## 2014-03-21

### eta-Earth

Today Dan Formeman-Mackey, Tim Morton (Princeton), and I measured the rate at which Sun-like stars host Earth-like planets! Foreman-Mackey did all the heavy lifting—it was a frantic hack day of sorts—but the work was based on the incredible data sent to us by Erik Petigura (Berkeley). Our measurement of "eta-Earth" involved building a period and radius distribution model which, when multiplied by survey completeness and transit probability, does a good job of modeling Petigura's scatter plot of data points.

IMHO, the correct definition of eta-Earth is the number of planets per star per natural-logarithmic interval in some pair of quantities, which could be radius and period, or which could be mass and insolation, evaluated at Earth's properties. That is, it involves an extrapolation or interpolation of any distribution function (measured from the observational data) to the location of Earth. From Petigura's data, and given assumptions (some listed yesterday), we get a number like five percent, plus-or-minus a percent or two. Importantly, this is five percent per natural-logarithmic interval of radius and per natural-logarithmic interval of period, so it should be compared to other papers and press releases with that clear definition in mind. Also, of course, it is conditioned on some pretty severe assumptions.

This blog post does not properly convey my Stoked-ness.

[Note added later: We found a bug in the code; the rate is probably a factor of two larger even!]

1. How can you *measure* eta-Earth when no habitable Earths are actually found?

You should call what it is: an extrapolated estimate of Eta-Earth.

And the lesson from the past on extrapolation is very clear: think about Jupiters -- there is a clear ~1 AU jump in dN/dlog(a) (see Wright et al., 2009 or Mayor et al., 2011). It would seriously underestimate eta-Jupiter if you had done similar extrapolations.

1. Agreed! Everything is conditional on assumptions. But, that said, uncertainties are being propagated fully.

2. That's what "eta" means. An extrapolation.

2. Eta Earth is the fraction of Earth-size planets in the HZ per star, using the original definition. Therefore, this is constrained by the definition of Earth-size, based either on mass or radius, and the HZ definition. Anything else with arbitrary ranges should be called just 'occurrence' as not to confuse it with the Eta Earth definition.