k=1 Z p( xk | wk) p( wk | ✓) d wk = K Y k=1 Z p( xk | wk) p( wk | ✓) p( wk | xk, ↵) p( wk | xk, ↵) d wk = Z↵ K Y k=1 Z p( wk | ✓) p( wk | ↵) p( wk | xk, ↵) d wk ⇡ Z↵ K Y k=1 1 N N X n=1 p( w (n) k | ✓) p( w (n) k | ↵)
inhomogeneous Poisson process p ( {wk } | ✓ ) = exp ✓ Z ˆ✓( w ) d w ◆ K Y k=1 ˆ✓( wk) ˆ ✓(w) ⌘ ✓(w) Qc(w) the observable rate density ✓(ln P, ln R) = dN d ln P d ln R for example
xk } | ↵) ⇡ exp ✓ Z ˆ✓(w) dw ◆ K Y k=1 1 Nk Nk X n=1 ˆ✓(w (n) k ) p (w (n) k | ↵) p({ xk } | ✓ ) = Z p({ xk } | { wk }) p({ wk } | ✓ ) d{ wk } The FML maths w (n) k ⇠ p( wk | xk, ↵ ) posterior samples
is piecewise constant the measurement uncertainties are non-negligible & known The probabilistic, censored “histogram” * also ignoring false positives, etc.
survey completeness for small planets is a complicated function of P and RP . It decreases with increasing P and decreasing both completeness factors). stars have a planet with pe between 1 and 2 R⊕ . 5 10 20 30 40 50 100 200 300 400 Orbital period (days) 0.5 1 2 3 4 5 10 20 Planet size (Earth-radii) 0 10 20 30 40 50 60 70 80 90 100 Survey Completeness (C) % Fig. 1. on a lo detected like star color sca injection photom complet missed rence. T orbital P graph). orbital favors d Petigura, Howard & Marcy (2013)
0.5 1 2 4 8 16 Planet size (Earth-radii) 4.9 0.6% 3.5 0.4% 0.3 0.1% 0.2 0.1% 6.6 0.9% 6.1 0.7% 0.8 0.2% 0.2 0.2% 7.7 1.3% 7.0 0.9% 0.4 0.2% 0.6 0.3% 5.8 1.6% 7.5 1.3% 1.3 0.6% 0.6 0.3% 3.2 1.6% 6.2 1.5% 2.0 0.8% 1.1 0.6% 5.0 2.1% 1.6 1.0% 1.3 0.6% 0% 1% 2% 3% 4% 5% 6% 7% 8% Planet Occurrence Fig. 2. Plan orbital perio d and RP = 0: are shown as in orbital pe in a cell is where the su each cell. He planets (for of the orbita completenes Best42k sam planet occur occurrence w where the c the small pl rence is cons entire range ports mild e RP = 1 − 2 R⊕ Petigura, Howard & Marcy (2013)
to Earth well as inferring the occurrence distribution of exoplanets, this dataset constrain the rate density of Earth analogs. Explicitly, we constrain the nsity of exoplanets orbiting “Sun-like” stars11, evaluated at the location = (ln P , ln R ) = dN d ln P d ln R R=R , P=P . is the rate density of exoplanets around a Sun-like star (expected per star per natural logarithm of period per natural logarithm of radius eriod and radius of Earth. Equation (23), we define “Earth analog” in terms of measurable quantit
with any orbital period up to a maximum period, P, on the horiz rly Earth size ð1 − 2 R⊕Þ are included. This cumulative distribution reaches 20.2% at P = 50 d, meaning 20.4% of Sun-like stars Extrapolation: Petigura, Howard & Marcy (2013)
ln 0.0 0.1 0.2 0.3 0.4 0.5 0.6 p(ln ) 10 2 10 1 100 101 [%] real data Foreman-Mackey, Hogg & Morton Petigura, Howard & Marcy on the rate density of Earth analogs (as defined h our result has large fractional uncertainty—with This is shown in Figure 9 where we compare the m for to the published value and uncertainty. Qu y of Earth analogs is = 0.017+0.018 0.009 nat 2 dicates that this quantity is a rate density, per natur mic radius. Converted to these units, Petigura et a the same quantity (indicated as the vertical lines in hat Petigura’s extrapolation model predicts but, for ferred rate density over their choice of “Earth-like”