The Formation of Low-Mass Transient X-Ray Binaries

We consider constraints on the formation of low-mass X-ray binaries containing neutron stars (NLMXBs) arising from the presence of soft X-ray transients among these systems. For a neutron star of mass M1 ≃ 1.4 M☉ at formation, we show that in short-period (≲1-2 day) systems driven by angular momentum loss these constraints require the secondary at the beginning of mass transfer to have a mass of 1.3 M☉ ≲ M2 ≲ 1.5 M☉ and to be significantly nuclear evolved, provided that supernova (SN) kick velocities are generally small compared with the pre-SN orbital velocity. As a consequence, a comparatively large fraction of such systems appear as soft X-ray transients even at short periods, as observed. Moreover, the large initial secondary masses account for the rarity of NLMXBs at periods P ≲ 3 hr. In contrast, NLMXB populations forming with large kick velocities would not have these properties, suggesting that the kick velocity is generally small compared with the pre-SN orbital velocity in a large fraction of systems, consistent with a recent reevaluation of pulsar proper motions. The results also place tight constraints on the strength of magnetic braking: if magnetic braking is significantly stronger than the standard form, too many unevolved NLMXBs would form; if it is slower by only a factor of ≃4, no short-period NLMXBs would form at all in the absence of a kick velocity. The narrow range for M2 found for negligible kick velocity implies restricted ranges near 4 M☉ for the helium star antecedent of the neutron star and near 18 M☉ for the original main-sequence progenitor. The pre-common-envelope period must lie near 4 yr, and we estimate the short-period NLMXB formation rate in the disk of the Galaxy as ~2 × 10-7 yr-1. Our results show that the neutron star mass at short-period NLMXB formation cannot be significantly larger than 1.4 M☉. Systems with formation masses of M1 ≲ 1.2 M☉ would have disrupted, so observations implying M1 ~ 1.4 M☉ in some NLMXBs suggest that much of the transferred mass is lost from these systems.


INTRODUCTION
In a recent paper Kolb, & Burderi hereafter (King, 1996, we considered the disk instability model for soft Paper I) X-ray transients (SXTs). Following Paradijs we van (1996) noted that X-ray irradiation of the disk surface tends to suppress the instability, so that SXT behavior requires that the mass transfer rate is below a critical limit, which is itself a rising function of orbital period P. For days, P Z 1È2 mass transfer is driven by the nuclear expansion of an evolved secondary and proceeds at low enough rates that almost all these systems appear as SXTs. For days, P [ 1È2 mass transfer is driven by angular momentum losses ; in we showed that the resulting rates are higher than Paper I the critical rate, hence too high for SXT behavior unless the secondary star is already somewhat nuclear evolved before mass transfer begins. This e †ect is particularly marked in a neutron star low-mass X-ray binary (neutron star LMXB, or NLMXB) : if we write (see for the ratio of the Paper I) m ü 2 secondaryÏs mass to that of a zero-age main-sequence (ZAMS) star Ðlling the Roche lobe at the same orbital period, SXT behavior requires By contrast, we m ü 2 [ 0.25. found a much weaker limit for SXT behavior in m ü 2 [ 0.75 an LMXB with a 10 black hole primary. These limits M _ agree with the extreme mass ratios always found in SXTs. In we suggested that these limits also o †ered a Paper I potential explanation for the prevalence of black hole systems among SXTs (eight out of 14 with known P) as compared with persistent LMXBs (one out of 29 with known P). As NLMXBs can be reliably identiÐed by the presence of X-ray bursts, we consider this problem further by investigating constraints on NLMXB formation.

NLMXB FORMATION
The requirement that short-period SXT secondaries should be signiÐcantly evolved is a powerful clue in investigating LMXB (and particularly NLMXB) formation. For example, the very fact that at least four out of 22 NLMXBs in the relevant period range are transients (0748-676, Cen X-4, 1658-298, Aql X-1 ; see Paradijs must imply van 1995) that NLMXB secondaries usually have masses M 2 Z 1 M _ at the onset of mass transfer. Lower initial mass secondaries would be essentially unevolved when mass transfer began, and a large population of them would require the four known SXTs among NLMXBs to be accompanied by a far greater number of persistent NLMXBs than observed. However, to get any NLMXBs at all at days simul-P [ 1È2 taneously requires that some of the secondaries initially have masses so that angular momentum loss M 2 [ 1.5 M _ , via magnetic braking can shrink the binary. In addition, for these NMLXBs with a main-sequence donor M 2 Z 1.5 M _ , will be unstable to thermal-timescale mass transfer once contact is reached ; they transfer mass at super-Eddington rates and do not appear as NLMXBs & (Kalogera Webbink Given these restrictions, all NLMXBs 1996a). with days (and indeed most at longer periods too) P [ 1È2 must apparently form with in a very restricted range of The properties and the very existence of this group depend sensitively on the efficiency of magnetic braking. Below we investigate in detail constraints on NLMXBs forming after a common-envelope (CE) phase and a subsequent helium star supernova (SN). We consider Ðrst the fundamental case of a spherically symmetrical SN (°2.1), and we shall show that in this case secondary stars in NLMXBs are indeed signiÐcantly nuclear evolved at the onset of mass transfer. A generalization to the more realistic case when the neutron star receives a kick velocity during the SN reveals that the fraction of systems with (°2.2) unevolved secondaries increases with increasing magnitude of the kick. For a given magnetic braking strength, this places limits on the magnitude of the kick velocity. Possible alternative evolutionary channels leading to NLMXB formation are discussed in°2.3.

NL MXBs from Spherically Symmetrical
Helium Star Supernovae To discover any limits on the initial mass of the M 2 secondary, we consider the constraints on the formation of LMXBs from helium star ] main-sequence (MS) binary remnants of common-envelope evolution in the case of a spherically symmetrical SN. list a number of requirements that the (1996a) various progenitor stages must satisfy in order to lead to LMXB formation. Of these, three turn out to be crucial in Ðxing M 2 : 1. The postÈcommon-envelope binary must be wide enough to allow the helium star to evolve to core collapse (requirement 6 of WK).
2. The binary must survive the supernova event resulting from the core collapse of the helium star (requirement 7 of WK).
3. After the helium star progenitor of the LMXB primary explodes as a supernova, the binary must be able to reach interaction within the age of the Galaxy (requirement 8 of WK).

If
is large enough and the post-SN separation is wide M 2 enough, requirement 3 will occur through the nuclear expansion of the secondary. Such systems require M 2 Z 1 to give main-sequence lifetimes less than the age M _ t MS t Gal of the Galaxy and must have days at the onset of P Z 1È2 mass transfer ; P will increase further as the secondary expands. However, if is less than this and/or the M 2 post-SN binary is narrower, requirement 3 can come about through orbital angular momentum loss. After reaching contact with an initial orbital period of order 1È2 days, such systems evolve to either longer day) or shorter (P Z 1 day) periods depending on the competition between (P [ 1 nuclear evolution and angular momentum loss (Pylyser & Savonije 1988a, 1988b.
For the remainder of this paper we concentrate on this latter group of systems, which are the only ones that can populate the short-period day) region. It is conve-(P [ 1È2 nient to apply constraints 1È3 in a di †erent order. From constraint 3 we require where are the timescales for detached orbital evolution under magnetic braking and gravitational radiation, with c as a dimensionless efficiency parameter for the former case. These shrink the post-SN binary from its initial period P to the value at which it Ðrst comes into contact with its P c Roche lobe etc.). If We assume that the secondary has a structure such that a magnetic stellar wind brakes its rotation ; since we are considering short-period LMXBs in which the secondaries are not too far from the main sequence, this requires 0.3 M _ [ Furthermore, we assume that the secondary M 2 [ 1.5 M _ . is tidally locked to the orbit such that magnetic braking removes orbital (rather than only rotational) angular momentum. Applying simple scalings for the tidal synchronization timescale (e.g., suggests that this is Tassoul 1995) the case for the detached systems in the period range under consideration (see below). The efficiency parameter c in allows us to test the sensitivity of our results to equation (1) the strength of magnetic braking. The standard case c \ 1 corresponds to the description by & Zwaan Verbunt (1981) when the radius of gyration is set to (0.2)1@2 and the calibration parameter to unity.
Requirement 3 thus gives the longest post-SN period compatible with short-period LMXB formation. It is easy to show that the longest possible value of this period is given by equating to with so that the term t MB t Gal P ? P c , in square brackets in is unity. KeplerÏs law now equation (1) gives a condition on the separation of the tidally a postSN circularized post-SN binary, i.e., where yr. t G \ t Gal /1010 Before exploiting further, we apply require-equation (3) ment 2. In a spherically symmetrical supernova explosion, it is well known that the remnant binary is disrupted if more than one-half of the binary mass is removed ; i.e., unless Obviously is equivalent to e \ 1. The pre-SN equation (5) separation and the post-SN separation of the tidally circularized orbit are related by (e.g., Verbunt 1993) Using equations and we Ðnd that the pre-SN (3) (7) Roche lobe radius of the He star must obey where we have assumed that occupies one-half of the R R pre-SN binary separation.
Finally, requirement 1 demands that the largest radius of the helium star must Ðt inside the pre-SN R max (He) binary. The evolution of the helium star is very rapid compared with the orbital evolution, so we assume that is a preSN constant and the condition is (If this condi-R max (He) \ R R . tion fails, it is possible that some LMXBs nevertheless form after a second common-envelope phase ; see et al. Iben 1995. We note, however, that these authors assume an unusually efficient envelope ejection compared with the standard common-envelope description, below. Application eq. [21] of the latter would almost always lead to a merger. Hence, we neglect this group of systems.) et al.
for (see also m He [ 2.5 M _ Habets 1986). Hence, combining with we see equation (9) equation (8) that requirements 1 and 3 demand thereby deÐning a lower limit for for given (or a m 2 m 1 , m He lower limit for for given m He m 1 , m 2 ).

Mass L imits for NL MXBs
Observational evidence strongly suggests that neutron stars emerge from the supernova that creates them with a mass shows the constraints (eqs. m 1 \ 1.4. Figure 1 [5], for this mass, with and c \ 1. (For where magnetic braking is thought not to operate, eq. [10] was replaced by the corresponding constraint using t GR instead of As can be seen, the two constraints cross at a t MB .) value Hence, º 1.22 for m 2 \ 1.22, m He \ 4.02. m 2 NLMXB formation through this channel. We now go one step further and ask under what conditions the binary can begin mass transfer with a secondary that has spent less than a fraction f (0 \ f ¹ 1) of its lifetime on the main t MS sequence, i.e., with Clearly this is a more stringent requirement than requirement 3, where the right-hand side is M _ star mass). These limits restrict the allowed range of NLMXB formation to the small triangular (shaded) area enclosed by these curves. Also shown is the lower limit for (or if the systemÏs age is less than 0.25t MS mass transfer turn-on. The allowed range for short-period NLMXBs with main-sequence donors is only the narrow-hatched area. and following the same line of reasoning that led to the constraint i.e., replacing in by (eq. [10]), t G equation (10) we Ðnd ( f/m 2 3), which must hold simultaneously with equation (10). is best understood as a lower limit for the He Condition (13) (13) f \ 1 6 . Figure  shows that the critical line corresponding to would 1 f \ 1 6 be to the right of the dash-dotted line ; hence, no f \ 1 4 parameter space would be left for short-period NLMXB formation. Physically this means that because of the correspondingly slower evolution the systems would not reach contact within the age of the Galaxy.
This underlines the sensitivity of our result to the strength of magnetic braking for main-sequence stars in the critical mass range where the convective envelope disappears. To illustrate this further, we consider the lower limit a crit we obtain for the post-SN orbital distance, a postSNŵ hen the requirement is com- Note that this limit does not depend on magnetic braking and that is essentially determined by a crit m 2 (m 1 \ 1.4, 0 ¹ e ¹ 1). For NLMXBs with main-sequence donors of mass to exist at all, the detached evolution time m 2 t MB needed to shrink the orbit from into contact must be a crit shorter than In contrast, to ensure that a large fraction t MS . of the secondaries are signiÐcantly evolved when mass transfer turns on, must be longer than a certain t MB minimum fraction of This places a tight limit on f^1 4 t MS . the allowed strength of magnetic braking for systems with secondary mass initial orbital distance and neutron m 2 , a crit , star mass m 1 \ 1.4 : independent of the functional dependence of on mass t MB and period.

Mass L imits for Progenitor Systems
The narrow range also implies a very restricted range for from m He : Figure 1 we see that this must obey Wind mass loss carrying the speciÐc angular momentum of the mass-losing component increases the orbital separation as the inverse of the total mass. Hence, the MS parent binary orbit is closer by a factor of^0.9 than the immediate pre-CE binary orbit.
Simple treatments of common-envelope evolution (e.g., lead to the relation WK) where are the binary separation before and a preCE , a postCE after the common-envelope phase, a is the fraction of the orbital binding energy used to drive o † the envelope, j is a weighting factor for the gravitational binding energy of the envelope to the core, and is the fraction- Since there is little orbital evolution before the helium star undergoes its supernova explosion, and the post-SN separation is within a factor of 2 of the pre-SN separation, we can translate this into limits on or, equivalently, the a preCE orbital period of the progenitor binary. We get an upper limit from requirement 3 (see and a lower limit from eq. [8]), requirement 1, using and the fact that equation (9) m He \ 4.3. (This limit prevents NLMXB progenitors emerging from the common-envelope phase with the MS secondary already close to Ðlling its Roche lobe, which is condi-WKÏs tion 6 ; the requirement here gives a tighter limit. This in turn justiÐes our assumption above that the postÈSNperiod P was much larger than the contact period The P c .) resulting constraints are close, i.e., D2300 and D1300 R _ /a respectively.
The formation channel we have considered requires that the progenitor Ðlls its Roche lobe after core-helium exhaustion but before it would explode as an SN where *x denotes the allowed range of the quantity x. relies on standard assumptions for the dis-Equation (25) (5) keep together binaries that would have disrupted according to this inequality. The latter case requires a kick velocity comparable to the pre-SN orbital velocity of the companion, directed almost parallel to its instantaneous motion, and is therefore rather rare unless the kick velocity is close to an optimum value near the pre-SN orbital velocity. Second, no longer holds, and can be relation (7) a postSN either smaller or larger than (but never larger than a preSN These two e †ects not only widen the allowed region 2a preSN ). of NLMXB formation in the plane of M 2 [ M He Figure 1 but, in particular, allow the formation of NLMXBs with almost unevolved secondaries.
In the absence of a theoretical understanding of the origin, magnitude, and direction of kick velocities, the standard approach in population synthesis considerations (e.g., & Podsiadlowski et al. Brandt 1995 ;Terman 1996 ;& Webbink is to assume that the kicks are Kalogera 1996b) isotropic and that their magnitude derives from a given distribution function, characterized by a certain rms value As demonstrated extensively by, e.g., p k . Kalogera (1996a), the resulting probability distributions can be expressed in terms of the two governing dimensionless parameters, the ratio m of kick velocity to relative orbital velocity in the pre-SN orbit, and the ratio b of post-SN and ). Given the stochastic nature of the problem, only a full population synthesis can provide a quantitative estimate of the fraction of NLMXBs with signiÐcantly nuclear evolved secondaries at turn-on of mass transfer. To illustrate the main e †ect of kick velocities and to gain a very rough estimate of the maximum kicks we can tolerate and still maintain the large fraction of systems with evolved secondaries found for spherically symmetrical SNs, we make use of the analytic expression for the distribution of binaries over derived by under the a c \ a postSN /a preSN Kalogera (1996a) assumption of a Maxwellian kick velocity distribution. In we show the same limits as plotted in but Figure 2 is equivalent to b \ 0.5Èwas replaced by the [5])Èwhich line b \ 0.4, as the survival probability of these systems is only a factor of 2 smaller than the one for the most stably bound systems (corresponding to b^0.75). The resulting enclosed area in the plane can be thought of as M 2 [ M He representative for the e †ective parameter space of NLMXB formation.
Assuming that the area is a measure of the corresponding relative formation rate, suggests that in the case of Figure 2 standard magnetic braking systems with unevolved (eq. [1]) secondaries still constitute only a small fraction of NLMXBs for m \ 0.1, represent the majority for m \ 0.3, and entirely dominate for m \ 1.0.
More efficient magnetic braking would increase the dominance of unevolved systems even further, whereas a favorable combination of weaker magnetic braking and a kick velocity distribution with small (or moderate) mean velocity could ensure both the formation of the (otherwise forbidden) class of short-period NLMXBs and the predominance of nuclear-evolved main-sequence donors. However, a population subject to a large mean kick velocity would necessarily contain a large fraction of systems where the secondary is close to contact in the post-SN orbit and hence essentially unevolved at mass transfer turn-onÈwhatever the strength of magnetic braking. The reason for this is twofold. First, the limit formally allows post-SN (eq. [14]) orbital periods shorter than the contact period for P c and with strong kicks a large fraction of such m He Z 4.7, binaries would survive the SN. Second, the SN-induced orbital reduction factor is very small for the majority of a c systems.
In view of this, the large fraction of soft X-ray transients observed among NLMXBs provides a strong argument not only against a magnetic braking stronger than our standard case but also against a mean kick velocity of order (eq. [1]) 350È400 km s~1 (m^1) invoked by & Lorimer Lyne (1994) from observed pulsar proper motions.
shows that Figure 2 kick velocities must on average be small compared with the pre-SN orbital velocity, probably This is consistent m [ 0.1. with a more recent reevaluation of the initial velocity distribution of radio pulsars that conÐrms the existence of the high-velocity tail found by Lyne & Lorimer but suggests that the distribution has its maximum at zero velocity, hence a smaller average value & Phinney (Hansen 1996 ; Hartmann 1996).

Alternative Evolutionary Channels
Short-period NLMXBs might also form from systems with initially fairly massive main-sequence secondaries above the limit for thermally stable mass transfer, 1.5È2 [ provided these survive the initial phase of thermal m 2 [ 3, timescale (hence super-Eddington) mass transfer. (Systems with would probably develop a delayed dynamical m 2 Z 3 instability ; see Kalogera & Webbink Hjellming 1989.) pointed out that such systems could reap- (1996a, 1996b) pear as stable NLMXBs with a sub-Eddington mass transfer rate and donor mass An analysis similar 1 [ m 2 [ 1.5. to the one in (using instead of reveals that a°2.1.2 t GR t MB ) signiÐcant fraction of them would emerge from the CE phase almost semidetached. They would turn on mass transfer with an unevolved secondary, again in conÑict with the observed comparatively large fraction of transients among NLMXBs. Hence we conclude that this channel cannot contribute signiÐcantly to the formation of shortperiod NLMXBs.
Recently, has described yet another for- Kalogera (1996b) mation mechanism for LMXBs where no CE phase is involved. Instead, the orbital shrinkage is achieved by a suitably directed kick when the primary star in the wide progenitor binary explodes as an SN before it reaches its Roche lobe. Population synthesis models show (Kalogera that the production of short-period LMXBs via this 1996b) channel is altogether negligible compared with the standard He-star SN case if the kick velocities are large of order (p k 300È400 km s~1) but might account for a formation rate comparable to the one derived in if is close to an°2.1.2 p k optimum value^50 km s~1. However, in the latter case a relatively large fraction of LMXBs would start the X-ray phase with a very small (below 0.3 donor mass, hence M _ ) with essentially unevolved secondaries. Again, this suggests that the direct SN mechanism represents only a minor channel for the formation of short-period NLMXBs (whereas it might be important to produce long-period systems with days) even if kick velocities are gen-P d Z 100 erally small.

NLMXBs
The di †erence between the period histograms of LMXBs and cataclysmic variables (CVs), in which the primary is a white dwarf, has long been remarked (e.g., & Mason White den Heuvel & van Paradijs & van 1985 ;van 1988 ; Verbunt den Heuvel The most prominent di †er-1995 ; Kolb 1996). ence is a total (or near total) lack of LMXBs in the 80 \ P \ 120 minute region below the famous CV period gap. A K-S test reveals that there are also statistically sig-niÐcant di †erences above the gap. The hypothesis that the LMXB and CV samples are drawn from the same underlying distribution can be rejected at a conÐdence level greater than 99.99% in the period range 80 minutes to 2 days top panel) and at^99.96% in the range 3 hr to (Fig. 3,  2 days bottom panel). This cannot be explained by (Fig. 3, selection e †ects discriminating against short orbital periods, since many of the X-ray periods were turned up by satellites such as EXOSAT , which had a 4 day orbit. The di †erences conÑict with the simple picture of CVs and LMXBs as essentially the same in terms of their secular evolution, apart from the substitution of a white dwarf by a neutron star or black hole primary. However, our arguments above show that this simple picture is inaccurate, particularly for NLMXBs. CVs emerge from common-envelope evolution with a full range of secondary masses down to m 2 D 0.1. The vast majority of these stars are essentially unevolved, and the post-CE separations are so small in many cases that the secondaries are close to their Roche lobes. The majority of CVs probably start mass transfer at periods below the gap (more than 67% according to et al. King 1994). these features hold for NLMXBs, as we have seen : the secondaries are conÐned to the narrow range 1.3 [ m 2 [ 1.5 initially, a large fraction of them must be signiÐcantly nuclear evolved, the post CE (and SN) Roche lobes are considerably larger than the secondaries, and they start mass transfer at periods in the range 10 hr hr.

None of
[ P [ 30 We investigate di †erences between the CV and NLMXB period histogram that arise alone from these e †ects in In particular, we test if the population of Figure 4. NLMXBs below P D 2 hr is much smaller than for CVs, as the lack of the enormous inÑux of newly formed systems boosting the CV distribution there would suggest. Despite this lack, the expected intrinsic period distribution (Fig. 4, middle panel) derived from a typical evolutionary sequence top panel) still predominantly populates the short- (Fig. 4 ) the text. The middle panel depicts the intrinsic discovery probability PP/P0 vs. log P, a quantity representing the intrinsic period distribution of a population of such binaries with similar initial conÐgurations at mass transfer turn-on. In the bottom panel we plot the corresponding discovery probability obtained by multiplying by a visibility factor to account PM 0 for observational selection.
slowly there (number density However, these short-PP0~1). period systems are suppressed for any visibility function with bottom panel). Such a visibility P([M 0 2 )a a Z 1 (Fig. 4, function corresponds, for example, to a Ñux-limited sample taken from a disklike population. In addition, the detection probability function obtained in this way shows a pronounced peak at long orbital periods (here P^8 hr), a feature consistent with the observed LMXB overpopulation at long orbital periods compared with CVs (see Fig. 3, bottom panel). This peak is a consequence of the high mass transfer rate immediately after contact is reached in NLMXBs, since the systems are close to instability (M 2 D The complexity of quantifying the relevant selection M 1 ). e †ects makes it difficult to decide if properly describes a Z 1 the NLMXB population. However, only if indeed a \ 1 is an additional mechanism needed to account for the lack of short-period systems. One such mechanism is the evaporation of the secondary star by pulsar irradiation from a rapidly rotating neutron star spun-up by accretion den (van Heuvel & van Paradijs Remarkably, this implicitly 1988). assumes that all LMXBs cross the gap, i.e., assumes the result we have demonstrated above.

CONCLUSIONS
We have shown that short-period neutron star low-mass X-ray binaries forming from helium-star supernovae without kick velocities must have secondaries with masses in the narrow range 1.
For standard strength magnetic braking, the secondary star is already signiÐcantly nuclear evolved when mass transfer begins, explaining why the resulting mass transfer rates are in many cases low enough for a substantial fraction of these systems to appear as soft X-ray transients even at short days) orbital periods (see In (P [ 1È2 Paper I). contrast, if the neutron star receives a strong kick velocity at birth, many NLMXBs would form with unevolved lowmass donors.
The observed large fraction of SXTs among NLMXBs then forces us to conclude that kick velocities must be small compared to the pre-SN orbital velocity i.e., ([10%, [50 km s~1) for a large fraction of progenitor systems. This is consistent with a recent reevaluation of observed pulsar proper motions that suggest that the distribution of neutron star velocities at birth has a maximum at zero.
Similarly, short-period NLMXBs forming from both initially thermally unstable systems (Kalogera & Webbink and via the direct SN channel 1996a, 1996b) (Kalogera would have a large fraction of (low-mass) unevolved 1996b) secondaries, suggesting that neither of these channels contributes signiÐcantly to the short-period NLMXB population.
Ignoring the uncertain survivors of thermally unstable mass transfer, the very special formation conditions for the case with negligible kick velocity also show that only a very restricted progenitor population (essentially 18 M _ ] 1.4 binaries with periods yr) can form NLMXBs, M _ P prog D 4 explaining their rarity in the Galaxy. We estimate a total formation rate D2 ] 10~7 yr~1. The formation conditions are much more restricted than for cataclysmic variables. In particular, short-period NLMXBs must all begin mass transfer at periods hr, in contrast to CVs, of which a Z12 majority start mass transfer at periods hr. The [1È2 resulting NLMXB period histogram has far fewer systems at short periods than the CV version, in agreement with observation.
The smallness of the area of the plane M He [ M 2 ( Fig. 1) allowing short-period NLMXB formation is very striking. The fact that the resulting population has several properties in good agreement with observation is implicit conÐrmation that the assumed formation conditions are realistic. In particular, it is clear that the neutron star mass at formation cannot be signiÐcantly larger than 1.4 if it were, the M _ : allowed area would become much larger, sharply decreasing the predicted relative population of transient systems. However, we can say nothing about the lower limit on the formation mass, as systems with would not appear m 1 [ 1.2 as NLMXBs (the allowed area in would disappear). Fig. 1 This result suggests another conclusion : with initial conditions evolution without 1. As there is no observational support for such 1993). masses, this suggests that a large fraction of the mass transferred in the NLMXB phase is lost from the binary. The most likely way for this to occur is through mass loss from the accretion disks in these systems (cf. McKee, Begelman, & Shields & King If the mass is lost at 1983 ; Czerny 1989). disk radii much greater than the size of the neutron star, the central accretion rate can in principle be smaller than the mass transfer rate, so it may be simplistic to infer the latter from the observed X-ray Ñux. This in turn would mean that the presence of transient behavior would pose a somewhat less stringent upper limit on the mass transfer rate than we inferred in Paper I. This paper has discussed the formation of LMXBs containing neutron stars. The constraints on the formation of black hole LMXBs appear to be weaker (cf., e.g., Romani as are the conditions for them to appear as 1994, 1996), transients We shall investigate this problem in a (Paper I). future paper.