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A&A
Volume 598, February 2017
Article Number L7
Number of page(s) 7
Section Letters
DOI https://doi.org/10.1051/0004-6361/201629930
Published online 30 January 2017

© ESO, 2017

1. Introduction

The visual triple star comprising α Centauri (WDS J14396-6050AB, GJ559AB) and Proxima (HIP 70890, GJ551) is the nearest stellar system to the Earth. The solar-like α Cen A (spectral type G2V, HD 128620) and the cooler dwarf α Cen B (HD 128621) are located at a distance of only d = 1.3384±0.0011 pc (Kervella et al. 2016b). The third star Proxima is a cool red dwarf (M5.5V), that is closer to Earth by approximately 7800 au, at d = 1.3008±0.0006 pc (Benedict et al. 1999). Owing to their similarity to the Sun, α Cen A and B are benchmarks for both stellar physics (Bazot et al. 2016) and extrasolar planet research (Demory et al. 2015). In August 2016, Anglada-Escudé et al. (2016) announced the discovery of a terrestrial-mass planet orbiting Proxima in its habitable zone (Proxima b). The presence of a potentially life-sustaining planet around our nearest stellar neighbor is a strong incentive for the Breakthrough Starshot1 initiative to send ultra-fast light-driven nanocrafts to α Centauri.

Proxima was discovered more than one century ago by Innes (1915), and the strong similarity of its proper motion and parallax with those of α Cen was immediately noticed (Innes 1926; Luyten 1925; Alden 1928). The question as to whether Proxima is gravitationally bound to α Cen has been discussed by several authors (Gasteyer 1966; Walke 1979; Matthews & Gilmore 1993; Anosova et al. 1994; Wertheimer & Laughlin 2006; Matvienko & Orlov 2014). Although statistical considerations are usually invoked to justify that Proxima is probably in a bound state, solid proof from dynamical arguments using astrometric and radial velocity (RV) measurements have never been obtained at a sufficient statistical significance level. As discussed by Worth & Sigurdsson (2016), if Proxima is indeed bound, its presence may have impacted planet formation around the main binary system.

2. Radial velocities

2.1. Observed radial velocities

We adopt the observed RV of the barycenter of α Cen A and B determined by Kervella et al. (2016b) that is statistically identical to the value obtained by Pourbaix & Boffin (2016) from the same RV dataset vr, obs [α Cen] = −22.393 ± 0.004 km s-1.

The main obstacle in demonstrating that Proxima is gravitationally bound to α Cen has historically been the lack of very-high-precision RVs of Proxima (see e.g., Thackeray 1967). This is a consequence of its relative faintness in the visible (mV = 11), but the exquisite accuracy and sensitivity achieved by modern planet-search spectrographs has overcome this limitation. We considered the possibility of adopting the RV of Proxima vr, obs [Proxima] = −22.345 km s-1 published by Barnes et al. (2014). However, the method they used to absolutely reference the velocity of their mask (GJ 1061) is uncertain. While the differential velocity between the mask and Proxima is measured with an accuracy of a few meters per second or better, the absolute value may be affected by large errors that could reach several hundred meters per second. In order to obtain the absolute velocity of Proxima, we thus returned to the original HARPS spectra (Lovis et al. 2006) obtained between 2004 and 2016. The details of our measurement technique are provided in Appendix A. We obtain a RV of vr, obs [Proxima] = −21.700 ± 0.027 km s-1. The RV variations induced by Proxima b are negligible (1.38 m s-1; Anglada-Escudé et al. 2016).

2.2. Convective blueshift

The convective blueshift (CB) is a systematic displacement of the wavelengths of a star’s spectral lines that is induced by the structure of its surface convection pattern (Dravins 1982; Shporer & Brown 2011). The upward moving material in convective cells usually occupies a larger surface and is hotter than the downward moving gas in the intergranular lanes. The net result is a systematic displacement of the spectral lines forming close to the photosphere of the star, in general toward the blue (but not systematically), that is, a negative RV shift. The CB is stronger for hotter stars with convective surfaces, and for larger convective cells (Bigot & Thévenin 2008).

The template that was used by Pourbaix & Boffin (2016) for the cross-correlation of the spectra of α Cen A and the derivation of the RV measurement is the Fourier transform spectrum of the Sun from Kurucz et al. (1984). The resulting RVs were corrected by Pourbaix & Boffin (2016) for the zero point determined by Molaro et al. (2013). The accuracy of this calibration was confirmed at a 2 m s-1 level by Haywood et al. (2016), so it is extremely solid. The solar spectrum is an excellent match to the spectrum of α Cen A and the cross correlation therefore automatically takes the CB into account. Their effective temperature Teff, that is, the flux emitted per unit surface of the photosphere, is identical to less than 20 K (Kervella et al. 2017). Their effective gravity log g is also very close: log g [α Cen A] = 4.3117±0.0015 (Kervella et al. 2017) whereas log g [ ⊙ ] = 4.4384, as the larger radius of α Cen A compensates for its higher mass. The surface convection thus operates in essentially identical conditions, and its properties in both stars are expected to be very similar. This similarity is essential to be insensitive to the CB uncertainty and to reach the highest absolute RV accuracy (Ramírez et al. 2008). As a remark, Pourbaix et al. (2002) measured a CB difference of only 72±26 m s-1 between α Cen A and B, although they have significantly different Teff and log g. The difference in CB between the Sun and α Cen A is thus probably one order of magnitude smaller. We therefore neglect the difference in CB between the Sun (−300 m s-1; Dravins 1999) and α Cen A.

The CB and gravitational redshift (GRS, see Sect. 2.3) of α Cen B are significantly different from the Sun. Their combined effect is taken into account in the orbital fits by Kervella et al. (2016b) and Pourbaix & Boffin (2016), through a constant differential velocity term ΔVB between α Cen A and B. Its amplitude is estimated at ΔVB = 322±5 m s-1(Kervella et al. 2016b). This correction term ΔVB also compensates for the mismatch of the template spectral mask. A small constant term is also adjusted for A (ΔVA = 8±5 m s-1) by Kervella et al. (2016b) to obtain a better quality fit of the full data set including astrometry, but it is marginally significant. In summary, the corrective term ΔVB brings the RV of α Cen B into the same barycentric referential as α Cen A, that is itself securely an absolute velocity thanks to the solar template used for the cross-correlation. This translates into an absolutely calibrated RV for the barycenter of α Cen A and B, that is by construction insensitive to the CB of both stars. However, we still have to correct the RV of the barycenter for the differential gravitational redshift (ΔvGRS) of α Cen A with respect to the Sun (Sect. 2.3).

For Proxima, the small expected size of the convective cells results in a very small predicted CB. In addition, our choice to measure its RV using emission lines that form essentially in the chromosphere is naturally less sensitive to CB. In the red dwarf Barnard’s star (GJ 699), whose properties are similar to Proxima, Kürster et al. (2003) proposed that the effect of convection is actually a redshift and not a blueshift. Through an analogy with the Sun, they estimate an upper limit of +33 m s-1, and variability with the magnetic field strength and location of stellar spots. The presence of magnetic field is likely to inhibit the convective flow (Kervella et al. 2016a), hence affecting the convective pattern at the surface. The velocity of Proxima, however, is stable at a level of a few m s-1 over long periods (Barnes et al. 2014; Anglada-Escudé et al. 2016). Owing to this stability, we neglect the effect of CB in Proxima’s RV with respect to the other uncertainties.

2.3. Gravitational redshift

According to general relativity, the wavelength of the photons emitted by a star are shifted to the red as they climb out of its gravitational well. This results in a shift of the wavelength of the spectral lines toward the red (i.e., longer wavelengths). The GRS effect on the RV is a function of the mass m of the star and its radius R through vGRS = Gm/ (Rc). The GRS has been observed by Takeda & Ueno (2012) in the Sun with an amplitude comparable to the expected value of + 633 m s-1, but its detection in main sequence and giant stars of the M 67 stellar cluster remains elusive (Pasquini et al. 2011). As it is a function of m/R, compact objects (white dwarfs, neutron stars, and black holes) create the strongest GRS, typically +40 km s-1 for white dwarfs (Falcon et al. 2012), while giants and supergiants exhibit very small GRS (Dravins 1999).

As the RV of α Cen A was determined by Pourbaix & Boffin (2016) from cross-correlation with a solar spectrum template, the GRS of the Sun is incorporated into the derived RV values (the velocity of the Sun is zero when the solar template is cross-correlated with a solar spectrum). However, we have to include a differential GRS term: (1)To estimate the GRS of α Cen A, its mass is taken from Kervella et al. (2016b) and its radius from Kervella et al. (2017) (Table 1). We obtain vGRS [α Cen A] = + 575.3 ± 2.4 m s-1. Subtracting the GRS of the Sun gives ΔvGRS(α Cen A) = −61.4 ± 2.7 m s-1. The ΔvGRS corrected, absolute velocity of the barycenter of α Cen is therefore vr, abs [α Cen] = −22.332 ± 0.005 km s-1.

We estimated the RV of Proxima from a direct comparison of the wavelengths of its emission lines to their laboratory wavelengths. Therefore, unlike for α Cen A, here we have to correct for the full amplitude of the GRS (vGRS). The mass of Proxima is not directly measured. Mann et al. (2015; see also Mann et al. 2016) used a large sample of M and K dwarfs to calibrate polynomial relations between the absolute Ks magnitude and the mass or the radius. The 2MASS Ks magnitude of Proxima is mKs = 4.384 ± 0.033(Cutri et al. 2003), corresponding to an absolute magnitude MKs = 8.813 ± 0.033. Using the mass-MKs relation from Mann et al. (2015), we obtain mProx = 0.1221 ± 0.0022 M. The derived mass is in perfect agreement with the value of mProx = 0.123 ± 0.006 M resulting from the mass-luminosity relation by Delfosse et al. (2000). The radius-MKs relation gives RProx = 0.1542 ± 0.0045 R, which is slightly larger (+ 1.6σ) than the interferometrically measured value of RProx = 0.141 ± 0.007 R from Demory et al. (2009). We adopt the radius predicted by the relation from Mann et al. (2015) as it is determined using the same underlying star sample as was the mass. In addition to being more precise than the interferometric measurement, this choice reduces the potential systematic error on m/R, that is the quantity of interest to determine the GRS. We obtain a GRS of vGRS [Proxima] = + 504 ± 17 m s-1. The GRS is an important source of uncertainty on the RV of Proxima, and is also a significant contributor for α Cen AB. We apply this correction to the measured RVs of Proxima and obtain vr, abs [Proxima] = −22.204 ± 0.032 km s-1.

3. Dynamics of the α Cen–Proxima system

3.1. Astrometry, proper motions, and parallaxes

We adopt the position of the barycenter of α Cen determined by Kervella et al. (2016b) at the Hipparcos epoch (1991.25), and the corresponding position of Proxima also from Hipparcos. The presence of occasional flares in Proxima is not expected to significantly affect its apparent position (Benedict et al. 1998). The parallax of α Cen is taken from Kervella et al. (2016b) and that of Proxima is adopted from Benedict et al. (1999), whose value is compatible with the measurement by Lurie et al. (2014) (ϖ = 768.13 ± 1.04 mas). An overview of the astrometric parameters is presented in Table 2. Proxima is closer to us than α Cen by 44.8 ± 1.5 light-days. This implies that Proxima’s position on the sky in the α Cen time referential is shifted by (+ 0.46″,−0.09″). We applied this correction to the apparent position of Proxima but the effect is negligible on the orbital parameters. It will, however, need to be considered when interpreting the coming Gaia (Gaia Collaboration 2016) observations of Proxima. The linear separation between the barycenter of α Cen AB and Proxima is dα−Prox = 12 947 ± 260 au. We neglect the change in differential RV between the time of the astrometric measurement by Hipparcos and the mean epoch of the HARPS spectra of Proxima (MJD ≈ 56 100). This is justified by the fact that no secular acceleration has been detected in Proxima. We also neglect the transverse Doppler redshift predicted by the special relativity theory (lower than 1 m s-1 and identical for α Cen and Proxima). From the coordinates, parallax, proper motion, and RVs of α Cen and Proxima, we compute their 3D heliocentric positions (X,Y,Z) and Galactic space velocity vectors (U,V,W) (Table B.1).

3.2. Orbital parameters

The relative velocity above which Proxima would not be gravitationally bound to α Cen is (2)where mtot = mA + mB + mProxima = 2.165 ± 0.008 M (Table 1). The difference of the space velocity vectors of α Cen AB and Proxima has a norm of vα−Prox = 273 ± 49 m s-1. The observed velocity is therefore lower than the unbound velocity limit by −5.4σ, corresponding to a theoretical probability of 4 × 10-8 that the stars are not gravitationally bound. This conclusion is robust with respect to the adopted GRS correction; the adoption of the interferometric radius value for Proxima (Demory et al. 2009) instead of the predicted value from Mann et al. (2015) results in a velocity of vα−Prox = 309 ± 55 m s-1 and a −4.2σ difference with the unbound velocity value.

Table 1

Adopted physical parameters of α Cen AB and Proxima.

Table 2

Positional data, parallax, and radial velocity of α Centauri AB (barycenter) and Proxima.

Table 3

Orbital parameters of Proxima.

We computed the orbital parameters of Proxima (Table 3) using, as inputs, the total mass of the system, the 3D position, and the Galactic space velocity of Proxima with respect to the barycenter of α Cen (mtot,X,Y,Z,U,V,W). The error bars were derived using a classical Monte Carlo approach. We drew a large number (100 000) of sets of input measurements with random fluctuations according to their error bars. The corresponding sets of orbital parameters were computed and the error bars were obtained from the 16th and 84th percentiles of their histograms (68% confidence interval; Fig. C.1). It is interesting to remark that the derived parameters are qualitatively similar to the range of possible values found by Gasteyer (1966). The orbit of Proxima is represented in projection on the plane of the sky in Fig. 1 and in cartesian Galactic coordinates in Fig. 2. Figure 3 shows the velocity and separation of Proxima with respect to α Cen over its orbit. The orbital plane of Proxima is inclined by ≈ 30° with respect to that of α Cen AB (i = 79°; Kervella et al. 2016b).

thumbnail Fig. 1

Best-fit orbit of Proxima around α Cen projected in the plane of the sky (black curve), with a set of 100 possible orbits from a Monte Carlo approach (thin gray curves). The blue dots represent the position of Proxima every 40 000 yr (labels in millenia) and the direction of the present velocity vector of Proxima is shown as a red arrow.

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thumbnail Fig. 2

Orbit in 3D cartesian Galactic coordinates, with the Sun direction shown as a green vector originating from α Cen. The labels are the same as in Fig. 1.

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thumbnail Fig. 3

Velocity (solid line) and separation (dotted line) of Proxima relative to α Cen. The present values are marked with crosses.

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4. Conclusions

Using high-accuracy RV measurements and astrometry, we show with a high level of confidence that Proxima is gravitationally bound to α Cen and orbits the pair on a moderately eccentric, very long-period orbit. This conclusion is particularly valuable for the modeling of this star as it means that the three stars are coeval and share the same initial metallicity. Due to the very weak gravitational interaction between Proxima and α Cen, Beech (2009, 2011, 2015) proposed that this system could be a test case for the modified Newtonian dynamics (MOND) theory (Milgrom 1983; Bekenstein 2004). Such a wide multiple system may have formed during the dissolution of their original star cluster (Kouwenhoven et al. 2010). In spite of its large semi-major axis, the statistical dissolution time of the Proxima-α Cen system is expected to be much longer than 10 Ga (Jiang & Tremaine 2010; Bahcall et al. 1985). The orbital motion of Proxima could have played a significant role in the formation and evolution of its planet. Barnes et al. (2016) proposed that a passage of Proxima close to α Cen may have destabilized the original orbit(s) of Proxima’s planet(s), resulting in the current position of Proxima b. Conversely, it may also have influenced circumbinary planet formation around α Cen (Worth & Sigurdsson 2016). Alternatively, Proxima b may also have formed as a distant circumbinary planet of α Cen, and was subsequently captured by Proxima. In these scenarios, it could be an ocean planet resulting from the meltdown of an icy body (Brugger et al. 2016). Proxima b may therefore not have been located in the habitable zone (Ribas et al. 2016) for as long as the age of the α Cen system (5 to 7 Ga; Miglio & Montalbán 2005; Eggenberger et al. 2004; Kervella et al. 2003; Thévenin et al. 2002).


Acknowledgments

We are grateful to Dr J. S. Jenkins for discussions that led to important improvements of this Letter. C.L. acknowledges the financial support of the Swiss National Science Foundation (SNSF). This research made use of Astropy2(Astropy Collaboration 2013), and of the SIMBAD (Wenger et al. 2000), VIZIER (CDS, Strasbourg, France) and NASA’s Astrophysics Data System databases.

References

Appendix A: Radial velocity of Proxima

The main difficulties in determining the absolute RV of red dwarfs from cross correlation with model templates are the extremely strong line blending and our incomplete knowledge of molecular line transitions. The high-resolution spectra produced by recent atmosphere models (e.g., PHOENIX, Husser et al. 2013) provide sufficient accuracy to determine the basic parameters of the stars, but they fail to reproduce many of the observed lines and the details of the line profiles. In addition, the wavelengths of the molecular lines are, in general, not known with sufficient accuracy to reach the m s-1 level. To overcome these limitations, we selected four strong very high signal-to-noise emission lines of Ca II and Na I, whose wavelengths are very accurately known (Wolf et al. 2008; Morton 2003, 2004). We subtracted the spectral background at the position of these emission lines from a PHOENIX3 model spectrum (Husser et al. 2013) that was scaled to the flux of the HARPS spectrum over a neighboring emission-free region. We checked that alternatemethods to estimate the background level (linear, constant) do not affect the derived RV at a ± 20 m s-1 level. Removing one of the lines from the sample also does not lead to biases beyond this level. We obtained the Doppler shift of each emission line separately from the measurement of the barycenter of their emission over a Δλ = ± 0.2 Å region (Fig. A.1).

Over the 271 HARPS spectra of Proxima present in the ESO Phase 3 archive, we kept 260 that provide a good consistency between the velocities estimated using the four emission lines (within 300 m s-1). The velocities of the four lines were averaged to obtain one measurement per HARPS epoch, whose time sequence is represented in Fig. A.2. The standard deviation of all epoch measurements is σ = 0.068 km s-1, and the histogram of the measurements is shown in Fig. A.3. We used a bootstrapping approach to estimate the statistical uncertainty of the resulting velocity (± 0.011 km s-1). We quadratically add a ± 0.025 km s-1 systematic uncertainty to account for the background and line selection dispersion. We thus obtain the barycentric velocity measure (Lindegren & Dravins 2003) of Proxima vr, obs [Proxima] = −21.700 ± 0.011 ± 0.025 km s-1.

thumbnail Fig. A.1

Average profiles of the selected emission lines of Proxima (blue curves), with the subtracted background level (green curves). The dashed red line marks the wavelength of the lines at a velocity of v = −21.700 km s-1.

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thumbnail Fig. A.2

Times series of the measured absolute radial velocities of Proxima over a period of 12 yr.

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thumbnail Fig. A.3

Histogram of the measured absolute radial velocities of Proxima.

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Appendix B: 3D positions and velocities

The heliocentric Galactic coordinates of the barycenter of α Cen and of Proxima are presented in Table B.1, together with their heliocentric space velocity. The differential position and velocity vectors between α Cen and Proxima are also listed. We followed the classical convention of X and U increasing toward the Galactic center, Y and V positive toward the Galactic direction of rotation, and Z and W positive toward the north Galactic pole.

Table B.1

Heliocentric coordinates (X,Y,Z) and space velocity vectors (U,V,W) of α Cen and Proxima in the Galactic frame.

Appendix C: Orbital parameter statistics

The histograms of the values of the orbital parameters of Proxima from our Monte Carlo simulations are presented in Fig. C.1.

thumbnail Fig. C.1

Histograms of the Monte Carlo simulations of the orbital parameters of Proxima. The solid blue line is the best fit parameter, and the dashed lines delimit the 68% confidence interval (1σ).

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All Tables

Table 1

Adopted physical parameters of α Cen AB and Proxima.

Table 2

Positional data, parallax, and radial velocity of α Centauri AB (barycenter) and Proxima.

Table 3

Orbital parameters of Proxima.

Table B.1

Heliocentric coordinates (X,Y,Z) and space velocity vectors (U,V,W) of α Cen and Proxima in the Galactic frame.

All Figures

thumbnail Fig. 1

Best-fit orbit of Proxima around α Cen projected in the plane of the sky (black curve), with a set of 100 possible orbits from a Monte Carlo approach (thin gray curves). The blue dots represent the position of Proxima every 40 000 yr (labels in millenia) and the direction of the present velocity vector of Proxima is shown as a red arrow.

Open with DEXTER
In the text
thumbnail Fig. 2

Orbit in 3D cartesian Galactic coordinates, with the Sun direction shown as a green vector originating from α Cen. The labels are the same as in Fig. 1.

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In the text
thumbnail Fig. 3

Velocity (solid line) and separation (dotted line) of Proxima relative to α Cen. The present values are marked with crosses.

Open with DEXTER
In the text
thumbnail Fig. A.1

Average profiles of the selected emission lines of Proxima (blue curves), with the subtracted background level (green curves). The dashed red line marks the wavelength of the lines at a velocity of v = −21.700 km s-1.

Open with DEXTER
In the text
thumbnail Fig. A.2

Times series of the measured absolute radial velocities of Proxima over a period of 12 yr.

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In the text
thumbnail Fig. A.3

Histogram of the measured absolute radial velocities of Proxima.

Open with DEXTER
In the text
thumbnail Fig. C.1

Histograms of the Monte Carlo simulations of the orbital parameters of Proxima. The solid blue line is the best fit parameter, and the dashed lines delimit the 68% confidence interval (1σ).

Open with DEXTER
In the text