Equations and methodology

Habitable-zone boundaries

Kopparapu, R. K. et al. (2013), Habitable Zones Around Main-Sequence Stars: New Estimates, ApJ 765, 131, doi:10.1088/0004-637X/765/2/131, with the corrected coefficients from the Erratum, ApJ 770, 82, doi:10.1088/0004-637X/770/1/82.

S_eff = S_eff,sun + a·T + b·T² + c·T³ + d·T⁴
T = T_eff − 5780 K
valid for 2600 K ≤ T_eff ≤ 7200 K

d = √(L / S_eff)   [au], L in solar luminosities

Five boundaries are computed (recent Venus, runaway greenhouse, moist greenhouse, maximum greenhouse, early Mars). The conservative habitable zone is runaway greenhouse to maximum greenhouse; the optimistic zone is recent Venus to early Mars. Outside the stated temperature range the model returns an explicit “extrapolated” flag rather than a silently clamped value.

Earth Similarity Index

Schulze-Makuch, D. et al. (2011), A Two-Tiered Approach to Assessing the Habitability of Exoplanets, Astrobiology 11(10), 1041–1052, doi:10.1089/ast.2010.0592.

F_x = 1 − |(x − x₀) / (x + x₀)|

ESI_global = F_radius^(w_r/4) · F_density^(w_d/4)
           · F_escape_velocity^(w_v/4) · F_temperature^(w_t/4)

  w_r = 0.57, w_d = 1.07, w_v = 0.70, w_t = 5.58

Computed in two tiers for reporting (plain product within each
tier, since w/2 is already applied per term -- one combining
sqrt only, not a sqrt per tier):

ESI_interior = F_radius^(w_r/2) · F_density^(w_d/2)
ESI_surface  = F_escape_velocity^(w_v/2) · F_temperature^(w_t/2)
ESI_global   = √(ESI_interior · ESI_surface)

The temperature term is referenced to Earth’s equilibrium temperature (254 K), not its surface temperature (288 K), because equilibrium temperature is the only temperature exoplanet catalogues provide. Consequence: Venus scores 0.92 on this metric. This is documented in Limitations.

Composite-source coherence

NASA’s pscomppars table chooses a preferred source separately for each parameter, so one row can mix radius, mass, orbit, and stellar values from different publications. The per-publication ps table is therefore retained as a second evidence layer. For each planet we report the number of distinct sources used by the composite, the overlap with the archive’s coherent default_flag=1 solution, and the median symmetric fractional difference over shared linear-valued parameters.

fractional difference = 2·|x_composite − x_default|
                        / (|x_composite| + |x_default|)

reported median over shared parameters; undefined when no overlap exists

This is a diagnostic rather than another ranking weight. Combining publications can improve completeness, so “mixed-source” is not itself evidence of poor quality; hiding the mixture would be.

Monte Carlo uncertainty propagation

Every parameter with a published asymmetric uncertainty is sampled from a two-piece (split) normal: draw z ~ N(0,1), scale by the upper sigma when z > 0 and the lower sigma when z < 0. Non-positive draws on positive-definite quantities are rejected and redrawn (bounded attempts), never clipped to zero.

x_sample = x + z·σ_upper   if z > 0
x_sample = x + z·σ_lower   if z ≤ 0

4,000 draws per planet, seed fixed for reproducibility

Rocky plausibility

Rogers, L. A. (2015), Most 1.6 Earth-Radius Planets are not Rocky, ApJ 801, 41, doi:10.1088/0004-637X/801/1/41; Fulton, B. J. et al. (2017), The California-Kepler Survey III, AJ 154, 109 (the radius-valley result).

p(rocky) = 1 / (1 + exp((R_p − 1.6) / 0.20))

A logistic centred on 1.6 R⊕, spanning the observed 1.5–2.0 R⊕ radius valley, rather than a hard cut the data does not support.

Characterisation metrics

Kempton, E. M.-R. et al. (2018), A Framework for Prioritizing the TESS Planetary Candidates Most Amenable to Atmospheric Characterization, PASP 130, 114401, doi:10.1088/1538-3873/aadf6f.

TSM = S · (R_p³ · T_eq) / (M_p · R_star²) · 10^(−m_J/5)
S = 0.190 / 1.26 / 1.28 / 1.15  by radius bin (<1.5 / 1.5–2.75 / 2.75–4.0 / >4.0 R⊕)

ESM = 4.29×10⁶ · (B₇.₅(T_day) / B₇.₅(T_star)) · (R_p/R_star)² · 10^(−m_K/5)

Follow-up readiness lanes

Observation planning is reported separately from the Earth-2.0 index. A target can be physically compelling but difficult to observe, or easy to observe but physically unlike Earth. The transit, radial-velocity, and reflected-light lanes therefore keep their native units and assumptions rather than collapsing into one detectability score.

Transit forecast at 2030-01-01:
σ_T(N) = √(σ_T0² + N²·σ_P²)

Nominal maximum angular separation:
θ_mas = 1000 · a_au / d_pc

Reflected-light contrast at quadrature:
C = A_g · (R_p/a)² · Φ(90°),  A_g = 0.30,  Φ_Lambert(90°) = 1/π

The timing formula assumes no published epoch–period covariance; transit-timing-variable systems carry a separate warning. Angular separation is an orbital scale, not an epoch-specific projected position. Reflected-light contrast is a stated albedo/phase scenario, not a measured contrast or instrument-detectability claim.

Composite Earth-2.0 index

index = exp( Σᵢ wᵢ · log(max(scoreᵢ, ε)) ),  Σwᵢ = 1,  ε = 0.01

Default weights: similarity 0.35, habitability 0.40,
                 confidence 0.25, characterisation 0.0

A weighted geometric mean, chosen specifically because it is non-compensatory: a near-zero component drags the whole index toward zero regardless of how strong the others are.

Radial-velocity semi-amplitude and minimum mass

K = 28.4329 m/s · (Mp sin i / M_Jup) · ((M* + Mp)/M_sun)^(−2/3)
                              · (P / 1 yr)^(−1/3) / √(1 − e²)

Inverted to recover M sin i from a fitted K. Reported only when a fit passes a three-criterion reliability gate: at least 20 velocities, amplitude significance ≥3σ, and residual scatter below five times the fitted amplitude.

Physical constants

IAU 2015 Resolution B3 nominal solar and terrestrial conversion constants; CODATA 2018 fundamental constants. Full values in src/earth2/constants.py in the repository, each with its source.