3 The distance of GRO J1655-40

In order to use our proper motion measurement to constrain the space velocity of the black hole, we need to know its distance. Unfortunately, as for most X-ray binaries, the distance to GRO J1655-40 is rather uncertain. In the following we discuss the observational constrains to the distance of this source.

3.1 The relativistic distance

It is widely believed that the distance of GRO J1655-40 was well determined solely from the kinematics of the two-sided radio jets. This is not true. The relativistic time delay of the motion of the ejecta in the sky is given by the two equations (Mirabel & Rodríguez 1994):

$$\mu_{{\rm r,a}} = \frac{\beta \sin(\theta)}{1 \pm \beta \cos(\theta)} \frac{c}{D},$$ (1)

where $\mu_{\rm r}$ and $\mu_{\rm a}$ are the proper motions of the receding and approaching jets. In this system of two equations there are three unknowns: the angle with the line of sight of the jet axis $\theta$, the velocity of the jets $\beta = \frac{v}{c}$, and the distance D. We point out that using solely the observations at radio wavelengths by Hjellming & Rupen (1995) it is not possible to solve these equations, unless one assumes a value for one of the three variables $\theta$, $\beta$, or D. Hjellming & Rupen (1995) assumed $\theta = 85^{\circ}\pm2^{\circ}$, from which one derives $\beta = \frac{v}{c} = 0.92$ and D = 3.2 kpc. As already noticed by Orosz & Bailyn (1997), in this case the axis of the jet and the axis of the orbital plane differ by $\sim$ $15^{\circ} \pm 2^{\circ}$.

We point out that the assumption that the jet axis is parallel to the axis of the orbital plane is equally consistent with the observations at radio wavelengths. The jet axis and the orbital plane must be coupled, since the period of rotation of the jets about the jet axis (Hjellming & Rupen 1995) is - within the uncertainties -, the same as the 2.6 day orbital period of the binary (Orosz & Bailyn 1997). If one assumes that the twin jets are perpendicular to the orbital plane of the binary, from $\mu_{\rm r} =$ 45 mas/day and $\mu_{\rm a} = 54$ mas/day (Hjellming & Rupen 1995), $\theta = 70.2^{\circ}\pm1.9^{\circ}$ (Greene et al. 2001) result $\beta = \frac{v}{c} = 0.27 \pm 0.03$ (where v is the velocity of the jets and c the speed of light) and a distance $D = 893 \pm 100$ pc. Under this assumption, the distance would be a factor 3.5 closer than commonly assumed, the jet velocity would be similar to that in SS433, and GRO J1655-40 would not be a superluminal source.

In summary, from the data at radio wavelengths and the system of two Eq. (1) with three unknowns one can only derive with certainty a relativistic upper limit (Mirabel & Rodríguez 1999) given by $D \leq c / (\mu_{\rm r} \mu_{\rm a} )^{-1/2} = 3.5$ kpc.

3.2 The distance and the interstellar matter along the line of sight

A distance for GRO J1655-40 was proposed on the basis of optical (Bailyn et al. 1995) and X-ray (Greiner et al. 1995; Ueda et al. 1998a,b) measurements of the column of interstellar matter in the line of sight, under the assumption that the absorbing material is distributed homogeneously between the source and the observer. However, GRO J1655-40 is at relatively high Galactic latitude (Galactic longitude and latitude $l = 345.0^{\circ}$, $b = +2.2^{\circ}$) in the Scorpius region of the sky which contains rather clumpy optical dark clouds in the foreground (see Fig. 1), that have $60~\mu$m and $100~\mu$m IRAS counterparts of dust emission. From a study of the reddening undergone by the stars that are at distances between 700 pc and 1900 pc, it is known that most of the reddening in this region of the sky occurs in the local arm within 700 pc from the Sun (Crawford et al. 1989).

On the other hand, a kinematic distance was proposed from the radial velocity of absorption features in the HI $\lambda$21 cm line spectrum (Tingay et al. 1995). However, it is known that in this region of the sky at distances $\leq$1900 pc there are clouds with anomalous velocities of up to -50 km s^-1(Crawford et al. 1989). Therefore, it is not possible to derive the distance of GRO J1655-40 only from the column density and/or kinematics of the interstellar matter in the line of sight.

3.3 Constrains on the distance from the properties of the secondary star

The main argument in favor of the canonical distance of $\sim$3.2 kpc is based on the flux, color and size of the secondary. The radius is inferred from the photometric light curves which provide evidence that the secondary fills its Roche lobe (Orosz & Bailyn 1997; Shahbaz et al. 1999; Soria et al. 2000; van der Hooft et al. 1997, 1998; Phillips et al. 1999). From the optical spectrum (Orosz & Bailyn 1997) and interstellar absorption (Horne et al. 1996) a temperature is derived, which together with the radius provides an intrinsic luminosity. In the most recent model by Beer & Podsiadlowski (2002) the secondary would have a luminosity of 21 $\pm$6.0 $L_{\odot}$, which is consistent with a distance $\geq$2 kpc. These authors estimate masses for the black hole $M_{\rm BH} = 5.4 \pm 0.3~M_{\odot}$ and for the donor star $M_{*} = 1.45 \pm 0.35~M_{\odot}$.

We point out that the secondary star in quiescence has an apparent magnitude m_V = 17.12, it has been classified as an F3 IV-F6 IV sub-giant (Orosz & Bailyn 1997), and along the line of sight there is an interstellar absorption $A_V = 3.1 \times {E(B-V)} = 4.03$ mag (Horne et al. 1996). A sub-giant star of this spectral type has a mean intrinsic magnitude $M_V \sim 3.2$ $\pm$ 0.2 mag (Popper 1980; Aller et al. 1982), and it would be at a distance $D = 950 \pm 150$ pc. Alternatively, if the secondary were a main sequence star of spectral type F5 V star (Regos et al. 1998), for an absorption A_V in the range of 3.3-4.3 mag the distance would be in the range of 800-1250 pc. However, it may be incorrect to attribute the absolute magnitudes of isolated stars to secondary stars in X-ray binaries with the same spectral type.

Figure 1: Position of GRO J1655-40 on a R band image from the Digitized Palomar Observatory Sky Survey II (POSS II), in Galactic coordinates. The arrow shows the direction of the motion at a rate of 5.2 $\pm$ 0.5 mas yr-1 measured with the Hubble Space Telescope. Most of the stars around l = 345.44$^{\circ }$, b = + 2.43$^{\circ }$ belong to the open cluster NGC 6242 which is at a distance of 1 $\pm$ 0.1 kpc from the sun. Because of the uncertainty in the distance to GRO J1655-40 the association with the cluster cannot be assessed.

Figure 2: Galactocentric orbits of GRO J1655-40 viewed from above the Galactic plane. The orbital plane is almost parallel to the Galactic disk and the source never reaches heights greater than 150 pc. On the left is shown the orbit obtained for an heliocentric distance D = 0.9 kpc, and on the right, for D = 3.2 kpc. GRO J1655-40 never penetrates the Galactic bulge. Plotted in dashed blue line is 1 Gyr of backwards integration, and in red the past 230 Myrs.

It was pointed out by Beer & Podsiadlowski (private communication) that the lower luminosity and temperature implied by a distance of $\sim$1 kpc would require a higher mass ratio with much reduced masses for the compact object ($\sim$3.2 $M_{\odot}$) and secondary star ($\sim$0.1 $M_{\odot}$). But the absorption spectrum of the secondary seems to rule out an M type star of such low mass or a K-type star with mass 0.6 $\leq$ $M_{\odot}$. The spectra of stars with $T_{{\rm eff}}$ $\leq$ 5000 K have different signatures, such as strong molecular bands. This was the case in GRS 1915+105 where K band spectroscopy (Greiner et al. 2001) revealed CO molecular bands rendering invalid the classification of the donor as a main sequence star.

We point out that there have been analogous uncertainties about the nature of the secondary in the X-ray binary LMX-3; the proposition that it is a main sequence star (Cowley et al. 1983; van der Klis et al. 1985) has recently been challenged by Soria et al. (2001) who argue that it is a sub-giant. Furthermore in GRO J1655-40, the contribution to the optical flux from: 1) the accretion disk detected in the X-rays at times when it was believed that the source was in optical quiescence (Garcia et al. 2001), and 2) possible non-thermal processes (e.g. synchrotron jets) that may be associated with the polarization observed in the optical flux (Gliozzi et al. 1998) is not known. In this context, for the scope of the present study we leave as an open question the issue of the distance of GRO J1655-40.