Context. Low-and intermediate-mass stars go through a period of intense mass-loss at the end of t... more Context. Low-and intermediate-mass stars go through a period of intense mass-loss at the end of their lives during the asymptotic giant branch (AGB) phase. While on the AGB a significant part, or even most, of their initial mass is expelled in a stellar wind. This process controls the final stages of the evolution of these stars and contributes to the chemical evolution of galaxies. However, the wind-driving mechanism of AGB stars is not yet well understood, especially so for oxygen-rich sources. Characterizing both the present-day mass-loss rate and wind structure and the evolution of the mass-loss rate of such stars is paramount to advancing our understanding of this processes. Aims. We study the dusty wind of the oxygen-rich AGB star W Hya to understand its composition and structure and shed light on the mass-loss mechanism. Methods. We modelled the dust envelope of W Hya using an advanced radiative transfer code. We analysed our dust model in the light of a previously calculated gas-phase wind model and compared it with measurements available in the literature, such as infrared spectra, infrared images, and optical scattered light fractions. Results. We find that the dust spectrum of W Hya can partly be explained by a gravitationally bound dust shell that probably is responsible for most of the amorphous Al 2 O 3 emission. The composition of the large (∼ 0.3 µm) grains needed to explain the scattered light cannot be constrained, but probably is dominated by silicates. Silicate emission in the thermal infrared was found to originate from beyond 40 AU from the star. In our model, the silicates need to have substantial near-infrared opacities to be visible at such large distances. The increase in near-infrared opacity of the dust at these distances roughly coincides with a sudden increase in expansion velocity as deduced from the gas-phase CO lines. The dust envelope of W Hya probably contains an important amount of calcium but we were not able to obtain a dust model that reproduces the observed emission while respecting the limit set by the gas mass-loss rate. Finally, the recent mass loss of W Hya is confirmed to be highly variable and we identify a strong peak in the mass-loss rate that occurred about 3500 years ago and lasted for a few hundred years.
During their asymptotic giant branch evolution, low-mass stars lose a significant fraction of the... more During their asymptotic giant branch evolution, low-mass stars lose a significant fraction of their mass through an intense wind, enriching the interstellar medium with products of nucleosynthesis. We observed the nearby oxygen-rich asymptotic giant branch star IK Tau using the highresolution HIFI spectrometer onboard Herschel. We report on the first detection of H 16 2 O and the rarer isotopologues H 17 2 O and H 18 2 O in both the ortho and para states. We deduce a total water content (relative to molecular hydrogen) of 6.6 × 10 −5 , and an ortho-to-para ratio of 3:1. These results are consistent with the formation of H 2 O in thermodynamical chemical equilibrium at photospheric temperatures, and does not require pulsationally induced non-equilibrium chemistry, vaporization of icy bodies or grain surface reactions. High-excitation lines of 12 CO, 13 CO, 28 SiO, 29 SiO, 30 SiO, HCN, and SO have also been detected. From the observed line widths, the acceleration region in the inner wind zone can be characterized, and we show that the wind acceleration is slower than hitherto anticipated.
The broad 30 µm feature in carbon stars is commonly attributed to MgS dust particles. However, re... more The broad 30 µm feature in carbon stars is commonly attributed to MgS dust particles. However, reproducing the 30 µm feature with homogeneous MgS grains would require much more sulfur relative to the solar abundance. Direct gas-phase condensation of MgS occurs at a low efficiency. Precipitation of MgS on SiC precursor grains provides a more efficient formation mechanism, such that the assumption of homogeneous MgS grains may not be correct. Using a Monte Carlo-based radiative transfer code, we aim to model the 30 µm feature of the extreme carbon star LL Peg with MgS dust particles. We find that for LL Peg this modeling is insensitive to the unknown MgS optical properties at λ < 10 µm. When MgS is allowed to be in thermal contact with amorphous carbon and SiC, the amount of MgS required to reproduce the strength of 30 µm feature agrees with the solar abundance of sulfur, thereby resolving the reported MgS mass problem. We conclude that MgS is a valid candidate to be the carrier of the 30 µm feature when it is part of a composite grain population that has optical properties representative of an ensemble of particle shapes.
With a luminosity > 10 5 L ⊙ and a mass-loss rate of ∼ 2.10 −4 M ⊙ yr −1 , the red supergiant VY ... more With a luminosity > 10 5 L ⊙ and a mass-loss rate of ∼ 2.10 −4 M ⊙ yr −1 , the red supergiant VY CMa truly is a spectacular object. Because of its extreme evolutionary state, it could explode as supernova any time. Studying its circumstellar material, into which the supernova blast will run, provides interesting constraints on supernova explosions and on the rich chemistry taking place in such complex circumstellar envelopes. We have obtained spectroscopy of VY CMa over the full wavelength range offered by the PACS and SPIRE instruments of Herschel, i.e. 55 -672 micron. The observations show the spectral fingerprints of more than 900 spectral lines, of which more than half belong to water. In total, we have identified 13 different molecules and some of their isotopologues. A first analysis shows that water is abundantly present, with an ortho-to-para ratio as low as ∼1.3:1, and that chemical non-equilibrium processes determine the abundance fractions in the inner envelope.
Context. The carbon-rich asymptotic giant branch star IRC +10216 undergoes strong mass loss, and ... more Context. The carbon-rich asymptotic giant branch star IRC +10216 undergoes strong mass loss, and quasi-periodic enhancements of the density of the circumstellar matter have previously been reported. The star's circumstellar environment is a well-studied, and complex astrochemical laboratory, with many molecular species proved to be present. CO is ubiquitous in the circumstellar envelope, while emission from the ethynyl (C 2 H) radical is detected in a spatially confined shell around IRC +10216. As reported in this article, we recently detected unexpectedly strong emission from the N = 4 − 3, 6 − 5, 7 − 6, 8 − 7, and 9 − 8 transitions of C 2 H with the IRAM 30 m telescope and with Herschel/HIFI, challenging the available chemical and physical models. Aims. We aim to constrain the physical properties of the circumstellar envelope of IRC +10216, including the effect of episodic mass loss on the observed emission lines. In particular, we aim to determine the excitation region and conditions of C 2 H, in order to explain the recent detections, and to reconcile these with interferometric maps of the N = 1 − 0 transition of C 2 H. Methods. Using radiative-transfer modelling, we provide a physical description of the circumstellar envelope of IRC +10216, constrained by the spectral-energy distribution and a sample of 20 high-resolution and 29 low-resolution CO lines -to date, the largest modelled range of CO lines towards an evolved star. We further present the most detailed radiative-transfer analysis of C 2 H that has been done so far. Results. Assuming a distance of 150 pc to IRC +10216, the spectral-energy distribution is modelled with a stellar luminosity of 11300 L ⊙ and a dust-mass-loss rate of 4.0 × 10 −8 M ⊙ yr −1 . Based on the analysis of the 20 high-frequency-resolution CO observations, an average gas-mass-loss rate for the last 1000 years of 1.5 × 10 −5 M ⊙ yr −1 is derived. This results in a gas-to-dust-mass ratio of 375, typical for this type of star. The kinetic temperature throughout the circumstellar envelope is characterised by three powerlaws: T kin (r) ∝ r −0.58 for radii r ≤ 9 stellar radii, T kin (r) ∝ r −0.40 for radii 9 ≤ r ≤ 65 stellar radii, and T kin (r) ∝ r −1.20 for radii r ≥ 65 stellar radii. This model successfully describes all 49 observed CO lines. We also show the effect of density enhancements in the wind of IRC +10216 on the C 2 H-abundance profile, and the close agreement we find of the model predictions with interferometric maps of the C 2 H N = 1 − 0 transition and with the rotational lines observed with the IRAM 30 m telescope and Herschel/HIFI. We report on the importance of radiative pumping to the vibrationally excited levels of C 2 H and the significant effect this pumping mechanism has on the excitation of all levels of the C 2 H-molecule.
Aims. Spectra of OH/IR stars show prominent spectral bands of crystalline olivine (Mg (2−2x) Fe (... more Aims. Spectra of OH/IR stars show prominent spectral bands of crystalline olivine (Mg (2−2x) Fe (2x) SiO 4 ). To learn more about the timescale of the outflows of OH/IR stars, we study the spectral band of crystalline olivine at 69 µm. Methods. The 69 µm band is of interest because its width and peak wavelength position are sensitive to the grain temperature and to the exact composition of the crystalline olivine. With Herschel/PACS, we observed the 69 µm band in the outflow of 14 OH/IR stars. By comparing the crystalline olivine features of our sample with those of model spectra, we determined the size of the outflow and its crystalline olivine abundance. Results. The temperature indicated by the observed 69 µm bands can only be reproduced by models with a geometrically compact superwind (R SW 2500 AU = 1400 R * ). This means that the superwind started less than 1200 years ago (assuming an outflow velocity of 10 km/s). The small amount of mass lost in one superwind and the high progenitor mass of the OH/IR stars introduce a mass loss and thus evolutionary problem for these objects, which has not yet been understood.
ABSTRACT Context. Infrared spectroscopy has been extensively used to determine the mineralogy of ... more ABSTRACT Context. Infrared spectroscopy has been extensively used to determine the mineralogy of circumstellar dust. The identification of dust species with featureless opacities, however, is still ambiguous. Here we present a method to lift the degeneracy using the combination of infrared spectroscopy and interferometry. Aims: The binary post-AGB star HR 4049 is surrounded by a circumbinary disk viewed at a high inclination angle. Apart from gaseous emission lines and molecular emission bands of polycyclic aromatic hydrocarbons (PAH), diamonds, and fullerenes, the 2-25 μm infrared spectrum is featureless. The goal of the paper is to identify the dust species responsible for the smooth spectrum. Methods: We gathered high-angular-resolution measurements in the near- and mid-infrared with the VLTI interferometric instruments AMBER and MIDI. The data set is expanded with archival Geneva optical photometry, ISO-SWS and Spitzer-IRS infrared spectroscopy, and VISIR N-band images and spectroscopy. We computed a grid of radiative-transfer models of the circumbinary disk of HR 4049 using the radiative-transfer code MCMax. We searched for models that provide good fits simultaneously to all available observations. Results: We find that the variable optical extinction towards the primary star is consistent with the presence of very small (0.01 μm) iron-bearing dust grains or amorphous carbon grains. The combination of the interferometric constraint on the disk extent and the shape of the infrared spectrum points to amorphous carbon as the dominant source of opacity in the circumbinary disk of HR 4049. The disk is optically thick to the stellar radiation in the radial direction. At infrared wavelengths it is optically thin. The PAH emission is spatially resolved in the VISIR data and emanates from a region with an extent of several hundreds of AU, with a projected photocenter displacement of several tens of AU from the disk center. The PAHs most likely reside in a bipolar outflow. Conclusions: Dust species with featureless opacity curves, such as metallic iron and amorphous carbon, can be identified by combining infrared spectroscopy and high-angular-resolution measurements. In essence, this is because the temperatures of the dust species are notably different at the same physical distance to the star. Appendices A and B are available in electronic form at http://www.aanda.org
Monthly Notices of the Royal Astronomical Society, 2012
We present a light curve analysis and radial velocity study of KOI-74, an eclipsing A star + whit... more We present a light curve analysis and radial velocity study of KOI-74, an eclipsing A star + white dwarf binary with a 5.2 day orbit. Aside from new spectroscopy covering the orbit of the system, we used 212 days of publicly available Kepler observations and present the first complete light curve fitting to these data, modelling the eclipses and transits, ellipsoidal modulation, reflection, and Doppler beaming. Markov Chain Monte Carlo simulations are used to determine the system parameters and uncertainty estimates. Our results are in agreement with earlier studies, except that we find an inclination of 87.0 ± 0.4 • , which is significantly lower than the previously published value. The altered inclination leads to different values for the relative radii of the two stars and therefore also the mass ratio deduced from the ellipsoidal modulations seen in this system. We find that the mass ratio derived from the radial velocity amplitude (q = 0.104 ± 0.004) disagrees with that derived from the ellipsoidal modulation (q = 0.052 ± 0.004 assuming corotation). This was found before, but with our smaller inclination, the discrepancy is even larger than previously reported. Accounting for the rapid rotation of the A-star, instead of assuming corotation with the binary orbit, is found to increase the discrepancy even further by lowering the mass ratio to q = 0.047 ± 0.004. These results indicate that one has to be extremely careful in using the amplitude of an ellipsoidal modulation signal in a close binary to determine the mass ratio, when a proof of corotation is not firmly established. The same problem could arise whenever an ellipsoidal modulation amplitude is used to derive the mass of a planet orbiting a host star that is not in corotation with the planet's orbit.
Context. Low-and intermediate-mass stars go through a period of intense mass-loss at the end of t... more Context. Low-and intermediate-mass stars go through a period of intense mass-loss at the end of their lives during the asymptotic giant branch (AGB) phase. While on the AGB a significant part, or even most, of their initial mass is expelled in a stellar wind. This process controls the final stages of the evolution of these stars and contributes to the chemical evolution of galaxies. However, the wind-driving mechanism of AGB stars is not yet well understood, especially so for oxygen-rich sources. Characterizing both the present-day mass-loss rate and wind structure and the evolution of the mass-loss rate of such stars is paramount to advancing our understanding of this processes. Aims. We study the dusty wind of the oxygen-rich AGB star W Hya to understand its composition and structure and shed light on the mass-loss mechanism. Methods. We modelled the dust envelope of W Hya using an advanced radiative transfer code. We analysed our dust model in the light of a previously calculated gas-phase wind model and compared it with measurements available in the literature, such as infrared spectra, infrared images, and optical scattered light fractions. Results. We find that the dust spectrum of W Hya can partly be explained by a gravitationally bound dust shell that probably is responsible for most of the amorphous Al 2 O 3 emission. The composition of the large (∼ 0.3 µm) grains needed to explain the scattered light cannot be constrained, but probably is dominated by silicates. Silicate emission in the thermal infrared was found to originate from beyond 40 AU from the star. In our model, the silicates need to have substantial near-infrared opacities to be visible at such large distances. The increase in near-infrared opacity of the dust at these distances roughly coincides with a sudden increase in expansion velocity as deduced from the gas-phase CO lines. The dust envelope of W Hya probably contains an important amount of calcium but we were not able to obtain a dust model that reproduces the observed emission while respecting the limit set by the gas mass-loss rate. Finally, the recent mass loss of W Hya is confirmed to be highly variable and we identify a strong peak in the mass-loss rate that occurred about 3500 years ago and lasted for a few hundred years.
During their asymptotic giant branch evolution, low-mass stars lose a significant fraction of the... more During their asymptotic giant branch evolution, low-mass stars lose a significant fraction of their mass through an intense wind, enriching the interstellar medium with products of nucleosynthesis. We observed the nearby oxygen-rich asymptotic giant branch star IK Tau using the highresolution HIFI spectrometer onboard Herschel. We report on the first detection of H 16 2 O and the rarer isotopologues H 17 2 O and H 18 2 O in both the ortho and para states. We deduce a total water content (relative to molecular hydrogen) of 6.6 × 10 −5 , and an ortho-to-para ratio of 3:1. These results are consistent with the formation of H 2 O in thermodynamical chemical equilibrium at photospheric temperatures, and does not require pulsationally induced non-equilibrium chemistry, vaporization of icy bodies or grain surface reactions. High-excitation lines of 12 CO, 13 CO, 28 SiO, 29 SiO, 30 SiO, HCN, and SO have also been detected. From the observed line widths, the acceleration region in the inner wind zone can be characterized, and we show that the wind acceleration is slower than hitherto anticipated.
The broad 30 µm feature in carbon stars is commonly attributed to MgS dust particles. However, re... more The broad 30 µm feature in carbon stars is commonly attributed to MgS dust particles. However, reproducing the 30 µm feature with homogeneous MgS grains would require much more sulfur relative to the solar abundance. Direct gas-phase condensation of MgS occurs at a low efficiency. Precipitation of MgS on SiC precursor grains provides a more efficient formation mechanism, such that the assumption of homogeneous MgS grains may not be correct. Using a Monte Carlo-based radiative transfer code, we aim to model the 30 µm feature of the extreme carbon star LL Peg with MgS dust particles. We find that for LL Peg this modeling is insensitive to the unknown MgS optical properties at λ < 10 µm. When MgS is allowed to be in thermal contact with amorphous carbon and SiC, the amount of MgS required to reproduce the strength of 30 µm feature agrees with the solar abundance of sulfur, thereby resolving the reported MgS mass problem. We conclude that MgS is a valid candidate to be the carrier of the 30 µm feature when it is part of a composite grain population that has optical properties representative of an ensemble of particle shapes.
With a luminosity > 10 5 L ⊙ and a mass-loss rate of ∼ 2.10 −4 M ⊙ yr −1 , the red supergiant VY ... more With a luminosity > 10 5 L ⊙ and a mass-loss rate of ∼ 2.10 −4 M ⊙ yr −1 , the red supergiant VY CMa truly is a spectacular object. Because of its extreme evolutionary state, it could explode as supernova any time. Studying its circumstellar material, into which the supernova blast will run, provides interesting constraints on supernova explosions and on the rich chemistry taking place in such complex circumstellar envelopes. We have obtained spectroscopy of VY CMa over the full wavelength range offered by the PACS and SPIRE instruments of Herschel, i.e. 55 -672 micron. The observations show the spectral fingerprints of more than 900 spectral lines, of which more than half belong to water. In total, we have identified 13 different molecules and some of their isotopologues. A first analysis shows that water is abundantly present, with an ortho-to-para ratio as low as ∼1.3:1, and that chemical non-equilibrium processes determine the abundance fractions in the inner envelope.
Context. The carbon-rich asymptotic giant branch star IRC +10216 undergoes strong mass loss, and ... more Context. The carbon-rich asymptotic giant branch star IRC +10216 undergoes strong mass loss, and quasi-periodic enhancements of the density of the circumstellar matter have previously been reported. The star's circumstellar environment is a well-studied, and complex astrochemical laboratory, with many molecular species proved to be present. CO is ubiquitous in the circumstellar envelope, while emission from the ethynyl (C 2 H) radical is detected in a spatially confined shell around IRC +10216. As reported in this article, we recently detected unexpectedly strong emission from the N = 4 − 3, 6 − 5, 7 − 6, 8 − 7, and 9 − 8 transitions of C 2 H with the IRAM 30 m telescope and with Herschel/HIFI, challenging the available chemical and physical models. Aims. We aim to constrain the physical properties of the circumstellar envelope of IRC +10216, including the effect of episodic mass loss on the observed emission lines. In particular, we aim to determine the excitation region and conditions of C 2 H, in order to explain the recent detections, and to reconcile these with interferometric maps of the N = 1 − 0 transition of C 2 H. Methods. Using radiative-transfer modelling, we provide a physical description of the circumstellar envelope of IRC +10216, constrained by the spectral-energy distribution and a sample of 20 high-resolution and 29 low-resolution CO lines -to date, the largest modelled range of CO lines towards an evolved star. We further present the most detailed radiative-transfer analysis of C 2 H that has been done so far. Results. Assuming a distance of 150 pc to IRC +10216, the spectral-energy distribution is modelled with a stellar luminosity of 11300 L ⊙ and a dust-mass-loss rate of 4.0 × 10 −8 M ⊙ yr −1 . Based on the analysis of the 20 high-frequency-resolution CO observations, an average gas-mass-loss rate for the last 1000 years of 1.5 × 10 −5 M ⊙ yr −1 is derived. This results in a gas-to-dust-mass ratio of 375, typical for this type of star. The kinetic temperature throughout the circumstellar envelope is characterised by three powerlaws: T kin (r) ∝ r −0.58 for radii r ≤ 9 stellar radii, T kin (r) ∝ r −0.40 for radii 9 ≤ r ≤ 65 stellar radii, and T kin (r) ∝ r −1.20 for radii r ≥ 65 stellar radii. This model successfully describes all 49 observed CO lines. We also show the effect of density enhancements in the wind of IRC +10216 on the C 2 H-abundance profile, and the close agreement we find of the model predictions with interferometric maps of the C 2 H N = 1 − 0 transition and with the rotational lines observed with the IRAM 30 m telescope and Herschel/HIFI. We report on the importance of radiative pumping to the vibrationally excited levels of C 2 H and the significant effect this pumping mechanism has on the excitation of all levels of the C 2 H-molecule.
Aims. Spectra of OH/IR stars show prominent spectral bands of crystalline olivine (Mg (2−2x) Fe (... more Aims. Spectra of OH/IR stars show prominent spectral bands of crystalline olivine (Mg (2−2x) Fe (2x) SiO 4 ). To learn more about the timescale of the outflows of OH/IR stars, we study the spectral band of crystalline olivine at 69 µm. Methods. The 69 µm band is of interest because its width and peak wavelength position are sensitive to the grain temperature and to the exact composition of the crystalline olivine. With Herschel/PACS, we observed the 69 µm band in the outflow of 14 OH/IR stars. By comparing the crystalline olivine features of our sample with those of model spectra, we determined the size of the outflow and its crystalline olivine abundance. Results. The temperature indicated by the observed 69 µm bands can only be reproduced by models with a geometrically compact superwind (R SW 2500 AU = 1400 R * ). This means that the superwind started less than 1200 years ago (assuming an outflow velocity of 10 km/s). The small amount of mass lost in one superwind and the high progenitor mass of the OH/IR stars introduce a mass loss and thus evolutionary problem for these objects, which has not yet been understood.
ABSTRACT Context. Infrared spectroscopy has been extensively used to determine the mineralogy of ... more ABSTRACT Context. Infrared spectroscopy has been extensively used to determine the mineralogy of circumstellar dust. The identification of dust species with featureless opacities, however, is still ambiguous. Here we present a method to lift the degeneracy using the combination of infrared spectroscopy and interferometry. Aims: The binary post-AGB star HR 4049 is surrounded by a circumbinary disk viewed at a high inclination angle. Apart from gaseous emission lines and molecular emission bands of polycyclic aromatic hydrocarbons (PAH), diamonds, and fullerenes, the 2-25 μm infrared spectrum is featureless. The goal of the paper is to identify the dust species responsible for the smooth spectrum. Methods: We gathered high-angular-resolution measurements in the near- and mid-infrared with the VLTI interferometric instruments AMBER and MIDI. The data set is expanded with archival Geneva optical photometry, ISO-SWS and Spitzer-IRS infrared spectroscopy, and VISIR N-band images and spectroscopy. We computed a grid of radiative-transfer models of the circumbinary disk of HR 4049 using the radiative-transfer code MCMax. We searched for models that provide good fits simultaneously to all available observations. Results: We find that the variable optical extinction towards the primary star is consistent with the presence of very small (0.01 μm) iron-bearing dust grains or amorphous carbon grains. The combination of the interferometric constraint on the disk extent and the shape of the infrared spectrum points to amorphous carbon as the dominant source of opacity in the circumbinary disk of HR 4049. The disk is optically thick to the stellar radiation in the radial direction. At infrared wavelengths it is optically thin. The PAH emission is spatially resolved in the VISIR data and emanates from a region with an extent of several hundreds of AU, with a projected photocenter displacement of several tens of AU from the disk center. The PAHs most likely reside in a bipolar outflow. Conclusions: Dust species with featureless opacity curves, such as metallic iron and amorphous carbon, can be identified by combining infrared spectroscopy and high-angular-resolution measurements. In essence, this is because the temperatures of the dust species are notably different at the same physical distance to the star. Appendices A and B are available in electronic form at http://www.aanda.org
Monthly Notices of the Royal Astronomical Society, 2012
We present a light curve analysis and radial velocity study of KOI-74, an eclipsing A star + whit... more We present a light curve analysis and radial velocity study of KOI-74, an eclipsing A star + white dwarf binary with a 5.2 day orbit. Aside from new spectroscopy covering the orbit of the system, we used 212 days of publicly available Kepler observations and present the first complete light curve fitting to these data, modelling the eclipses and transits, ellipsoidal modulation, reflection, and Doppler beaming. Markov Chain Monte Carlo simulations are used to determine the system parameters and uncertainty estimates. Our results are in agreement with earlier studies, except that we find an inclination of 87.0 ± 0.4 • , which is significantly lower than the previously published value. The altered inclination leads to different values for the relative radii of the two stars and therefore also the mass ratio deduced from the ellipsoidal modulations seen in this system. We find that the mass ratio derived from the radial velocity amplitude (q = 0.104 ± 0.004) disagrees with that derived from the ellipsoidal modulation (q = 0.052 ± 0.004 assuming corotation). This was found before, but with our smaller inclination, the discrepancy is even larger than previously reported. Accounting for the rapid rotation of the A-star, instead of assuming corotation with the binary orbit, is found to increase the discrepancy even further by lowering the mass ratio to q = 0.047 ± 0.004. These results indicate that one has to be extremely careful in using the amplitude of an ellipsoidal modulation signal in a close binary to determine the mass ratio, when a proof of corotation is not firmly established. The same problem could arise whenever an ellipsoidal modulation amplitude is used to derive the mass of a planet orbiting a host star that is not in corotation with the planet's orbit.
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Papers by R. Lombaert