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Planet-Population Densities and Throughputs

This series of plots shows where in the radius/SMA plane a mission's planets are, and what fraction of them it reaches. Every plot in the series lives on the same axes as the Kopparapu bin plots: planet radius against luminosity-scaled semi-major axis, both logarithmic. The 5x3 bins and the outline of the earthlike region are drawn behind the data as context.

The data source is reduce-planet-population.csv.gz, one row per planet per simulation, produced by per_planet_yield() in reduce_drms.py. The columns used here are sma_scaled, radius, and four flags: det_ok, char_ok, star_det_obs, star_char_obs.

Luminosity-scaled SMA. The x coordinate is a / sqrt(L_star), because planets in our simulated universes are chosen in these scaled coordinates to begin with. This makes, for instance, the all-planet density plot show the power law that defines it.

The populations

Four planet populations, and one star variable that sits behind them:

  • A -- all planets in the table. Note this is not the whole simulated universe: the reduction keeps only planets around stars the mission actually visited, because the remainder would all read det_ok = char_ok = 0.
  • D -- det_ok: the planet was successfully detected at least once.
  • C -- char_ok: the planet was successfully characterized at least once.
  • B -- incidental characterizations, C \ D: characterized but never detected. These planets are swept up as a side effect of working a star for other reasons. (Mnemonic: B = byproduct)
  • S -- the star. star_det_obs and star_char_obs say whether the star was observed that way, whatever came of it. S is not a planet property, and it is the scheduler's choice.

The funnel is not a chain. A -> D -> C would be two filters in series, but characterization also draws directly from planets that were never detected, so C has two feeder streams:


                 +--------------------------------------------+
                 |  S : the star                              |
                 |  star_det_obs / star_char_obs              |
                 |  (scheduler's choice; not a planet         |
                 |   property, but it gates both paths)       |
                 +---------------------+----------------------+
                        |                            |
                        v                            v
     A  ------------->  D  ---------------------->  C n D
    all    P(det|A)   det_ok      P(char|D)         "targeted"
  planets    |                                        ^
             |  1 - P(det|A)                          |  both are in C
             v                                        v
            A \ D  ---------------------------->  B = C \ D
         not detected      P(char|not det)       "incidental"

Because S gates both arrows out of the star box, a plot of P(char | planet present) mixes two unrelated processes: whether the scheduler chose to work this star at all, and whether a planet of this radius and SMA shows up once it did. The two are separable, and separating them is the point of the extra plots below.

Coverage

Every planet falls in exactly one of four cells. The table gives the cell names, which population each belongs to, and -- for orientation -- typical shares from one 100-simulation ensemble:

det_ok char_ok cell in A in D in C example share
0 0 missed x 0.713
1 0 detected only x x 0.194
0 1 incidental (B) x x 0.058
1 1 targeted x x x 0.035

Read across: the density plots show where each column's population lies. The throughput plots show the ratio between two columns, as a function of position. The incidental row is the one that A -> D -> C reasoning misses; in the example ensemble it is 62% of all characterizations.

Densities and throughputs

Two kinds of plot, with different estimators.

Density. A kernel density estimate of one population, in units of probability per dex^2. It answers "where are these planets?" and integrates to one, so densities of different populations are not directly comparable in height -- each is normalized to its own count.

Throughput. A conditional probability P(numerator | position), also called a throughput here: of the planets at this radius and SMA in the denominator population, what fraction are also in the numerator population? These are bounded in [0, 1], and all such plots use the same absolute 0-1 color scale, so their colors mean one thing across the whole series.

A throughput is not computed as a ratio of two densities. This is because two separate kernel fits would choose their bandwidth from their own samples, making their ratio unbounded, potentially blowing up where the denominator thins out. Instead the numerator and denominator are kernel sums over the same sample with the same kernel,

    P(num | x)  =  sum_i w_i(x) num_i  /  sum_i w_i(x) .

This is the kernel regression (Nadaraya-Watson) estimator of the indicator and lies in [0, 1] by construction. Where the denominator holds less than a set fraction of its peak kernel weight, the cell is left blank rather than shown as a ratio of two nearly-zero numbers.

The plots

Five are always produced; five more with mode_op set to + (make S=... graphics-extra). All are named det-planet-pop-*.png and appear in the Radius/SMA Density Plots section of the ensemble page.

Customize either the scenario or the parent family/experiment to always make extra plots by adding this entry to config-reduce.json:


   "graphics": {
     "mode_op": {
       "planet_pop": "+"
     }
   }
Plot Quantity Extra?
density-all density of A, all planets in the table
density-det density of D, detected planets
density-char density of C, characterized planets
tput-all2det P(detected | planet present)
tput-all2char P(characterized | planet present)
tput-det2char P(characterized | detected) -- the targeted path only x
tput-all2starchar P(star observed for char. | planet present) -- targeting x
tput-starchar2char P(characterized | star observed for char.) -- response x
tput-nodet2char P(characterized | not detected) -- the incidental rate x
tput-char2det P(detected | characterized) -- provenance of the chars x

Two identities tie the series together, and are worth checking on any new scenario:

  • all2starchar x starchar2char = all2char, in the aggregate. The first factor is scheduling, the second is response to a planet at that radius and SMA. If the targeting map is flat, target selection is planet-agnostic; if it is not, some planet property is driving the scheduler.
  • all2char = det2char x all2det + nodet2char x (1 - all2det), in the aggregate: the targeted and incidental channels, weighted by how much of the population each draws from.

Caveats

  • The population is restricted to observed stars. "All planets" is what the mission had the chance to observe, not the simulated universe. Totals from this file are not population totals.
  • The flags are cumulative over the mission. det_ok means ever detected, so P(char | det) does not imply the detection came first. A question about ordering needs first-detection and first-char times, which this table does not carry.
  • Large populations are sampled. The estimate is capped at a fixed number of points, drawn with a fixed seed so plots reproduce; the title says when this happened. Broad structure is stable across seeds, fine detail in thin regions is not.
  • The kernel smooths across boundaries. Density appears just outside the earthlike region and outside the populated part of the plane. The outline is drawn over the density so the boundary stays visible.