Mon. Not. R. Astron. Soc. 000, 1–5 (2007) Printed 17 January 2014 (MN LATEX style file v2.2) Dynamical parallax of σ Ori AB: mass, distance and age arXiv:0710.3541v1 [astro-ph] 18 Oct 2007 José Antonio Caballero1⋆ 1 Dpto. de Astrofı́sica y Ciencias de la Atmósfera, Facultad de Ciencias Fı́sicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain. E-mail: caballero@astrax.fis.ucm.es Accepted 2007 October 18. Received 2007 October 01; in original form 2007 August 21 ABSTRACT The massive OB-type binary σ Ori AB is in the centre of the very young σ Orionis cluster. I have computed the most probable distances and masses of the binary for several ages using a dynamical parallax-like method. It incorporates the BV RIH-band apparent magnitudes of both components, precise orbital parameters, interstellar extinction and a widely used grid of stellar models from the literature, the Kepler’s third law and a χ2 minimisation. The derived distance is 334+25 −22 pc for an age of 3±2 Ma; larger ages and distances are unlikely. The masses of the primary and the secondary lie on the approximate intervals 16–20 and 10–12 M⊙ , respectively. I also discuss the possibility of σ Ori AB being a triple system at ∼385 pc. These results will help to constrain the properties of young stars and substellar objects in the σ Orionis cluster. Key words: stars: individual: σ Ori – stars: binaries: close – open clusters and associations: individual: σ Orionis 1 INTRODUCTION The Trapezium-like system σ Ori, that illuminates the encolure of the Horsehead Nebula, is the fourth brightest star in the young Ori OB 1 b association. The multiple system is composed of at least five early-type stars (Burnham 1892; Greenstein & Wallerstein 1958; van Loon & Oliveira 2003; Caballero 2007b). The two hottest components, σ Ori A and B (O9.5V and B0.5V), are separated by only ∼0.25 arcsec and were for a long time “the most massive visual binary known” (MA + MB ∼ 25 + 15 M⊙ ; Heintz 1974). Although the binary has not yet completed a whole revolution, the orbital parameters are relatively well determined (Hartkopf, Mason & McCalister 1996; Heintz 1997; Horch et al. 2002). It has been suggested that σ Ori AB is a hyerarchical triple, being the primary a short-period, double-line spectroscopic binary (Frost & Adams 1904; Henroteau 1921; Miczaika 1950; Bolton 1974; Morrell & Levato 1991). However, a large amount of accurate, comprehensive spectroscopic investigations have failed to confirm this hypothesis (Heard 1949; Conti & Leep 1974; Humphreys 1978; Bohannan & Garmany 1978; Garmany, Conti & Massey 1980; Simón-Dı́az & Lennon, priv. comm.). The σ Ori system is located in the centre of the wellknown σ Orionis open cluster. The proper motions, radial velocities and spacial distribution of stars in this cluster strongly suggests a physical association between σ Ori itself and the young cluster (Zapatero Osorio et al. 2002a; Caballero 2007a, 2007c – see also Jeffries et al. [2006], who ⋆ Investigador Juan de la Cierva at the UCM. discovered a second older and kinematically and spacially distinct population). Because of its youth, comparative nearness and low extinction, the cluster has become the richest hunting ground for brown dwarfs and planetary-mass objects in the whole sky (Béjar et al. 1999; Zapatero Osorio et al. 2000, 2002b, 2007). There is plentiful material in the literature about this cluster, covering topics like the initial mass function down to a few Jupiter masses (Béjar et al. 2001; González-Garcı́a et al. 2006; Caballero et al. 2007), jets and Herbig-Haro objects (Reipurth et al. 1998; Andrews et al. 2004), the frequency of accretors and discs (Zapatero Osorio et al. 2002a; Oliveira, Jeffries & van Loon 2004; Kenyon et al. 2005; Oliveira et al. 2006; Hernández et al. 2007; Caballero et al. 2007), the X-ray emission from young objects (Walter et al. 1997; Sanz-Forcada et al. 2004; Franciosini, Pallavicini & Sanz-Forcada 2006) or their photometric variability (Caballero et al. 2004; Scholz & Eislöffel 2004). The most used values of heliocentric distance and age of the σ Orionis cluster are d ∼ 360 pc and ∼3 Ma (Brown, de Geus & de Zeeuw 1994; Perryman et al. 1997; Zapatero Osorio et al. 2002a; Oliveira et al. 2002). There is a consensus in the literature that the cluster is younger than 8 Ma and older than 1 Ma. There is, however, a strong divergence of opinion on the heliocentric distance. Caballero (2007a) compiled determinations in the literature of the distance to the σ Orionis cluster from the 352+166 −168 pc from Hipparcos parallax to almost 500 pc from colour-magnitude diagrams. Apart from the uncertainties of theoretical isochrones at very young ages, the derivation of the initial mass function of the cluster is strongly affected by the uncertainty in the actual age and heliocentric distance (Jeffries et al. 2 José A. Caballero measurement, that was estimated from the Tycho-2 BT VT magnitudes, all the data come from adaptive optics observations. The Hipparcos catalogue also tabulated the nonstandard HP -band magnitudes. The magnitudes and colours of both stars correspond to what it was expected of an O9.5V and a B0.5V at d ∼ 350 pc. I have computed through a simple minimisation method which are the most probable heliocentric distances for several cluster ages. In particular, I have looked for the minima of the following chi-square distributions: Table 1. Photometry of σ Ori A and B from the literature. Band B V R I H [mag] [mag] [mag] [mag] [mag] σ Ori A σ Ori B 3.85±0.05 4.10±0.03 4.15±0.04 4.41±0.04 4.81±0.10 5.18±0.05 5.34±0.10 5.49±0.13 5.66±0.16 6.02±0.10 References Ca07a tBr00 tBr00 tBr00 Ca06 2006; Caballero et al. 2007). There are other investigations that require a precise age determination, such as the evolution of the angular momentum due to discs and stellar winds (Eislöffel & Scholz 2007), disc dissipation (Hernández et al. 2007) and evolution of hot massive stars. In this Letter, I revisit a well known method for distance determination: the dynamical parallax (e.g. Russell 1928). I apply it to the σ Ori AB binary using state-of-the-art data and tools to determine its mass, age and heliocentric distance. 2 ANALYSIS AND RESULTS The determination of the dynamical parallax of a binary of known orbital period, P , and angular semimajor axis, α, uses the Kepler’s third law and a power-law mass-luminosity relation (e.g. Reed 1984). In this Letter, I instead use the grids of theoretical models from the Geneva group (Schaller et al. 1992). On the contray to other widely used grids, like those by the Lyon and Padova groups (Baraffe et al. 1998; Girardi et al. 2000), the Geneva grids tabulate absolute magnitudes in a large number of passbands and ages down to 1 Ma and are valid up to very high masses (i.e. M > 10 M⊙ ). Although there are more binaries and binary candidates in the cluster (Caballero 2005; Kenyon et al. 2005; Caballero et al. 2006 and references therein), σ Ori AB is the only pair whose orbital parameters are known. Here, I have employed the parameters given by Hartkopf et al. (1996). The almost face-on orbit is characterised by a long period (P = 155.3±7.5 a), a close angular separation (α = 0.2642±0.0052 arcsec) and a low eccentricity (e = 0.051±0.015). The orbital parameters are consistent with those of Heintz (1997; P = 158 a, α = 0.265 arcsec, e = 0.06). Using the standard units M⊙ , a (annum) and AU for MA + MB , P and the physical semimajor axis a, respectively, the Kepler’s third law takes the simple expression (MA + MB )P 2 = a3 . Accounting for a = d tan α ≈ dα, where d is the heliocentric distance in parsecs, and replacing the values of P and α from Hartkopf et al. (1996), then the third law for σ Ori AB can be written as: MA + MB = 7.45 10−7 d3 (1) (Mtotal ≡ MA +MB ). At the Hipparcos distance d = 352 pc, the total mass of the binary would be about 32 M⊙ , that is less than the classical value of Mtotal ∼ 40 M⊙ , but is consistent with the value of Mtotal ∼ 30 M⊙ estimated by Caballero (2007a). I tabulate in Table 1 the BV RIH-band magnitudes of both components in σ Ori AB, taken from the literature (ten Brummelaar et al. 2000 –tBr00–; Caballero 2006 –Ca06–; Caballero 2007a –Ca07a–). Except for the B-band χ2 (d, MA , MB ) = χ2A (d, MA ) + χ2B (d, MB ) (2) (for fixed ages and metallicities), where χ2A and χ2B are: χ2A = X (mλ,A − m∗λ,A )2 δm2λ,A , χ2B = X (mλ,B − m∗λ,B )2 δm2λ,B (3) (λ ≡ B, V, R, I, H). mλ,[A,B] and δmλ,[A,B] are the observed apparent magnitudes and corresponding uncertainties of σ Ori A and B in Table 1, and m∗λ,[A,B] = m∗λ,[A,B] (d, M[A,B] ) are the theoretical apparent magnitudes that a hypothetical star of mass M[A,B] would have at an heliocentric distance d. To compute m∗λ,[A,B] , I have used: (i) the theoretical absolute magnitudes Mλ,[A,B] from the basic grids of non-rotating stellar models with solar metallicity (Z = 0.020), overshooting and OPAL opacities of the Geneva group, (ii) the colour excess E(B −V ) = 0.05 mag of σ Ori AB from Lee (1968) and (iii) the interstellar extinction law parameters Aλ /AV and RV from Rieke & Lebofski (1985). In detail, I have used the grids with standard mass loss Ṁ and ages 1.0, 2.0, 3.2, 4.9 and 10.0 Ma (Schaller et al. 1992 – Sc92) and high mass loss 2 × Ṁ and age 3.2 Ma (Meynet et al. 1994 – Me94). The models do not provide data for other ages less than 32 Ma. Caballero (2006) measured an average solar metallicity of solar-like stars in the σ Orionis cluster, [Fe/H] = 0.0±0.1 dex, which justifies the use of Z = 0.020. For a fixed age (and metallicity), there is a one-to-one correspondence in the grids between Mλ,[A,B] and M[A,B] if both A and B stars are in the main sequence. For each heliocentric distance, there is only one corresponding total mass of the hypothetical binary, Mtotal ∝ d3 (Eq. 1). If the mass of the primary, MA , is fixed, then the mass of the secondary is obtained from the simple expression MB (d, MA ) = Mtotal (d) − MA . To sum up, there is a value of χ2 for each trio (d, MA , age). Because of the known spectral types of σ Ori A and B, I conservatively imposed that the masses of the primary and of the secondary should lie on the intervals 10 M⊙ 6 MA 6 50 M⊙ and 1.5 M⊙ 6 MB 6 MA , respectively. These constraints speed up the minimization but do not influence the results (see below). Fig. 1 illustrates the χ2 minimisation for an age of 3.2 Ma and standard mass loss. Top and bottom windows display different coverage resolutions of the (d, MA ) plane. The results for normal and high mass loss for 3.2 Ma are almost identical. The plots for the other ages except for 10.0 Ma are quite similar, with the minimum shifted along the X axis (distance). There are no solutions (i.e. the masses of A and B simultaneouly satisfy Eq. 1 and the above constraints) for d . 250 pc and d & 500 pc. In addition, the 4.9 Ma models do not provide solutions for d & 370 pc. Finally, there are no solutions at all for 10.0 Ma. At this Dynamical parallax of σ Ori AB Age = 3.2 Ma 3 Table 2. Best fits of the dynamical parallax of σ Ori AB. 4 10 3 χ 2 10 Mass loss Age [Ma] d [pc] MA [M⊙ ] MB [M⊙ ] χ2 Ṁ Ṁ Ṁ Ṁ Ṁ 2 × Ṁ 1.0 2.0 3.2 4.9 10.0 3.2 346±13 337±13 334±13 325±13 ... 333±13 20.1±0.2 18.3±0.2 17.6±0.2 16.0±0.2 ... 17.4±0.2 11.7±0.2 11.1±0.2 10.8±0.2 10.2±0.2 ... 10.8±0.2 8.70 9.38 9.76 10.56 ... 9.72 2 10 1 10 200 250 300 350 400 450 500 550 d [pc] for all ages. Finally, I have determined the most probable masses of A and B for the Hipparcos parallax distance (d = 352 pc): MA = 21.4 M⊙ , MB = 12.0 M⊙ for an age of 1 Ma. The minimum χ2 for 1 Ma is an order of magnitude smaller than for the other ages tested, meaning that a binary age older than 1 Ma would be unlikely at the Hipparcos distance. Age = 3.2 Ma 3 DISCUSSION 4 10 3 χ 2 10 2 10 1 10 315 320 325 330 335 340 345 350 355 d [pc] Figure 1. χ2 vs. heliocentric distance for several masses of σ Ori A. The dotted lines indicate curves of constant mass. The basic grid of models is from Schaller et al. (1992), the age is 3.2 Ma and the metallicity is Z = 0.020. Top window: whole interval of heliocentric distance in steps of 1 pc from 200 to 550 pc and of mass of the primary in steps of 2 M⊙ from 10 to 50 M⊙ . The minimum χ2 is for MA = 18 M⊙ (d = 336 pc). Bottom window: same as top window, but for intervals of heliocentric distance in steps of 0.2 pc and of mass of the primary in steps of 0.2 M⊙ . The minimum in χ2 is better constrained (cf. Table 2). age, the primary has left the main sequence and got much brighter than the secondary. The values of (d, MA ) that minimise χ2 for a given age and mass loss are provided in Table 2. The uncertainties in the values of d account for the error bars in the photometric data in Table 1 and in the orbital parameters P (5 %) and α (2 %) in Hartkopf et al. (1996), and the size of the steps in the high resolution minimisation (∆d = 0.2 pc). The uncertainty in the masses is set to the step size, ∆MA = 0.2 M⊙ . The results would be identical if no mass constraints were set. For an age interval of 3±2 Ma, the corresponding heliocentric distance interval is d = 334+25 −22 pc. Distances larger than 400 pc and less than 290 pc are less likely for the 1–5 Ma age range; distances larger than 450 pc are highly unlikely The theoretical effective temperatures that correspond to the optimal fits lie on the intervals Teff = 30.4–34.6 kK for the primary and Teff = 25.2–27.5 kK for the secondary. The hottest temperatures are for the youngest ages. These values are consistent with the expected Teff for O9.5V and B0.5V stars, respectively (e.g. O9.5V: 30–35 kK – Popper 1980; Gulati, Malagnini & Morossi 1989; Castelli 1991; Vacca, Garmany & Shull 1996; Martins, Schaerer & Hillier 2005), and with previous measurements of the Teff of σ Ori A (30.0– 33.0 kK – Morrison 1975; Underhill et al. 1979; Morossi & Crivellari 1980; Repolust et al. 2005). The corresponding theoretical gravities (log g ∼ 4.00–4.22) are also normal for class V at such temperatures. The minima of χ2 in Table 2 are very sensitive to the variations of heliocentric distance and of mass: on the one hand, at fixed mass and age, a fluctuation of d of barely 30 pc results in a change of three orders of magnitude in χ2 ; on the other hand, at fixed distance and age, a fluctuation of MA of barely 5 M⊙ results in a change of almost two orders of magnitude in χ2 . The minima of χ2 are, however, quite unresponsive to the variations of the age between 1.0 and 4.9 Ma (see last column in Table 2). A younger age gives a slightly better fit results and that favours a slightly larger distance (d ∼ 350 pc). The results do not strongly suggest a younger age of 1 Ma, but they are useful in excluding an older age. The absence of a solution at 10 Ma agrees with previous upper limits on the ages of the Ori OB 1 b association from the presence of very early-type stars in the main sequence (Blaauw 1964) and of the σ Orionis cluster from spectral synthesis surrounding the Li i λ670.78 nm line (Zapatero Osorio et al. 2002a). The derived distance interval for 3±2 Ma, d = 334+25 −22 pc, is consistent with the canonical distance to the σ Orionis cluster of d ∼ 360 pc, but is difficult to conciliate with the distance of 440 pc for 2.5 Ma that Sherry, Walter & Wolk (2004) used. The derived distance interval also deviates from very recent determinations of the distance to some elements in the Ori OB 1 complex. In particular, Terrell, Munari & Siviero (2007), using the eclipsing spectroscopic binary VV Ori close to Alnilam (ǫ Ori) in Ori OB 1 b, and 4 José A. Caballero Sandstrom et al. (2007), employing the Very Large Baseline Array in the Orion Nebula Cluster in Ori OB 1 a, have determined very accurate heliocentric distances of 388–389 pc (see also Menten et al. 2007). These values are also lower than the classical distance to the Ori OB 1 complex of ∼440 pc from average Hipparcos parallax (Brown, Walter & Blaauw 1999; de Zeeuw et al. 1999). Because of projection effects and the large physical size of Ori OB 1 (Reynolds & Ogden 1979), σ Ori could be easily contained within the complex. Given their kinematic and spacial association, the σ Ori system and the young σ Orionis cluster are likely at the same heliocentric distance and also age, if one assumes that massive and low mass star formation in a cluster is coeval (Prosser et al. 1994; Massey, Johnson & Degioia-Eastwood 1995; Stauffer et al. 1997; see, however, Sacco et al. 2007). Recently, it has been suggested that the σ Orionis cluster is actually kinematically distinct from the Ori OB 1 b association (Jeffries et al. 2006), just as 25 Ori is distinct from Ori OB 1 a (Briceño et al. 2007). If σ Ori AB were a hyerarchical triple, as described in Section 1, it would be located at a larger heliocentric distance. The hypothetical companion to σ Ori A, to which I tentatively call σ Ori F, would be 0.5 mag fainter than the primary in the 370–493 nm interval according to Bolton (1974). This wavelength interval corresponds to the U, B Johnson bands. Taking into account mA = mA+F +   2.5 log 1 + 10− mA −mF 2.5 and the difference BA+F − BB in Table 1, then the apparent magnitudes in the blue band of the three components would be related to the combined magnitude BA+F through: BA ≈ BA+F + 0.53 mag, BB ≈ BA+F + 1.33 mag and BF ≈ BA+F + 1.03 mag. I have looked for the distances and theoretical masses whose corresponding apparent magnitudes match the B[A,B,F] relations and the Kepler’s third law, (MA + MF ) + MB = 7.45 10−7 d3 . There are only a few solutions that simultaneously verify Teff,A ≈ 30.0–33.0 kK and Teff,B ≈ 26.0–30.0 kK, as expected from the spectral types of σ Ori A+F and σ Ori B. In the triple scenario, the F component would have an intermediate temperature between the A and B stars (roughly B0.0V) and would orbit very close to σ Ori A. The valid solutions are found for the narrow distance interval 370–400 pc and only for ages between 1.0 and 4.9 Ma. Although distances less than 290 pc and larger than 450 pc are ruled out again, the most probable distance to σ Ori under the triple hypothesis, d ∼ 385 pc, agrees very well with those of VV Ori and the Orion Nebula Cluster. Further high-resolution spectroscopic studies are needed to ascertain the existence and characteristics of σ Ori F. 4 SUMMARY I have determined the most probable heliocentric distances and masses of the components in the young binary σ Ori AB. The used methodology is an improvement of the dynamical parallax method using a χ2 minimisation of observed and theoretical apparent magnitudes of A and B. The values range in the intervals 325 pc . d . 346 pc, 16 M⊙ . MA . 20 M⊙ , 10 M⊙ . MB . 12 M⊙ for 1–5 Ma. Ages and distances larger than or equal to 10 Ma and ∼450 pc are excluded from the minimisation. The theoretical effective tem- peratures of both components are consistent with the observed spectral types. Accounting for uncertainties in the orbital parameters and photometric data from the literature, the derived distance interval for age = 3±2 Ma is d = 2 334+25 −22 pc. The values of χ are minimum for the largest distances within this interval, that translate into the youngest ages. If there were a third component, σ Ori F, in a tight orbit to σ Ori A, then the system could be at a larger heliocentric distance of about 385 pc. The σ Ori star system is in the centre of the young σ Orionis cluster. The knowledge of the age and heliocentric distance of the cluster is fundamental for the study of the initial mass function down to the planetary regime and the evolution of discs and angular momentum in very young objects. ACKNOWLEDGMENTS I thank the anonymous referee for improving comments and suggestions, and D. Montes for revising the manuscript. 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