1 Introduction

1.1 Type I X-ray bursts

Type I X-ray bursts (hereafter X-ray bursts, except when needed) in low-mass X-ray binaries are thermonuclear runaways in freshly accreted material on the surface of a neutron star (for a review see Lewin et al. 1993). The freshly accreted matter is compressed and heated, on a time scale of hours to days, to densities and temperatures adequate for thermonuclear ignition. Due to the strong temperature sensitivity of the nuclear reactions, the ignition nearly always results in a runaway process that leads to a sudden, rapid (between about 1 to 10 s) increase in the X-ray flux. These X-ray bursts last generally for tens to hundreds of seconds, and recur with a frequency (partly) set by the supply rate of fresh fuel. The spectral hardening during the X-ray burst rise and subsequent softening during the decay reflect the heating and subsequent cooling of the neutron star surface. The X-ray spectra during the X-ray bursts are consistent with black-body emission from a compact object with a radius of approximately 10 km and temperature between 1 and 2.5 keV.

Spikes with similar duration in X-ray light curves have been seen in MXB 1730-335, also known as "The Rapid Burster'' (e.g. Lewin et al. 1993, and references therein; this source also shows type I X-ray bursts), and GRO J1744-28, or "The Bursting Pulsar'' (Lewin et al. 1996; Kommers et al. 1997), but these do not show the characteristics of type I X-ray bursts (in particular, the cooling during the decay). They are thought to be due to sudden accretion events and are referred to as type II X-ray bursts (Hoffman et al. 1978).

During some type I X-ray bursts the energy release is high enough that the luminosity at the surface of the neutron star reaches the Eddington limit. At that point the neutron star atmosphere expands due to radiation pressure. During expansion and subsequent contraction the luminosity is expected to remain almost constant near the Eddington limit. Since the luminosity scales as $L_{\rm b}\propto R^2T^4$for pure black-body radiation, the effective temperature, T, drops when the radius of the photosphere, R, expands. X-ray bursts with a photospheric radius expansion/contraction phase are therefore recognized by an increase in the inferred radius with a simultaneous decrease in the effective temperature near the peak of an X-ray burst, at approximately constant observed flux. The point at which the neutron star atmosphere reaches the surface again (i.e. at the highest effective temperatures) is called "touch-down''. When the expansion is large the temperature may become so low that the peak of the radiation shifts to the UV wavelengths, and no or little X-rays are emitted. Such photospheric radius expansion events (hereafter radius expansion bursts, except when needed) are recognizable by so-called "precursors'' in the X-ray light curves followed by a "main'' burst (Tawara et al. 1984; Lewin et al. 1984).

We note that, although the X-ray burst spectra seem to be adequately described by black-body emission, the "true'' emission from the neutron star and its close environment is expected to be more complex. Electron scattering dominates the opacity in the neutron star atmosphere, leading to deviations from the original Planckian spectrum (van Paradijs 1982; London et al. 1984, 1986; see also Titarchuk 1994; Madej 1997, and references therein). Similarly, electron scattering in a disk corona (e.g. Melia 1987) and/or winds or outflows from the neutron star surface interacting with the accretion disk (e.g. Melia & Joss 1985; Stollman & van Paradijs 1985) can affect the original X-ray spectrum.

1.2 A standard candle?

Shortly after the discovery of X-ray bursts, van Paradijs (1978) introduced the idea that the average peak luminosity of type I X-ray bursts is a standard candle, presumably the Eddington luminosity limit for the neutron star photosphere. With that assumption, approximate neutron star radii and distances could be derived for a sample of X-ray bursters. Scaling the derived distances so that they are symmetrically distributed around the Galactic center at $\simeq$9 kpc, van Paradijs (1981) found the average peak (bolometric) luminosity to be $3 \times 10 ^{38}$ erg s^-1. Van Paradijs (1981) realized that X-ray bursters residing in globular clusters are important calibrators, since their distances can be determined independently. Their peak luminosity agreed with the above quoted value of the average peak luminosity of Galactic center X-ray bursters. Verbunt et al. (1984) redid this work, based upon a more extensive sample of X-ray bursters. They found the average peak luminosity of X-ray bursters not located in globular clusters to be $3.5 \pm 1.0 \times 10 ^{38}$ erg s^-1, when taking for the distance to the Galactic center a value of 8.5 kpc. For the X-ray bursters in globular clusters a luminosity value of $4.0 \pm 0.9 \times 10 ^{38}$ erg s^-1was derived, i.e. consistent with the above value. A "standard candle'' value of $3.7 \times 10 ^{38}$ erg s^-1 was adopted. Cominsky (1981), on the other hand, assumed that the black-body radius is the same for all X-ray bursts and subsequently derived source distances and X-ray burst peak luminosities. Again scaling the average source distance to 9 kpc, peak luminosities between $\simeq$10³⁸ and $\simeq$ $5 \times 10 ^{38}$ erg s^-1 were found.

Van Paradijs (1978, 1981) cautioned, however, that in a few individual sources the scatter in the X-ray burst peak luminosity was appreciable. It was therefore suggested by Lewin (1982) to use instead only the brightest X-ray bursts. In this way the standard candle became $\mbox{$\ga$ }$5. $5 \times 10 ^{38}$ erg s^-1. Of course, the problem arises that one does not know whether the strongest X-ray bursts observed are also the strongest ones possible. It seemed, however, that those X-ray bursts which showed clear photospheric radius expansion do reach a "true'' critical luminosity. But whether this critical luminosity is also a standard candle was still questioned (Basinska et al. 1984). If that would turn out to be the case one can in principle determine the distances to sources which show clear radius expansion bursts; for those sources whose X-ray bursts do not show a radius expansion phase we can still determine an upper limit to the distance using the maximum observed peak flux.

The thirteen persistent or transient luminous ( $\mbox{$\ga$ }$10³⁶ erg s^-1) X-ray sources in twelve globular clusters are all low-mass X-ray binaries (e.g. Verbunt et al. 1984; Hut et al. 1992; White & Angelini 2001; in 't Zand et al. 2001b). All but one (AC 211 in NGC 7078) have shown type I X-ray bursts, proving they harbour neutron stars. Several of these X-ray bursts exhibited a photospheric radius expansion phase (see, e.g., Lewin et al. 1993, and references therein). We here show that the peak luminosity of radius expansion bursts from about two third of the X-ray bursters in globular clusters do indeed reach an empirical critical luminosity, and may be used as approximate standard candles for other sources.