Abstract
V* PX Vir is a nearby single-lined spectroscopic binary with the primary being a variable of a BY Dra-type star. It has a period of ∼216 d and the components of the system are two low-mass pre-main-sequence stars. By a simultaneous least-squares fit to the radial velocities, the relative positions, and the revised Hipparcos intermediate astrometric data, the complete orbital solution is derived and the precision is improved significantly. In addition, the Hipparcos parallax is revised as 46.06±0.69 mas. Taking advantage of the resulting orbital elements and parallax, the component masses are calculated to be 0.74±0.06 M⊙ and 0.47±0.02 M⊙, respectively. The absolute magnitudes in the K and H bands are determined as 4.32±0.14, 4.42±0.03 for the primary and 5.63±0.46, 6.08±0.03 for the secondary, respectively. According to the derived physical parameters, the two components of the system lie very close to the same isochrone age 0.1–0.14 Gyr by comparison with pre-main-sequence evolution tracks.
1 Introduction
The determination of stellar mass is fundamental for studying stellar physics. Binary stars offer the best opportunity for determing the masses of stars, which is essential to verify stellar evolution theory (Andersen 1991; Torres et al. 2010). Compared with double-lined spectroscopic binaries (SB2s), the masses and physical properties of the two components are usually very different for single lined-spectroscopic binaries (SB1s). Therefore, SB1s can provide much more stringent constraints on stellar models at a single age. However, the component masses of SB1 systems are difficult to derive by orbital fitting without a reliable measure of the distance. Only a few nearby SB1 systems with relative astrometric measurements and reliable distance have known component masses, such as ε Hydrae and 10 Ursae Majoris (Underhill 1963). The data released by Hipparcos and the Gaia mission provide an opportunity to determine the component masses of the SB1s with relative astrometric measurements (Perryman et al. 1997; Malkov et al. 2012; Gaia Collaboration 2016).
For observational limitations, the binaries with low-mass pre-main-sequence components and fundamental stellar parameters derived directly from observation are extremely rare. The lack of direct information on fundamental parameters for low-mass pre-main-sequence stars prevent the stellar models being tested and constrained (Torres & Ribas 2002). V* PX Vir (HIP 63742, HD 113449) is a single-lined spectroscopic binary that is also a visual binary with two low-mass pre-main-sequence stars. Its orbital period is ∼216 d (Cusano et al. 2009; Griffin 2010; Evans et al. 2012), and the combined spectral type of the system is K1V (Houk & Swift 1999). The variable amplitude of the primary for the system is 0.04 mag (Kazarovets et al. 2006). Strassmeier et al. (2000) found a rotation period of 6.47 d for the primary star and gave two radial velocity measurements of 5.8 km s−1 and 8.2 km s−1 obtained a week apart. The discrepancy between the two radial velocities is not enough to demonstrate real variation. Paulson and Yelda (2006) have provided 15 radial velocities spread over eight nights, and found that the rms dispersion was too small to indicate orbital motion. The preliminary astrometry orbit was first given in part O of volume 10 of the Hipparcos and Tycho catalogues (ESA 1997). Then, the system was placed on the Cambridge spectroscopic-binary observing program in 2003, and the spectroscopic orbital solution was given by Griffin (2010). Using the High Accuracy Radial velocity Planet Searcher (HARPS) spectrograph, the system has been observed, and another spectroscopic orbital solution was given by Cusano et al. (2009). However, for particular orbital elements (the orbital period P and eccentricity e), the two spectroscopic orbits are inconsistent with each other with three times uncertainty. From a first analysis of the CRIRES spectra, Cusano et al. (2009) determined the mass ratio of the primary to the secondary as 0.57 ± 0.05. Cusano et al. (2010) gave an estimation of the inclination of the system with interferometric data acquired with VLTI/AMBER, then the masses of the two components were derived to be 0.88 ± 0.13 M⊙ and 0.50 ± 0.07 M⊙ combining the inclination and the mass function. The relative astrometric orbit was given by Evans et al. (2012), and the total mass of the system was given as 1.10 ± 0.09 M⊙, while the isochrone-determined masses for the components are 0.84 ± 0.08 M⊙ and 0.51 ± 0.01 M⊙. The launch of space missions allows high-precision parallax measurement for nearby stars, and the revised parallax given by the Hipparcos mission was 46.10 ± 0.81 mas (van Leeuwen 2007a). The latest parallax given by the Gaia mission was 48.8829 ± 0.8872 (Luri et al. 2018).
Torres (2004) emphasized the merging of observations from different techniques (e.g., radial velocity data and astrometric data) into a single fit that is usually better than separate solutions using either of those kinds of data. In the present paper, a three-dimensional orbital solution with high precision as well as the parallax of the system is given via a simultaneous adjustment including radial velocity data, astrometric measurements, and the Hipparcos Intermediate Astrometric Data (rHIAD). Then, the component masses are determined, and the age of the system is estimated in comparison with the pre-main-sequence evolution tracks.
All of the observations of the system are introduced in section 2. The orbital determination is presented in section 3. The discussion and conclusion are provided in section 4.
2 Observations
V* PX Vir is a nearby SB1 with astrometric orbit, and the two components are identified as pre-main-sequence stars. Kazarovets et al. (2006) noted that the primary of the system considered is a BY Dra variable star with a magnitude amplitude of 0.04 mag.
Its parallax given by the Hipparcos and Gaia missions is 46.10 ± 0.81 mas and 48.8829 ± 0.8872 mas, respectively (van Leeuwen 2007a; Luri et al. 2018). The Hipparcos and Tycho catalogues first provided the astrometric orbit of the system, but the orbital elements are at low precision (ESA 1997). In addition, a new reduction of the Hipparcos catalogue by van Leeuwen, including the improved individual measurements, is available (van Leeuwen 2007a, 2007b).
A total of 15 measurements spread over eight nights was provided by Paulson and Yelda (2006), and the rms dispersion of all measurements was found to be only 0.14 km s−1. Two radial velocity measurements obtained a week apart were given by Strassmeier et al. (2000). In 2003, V* PX Vir was placed on the Cambridge spectroscopic-binary observing program (Griffin 2010), and 41 available radial velocity measurements were provided. In addition, the system was observed with HARPS between 2004 and 2009, and 20 radial velocity measurements with extremely high precision became available (Trifonov et al. 2020).
The Gemini North telescope with adaptive optics has been used to look for close companions, but nothing was seen. The companion that we report here was first announced by Cusano et al. (2009, 2010). Then, six sets of relative astrometric measurements with high precision were provided using the adaptive optics facilities of the Keck II and Palomar Hale Telescopes by Evans et al. (2012).
The total magnitudes in the K, H bands for V* PX Vir are 5.89 and 5.72 (Skrutskie et al. 2006), and the magnitude differences near the K, H bands of the two components were derived as 1.3 ± 0.6 and 1.66 ± 0.01 from the Fourth Catalog of Interferometric Measurements of Binary Stars (Hartkopf et al. 2001), respectively.
3 Orbital determination
Considering that the small amplitude variation of the primaries of V* PX Vir has very little impact on the photocentric orbit, rHIAD were also used to determine the orbital solution together with the relative position data and radial velocities available. rHIAD describe the motion of the reference point and the photocentric motion on the plane of sky, the relative position data describe the motion of the secondary relative to the primary, and the radial velocities describe the motion on the line of sight. In general, simultaneous fits strengthen the determination of the elements because they fully exploit the constraints available from both types of data. The data used to determine the orbital solution include rHIAD (van Leeuwen 2007b), radial velocity measurements (Griffin 2010; Trifonov et al. 2020), and relative astrometric measurements (Evans et al. 2012).
A modified grid method and the Levenberg–Marquardt method described in Ren and Fu (2010) is used to find the global minimum of the objective function (1). Each measurement is weighted by the inverse square of its own error, and the uncertainties for the derived parameters are determined from the covariance matrix. The derived orbital elements are shown in table 1, and the published ones are also listed in the table for comparison. From the table, we can see that the precision of the orbital solutions is improved significantly compared with the published ones. The radial velocities and the fitted radial velocity curve are shown in figure 1, and all the radial velocities are adjusted to share the same systemic velocity as Griffin (2010) for the sake of plotting. The black and red symbols with error bars indicate the radial velocities supplied by Griffin (2010) and Trifonov et al. (2020), respectively. The relative position data and the fitted apparent orbit are shown in figure 2. The straight dotted line indicates the periastron. The black dots with error bars represent the observed data and the red triangles mark the corresponding epochs of the orbital solution. The newly derived parallax π is 46.06 ± 0.69 mas with full consideration of orbital motion, and the precision is higher than that given by the Hipparcos and Gaia missions (Perryman et al. 1997; Gaia Collaboration 2016).
Calculated radial velocity curve and the observed RV data of V* PX Vir. For the sake of plotting, all the radial velocities are adjusted to share the same systemic velocity as Griffin (2010). The black and red dots with error bars present radial velocities provided by Griffin (2010) and Trifonov et al. (2020), respectively.
Calculated apparent orbit and relative position data of V* PX Vir from the Fourth Catalog of Interferometric Measurements of Binary Stars (Hartkopf et al. 2001). The straight dotted line indicates the periastron. The black dots with error bars represent the observed data and the red triangles mark the corresponding epochs of the orbital solution.
Newly derived and historical orbital parameters and masses of V* PX Vir.
| Parameter | Present work | Cusano et al. (2009) | Griffin (2010) | Evans et al. (2012) |
|---|---|---|---|---|
| K1 (km s−1) | 12.8978 ± 0.0003 | 13.40 ± 0.02 | 13.03 ± 0.10 | |
| ωA (°) | 295.6 ± 0.2 | 114.6 ± 0.5 | 294.5 ± 1.9 | 294.5 ± 0.5 |
| e | 0.2543 ± 0.0002 | 0.300 ± 0.005 | 0.261 ± 0.007 | 0.300 ± 0.005 |
| P (d) | 216.48 ± 0.01 | 215.9 ± 0.1 | 216.48 ± 0.06 | 216.9 ± 0.2 |
| T (JD 2400000) | 53846.4 ± 0.1 | 53411 ± 1 | 53845.3 ± 1.0 | 53410.5 ± 1 |
| a″ (mas) | 34.6 ± 0.001 | 33.7 ± 0.004 | ||
| i (°) | 121.2 ± 0.2 | 63 ± 3 | 57.5 ± 1.5 | |
| Ω (°) | 204.5 ± 0.2 | 102 ± 8 | 201.8 ± 1.6 | |
| m1 (M⊙) | 0.74 ± 0.06 | 0.88 ± 0.13 | 0.84 ± 0.08 | |
| m2 (M⊙) | 0.47 ± 0.02 | 0.50 ± 0.07 | 0.51 ± 0.01 |
| Parameter | Present work | Cusano et al. (2009) | Griffin (2010) | Evans et al. (2012) |
|---|---|---|---|---|
| K1 (km s−1) | 12.8978 ± 0.0003 | 13.40 ± 0.02 | 13.03 ± 0.10 | |
| ωA (°) | 295.6 ± 0.2 | 114.6 ± 0.5 | 294.5 ± 1.9 | 294.5 ± 0.5 |
| e | 0.2543 ± 0.0002 | 0.300 ± 0.005 | 0.261 ± 0.007 | 0.300 ± 0.005 |
| P (d) | 216.48 ± 0.01 | 215.9 ± 0.1 | 216.48 ± 0.06 | 216.9 ± 0.2 |
| T (JD 2400000) | 53846.4 ± 0.1 | 53411 ± 1 | 53845.3 ± 1.0 | 53410.5 ± 1 |
| a″ (mas) | 34.6 ± 0.001 | 33.7 ± 0.004 | ||
| i (°) | 121.2 ± 0.2 | 63 ± 3 | 57.5 ± 1.5 | |
| Ω (°) | 204.5 ± 0.2 | 102 ± 8 | 201.8 ± 1.6 | |
| m1 (M⊙) | 0.74 ± 0.06 | 0.88 ± 0.13 | 0.84 ± 0.08 | |
| m2 (M⊙) | 0.47 ± 0.02 | 0.50 ± 0.07 | 0.51 ± 0.01 |
Newly derived and historical orbital parameters and masses of V* PX Vir.
| Parameter | Present work | Cusano et al. (2009) | Griffin (2010) | Evans et al. (2012) |
|---|---|---|---|---|
| K1 (km s−1) | 12.8978 ± 0.0003 | 13.40 ± 0.02 | 13.03 ± 0.10 | |
| ωA (°) | 295.6 ± 0.2 | 114.6 ± 0.5 | 294.5 ± 1.9 | 294.5 ± 0.5 |
| e | 0.2543 ± 0.0002 | 0.300 ± 0.005 | 0.261 ± 0.007 | 0.300 ± 0.005 |
| P (d) | 216.48 ± 0.01 | 215.9 ± 0.1 | 216.48 ± 0.06 | 216.9 ± 0.2 |
| T (JD 2400000) | 53846.4 ± 0.1 | 53411 ± 1 | 53845.3 ± 1.0 | 53410.5 ± 1 |
| a″ (mas) | 34.6 ± 0.001 | 33.7 ± 0.004 | ||
| i (°) | 121.2 ± 0.2 | 63 ± 3 | 57.5 ± 1.5 | |
| Ω (°) | 204.5 ± 0.2 | 102 ± 8 | 201.8 ± 1.6 | |
| m1 (M⊙) | 0.74 ± 0.06 | 0.88 ± 0.13 | 0.84 ± 0.08 | |
| m2 (M⊙) | 0.47 ± 0.02 | 0.50 ± 0.07 | 0.51 ± 0.01 |
| Parameter | Present work | Cusano et al. (2009) | Griffin (2010) | Evans et al. (2012) |
|---|---|---|---|---|
| K1 (km s−1) | 12.8978 ± 0.0003 | 13.40 ± 0.02 | 13.03 ± 0.10 | |
| ωA (°) | 295.6 ± 0.2 | 114.6 ± 0.5 | 294.5 ± 1.9 | 294.5 ± 0.5 |
| e | 0.2543 ± 0.0002 | 0.300 ± 0.005 | 0.261 ± 0.007 | 0.300 ± 0.005 |
| P (d) | 216.48 ± 0.01 | 215.9 ± 0.1 | 216.48 ± 0.06 | 216.9 ± 0.2 |
| T (JD 2400000) | 53846.4 ± 0.1 | 53411 ± 1 | 53845.3 ± 1.0 | 53410.5 ± 1 |
| a″ (mas) | 34.6 ± 0.001 | 33.7 ± 0.004 | ||
| i (°) | 121.2 ± 0.2 | 63 ± 3 | 57.5 ± 1.5 | |
| Ω (°) | 204.5 ± 0.2 | 102 ± 8 | 201.8 ± 1.6 | |
| m1 (M⊙) | 0.74 ± 0.06 | 0.88 ± 0.13 | 0.84 ± 0.08 | |
| m2 (M⊙) | 0.47 ± 0.02 | 0.50 ± 0.07 | 0.51 ± 0.01 |
4 Discussion and conclusion
The magnitude differences of the two components near the K, H bands are about 1.3 ± 0.6 and 1.66 ± 0.01 (Hartkopf et al. 2001), and the total magnitudes in the corresponding bands are 5.89 and 5.72 (Skrutskie et al. 2006), respectively. Therefore, the apparent magnitude in the K, H bands can be calculated as 6.007 ± 0.139, 6.103 ± 0.002 for the primary and 7.306 ± 0.461, 7.763 ± 0.008 for the secondary, respectively. With the parallax given in our work, the absolute magnitudes for the K, H bands of the individual stars are derived as 4.32 ± 0.14, 4.42 ± 0.03 for the primary and 5.63 ± 0.46, 6.08 ± 0.03 for the secondary, respectively.
The location of the V* PX Vir components in the mass–magnitude diagram in K and H is shown in figure 3. The black, red, green, blue, and magenta lines in the figure are theoretical isochrones of age 0.06, 0.08, 0.01, 0.12, and 0.14 Gyr from the pre-main-sequence evolution tracks (Baraffe et al. 1998) for initial metallicity [M/H] = −0.05 (Luck 2018). From the K-band mass relation, the isochrone of age of the system is between 0.6 and 1.4 Gyr. From the H-band mass relation, the isochrone of age of the system is bigger than 0.8 Gyr. Then, the isochrone of age for the system is estimated to be about 0.1–0.14 Gyr.
Isochrones of V* PX Vir in the K and H bands. The black, red, green, blue, and magenta lines indicate isochrones of 0.06, 0.08, 0.01, 0.12, and 0.14 Gyr from the pre-main-sequence evolution tracks (Baraffe et al. 1998).
The determination of physical parameters provides a mass calibration as well as a relative age calibration for the theoretical isochrones of pre-main-sequence tracks. The precision of the component masses is influenced by the precision of the orbital solution as well as the parallax. The Gaia mission is running now and its astrometric data have a much higher precision than those of Hipparcos. We look forward to merging the time series data of Gaia to improve the precision of parallax as well as the orbital solution. Once the uncertainty of the determined parallax is higher than ∼100 μ as, the precision of both the component masses will be better than |$3\%$|.
Acknowledgements
We acknowledge the Natural Science Foundation of Shandong Province, China (Grant No. ZR202102220686), the National Natural Science Foundation of China (Grant Nos. 11603072, 11727806, 11673071, and 12073008), and the science research grants from the China Manned Space Project (Nos. CMS-CSST-2021-A12 and CMS-CSST-2021-B10). This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, the Washington Double Star Catalog maintained at the US Naval Observatory, and NASA’s Astrophysics Data System Bibliographic Services.
