VLBA DETERMINATION OF THE DISTANCE TO NEARBY STAR-FORMING REGIONS. III. HP TAU/G2 AND THE THREE-DIMENSIONAL STRUCTURE OF TAURUS

The flux of HP Tau/G2 was fairly constant around 0.8 mJy at eight of our nine observations. During the fifth observation (2006 June), however, HP Tau/G2 clearly underwent a flaring event, reaching a flux 3–4 times higher than that at the other epochs (Figure 2). This type of variability is not unexpected for nonthermal sources such as HP Tau/G2 (e.g., Feigelson & Montmerle 1999; Loinard et al. 2008), and is consistent with previous radio measurements (Bieging et al. 1984; Cohen & Bieging 1986; Phillips et al. 1991; see Section 1). The radio source was found to be unresolved at all epochs except the seventh (obtained in 2007 June) when it was clearly extended in the north–south direction, with a deconvolved size in that direction of about 2.5 mas (Figure 3). This increase in the source size might be due to changes in the structure of the active magnetosphere of HP Tau/G2. Interestingly, however, this observation does not correspond to an epoch when the source was particularly bright, nor particularly dim. In any event, the determination of the source position for that epoch is adversely affected by the fact that the source is extended (see below).

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The displacement of HP Tau/G2 on the celestial sphere is the combination of its trigonometric parallax (π) and proper motion (μ). Since HP Tau/G2 is a member of a triple system (see Section 1), we should in principle describe its proper motion as the combination of the uniform motion of the center of mass and a Keplerian orbit. This is not necessary, however, because the orbital period of the system must be very much longer than the time span covered by our observations. If we assume that the total mass of the HP Tau/G2–HP Tau/G3 system is 2–3 M_☉ and that the current observed separation is a good estimate of the system's semimajor axis, then the orbital period is expected to be 35,000–45,000 yr. This is indeed very much longer that the 2 yr covered by our observations, and the acceleration terms can be safely ignored. The astrometric parameters were calculated using the SVD-decomposition fitting scheme described by Loinard et al. (2007). The necessary barycentric coordinates of the Earth, as well as the Julian date of each observation, were calculated using the Multi-year Interactive Computer Almanac (MICA) distributed as a CDROM by the US Naval Observatory. The reference epoch was taken at the mean of our observations: JD 2454029.90 ≡ J2006.81.

Since the source was elongated in the north–south direction during the seventh observation, two different fits were made: one where the seventh epoch was included, and one where it was ignored. When the seventh epoch is included, we obtain the following astrometric parameters:

$$\begin{eqnarray} \alpha _{J2006.81} & = & \mbox{ $04^{\rm h}35^{\rm m}54\mbox{$\fs$}162033$} \pm \mbox{$0\mbox{${\fs}$}000003$ } \nonumber \\ \delta _{J2006.81} & = & \mbox{ $22^{\circ }54^{\prime }13\mbox{$\farcs$}49345$} \pm \mbox{$0\mbox{$\farcs$}000020$ } \nonumber \\ \mu _\alpha \cos \delta & = & 13.90 \pm 0.06 \mbox{ mas yr$^{-1}$} \nonumber \\ \mu _\delta & = & {-}15.6 \pm 0.3 \mbox{ mas yr$^{-1}$} \nonumber \\ \pi & = & 6.19 \pm 0.07 \mbox{ mas.} \nonumber \end{eqnarray}$$

This corresponds to a distance of 161.6 ± 1.7 pc. The post-fit rms in this case is 0.12 mas in right ascension and 0.51 mas in declination.³ To obtain a reduced χ² of one in both right ascension and declination, one must add quadratically 8.8 μs and 0.59 mas in right ascension and declination, respectively, to the errors listed in Table 1. The uncertainties on the parameters quoted above include these systematic contributions. Note that the seventh epoch contributes significantly to the total post-fit rms since the position corresponding to that observation is farther from the fit (both in right ascension and declination) than that at any other epoch (Figure 4). If the seventh observation is ignored, the best fit yields the following parameters:

$$\begin{eqnarray} \alpha _{J2006.81} & = & \mbox{ $04^{\rm h}35^{\rm m}54\mbox{$\fs$}162030$} \pm \mbox{$0\fs 000002$ } \nonumber \\ \delta _{J2006.81} & = & \mbox{ $22^{\circ }54^{\prime }13\mbox{$\farcs$}49362$} \pm \mbox{$0\mbox{$\farcs$}000014$ } \nonumber \\ \mu _\alpha \cos \delta & = & 13.85 \pm 0.03 \mbox{ mas yr$^{-1}$} \nonumber \\ \mu _\delta & = & {-}15.4 \pm 0.2 \mbox{ mas yr$^{-1}$} \nonumber \\ \pi & = & 6.20 \pm 0.03 \mbox{ mas.} \nonumber \end{eqnarray}$$

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All these parameters are consistent within 1σ with those obtained when the seventh observation is included. The corresponding distance in this case is 161.2 ± 0.9 pc, and the post-fit rms is 0.058 mas in right ascension and 0.33 mas in declination, significantly better than in the previous fit. Indeed, to obtain a reduced χ² of one in both right ascension and declination, one must only add quadratically 3.65 μs and 0.38 mas to the formal errors delivered by JMFIT. Again, the uncertainties on the parameters quoted above include these systematic contributions.

As mentioned earlier, the source during the seventh epoch was extended, and the astrometry consequently less reliable. Since the fit when it is ignored is clearly much better than that when it is included, we consider the second fit above our best result.

It is noteworthy that, whether or not the seventh epoch is included, the post-fit rms and the systematic error contribution that must be added to the uncertainties quoted in Table 1 are much larger in declination than in right ascension. Fortunately, this large declination contribution does not strongly affect the distance determination, because the strongest constraints on the parallax come from the right ascension measurements. Interestingly, this is, with Hubble 4 (Torres et al. 2007), the second source for which we find large systematic declination residuals. Astrometric fitting of phase-referenced VLBI observations is usually worse in declination than in right ascension (e.g., Figure 1 in Chatterjee et al. 2004) as a result of residual zenith phase delay errors (Reid et al. 1999). In the case of Hubble 4, however, we argued that the large post-fit declination rms might trace the reflex motion caused by an unseen companion, because a periodicity of about 1.2 yr could be discerned in the residuals. In the present source, the case for a periodicity is less clear (Figure 4, inset), but the residuals are clearly not random. Interestingly, the large residuals are in the same north–south direction as the extension of the source seen during our seventh observation. This orientation might, therefore, correspond to a preferred direction of the system along which it tends to vary more strongly. Additional observations will clearly be necessary to settle this issue.

For Galactic sources, it is interesting to express the proper motions in Galactic coordinates rather than in the equatorial system naturally delivered by the VLBA. The results for HP Tau, and the three sources previously observed with the VLBA are given in Columns 3 and 4 of Table 2. Interestingly, the proper motion of HP Tau/G2 is very similar to that of T Tau, but significantly different from those of Hubble 4 and HDE 283572 (which are themselves very similar to each other). HP Tau/G2 and T Tau also happen to both be located on the eastern side of the Taurus complex, whereas Hubble 4 and HDE 283572 are both around Lynds 1495 near the center of the complex (Figure 5).

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There is a fifth star in Taurus (V773 Tau) with VLBI-based proper motion and trigonometric parallax measurements (Lestrade et al. 1999). V773 Tau is located about a degree southwest of Hubble 4, and the proper motions reported by Lestrade et al. (1999) are similar to those of Hubble 4 and HDE 283572 (see Figure 4 in Torres et al. 2007). It is now known that V773 Tau is a quadruple system composed of a tight spectroscopic binary orbited by two companions. The radio source observed by Lestrade et al. (1999) is associated with the spectroscopic binary (the other two stars do not appear to be detectable radio emitters). In several VLBI observations (e.g., Phillips et al. 1991; Boden et al. 2007; Torres et al. 2008), the radio source has been reported to be double, with each component tracing one of the stars in the spectroscopic binary. This binarity was not taken into account by Lestrade et al. (1999), and may have affected their parallax measurement. Indeed, Boden et al. (2007) recently modeled the orbit of V773 Tau combining spectroscopic observations, Keck Interferometer data and VLBA imaging. The distance to V773 Tau that they obtain (136.2 ± 3.7 pc) is somewhat smaller than the value (148.4^+5.7_−5.3 pc) reported by Lestrade et al. (1999). Because of this slight discrepancy, we will not include V773 Tau in the present analysis. It should be mentioned that we are currently analyzing new multiepoch VLBA observations of V773 Tau designed to constrain both its distance and orbital motions. These data will be published in a forthcoming paper.

Knowing the distance to the sources with high accuracy, it is possible to transform the observed proper motions into transverse velocities. Combining this information with radial (Heliocentric) velocities taken from the literature (the second column of Table 2), it becomes possible to construct the three-dimensional velocity vectors. It is common to express these vectors on a rectangular (X, Y, Z) coordinate system centered in the Sun, with X pointing toward the Galactic center, Y in the direction of Galactic rotation, and Z toward the Galactic North Pole. In this system, the coordinates of the Heliocentric velocities will be written (U, V, W). As a final step, it is also possible to calculate the peculiar velocity of the stars. This involves two stages: first, the peculiar motion of the Sun must be removed to transform the Heliocentric velocities into velocities relative to the LSR. Following Dehnen & Binney (1998), we will use u₀ = +10.00 km s⁻¹, v₀ = +5.25 km s⁻¹, and w₀ = +7.17 km s⁻¹ for the peculiar velocity of the Sun expressed in the coordinate system defined above. The second stage consists in estimating the difference in circular velocity between Taurus and the Sun, so the peculiar velocities are expressed relative to the LSR appropriate for Taurus, rather than relative to the LSR of the Sun. This was done assuming the rotation curve of Brand & Blitz (1993), and represents a small correction of only about 0.3 km s⁻¹. We will write (u, v, w) the coordinates of the peculiar velocity of the sources. Both (U, V, W) and (u, v, w) are given in Table 2 for the four sources considered here. Their projections onto the (X, Y), (X, Z), and (Y, Z) planes are shown in Figure 6.

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The mean heliocentric velocity and the velocity dispersion of the four sources are

$$\begin{equation} U = {-}17.1 \pm 1.7 \mbox{ km s$^{-1}$} \end{equation} \tag{ 1 }$$

$$\begin{equation} V = {-}12.9 \pm 1.2 \mbox{ km s$^{-1}$} \end{equation} \tag{ 2 }$$

$$\begin{equation} W = {-}9.0 \pm 3.8 \mbox{ km s$^{-1}$.} \end{equation} \tag{ 3 }$$

These values are similar to those reported by Bertout & Genova (2006) for a larger sample of young stars in Taurus with optically measured proper motions. Note that the velocity dispersion in the W direction is somewhat artificially high because (as noted earlier) Hubble 4 and HDE 283572 on the one hand, and T Tau and HP Tau/G2 on the other, clearly have different vertical velocities. They likely belong to two different kinematic subgroups.

The mean peculiar velocity of the four sources considered here is

$$\begin{equation} u = {-}7.1 \pm 1.7 \mbox{ km s$^{-1}$} \end{equation} \tag{ 4 }$$

$$\begin{equation} v = {-}7.7 \pm 1.2 \mbox{ km s$^{-1}$} \end{equation} \tag{ 5 }$$

$$\begin{equation} w = {-}1.8 \pm 3.8 \mbox{ km s$^{-1}$.} \end{equation} \tag{ 6 }$$

We argue that this is a good estimate of the mean peculiar velocity of the Taurus complex. This velocity is almost entirely in the (X, Y) plane. Thus, although Taurus is located significantly out of the midplane of the Galaxy (about 40 pc to its south), it appears to be moving very little in the vertical direction. The motion in the (X, Y) plane, on the other hand, is fairly large, leading to a total peculiar velocity (u² + v² + w²)^0.5 = 10.6 km s⁻¹. According to Stark & Brand (1989), the one-dimensional velocity dispersion of giant molecular clouds within 3 kpc of the Sun is about 8 km s⁻¹. As a consequence, each component of the peculiar velocity of a given molecular cloud is expected to be of that order, and our determination of the mean peculiar velocity of Taurus is in reasonable agreement with that prediction. Another useful comparison is with the velocity dispersion of young main-sequence stars. For the bluest stars in their sample (corresponding to early A stars), Dehnen & Binney (1998) found velocity dispersions of about 6 km s⁻¹ in the vertical direction, and of 10–14 km s⁻¹ in the X and Y directions. The young stars in Taurus are significantly younger than typical main-sequence early A stars, so one would expect young stars in Taurus to have peculiar velocities somewhat smaller than 6 km s⁻¹ in the vertical direction, and than 10–14 km s⁻¹ in the X and Y directions. This is indeed what is observed. Note, however, that Taurus is not among the star-forming regions with the smallest peculiar velocities. In Orion, Gómez et al. (2005) found a difference between expected and observed proper motions smaller than 0.5 km s⁻¹.

One last comment should be made here. The data presented here and in the other papers of this series yield proper motions and trigonometric parallaxes that, together, enable the measurement of transverse velocities with an accuracy of about 1%. For sources in Taurus, this corresponds to an absolute error better than 0.1 km s⁻¹. In comparison, the radial velocity measurements available in the literature have typical uncertainties of 1–2 km s⁻¹. To take full advantage of the VLBA data, it will become important to measure radial velocities with a significantly improved accuracy.

Taking the mean of the four VLBA-based parallax measurements available (HP Tau/G2, T Tau, Hubble 4, and HDE 283572), we can estimate the mean parallax to the Taurus complex to be $\bar{\pi }$ = 7.08 mas. This corresponds to a mean distance $\bar{d}$ of 141.2 pc, in good agreement with previous estimates (Kenyon et al. 1994).

The angular size of Taurus is about 10°, corresponding to a physical size of roughly 25 pc. It would be natural to expect that the depth of Taurus might be similar, and that different sources may be found at significantly different distances from us. The observations presented here and in the previous papers of this series are, however, the first ones with enough accuracy to directly probe the depth of the Taurus complex. They reveal that HP Tau is about 30 pc farther than Hubble 4 and HDE 283572, and that Taurus is at least as deep as it is wide. A trivial but important consequence is that using the mean distance indiscriminately for all the stars in the complex will result in systematic errors at the levels of about 10%. To reach higher accuracy, one will have to reconstruct the complete three-dimensional structure of Taurus. The number of sources considered so far is obviously too limited to obtain such a complete view. It is interesting to note, however, that Hubble 4 and HDE 283572, which are very near one another in projection and share the same kinematics (See Section 4.2), are also found to be at similar distances from us (∼130 pc). This suggests that there exist in that region (corresponding to the surroundings of the dark cloud Lynds 1495) a coherent spatio-kinematical structure at about 130 pc. Observations with an astrometric precision similar to that of the data presented here for several dozen young stars would allow the identification of several such coherent groups across the complex. This, in turn, would allow a fairly accurate reconstruction of a three-dimensional structure of Taurus. Currently, very long baseline interferometry is the only technique with sufficient accuracy to carry out the necessary observations.

As mentioned in Section 1, HP Tau/G2 is a member of a compact group of four young stars, comprising HP Tau itself, HP Tau/G1, G2, and G3. Given the small angular separations between them, the members of this group are very likely to be physically associated—-indeed, HP Tau/G2 and G3 are thought to form a bound system. They are, therefore, very likely to be at the same distance from the Sun. Using our accurate estimate of the distance to HP Tau/G2, we are now in a position to refine the determination of the luminosities of all four stars. Little is known about HP Tau/G1, but the effective temperature and the bolometric luminosity (obtained assuming d = 142 pc) of the other three members are given in Briceño et al. (2002). Those values (corrected to the new distance) allow us to place the stars accurately on an H–R diagram (Figure 7).

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From their position on the H–R diagram, one can (at least in principle) derive the mass and age of the stars using theoretical pre-main-sequence evolutionary codes. Several such models are available, and we will use four of them here⁴: those of Siess et al. (2000); Demarque et al. (2004; known as the Yonsei-Yale Y² models); D'Antona & Mazzitelli (1997); and Palla & Stahler (1999). The isochrones for those four models at 1, 3, 5, 7, and 10 Myr are shown as solid black lines in Figure 7. Also shown are the evolutionary tracks (from the same models) for stars of 1.0, 1.5, and 2.0 M_☉. The three HP Tau members are shown as blue symbols, and HDE 283572 (from Torres et al. 2007) is shown as a red symbol (we will discuss momentarily the reason for incorporating that source in the present analysis).

A number of interesting points can be seen from Figure 7. First, there is reasonable agreement (within 40%, see below) between the masses predicted by different models. The best case is that of HP Tau/G2, for which the different models predict masses consistent with each other at the 10% level (between 1.7 and 1.9 M_☉). The situation for HP Tau is somewhat less favorable, since the models of Siess et al. (2000) or Palla & Stahler (1999) predict a mass of ∼1.5 M_☉, whereas those of D'Antona & Mazzitelli (1997) predicts a significantly smaller mass of ∼1.0 M_☉. Thus, there is a 35% spread in the values predicted by different models for the mass of that source. The least favorable situation is for HP Tau/G3. The mass of that source is about 0.8 M_☉ according to the models of Siess et al. (2000), but slightly less than 0.5 M_☉ according to those D'Antona & Mazzitelli (1997). This is a 40% discrepancy. This tendency for pre-main-sequence evolutionary models to become more discrepant at lower mass had been noticed before, and is discussed at length in Hillenbrand et al. (2008). In the absence of dynamically measured masses, it is impossible to assess which of the models used here provides the "best" answer.

Another interesting issue is related to the age predictions of the different models. Since the different members of the HP Tau group are likely to be physically associated, they are expected to be nearly coeval. This is particularly true of HP Tau/G2 and HP Tau/G3 which are believed to form a loose binary system. Interestingly, most models predict significantly different ages for the three sources (see Figure 7). The models by Siess et al. (2000) predict ages of about 8 Myr and 3 Myr for HP Tau/G2 and HP Tau/G3, respectively. A similar 5 Myr age difference is found for the models of Demarque et al. (2004) and D'Antona & Mazzitelli (1997): both predict ages slightly smaller than 1 Myr for HP Tau/G3, and somewhat larger than 5 Myr for HP Tau/G2. In principle, those differences could be real. In should be noticed, however, that the vast majority of low-mass stars in Taurus (with spectral types M and late K) have ages smaller than 3 Myr (Briceño et al. 2002). Moreover, mass-dependent systematic effects in the age predictions made by evolutionary tracks have been reported before. In particular, Hillenbrand et al. (2008) argued that existing models could significantly over-predict the age of relatively massive stars (M ≳ 1.5 M_☉). HP Tau/G2 is precisely such a fairly massive star. So is HDE 283572, another young star in Taurus with a recently measured accurate distance (Torres et al. 2007). The age estimate for that star based on the models by Siess et al. (2000), Demarque et al. (2004), and D'Antona & Mazzitelli (1997) is 6–10 Myr (Figure 7), somewhat larger than would be expected for Taurus. The only of the four models considered here to predict similar ages for the three members of the HP Tau group is that of Palla & Stahler (1999). Within the errors, all three stars fall on the 3 Myr isochrone. Note that this value is also consistent with the ages of lower mass stars in Taurus (Briceño et al. 2002; see above).