3 Results

   
3.1 Relative motion of the components

In Fig. A.1 the relative positions of the components at different epochs are shown in cartesian and polar coordinates. These plots are only given for the 23 out of our 34 systems for which there are at least three data points. To obtain a simple approximation of the relative velocity we applied weighted linear fits to this data. For the 11 systems with only two observations we simply connect the two data points and derive the error of the relative velocity from the uncertainties of the two separations and position angles.

For a quantitative analysis, the relative velocities must be transformed to an absolute length scale. This requires knowing the distances of the discussed objects. We adopt distances to the SFRs that are the mean of all Hipparcos distances derived for members of the respective association. The values and references are given in Table 1. There remains, however, an uncertainty, because distances of individual objects may be different from these mean values. To take this into account, we assume that the radial diameters of the SFRs are as large as their projected diameters on the sky. The latter quantity can be estimated to be $\approx$ $20^{\circ}$ for Taurus-Auriga (see Fig. 1 in Köhler & Leinert 1998) as well as Scorpius-Centaurus (see Fig. 1 in Köhler et al. 2000). Concerning the mean distances from Table 1, this corresponds to a diameter of 50pc. Thus we will assume $\pm 25\,{\rm pc}$ as the uncertainty for the distance of an individual system, which is an upper limit: more than two thirds of the stars will be within $\pm$15pc for an even distribution.

The velocities of the companions relative to the primaries derived by applying the assumed distances are given in Table 2. They are also plotted in Fig. 2 in cartesian and polar coordinates (similar to Fig. 3 in G95). Our measurements can only cover the projection of motion onto the sky, so the $v_{\rho }$ are given with respect to the main component, not to the observer. The adopted v is the mean of the respective values derived from the fits in cartesian and polar coordinates. In 3 out of 34 systems v is different from zero on the 3$\sigma$ level, in 9 systems on the 2$\sigma$ level and in 18 systems on the $1\sigma$ level. Thus, we are fairly confident that there really is relative motion of the components in most systems.

Figure 2: Relative velocities of the components in T Tauri binary systems in cartesian coordinates (left panel) and polar coordinates (right panel). The dashed line in the right panel points towards the locus of Herbig-Haro objects that is far outside this plot. RXJ1546.1-2804 is not plotted in this figure. Its locus is out of the picture in both panels. This plot is similar to Fig. 3 in G95

   
3.2 Origin of the relative motion

We now examine the origin of this relative motion. For this purpose, we must discriminate orbital motion from an apparent relative motion that can be caused by the proper motion of a T Tauri star with respect to a background star projected by chance or by the proper motions of two T Tauri stars projected by chance. One has further to consider that "companions'' to T Tauri stars detected with only one observation in one filter are not necessarily stellar and may be Herbig-Haro objects. We will also examine the possible influence of unresolved additional components on the observed motion.

   
3.2.1 Orbital motion

First we will derive an upper limit for relative velocities caused by orbital motion. That limit is given by the condition that in the case of orbital motion the kinetic energy of the system is less than its (negative) potential energy and is equal to it in the extreme case of a parabolic orbit, i.e.

$$v^2\le\frac{2GM_{{\rm Sys}}}{r},$$ (1)

where r is the instantaneous separation of the components.

We adopt $M_{\rm TTS} \le 2~M_{\odot}$ as upper mass limit for one T Tauri star (Hartmann 1998) and take the mean of the measured projected separations as estimate for r. There is only one companion with a relative velocity that is larger than the value derived from Eq. (1), namely that of RXJ1546.1-2804. The relative velocity of this companion is, however, still consistent with orbital motion, considering the $1\sigma$ error in its v. The lower limit for v is zero because in short pieces of orbit as discussed here the orbital motion may occur purely radial to the observer. Thus, the large majority of the velocities from Table 2 are consistent with orbital motion.

Furthermore, it is interesting to examine the relationship between separation and relative velocity. In the special case of a circular orbit observed face-on this relation will be

$$v\propto \frac{1}{\sqrt{a}},$$ (2)

for the orbital velocity v and the semimajor axis a. Equation (2) will be modified by projection effects and the actual orbital parameters, and the unknown system mass is an additional parameter in this relation. However, close companions should, on average, move faster than distant ones. In Fig. 3 the measured projected velocities, v, are plotted as a function of the components' projected separation, d. The correlation is weak for the reasons mentioned above, but there is at least a tendency to fulfil the prediction of Eq. (2). Among the closest companions with $d< 25\,{\rm AU}$, relative velocities with $v\ge 5~{\rm km\,s^{-1}}$ are more frequent than for the companions with larger separations (Table 3).

We conclude from this section that the observed relative velocities in almost all cases are not in contradiction to orbital motion. For a final classification, other possible origins of the relative motion must be considered.

 

 

Table 3: Distribution of the companions with respect to certain relative velocities and separations

	$d< 25\,{\rm AU}$	$d \ge 25\,{\rm AU}$
$v\ge 5~{\rm km}\,{\rm s^{-1}}$	$7\pm 2.6$	$11\pm 3.3$
$v< 5~{\rm km}\,{\rm s^{-1}}$	$3\pm 1.7$	$13\pm 3.6$

Figure 3: Relative velocities of the components as function of their mean projected separation

   
3.2.2 Chance projected background stars

A background star projected by chance will probably be located at a much larger distance than the observed T Tauri star, so its proper motion can be neglected. The measured relative motion is thus expected to be the proper motion of the T Tauri star. In Taurus-Auriga there are subgroups with different directions of proper motion and a mean proper motion of $15.8~{\rm km\,s^{-1}}$ (Jones & Herbig 1979). For Scorpius-Centaurus, de Zeeuw et al. (1999) give a mean proper motion of $17.6~{\rm km\,s^{-1}}$ with a mean direction of $v_x = -17.4~{\rm km\,s^{-1}}$ and $v_y = -2.8~{\rm km\,s^{-1}}$. For both SFRs the distances given in Table 1 were adopted. There are only two systems with relative velocities in this order of magnitude, namely V 773 Tau ( $v = 13.78\pm 7.34~{\rm km\,s^{-1}}$) and RXJ1546.1-2804 ( $v = 23.60\pm 6.28~{\rm km\,s^{-1}}$).

For V 773 Tau, the Hipparcos catalogue gives proper motions of $\mu_x = -24.89 \pm 1.89\,{\rm mas/yr}$ and $\mu_y = 0.65 \pm 2.83~{\rm mas/yr}$. The resultant $v = 16.68\pm 2.28~{\rm km\,s^{-1}}$ is comparable with the observed relative velocity of the components, but the proper motion of V 773 Tau happens almost only in declination (X in Fig. A.1) which is contradictory to our observations. Furthermore, for V 773 Tau the relative motion is still explainable with orbital motion, considering the upper limit derived in Sect. 3.2.1.

In the case of RXJ1546.1-2804 there are no Hipparcos data for this individual object, so we adopt the mean proper motion for the OB association Upper Scorpius given by de Zeeuw et al. (1999). These values are $\mu = 17.7~{\rm km\,s^{-1}}$, $\mu_x = -17.4~{\rm km\,s^{-1}}$ and $\mu_y = -2.8~{\rm km\,s^{-1}}$ with (formal) errors of $\approx$ $0.1~{\rm km\,s^{-1}}$. The respective values from Table 2 are $v = 23.6\pm 6.3~{\rm km\,s^{-1}}$, $v_x = 22.3\pm 1.6~{\rm km\,s^{-1}}$ and $v_y = 2.9\pm 4.9~{\rm km\,s^{-1}}$. This is a close correspondance, given that the the proper motion presented by de Zeeuw et al. (1999) is not for this single object, but for the whole association. Note that $\vec{\mu}$ and $\vec{v}$ must be antiparallel if the companion is a chance-projected background star, because in that case $\vec{\mu}$ is the motion of the primary with respect to the "companion''. Another argument that supports the idea that the companion of RXJ1546.1-2804 is a chance-projected background star is that its relative motion is above the limit given by Eq. (1). It is, however, still consistent with orbital motion at the $1\sigma$ level.

One must take into account that in both cases the measured projected separations are $d\approx 0\hbox{$.\!\!^{\prime\prime}$ }1$, which makes any chance projections very unlikely (see Sect. 3.2.3 for the probability of chance projections). Thus, one must consider other origins of these high relative velocities. One solution may be that the distances of individual systems are largely different from the adopted values (Table 1) for the SFRs as a whole. The unusually high derived values for relative velocities would then be caused by overestimating the objects' distances. Another possibile explanation, namely the presence of unresolved additional companions, will be discussed in Sect. 3.2.5.

The companion of RXJ1546.1-2804 may be such a chance-projected background star. In the case of V 773 Tau this seems to be unlikely because the direction of proper motion does not match, however, the high relative velocity of the components remains problematic. Further observations of these systems will be necessary to determine whether there is a curvature in the relative motion which would undoubtedly classify it as orbital motion. In general, chance-projected background stars are not frequent among close visual companions to T Tauri stars.

   
3.2.3 Chance projection of two T Tauri stars

There may also be companions that are in fact objects projected by chance belonging to the same SFR as the "primary''. In that case both components would have roughly the same proper motion. Any relative motion would then be caused by the velocity dispersion within the respective SFR or subgroup, as was mentioned by G95. These velocity dispersions are between 1 and 2 ${\rm km\,s^{-1}}$ in each coordinate for different subgroups in Taurus-Auriga (Jones & Herbig 1979). For Scorpius-Centaurus, de Zeeuw et al. (1999) formally derived a velocity dispersion of only $0.1~{\rm km\,s^{-1}}$. One cannot distinguish this kind of relative motion from slow orbital motion by analyzing relative velocities.

It is, however, not probable that chance projections within the same SFR are a frequent phenomenon, because of the low stellar density in the SFRs discussed here. Leinert et al. (1993) concluded that there are less than 4 10^-5 objects brighter than $K = 12\,{\rm mag}$ per ${\rm arcsec}^2$ in Taurus-Auriga. This includes association members and field stars. All companions discussed here are brighter than $K = 12\,{\rm mag}$ and have projected separations of less than $0\hbox{$.\!\!^{\prime\prime}$ }5$. The mean number of chance-projected companions within a radius of $0\hbox{$.\!\!^{\prime\prime}$ }5$ around our 21 objects in Taurus-Auriga is thus

$$N_{{\rm proj}} = 21 \cdot 0.5^2\cdot \pi~ \cdot ~4~10^{-5} = 6.6~10^{-4}.$$ (3)

For Scorpius-Centaurus, Köhler et al. (2000) derived a stellar density of $(6.64\pm 0.45)~10^{-4}\,{\rm arcsec}^{-2}$from star counts in the vicinity of X-ray selected T Tauri stars. Applying Eq. (3) yields a mean number of 5.7 10^-3chance-projected companions around 11 objects in Scorpius-Centaurus. These values are low enough to rule out that chance projections within the same SFR cause the detection of an appreciable number of close "companions''. The low observed relative velocities in some systems are probably due to the fact that we can only detect the projected motion on the sky. If most of the orbital motion happens radial to the observer in the time span covered by our data it will not be measurable by our observations.

   
3.2.4 Herbig-Haro objects

G95 raised the question of whether an appreciable number of companions of young stars detected with only one observation in one broad band filter may in fact not be stellar, but rather condensations in gaseous nebulae. Such companions would appear as Herbig-Haro objects that are driven by strong winds of active T Tauri stars. They usually have velocities of some $100~{\rm km\,s^{-1}}$ (e.g. Schwartz 1983). Moreover, their relative motion with respect to their driving source can be only radial, not tangential. Thus, the locus of Herbig-Haro objects in Fig. 2 (right panel) is in the direction of the dashed line in this figure, but far outside the plot. As was reported by G95, we find no system where the observed relative velocity is consistent with these restrictions.

Furthermore, in the case of HVTau, where companion C has been declared as a Herbig-Haro object, we could show that in fact it is a stellar companion (Woitas & Leinert 1998).

   
3.2.5 Apparent relative motion caused by unresolved companions

Another problem with observed relative velocities in multiple systems has been pointed out by G95. If the main component has an additional unresolved companion, orbital motion in this close pair can shift the photocenter of the "primary''. This will be misinterpreted as motion of the visual secondary if only relative astrometry is measured, as is the case here. G95 have used this effect to explain the surprisingly high relative velocity in the Elias 12 system where the visual companion is moving with $10.73~{\rm km\,s^{-1}}$ at a separation of $49.3\,{\rm AU}$(Table 2). This is only slightly below the upper limit for orbital motion, which is $12.0~{\rm km\,s^{-1}}$ in this case (Eq. (1)). In this system, the primary has another companion with a separation of $d = 23\,{\rm mas}$, detected by Simon et al. (1995) using lunar occultations. If the system mass of the close pair is assumed to be $1\ M_{\odot}$ and a relation $<d>\ = 0.95\,a$ between the mean projected separation <d> and the semi-major axis a(Leinert et al. 1993) is adopted, one would expect the period of the close pair to be $P\approx 6.4\,{\rm yr}$. This is roughly the same timespan that is covered by our observations (Table A.1). The shift of the photocenter of the close pair with respect to the visual secondary will cancel, and thus the derived secondary's velocity is not affected by the tertiary.

Also in the V 773 Tau system, where we have noticed an unusually high relative velocity of the visual secondary (Sect. 3.2.2) there is an additional spectroscopic companion (Welty 1995). The period of this close pair is 51.1 days, so possible shifts of the photocenter are less than 2mas and thus not measurable by our observations.

A candidate for a system where the observed relative motion may be influenced by an unresolved tertiary is BD+26718B Aa. In this system $v = 8.55~{\rm km\,s^{-1}}$ at a separation of $67.5\,{\rm AU}$. Similar to Elias 12, this value is close to the upper limit for orbital motion ( $v\le 10.2~{\rm km\,s^{-1}}$from Eq. (1)), but also far below the relative velocity expected for a background star projected by chance (Sect. 3.2.2).

3.2.6 Conclusions

The observed relative velocties can be explained by orbital motion. Together with the result that other origins of relative motion can be ruled out at a high confidence level for most systems, we conclude that the observed motion is orbital motion in nearly all binaries discussed.

   
3.3 Estimation of an empirical average system mass

For all binary systems discussed here the available portions of the orbit are too short to calculate orbital parameters. The results presented in Table A.1, however, remain valuable for future orbit determinations that will yield empirical masses for T Tauri binary systems. Furthermore, it is already possible to estimate an average system mass from this database. This average mass is not dependent on theoretical assumptions about the physics of PMS evolution and should therefore be a reliable empirical estimation of the masses of T Tauri stars. To derive this mass we follow the approach of G95, but improve it in some important aspects.

First, we write Kepler's third law in the natural units $M/M_{\odot}$, $a/1\,{\rm AU}$ and $P/1\,{\rm yr}$:

$$M_{{\rm Sys}} = \frac{a^3}{P^2}$$ (4)

and assume a face-on circular orbit where the total orbital motion happens tangential to the observer and therefore equals the observed velocity:

$$v_{\rm face-on, circ} = \frac{2\pi a}{P}\cdot$$ (5)

In this special case ( $i=0^{\circ}$, e=0) Eq. (4) becomes

$$M_{{\rm Sys}} = \frac{v_{\rm face-on, circ}^2\,a}{4\pi^2}\cdot$$ (6)

Unfortunately, we don't know a or $v_{\rm face-on, circ}$ of our systems. G95 used computer simulations of binary orbits to derive statistical relations between the observed quantities $d_{\rm proj}$ and $v_{\rm obs}$ on the one hand and a and $v_{\rm face-on, circ}$ on the other hand. They assumed an eccentricity distribution of f(e) = 2e and an isotropic distribution of inclinations and obtained the following results:

$$<d_{{\rm proj}}>\ = 0.91\,a\;.$$ (7)

$$<v_{{\rm obs}}>\ = 0.72\,v_{\rm face-on, circ}\;.$$ (8)

However, $d_{\rm proj}$ and $v_{\rm obs}$ are not uncorrelated. For example, if we observe a companion in the outer part of its eccentric orbit, $d_{\rm proj}$ will be larger, and $v_{\rm obs}$ smaller than their average values. Therefore, we cannot insert these equations into Eq. (6) to obtain a statistical relation between $d_{\rm proj}$, $v_{\rm obs}$, and $M_{\rm Sys}$.

To overcome this problem, we performed more sophisticated computer simulations. Each simulation contains 10million binaries with a fixed system mass and randomly distributed orbital parameters. The periods follow the distribution of periods of main-sequence stars (Duquennoy & Mayor 1991), the distribution of eccentricities is f(e)=2e and the inclinations are distributed isotropically, while all the other parameters have uniform distributions. The distances to the observer are varied within a range of $143\pm25$pc. We chose two observation dates separated by a random timespan between 4 and 10 years and computed the average projected separation and orbital velocity, in much the same way as we did for the real data. We then select binaries in the projected separation range from 10 to 70AU. For these binaries, we compute $<v^2\cdot d>/M_{\rm Sys}$. These simulations are repeated for different system masses. For $M_{\rm Sys}$ between $0.5~M_{\odot}$ and $2.5~M_{\odot}$, the results vary from 18.4 to $18{\rm\,AU^3/(yr}^2\,M_{\odot})$. This gives us the following relation:

$$M_{\rm Sys}\approx\frac{1}{18.2}\cdot <v^2\cdot d>\cdot$$ (9)

The term on the right hand side of Eq. (9) contains v². Since the measured velocities and separations differ from the real values by some unknown measurement error, this leads to an additional bias term:

 

$$<v_{\rm obs}^2\cdot d_{\rm obs}> <(v_{\rm real}+\delta v)^2\cdot (d_{\rm real}+\delta d)>$$ =

$$<v_{\rm real}^2\cdot d_{\rm real}> + <\delta v^2\cdot d_{\rm real}>\cdot$$ = (10)

Combining Eqs. (9) and (10), and using $\Delta v$ as an estimate for $\delta v$, we obtain a relation that allows us to use the observed values to obtain an average system mass:

$$M_{{\rm Sys}} \approx \frac{1}{18.2}\left(<v_{\rm obs}^2\cdo... ...\rm proj}> - <\Delta v_{\rm obs}^2\cdot d_{\rm proj}>\right).$$ (11)

This is only an estimate for the mass in an ensemble of systems with identical system masses. The stars in our sample will probably not all have the same mass. However, we expect them to have similar masses, and for a sufficiently large number of systems, the uncertainties will cancel. The result should yield a reliable average mass for these systems.

If we use the data of all stars in our sample, Eq. (11) yields a system mass of $2.5~M_{\odot}$. Excluding RXJ1546.1-2804 lowers the result to $2.0~M_{\odot}$. If we further exclude V 773 Tau, Elias 12, and BD+26718BAa, we arrive at a system mass of $1.3~M_{\odot}$. We have reason to assume that the companion to RXJ1546.1-2804 is a chance projected background star. For the three other systems mentioned, the velocity is puzzling at first sight, but is still consistent with orbital motion. Furthermore, other possible explanations fail for V 773 Tau and Elias 12 (Sects. 3.2.2 and 3.2.5). Thus, it does not seem justified to exclude them from the sample, and we will adopt $2~M_{\odot}$ as the result for the average dynamical system mass.

Given the statistical uncertainties, it is difficult to estimate the error of the system mass. Using the standard deviation of the quantities averaged in Eq. (11) to estimate the error of the mean yields $\approx$ $0.7~M_{\odot}$. This is in agreement with the scatter we obtain if we exclude or include the stars mentioned in the last paragraph.

We conclude that the average mass of the systems in our sample is in the range 1.3...2.5 $~M_{\odot}$, with a most probable system mass of $2~M_{\odot}$. Our result is thus consistent with the expectation that T Tauri stars' masses are around $M\approx 1~M_{\odot}$ and also with the average mass of $1.7~M_{\odot}$ that G95 derived for their sample.