{
  "schema_version": 1,
  "languages": {
    "zh": {
      "title": "MW 与 M31：stacking、短轴及 Galactic 坐标比较",
      "description": "14 个 XMM-Newton CGMsum 视场与 Zhang、He & Li stacking 模板的条件性比较草稿。",
      "eyebrow": "比较草稿 · 2026-10-13",
      "heading": "从 M31 短轴到 MW 天空坐标",
      "intro": [
        "观测点是 14 个主样本视场的 MW+M31 约 0.175 keV 热气体视线总分量（CGMsum），不是分离出的 M31 发射。这里采用已保存的吸收后 0.5–2.0 keV 表面亮度及局部协方差统计误差。",
        "先比较各文献的原始质量档，再检查质量缩放；第三张图使用真正的投影短轴坐标。所有曲线是条件模板，没有加入一个任意的 MW 常数来贴合数据，也没有执行新的谱拟合或成分分解。",
        "新增 MW 比较采用同一批视场的 Galactic b 和距银心角距离 ψ，并沿用已审计的逐视线 MW 模型预测。银河内的气体沿视线分布，每个观测亮度点没有唯一的三维银心距离。"
      ],
      "sections": [
        {
          "id": "native",
          "kicker": "01 · 原始质量档",
          "heading": "版本差异与群体差异分别看",
          "paragraphs": [
            "上图比较 Zhang MW-mass 的早期 v1、最终 v3，以及作者于 2026-10-08 提供的本征 β-model 后验数值表。下图并列 Zhang v3 M31-mass 与 He & Li v2 的全样本、高质量子样本。每张图都重复显示同一组真实 CGMsum 数据。",
            "作者表属于 MW-mass；蓝色带为 500 条模型后验曲线逐点的 16–84 百分位，不是星系间散布，也不包含完整径向协方差。5、15、25…kpc 网格之间采用 log-log PCHIP 插值。v1 仅用于追踪历史变化。",
            "He & Li 的两条本征曲线由 v2 Table 1 的 rc、β 和四舍五入后的 log L(10–200 kpc) 重建，因此是近似代表模型，不能称为作者的精确最佳拟合曲线。其主要约束范围为 10–50 kpc；下载表中大于 50 kpc 的点单独标为外推。",
            "内侧范围按来源区分：Zhang v3 第 2.2 节给出的原始分箱从 0–10、10–30、30–50 kpc 开始；作者更新表的 5 kpc 是本征模型求值位置，并不是一个新增、独立分辨的观测点。He & Li Figure 4 的最内数据点约在 1–2 kpc，但其 CGM 拟合主要使用 10–50 kpc，中心受恒星成分影响较大。",
            "本页每侧画到 40 kpc。已知参数的解析 β-model 从 0 kpc 起画，R<10 kpc 用点线表示模型向内延拓；对 He & Li，这段位于主要拟合范围之外。Zhang 中央分箱本身包含 0–10 kpc，但其中更细的曲线形状仍来自模型，而非独立分辨的测量。作者后验数值表只画 5–40 kpc，不在 0–5 kpc 外推曲线或误差带。"
          ],
          "figures": [
            {
              "stem": "stack_comparison_native",
              "caption": "各自原始质量档，不进行质量缩放。横轴为保留自测量表的带符号圆投影半径：NW 为正、SE 为负。误差棒为局部协方差统计误差；水平范围表示名义 15′ 视场尺度，不是统计误差或精确掩膜范围。曲线在固定 NH 下转换；观测谱各自使用其视场的 HI4PI NH。 解析曲线在 0–10 kpc 为点线；作者表从 5 kpc 开始。"
            }
          ]
        },
        {
          "id": "mass",
          "kicker": "02 · 条件性质量缩放",
          "heading": "归一化不是只选一个“典型质量”",
          "paragraphs": [
            "上图将模板按 (1.3×10^11 M☉ / M★,median)^2.4 在固定 kpc 半径上作纵向缩放。目标是 Zhang 的 M31 质量档中位数，不是本工作重新测得的 M31 恒星质量。Zhang M31 曲线因此不变，观测数据在所有图中均不缩放。",
            "指数 2.4 来自 R500c 内的积分光度关系。把它用于每个固定半径的表面亮度，是额外的共同形状假设；这里没有做 halo 半径缩放，也没有统一各论文的质量标定。不能把这组曲线当作质量匹配后重新 stacking 的测量。",
            "下图仅对 He & Li 加入等星系数的质量矩敏感性计算：因子为 M★,target^2.4 / mean(M★^2.4)。实际 stacking 的径向权重未知，这不是更准确的校正，也不是置信区间。两种结果的差异显示归一化仍然依赖样本分布和权重。"
          ],
          "figures": [
            {
              "stem": "stack_comparison_mass",
              "caption": "相同 CGMsum 数据与条件性质量缩放。上：按样本中位质量缩放；下：He & Li 中位质量缩放（虚线）与等星系数质量矩演示（实线）。作者后验带只传播原表逐点区间，不包括质量关系和谱形系统误差。不能从这张比较图直接报告某个模型被排除。 所有解析曲线的 R<10 kpc 部分统一为点线，标示内侧模型延拓；作者表仍只从 5 kpc 起画。"
            }
          ],
          "table": {
            "caption": "采用的样本质量、profile 参数和缩放因子；MW 与 M31 是 stacking 质量档标签。",
            "columns": [
              "模板",
              "M★中位数 / 10^10 M☉",
              "rc / kpc",
              "β",
              "中位质量因子",
              "等数质量矩因子"
            ],
            "rows": [
              [
                "Zhang v1 MW（历史）",
                "5.5",
                "30.0",
                "0.42",
                "7.88",
                "—"
              ],
              [
                "Zhang v3 MW",
                "5.5",
                "6.0",
                "0.43",
                "7.88",
                "—"
              ],
              [
                "Zhang 作者 MW（10 月 8 日）",
                "5.5",
                "posterior",
                "posterior",
                "7.88",
                "—"
              ],
              [
                "Zhang v3 M31 质量档",
                "13",
                "4.0",
                "0.37",
                "1",
                "—"
              ],
              [
                "He & Li v2 全样本",
                "2.41",
                "8.7",
                "0.52",
                "57.2",
                "7.94"
              ],
              [
                "He & Li v2 高质量样本",
                "5.05",
                "9.1",
                "0.55",
                "9.66",
                "4.05"
              ]
            ]
          }
        },
        {
          "id": "vertical",
          "kicker": "03 · 垂直方向",
          "heading": "用实际短轴位置，但在正确的半径计算模板",
          "paragraphs": [
            "以 M31 中心为原点作 TAN 投影，采用 D=780 kpc、盘长轴 PA=37.3°（北向东量）；短轴坐标朝 NW 为正。这是天球平面的投影坐标，没有把它去投影为气体的真实离盘高度。",
            "图中的加号是每个观测视场的模型预测：把该视场的圆投影半径 Rproj 代入球对称 β-model，再画到该视场的短轴坐标。没有把所有视场都当作严格位于短轴上，也没有用 |zproj| 代替 Rproj。",
            "这些仍是视场中心的固定 NH 模板值，而非按源掩膜、有效面积和仪器响应积分后的预测。原测量表的圆半径保持原值；新短轴坐标使用 780 kpc，两套半径的差别小于 1%。"
          ],
          "figures": [
            {
              "stem": "stack_comparison_vertical",
              "caption": "实际短轴方向上的观测与模板。加号表示在各视场 Rproj 处计算的原始质量档模板，不构成额外观测点；不同视场的模型点不连成一条虚构的垂直剖面。"
            }
          ]
        },
        {
          "id": "geometry",
          "kicker": "04 · 视场分布",
          "heading": "两侧视场并不组成完整环带",
          "paragraphs": [
            "圆环为 10、20、30 kpc 的圆投影半径；彩色小圆是名义 15′ 半径视场的示意，不包含真实相机边界和点源掩膜。NW/SE 两组共 14 个视场。"
          ],
          "figures": [
            {
              "stem": "stack_comparison_geometry",
              "caption": "同一批数据的天空投影几何。离短轴的偏移解释了为什么圆投影半径与垂直方向坐标不能直接互换。"
            }
          ]
        },
        {
          "id": "mw-coordinates",
          "kicker": "05 · MW 坐标",
          "heading": "先在观测者的天空坐标中比较 MW 模型",
          "paragraphs": [
            "盘状或扁平的 MW 热气体通常写成 n(R,z)，最自然的可视化坐标是 Galactic b，同时保留每个视场实际的 l 和 NH。球对称 halo 写成 n(r)，它预测从太阳向不同方向积分得到的亮度；合适的天空横轴是 ψ=acos[cos b cos l]，即距理想银心方向 (l,b)=(0,0) 的角距离，单位为度。ψ 不是气体到银心的 kpc 距离。",
            "设太阳位于 R☉=8.2 kpc、z☉=0，沿视线的距离为 s≥0，则 r(s)²=R☉²+s²−2R☉s cos b cos l；盘模型还使用 R(s)²=R☉²+(s cos b)²−2R☉s cos b cos l 和 z(s)=s sin b。我们的 14 个视场覆盖 l=119.51–122.30°、b=−23.17 至 −19.89°、ψ=117.54–119.72°，全都朝银心外侧看。正向视线上的最近银心半径因此均为 8.2 kpc，而无限直线的切点在太阳背后，不能把 R☉sinψ 当成这些气体的距离。",
            "加号是此前冻结审计的 Locatelli reference、Ueda disk 和 HaloSat combined 逐视场名义预测，沿用各研究的温度、丰度、谱转换及该视场 HI4PI NH，统一到吸收后 0.5–2 keV 单位。没有重新拟合或强行统一到 0.175 keV。Locatelli 从 western-sky 模型外推到 M31，HaloSat 从 b<−30° 拟合域外推到这里；Ueda 的视场在所采用的角域内。模型数值是视场中心的视线积分，并非真实相机掩膜上的二维积分，图中不画缺少后验支持的误差带，也不把不同 l、NH 的预测连成一条通用 b-only 曲线。",
            "第二幅图把 Henley & Shelton 的 86 个可转换高纬检测放到相同吸收后能段，显示其与我们的低纬视场在天空位置和亮度上的关系。这是有检测筛选的星系内天空群体背景，不是 M31 方向的 foreground prior；没有画成一条通用纬度模型。eFEDS 是另一个天空区域的谱测量，同样不适合作为整个 b 轴上的水平 profile。",
            "这次换坐标提供新的观察方式，并没有增加独立观测信息。我们只有约 3.3° 的 b 跨度和 2.2° 的 ψ 跨度，各 MW 模型在这片天空上变化较小；因此局部归一化和 patchiness 通常比全局形状更重要。密度或吸收柱模型 n(r)、N(O VII) 必须先预测同一视线的发射（S∝∫n_e n_H Λ[T,Z] ds，并处理吸收），才能与这些亮度点比较。"
          ],
          "figures": [
            {
              "stem": "stack_comparison_mw_coordinates",
              "caption": "同一组 MW+M31 CGMsum 数据，分别改用 Galactic b 和距银心角距离 ψ。加号为三种已有 MW 模型在实际 (l,b,NH) 下的名义预测，不能当作独立测量。水平棒仅表示名义 15′ 视场半径。不同模型保留各自的原生谱转换；星系内 MW halo/disk 和外部 M31 stack 采用各自的几何。"
            },
            {
              "stem": "stack_comparison_mw_population",
              "caption": "Henley & Shelton 2013 的 86 个高纬可转换检测（灰点）与我们的 MW+M31 总分量（蓝/红）按真实 Galactic b 并列。灰点没有在 M31 方向重投影，其离散不是我们的观测误差或当地 MW 先验；蓝色竖带显示本项目的纬度覆盖。"
            }
          ],
          "links": [
            {
              "label": "Locatelli et al. 2024",
              "url": "https://arxiv.org/abs/2310.10715"
            },
            {
              "label": "Ueda et al. 2022",
              "url": "https://arxiv.org/abs/2209.01698"
            },
            {
              "label": "Kaaret et al. 2020 / HaloSat",
              "url": "https://arxiv.org/abs/2011.00126"
            },
            {
              "label": "Henley & Shelton 2013",
              "url": "https://arxiv.org/abs/1306.2312"
            }
          ],
          "table": {
            "caption": "横坐标应跟随模型的原生几何和实际观测量。",
            "columns": [
              "资料或模型",
              "原生坐标 / 观测量",
              "与我们的亮度如何比较"
            ],
            "rows": [
              [
                "MW 盘 / 扁平模型",
                "三维 R,z；天空 l,b",
                "先沿实际 (l,b) 视线积分，再以 b 展示并保留 l、NH。"
              ],
              [
                "MW 球形 halo 模型",
                "三维 r；视线发射积分",
                "先从太阳位置投影为 S(l,b)，可用 ψ 作天空横轴；观测点不直接放在 r 上。"
              ],
              [
                "Locatelli 2024",
                "O VIII 天空图；disk + halo",
                "沿用 O VIII 归一化桥接到宽能段；M31 在原拟合经度域之外。"
              ],
              [
                "Ueda 2022 / HaloSat 2020",
                "各方向 EM；n(R,z) 或 disk+halo",
                "沿用已有 14 条视线预测；HaloSat 在原南天高纬域之外。"
              ],
              [
                "Henley & Shelton 2013",
                "高纬每条视线的热气体亮度",
                "按实际 b 或 ψ 作散点群体对照；不能当 M31 方向的径向 profile。"
              ],
              [
                "eFEDS / MW 吸收柱研究",
                "天空 patch 的谱 / ∫n f_ion ds",
                "分别保留天空区域与观测量；密度、吸收柱和发射亮度需要模型转换。"
              ]
            ]
          }
        },
        {
          "id": "interpretation",
          "kicker": "06 · 如何使用",
          "heading": "内晕形状优先参考近距离样本，幅度仍需单独约束",
          "paragraphs": [
            "He & Li 样本在 15–50 Mpc；Yi 给出的 Zhang 样本中位距离约 400 Mpc。以 30″ 作小角尺度估算，两者约为 2–7 kpc 与 58 kpc。He & Li 对 10–30 kpc 形状有更直接的分辨能力；Zhang 的本征内晕曲线仍依赖 β-model 和点源处理。",
            "共同纵轴的转换为 Sabs = ΣX × [π/(180×60)]² / [4π × (1 kpc in cm)²] × 0.678988916。吸收比来自已保存的项目 APEC 模板：kT=0.175306 keV、Z=0.3 Z☉、NH=5.6×10^20 cm−2、angr 丰度表和 vern 截面。",
            "这是统一谱形下的显示转换；它没有重新拟合文献计数谱，特别是 He & Li 原本采用的温度分布。数据的总分量也不能在两种温度相近的组分之间唯一分配。模板高于某些总量点，是需要检查的条件性张力；不能据此宣称负的 MW 或已经分离出 M31。",
            "正式的 MW/M31 约束还需要真实掩膜上的模板积分、逐视场吸收与响应、MW 空间模型，以及质量和谱形系统误差。当前页面提供可审阅的比较，不给出新成分检测或排除显著性。"
          ],
          "links": [
            {
              "label": "Zhang 2024 I：早期 v1",
              "url": "https://arxiv.org/html/2401.17308v1"
            },
            {
              "label": "Zhang 2024 I：最终 v3",
              "url": "https://arxiv.org/html/2401.17308v3"
            },
            {
              "label": "Zhang 2024 II：积分光度–质量关系",
              "url": "https://arxiv.org/abs/2401.17309v3"
            },
            {
              "label": "He & Li 2026 v2",
              "url": "https://arxiv.org/html/2601.16499v2"
            },
            {
              "label": "He & Li 公开样本",
              "url": "https://zenodo.org/records/21216062"
            }
          ]
        }
      ],
      "downloads": [
        {
          "path": "assets/data/stack_comparison_fields.csv",
          "label": "14 个观测视场与两种坐标"
        },
        {
          "path": "assets/data/stack_comparison_models.csv",
          "label": "六个模板与质量因子"
        },
        {
          "path": "assets/data/stack_comparison_curves.csv",
          "label": "连续曲线、场景及作者逐点区间"
        },
        {
          "path": "assets/data/stack_comparison_field_predictions.csv",
          "label": "各视场中心的模板值"
        },
        {
          "path": "assets/data/stack_comparison_summary.json",
          "label": "计算约定、输入哈希与数值检查"
        },
        {
          "path": "assets/data/stack_comparison_mw_fields.csv",
          "label": "MW 坐标与原观测值"
        },
        {
          "path": "assets/data/stack_comparison_mw_predictions.csv",
          "label": "逐视场 MW 模型预测"
        },
        {
          "path": "assets/data/stack_comparison_mw_population.csv",
          "label": "H&S 高纬群体对照"
        },
        {
          "path": "assets/data/stack_comparison_mw_summary.json",
          "label": "MW 比较的来源与边界"
        }
      ]
    },
    "en": {
      "title": "MW and M31: stacked profiles and Galactic-coordinate comparisons",
      "description": "A conditional draft comparison of 14 XMM-Newton CGMsum fields with Zhang and He & Li stacking profiles.",
      "eyebrow": "COMPARISON DRAFT · 2026-10-13",
      "heading": "M31 profiles and the Milky Way foreground along our sightlines",
      "intro": [
        "The 14 primary data points measure the combined cool line-of-sight component, CGMsum = MW + M31. They are not a separated M31 measurement. We use the saved absorbed 0.5–2.0 keV surface brightnesses and local-covariance statistical errors.",
        "Compare the native sample masses first, then inspect conditional mass scaling. The third figure uses the actual projected minor-axis coordinate. Every curve is a conditional template; no arbitrary MW pedestal was fitted and no new spectral fit or component decomposition was performed.",
        "The added Milky Way comparison uses Galactic latitude b and angular separation from the Galactic centre, psi, for the same fields and retains the previously audited MW sightline predictions. Foreground gas is distributed along each sightline, so an observed brightness has no unique three-dimensional Galactocentric radius."
      ],
      "sections": [
        {
          "id": "native",
          "kicker": "01 · NATIVE SAMPLE MASSES",
          "heading": "Distinguish revisions from population differences",
          "paragraphs": [
            "The upper panel compares the historical Zhang MW-mass v1 model, the final v3 model, and the intrinsic beta-model posterior table supplied by the author on 2026-10-08. The lower panel compares the Zhang v3 M31-mass bin with the He & Li v2 full and high-mass samples. The same CGMsum measurements appear in every panel.",
            "The author table is for MW-mass galaxies. Its blue band contains the pointwise 16th–84th percentiles of 500 posterior model curves, not galaxy-to-galaxy scatter or a full radial covariance. Log-log PCHIP interpolates the 5, 15, 25… kpc grid. The v1 curve is historical context.",
            "The intrinsic He & Li curves are approximate representative reconstructions from the marginal rc, beta, and rounded log L(10–200 kpc) in v2 Table 1. They are not exact author best-fit curves. The main fitting interval is 10–50 kpc; downloadable values beyond 50 kpc are flagged as extrapolation.",
            "The inner radial coverage differs by source. Section 2.2 of Zhang v3 lists native bins beginning at 0–10, 10–30 and 30–50 kpc. The 5 kpc entry in the updated author table is an intrinsic-model evaluation, not an additional independently resolved measurement. The innermost data point in He & Li Figure 4 is at roughly 1–2 kpc, but the CGM fit mainly uses 10–50 kpc; stellar emission is important closer to the centre.",
            "Each side is plotted out to 40 kpc. Analytic beta models with published parameters extend to 0 kpc, with dotted lines below 10 kpc marking inward model continuation. For He & Li this lies outside the main fitting range. Although the central Zhang bin covers 0–10 kpc, finer structure within it still comes from the model rather than independently resolved measurements. The author posterior table is shown only over 5–40 kpc; neither the curve nor its uncertainty band is extrapolated into 0–5 kpc."
          ],
          "figures": [
            {
              "stem": "stack_comparison_native",
              "caption": "Native sample masses, without mass rescaling. The signed circular projected radius is retained from the measurement table: NW positive, SE negative. Vertical bars are local-covariance statistical errors. Horizontal extents show a nominal 15-arcmin field radius, not statistical errors or exact masks. Templates use a fixed NH; the observed spectra used their individual HI4PI columns. Analytic curves are dotted over 0–10 kpc; the author table begins at 5 kpc."
            }
          ]
        },
        {
          "id": "mass",
          "kicker": "02 · CONDITIONAL MASS SCALING",
          "heading": "Normalization depends on the sample and its weights",
          "paragraphs": [
            "The upper panel multiplies each template by (1.3×10^11 solar masses / median stellar mass)^2.4 at fixed physical radius. The target is the Zhang M31-bin median, not a new stellar-mass measurement for M31. The native Zhang M31 curve is unchanged. Observed data are never rescaled.",
            "The exponent 2.4 describes luminosity integrated within R500c. Applying it to surface brightness at every fixed radius requires an additional common-shape assumption. Halo radii are not dilated and stellar-mass calibrations are not homogenized. These curves are not measurements from a mass-matched re-stack.",
            "The lower panel adds a He & Li sensitivity calculation with equal galaxy weights: target_mass^2.4 / mean(stellar_mass^2.4). Actual radial stacking weights are unknown. This is neither a preferred correction nor a confidence interval; the difference illustrates the dependence on mass distribution and weighting."
          ],
          "figures": [
            {
              "stem": "stack_comparison_mass",
              "caption": "The same CGMsum data with conditional mass scaling. Top: median-mass factors. Bottom: He & Li median factors (dashed) and equal-number mass-moment illustrations (solid). The author band propagates only its pointwise interval, without mass-scaling or spectral systematics. This display does not establish a model-exclusion significance. All analytic curves use dotted lines below 10 kpc to mark inward model continuation; the author table still begins at 5 kpc."
            }
          ],
          "table": {
            "caption": "Adopted sample masses, profile parameters, and scale factors. MW and M31 denote stacking mass bins.",
            "columns": [
              "Template",
              "Median M★ / 10^10 solar masses",
              "rc / kpc",
              "Beta",
              "Median factor",
              "Equal-number moment factor"
            ],
            "rows": [
              [
                "Zhang v1 MW (historical)",
                "5.5",
                "30.0",
                "0.42",
                "7.88",
                "—"
              ],
              [
                "Zhang v3 MW",
                "5.5",
                "6.0",
                "0.43",
                "7.88",
                "—"
              ],
              [
                "Zhang author MW (Oct 8)",
                "5.5",
                "posterior",
                "posterior",
                "7.88",
                "—"
              ],
              [
                "Zhang v3 M31 bin",
                "13",
                "4.0",
                "0.37",
                "1",
                "—"
              ],
              [
                "He & Li v2 full",
                "2.41",
                "8.7",
                "0.52",
                "57.2",
                "7.94"
              ],
              [
                "He & Li v2 high mass",
                "5.05",
                "9.1",
                "0.55",
                "9.66",
                "4.05"
              ]
            ]
          }
        },
        {
          "id": "vertical",
          "kicker": "03 · VERTICAL DIRECTION",
          "heading": "Use the actual minor-axis position and the correct model radius",
          "paragraphs": [
            "We use a TAN projection about M31 with D=780 kpc and major-axis PA=37.3 degrees east of north. The projected minor coordinate is positive toward NW. It is a sky-plane coordinate, not a deprojected gas height above the disk.",
            "Each model cross is evaluated at that field’s circular projected radius Rproj and displayed at its actual minor-axis coordinate. Fields are not assumed to lie exactly on the minor axis, and absolute minor-axis distance does not replace Rproj.",
            "Predictions are fixed-NH field-center values, not integrals over source masks, effective area, and response. Saved circular radii are preserved; new minor-axis coordinates adopt 780 kpc. The two radius conventions differ by less than one percent."
          ],
          "figures": [
            {
              "stem": "stack_comparison_vertical",
              "caption": "Observed data and native-mass templates at the actual projected minor-axis positions. Crosses are model evaluations at each field’s Rproj, not additional measurements. They are not joined into an artificial continuous vertical profile."
            }
          ]
        },
        {
          "id": "geometry",
          "kicker": "04 · FIELD GEOMETRY",
          "heading": "The two chains do not form a complete annulus",
          "paragraphs": [
            "The rings mark circular projected radii of 10, 20, and 30 kpc. Colored circles illustrate nominal 15-arcmin field radii, without actual camera boundaries or source masks. The NW/SE groups contain 14 fields in total."
          ],
          "figures": [
            {
              "stem": "stack_comparison_geometry",
              "caption": "Sky-plane geometry of the same measurements. Offsets from the minor axis explain why circular projected radius and the vertical coordinate are not interchangeable."
            }
          ]
        },
        {
          "id": "mw-coordinates",
          "kicker": "05 · MW coordinates",
          "heading": "Compare Milky Way models in the observer’s sky coordinates first",
          "paragraphs": [
            "Disk-like or flattened MW gas is usually described by n(R,z). Galactic latitude b is a useful display coordinate while retaining each field’s actual longitude l and NH. A spherical halo n(r) predicts brightness integrated from the Solar position along a sightline; a useful sky coordinate is psi=acos[cos b cos l], the angular separation from the ideal Galactic-centre direction (l,b)=(0,0), in degrees. Psi is not the gas’s Galactocentric distance in kpc.",
            "With the Sun at R_sun=8.2 kpc and z_sun=0, and distance s>=0 along the sightline, r(s)^2=R_sun^2+s^2−2 R_sun s cos b cos l. Disk models additionally use R(s)^2=R_sun^2+(s cos b)^2−2 R_sun s cos b cos l and z(s)=s sin b. Our 14 fields span l=119.51–122.30 deg, b=−23.17 to −19.89 deg and psi=117.54–119.72 deg, all looking outward from the Galactic centre. The minimum Galactocentric radius on each forward ray is therefore 8.2 kpc; the infinite-line tangent point lies behind the Sun, so R_sun sin(psi) cannot be assigned as the gas distance.",
            "The crosses retain the previously audited nominal per-field predictions from the Locatelli reference, Ueda disk and HaloSat combined models, including each study’s temperature, abundance and spectral conversion and each field’s HI4PI NH, in absorbed 0.5–2 keV units. Nothing was refitted or forced to 0.175 keV. Locatelli is extrapolated from its western-sky fit and HaloSat from its b<−30 deg fit domain; the Ueda fields lie within the adopted angular selection. These are field-centre sightline integrations, not integrations over the actual masked camera footprints. No unsupported posterior bands are drawn, and predictions at different l and NH are not connected as a universal b-only profile.",
            "The second figure places the 86 convertible high-latitude detections from Henley & Shelton beside our lower-latitude fields in the same absorbed energy band. This is detection-selected population context, not a foreground prior toward M31 or a universal latitude model. The eFEDS spectrum comes from another sky patch and likewise should not become a horizontal profile across the whole b axis.",
            "A coordinate change offers another view without adding independent observations. Our span is only about 3.3 deg in b and 2.2 deg in psi, over which the MW predictions vary little; local normalization and patchiness are generally more consequential than global profile shape. Density or absorption-column models n(r) and N(O VII) must first predict the corresponding sightline emission, S proportional to the integral of n_e n_H Lambda[T,Z] along s, including absorption, before being compared with these brightness measurements."
          ],
          "figures": [
            {
              "stem": "stack_comparison_mw_coordinates",
              "caption": "The same MW+M31 CGMsum measurements displayed against Galactic latitude b and angular separation from the Galactic centre, psi. Crosses are nominal predictions of three existing MW models at the actual (l,b,NH), not independent measurements. Horizontal bars indicate only the nominal 15 arcmin field radius. Each model retains its original spectral transfer and appropriate internal-MW geometry."
            },
            {
              "stem": "stack_comparison_mw_population",
              "caption": "The 86 convertible high-latitude Henley & Shelton 2013 detections (grey) and our MW+M31 total component (blue/red) at their actual Galactic latitudes. Grey points have not been relocated toward M31; their scatter is neither our measurement error nor a local MW prior. The blue vertical strip marks our latitude coverage."
            }
          ],
          "links": [
            {
              "label": "Locatelli et al. 2024",
              "url": "https://arxiv.org/abs/2310.10715"
            },
            {
              "label": "Ueda et al. 2022",
              "url": "https://arxiv.org/abs/2209.01698"
            },
            {
              "label": "Kaaret et al. 2020 / HaloSat",
              "url": "https://arxiv.org/abs/2011.00126"
            },
            {
              "label": "Henley & Shelton 2013",
              "url": "https://arxiv.org/abs/1306.2312"
            }
          ],
          "table": {
            "caption": "Choose the coordinate to match the native model geometry and observable.",
            "columns": [
              "Data or model",
              "Native coordinates / observable",
              "Comparison with our brightness"
            ],
            "rows": [
              [
                "MW disk / flattened model",
                "Three-dimensional R,z; sky l,b",
                "Integrate along each actual (l,b), then display against b while retaining l and NH."
              ],
              [
                "MW spherical halo model",
                "Three-dimensional r; sightline emission",
                "Project from the Sun to S(l,b); psi is a useful sky axis. Do not assign an observed brightness a unique r."
              ],
              [
                "Locatelli 2024",
                "O VIII sky map; disk + halo",
                "Retain the O VIII normalization bridge to the broad band; M31 lies outside the fitted longitude domain."
              ],
              [
                "Ueda 2022 / HaloSat 2020",
                "Directional EM; n(R,z) or disk+halo",
                "Retain audited predictions for the 14 sightlines; HaloSat is outside its original southern high-latitude domain."
              ],
              [
                "Henley & Shelton 2013",
                "High-latitude sightline thermal brightness",
                "Display population points at actual b or psi; not a radial foreground profile toward M31."
              ],
              [
                "eFEDS / MW absorption studies",
                "Sky-patch spectrum / integral n f_ion ds",
                "Retain sky region and observable; density, absorption columns and emission brightness require model conversion."
              ]
            ]
          }
        },
        {
          "id": "interpretation",
          "kicker": "06 · Interpretation",
          "heading": "Nearby stacks better resolve inner shape; amplitude remains conditional",
          "paragraphs": [
            "The He & Li galaxies lie at 15–50 Mpc; Yi quoted about 400 Mpc for the median distance of her stack. A 30-arcsec angular scale corresponds approximately to 2–7 versus 58 kpc. He & Li more directly resolve the 10–30 kpc shape. The intrinsic Zhang inner profile still depends on the beta model and compact-source treatment.",
            "The common ordinate uses Sabs = SigmaX × [pi/(180×60)]^2 / [4pi × (1 kpc in cm)^2] × 0.678988916. This absorption ratio is from the saved project APEC template: kT=0.175306 keV, Z=0.3 solar, NH=5.6×10^20 cm−2, angr abundances, and vern cross-sections.",
            "This is a common-spectrum display conversion. It does not refit the literature count spectra, including the temperature distribution adopted by He & Li. The measured total cannot uniquely be allocated to two similarly cool components. A template above some total measurements indicates conditional tension to investigate; it does not imply negative MW emission or a separated M31 detection.",
            "A formal MW/M31 constraint requires integration over the actual masks, field-specific absorption and responses, a spatial MW model, and mass/spectral systematics. This page is a reviewable comparison, with no new component detection or exclusion significance."
          ],
          "links": [
            {
              "label": "Zhang 2024 I: historical v1",
              "url": "https://arxiv.org/html/2401.17308v1"
            },
            {
              "label": "Zhang 2024 I: final v3",
              "url": "https://arxiv.org/html/2401.17308v3"
            },
            {
              "label": "Zhang 2024 II: integrated luminosity–mass scaling",
              "url": "https://arxiv.org/abs/2401.17309v3"
            },
            {
              "label": "He & Li 2026 v2",
              "url": "https://arxiv.org/html/2601.16499v2"
            },
            {
              "label": "He & Li public sample",
              "url": "https://zenodo.org/records/21216062"
            }
          ]
        }
      ],
      "downloads": [
        {
          "path": "assets/data/stack_comparison_fields.csv",
          "label": "14 observed fields and both coordinates"
        },
        {
          "path": "assets/data/stack_comparison_models.csv",
          "label": "Six templates and mass factors"
        },
        {
          "path": "assets/data/stack_comparison_curves.csv",
          "label": "Curves, scenarios, and author pointwise intervals"
        },
        {
          "path": "assets/data/stack_comparison_field_predictions.csv",
          "label": "Field-center template evaluations"
        },
        {
          "path": "assets/data/stack_comparison_summary.json",
          "label": "Conventions, input hashes, and numerical checks"
        },
        {
          "path": "assets/data/stack_comparison_mw_fields.csv",
          "label": "MW coordinates and original measurements"
        },
        {
          "path": "assets/data/stack_comparison_mw_predictions.csv",
          "label": "MW predictions at each field"
        },
        {
          "path": "assets/data/stack_comparison_mw_population.csv",
          "label": "H&S high-latitude population"
        },
        {
          "path": "assets/data/stack_comparison_mw_summary.json",
          "label": "MW comparison provenance and assumptions"
        }
      ]
    }
  }
}