============================================================================ NEPTUNE'S AURORA: THE CLUB'S OWN CALCULATION Science Journaling Club | seed = 20250326 | numpy 2.4.2 Every number below is produced by this script. None of it is a measurement, and none of it is the paper's own analysis. ============================================================================ MODEL A -- WHERE A TILTED, OFFSET DIPOLE PUTS THE OVAL ---------------------------------------------------------------------------- Field lines traced by RK4 from the magnetic equator at fixed L down to r = 1 R_N. Southern footpoints only. Latitudes are planetocentric and negative in the southern hemisphere. A0. Sanity check, centred dipole, zero tilt. Textbook: cos^2(lat) = 1/L, so lat = -arccos(sqrt(1/L)). L traced analytic diff 5.0 -63.4349 -63.4349 +0.0000 deg 10.0 -71.5650 -71.5651 +0.0000 deg 20.0 -77.0790 -77.0790 +0.0000 deg 30.0 -79.4803 -79.4803 +0.0000 deg The tracer agrees with the closed-form dipole to better than 0.001 degrees, so the integrator is not the source of any error. A1. The migration. Southern oval at L = 20 R_N, centred dipole. The oval is a ring of constant magnetic colatitude, so the tilt simply carries it down the planet. Nothing subtle happens here. tilt poleward centre equatorward width (deg) edge (deg) (deg) edge (deg) (deg) 0.0 -77.08 -77.08 -77.08 0.00 5.0 -82.08 -76.60 -72.08 10.00 10.0 -87.08 -75.09 -67.08 20.00 15.0 -87.92 -72.11 -62.08 25.84 20.0 -82.92 -67.93 -57.08 25.84 25.0 -77.92 -63.40 -52.08 25.84 30.0 -72.92 -58.72 -47.08 25.84 35.0 -67.92 -53.95 -42.08 25.84 40.0 -62.92 -49.12 -37.08 25.84 43.0 -59.92 -46.21 -34.08 25.84 46.9 -56.02 -42.41 -30.18 25.84 At zero tilt the ring sits at a single latitude, -77.08 deg, 12.92 deg from the pole, because a circle centred on the pole is a line of latitude. Tilt it and the ring keeps its shape but no longer lies flat: its equatorward edge runs from -77.08 deg to -30.18 deg, a walk of 46.90 degrees, which is the tilt angle to the second decimal place. In a centred dipole that is all a tilt can do, and it is enough. A2. Add the offset and the ring stops lying flat against the lines of latitude. Same sweep, L = 20, offset 0.55 R_N held across the dipole axis. tilt poleward centre equatorward width 0.0 -70.24 -56.46 -46.27 23.97 5.0 -75.24 -61.29 -51.27 23.97 10.0 -80.24 -66.05 -56.27 23.97 15.0 -85.24 -70.66 -61.27 23.97 20.0 -89.76 -74.78 -66.27 23.49 25.0 -84.76 -77.35 -71.27 13.49 30.0 -80.26 -78.73 -76.27 3.99 35.0 -81.34 -79.05 -74.76 6.58 40.0 -86.27 -78.21 -69.76 16.51 43.0 -89.27 -76.99 -66.76 22.51 46.9 -86.83 -74.12 -62.86 23.97 The latitude width swings from 24.0 deg down to 4.0 deg and back again as the ring is carried across the pole and down the far side. The ring itself never changes shape. What changes is how it lies against the lines of latitude, and that is the distortion a reader sees in Figure 3. A3. The offset direction, which nobody has published in a form we could check. Tilt fixed at 46.9 deg, L = 20, offset 0.55 R_N, swept from pointing at the southern magnetic pole (beta = 0) round to pointing at the northern one (beta = 180). beta poleward centre equatorward width miss overlap 0 -47.01 -43.04 -39.19 7.81 1.96 26% 15 -55.23 -51.15 -47.11 8.13 6.15 27% 30 -63.37 -58.82 -54.24 9.13 13.82 19% 45 -71.02 -65.55 -60.03 10.99 20.55 0% 60 -77.77 -70.77 -63.78 13.99 25.77 0% 75 -83.18 -73.79 -64.84 18.34 28.79 0% 90 -86.83 -74.12 -62.86 23.97 29.12 0% 120 -87.83 -67.62 -51.24 36.58 22.62 29% 150 -80.76 -55.33 -34.70 46.06 10.33 84% 180 -67.80 -40.53 -18.40 49.40 4.47 100% 'miss' is the distance from our band centre to the centre of the observed band. Prefer it to the overlap column, which rewards a model for being vague: a wide enough oval overlaps everything. Smallest miss: beta = 0 deg, 1.96 degrees out. A displacement lying across the axis, beta near 90, throws the oval 29 degrees away from the emission and is ruled out on these latitudes alone. Displacements close to the axis, in either direction, keep the oval where JWST found it. So the observation does constrain the offset geometry, and it says the displacement runs near the dipole axis. We did not put that in. A4. Against the observation, centred dipole, tilt 46.9 deg. JWST enhanced H3+ column, southern hemisphere [1]: -60.0 to -30.0 degrees latitude Our L = 20 oval: -56.0 to -30.2 degrees latitude Overlap: 25.8 degrees of the observed 30 degree band, which is 86% of it. Our band centre -42.4 deg; observed band centre -45.0 deg; miss distance 2.6 degrees. For contrast, the same model with the tilt set to zero, which is where every textbook picture of a giant-planet aurora puts it: -77.1 to -77.1 degrees. That is 35 degrees of latitude poleward of where the emission actually is. And the L shells. The paper maps the southern emission to field lines running from under 3 out to 20 R_N [1]. Here is what our model does with that span, tilt 46.9 deg, centred: L (R_N) mag. latitude poleward equatorward 3 54.74 -78.36 -7.84 5 63.43 -69.67 -16.53 10 71.57 -61.53 -24.67 15 75.04 -58.06 -28.14 20 77.08 -56.02 -30.18 30 79.48 -53.62 -32.58 A5. Monte Carlo over the things we do not know exactly. tilt ~ N(46.9, 1.0) deg offset ~ U(0, 0.55) R_N beta ~ U(0, 180) deg L ~ U(15, 25) R_N azimuth around the oval ~ U(0, 360) deg 2000 seeded draws, one traced field line each. The offset direction is drawn flat because we genuinely do not know it. draws that reached the surface: 2000 of 2000 median footpoint latitude : -51.89 deg 68% interval : -64.38 to -37.91 deg 95% interval : -76.71 to -29.60 deg fraction landing inside the observed -60 to -30 deg band: 70.3% Now the same run with the offset held at zero, which is the model the article headlines: median -41.79 deg, 68% -53.88 to -31.61, 95% -56.96 to -29.06 fraction inside the observed band: 94.2% Read the two together. Knowing the tilt alone puts you in the right latitudes 94 times out of a hundred. Letting the offset point anywhere drops that to 70. The honest statement is that the 47 degree tilt is what predicts a mid-latitude aurora, and the offset is a source of error we cannot close from here. MODEL B -- WHY THE GLOW WENT OUT ---------------------------------------------------------------------------- LTE nu2 band emission per H3+ ion, rigid symmetric top, levels to J = 22, nuclear spin weights 4 for K divisible by 3 and 2 otherwise. T (K) Q(T) eps(T) rel. vs 750 K 200 34.098 1.689087e-06 0.0002% 250 48.294 6.346875e-05 0.0063% 300 64.115 7.124711e-04 0.0712% 350 81.423 4.008117e-03 0.4008% 358 84.323 5.052666e-03 0.5053% 400 100.115 1.464011e-02 1.4640% 450 120.113 4.009254e-02 4.0093% 500 141.363 8.973775e-02 8.9738% 550 163.830 1.734260e-01 0.173 600 187.496 3.001662e-01 0.300 650 212.357 4.772082e-01 0.477 700 238.416 7.095800e-01 0.710 750 265.687 1.000000e+00 1.000 800 294.189 1.349041e+00 1.349 900 354.980 2.216453e+00 2.216 1000 420.992 3.286223e+00 3.286 Cooling from 750 K to 358 K: emission per ion falls to 0.5053% of its old value, a factor of 197.9 fainter. Cooling from 750 K to 350 K: 0.4008%, a factor of 249.5. The paper states 0.8% at 358 K for the same density [1]. We get 0.5053%. Our answer is a factor of 1.58 from theirs, which for a rigid-rotor model with a made-up band-average Einstein coefficient is about as well as we deserve to do. B1. How much of that is the Boltzmann factor and how much is Q? population of the emitting levels, 358 K over 750 K : 1.6036e-03 which on its own is a fall by a factor of : 624 partition function Q(750)/Q(358) : 3.1508 net emission ratio, the product : 5.0527e-03 The two effects pull against each other. Cooling empties the emitting levels by a factor of 624. Cooling also crowds the ions into fewer states, which hands back a factor of 3.15. The fight is not close. Net, a factor of 198. B2. Sensitivity of the answer to the temperatures themselves. Voyager's number carries +/- 150 K, which is the wide one. case ratio factor 750 -> 358 (nominal) 0.5053% 197.9 600 -> 358 (cool 1989) 1.6833% 59.4 900 -> 358 (hot 1989) 0.2280% 438.7 750 -> 350 0.4008% 249.5 750 -> 366 0.6305% 158.6 MODEL C -- THE SENSITIVITY LADDER (our crudest model, read the docstring before believing any of it) ---------------------------------------------------------------------------- Calibration: JWST's floor is set to half the brightness the paper actually measured, 0.2526% of the 750 K expectation. That fixes k = 0.8511. Everything else is then predicted. instrument FOM floor floor as % Voyager 2 IRIS (1989 flyby) 10.9 7.84e-02 7.841% IRTF 3.0 m / iSHELL (2017) 42.2 2.02e-02 2.016% Keck II 10 m / NIRSPEC (2009) 75.2 1.13e-02 1.132% JWST 6.5 m / NIRSpec IFU (2023) 336.9 2.53e-03 0.253% Now run each one against a hot Neptune and a cold one. Emission is in units of the 750 K expectation, so a hot Neptune sits at 1.000 and the Neptune JWST actually saw sits at 0.00505. instrument at 750 K at 358 K Voyager 2 IRIS (1989 flyby) DETECT (13x floor) miss (0.06x floor) IRTF 3.0 m / iSHELL (2017) DETECT (50x floor) miss (0.25x floor) Keck II 10 m / NIRSPEC (2009) DETECT (88x floor) miss (0.45x floor) JWST 6.5 m / NIRSpec IFU (2023) DETECT (396x floor) DETECT (2.00x floor) Read that table honestly. Three of the four rows are the story: every ground-based attempt had the sensitivity to find H3+ on a 750 K Neptune and none of them had the sensitivity to find it on a 358 K one. The fourth row is where our ladder breaks. It says Voyager's IRIS should have seen the ion during the flyby, and no such detection was ever reported. IRIS was a low-resolution interferometer whose H3+ lines would have been blended into the continuum, which our figure of merit knows nothing about. We are leaving the wrong answer in the table rather than tuning it out. The exposure penalty. Background-limited signal to noise scales as brightness times sqrt(area x time), so recovering a factor of 197.9 in brightness costs a factor of 39170 in area x time. Keck would have needed roughly 9793 nights of pure integration. FIGURE DATA (transcribed by hand into the article's SVGs) ---------------------------------------------------------------------------- F2. centred dipole: tilt, poleward, centre, equatorward, width [L=20] 0.0 -77.08 -77.08 -77.08 0.00 5.0 -82.08 -76.60 -72.08 10.00 10.0 -87.08 -75.09 -67.08 20.00 15.0 -87.92 -72.11 -62.08 25.84 20.0 -82.92 -67.93 -57.08 25.84 25.0 -77.92 -63.40 -52.08 25.84 30.0 -72.92 -58.72 -47.08 25.84 35.0 -67.92 -53.95 -42.08 25.84 40.0 -62.92 -49.12 -37.08 25.84 43.0 -59.92 -46.21 -34.08 25.84 46.9 -56.02 -42.41 -30.18 25.84 F2b. with the 0.55 R_N offset across the axis, same columns 0.0 -70.24 -56.46 -46.27 23.97 5.0 -75.24 -61.29 -51.27 23.97 10.0 -80.24 -66.05 -56.27 23.97 15.0 -85.24 -70.66 -61.27 23.97 20.0 -89.76 -74.78 -66.27 23.49 25.0 -84.76 -77.35 -71.27 13.49 30.0 -80.26 -78.73 -76.27 3.99 35.0 -81.34 -79.05 -74.76 6.58 40.0 -86.27 -78.21 -69.76 16.51 43.0 -89.27 -76.99 -66.76 22.51 46.9 -86.83 -74.12 -62.86 23.97 F2c. beta scan at full tilt: beta, poleward, centre, equatorward, width, overlap with the observed band, miss (all degrees) 0 -47.01 -43.04 -39.19 7.81 7.81 1.96 15 -55.23 -51.15 -47.11 8.13 8.13 6.15 30 -63.37 -58.82 -54.24 9.13 5.76 13.82 45 -71.02 -65.55 -60.03 10.99 0.00 20.55 60 -77.77 -70.77 -63.78 13.99 0.00 25.77 75 -83.18 -73.79 -64.84 18.34 0.00 28.79 90 -86.83 -74.12 -62.86 23.97 0.00 29.12 120 -87.83 -67.62 -51.24 36.58 8.76 22.62 150 -80.76 -55.33 -34.70 46.06 25.30 10.33 180 -67.80 -40.53 -18.40 49.40 30.00 4.47 F2d. L shell, magnetic latitude, poleward, equatorward [tilt 46.9] 3 54.74 -78.36 -7.84 5 63.43 -69.67 -16.53 10 71.57 -61.53 -24.67 15 75.04 -58.06 -28.14 20 77.08 -56.02 -30.18 30 79.48 -53.62 -32.58 F3. oval shape at four tilts: azimuth-ordered (lon, lat) pairs, 24 points each, L = 20, centred dipole tilt 0.0 lon 0.0 15.0 30.0 45.0 60.0 75.0 90.0 105.0 120.0 135.0 150.0 165.0 180.0 195.0 210.0 225.0 240.0 255.0 270.0 285.0 300.0 315.0 330.0 345.0 lat -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 -77.1 tilt 20.0 lon 180.0 156.1 143.6 139.4 139.7 142.3 146.1 150.9 156.2 161.8 167.8 173.8 180.0 186.2 192.2 198.2 203.8 209.1 213.9 217.7 220.3 220.6 216.4 203.9 lat -82.9 -81.8 -79.2 -75.9 -72.6 -69.3 -66.3 -63.7 -61.4 -59.5 -58.2 -57.4 -57.1 -57.4 -58.2 -59.5 -61.4 -63.7 -66.3 -69.3 -72.6 -75.9 -79.2 -81.8 tilt 35.0 lon 180.0 171.4 164.4 159.8 157.5 157.1 158.2 160.4 163.4 167.1 171.1 175.5 180.0 184.5 188.9 192.9 196.6 199.6 201.8 202.9 202.5 200.2 195.6 188.6 lat -67.9 -67.3 -65.4 -62.8 -59.6 -56.3 -53.0 -49.9 -47.2 -45.0 -43.4 -42.4 -42.1 -42.4 -43.4 -45.0 -47.2 -49.9 -53.0 -56.3 -59.6 -62.8 -65.4 -67.3 tilt 46.9 lon 180.0 174.1 169.1 165.3 163.0 162.2 162.6 164.0 166.2 169.1 172.5 176.1 180.0 183.9 187.5 190.9 193.8 196.0 197.4 197.8 197.0 194.7 190.9 185.9 lat -56.0 -55.5 -53.8 -51.4 -48.4 -45.1 -41.8 -38.6 -35.8 -33.4 -31.6 -30.5 -30.2 -30.5 -31.6 -33.4 -35.8 -38.6 -41.8 -45.1 -48.4 -51.4 -53.8 -55.5 and the same four with the 0.55 R_N offset held across the axis tilt 0.0 (offset, beta = 90) lon 0.0 3.1 6.1 9.1 12.1 14.8 17.2 19.1 19.8 18.9 15.4 8.9 0.0 351.1 344.6 341.1 340.2 340.9 342.8 345.2 347.9 350.9 353.9 356.9 lat -46.3 -46.5 -47.2 -48.4 -50.1 -52.2 -54.8 -57.8 -61.1 -64.3 -67.3 -69.4 -70.2 -69.4 -67.3 -64.3 -61.1 -57.8 -54.8 -52.2 -50.1 -48.4 -47.2 -46.5 tilt 20.0 (offset, beta = 90) lon 0.0 5.3 10.7 16.3 22.3 28.7 35.7 43.4 51.9 61.3 71.6 83.9 180.0 276.1 288.4 298.7 308.1 316.6 324.3 331.3 337.7 343.7 349.3 354.7 lat -66.3 -66.5 -67.0 -68.0 -69.3 -71.0 -73.0 -75.4 -78.0 -80.8 -83.8 -86.9 -89.8 -86.9 -83.8 -80.8 -78.0 -75.4 -73.0 -71.0 -69.3 -68.0 -67.0 -66.5 tilt 35.0 (offset, beta = 90) lon 0.0 14.1 28.7 44.3 61.1 78.7 96.1 112.8 128.2 142.3 155.4 167.9 180.0 192.1 204.6 217.7 231.8 247.2 263.9 281.3 298.9 315.7 331.3 345.9 lat -81.3 -81.3 -81.3 -81.3 -81.2 -80.8 -80.1 -79.1 -78.0 -76.8 -75.7 -75.0 -74.8 -75.0 -75.7 -76.8 -78.0 -79.1 -80.1 -80.8 -81.2 -81.3 -81.3 -81.3 tilt 46.9 (offset, beta = 90) lon 180.0 148.5 135.8 133.1 134.4 137.8 142.3 147.6 153.4 159.7 166.2 173.1 180.0 186.9 193.8 200.3 206.6 212.4 217.7 222.2 225.6 226.9 224.2 211.5 lat -86.8 -86.0 -84.0 -81.7 -79.2 -76.5 -73.8 -71.1 -68.5 -66.2 -64.4 -63.3 -62.9 -63.3 -64.4 -66.2 -68.5 -71.1 -73.8 -76.5 -79.2 -81.7 -84.0 -86.0 F4. temperature, relative emission (750 K = 1) 200 1.689087e-06 225 1.266221e-05 250 6.346875e-05 275 2.373499e-04 300 7.124711e-04 325 1.805959e-03 350 4.008117e-03 375 7.998374e-03 400 1.464011e-02 425 2.495648e-02 450 4.009254e-02 475 6.126961e-02 500 8.973775e-02 525 1.267307e-01 550 1.734260e-01 575 2.309128e-01 600 3.001662e-01 625 3.820306e-01 650 4.772082e-01 675 5.862548e-01 700 7.095800e-01 725 8.474511e-01 750 1.000000e+00 775 1.167233e+00 800 1.349041e+00 825 1.545211e+00 850 1.755438e+00 875 1.979336e+00 900 2.216453e+00 925 2.466277e+00 950 2.728249e+00 975 3.001773e+00 1000 3.286223e+00 F4b. Monte Carlo latitude histogram, 30 bins from -75 to -15 deg -75.00 -73.00 35 -73.00 -71.00 32 -71.00 -69.00 51 -69.00 -67.00 48 -67.00 -65.00 60 -65.00 -63.00 83 -63.00 -61.00 105 -61.00 -59.00 103 -59.00 -57.00 114 -57.00 -55.00 111 -55.00 -53.00 119 -53.00 -51.00 129 -51.00 -49.00 111 -49.00 -47.00 89 -47.00 -45.00 93 -45.00 -43.00 96 -43.00 -41.00 79 -41.00 -39.00 99 -39.00 -37.00 85 -37.00 -35.00 80 -35.00 -33.00 61 -33.00 -31.00 62 -31.00 -29.00 37 -29.00 -27.00 21 -27.00 -25.00 8 -25.00 -23.00 7 -23.00 -21.00 3 -21.00 -19.00 5 -19.00 -17.00 0 -17.00 -15.00 0 ============================================================================ THE FOUR NUMBERS THE ARTICLE QUOTES ============================================================================ 1. Oval latitude at 46.9 deg tilt, L=20 : -56.0 to -30.2 deg (centred; the offset sweep spans : -87.8 to -18.4 deg) 2. Observed JWST auroral band : -60.0 to -30.0 deg 3. H3+ emission, 750 K -> 358 K : 0.5053%, factor 197.9 4. Area x time penalty to recover it : 39170 Every one of those is the club's own arithmetic on published constants. The measurements belong to Melin and colleagues [1]. ============================================================================