) applied to LMXBs (Meyer-Hofmeister & Meyer 1999).
With evaporation the evolution of the surface density is governed by conservation of mass and angular momentum, where ν is the kinematic viscosity, ṁevap is the evaporation rate per unit surface area, which is related to the integrated evaporation rate Ṁevap (see Liu et al. 1997).When the surface density reaches Σ B an outburst is triggered.Additional effects in disc evolution are caused by irradiation or a magnetic field of the neutron star.
The effect of irradiation from the neutron star surface on the outburst behaviour is difficult to include as shown in the detailed review by Dubus et al. (2001).It was shown that irradiation is negligi-ble for the accumulation of matter in quiescence with low accretion rates.But irradiation has an effect on the outburst behavior and on the recurrence time because after an irradiation-controlled outburst the amount of matter in the disc is low (Dubus et al. 2001).We assume for our computations that only a small amount of matter is left over in the disc after the outburst.We take the initial distribution Σ(r) = 0.1 Σ A (r).For low mass transfer rates the surface density in the outermost region then can be too low for the propagation of the heating front to the outer edge of the disc, and due to the matter remaining in the cool outermost part the quiescent time until the next outburst would be somewhat shorter.
An additional feature of neutron star LMXBs is a magnetic field of the neutron star, which also can cause a truncation of the disc within the Alfven radius (e.g.Frank et al. 2002).Such a disc truncation was discussed for 4U 1608-52 by van den Eijnden et al. (2020).Moreover it is possible that the accreting matter is forced to follow the magnetic field lines of a rapidly rotating magnetic neutron star known as propeller effect (Illarionov & Sunyaev 1975).The efficiency of the propeller effect was discussed by Menou et al. (1999, and references therein) for neutron star and black hole sources.It depends on the magnetic field strength and the spin velocity of the neutron star and further assumptions of the properties of the binary.The fraction of matter finally accreted on the surface of the neutron star could be so low that this effect is more important than evaporation.For black hole sources without a solid surface only the possibility remains that the wind existing with the coronal flow reduces the matter accretion onto the compact object.A further process limiting the accretion of matter onto the neutron star could be, together with a stop of the mass transfer from the secondary star, the ignition of the radio pulsar which sweeps away the mass at the inner Lagrangian point as discussed by Burderi et al. (2001).

COMPUTATION OF THE RECURRENCE TIME
The accumulation of matter in the disc depends on the rate of mass transfer from the secondary star and only low or very low mass transfer rates will lead to long quiescent phases (see Dubus et al. (2001) for a LMXB example).Low mass transfer rates are confirmed theoretically as shown by Ritter (1999) who gave an analytical formula for the rate of nuclear driven mass transfer from a Roche lobe filling red giant.Observations document very low accretion rates in some cases as listed by Watts et al. (2008) in their investigation of LMXBs on the selection for the Advanced LIGO observing run.
For the computation of the evolution of the accretion disc during quiescence we have chosen the parameters neutron star mass and disc size derived for MXB 1659-298, a source for which a very low X-ray flux was clearly detected by Wijnands et al. (2003).MXB 1659-298 is a source with a long recurrence time of 14 years: After the launch of RXTE (Lin et al. 2019) two outbursts 1999/2001 (Wijnands et al. 2003) and 2015/2017 (Iaria et al. 2019) occurred.The orbital period is 7.11 hours, typical within the range of neutron star LMXBs as described by Lin et al. (2019).For our computations we take 1.4 M ⊙ for the mass of the neutron star, 0.7 M ⊙ for that of the secondary star, and an orbital period of 7 hours, with which the size of disc is determined.The neutron star mass could be higher as pointed out by Özel & Freire (2016) and by Watts et al. (2008).The secondary star mass of 0.7M ⊙ is derived from the study of Piro & Bildsten (2002), which was also considered by Watts et al. (2008) to lie in the range between 0.1 and 0.78 M ⊙ and by Iaria et al. (2018) in the range between 0.3 and 1.2 M ⊙ .
With the chosen parameters for the neutron star and secondary star mass, the orbital period, and the mass transfer rate at the outer boundary, the diffusion equation ( 3) is integrated with an assumed initial radial distribution of surface density (for details see Liu et al. 1997), and thus the evolution of the matter distribution in the disc can be determined as shown in the following figures.Fig. 1 shows how the matter distribution in the disc evolves until the surface density reaches the critical value Σ B for the onset of the instability.For a low accretion rate the evaporation of matter to the coronal flow causes a quite large inner hole filled with the ADAF.With the accretion rate of Ṁtransfer = 5.52 10 −11 M ⊙ /yr the outburst is triggered after a recurrence time of 14 years at the distance of r = 10 9.7 cm.In the inner region, r < ∼ 10 9.4 cm, only a corona/ADAF flows toward the neutron star as a consequence of complete evaporation.
Fig. 2 shows the accumulation of matter for the lower accretion rate of Ṁtransfer = 4.6 10 −11 M ⊙ /yr.The balance between storage of matter in the disc and evaporation establishes an inner hole of about the same extension as for the higher rate.The critical surface density to trigger an outburst is reached at about the same distance as with the higher rate.The recurrence time is longer, about 31 years.The time needed to reach the instability is very long since the chosen mass transfer rate is only slightly higher than the evaporation rate at the truncation radius.
In neutron star LMXBs with shorter orbital periods the accumulation of matter can lead earlier to an outburst.To study this ef- fect we compute the disc evolution also for an orbital period of 3 hours and and we take again the mass transfer rate of Ṁtransfer = 5.52 10 −11 M ⊙ /yr (used for the accumulation shown in Fig. 1).We take a smaller secondary star mass, 0.3M ⊙ .The smaller mass ratio leads to a limitation of the disc size due to the 3:1 resonance (see Sect.3).The recurrence time then is much shorter, only 6.16 years.
The accumulation of matter for this case is shown in Fig. 3. Generally for binaries with shorter orbital periods, where the two components are closer and the discs are smaller, enough matter to trigger an outburst can be accumulated within shorter time.
For a binary with a longer orbital period, e.g. 12 hours, the recurrence times are longer since more matter is stored in the larger disc until an outburst is triggered.But independent of the disc size evaporation reduces the accumulation of matter.
Fig. 4 shows how the recurrence times for different orbital periods depend on the mass transfer rate.A recurrence time of a few years would be expected for quite a range of mass transfer rates, but for low rates a much longer time since only a small fraction of the transferred mass is accumulated in the disc.For even lower rates the transferred mass is balanced by the evaporation and thus the source could stay in the stationary cool state without an outburst.The vertical lines in Fig. 4 give the limiting rate for the occurrence of the ionization instability.
The onset of an outburst from a stationary cool state might in rare cases be triggered in a different way: For sources with mass transfer rates so low that they remain always in the cool state, random fluctuations of mass transfer could cause a transition to the hot state.Such a change of the disc structure then appears as an outburst after long recurrence time.
We have not included magnetic fields of neutron stars for our modelling of the accretion disc structure.Using observations Asai et al. (2016) investigated the effect of neutron star magnetic fields of LMXBs and found that in general the magnetic fields are weak.X-ray spectral observations, especially new data from NuS-TAR (Nuclear Spectroscopic Telescope Array) and NICER (Neutron star Interior Composition Explorer) for several sources, e.g. for atoll sources (Ludlam et al. 2019), led to the determination of the inner edge of the accretion disc and the determination of the magnetic field strength responsible for this truncation.The computation of the disc evolution for the parameters corresponding to a considered binary then would allow to investigate whether evaporation or the effect of the magnetic field are more important.Degenaar et al. (2017) pointed out that the disc truncation determined by modelling the reflection spectrum of J17062-6143 could be due to either evap-

DISTINCT OUTBURST PROPERTIES PREDICTED BY THE EVAPORATION MODEL
The evaporation is an important process that influences the accumulation of matter during quiescence and the outburst behaviors.The most distinct property is that during quiescence matter continually evaporates from the disc, leading to a longer accumulation time until an outburst is triggered.This effect is particularly important at low transfer rates as it takes away a significant fraction of the transferred gas flow during the evolution.If the rate is so low that all the transferred gas can be evaporated, no outburst can be triggered.On the other hand short recurrence times are expected for high transfer rates where the evaporation only takes away small fraction of the transferred matter and the influence on the accumulation is small.The effect of evaporation on the total amount of matter stored in the disc when the instability sets in is essential for the outburst behavior of LMXBs with low mass transfer rates.Our computations result in the following amount of accumulated matter: For the higher mass transfer rate of 5.52 10 −11 M ⊙ /yr an amount of 9.16 10 23 g, for the lower rate of 4.6 10 −11 M ⊙ /yr an amount of 9.52 10 23 g.It is interesting that about the same amount of matter is accumulated in the disc for the different mass transfer rates (see the distributions of the surface density at ignition in Figs. 1 and 2).Otherwise one finds the total amount of matter transferred from the secondary star to the disc, 1.5 10 24 g for the higher rate and about double the amount for the lower rate during the longer quiescence.This means that a fraction of 61% of the transferred matter is accumulated in the disc for the higher rate and only 34% for the lower rate.The evaporated gas flows inwards as an ADAF, of which a significant fraction is lost in wind at large distances (Yuan & Narayan 2014) and the remaining part could flow along magnetic lines at small distances or accrete directly to the neutron star depending on the strength of magnetic field and the spin.Eventually a propeller effect can determine the flow of the matter.
The accumulation of matter in black hole LMXBs is reduced by evaporation in the same way, during also long quiescent phases as computations for A0620-00 show (Meyer-Hofmeister & Meyer 2000), with a similar fraction of finally accumulated matter in the disc though for different parameters.Due to a higher mass transfer rate and the larger size of the disc then a bright outburst occurs (Lloyd et al. 1977).
The outcome of the same amount of matter accumulated in the discs of different neutron star LMXBs at the onset of the instability in spite of different (low) transfer rates implies about the same outburst strength, peak luminosity and duration, contrary to what might be expected due to the long lasting accumulation of matter.This means that the systems with long recurrence times are visible in outburst only during a shorter time and one might expect that many LMXBs with long recurrence time are not yet detected.
We want to point out that the effect of the viscosity parameter is distinct from that of evaporation.A smaller viscous parameter means slower accretion and thus more gas accumulation in the outer region until an outburst can start.With a low transfer rate the accumulation takes also a long time, but the large amount of gas accumulated in the outer disc could lead to outburst properties different from those caused by evaporation, though both of which can predict a long recurrence time.
In the context of viscosity one might ask whether the long recurrence times in LMXBs could be caused in a similar way as those in WZ Sagittae stars, a subgroup of cataclysmic variables, where the primary star is a white dwarf.But the large amount of matter accumulated in these binaries demands a very low viscosity.A sharp enhancement of mass transfer is excluded (for a discussion see Smak (1993); Meyer-Hofmeister et al. (1998); Meyer & Meyer-Hofmeister (1999); Hameury et al. (2000)).

COMPARISON WITH OBSERVATIONS
We want to compare with observations for long recurrence times and with the properties of the outbursts after a long recurrence time.Maccarone et al. (2022) selected a sample of 20 transients and came to the conclusion that at least 40% of transient LMXBs have recurrence times longer than 10 years.Lin et al. (2019) investigated the outburst rate based on the X-ray monitoring data from Swift/BAT, RXTE/ASM and MAXI in the past few decades and found that 10 of the 19 neutron star LMXBs had only one outburst.Watts et al. (2008) summarized the properties of 35 accreting neutron stars in our Galaxy for the detection of gravitational wave emission and derived estimates for the average long-term and outburst flux.They found for most of the sources with rare outbursts an outburst duration not particularly long, only 10 to 60 days (as also observed for other transient sources (Muñoz-Darias et al. 2014)) and peak luminosities not especially high, only around 1/10 of the peak luminosity of the well-known source Aql X-1.The average outburst flux is 1.3 10 −8 erg cm −2 s −1 (Watts et al. 2008, Table 1) which corresponds to a luminosity of 3.2 -3.7 10 37 erg s −1 for the usually assumed distance of 4.5-5 kpc (the outburst history is shown in the work of Ootes et al. (2018)).
A subgroup of LMXBs with, for most sources, only one observed outburst since their discovery during the last 20 years are the accretion powered millisecond pulsars (AMXP).The general properties were reviewed by Patruno & Watts (2021) and Di Salvo & Sanna (2022).The outburst luminosities were faint and most sources did not not reach a soft spectral state.The outbursts of AMXPs last from less than a week up to about 60 days in most sources (Marino et al. 2019).These observations are in good agreement with the theoretically derived expectations for the outburst behaviour for the amount of accumulated matter in the disc during quiescence.Only the source HETE J1900.1-2455, also an AMXP, seems to have a remarkable complex activity different from the other sources.
It is possible that the mass transfer is non-conservative in LMXBs.Marino et al. (2019) investigated such a situation for accreting millisecond pulsars and found strong evidence for this in five sources out of his sample of 18 sources.

THE DISC STRUCTURE IN QUASI-PERSISTENT TRANSIENTS
Besides outburst cycles with long recurrence time a few neutron star LMXBs show a definitely different long-term behaviour.These sources, designated as quasi-persistent sources (Wijnands et al. 2003), are observed in a soft spectral state for years with an about constant luminosity.Detailed observations document the long outbursts of the three sources EXO 0748-676, KS 1731-260 and HETE J1900.1-2455lasting 23, 12.5 and 10 years (the outburst duration might even be longer since the sources were discovered already in outburst).With the final transition to a lower luminosity the sources then enter a continuous quiescent state.The luminosity decrease between these two states is less than the decay in regular outburst cycles of neutron star LMXBs.It was for EXO 0748-676 a factor of 4 (Muñoz-Darias et al. 2014, Fig. 1).For the understanding of the accretion process also the quiescent state is of interest, for HETE J1900.1-2455discussed by Degenaar et al. (2021) and for EXO 0748-676 by Parikh et al. (2021).
What is the nature of these sources?The computation of matter accumulation in the disc shows that during quiescence due to evaporation only a limited amount can be stored, insufficient for a very long outburst.Therefore continually enough matter must be transferred from the secondary star to the disc at a high rate, i.e. that the sources are in a quasi-stationary hot state.This is the case also in other sources within the regime of neutron star LMXBs, known as bright atolls or Z sources, which remain almost permanently in a bright soft state.The source XTE J1701-462 even displayed a change from Z and atoll phenomenology to a regular outburst (Homan et al. 2007;Homan et al. 2010).
A steady soft state is possible if the mass transfer rate from the secondary star is high enough so that the entire disc is at the upper, stable branch of the S-shaped curve of the viscosity-surface density relation (Ludwig et al. 1994).A limit cycle behavior appears when the mass transfer rate lies within some range.At this range the local unstable behavior of some annuli influences its neighbors and produces a coherent behavior of the entire disc, leading to outburst cycles with no need of variation of the transfer rate.A change from a soft spectral state to a hard state can also be triggered if the mass transfer rate decreases from sufficiently high to low values.When the surface density decreases with the decreasing transfer rate to a value below the critical surface density Σ A (r), usually first fulfilled at the outer disc edge, a cooling front moves inward and establishes a cold disc structure.This probably happens in quasipersistent sources and is caused by a temporary decrease of the mass transfer rate after a long lasting hot state.
In the long lasting hot state the surface density Σ(r) increases inward and the disc extends down to the ISCO.The surface density distribution during the quasi-stationary hot, soft state is different from that in the hard state (in quiescence) shown in Figs. 1 to 3, where the inner region is filled by an ADAF.A minimal mass transfer rate ṀA (r) which allows a stationary hot state can be determined as in the work of Ludwig et al. (1994, Fig.13) corresponding to the relations for the critical values of surface density.The limiting rate depends on the disc size (therefore on the mass of the secondary star).
For a binary with a short orbital period of 3 hrs and a secondary star mass of 0.3M ⊙ (example in Sect.3) the disc size is limited by the 3:1 resonance and the limiting mass transfer rate is lower.Then for low transfer rates more likely a quasi-stationary state can exist.
This might be the case for the source with the shortest orbital period HETE J1900.1-2455(Šimon 2018) and an average value of the mass transfer rate during the 10 years of outburst of 3.5 10 −10 M ⊙ /yr (Degenaar et al. 2017).It was claimed by Papitto et al. (2013) that the source behaves as a typical atoll source.The properties of EXO 0748-676 are similar, with the longest observed soft state of 23 years, an orbital period of 3.82 hrs and an accretion rate of 0.022 ṀEdd over the course of the RXTE observations, which is about 7 10 −10 M ⊙ /yr (Galloway et al. 2008).For both of these sources the rates determined from the observations are probably high enough to allow a stationary hot state.

CONCLUSIONS
We investigate the evolution of the accretion disc until the onset of an outburst for binaries with long recurrence time and therefore low mass transfer rates from the secondary star.The existence of low rates is clearly confirmed by observations (Watts et al. 2008;Jonker et al. 2006).A significant result of our computations of disc evolution for low mass transfer rates is the essentially decreased accumulation of matter in the disc due to evaporation of matter from the geometrically thin, optically thick disc to a coronal flow toward the neutron star.
It is important that different mass transfer rates lead to nearly the same amount of finally accumulated matter for the outburst.This means that the outburst after the long quiescence has only a standard peak luminosity and duration in most sources similar or less than of the outbursts in sources with shorter recurrence time, e.g.Aql X-1 (Campana et al. 2013).Due to the occurrence of only standard outbursts after long time, maybe after 10 or 20 years, less such sources will be detected and quite a number remain undetected and it is difficult to determine the extent of the population of neutron star LMXBs.
Since in the context of the recycling scenario (Di Salvo & Sanna 2022) neutron stars are spun up to millisecond pulsars (MSP) the number of observed LMXBs as progenitors of MSPs (taken the birthrate of LMXBs) should be comparable with this population.But the population is smaller at least a factor of 2.5 (D'Antona & Tailo 2022) or even a factor of 5 (Maccarone et al. 2022).This "birthrate problem" would be dimineshed if a large number of undetected LMXBs exist as already claimed by Maccarone et al. (2022) and as our computations suggest.
The main differences between the neutron star LMXBs and the black hole LMXBs are whether there exist a solid surface and magnetic fields.In addition to the relevant effects discussed in Sect.2, there is a new effect concerning the evaporation.During quiescence the evaporated gas flows inwards as an ADAF with an accretion rate ∼ 10 −11 M ⊙ /yr.This hot flow is mostly lost in wind at large distances as shown by MHD simulations (for a review see Sect.3.4 of Yuan & Narayan 2014), with the accretion rate decreasing with decreasing distance in a power law.Further diversion of the accreting flow occurs in the inner region, affected by the magnetic field.Therefore, the mass flows to the neutron star surface could be at a rate less than ∼ 10 −13 M ⊙ /yr, which results in a luminosity from the boundary layer of the solid surface within the observational range.Uncertainty in this estimation of the quiescent luminosity lies in the true fraction of evaporated gas flowing to the neutron star surface.
Concerning the year-long outbursts of quasi-persistent X-ray transients we want to point out that the so-called outbursts are a feature very different from long recurrence times, so to speak an opposite state.In quasi-persistent sources the mass transfer from the secondary star seems to be strong enough to maintain a soft spectral state for a long time, that is to keep the surface density above the critical value Σ A in the curve of the viscosity-surface density relation everywhere in the disc.As shown such a quasi-stationary hot state is more likely possible for sources with a short orbital period where the disc is smaller.The source HETE J1900.1-2455 with the shortest orbital period of 1.39 hrs and EXO 0748-676 with an orbital period of 3.82 hrs might be examples.Further observations of long-term phenomena might help to better understand these features.

Figure 4 .
Figure 4. Dependence of the recurrence time on mass transfer rate and orbital period