Measuring the Mass of an Earth-size Planet Using a Gaussian Process Model of Precise Radial Velocities

Application of a new nonparametric model to combine two independent datasets, remove a stellar activity signal, and uncover the much weaker planet signal hiding underneath.

Transcript

1. Measuring the Mass of an Earth-size Planet Using a Gaussian Process Model of Precise Radial Velocities Sam Grunblatt 699 Advisor: Andrew Howard
2. Kepler-78b Discovery Sanchis-Ojeda et al. (2013) Porb = 8.55hr Rpl = 1.16 ± 0.19R
3. Porb Prot Howard et al. (2013) Stellar activity model: Keplerian planet model: vKep(t) = K sin[2⇡(t tc)/Porb] vspot(t, ) = 3 X i=1 ai sin[ i + i2⇡t/Prot] + K = 1.66 ± 0.40 m s 1
4. Floating Chunk Offset model: Nightly RV zero point established Pepe et al. (2013) K = 1.96 ± 0.32 m s 1
5. Howard et al. (2013) M HIRES = 1.69 ± 0.41M M HARPS N = 1.86+0.38 0.25 M
6. Howard et al. (2013) Consistent within errors! M HIRES = 1.69 ± 0.41M M HARPS N = 1.86+0.38 0.25 M
7. • Build Gaussian process model of stellar activity in RV measurements. • Measure a more precise and accurate mass, and calculate a robust composition of Kepler-78b. Project Goals
8. What is a Gaussian process regression? • Given a set of input data and a choice of covariance kernel… ⌃ij = k(ti, tj) = h2exp  ⇣ti tj ⌘2
9. What is a Gaussian process regression? • …find the best-fit hyperparameters by optimizing the appropriate likelihood function: log[L(r)] = 1 2 rT⌃ 1r 1 2 log|⌃| N 2 log(2⇡) ⌃ij = k(ti, tj) = h2exp  ⇣ti tj ⌘2
10. Covariance Kernel • Simplest kernel: squared exponential. Described by: • where (h, λ) are the hyperparameters: parameters of the kernel. Roberts et al. (2012) λ = 0.1 λ = 1 λ = 10 ⌃ij = k(ti, tj) = h2exp  ⇣ti tj ⌘2
11. Why use Gaussian process regression? • Non-parametric: f(x) not needed • previously used in exoplanet studies (Gibson et al. 2012, Petigura et al. 2013, concurrently with Haywood et al. 2014) • Depends only on choice of covariance kernel Gibson et al. (2012)
12. Gaussian Process Model Evaluate choice of mean function parameters and hyperparameters with likelihood function. (for squared exponential case) log[L(r)] = 1 2 rT⌃ 1r 1 2 log|⌃| N 2 log(2⇡) r = v Ksin ⇣2⇡(t tc) Porb ⌘ ⌃ij = k(ti, tj) = h2exp  ⇣ti tj ⌘2
13. Modeling Procedure
14. Modeling Procedure
15. Modeling Procedure Porb << Prot
16. Modeling Procedure
17. Modeling Procedure
18. Modeling Procedure
19. Gaussian Process Model Evaluate choice of mean function parameters and hyperparameters with likelihood function. (for squared exponential case) log[L(r)] = 1 2 rT⌃ 1r 1 2 log|⌃| N 2 log(2⇡) r = v Ksin ⇣2⇡(t tc) Porb ⌘ ⌃ij = k(ti, tj) = h2exp  ⇣ti tj ⌘2
20. Models Tested Squared Exponential Periodic Quasiperiodic
21. GP Model of Kepler Photometry
22. Analysis: HIRES & HARPS-N
23. Results Note: priors included on the parameters with *s K = 1.92 ± 0.25 m s 1 Adopted Quasiperiodic Model Parameters HYPERPARAMETER HIRES HARPS-N h: amplitude -8.2 +27/-11 m/s 4.4 +2.9/-10.5 m/s θ*: period 12.78 ± 0.03 days 12.78 ± 0.04 days w: roughness 0.32 +0.11/-0.07 0.30 +0.10/-0.07 λ*: lengthscale 1.8e23 +3.6e30/-1.8e23 days 23 +22/-17 days σ: stellar jitter 2.1 ± 0.3 m/s -0.02 ± 1.3 m/s
24. Results 23+23 17 days 8.2+27 11 m s 1 0.32+0.11 0.07 1.8e23+3.6e30 1.8e23 days 0.30+0.10 0.07 h = ✓ = w = = jitter = jitter2 = 2 = w2 = ✓2 = h2 = K =
25. Discussion • , an improvement of 2.5-σ over Howard et al. (2013) • • Iron fraction 0.32 ± 0.26, suggesting formation possibly similar to Earth’s • Results broadly consistent with Hatzes (2014), who analyzed both datasets with variant on Floating Chunk Offset model ⇢pl = 6.1+1.9 1.4 g cm 3 Black cross: previous estimate Red point: This work Gray points: Other small known exoplanets Green points: Earth/Venus/Mars Mpl = 1.83 ± 0.27M M HIRES = 1.69 ± 0.41M M HARPS N = 1.86+0.38 0.25 M Mpl = 1.83 ± 0.27M
26. Summary • Performed combined analysis of RV data of Kepler-78 to better extract planetary Doppler amplitude • Found separate quasiperiodic kernel with stellar jitter terms describe datasets best • Converted Doppler amplitude into 2.5-σ improvement in mass calculation over Howard et al. (2013) • Iron mass fraction of 0.32 ± 0.26 suggests Kepler-78b is most likely Earthlike in composition. • True benefit of this technique: nonparametric model of stellar activity.
27. Conclusions • Iron fraction suggests formation of Kepler-78b might have been similar to Earth’s. • • Quasiperiodic kernel nonparametrically describes stellar RV activity precisely. Mpl = 1.83 ± 0.27M
28. Next steps: • Test on alternate datasets: longer time baselines • Refine choice of priors • What is the smallest RV signal for which we can characterize stellar noise?
29. Context in planet radius-density relation Weiss et al. (2014) ⇢pl = 6.1+1.9 1.4 g cm 3 Rpl = 1.20 ± 0.09R