Probabilistic modeling and Inference in Astronomy Dan Foreman-Mackey Sagan Fellow, University of Washington github.com/dfm // @exoplaneteer // dfm.io 

Dan Foreman-Mackey Sagan Fellow, University of Washington github.com/dfm // @exoplaneteer // dfm.io 

I study astronomy. Photo credit NASA Ames/SETI Institute/JPL-Caltech 

I study astronomy. Photo credit NASA Ames/SETI Institute/JPL-Caltech this isn't what my data look like 

Why Astronomy? simple but interesting physical models precise open-access data observational only 

Why Astronomy? simple but interesting physical models precise open-access data observational only no chance of financial gain ever 

ex·o·plan·et ˈeksōˌplanət/ noun. a planet that orbits a star outside the solar system. Credit Google 

How do we find & study exoplanets? 

transit radial velocity direct imaging microlensing timing astrometry 1281 616 45 32 20 0 Data from Open Exoplanet Catalogue 

Data from Open Exoplanet Catalogue 2000 2005 2010 2015 year of discovery 0 200 400 600 800 1000 number of exoplanets transit RV microlensing direct imaging timing 

Data from Open Exoplanet Catalogue 2000 2005 2010 2015 year of discovery 0 200 400 600 800 1000 number of exoplanets transit RV microlensing direct imaging timing first public data release from Kepler 

the transit method 

Credit NASA/European Space Agency 

Credit NASA/European Space Agency Jupiter 

Credit NASA/European Space Agency Jupiter Earth 

that's not what most stars look like! 

No content

1.0 0.5 0.0 0.5 1.0 time since transit [days] 100 50 0 relative brightness [ppm] 

everything is against us! 

Fig. 3.— Calculation of the transit probability. Left.—Transits are visible by observers within the penumbra of the planet, a cone with opening angle Θ with sin Θ = (R⋆ +Rp )/r, where r is the instantaneous star-planet distance. Right.—Close-up showing the penumbra (thick lines) as well as the antumbra (thin lines) within which the transits are full, as opposed to grazing. are tangent at four contact times tI –tIV , illustrated in Fig- ure 2. (In a grazing eclipse, second and third contact do not occur.) The total duration is Ttot = tIV − tI , the full duration is Tfull = tIII − tII , the ingress duration is ingress and egress. In practice the difference is slight; to leading order in R⋆/a and e, τe − τi ∼ e cosω R⋆ 3 1 − b2 3/2 , (17) Credit Winn (2010) arXiv:1001.2010 

need to look at the right place at the right time and measure extremely precise photometry 

Credit NASA Kepler 

Credit NASA 

Credit Carter Roberts 

Credit NASA 

Kepler-32 

Kepler-32 

Kepler-32 

Kepler-32 

Credit Fabrycky et al. (2012) 12 Fabrycky et al. Figure 16. Kepler-31 phase curves, in the style of figure 3. For the small inner candidate KOI-952.05, the phase is with respect to terest. The Kepler Follow-up spectra of Kepler-32: one sp servatory and one from Keck are weak due to the faintness cross correlation function be and available models is max ∼ 3900 K and ∼ 3600 K, atmospheric parameters are star is cooler than the library able. Both spectra are con sification as a cool dwarf ( [M/H]=0.172). We conserva Teff and log g with uncertain a [M/H] of 0± 0.4 based on t By comparing to the Yonse values for the stellar mass ( (0.53 ± 0.04R⊙ ) that are sli the KIC. We estimate a lum and an age of ≤ 9Gyr. Muirhead et al. (2011) h resolution IR spectrum of K a stellar Teff = 3726+73 −67 , [Fe ing their data via Padova m they inferred a considerably l We encourage further detail properties, as these have con directly affect the sizes and The probability of a star u being the actual host is only ity of a physical companion h This latter number is relative all the transit depths are sma be much larger planets hoste ically diluted. This opens up 

101 102 orbital period [days] 100 101 planet radius [R ] Data from NASA Exoplanet Archive 

that looks pretty good… 

101 102 orbital period [days] 100 101 planet radius [R ] Data from NASA Exoplanet Archive 

101 102 orbital period [days] 100 101 planet radius [R ] Data from NASA Exoplanet Archive 

100 101 102 103 104 105 orbital period [days] 100 101 planet radius [R ] Data from NASA Exoplanet Archive 

May 2013 The Kepler Mission goes up in flames * not exactly 

Credit NASA Kepler RIP 

cbna Flickr user Aamir Choudhry introducing: K2 

Credit NASA K2 

No content

No content

4000 2000 0 2000 4000 raw: 301 ppm EPIC 201374602; Kp = 11.5 mag 10 20 30 40 50 60 70 80 time [BJD - 2456808] 400 0 400 residuals: 35 ppm relative brightness [ppm] 4000 2000 0 2000 4000 raw: 301 ppm EPIC 201374602; Kp = 11.5 mag 10 20 30 40 50 60 70 80 time [BJD - 2456808] 400 0 400 residuals: 35 ppm relative brightness [ppm] 

7.2 7.4 7.6 x [pix] 10 20 30 40 50 60 70 80 time [BJD - 2456808] 9.05 9.10 9.15 9.20 y [pix] 

Can we find planets using K2? 

Anatomy of a transit signal + + + = planet star space craft detector signal 

Designing the probabilistic model n Pn xn K Sn stars: n = 1, · · · , N n Pn xn K Sn planet star space craft detector 

Designing the probabilistic model planet: star: noise: space craft: physics and geometry continuous in time → GP CCD, photon noise → Poisson ?? representation: 

The planet orbit model cba Wikipedia user Gonfer Kepler's Laws of Planetary Motion 

The planet orbit model cba Wikipedia user Gonfer Kepler's Laws of Planetary Motion 

TRANSIT LIGHT CURVES Vol. 580 mb darkening. The star is seen edge-on, with the observer off the top of the page. The star has radius , and v is defined as the r∗ d the normal to the stellar surface, while . (b) Transit geometry from the perspective of the observer. m p cos v 3. NONLINEAR LIMB DARKENING s a star to be more centrally peaked in brightness compared to a uniform source. This leads to more ng eclipse and creates curvature in the trough. Thus, including limb darkening is important for computing Reference Mandel & Agol (2002); arXiv:astro-ph/0210099 The planet transit model 

TRANSIT LIGHT CURVES Vol. 580 mb darkening. The star is seen edge-on, with the observer off the top of the page. The star has radius , and v is defined as the r∗ d the normal to the stellar surface, while . (b) Transit geometry from the perspective of the observer. m p cos v 3. NONLINEAR LIMB DARKENING s a star to be more centrally peaked in brightness compared to a uniform source. This leads to more ng eclipse and creates curvature in the trough. Thus, including limb darkening is important for computing Reference Mandel & Agol (2002); arXiv:astro-ph/0210099 The planet transit model 

TRANSIT LIGHT CURVES Vol. 580 mb darkening. The star is seen edge-on, with the observer off the top of the page. The star has radius , and v is defined as the r∗ d the normal to the stellar surface, while . (b) Transit geometry from the perspective of the observer. m p cos v 3. NONLINEAR LIMB DARKENING s a star to be more centrally peaked in brightness compared to a uniform source. This leads to more ng eclipse and creates curvature in the trough. Thus, including limb darkening is important for computing Reference Mandel & Agol (2002); arXiv:astro-ph/0210099 The planet transit model "…elliptic integral of the third kind…" 

TRANSIT LIGHT CURVES Vol. 580 mb darkening. The star is seen edge-on, with the observer off the top of the page. The star has radius , and v is defined as the r∗ d the normal to the stellar surface, while . (b) Transit geometry from the perspective of the observer. m p cos v 3. NONLINEAR LIMB DARKENING s a star to be more centrally peaked in brightness compared to a uniform source. This leads to more ng eclipse and creates curvature in the trough. Thus, including limb darkening is important for computing Reference Mandel & Agol (2002); arXiv:astro-ph/0210099 The planet transit model "…elliptic integral of the third kind…" 1.0 0.5 0.0 0.5 1.0 time since transit [days] 100 50 0 relative brightness [ppm] 

Designing the probabilistic model planet: star: noise: space craft: physics and geometry continuous in time → GP CCD, photon noise → Poisson ?? representation: 

The stellar variability model 

The stellar variability model y ⇠ N(f✓ (t), K↵(t)) Gaussian Mean Covariance 

0 1 2 3 4 5 x 6 4 2 0 2 4 6 y y ⇠ N (f✓ (t), K↵(t)) 

0 1 2 3 4 5 x 6 4 2 0 2 4 6 y y ⇠ N (f✓ (t), K↵(t)) 

0 1 2 3 4 5 x 6 4 2 0 2 4 6 y y ⇠ N (f✓ (t), K↵(t)) 

0 1 2 3 4 5 x 6 4 2 0 2 4 6 y y ⇠ N (f✓ (t), K↵(t)) 

The stellar variability model 

Designing the probabilistic model planet: star: noise: space craft: physics and geometry continuous in time → GP CCD, photon noise → Poisson ?? representation: 

Credit NASA The noise model 

Designing the probabilistic model planet: star: noise: space craft: physics and geometry continuous in time → GP CCD, photon noise → Poisson ?? representation: 

Designing the probabilistic model n Pn xn K Sn stars: n = 1, · · · , N n Pn xn K Sn 

stars: n = 1, · · · , N n xn K Designing the probabilistic model simple space craft assumption: 

4000 2000 0 2000 4000 raw: 301 ppm EPIC 201374602; Kp = 11.5 mag 10 20 30 40 50 60 70 80 time [BJD - 2456808] 400 0 400 residuals: 35 ppm relative brightness [ppm] 4000 2000 0 2000 4000 raw: 301 ppm EPIC 201374602; Kp = 11.5 mag 10 20 30 40 50 60 70 80 time [BJD - 2456808] 400 0 400 residuals: 35 ppm relative brightness [ppm] 

20 40 60 80 time [BJD - 2456808] 

60 62 64 66 68 70 time [BJD - 2456808] 

4000 2000 0 2000 4000 raw: 301 ppm EPIC 201374602; Kp = 11.5 mag 10 20 30 40 50 60 70 80 time [BJD - 2456808] 400 0 400 residuals: 35 ppm relative brightness [ppm] 

Designing the probabilistic model planet: star: noise: space craft: physics and geometry continuous in time → GP CCD, photon noise → Poisson data-driven linear model representation: 

Designing the probabilistic model n Pn xn K Sn stars: n = 1, · · · , N n Pn xn K Sn 

n Pn xn K Sn Designing the probabilistic model 

2 0 2 4 (a) raw 4 2 0 (b) 10 ELCs depth: 3.2 ppt 4 2 0 (c) 150 ELCs depth: 2.7 ppt 63 64 65 66 67 time [BJD - 2456808] 4 2 0 (d) conditional depth: 3.7 ppt relative brightness [ppt] 

Can we find planets using K2? 

Yes. 

stars days of data planet candidates confirmed planets 21,703 80 36 18 K2 Campaign 1 exoplanet discoveries Published: Foreman-Mackey, Montet, Hogg, et al. (arXiv:1502.04715) Montet, Morton, Foreman-Mackey, et al. (arXiv:1503.07866) Schölkopf, Hogg, Wang, Foreman-Mackey, et al. (arXiv:1505.03036) 

XKCD/1555 

XKCD/1555 

Foreman-Mackey, Montet, Hogg, et al. (arXiv:1502.04715) Montet, Morton, Foreman-Mackey, et al. (arXiv:1503.07866) Schölkopf, Hogg, Wang, Foreman-Mackey, et al. (arXiv:1505.03036) Probabilistic modeling—combining physical and data-driven models—enables the discovery of new planets using open data and open source software