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Combining astrometric and radial velocity data for exoplanet detection

Holmelin, Viktor LU (2013) In Lund Observatory Examensarbeten ASTM31 20131
Department of Astronomy and Theoretical Physics
Lund Observatory
Abstract
Context: The Gaia satellite will be provide astrometry with micro-arcsecond accuracy. This will allow for the determination of the orbital elements of exoplanets with measurable astrometric signature. Astrometry alone is not be able to disentangle an ambiguity in the orientation of planetary orbits, just as radial velocity is only able to give the mass of orbiting planets combined with multiplicative factor. Combining astrometry and radial velocity this can be resolved. Many different approaches exist of how to parameterize and solve the problem of finding the orbital parameters. This text takes a look at one parametrization and one method for solving the problem.
Aim: Define a model for the parametrization of the problem of finding the... (More)
Context: The Gaia satellite will be provide astrometry with micro-arcsecond accuracy. This will allow for the determination of the orbital elements of exoplanets with measurable astrometric signature. Astrometry alone is not be able to disentangle an ambiguity in the orientation of planetary orbits, just as radial velocity is only able to give the mass of orbiting planets combined with multiplicative factor. Combining astrometry and radial velocity this can be resolved. Many different approaches exist of how to parameterize and solve the problem of finding the orbital parameters. This text takes a look at one parametrization and one method for solving the problem.
Aim: Define a model for the parametrization of the problem of finding the orbital elements for an exoplanet. This model should both describe the astrometric and radial velocity. Implement the model in AGISLab and determine how well it is able to retrieve the orbital elements.
Method: The problem was parameterized using Thiele-Innes parameters for describing the orientation and semi-major axis of the system. Further parameters were; mass ratio of the planet to the star, time of periapsis passage, eccentricity of the orbit and period. In addition, the astrometric parameters were also included; position, parallax and proper motion. Radial motion was not included. The Levenberg-Marquardt algorithm was used for the optimization. Boundaries on the planetary parameters were introduced, in order to prevent unrealistic solutions, through transformations. The transformation of the parameters was a standard trigonometric function. To estimate the formal errors on the optimized parameters, parametric bootstrapping was performed.
Result: The optimization works and provides sensible parameters, though the solution is very sensitive to the initial selection of parameters. Singular value decomposition of the matrix J|WJ, the Hessian of the merit function, indicates that the problem is ill-conditioned so bootstrapping may be a better solution for computing formal errors on the parameters than calculating them from the square root of the diagonal elements of its inverse. Bootstrapping also indicates that the formal errors of the parameters are not normally distributed.
Conclusion: The method does work and allows for very easy combination of radial velocity and astrometric data. Due to the choice of parametrization and/or optimization procedure the method is unstable. (Less)
Abstract (Swedish)
I höst kommer rymdobservatoriet Gaia att sändas upp. Ett av dess mål är att mätapositionen av stjärnor med väldigt hög noggrannhet, denna positionsbestämning kallas astrometri. Gaias noggrannhet är såpass bra att det kommer bli möjligt att även mäta hur planeter när de färdas i sina banor runt stjärnor påverkar positionen av stjärnan.
Astrometri kan inte ge en fullständig bild av hur planeten rör sig runt stjärnan utan kan endast se en projektion,
En annan metod för planetdetektion är radiell hastighet. Detta går ut på att man mäter hur snabbt en stjärna rör sig bort eller emot oss på grund av en planet i om- loppsbana. Denna metod kan inte heller ge en fullständig bild av hur planeten rör sig.
Genom att kombinera astrometri och... (More)
I höst kommer rymdobservatoriet Gaia att sändas upp. Ett av dess mål är att mätapositionen av stjärnor med väldigt hög noggrannhet, denna positionsbestämning kallas astrometri. Gaias noggrannhet är såpass bra att det kommer bli möjligt att även mäta hur planeter när de färdas i sina banor runt stjärnor påverkar positionen av stjärnan.
Astrometri kan inte ge en fullständig bild av hur planeten rör sig runt stjärnan utan kan endast se en projektion,
En annan metod för planetdetektion är radiell hastighet. Detta går ut på att man mäter hur snabbt en stjärna rör sig bort eller emot oss på grund av en planet i om- loppsbana. Denna metod kan inte heller ge en fullständig bild av hur planeten rör sig.
Genom att kombinera astrometri och radiell hastighet är det möjligt att ge en fullständig beskrivning av deras bana.
Här presenteras en metod att kombinera radiell hastighet och astrometri genom att utnyttja att några av parametrarna är desamma för de två tillvägagångssätten. Dessvärre visar det sig att denna metod lider av några brister vilket gör att även om en beskrivning av planetbanan hittas är den väldigt osäker. (Less)
Please use this url to cite or link to this publication:
author
Holmelin, Viktor LU
supervisor
organization
course
ASTM31 20131
year
type
H2 - Master's Degree (Two Years)
subject
publication/series
Lund Observatory Examensarbeten
report number
2013-EXA75
language
English
id
4001765
date added to LUP
2013-08-26 16:24:29
date last changed
2013-08-26 16:24:29
@misc{4001765,
  abstract     = {Context: The Gaia satellite will be provide astrometry with micro-arcsecond accuracy. This will allow for the determination of the orbital elements of exoplanets with measurable astrometric signature. Astrometry alone is not be able to disentangle an ambiguity in the orientation of planetary orbits, just as radial velocity is only able to give the mass of orbiting planets combined with multiplicative factor. Combining astrometry and radial velocity this can be resolved. Many different approaches exist of how to parameterize and solve the problem of finding the orbital parameters. This text takes a look at one parametrization and one method for solving the problem. 
Aim: Define a model for the parametrization of the problem of finding the orbital elements for an exoplanet. This model should both describe the astrometric and radial velocity. Implement the model in AGISLab and determine how well it is able to retrieve the orbital elements.
Method: The problem was parameterized using Thiele-Innes parameters for describing the orientation and semi-major axis of the system. Further parameters were; mass ratio of the planet to the star, time of periapsis passage, eccentricity of the orbit and period. In addition, the astrometric parameters were also included; position, parallax and proper motion. Radial motion was not included. The Levenberg-Marquardt algorithm was used for the optimization. Boundaries on the planetary parameters were introduced, in order to prevent unrealistic solutions, through transformations. The transformation of the parameters was a standard trigonometric function. To estimate the formal errors on the optimized parameters, parametric bootstrapping was performed.
Result: The optimization works and provides sensible parameters, though the solution is very sensitive to the initial selection of parameters. Singular value decomposition of the matrix J|WJ, the Hessian of the merit function, indicates that the problem is ill-conditioned so bootstrapping may be a better solution for computing formal errors on the parameters than calculating them from the square root of the diagonal elements of its inverse. Bootstrapping also indicates that the formal errors of the parameters are not normally distributed.
Conclusion: The method does work and allows for very easy combination of radial velocity and astrometric data. Due to the choice of parametrization and/or optimization procedure the method is unstable.},
  author       = {Holmelin, Viktor},
  language     = {eng},
  note         = {Student Paper},
  series       = {Lund Observatory Examensarbeten},
  title        = {Combining astrometric and radial velocity data for exoplanet detection},
  year         = {2013},
}