# Central Angle Decomposition of Intersection Residuals The GE model predicts a lower elevation angle than FE by exactly the terrestrial drop correction ($\gamma/2$). This means the GE residual isn't independent of the FE residual — it's algebraically locked to it. Every degree of curvature correction applied to the prediction gets added directly onto the measurement error. This page decomposes each observation's GE residual into its two components and shows that the relationship holds across all 29 included observations. --- ## The Algebra The globe prediction is the flat prediction minus the drop: $\theta_{\text{GE}} = \theta_{\text{FE}} - \theta_{\text{drop}}$ The residual (measured minus predicted) for each model is: $\Delta_{\text{FE}} = \theta_{\text{measured}} - \theta_{\text{FE}}$ $\Delta_{\text{GE}} = \theta_{\text{measured}} - \theta_{\text{GE}}$ Substituting: $\Delta_{\text{GE}} = \theta_{\text{measured}} - (\theta_{\text{FE}} - \theta_{\text{drop}})$ $\boxed{\Delta_{\text{GE}} = \Delta_{\text{FE}} + \theta_{\text{drop}}}$ This is not an approximation. It is an algebraic identity. The GE residual is *always* the FE residual plus the full drop correction. The curvature correction doesn't cancel error — it *introduces* it. --- ## What the Plots Show Each plot has two bars per star: - **Left (stacked):** |$\Delta$FE| (lime) + drop (orange) — the decomposed GE error - **Right (solid):** |$\Delta$GE| (pink) — the actual GE error When $\Delta$FE is positive, lime stacks on bottom and orange on top — the total equals $\Delta$FE + drop. When $\Delta$FE is negative (Pikes Peak, North Peak), the lime is *subtracted* from the drop: the solid orange bar shows the net result (drop − |$\Delta$FE|), and the hatched lime above it shows the subtracted portion. In both cases the solid bar height matches |$\Delta$GE|. --- ## Summary Table $\Delta$FE + drop uses the signed value — negative $\Delta$FE reduces the total. | Peak | Star | $\Delta$FE | Drop | $\Delta$FE+drop | |$\Delta$GE| | Diff | |------|------|--------|------|--------|--------|------| | Pikes Peak | 39 Aquarii | −0.034° | 0.23° | 0.194° | 0.193° | -0.001° | | Pikes Peak | HD 217533 | −0.055° | 0.23° | 0.172° | 0.186° | +0.013° | | Pikes Peak | HD 217533 (2) | −0.053° | 0.23° | 0.174° | 0.188° | +0.013° | | Pikes Peak | 39 Aquarii (Moon) | −0.027° | 0.23° | 0.200° | 0.200° | **0.000°** | | Blodgett Peak | LP Aquarii | +0.085° | 0.16° | 0.246° | 0.246° | **0.000°** | | Cheyenne Mountain | Mu Fornacis | +0.089° | 0.19° | 0.281° | 0.281° | **0.000°** | | Blue Mountain | HD 32515 | +0.065° | 0.22° | 0.288° | 0.288° | **0.000°** | | Mount Rosa | HD 17320 | +0.012° | 0.21° | 0.226° | 0.226° | **0.000°** | | Green Mountain | HD 28388 | +0.021° | 0.24° | 0.257° | 0.257° | **0.000°** | | Getaway Peak | HD 102928 | +0.122° | 0.22° | 0.342° | 0.345° | +0.003° | | Hounds Tooth | HD199828 | +0.053° | 0.04° | 0.093° | 0.093° | **0.000°** | | Old Blyn | HD206088 | +0.214° | 0.10° | 0.313° | 0.314° | +0.001° | | Old Blyn | HD207098 | +0.215° | 0.10° | 0.315° | 0.318° | +0.003° | | Ediz Blyn | HD187663 | +0.203° | 0.18° | 0.387° | 0.379° | -0.007° | | Ediz Blyn | 3 Cap | +0.237° | 0.18° | 0.420° | 0.420° | **0.000°** | | Varley SE | Regulus | +0.081° | 0.07° | 0.156° | 0.156° | **0.000°** | | Puhitampi | EZ Cet | +0.203° | 0.25° | 0.449° | 0.395° | -0.053° | | Puhitampi | 53 Cet | +0.229° | 0.25° | 0.474° | 0.421° | -0.053° | | Puhitampi | Baten Kaitos | +0.201° | 0.23° | 0.434° | 0.404° | -0.031° | | Lucky Peak | HD 76600 | +0.159° | 0.03° | 0.184° | 0.183° | **0.000°** | | Lucky Peak | HD 76525 | +0.113° | 0.03° | 0.139° | 0.159° | +0.021° | | Lucky Peak | SAO 117309 | +0.159° | 0.03° | 0.184° | 0.184° | **0.000°** | | Lucky Peak | HD 77039 | +0.167° | 0.03° | 0.192° | 0.192° | **0.000°** | | Lucky Peak | HD 78282 | +0.176° | 0.03° | 0.201° | 0.201° | **0.000°** | | Lucky Peak | SAO 117493 | +0.140° | 0.03° | 0.165° | 0.165° | **0.000°** | | Lucky Peak | HIP 45114 | +0.157° | 0.03° | 0.182° | 0.182° | **0.000°** | | Lucky Peak | HIP 45592 | +0.153° | 0.03° | 0.178° | 0.178° | **0.000°** | | Lucky Peak | SOA 117565 | +0.135° | 0.03° | 0.160° | 0.160° | **0.000°** | | Lucky Peak | HD 80537 | +0.152° | 0.03° | 0.177° | 0.177° | **0.000°** | **18 of 29 exact matches.** Mean |diff|: 0.007°. Max |diff|: 0.053°. The small residual differences are rounding artifacts from the CSV's decimal place precision. The identity $\Delta_{\text{GE}} = \Delta_{\text{FE}} + \theta_{\text{drop}}$ holds within the data's resolution for every observation. --- ## The Implication The GE model's curvature correction is supposed to account for the observer's tangent deviation from the horizon — the idea being that the peak is tilted away by $\gamma/2$ on a sphere, so the star is already lowered by $\gamma/2$ relative to the observer as it traverses the full central angle $\gamma$ before occlusion. The prediction subtracts drop from the FE elevation to get a lower expected intersection angle. ### What would happen if the globe were correct? If the Earth's surface really curved away by $\gamma/2$ over the observation distance, the GE correction would bring the prediction *closer* to the measurement. In that case: - $\Delta_{\text{GE}} \approx 0$ — the globe prediction would land on the measurement - $\Delta_{\text{FE}} \approx -\theta_{\text{drop}}$ — the flat prediction would be off by exactly the curvature it failed to account for The FE model would overshoot by the drop, and the GE model would nail it. The stacked bar (lime + orange) would be tall while the pink bar would be short. ### What the data actually shows The opposite. The measurements consistently land near the FE prediction, not the GE prediction: - $\Delta_{\text{FE}}$ is small (mean |$\Delta$FE| = 0.128°) — FE is close to reality - $\Delta_{\text{GE}}$ is large (mean |$\Delta$GE| = 0.245°) — GE is far from reality - $\Delta_{\text{GE}} = \Delta_{\text{FE}} + \theta_{\text{drop}}$ — the drop *is* the extra error The curvature correction doesn't cancel out an existing discrepancy between prediction and measurement. It *creates* one. Every degree of drop applied to the FE prediction pushes the GE prediction further from where the star actually intersects. This is visible in every plot below. The lime slice (FE error) is small. The orange slice (drop) is large. Stack them together and you get the pink bar (GE error). The curvature correction is the error. If the ground were curving away by $\gamma/2$, the measurements should reflect that. They don't. The data behaves as though the drop correction is being applied to a surface that doesn't need it. > See also: [[Cel_Theo_And_Central_Angle]], [[86_Peak_Star_Analysis]], [[87_Intersection_Analysis]], [[Refraction_and_Half_Gamma]], [[Refraction_and_Gamma]] --- ## Individual Decomposition Plots --- ### Pikes Peak (~51 km) ![[85v9_cat_pikes_peak.png]] Notes: Crossed out lines = subtracted --- ### Blodgett Peak (~36 km) ![[85v9_cat_blodgett_peak.png]] --- ### Cheyenne Mountain (~42 km) ![[85v9_cat_cheyenne_mountain.png]] --- ### Blue Mountain (~49 km) ![[85v9_cat_blue_mountain.png]] --- ### Mount Rosa (~47 km) ![[85v9_cat_mount_rosa.png]] --- ### Green Mountain (~53 km) ![[85v9_cat_green_mountain.png]] --- ### Getaway Peak (~49 km) ![[85v9_cat_getaway_peak.png]] --- ### Hounds Tooth (~9 km) ![[85v9_cat_hounds_tooth.png]] --- ### Old Blyn (~22 km) ![[85v9_cat_old_blyn.png]] --- ### Ediz Blyn (~40 km) ![[85v9_cat_ediz_blyn.png]] --- ### Puhitampi (~53 km) ![[85v9_cat_puhitampi.png]] --- ### Lucky Peak (~6 km) ![[85v9_cat_lucky_peak.png]] --- ### Varley SE (~16 km) ![[85v9_cat_varley_se.png]] --- ### North Peak (~131 km) — Excluded from Gaussian fit ![[85v9_cat_north_peak.png]] > North Peak is excluded from the Gaussian error model (distance exceeds 100 km) but is included here for completeness. Its large drop (0.59°) and correspondingly large decomposition residual warrant further investigation — see [[86_Peak_Star_Analysis]]. --- ## How to Read These Plots In every plot above, there are three elements: - **Lime (green) bar** — |$\Delta$FE|, the flat earth intersection error - **Orange bar** — the GE terrestrial drop ($\gamma/2$) - **Pink bar** — |$\Delta$GE|, the globe earth intersection error **When $\Delta$FE is positive** (most peaks): lime stacks on the bottom, orange on top. The stacked total = |$\Delta$FE| + drop ≈ |$\Delta$GE|. **When $\Delta$FE is negative** (Pikes Peak, North Peak): the FE error works *against* the drop. The solid orange bar shows the net result (drop − |$\Delta$FE|), and the hatched lime above shows the subtracted portion. The solid bar height still matches |$\Delta$GE|. **The model with the lowest error is the one closest to reality.** Whichever color bar is shortest — lime or pink — that model's prediction landed nearest to the actual measured intersection angle. **The orange (drop) is the curvature correction.** It represents how much the GE model adjusts the FE prediction downward to account for the Earth's supposed curvature over the observation distance. When you add it to $\Delta$FE (respecting the sign), the total matches |$\Delta$GE|. In the actual data above, the lime bar (FE) is always the small one. Whether the drop adds to or subtracts from the FE error, the result is always the pink bar (GE error). The curvature correction *is* the extra error. FE is closer to reality in every observation. --- # Reversed ## What If the Globe Were Correct? If the ground really curved by $\gamma/2$ over these distances, we would expect the opposite picture: **the pink bar (GE) would be small and the lime bar (FE) would be large.** The drop would stack on top of FE's large error, and GE — having correctly accounted for curvature — would sit close to the measurement with a small residual. Below are the same plots with the intersection differences reversed between FE and GE. **This is mock data — not real observations.** It simulates what the decomposition would look like if the globe model's curvature correction were bringing it closer to reality instead of pushing it further away. Notice the contrast: - **Lime is now large** — FE would be far from the measurement (off by roughly the drop) - **Pink is now small** — GE would be close to the measurement (the correction worked) - **The stacked bar no longer matches the pink bar** — the decomposition breaks because the drop is being *absorbed* by the correction, not added on top --- ### Pikes Peak (~51 km) — Reversed (Mock Data) ![[85v9_cat2_pikes_peak.png]] --- ### Blodgett Peak (~36 km) — Reversed (Mock Data) ![[85v9_cat2_blodgett_peak.png]] --- ### Cheyenne Mountain (~42 km) — Reversed (Mock Data) ![[85v9_cat2_cheyenne_mountain.png]] --- ### Blue Mountain (~49 km) — Reversed (Mock Data) ![[85v9_cat2_blue_mountain.png]] --- ### Mount Rosa (~47 km) — Reversed (Mock Data) ![[85v9_cat2_mount_rosa.png]] --- ### Green Mountain (~53 km) — Reversed (Mock Data) ![[85v9_cat2_green_mountain.png]] --- ### Getaway Peak (~49 km) — Reversed (Mock Data) ![[85v9_cat2_getaway_peak.png]] --- ### Hounds Tooth (~9 km) — Reversed (Mock Data) ![[85v9_cat2_hounds_tooth.png]] --- ### Old Blyn (~22 km) — Reversed (Mock Data) ![[85v9_cat2_old_blyn.png]] --- ### Ediz Blyn (~40 km) — Reversed (Mock Data) ![[85v9_cat2_ediz_blyn.png]] --- ### Puhitampi (~53 km) — Reversed (Mock Data) ![[85v9_cat2_puhitampi.png]] --- ### Lucky Peak (~6 km) — Reversed (Mock Data) ![[85v9_cat2_lucky_peak.png]] --- ### Varley SE (~16 km) — Reversed (Mock Data) ![[85v9_cat2_varley_se.png]] --- ### North Peak (~131 km) — Reversed (Mock Data) ![[85v9_cat2_north_peak.png]]