============================================================================== A GREENHOUSE MODEL YOU CAN CHECK BY HAND, AND WHERE IT STOPS BEING HONEST Science Journaling Club, Volume 2 Issue 3, Spring 2026 ============================================================================== python : 3.12.3 numpy : 2.4.2 master seed : 20260321 This file computes. It does not observe. Every physical input below is a published number; every other number is arithmetic done here. ============================================================================== 1. INPUTS AS USED ============================================================================== solar constant S = 1361.0000 W/m^2 (sd 0.50) planetary albedo alpha = 0.2930 (sd 0.0050) textbook albedo = 0.3000 absorbed solar F=S(1-a)/4 = 240.5568 W/m^2 observed T_surface = 288.0000 K (sd 0.50) observed surface LW up = 398.2000 W/m^2 observed OLR = 239.0000 W/m^2 observed window escape = 22.0000 W/m^2 Stefan-Boltzmann sigma = 5.67037441900e-08 W m^-2 K^-4 ============================================================================== 2. VALIDATION A. THE ZERO-LAYER CASE AGAINST THE ACCEPTED 255 K ============================================================================== With no atmosphere in the longwave, the surface alone balances the absorbed sunlight: sigma T^4 = S(1-alpha)/4. Every textbook quotes about 255 K. case club accepted diff (K) albedo 0.293 (CERES, Loeb 2018) 255.2122 255.0 +0.2122 albedo 0.30 (round textbook value) 254.5781 255.0 -0.4219 Both land within 0.42 K of the quoted 255 K. The textbook figure is a rounding of a number that depends on which albedo you adopt; the spread between the two albedos above is 0.634 K, which is larger than the rounding error in either. VALIDATION A PASSES. Working value used from here on: T_e = 255.2122 K, F = 240.5568 W/m^2 ============================================================================== 3. VALIDATION B. NUMERIC AGAINST THE EXACT ANALYTIC SOLUTION ============================================================================== The system has a closed-form answer: T_s = T_e (N+1)^(1/4), and the i-th layer counting up from the ground sits at T_i = T_e (N+1-i)^(1/4). Below, the numerically solved surface temperature is printed beside it. N T_s numeric (K) T_s analytic (K) difference rel. diff solve resid 0 255.2122126240 255.2122126240 +0.000e+00 0.000e+00 0.00e+00 1 303.5001790881 303.5001790881 +0.000e+00 0.000e+00 0.00e+00 2 335.8781608226 335.8781608226 +0.000e+00 0.000e+00 5.68e-14 3 360.9245723762 360.9245723762 +0.000e+00 0.000e+00 5.68e-14 4 381.6312711001 381.6312711001 +0.000e+00 0.000e+00 5.68e-14 5 399.4286986243 399.4286986243 +0.000e+00 0.000e+00 1.14e-13 6 415.1222033133 415.1222033133 +0.000e+00 0.000e+00 1.14e-13 7 429.2140694490 429.2140694490 +0.000e+00 0.000e+00 5.68e-14 8 442.0405189769 442.0405189769 +0.000e+00 0.000e+00 5.68e-14 9 453.8386228998 453.8386228998 +0.000e+00 0.000e+00 1.71e-13 10 464.7823463469 464.7823463469 +5.684e-14 1.223e-16 1.71e-13 max |numeric - analytic| over N=0..10 : 5.684e-14 K max relative difference : 1.223e-16 double-precision epsilon : 2.220e-16 ratio of max rel. diff to eps : 0.55 Agreement is at machine precision. VALIDATION B PASSES. ============================================================================== 4. VALIDATION B2. EVERY LEVEL, NOT JUST THE SURFACE, AND LARGE N ============================================================================== For N = 5 the whole column is printed. Level 0 is the ground. level T numeric (K) T analytic (K) difference sigma T^4 0 399.4286986243 399.4286986243 +0.000e+00 1443.3405 1 381.6312711001 381.6312711001 +0.000e+00 1202.7838 2 360.9245723762 360.9245723762 +0.000e+00 962.2270 3 335.8781608226 335.8781608226 +0.000e+00 721.6703 4 303.5001790881 303.5001790881 +0.000e+00 481.1135 5 255.2122126240 255.2122126240 +0.000e+00 240.5568 top-of-atmosphere check: flux leaving the top layer = 240.556750 W/m^2 absorbed solar = 240.556750 W/m^2 imbalance = +0.000e+00 W/m^2 Pushing N further, to see where the dense linear solve starts to hurt. N T_s numeric T_s analytic rel. diff cond(A) 20 546.331543980 546.331543980 2.081e-16 7.457e+02 50 682.015110405 682.015110405 1.667e-16 4.296e+03 100 809.061986003 809.061986003 1.405e-16 1.670e+04 200 960.949283103 960.949283103 9.465e-16 6.582e+04 500 1207.427003011 1207.427003011 2.448e-15 4.077e+05 The condition number grows roughly as N^2, so the relative error grows with it. Even at N = 500 it stays far below anything that could matter physically; we print it because it is the honest place where 'exact' turns into 'exact enough'. ============================================================================== 5. SURFACE TEMPERATURE AGAINST LAYER COUNT ============================================================================== Observed global mean surface temperature: 288.00 K No-atmosphere effective temperature : 255.2122 K N T_s (K) T_s (C) T_s - 288 (K) T_s - T_e (K) 0 255.2122 -17.9378 -32.7878 +0.0000 1 303.5002 30.3502 +15.5002 +48.2880 2 335.8782 62.7282 +47.8782 +80.6659 3 360.9246 87.7746 +72.9246 +105.7124 4 381.6313 108.4813 +93.6313 +126.4191 5 399.4287 126.2787 +111.4287 +144.2165 6 415.1222 141.9722 +127.1222 +159.9100 7 429.2141 156.0641 +141.2141 +174.0019 8 442.0405 168.8905 +154.0405 +186.8283 9 453.8386 180.6886 +165.8386 +198.6264 10 464.7823 191.6323 +176.7823 +209.5701 The observed 288 K sits between N = 0 (255.21 K, -32.79 K too cold) and N = 1 (303.50 K, 15.50 K too hot). Solving T_e (N+1)^(1/4) = 288.00 for a real-valued N: N* = (T_obs/T_e)^4 - 1 = 0.621676 So the answer to the question in the title is: about two thirds of one layer. No whole number of perfectly absorbing layers reproduces Earth. One layer overshoots by 15.50 K, which is 47% of the entire 33 K greenhouse effect the model is trying to explain. Greenhouse effect as this model defines it: 32.7878 K Greenhouse effect as a flux, sigma T_obs^4 - F: 149.548 W/m^2 Same quantity from observations, LW_up - OLR: 159.200 W/m^2 difference -9.652 W/m^2 (-6.1%) That gap is not a bug. sigma*288^4 = 390.11 W/m^2 but the measured surface emission is 398.2 W/m^2. Section 8 explains where the missing 8.1 W/m^2 comes from. ============================================================================== 6. THE SAME QUESTION FOR TWO OTHER PLANETS ============================================================================== The model is not about Earth. It is about any transparent-to-sunlight, grey-in-the-infrared atmosphere. Published solar constants, albedos and mean surface temperatures, run through the same one-line formula. body S (W/m2) albedo T_e club T_e NASA diff T mean N* layers Mars 586.2 0.250 209.826 209.8 +0.026 214.0 0.0820 Earth 1361.0 0.294 255.122 254.0 +1.122 288.0 0.6240 Venus 2601.3 0.770 226.627 226.6 +0.027 737.0 110.8468 Two of the three agree with NASA's own blackbody temperature to better than 0.03 K, which is a real check: their number and ours were computed independently from the same two inputs. Earth does not agree, and the discrepancy is +1.12 K. That is not a bug in our code. Solving backwards, the albedo that reproduces NASA's listed 254.0 K from their listed 1361.0 W/m^2 is: alpha = 0.3063 against the 0.294 printed on the same page. The older Earth Bond albedo was 0.306. The fact sheet appears to carry an updated albedo beside a blackbody temperature computed from the previous one. We mention it because it is exactly the kind of thing a reader can check in one line, and because our own numbers deserve the same treatment. Venus needs about 111 layers of perfect absorber; Mars needs 0.082. The model spans two orders of magnitude in solar flux and a factor of three in surface temperature, which is the reason it earns a blackboard. The Mars figure is the weakest row: the fact sheet gives an average atmospheric temperature rather than a mean surface temperature, and we have used it as though the two were the same thing. They are not. ============================================================================== 7. WHAT 'NO ATMOSPHERE' ACTUALLY MEANS ============================================================================== The 33 K figure quietly assumes that removing the greenhouse effect leaves the albedo untouched. It would not. Three different airless Earths: case T (K) 'greenhouse' albedo 0.293 held fixed (the standard 33 K claim) 255.212 +32.788 albedo 0.15, surface only, clouds gone with the air 267.240 +20.760 albedo 0.00, a perfect blackbody sphere 278.321 +9.679 The famous 33 K becomes 20.8 K if you take the clouds away with the atmosphere, which you would have to. That is a 37% change in the headline number produced entirely by a modelling choice. ============================================================================== 8. TWO PLACES THE GLOBAL MEAN LIES, MEASURED ============================================================================== (a) sigma*mean(T)^4 is not mean(sigma*T^4). Jensen's inequality guarantees the second is larger. Surface temperature varies across the planet with a spatial-plus-seasonal standard deviation of roughly 15 to 20 K. sd (K) mean(sigma T^4) sigma mean(T)^4 excess 10.0 392.929 390.105 +2.824 15.0 396.463 390.105 +6.358 20.0 401.420 390.105 +11.315 25.0 407.809 390.105 +17.704 The measured surface emission of 398.2 W/m^2 is recovered at sd = 16.92 K, which is squarely inside the plausible range. So the 8.1 W/m^2 gap in section 5 is mostly Jensen's inequality, not a broken model. Expressed as a temperature, the planet radiates like a 289.48 K surface while its mean temperature is 288.00 K, a gap of 1.48 K. (b) The same inequality, run the other way, on an airless body. The Moon receives the same sunlight as Earth and has no atmosphere at all, so it is the closest thing to a control the Solar System offers. Moon T_e from S and albedo, computed here : 270.330 K Moon T_e listed by NASA : 270.4 K difference : -0.070 K Moon subsolar temperature : 382.304 K zero-thermal-inertia area-mean (computed) : 152.922 K analytic check, 0.4 * T_subsolar : 152.922 K difference : -8.144e-07 K NASA equatorial diurnal range : 95 to 390 K Our T_e reproduces NASA's to 0.07 K, so the inputs are being used the same way. Now look at the third line. A body with no atmosphere and no thermal inertia would average 152.9 K, which is 117.4 K BELOW the effective temperature that supposedly describes it. No greenhouse gas is involved. The whole gap is Jensen's inequality: the Moon balances its energy budget in sigma T^4, and the mean of T^4 has almost nothing to do with the fourth power of the mean of T when T runs from 95 to 390 K. The real Moon has thermal inertia and so sits above 153 K, but well below 270 K. Which means the 33 K we keep quoting for Earth is the gap to a fictitious uniform-temperature airless Earth, not to any airless planet that could actually exist. ============================================================================== 9. THE SINGLE LAYER WITH A TUNABLE EMISSIVITY ============================================================================== Drop the perfect-absorber assumption. One layer, emissivity eps in the longwave, still transparent to sunlight. layer: eps sigma T_s^4 = 2 eps sigma T_a^4 -> T_a^4 = T_s^4 / 2 surface: F + eps sigma T_a^4 = sigma T_s^4 => sigma T_s^4 = F / (1 - eps/2), T_s = T_e (1 - eps/2)^(-1/4) Note what the first line says: the layer temperature does not depend on eps at all. A thin veil and an opaque slab sit at the same temperature. They differ only in how much of the ground they can see. eps T_s (K) T_s - T_e (K) T_a (K) 0.0000 255.2122 +0.0000 214.6070 0.1000 258.5060 +3.2937 217.3767 0.2000 262.0239 +6.8116 220.3349 0.3000 265.7949 +10.5827 223.5060 0.4000 269.8541 +14.6418 226.9193 0.5000 274.2434 +19.0312 230.6103 0.6000 279.0146 +23.8024 234.6224 0.7000 284.2321 +29.0199 239.0097 0.7667 288.0000 +32.7878 242.1782 0.8000 289.9771 +34.7648 243.8407 0.9000 296.3540 +41.1418 249.2030 1.0000 303.5002 +48.2880 255.2122 Emissivity that reproduces T_obs = 288.00 K by inverting the formula : 0.7667081667 by bisection on T_s(eps) : 0.7667081667 difference : +6.661e-16 eps* = 0.7667. The layer temperature it implies is T_a = 242.178 K, which corresponds to an emission altitude of about 7.05 km at a lapse rate of 6.5 K/km. The real effective emission level is around 5 km, so this is the right order and about 41% off. Sensitivity: d T_s / d eps at eps* = 58.380 K per unit emissivity, so one percentage point of emissivity is worth 0.584 K here. ============================================================================== 10. MONTE CARLO ON eps* AND N* ============================================================================== Published uncertainties propagated to the two inferred quantities. S ~ Normal(1361.0, 0.50) W/m^2 Kopp & Lean 2011 albedo ~ Normal(0.293, 0.005) Loeb et al. 2018 T_obs ~ Normal(288.0, 0.50) K trials: 200000 stream: PCG64 spawned from SeedSequence(20260321), child 0 quantity mean sd SE 95% interval eps* 0.766666 0.012213 2.731e-05 0.7426- 0.7904 N* (layers) 0.621780 0.016059 3.591e-05 0.5906- 0.6534 T_e (K) 255.212161 0.452187 1.011e-03 254.3215-256.0900 greenhouse (K) 32.788857 0.672771 1.504e-03 31.4687-34.1040 Point estimate from the central inputs: eps* = 0.766708 Monte Carlo mean : eps* = 0.766666 difference : -4.173e-05 (1.53 standard errors) The Monte Carlo mean sits a few standard errors off the point estimate because eps* is a nonlinear function of the inputs, so the mean of the function is not the function of the mean. The offset, 4.17e-05, is orders of magnitude below the spread, so it changes nothing physical. We print it rather than round it away. Which input dominates? One-at-a-time, each varied by its own sd: S +1 sd moves eps* by -0.00045 albedo +1 sd moves eps* by +0.00872 T_obs +1 sd moves eps* by +0.00853 Convergence of the running mean of eps*: trials running mean running SE dev from final 100 0.768514 0.001313 +1.848e-03 300 0.767027 0.000695 +3.604e-04 1000 0.766493 0.000387 -1.731e-04 3000 0.766787 0.000229 +1.207e-04 10000 0.766683 0.000123 +1.636e-05 30000 0.766629 0.000071 -3.709e-05 60000 0.766626 0.000050 -4.058e-05 100000 0.766639 0.000039 -2.740e-05 150000 0.766643 0.000032 -2.348e-05 200000 0.766666 0.000027 -1.443e-15 SE falls as 1/sqrt(n) as it must: the ratio of the SE at 100 trials to the SE at 200000 is 48.10, against sqrt(200000/100) = 44.72. ============================================================================== 11. PLANCK BAND FRACTIONS, COMPUTED ============================================================================== The two-band model needs to know what fraction of a blackbody's emission falls in the atmospheric window. That is an integral of the Planck function, done here by Simpson's rule on a fine wavelength grid. VALIDATION C. The Simpson integrator against the exact series solution for the blackbody fraction function. The series is closed form; no quadrature is involved in it at all. T (K) lo (um) hi (um) Simpson exact series difference 288.0 8.0 12.0 0.252827765636 0.252827765628 +8.257e-12 255.0 8.0 12.0 0.213713967671 0.213713967664 +7.057e-12 242.0 8.0 12.0 0.194302379018 0.194302379011 +6.441e-12 737.0 8.0 12.0 0.156673461155 0.156673461150 +5.139e-12 93.7 8.0 12.0 0.001133443437 0.001133443437 +3.666e-14 max |Simpson - exact series| over the bands the model uses : 8.257e-12 Exact to better than one part in 10^9. VALIDATION C PASSES. The default grid is 401 points across the band, which is plenty for a 4 um interval. It is not plenty for a wide one, and pretending otherwise would be the easy dishonesty here. The same integrator over 4 to 100 um at 288 K, refined: grid points Simpson error vs series 401 0.993271129138 -3.703e-08 1601 0.993271166059 -1.113e-10 6401 0.993271166202 +3.175e-11 25601 0.993271166203 +3.230e-11 102401 0.993271166203 +3.231e-11 exact 0.993271166171 Fourth-order convergence, as Simpson's rule promises: each fourfold refinement cuts the error by about 256. Nothing in this study integrates a range wider than 5.5 um, so the 401-point grid stands. Window fraction (8.0 to 12.0 um) against temperature: T (K) window frac absorbing frac 210.0 0.139050 0.860950 240.0 0.191134 0.808866 255.0 0.213714 0.786286 270.0 0.233356 0.766644 288.0 0.252828 0.747172 300.0 0.263341 0.736659 320.0 0.276709 0.723291 At the observed surface temperature the window carries 0.2528 of the emission, which is 98.63 W/m^2 out of sigma*288^4 = 390.11 W/m^2. Wien peak at 288 K: 10.062 um, inside the window. That is the awkward fact the grey model cannot represent: the window sits on top of the emission peak, not off in the tail. ============================================================================== 12. THE TWO-BAND MODEL, AND THE CONTRADICTION IT EXPOSES ============================================================================== One layer, two spectral bands. The window (8-12 um) has optical depth tau_w, the rest of the spectrum has tau_b. Emissivity in each band is 1 - exp(-tau). Band fractions come from section 11 and move with T_s, so the surface temperature is found by fixed-point iteration. Grey calibration. eps* = 0.766708 corresponds to tau = -ln(1-eps*) = 1.455465 check: T_s(tau) = 288.000000 K against target 288.00 K, diff +0.000e+00 Two-band calibration with the window held at tau_w = 0.30. Solve for tau_b such that T_s = 288.00 K. First, a feasibility question the grey model never has to face. Even with the absorbing band made completely opaque, the band only covers 0.7472 of the emission. To reach 288 K the window must supply the rest: required window emissivity at least : 0.077270 i.e. window optical depth at least : 0.080418 A perfectly clear window makes 288 K unreachable with one layer, whatever you do to the rest of the spectrum. The classroom picture of 'a window that lets radiation straight out' is quantitatively too generous by itself. Calibration cross-check, closed form against bisection: tau_b* closed form : 2.7878191732 tau_b* bisection : 2.7878191732 difference : +2.087e-14 tau_b* = 2.787819 band emissivity= 0.938445 window emissiv.= 0.259182 window fraction= 0.252828 A_eff = 0.766708 (compare grey eps* = 0.766708) T_s = 288.000000 K (target 288.00, diff +5.684e-14, 11 iterations) The two models agree exactly on the present-day state, by construction. A_eff and eps* are the same number to 0.0e+00. The equilibrium equations are identical once you collapse the bands into one absorptivity. Now the first thing the grey model cannot tell you. Surface emission reaching space without being absorbed: grey model, tau = 1.4555 : 91.008 W/m^2 two-band model, through window : 73.066 W/m^2 two-band model, total : 91.008 W/m^2 OBSERVED (Costa & Shine 2012) : 22.000 W/m^2 grey overshoots observation by : +69.008 W/m^2 (4.1 times too much) two-band overshoots by : +69.008 W/m^2 (4.1 times too much) Both models leak far too much radiation straight to space. Turn it around: what tau_w reproduces the measured 22 W/m^2 of window escape, with the absorbing band taken as completely opaque? tau_w needed = 1.703858 window escape = 22.0000 W/m^2 (target 22.0) resulting T_s = 299.9695 K against 288 K = +11.9695 K THE DOUBLE BIND. A single-layer two-band model can match the observed surface temperature or the observed window escape. Not both. tuned to T_s = 288 K -> window escape 91.0 W/m^2, 4.1 times measured tuned to escape 22 W/m^2 -> T_s = 300.0 K, 12.0 K too hot One more thing, which is sharper than either number above. Rerun the calibration at several window opacities. Each time, tau_b is re-solved so that the surface still lands on 288 K. tau_w tau_b* window escape total escape T_s 0.30 2.7878 73.066 91.008 288.000 0.60 2.0673 54.129 91.008 288.000 1.00 1.6726 36.284 91.008 288.000 1.50 1.4408 22.007 91.008 288.000 2.00 1.3226 13.348 91.008 288.000 The window escape falls by a factor of 5.5 across that range and the total escape does not move at all: 91.008 W/m^2 in every row, to the last printed digit. It cannot move. Once the model is pinned to 288 K the effective absorptivity is fixed at eps* = 0.766708, so the untouched fraction leaving the top is (1 - eps*) sigma T_s^4 = 91.008 W/m^2 however you divide the opacity between the bands. Closing the window forces the absorbing band open by exactly the compensating amount. At tau_w = 1.50 the window escape alone reads 22.007 W/m^2, which is the measured figure, and the model is still leaking 91.0 W/m^2 in total. The reason is physical and the model cannot fix it. One layer has one temperature. The real atmosphere emits from many altitudes: the window leaks from near the ground where it is warm, the band centres radiate from high and cold. A single slab has to pick one, and whichever it picks, the other quantity is off by a factor of four, or by twelve kelvin. Adding layers helps; making them grey does not. ============================================================================== 13. HOW WRONG IS GREY? THE RESPONSE TO ADDED ABSORBER ============================================================================== The present-day state is not where the grey assumption fails. Both models were forced through the same point. It fails in the derivative. Experiment: multiply the optical depth by a factor and recompute T_s. Grey scales one tau; the two-band scales tau_w and tau_b together. multiplier grey T_s two-band T_s grey dT 2-band dT ratio 1.00 288.0000 288.0000 +0.0000 +0.0000 -- 1.25 292.3108 290.0995 +4.3108 +2.0995 2.053 1.50 295.5060 291.5278 +7.5060 +3.5278 2.128 2.00 299.5057 293.4422 +11.5057 +5.4422 2.114 3.00 302.5444 295.9588 +14.5444 +7.9588 1.827 4.00 303.2758 297.8000 +15.2758 +9.8000 1.559 8.00 303.4995 301.7052 +15.4995 +13.7052 1.131 Doubling the absorber: grey model warms by +11.506 K two-band model warms by +5.442 K grey overstates by a factor of 2.11 Why: the grey model can always absorb more, because a single band with tau = 1.46 is only 76.7% opaque. The two-band model's absorbing band is already 93.84% opaque at present day, so doubling it buys almost nothing and the warming has to come from the window alone. A harder version, closer to what CO2 actually does. CO2 absorbs in the band and barely touches the 8-12 um window, so scale tau_b only: tau_b mult two-band T_s dT (K) 1.0 288.0000 +0.0000 2.0 290.4719 +2.4719 4.0 290.6366 +2.6366 8.0 290.6372 +2.6372 100.0 290.6372 +2.6372 A hundredfold increase in band opacity is worth 2.6372 K. The band is saturated. This is the single most important thing the grey model hides, and it is part of why the real CO2 forcing grows as the logarithm of the concentration rather than linearly. Saturation ceiling of the two-band model (tau_b -> infinity): T_s = 290.6372 K, i.e. +2.6372 K above present day, no matter how much absorber you add to that band. Sensitivity to the window optical depth, which is the softest number in the whole exercise: tau_w tau_b* grey dT (2x) 2-band dT (2x) ratio 0.10 5.1069 +11.5057 +1.4852 7.747 0.20 3.3470 +11.5057 +3.6515 3.151 0.30 2.7878 +11.5057 +5.4422 2.114 0.40 2.4603 +11.5057 +6.9016 1.667 0.60 2.0673 +11.5057 +9.0033 1.278 0.90 1.7449 +11.5057 +10.6990 1.075 The ratio stays above 1 everywhere in this range, so the direction of the conclusion is safe. Its size is not: the ratio runs from 1.08 to 7.75 across these six values of tau_w. The safe reading is that grey overstates the response; the factor of two is an estimate that carries the window optical depth's uncertainty along with it. ============================================================================== 14. MONTE CARLO ON THE GREY-VERSUS-TWO-BAND RATIO ============================================================================== The window edges are not sharp and its optical depth is not well known. Sampling them: window low edge ~ Uniform(7.5, 8.5) um window high edge ~ Uniform(11.5, 13.0) um tau_w ~ Uniform(0.15, 0.60) trials: 4000 stream: PCG64 spawned from SeedSequence(20260321), child 1 Each trial recalibrates tau_b to hit 288 K, then doubles both depths. trials where no tau_b can reach 288 K: 323 of 4000 those are dropped; 3677 usable trials remain quantity mean sd SE two-band dT for doubling (K) 6.3179 1.7914 0.0295 grey / two-band ratio 2.0204 0.7523 0.0124 95% of trials give a ratio between 1.268 and 4.052. Fraction of trials with ratio > 2 : 0.3824 Fraction of trials with ratio > 1 : 1.0000 Convergence of the running mean of the ratio: trials running mean running SE dev from final 50 1.99489 0.08785 -0.02555 100 1.94720 0.07166 -0.07325 250 2.02938 0.04845 +0.00893 500 2.00597 0.03210 -0.01447 1000 2.02991 0.02405 +0.00947 2000 2.03470 0.01707 +0.01425 3000 2.01766 0.01355 -0.00278 3677 2.02045 0.01241 +0.00000 The grey model overstates the response to doubled absorber by 2.02 +/- 0.01 (SE). The 2.5th percentile is 1.27, so even the most forgiving window we sampled still leaves grey overstating by a quarter. One caution about the 323 dropped trials. They were dropped because a wide, clear window makes 288 K unreachable, and those are exactly the cases where the grey model would have looked worst. Dropping them pulls the reported ratio down, so 2.02 is a conservative figure rather than a flattering one. ============================================================================== 15. WHERE A DIFFERENT MODELLING CHOICE WOULD CHANGE THE ANSWER ============================================================================== Each row swaps one decision and reports the headline numbers. choice T_e (K) eps* baseline: S=1361, albedo=0.293, T_obs=288 255.212 0.7667 textbook albedo 0.30 254.578 0.7789 albedo 0.29 (Stephens et al. 2012) 255.483 0.7615 older S=1366 (pre-SORCE value) 255.446 0.7622 T_obs = 287.0 K (a cooler baseline period) 255.212 0.7494 T_obs = 289.0 K (a warmer baseline period) 255.212 0.7837 T_obs = 289.4 K (from measured LW, sect. 8) 255.212 0.7918 eps* spread across these choices: 0.7494 to 0.7918, a range of 0.0423, which is 3.5 times the Monte Carlo standard deviation of 0.0122. The modelling choices matter more than the measurement uncertainty. Anyone quoting eps* to three decimals without saying which albedo they used is quoting noise. And the choice that matters most of all is not on this list: whether the atmosphere is allowed to convect. Pure radiative equilibrium with a realistic absorber gives a surface far hotter than 288 K and a lapse rate steeper than any real air column can hold, which is what Manabe and Strickler found in 1964; convection has to be added by hand before the answer becomes Earth-like. None of that is in this file. ============================================================================== 16. FIGURE DATA ============================================================================== Every number the article plots, printed so the figures can be checked. [FIG1] surface temperature against layer count N, T_s(K) 0, 255.2122 1, 303.5002 2, 335.8782 3, 360.9246 4, 381.6313 5, 399.4287 6, 415.1222 7, 429.2141 8, 442.0405 9, 453.8386 10, 464.7823 T_e = 255.2122 ; T_obs = 288.0000 ; N* = 0.621676 [FIG2] vertical temperature profile for several N N=1: 303.500, 255.212 N=2: 335.878, 303.500, 255.212 N=3: 360.925, 335.878, 303.500, 255.212 N=5: 399.429, 381.631, 360.925, 335.878, 303.500, 255.212 N=10: 464.782, 453.839, 442.041, 429.214, 415.122, 399.429, 381.631, 360.925, 335.878, 303.500, 255.212 [FIG3] surface temperature against single-layer emissivity eps, T_s(K) 0.00, 255.2122 0.05, 256.8327 0.10, 258.5060 0.15, 260.2352 0.20, 262.0239 0.25, 263.8757 0.30, 265.7949 0.35, 267.7861 0.40, 269.8541 0.45, 272.0045 0.50, 274.2434 0.55, 276.5776 0.60, 279.0146 0.65, 281.5630 0.70, 284.2321 0.75, 287.0327 0.80, 289.9771 0.85, 293.0789 0.90, 296.3540 0.95, 299.8207 1.00, 303.5002 eps* = 0.766708 -> T_s = 288.0000 [FIG4] response to scaled optical depth, grey against two-band multiplier, grey T_s, two-band T_s, band-only T_s 0.50, 275.0258, 279.1413, 280.5814 0.75, 282.3146, 284.6700, 285.4421 1.00, 288.0000, 288.0000, 288.0000 1.25, 292.3108, 290.0995, 289.3109 1.50, 295.5060, 291.5278, 289.9733 2.00, 299.5057, 293.4422, 290.4719 2.50, 301.5378, 294.8209, 290.5961 3.00, 302.5444, 295.9588, 290.6270 4.00, 303.2758, 297.8000, 290.6366 6.00, 303.4879, 300.2817, 290.6372 8.00, 303.4995, 301.7052, 290.6372 [FIG5] Monte Carlo convergence of eps* trials, running mean, running SE 100, 0.768514, 0.001313 300, 0.767027, 0.000695 1000, 0.766493, 0.000387 3000, 0.766787, 0.000229 10000, 0.766683, 0.000123 30000, 0.766629, 0.000071 60000, 0.766626, 0.000050 100000, 0.766639, 0.000039 150000, 0.766643, 0.000032 200000, 0.766666, 0.000027 [FIG5b] Monte Carlo convergence of the grey/two-band ratio trials, running mean, running SE 50, 1.99489, 0.08785 100, 1.94720, 0.07166 250, 2.02938, 0.04845 500, 2.00597, 0.03210 1000, 2.02991, 0.02405 2000, 2.03470, 0.01707 3000, 2.01766, 0.01355 3677, 2.02045, 0.01241 [TABLE] the results table in the article N, T_s analytic, T_s numeric, difference, T_s - T_obs, equivalent eps 0, 255.212213, 255.212213, +0.000e+00, -32.7878, 0.0000 1, 303.500179, 303.500179, +0.000e+00, +15.5002, 1.0000 2, 335.878161, 335.878161, +0.000e+00, +47.8782, 1.3333 3, 360.924572, 360.924572, +0.000e+00, +72.9246, 1.5000 4, 381.631271, 381.631271, +0.000e+00, +93.6313, 1.6000 5, 399.428699, 399.428699, +0.000e+00, +111.4287, 1.6667 6, 415.122203, 415.122203, +0.000e+00, +127.1222, 1.7143 7, 429.214069, 429.214069, +0.000e+00, +141.2141, 1.7500 8, 442.040519, 442.040519, +0.000e+00, +154.0405, 1.7778 9, 453.838623, 453.838623, +0.000e+00, +165.8386, 1.8000 10, 464.782346, 464.782346, +5.684e-14, +176.7823, 1.8182 ============================================================================== 17. HEADLINE NUMBERS ============================================================================== effective temperature T_e 255.212 K observed surface temperature 288.000 K greenhouse effect (this model's definition) 32.788 K layers needed, N* 0.6217 T_s at N = 1 303.500 K (+15.50 K) single-layer emissivity eps* 0.7667 +/- 0.0122 (sd) grey optical depth tau* 1.4555 two-band tau_b* at tau_w = 0.30 2.7878 window fraction at 288 K 0.2528 window escape, model 91.01 W/m^2 window escape, observed 22.00 W/m^2 T_s if escape is forced to 22 W/m^2 299.97 K (+11.97 K) doubling response, grey 11.506 K doubling response, two-band 5.442 K grey overstatement factor 2.02 +/- 0.01 (SE) band-only 100x response 2.6372 K Runtime: 8.82 s Seed: 20260321. Rerunning this file reproduces every digit above.