A PARALLAX DISTANCE TO THE MICROQUASAR GRS 1915+105 AND A REVISED ESTIMATE OF ITS BLACK HOLE MASS

On 2013 May 24, GRS 1915+105 was strongly variable and an image showed very complex structure extended over about 5 mas in the NW–SE direction. This orientation is similar to that of the jet seen on ∼1'' scales with interferometric imaging by Mirabel & Rodríguez (1994) using the Very Large Array, on 250 mas scales by Fender et al. (1999) using the MERLIN array, and on 5–50 mas scales by Dhawan et al. (2000) using the VLBA, when GRS 1915+105 exhibited superluminal expansion. Since jet motion is expected to be of order a synthesized beam size per hour, we re-imaged the data from individual 20 minute spans. We used the eight continental antennas of the VLBA (excluding the very long baselines to the Mauna Kea antenna and the St. Croix antenna, which generally suffers from high water vapor opacity) to produce images. Owing to the limited interferometer (u, v)-coverage afforded by these "snap shot" images, we could only reliably measure the position of the peak emission, which we restricted to be within ±10 mas of the expected core position based on the parallax and proper motion fitting results (see Section 2.2). The position offsets, relative to the expected position of the core of GRS 1915+105, are shown in Figure 2.

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Both the eastward and northward offsets of the peak emission versus time display linear motion, with one apparent outlier at UT = 9.3 hr. Removing the outlier and fitting a constant motion to these data yields 23.6 ± 0.5 mas day⁻¹ toward 1305 ± 11 east of north. Since, these positions were registered relative to the expected position of the core (black hole) of GRS 1915+105, the zero crossings should indicate the time of the expulsion of the radiating plasma. This occurs near UT 6.5 hr and 6.9 hr for the easterly and northerly measurements, respectively, suggesting expulsion at UT 6.7 ± 0.2 hr.

Parallax measurements at centimeter wavelengths with very long baseline interferometry (VLBI) involve rapid switching between the target source and one or more background quasars. This serves two purposes: extending the coherence time of the interferometer by removing short term (∼minute) fluctuations of propagation delay through the troposphere (usually caused by variable water vapor) and providing a stationary position reference for the parallax fitting (Reid & Honma 2014). While the background quasar can serve both purposes, one can use a different source to extend coherence times. For GRS 1915+105 we used an unusually stable Galactic water maser, G045.07+0.13, which was fortuitously close on the sky, projected within $0{{\rlap{.}}^\circ }5$. Switching between these two sources provided excellent phase calibration and relative parallax data.

Preliminary fitting for a relative parallax and proper motion to the eastward and northward offsets revealed two significant (>4σ in each coordinate) outliers at epochs 2010.698 and 2010.873. For each case, compared to the previous epoch's position, these points were offset along a position angle (P.A.) of ≈130° ± 4° east of north, suggesting that low-level jet activity was affecting the apparent core position. This was dramatically confirmed on 2013 May 24 when a superluminal jet component moving along position angle 1305 ± 11 dominated the emission (see Section 2.1).

Generally for VLBI measurements, formal position precision does not fully represent astrometric accuracy. Typically, slowly varying (∼hours) uncompensated differences in tropospheric propagation delays among antennas and sometimes structural changes in the radio sources among epochs lead to degraded position accuracy (Reid & Honma 2014). In order to account for these effects, one generally adds "error floors" in quadrature with the formal measurement uncertainties in both coordinates when fitting for parallax. For the GRS 1915+105 data, owing to the likelihood of increased "jitter" in the apparent position of the core⁶ along the jet axis at some epochs, we rotated the east and north offset data by 40° counterclockwise on the sky, orienting the approaching jet toward the negative y-axis. The rotation angle represents an average jet direction for our observations, based on the measured motion in Section 2.1, and those of Mirabel & Rodríguez (1994) and Fender et al. (1999).

Rotating the data allows us to fit for parallax and proper motion with data perpendicular and parallel to the jet. We can then add error-floor values in quadrature with the formal measurement uncertainties and adjust their values to achieve component chi-squares per degree of freedom ($\chi ^2_\nu$) near unity in each coordinate. In this way, the more jittery data parallel to the jet receives a larger total uncertainty and is appropriately down-weighted compared to the more stable data perpendicular to the jet. After removing the two outliers discussed above, we achieved $\chi ^2_\nu =1.0$ for each component of the data with error floors of 0.09 and 0.19 mas for the position residuals perpendicular and parallel to the jet, respectively. Given the (expected) larger uncertainties in the parallel compared to the perpendicular data, and the shape of the parallax ellipse which has a smaller amplitude in the parallel compared to the perpendicular direction, the parallax for GRS 1915+105 is nearly totally constrained by the perpendicular-to-the-jet data.

Fitting for the parallax of GRS 1915+105 relative to the Galactic water maser source G045.07+0.13, with no prior constraints on distance, gives 0.005 ± 0.030 mas, which places both sources at nearly the same distance. Also, the small relative proper motion of 0.3 mas yr⁻¹ is consistent with a similar distance. Figure 3 displays the relative parallax and proper motion for both the data and the model. Changing the position angle rotation of the data by ±5° changes the relative parallax by about ∓0.008 mas. Thus, the relative parallax is not very sensitive to the adopted orientation of the jet axis.

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In order to obtain an absolute parallax, we also switched between the water maser G045.07+0.13 and two quasars, J1913+0932 and J1913+1220, during the same observations as for GRS 1915+105. These data were originally presented by Wu et al. (2014), who reported a parallax for G045.07+0.13 of 0.129 ± 0.007 mas measured relative to J1913+0932. The parallax fitting for G045.07+0.13 relative to J1913+1220 revealed small systematic deviations from the model parallax and proper motion. These deviations could be removed by adding position acceleration parameters (∼0.1 mas yr⁻²) and gave a parallax estimate of 0.111 ± 0.007 mas. We reproduce the parallax fit results of Wu et al. in Figure 4. Wu et al. chose to discard this alternative parallax estimate. Since the two quasars symmetrically straddle G045.07+0.13 on the sky (see Figure 1), and hence would well cancel most sources of systematic error, we instead adopt the unweighted average of the two results. This yields a parallax for the water maser of 0.120 ± 0.009 mas.

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As shown in Figure 1, there is a second Galactic water maser, G045.45+0.05, projected within $0{{\rlap{.}}^\circ }4$ (≈50 pc) on the sky of G045.07+0.13. G045.45+0.05 has a similar LSR velocity as G045.07+0.13, and both masers can be confidently associated with the Sagittarius spiral arm of the Milky Way, based on identification of the masers with a continuous arc of CO emission on Galactic longitude–velocity plots (Wu et al. 2014). Thus, both masers are likely associated with the same giant molecular cloud (or adjacent clouds in the same spiral arm) and are very likely within about 100 pc of each other. The parallax of G045.45+0.05 is 0.119 ± 0.017 mas, measured with respect to the four quasars shown in Figure 1, then provides an independent check on the distance to G045.07+0.13. As such, we confidently adopt the parallax of G045.07+0.13 of 0.120 ± 0.009 mas, corresponding to a distance of $8.33^{+0.68}_{-0.58}$ kpc to this water maser.

For our final estimate of the distance to GRS 1915+105, we fit the data with a Bayesian Markov chain Monte Carlo approach using the Metropolis–Hastings algorithm to accept or reject trials. We adopt as prior information two constraints on distance (d): (1) d greater than the Galactic tangent point distance of 5.9 kpc, as discussed in Section 1; and (2) d less than 12.0 kpc, where the observed jet motions would require bulk plasma flow speeds exceeding that of light. Rather than use the inclination-dependent constraint of Fender et al. (1999) of 12.5 kpc (95% confidence), here we adopt the slightly stronger and more direct (inclination independent) limit given by $d<c/\sqrt{(\mu _a \mu _r)}$, where c is the speed of light and μ_a and μ_r are the apparent angular speeds of jet components in the approaching and receding directions. Thus, we adopt 5.9 < d < 12.0 kpc with uniform probability as prior information when fitting the data.

Traditionally, when fitting relative position as a function of time, one parameterizes the model with a parallax parameter (and proper motion terms). Owing to the inverse relation between distance and parallax, π, a flat distance prior transforms to a quadratic parallax prior: Prior(π) = Prior(d) × d² (where Prior(d) = 1 for 5.9 < d < 12.0 kpc and zero elsewhere). Then after fitting, the posteriori probability density function (PDF) for parallax should be transformed to the desired distance PDF, rescaling by Prob(d) = Prob(π) × π² (in order to account for the nonlinear stretching of d with decreasing π). This latter scaling is important only if the parallax uncertainty is a significant fraction (e.g., ≳ 15%) of the parallax. Alternatively, one can fit the data with a model parameterized with distance instead of parallax. This is more convenient when applying prior information on distance and gives the desired distance PDF directly without rescaling. Of course, both approaches are equivalent.

We adopted the distance parameterization for our final fits and obtain the posteriori PDF for the distance of GRS 1915+105 shown in Figure 5. The peak of the PDF is at 7.8 kpc, and the expectation value for distance is 8.6 kpc, reflecting the asymmetric tail of the PDF toward larger distances. The minimum-width 68% confidence interval is 7.0–10.6 kpc, which includes the small contribution from the distance uncertainty of the maser G045.07+0.13. Adopting the expectation value as the best estimate of distance, we therefore find GRS 1915+105 at a distance of $8.6^{+2.0}_{-1.6}$ kpc.

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After removing the effects of the parallax of GRS 1915+105, we fit the position offsets of GRS 1915+105 with respect to J1913+0932 (without rotating them to align with the jet axis) to estimate the proper motion of GRS 1915+105 in the easterly and northerly directions to be −3.19 ± 0.03 mas yr⁻¹ and −6.24 ± 0.05 mas yr⁻¹. This compares reasonably with the measurement of −2.86 ± 0.07 mas yr⁻¹ and −6.20 ± 0.09 mas yr⁻¹ from 10 yr of monitoring with the VLBA by Dhawan et al. (2007).

We revisited the systemic velocity of the system whose accuracy, as reported in Steeghs et al. (2013), is limited by the published radial velocity values of the template stars used. We retrieved several archival spectra of one of our templates, HD 176354, which was observed with the high-resolution HARPS spectrograph at ESO. Four observations that span 4 yr show a very stable velocity for this template of −30.8 ± 0.8 km s⁻¹. This translates into a refined heliocentric systemic velocity for GRS 1915+105 of γ = +12.3 ± 1.0 km s⁻¹. Adopting this heliocentric velocity of the GRS 1915+105 binary completes the three-dimensional velocity information.

Using the Galactic and solar motion parameters from Reid et al. (2014; R₀ = 8.34 kpc, Θ₀ = 240 km s⁻¹, U_☉ = 10.7, V_☉ = 15.6, W_☉ = 8.9 km s⁻¹), we calculate the peculiar motion of GRS 1915+105 with respect to a circular Galactic orbit. We find (U_s,V_s,W_s) = (19 ± 3, −10 ± 24, 6 ± 2) km s⁻¹, where U_s is toward the Galactic center at the location of GRS 1915+105, V_s is in the direction of Galactic rotation, and W_s is toward the north Galactic pole. This translates to a non-circular (peculiar) speed of 22 ± 24 km s⁻¹ and a 95% confidence upper limit of 61 km s⁻¹.

The Lorentz factors of ballistic jets from black hole microquasars are generally assumed to be in the vicinity 2 ≲ Γ ≲ 10 (Fender et al. 2004; Fender 2006), although there is a dearth of strong constraints. Specifically, the bulk of observational data provide only lower limits on the Lorentz factor, when derived directly from kinematic measurements and flux ratios. Generally, these lower limits fall near Γ_min ∼ 1.5–2 (Steiner & McClintock 2012; Steiner et al. 2012; Fender et al. 2004). As pointed out by Fender (2003), proper motion data are frequently unable to place significant upper bounds on Γ. This is due to the strong, nonlinear degeneracy between Γ and distance, particularly when Γ is large and the distance estimate approaches the maximum limit, d_max, at which β → 1. Only when Γ is relatively modest and the distance is well constrained below d_max can the Lorenz factor be reliably bounded from above via kinematic data.

Although direct measurements provide only limited upper bounds for Γ, there are indirect arguments which suggest that Lorentz factors are modest, usually ≲ 5, e.g., based on observations of radio–X-ray coupling (Fender 2006) and of the deceleration of jets at late times as they expand (Miller-Jones et al. 2006). Alternatively, it has been argued that microquasar jets may have Γ ≳ 10, based on observations of very narrow jet opening angles and the assumption that the jet plasma expands freely at nearly the speed of light (Miller-Jones et al. 2006, 2007). However, large values of Γ may not be required to explain narrow jets if they are confined externally; indeed, Miller-Jones et al. have provided evidence for such confinement in some systems.

During one observation (see Section 2.1), we observed a motion of 23.6 ± 0.5 mas day⁻¹ along a position angle of 130°, consistent with the approaching jet in GRS 1915+105. At the parallax distance of 8.6 kpc, this corresponds to 1.2c, where c is the speed of light, making this source just marginally superluminal. The magnitude of the motion we measured agrees (exactly) with that measured by Fender et al. (1999) for the approaching jet of 23.6 ± 0.5 mas day⁻¹, but our position angle differs significantly from their value of 142°  ±  2°, estimated from the orientation on the sky of multiple jet components separated by ∼300 mas in 1997 November. Our measured motion also agrees with the average of those of Dhawan et al. (2000) of 22.2 ± 1.3 mas day⁻¹, but differs from their average position angle of 141° ± 3°. However, the magnitude of our measured motion is greater than the average of 17.3 ± 0.2 mas day⁻¹ toward 151° ± 3° for the approaching jet observed from components separated by ≈1000 mas by Mirabel & Rodríguez (1994) and Rodríguez & Mirabel (1999) in 1994/1995. These jet parameters are summarized in Table 2.

Combining the measurements of the motion of the approaching and receding jet components with the source distance allows one to calculate the inclination angle of the jet from the line of sight, i, and the bulk flow speed in the jet, β, in units of the speed of light (Fender et al. 1999). Using the parallax distance of $8.6^{+2.0}_{-1.6}$ kpc (Section 2.2) and the apparent speeds of the approaching (23.6 ± 0.5 mas yr⁻¹) and receding (10.0 ± 0.5 mas yr⁻¹) jet components from Fender et al., we find i = 60° ± 5° and β = 0.81 ± 0.04. This constrains the Lorentz factor γ = (1 − β²)^−1/2 to have values between 1.6 and 1.9.

There are now measurements of jet orientation and bulk flow speed over a wide range of angular scales for GRS 1915+105 (Table 2). Clearly the jet appears to bend through a position angle of 20° on the sky between scales of 5 mas (40 AU) and 10 (0.04 pc) from the black hole. The apparent speed of the approaching jet was the same for our observations and that of Fender et al. (1999). However, Rodríguez & Mirabel (1999) measured lower apparent speeds for both the approaching and receding jets compared to Fender et al. (see Table 2). While both jet orientation and internal flow speed can affect apparent motion, the ratio of approaching to receding motions measured by Rodríguez & Mirabel and Fender et al. are similar and most readily explained with a nearly constant inclination of ≈60°, but with different internal flow speeds of β ≈ 0.65 and β ≈ 0.81, respectively. Therefore, we conclude that the jet bends and decelerates between radii of 0.01–0.04 pc, and/or plasma is ejected with different speeds at different times.

Our parallax distance of $8.6^{+2.0}_{-1.6}$ kpc is lower than the value of 11 kpc often adopted for GRS 1915+105. However, it is consistent with the estimate of ≲ 10 kpc of Zdziarski (2014), which is based on a relation between jet power and luminosity. Distance directly impacts estimates of the black hole mass M, based on orbital period and companion radial velocity measurements, which constrain Msin ³i. Since the inclination of the system (i), as inferred from the apparent speeds of the approaching (μ_a) and receding (μ_r) radio jets, is distance dependent, via tan i = (2d/c)μ_aμ_r/(μ_a − μ_r), a lower distance leads to a smaller binary inclination and a higher mass.

Steeghs et al. (2013) assumed a uniform distance prior of 10 < d < 12 kpc (and a direct relation between distance and inclination, which did not incorporate uncertainty in the apparent jet motions), leading to an estimate of M = 10.1 ± 0.6 M_☉. In order to quantify the impact of our distance on the estimate of black hole mass, we revisited the Monte Carlo simulations of Steeghs et al. and used our parallax distance PDF (see Figure 5) as prior information. Each trial distance was combined with a trial for the observed jet velocity, based on the values and uncertainties of Fender et al. (1999), to calculate the implied inclination. This inclination was then combined with the spectroscopic parameters to calculate a black hole mass, contributing to the posteriori mass distribution. The resulting mass and inclination distribution functions are shown in Figure 6. Both PDFs are significantly asymmetric, with extended tails towards higher mass and lower inclination. The peak of the inclination PDF is at 60° with an expectation value of 59° and a minimum-width 68% confidence interval of 55°–64°. The peak of the mass PDF is at 11.2 M_☉ with an expectation value of 12.4 M_☉ and a minimum-width 68% confidence interval of 10.5–14.3 M_☉, suggesting a slightly higher mass than quoted in Steeghs et al.

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Concerning X-ray luminosity, we note that our 8.6 kpc distance estimate deflates claims for extreme values (e.g., ∼2–3 L_Eddington for an assumed distance of d = 12.5 kpc by Done et al. 2004) and indicates that the maximum luminosity of GRS 1915+105 is at or near the Eddington limit.

McClintock et al. (2006) estimated the dimensionless spin parameter,⁷a_*, for the black hole in GRS 1915+105 using the continuum-fitting method (McClintock et al. 2014) and a sample of seven selected spectra. In applying this method, one fits a thermal continuum spectrum to the thin disk model of Novikov & Thorne (1973); the key parameter is the radius of the inner edge of the accretion disk, assumed to be the innermost stable circular orbit for a black hole, which is directly related to M and a_*. The continuum-fitting method relies upon having independent and accurate estimates of distance, the black hole mass, and the inclination of its spin axis to the line of sight. McClintock et al. (2006) assumed a distance of 11 kpc and adopted M = 14.0 ± 4.4 M_☉ and i = 66° ± 2°, which yielded an extreme value for the spin parameter of a_* > 0.98.

Using our improved constraints on d, M, and i, we present a preliminary analysis for the spin of GRS 1915+105 (details will appear in a forthcoming paper). We have analyzed a large sample of ∼2500 RXTE spectra obtained with the PCU-2 detector, for which the quality of the data calibration has been significantly enhanced using the tool pcacorr (Garcia et al. 2014). We modeled the spectra using the relativistic disk model kerrbb2 (McClintock et al. 2006; Li et al. 2005; Davis & Hubeny 2006) and a cutoff Compton power-law component simplcut, a variant of simpl (Steiner et al. 2009b). For a given set of values of d, M, and i, a subset of the data is selected that meets standard criteria (e.g., Steiner et al. 2011): goodness of fit $(\chi ^2_\nu <2)$, low-to-moderate luminosity (L_disk < 0.3 L_Eddington), and a dominant thermal component with scattering fraction <25% (Steiner et al. 2009a). The size of the data sample, selected over the broad grid of values of d, M, and i, ranges from 8 to 167 spectra. (The luminosity criterion, based on the values of d, M, and i, determines the number of selected spectra.) For each set of parameters, a weighted mean value of spin is computed. Our preliminary estimate is a_* = 0.98 ± 0.01, where the uncertainty is statistical only. Our result is consistent with the original estimate of McClintock et al. (2006), but it is now based firmly on a parallax measurement of distance and a much improved estimate of the black hole's mass. Including sources of systematic uncertainty (Steiner et al. 2011), we estimate a_* > 0.92 at 3σ confidence.

Rodríguez & Mirabel (1998) and Chaty et al. (2001) mapped a field of 1° centered near GRS 1915+105 at centimeter wavelengths and found two H ii regions (IRAS 19132+1035 and IRAS 19124+1106) symmetrically offset by 17' about GRS 1915+105 along a position angle of about 157° ± 1° (our measurement from their published figures). They also found a nonthermal "trail" of emission near IRAS 19132+1035 along a position angle of about 152° ± 5°, which points back toward GRS 1915+105. However, since the H ii regions have (far) kinematic distances between about 6 and 8 kpc, and they assumed a distance of 12.5 kpc for GRS 1915+105, Rodríguez & Mirabel reluctantly concluded that these sources were only a chance projection. However, our near-zero parallax of GRS 1915+105 relative to G045.07+0.13, and the likely association of the H ii regions (IRAS 19132+1035 and IRAS 19124+1106) and the water masers (G045.07+0.13 and G045.45+0.05) with massive star formation in the Sagittarius spiral arm of the Galaxy (Wu et al. 2014), now suggests that most or all of these sources are spatially related. In this case, the nonthermal trail of emission near IRAS 19132+1035 pointing toward GRS 1915+105 could indeed be related to its jet emission, as modeled by Kaiser et al. (2004).

The peculiar velocity components for GRS 1915+105 estimated in Section 2.3 are relatively modest and consistent with an old stellar system that has orbited the Galaxy many times, having responded to random gravitational perturbations from encounters with spiral arms and giant molecular clouds, and acquired a peculiar velocity of 20–30 km s⁻¹ as observed. This suggests that the GRS 1915+105 binary system did not receive a large velocity "kick," owing to mass loss from the primary as it evolved and became a black hole (e.g., Katz 1975).