Abstract
Research in extrasolar-planet science is data-driven. With the advent of radial-velocity instruments like HARPS and HARPS-N, and transit space missions like Kepler, our ability to discover and characterise extrasolar planets is no longer limited by instrumental precision but by our ability to model the data accurately. This chapter presents the models that describe radial-velocity measurements and transit light curves. I begin by deriving the solution of the two-body problem and from there, the equations describing the radial velocity of a planet-host star and the distance between star and planet centres, necessary to model transit light curves. Stochastic models are then presented and I delineate how they are used to model complex physical phenomena affecting the exoplanet data sets, such as stellar activity. Finally, I give a brief overview of the processes of Bayesian inference, focussing on the construction of likelihood functions and prior probability distributions. In particular, I describe different methods to specify ignorance priors.
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Notes
- 1.
As we are describing the relative motions of the bodies, the actual angular momentum of the system contains a term related to the motion of the star. Whenever m 2 ≪ m 1, this term can be neglected and h is equal to the total angular momentum of the system per unit mass of the body m 2.
- 2.
Of course, exactly as we have redefined the polar angle so that ν is 0 at periapsis, we could also measure time starting at the moment of periastron passage, and get rid of τ in the definition of the mean anomaly. However, usually τ is unknown and including it as a model parameter allows us to measure it.
- 3.
A Python code to compute the true anomaly from the mean anomaly and eccentricity has been made available at https://github.com/exord/faial/blob/master/trueanomaly.py.
- 4.
Special care must be taken when the timescale of the variability is comparable to the integration time of individual points (Kipping 2010).
- 5.
The derivation is not as trivial as one may think from considering Eq. (1).
- 6.
Real data D are not available at the moment of specifying priors for the model parameters.
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Acknowledgements
The author thanks the organisers of the IVth Azores International Advanced School in Space Sciences and acknowledges the participants—both lecturers and students—for the quality of their work. The preparation of this lecture was carried out within the frame of the Swiss National Centre for Competence in Research “PlanetS” funded by the Swiss National Science Foundation (SNSF). The author acknowledges support by the Argentinian National Council for Research and Technology (CONICET).
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Díaz, R.F. (2018). Modelling Light and Velocity Curves of Exoplanet Hosts. In: Campante, T., Santos, N., Monteiro, M. (eds) Asteroseismology and Exoplanets: Listening to the Stars and Searching for New Worlds. Astrophysics and Space Science Proceedings, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-319-59315-9_11
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