MW参考模型如何进入M31 Figure 3

01 · measurement before decomposition

先问“测到了什么”，再问“属于谁”

Ask what was measured before assigning it

14个 primary fields 给出的是约0.18 keV phenomenological thermal component 的 absorbed surface brightness。当前 fit 中该分量没有独立的 MW/M31 distance tag。

The 14 primary fields measure the absorbed surface brightness of an approximately 0.18 keV phenomenological thermal component. The fit does not attach an independent MW/M31 distance tag.

14 fields

quality-selected dual-MOS primary sample；不是全部22个 pointings。

Quality-selected dual-MOS primary sample, not all 22 pointings.

0.5–2.0 keV

Figure 3及本站所有一致性检验统一使用absorbed标准比较band。

Figure 3 and all consistency tests on this page use the absorbed standard comparison band.

CGMsum

是数据产品名，不是纯 MW，也不是直接测得的 M31 contribution。

A data-product name: neither pure MW nor a directly measured M31 contribution.

为什么Figure 3改用0.5–2.0 keV？Why does Figure 3 use 0.5–2.0 keV?

0.5–2.0 keV是常见的跨样本文献比较band，因此D42将Figure 3及相连的一致性检验统一到这一convention。它不是v19的直接component-sensitive primary band：其中1.4–2.0 keV跨越了被排除的MOS仪器谱线区间，因此这部分依赖模型外推。直接基线仍是0.4–1.25 keV；对kT=0.175–0.20 keV、Z=0.1–0.3、NH=(4–8)×10²⁰ cm⁻²的APEC grid，该band保留0.4–2.0 absorbed model flux的98.8–99.5%。

0.5–2.0 keV is the common cross-sample literature band, so D42 uses that convention for Figure 3 and its linked consistency diagnostics. It is not the direct component-sensitive primary band of v19: the 1.4–2.0 keV part crosses the excluded MOS instrumental-line interval and therefore depends on model extrapolation. The direct baseline remains 0.4–1.25 keV, which retains 98.8–99.5% of the absorbed 0.4–2.0 model flux across the audited kT=0.175–0.20 keV, Z=0.1–0.3, NH=(4–8)×10²⁰ cm⁻² APEC grid.

外部reference并不共享同一原始observable：Locatelli拟合0.614–0.694 keV O VIII窄带；Zhang使用rest-frame 0.5–2.0 keV；Grayson的emissivity图和比较语境支持intended 0.5–2.0 keV，但profile方法段没有明写event cut。本站先在各自原始band重建，再显式转换到absorbed 0.5–2.0 convention。旧0.4–1.25值仍作为v19 component-sensitive provenance保留，但不再用于Figure 3的一致性比较。

The external references do not share one native observable: Locatelli fits the 0.614–0.694 keV O VIII narrow band; Zhang uses rest-frame 0.5–2.0 keV; and Grayson's emissivity figure and comparison context support an intended 0.5–2.0 keV band, although its profile-method paragraph does not state the event cut explicitly. This site reconstructs each native observable before explicitly converting it to an absorbed 0.5–2.0 convention. The old 0.4–1.25 values remain available as v19 component-sensitive provenance but are no longer used in Figure 3's consistency comparison.

禁止的读法：Forbidden reading: 把 Panel C 的14个点直接叫作“M31 surface-brightness profile”。它们必须标成 observed CGMsum versus M31 projected radius。 Calling the 14 points an “M31 surface-brightness profile.” They are observed CGMsum versus M31 projected radius.

02 · three different coordinates

三个横坐标，回答三个不同问题

Three horizontal coordinates answer three questions

把它们混在一起会把 MW latitude dependence、M31-centric radius 和不规则 XMM footprint 误当成同一种“径向变化”。

Mixing them would turn MW latitude dependence, M31-centric radius, and an irregular XMM footprint into one misleading “radial trend.”

|b| at l=l_M31

MW 纬度切片

MW latitude slice

固定 Galactic longitude，改变南半球 b=−|b|。这是 internal-observer sky slice，不是 MW 三维半径。

Hold Galactic longitude fixed and vary southern b=−|b|. This is an internal-observer sky slice, not a 3D MW radius.

R_∥

M31→Galactic-center 天球轴

M31→Galactic-center sky axis

以 M31 为零点；正值指向 Galactic center，负值背离。R_∥=D_M31tanθ。

Zero at M31; positive toward the Galactic center, negative away. R_∥=D_M31tanθ.

R_proj,M31

M31 圆形投影半径

M31 circular projected radius

用于放置14个观测 CGMsum points 和外部 M31-mass conditional templates；不携带 MW/M31 decomposition。

Places the 14 observed CGMsum points beside external M31-mass conditional templates; it carries no MW/M31 decomposition.

概念图，不按比例。左侧的(l,b)属于 MW internal-observer geometry；右侧的 R∥ 和 Rproj 都以 M31 为原点，但一个有方向符号，一个是非负圆形距离。

Conceptual and not to scale. The left-hand (l,b) belongs to the MW internal-observer geometry. On the right, R∥ and Rproj both originate at M31, but one is signed and directional while the other is a non-negative circular distance.

M31中心采用 (l,b)=(121.174329°,−21.573309°)，D_M31=780 kpc。M31 指向 Galactic center 的 ICRS position angle 为255.377112° east of north。The M31 center is (l,b)=(121.174329°,−21.573309°), with D_M31=780 kpc. The ICRS position angle from M31 to the Galactic center is 255.377112° east of north.

03 · model → observable

Figure 3 前面的完整转换链

The complete conversion before Figure 3

橙色 band 不是从论文表格抄来的 broadband prior。它是把一个 O VIII-constrained MW density geometry 投影到14个真实 sightlines，再固定O VIII line normalization、转换到Z=0.3 APEC的absorbed 0.5–2.0 keV broadband flux，并应用每场 HI4PI absorption 得到的。

The orange band is not a broadband prior copied from a table. It projects an O VIII-constrained MW density geometry through 14 real sightlines, holds the O VIII line normalization fixed, converts to the absorbed 0.5–2.0 keV broadband flux of a Z=0.3 APEC model, and applies field-specific HI4PI absorption.

n_h, n_dspherical halo + exponential diskspherical halo + exponential disk

∫(n_h²+n_d²)ds逐 sightline emission measureemission measure by sightline

O VIII L.U.Locatelli 0.614–0.694 keV observableLocatelli 0.614–0.694 keV observable

f_line,Z=0.3=0.352343固定O VIII line normalizationfixed O VIII line normalization

phabs × APECabsorbed 0.5–2.0 keV surface brightnessabsorbed 0.5–2.0 keV surface brightness

采用 named reference geometry

Adopt the named reference geometry

使用 Locatelli et al. reference Combined β=0.5，而不是旧 combined+SWCX systematics branch。

Use the Locatelli et al. reference Combined β=0.5 model, not the old combined+SWCX systematics branch.

n_h(r)=C r^−3β ; n_d(R,z)=n₀ exp(−R/R_h) exp(−|z|/z_h)

β	C	n₀	R_h	z_h	kT	Z
0.5	0.046	0.032 cm⁻³	6.2 kpc	1.1 kpc	0.15 keV	0.1 Z_⊙

从太阳位置沿每条视线积分

Integrate each sightline from the Solar position

R²(s)=R₀²+(s cos b)²−2R₀s cos b cos l ; z=s sin b ; r=(R²+z²)^1/2

取 R₀=8.2 kpc，s=0.001–350 kpc。实现积分的是 n_h²+n_d²；没有使用 (n_h+n_d)²，因此没有人为加入2n_hn_d cross term。

Use R₀=8.2 kpc and s=0.001–350 kpc. The implementation integrates n_h²+n_d², not (n_h+n_d)², so it does not insert a 2n_hn_d cross term.

先回到原模型实际拟合的 O VIII observable

Return first to the fitted O VIII observable

I_{O VIII} = EM × ε_Loc / 4π ; ; ε_Loc=1.649×10⁻¹⁶ ph cm³ s⁻¹

14个 reference sightlines 的 intrinsic O VIII intensity 约4.14–4.56 L.U.。这里还没有应用 photoelectric absorption，也还不是 broadband energy flux。

The 14 reference sightlines have intrinsic O VIII intensities of about 4.14–4.56 L.U. This is before photoelectric absorption and is not yet a broadband energy flux.

以约80 eV FWHM Gaussian smoothing匹配原始map能量分辨

Match the map's approximate 80-eV FWHM with Gaussian smoothing

σ_E = 0.080 keV / [2(2 ln 2)^1/2] = 0.03397 keV

f_line,Z=0.3=ε_Loc/ε_{APEC,smoothed,Z=0.3}=0.352343

Figure 3把Locatelli的published O VIII line normalization固定，再用Z=0.3 target APEC转换到宽带。80 eV smoothing必须在窄带emissivity匹配前应用；跳过它会错误改变line-to-broadband normalization。这里没有使用真实 MOS RMF/ARF，因此不是 detector response-folded spectrum。

Figure 3 holds the published Locatelli O VIII line normalization fixed and converts it to a Z=0.3 target APEC broadband flux. The 80-eV smoothing is applied before matching narrow-band emissivities; omitting it would change the line-to-broadband normalization incorrectly. No real MOS RMF/ARF is used, so this is not a detector-response-folded spectrum.

逐 pointing 应用 HI4PI absorption 和 v19 band

Apply HI4PI absorption and the v19 band by pointing

S_direct,i = Λ_abs,i EM_i (1 kpc in cm) / [4π × (arcmin² per sr) × 10⁻¹⁵]

F_ref,i=f_line,Z=0.3 × F[phabs(N_H,i) × APEC(kT=0.15,Z=0.3)]_{0.5–2.0 keV}

其中1 sr=(180×60/π)² arcmin²，最后除以10⁻¹⁵得到网页和 Figure 3 的units。采用 Anders & Grevesse abundance、Verner cross-sections。连续 profile 曲线只使用明确标注的固定 N_H；只有14个实测 footprint points 使用各自 HI4PI column。

Here 1 sr=(180×60/π)² arcmin², and division by 10⁻¹⁵ gives the units used on this page and in Figure 3. We use Anders & Grevesse abundances and Verner cross-sections. Continuous profile curves use an explicitly labeled fixed N_H; only the 14 real footprint points use their individual HI4PI columns.

最后才计算 Figure 3 estimators

Only then compute the Figure 3 estimators

field min–max、North/South inverse-variance means、side-balanced central 和 all-field inverse-variance central 是四个不同 summaries，不能互换。

Field min–max, North/South inverse-variance means, the side-balanced central, and the all-field inverse-variance central are four different summaries and are not interchangeable.

04 · fixed-longitude latitude slice

MW reference 随 |b| 怎样变化？

How does the MW reference vary with |b|?

固定 l=l_M31=121.174329°，沿南半球 b=−|b| 改变视线。这里用固定 N_H=0.056×10²² cm⁻²，目的是单独观察 density geometry，不是假装拥有连续 HI4PI map。

Hold l=l_M31=121.174329° and vary the southern sightline b=−|b|. The fixed N_H=0.056×10²² cm⁻² isolates density geometry; it does not pretend to use a continuous HI4PI map.

MW reference surface brightness versus absolute Galactic latitude — Halo、disk 和 total 分开显示。低 |b| 增强不是 M31 radial signal，而是我们从 MW 内部观察 disk+halo geometry 的结果；极区轻微回升同样是 named reference geometry 的性质。

读取几个具体纬度值Read several latitude checkpoints

\|b\| (deg)	MW absorbed 0.5–2.0 keV
10.0000	1.930
20.0000	1.336
21.5000 (nearest M31 grid point)	1.284
30.0000	1.073
45.0000	0.892
60.0000	0.823
90.0000	0.855

单位均为10⁻¹⁵ erg cm⁻² s⁻¹ arcmin⁻²，固定 N_H=0.056×10²² cm⁻²。90°附近的轻微回升是该 named geometry 的性质，不是观测。

Units are 10⁻¹⁵ erg cm⁻² s⁻¹ arcmin⁻² at fixed N_H=0.056×10²² cm⁻². The mild upturn near 90° is a property of the named geometry, not an observation.

为什么用 |b| 仍要写 b=−|b|？Why specify b=−|b| if the axis is |b|?

density geometry 对 z 使用 |z|，但真实 Galactic absorption 和其他 foreground 并不保证南北对称。本页连续曲线固定在 M31 所在的 southern branch；若以后接入连续 HI4PI map，必须分别计算正负 b，而不能只靠 |b|。

The density geometry uses |z|, but real Galactic absorption and other foregrounds need not be north–south symmetric. The continuous curve stays on M31's southern branch. A future continuous HI4PI version must compute positive and negative b separately.

05 · signed sky-plane axis

沿 M31→Galactic center，MW reference 几乎不变

The MW reference is nearly flat along M31→Galactic center

这条轴不是 MW Galactocentric radius。它只是穿过 M31 的天球切线方向，用来检验在 M31 的10–30 kpc角尺度内，MW geometry 是否会制造明显梯度。

This is not a MW Galactocentric radius. It is the tangent-plane direction through M31 used to test whether MW geometry creates a substantial gradient across M31's 10–30 kpc angular scale.

1.2814M31中心，固定NHAt M31, fixed NH

1.2828+30 kpc toward GC+30 kpc toward GC

0.12%0→30 kpc 变化0→30 kpc change

255.377112°ICRS position angleICRS position angle

MW reference along the signed M31 to Galactic-center sky axis — 连续线是固定NH的几何曲线；14个 field markers 使用各自坐标，并将实际 HI4PI absorbed prediction 与 fixed-NH geometry-only prediction 分开。正值指向 Galactic center。

结论不是“MW foreground 完全均匀”，而是：在这个 named reference geometry 中，M31附近±几度的连续 density gradient 很弱。实际 pointing 间变化仍会受到 l、b 和 N_H 的共同影响。The conclusion is not “the MW foreground is perfectly uniform.” It is that the continuous density gradient is weak across a few degrees around M31 in this named reference geometry. Actual pointings still vary jointly with l, b, and N_H.

06 · data versus conditional templates

10–30 kpc：观测 CGMsum 与 M31 条件模板

10–30 kpc: observed CGMsum and conditional M31 templates

横坐标现在改为 circular M31 projected radius。点是数据；曲线和横带是外部假设。它们可以比较，但不能在图例中混成同一种 measurement。

The horizontal coordinate is now circular M31 projected radius. Points are data; curves and horizontal bands are external assumptions. They can be compared, but not merged into one kind of measurement.

Observed CGMsum versus M31 projected radius with conditional M31 templates — 14个 points 使用 local-covariance y errors，并按 North/NW 与 South/SE 区分；horizontal bars 表示 nominal 15′ aperture 的±3.403 kpc径向范围，不是统计误差或 exact camera-mask-weighted radius。Zhang 是归一化到20 kpc的 M31-mass population stack 条件模板；Grayson values 只表示10–30 kpc bin averages，因此画成横带而不是 radial curves。

如果做减法：If a subtraction is shown: S_CGMsum−S_MW,Locatelli 只能叫 模型张力残差 / model-tension residual。负值表示 model+conversion 与 measured total 不相容，绝不是负的物理 M31 emission。S_CGMsum−S_MW,Locatelli is only a model-tension residual. A negative value signals incompatibility between model+conversion and the measured total, never negative physical M31 emission.

Zhang profile 的数值是什么？What are the Zhang-profile values?

采用 r_c=7.39 kpc、β=0.37，并令 absorbed 0.5–2.0 keV S(20 kpc)=0.8285。条件曲线在10/15/20/25/30 kpc约为1.600/1.114/0.828/0.648/0.527。它是 population template，不是本项目从14个 XMM spectra 单独恢复的 M31 profile。

Use r_c=7.39 kpc and β=0.37 with absorbed 0.5–2.0 keV S(20 kpc)=0.8285. The conditional curve is approximately 1.600/1.114/0.828/0.648/0.527 at 10/15/20/25/30 kpc. It is a population template, not an M31 profile recovered independently from the 14 XMM spectra.

07 · MW-prior consistency test

Figure 3检验什么：不同MW估计与视线总和是否一致？

What Figure 3 tests: are different MW estimates consistent with the line-of-sight sum?

Figure 3 的横轴不是天空位置，而是假定的 MW contribution；纵轴是假定的 M31 contribution。数据本身只给出 x+y=total 的斜带。所有彩色竖向信息都是MW轴上的外部reference，不是“正确答案”。

Figure 3's horizontal axis is not a sky coordinate but an assumed MW contribution; the vertical axis is an assumed M31 contribution. The data provide only diagonal x+y=total bands. Every colored vertical feature is an external MW-axis reference, not a privileged answer.

Step 1先只看非负 x=S_MW、y=S_M31 和 x+y=S_obs；这是named-prior consistency test，不是空间坐标，也不是自动分解。Start with non-negative x=S_MW, y=S_M31, and x+y=S_obs. This is a named-prior consistency test, not a spatial coordinate or an automatic decomposition.
Step 2蓝、红和黑色斜约束分别加入 North/NW、South/SE 和 all-field observed totals。Add the blue, red, and black diagonal constraints for North/NW, South/SE, and the all-field observed total.
Step 3水平线与error bars加入 Zhang/Grayson 的 M31 conditional templates；它们不是本项目直接测得的 M31 emission。Horizontal lines and error bars add the Zhang/Grayson conditional M31 templates; they are not direct M31 measurements from this project.
Step 4五组MW结果现在都画成贯穿绘图区的竖向x-reference：绿色Ueda+22是M31足迹条件模型，青色HaloSat+20仅是M31方向域外supplemental extrapolation，紫色eFEDS与灰色H&S13是非同向天空参照，橙色Locatelli是另一套M31足迹模型。All five MW results are now drawn as full-height vertical x references: green Ueda+22 is an M31-footprint conditional model, cyan HaloSat+20 is only a supplemental out-of-domain M31-direction extrapolation, purple eFEDS and gray H&S13 are non-local sky context, and orange Locatelli is a second M31-footprint model.
Step 5这些span不是同一种uncertainty：Locatelli/Ueda/HaloSat填充区是footprint spread，H&S13点纹区是population IQR，eFEDS只有两条fixed-SWCX scenario线而没有填充区；HaloSat外点线是patchiness predictor scatter，其他点线另表示parameter sensitivity或population percentiles。These spans do not share one uncertainty definition: the Locatelli/Ueda/HaloSat fills are footprint spreads, the H&S13 dotted span is a population IQR, and eFEDS has only two fixed-SWCX scenario lines with no filled interval. HaloSat's outer dotted lines are patchiness predictor scatter; other dotted lines encode parameter sensitivity or population percentiles.
Step 6只有Locatelli all-field projection比observed total高0.326。Ueda同足迹模型给出0.312，HaloSat域外supplemental外推给出0.565，二者都允许正的条件M31 residual；eFEDS/H&S13天空参照进一步说明这是Locatelli-specific tension。Only the all-field Locatelli projection exceeds the observed total by 0.326. The footprint-matched Ueda model gives 0.312 and the supplemental out-of-domain HaloSat extrapolation gives 0.565; both permit a positive conditional M31 residual. The eFEDS/H&S13 sky context further shows that this is a Locatelli-specific tension.

Draft Figure 3 MW-M31 consistency test — 所有量均为absorbed 0.5–2.0 keV。五类MW结果都画成全高竖向reference；线的x位置才是MW brightness。绿色/青色/橙色hatched spans分别是Ueda/HaloSat/Locatelli footprint spread，紫色两条虚线是eFEDS fixed-SWCX scenarios，灰色点纹span是H&S13 population IQR。青色HaloSat是域外supplemental extrapolation，不是primary prior；这些span也不是同一种uncertainty。

Positive-quadrant log-log MW-M31 consistency test — log轴不能显示0或负值，因此只画非负物理解；SIMBA-NoAGN现在进入视野。绿色、青色、紫色、灰色与橙色MW references都贯穿绘图区，没有人为纵向shift。底边三角只表示Locatelli all-field reference与all-field total没有非负M31交点，不表示该模型更正确。

这张 log–log 图上的 MW 竖线和span应该怎样读？How should the vertical MW lines and spans in the log-log figure be read?

五组MW结果都采用完全相同的几何语法：中心值或scenario画成贯穿绘图区的竖线，范围画成竖向span或边界线。它们只固定x=S_MW，并不沿y轴提供额外测量；全高只是让每个study拥有相同视觉权重。真正的y=S_M31条件模板仍是Zhang/Grayson水平线。

All five MW results use the same geometric grammar: central values or scenarios are full-height vertical lines, while ranges are vertical spans or boundary lines. They constrain only x=S_MW and add no measurement along y; full height gives each study equal visual status. The true y=S_M31 conditional templates remain the horizontal Zhang/Grayson lines.

Locatelli 的高值来自一个具体转换链：采用论文指定的 reference Combined β=0.5 geometry，将 n_h²+n_d² 沿14条真实视线从太阳位置积分，得到4.14–4.56 L.U. 的 intrinsic O VIII 0.614–0.694 keV intensity；再用80 eV FWHM Gaussian smoothing 匹配原始 O VIII map 的能量处理。冻结CSV中的 response-matched self-check scale 为0.9498760469603238。最终 Figure 3 不是继续使用 published Z=0.1 broadband，而是固定这条 O VIII line normalization，换成Z=0.3 target APEC，并用每个 pointing 的 HI4PI N_H 做 full foreground screen phabs，得到absorbed 0.5–2.0 keV。

The high Locatelli value follows a specific conversion chain: use the paper-designated reference Combined β=0.5 geometry, integrate n_h²+n_d² from the Solar position through the 14 real sightlines, obtain intrinsic O VIII 0.614–0.694 keV intensities of 4.14–4.56 L.U., and match the original O VIII map treatment with an 80-eV-FWHM Gaussian smoothing. The frozen CSV gives a response-matched self-check scale of 0.9498760469603238. Figure 3 then does not reuse a published Z=0.1 broadband value; it holds that O VIII line normalization fixed, converts to a Z=0.3 target APEC model, and applies each pointing's HI4PI N_H as a full foreground screen phabs to obtain absorbed 0.5–2.0 keV.

重算结果闭合：14场final values为1.158–1.341，North/South inverse-variance means为1.296/1.213，side-balanced辅助summary为1.255，all-field inverse-variance estimator为1.290。图中实心橙线采用1.290，因为它应与观测all-field total 0.965±0.044比较；因此超过all-field观测总量0.326，即约34%。这不是“测得的MW一定这么亮”，而是说明这个 named Locatelli conversion 与我们的 observed CGMsum total 不相容，或者原模型normalization/line-to-broadband/foreground-screen假设在M31低纬视线方向过高。它也不是posterior interval；14-field min–max只是footprint spread。

The recomputation closes: the 14 final field values are 1.158–1.341, the North/South inverse-variance means are 1.296/1.213, the side-balanced auxiliary summary is 1.255, and the all-field inverse-variance estimator is 1.290. The solid orange line uses 1.290 because it is compared with the observed all-field total, 0.965±0.044; it therefore exceeds the all-field observed total by 0.326, or about 34%. This does not mean the MW brightness has been measured to be exactly that high; it means this named Locatelli conversion is inconsistent with our observed CGMsum total, or that its normalization, line-to-broadband conversion, or foreground-screen assumption is too high for this low-latitude M31 direction. It is not a posterior interval; the 14-field min–max is only footprint spread.

HaloSat真的有spherical+disk model吗？M31预测是多少？Does HaloSat actually provide a spherical+disk model, and what does it predict toward M31?

有，但要准确命名：Kaaret+20给出的是empirical disk + adiabatic halo，halo是在NFW势中静水平衡的多方气体，不是一个由全天数据拟合的通用spherical β component。73-field parent fit只使用南天 b<−30° 的HaloSat EM；M31十四场约在b=−21.6°，全部在拟合域外。最近的拟合场中心仍距M31 17.70°，超过HaloSat 7°零响应半径，因此没有观测重叠。模型几何可以算到M31方向，但结果只能作为supplemental extrapolation。

Yes, but it must be named precisely. Kaaret+20 fit an empirical disk plus an adiabatic halo, with the halo a hydrostatic polytrope in an NFW potential, not a generic spherical beta component calibrated by full-sky data. The 73-field parent fit used only southern HaloSat emission measures at b<−30°. All 14 M31 fields are outside the original southern-sky fit domain near b=−21.6°. The nearest fitted field center is still 17.70° from M31, beyond HaloSat's 7° zero-response radius, so there is no observational overlap. The result is only a supplemental extrapolation.

本页采用论文best fit：n_d=0.0081 Σ(R) exp(−|z|/1.60 kpc)，halo使用Fang+13 adiabatic profile和ρ_V=4.8×10⁻⁵ cm⁻³，积分到260 kpc。对73场复现published χ²要求published-fit n² convention：EM_model=∫(n_d+n_h)²ds并保留cross term；论文没有说明fitted n是n_e、n_H或其他particle density，因此不再额外乘1.2。band conversion明确补充sample median kT=0.225 keV、Z=0.3、Wilms+Verner TBabs和逐场HI4PI。得到all-field 0.565，14场0.526–0.578，patchiness predictor scale为±0.165；条件M31 complement为0.399。若去掉cross term，structural check为0.439。

The executable uses n_d=0.0081 Σ(R) exp(−|z|/1.60 kpc), the Fang+13 adiabatic profile with ρ_V=4.8×10⁻⁵ cm⁻³, and integration to 260 kpc. Reproducing the published χ² for all 73 fields requires the published-fit n^2 convention, EM_model=∫(n_d+n_h)²ds with the cross term retained. The paper does not identify fitted n as n_e, n_H, or another particle density, so no extra factor of 1.2 is applied. The band conversion adds sample-median kT=0.225 keV, Z=0.3, Wilms+Verner TBabs, and field-specific HI4PI. The result is 0.565 all-field, 0.526–0.578 across 14 fields, a ±0.165 patchiness predictor, and conditional M31=0.399. A no-cross-term structural check gives 0.439.

Bluem+22后来分析156个|b|>30°场，但没有为双温模型重新拟合spherical+disk geometry；gain correction使旧单温EM中位变化约15%。M31附近虽有HaloSat archive target，但未进入已发表高纬光谱目录，而且非成像大视场会包含M31弥散辐射，不能作为独立MW先验。

Bluem+22 later analyzed 156 fields at |b|>30° but did not refit a spherical+disk geometry for the two-temperature model; the gain correction changed the median legacy one-temperature EM by about 15%. A HaloSat archive target exists near M31, but it is absent from the published high-latitude spectral catalogs and its non-imaging field would contain M31 diffuse emission, so it is not an independent MW prior.

1.158–1.34114-field min–max；不是interval14-field min–max; not an interval

1.296 / 1.213North / South model estimatorsNorth / South model estimators

1.255side-balanced辅助summary；不作为主竖线Side-balanced auxiliary summary; not the main line

1.290all-field inverse-variance；图中实心橙线All-field inverse-variance; solid orange line

0.312Ueda+22 all-field inverse-varianceUeda+22 all-field inverse-variance

0.289–0.321Ueda nominal 14-field min–maxUeda nominal 14-field min–max

0.273–0.340仅展开Ueda n₀边际误差；不是posteriorUeda marginal-n₀ sensitivity; not a posterior

0.653Ueda条件下的正M31 complementPositive conditional M31 complement under Ueda

0.565HaloSat+20 M31方向域外supplemental外推HaloSat M31-direction extrapolation (Kaaret+20)

0.526–0.578HaloSat nominal 14-field footprintHaloSat nominal 14-field footprint

±0.165converted patchiness predictor；非posteriorConverted patchiness predictor; not a posterior

0.399HaloSat条件下M31 complementConditional M31 complement under HaloSat

0.291–0.407eFEDS两个固定SWCX scenarios；不是置信区间Two fixed eFEDS SWCX scenarios; not a confidence interval

0.385H&S13 absorbed population medianH&S13 absorbed population median

0.274–0.549H&S13 IQR；不是measurement errorH&S13 IQR; not a measurement error

>105°eFEDS到M31的最小角距Minimum eFEDS–M31 separation

Field-by-field decomposition of the Figure 3 MW reference prediction — fixed-NH geometry-only、逐场HI4PI的Z=0.1 direct-EM自检和Z=0.3 line-normalized values分开显示。这个图解释橙色footprint band的宽度来自哪里。

逐场数值

Field-level values

表中 observed 一列始终是 CGMsum。MW reference 列是 named model projection；二者没有被当成同一种数据。

The observed column remains CGMsum. The MW-reference columns are named model projections; they are not treated as the same kind of data.

面亮度：10⁻¹⁵ erg cm⁻² s⁻¹ arcmin⁻²；N_H：10²² cm⁻²。Surface brightness: 10⁻¹⁵ erg cm⁻² s⁻¹ arcmin⁻²; N_H: 10²² cm⁻².
OBSID	分区Side	R_proj (kpc)	R_∥ (kpc)	N_H	观测 CGMsumObserved CGMsum	MW固定NHMW fixed-NH	MW实际HI4PIMW actual HI4PI
800731501	North/NW	10.52	+8.84	0.0585	0.953 ± 0.126	1.296	1.272
800731901	South/SE	11.17	-8.61	0.0648	0.820 ± 0.605	1.264	1.185
800731601	North/NW	13.07	+12.69	0.0630	0.824 ± 0.094	1.289	1.225
800732001	South/SE	13.21	-12.84	0.0692	0.869 ± 0.244	1.275	1.158
800731101	North/NW	13.45	+5.85	0.0567	0.887 ± 0.632	1.312	1.305
800732301	South/SE	16.73	-3.20	0.0529	0.745 ± 0.561	1.243	1.271
800730801	North/NW	18.92	+10.71	0.0560	0.794 ± 0.435	1.321	1.322
800730901	North/NW	19.01	+14.26	0.0559	0.931 ± 0.121	1.313	1.314
800730701	North/NW	19.54	+6.64	0.0573	0.816 ± 0.175	1.329	1.317
800730501	North/NW	23.40	+11.39	0.0553	1.154 ± 0.141	1.334	1.341
800730601	North/NW	25.03	+7.19	0.0579	1.381 ± 0.187	1.344	1.325
800732801	South/SE	26.10	-9.07	0.0481	0.725 ± 0.319	1.225	1.298
800730301	North/NW	28.69	+16.68	0.0576	0.975 ± 0.123	1.343	1.327
800730201	North/NW	28.79	+11.82	0.0578	1.109 ± 0.137	1.351	1.333

为什么1.255和1.290同时出现？Why do 1.255 and 1.290 both appear?

1.255先分别计算North与South的inverse-variance mean，再给两个sides等权，是side-balanced辅助summary。1.290直接对全部14场使用观测inverse-variance weights，并与all-field measured total 0.965配对；新版Figure 3用它作为实心橙色主竖线。它们是不同estimator。

1.255 first computes inverse-variance means within North and South, then weights the two sides equally; it is a side-balanced auxiliary summary. 1.290 applies observational inverse-variance weights to all 14 fields and pairs with the all-field total 0.965; the revised Figure 3 uses it as the solid orange main line. They are different estimators.

0.934–1.575 是 credible interval 吗？Is 0.934–1.575 a credible interval?

不是。它只用published marginal parameter errors做diagonal first-order propagation；Locatelli posterior covariance不可用。因此它只能叫parameter-sensitivity boundaries，不能叫posterior、confidence或credible interval。

No. It uses published marginal parameter errors in a diagonal first-order propagation without the Locatelli posterior covariance. They are parameter-sensitivity boundaries, not a posterior, confidence, or credible interval.

为什么Ueda+22可以投影到M31，而eFEDS/H&S13不行？Why can Ueda+22 be projected to M31 while eFEDS/H&S13 cannot?

Ueda+22的75°<l<285°、|b|>15°描述130场parent sample；本页采用的n_e0=3.4×10⁻³ cm⁻³来自Table 3的2005–2009、|l|>105°、N=36 disk row。M31的14场仍在这个selected angular domain内，因此可以沿每条真实视线积分并应用逐场HI4PI。得到的0.312是与观测采用相同inverse-variance weights的条件模型值。绿色hatched span是14场nominal spread，绿色点线边界仅展开n₀边际误差；没有参数covariance，不能叫posterior interval。eFEDS和H&S13只提供其他天空方向的测量分布，不能局部化到M31。

Ueda+22 used a 130-field parent sample over 75°<l<285° and |b|>15°. The n_e0=3.4×10⁻³ cm⁻³ normalization used here comes specifically from the Table 3 2005–2009, |l|>105°, N=36 disk row. All 14 M31 fields remain inside that selected angular domain, so the model can be integrated along each actual sightline and converted with field-specific HI4PI. Its 0.312 value uses the same inverse-variance weights as the observed total. The green hatched span is nominal footprint spread and the green dotted boundaries vary only marginal n₀; without parameter covariance this is not a posterior interval. eFEDS and H&S13 instead provide measurements in other sky directions and cannot be localized to M31.

X-LEAP / Zhijie Qu在M31方向有测量吗？Does X-LEAP / Zhijie Qu provide measurements toward M31?

有line catalog，但不能直接变成独立MW broadband prior。X-LEAP I在M31 4.4°内有52条catalog rows，全部被标为M31-contaminated且没有一条属于clean sample；作者还报告约10°–20°的M31-associated enhancement。clean 10°–20°和20°–30° annuli的raw detected O VIII median都约1.41 L.U.，但这是observed line intensity，仍混有方向依赖的SWCX/absorption，也没有MW/M31 component separation。

X-LEAP provides a line catalog, but not an independent MW broadband prior. X-LEAP I has 52 catalog rows within 4.4° of M31; all are flagged as M31-contaminated and none belongs to the clean sample, and the paper reports an M31-associated enhancement over roughly 10°–20°. The clean 10°–20° and 20°–30° annuli both have a raw detected O VIII median near 1.41 L.U., but these are observed line intensities with direction-dependent SWCX/absorption and no MW/M31 component separation.

X-LEAP II的Eq. 3机械代入M31距Galactic center的118.78°得到EM=−0.0011 pc cm⁻⁶，但Figure 5和正文在同一角距却约为17–18×10⁻³，并称负值只在>150°出现；公式、图和正文不闭合，所以不存在唯一可复现的M31原生EM。该文也明确把3D density decomposition留给未来工作。因此X-LEAP只保留为line/temperature context，不加成第六条broadband竖线。

Literal substitution into X-LEAP II Equation 3 gives EM=−0.0011 pc cm⁻⁶ at M31's 118.78° separation from the Galactic center, whereas Figure 5 and the text imply roughly 17–18×10⁻³ there and say negative values appear only beyond 150°. Equation 3 conflicts with Figure 5 and the prose, so there is no unique reproducible native M31 EM. X-LEAP II also leaves the 3D density decomposition to future work; it remains line/temperature context rather than a sixth broadband reference.

eROSITA不是已经测了西半球吗？Did eROSITA not already measure the western hemisphere?

是，但要区分两项结果。Ponti+23在107.5 deg² eFEDS区域发表了可分解的CGM component flux，所以能转换成图中的两条紫色scenario竖线；它离M31至少105°，不是M31同向先验。Yeung+24 Paper I覆盖西半球约2000个spectral bins，但文章主题是Local Hot Bubble，并明确把CGM结果留给Paper II；不能从Paper I虚构一个western-hemisphere broadband CGM分布。

Yes, but two results must be separated. Ponti+23 published a decomposed CGM-component flux for the 107.5 deg² eFEDS field, supporting the two purple vertical scenario lines; it remains at least 105° from M31 and is not M31-matched. Yeung+24 Paper I analyzed roughly 2000 western-hemisphere spectral bins but focused on the Local Hot Bubble and explicitly deferred the CGM results to Paper II. Paper I therefore does not support inventing a western-hemisphere broadband CGM distribution.

08 · what this page does not claim

解释边界

Interpretation boundaries

连续曲线是固定NH的几何曲线，不是 continuous HI4PI prediction。
Locatelli MW emission 使用完整 foreground screen；真实 distributed absorber 会让更多 MW photons 透过，因此当前处理不是降低张力的捷径。
14-field min–max 是 footprint spread，不是 model uncertainty。
MW projection 在每个 pointing center 计算，没有对15′ detector/camera-mask footprint 做二维角平均。
Zhang/Grayson 是条件模板，不是对 M31 contribution 的直接测量。
当前 statistical errors 是 local covariance；尚无完整 N_H/SWCX/CXB/SP/abundance/band systematic budget。
最终 component constraints 需要 shared radial-plus-side likelihood 和 boundary-aware intervals。

The continuous curve is a fixed-NH geometry curve, not a continuous HI4PI prediction.
The Locatelli MW emission uses a full foreground screen. A distributed absorber would transmit more MW photons, so it is not a shortcut that removes the tension.
The 14-field min–max is footprint spread, not model uncertainty.
The MW projection is evaluated at each pointing center; it is not a two-dimensional angular average over the 15′ detector/camera-mask footprint.
Zhang/Grayson are conditional templates, not direct measurements of the M31 contribution.
Current statistical errors are local covariance errors; a complete NH/SWCX/CXB/SP/abundance/band systematic budget is not yet available.
Final component constraints require a shared radial-plus-side likelihood and boundary-aware intervals.

09 · reproduce every step

数据、图、代码与 provenance

Data, figures, code, and provenance

所有页面资产进入 SHA-256 manifest。CSV 保存连续 profiles 和逐场 conversion；JSON 保存参数、单位、estimator 和 build metadata。

Every page asset enters a SHA-256 manifest. CSV files preserve continuous profiles and field-level conversions; JSON preserves parameters, units, estimators, and build metadata.

data/conditional_m31_templates.csv13.8 KB data/explorer_summary.json6.6 KB data/m31_cgmsum_efeds2023_absorbed_0p5_2p0.csv529 B data/m31_cgmsum_external_mw_comparator_audit.json7.9 KB data/m31_cgmsum_halosat_kaaret2020_disk_adiabatic_halo_m31_extrapolation.csv3.1 KB data/m31_cgmsum_henley_shelton2013_absorbed_0p5_2p0.csv9.4 KB data/m31_cgmsum_henley_shelton2013_vizier_main_sample.csv7.7 KB data/m31_cgmsum_ueda2022_disk_m31_footprint_absorbed_0p5_2p0.csv2.7 KB data/m31_field_profile_decomposition.csv21.5 KB data/mw_latitude_profile.csv52.2 KB data/mw_m31_gc_signed_axis_profile.csv72.2 KB figures/cgmsum_m31_radius_profile_1col.pdf30.7 KB figures/cgmsum_m31_radius_profile_1col.png101.9 KB figures/cgmsum_m31_radius_profile_2col.pdf30.8 KB figures/cgmsum_m31_radius_profile_2col.png175.1 KB figures/draft_figure3_consistency_test.png538.7 KB figures/draft_figure3_consistency_test_loglog_1col.pdf96.7 KB figures/draft_figure3_consistency_test_loglog_1col.png353.6 KB figures/draft_figure3_consistency_test_loglog_2col.pdf97.6 KB figures/draft_figure3_consistency_test_loglog_2col.png520.7 KB figures/figure3_field_conversion_1col.pdf34.7 KB figures/figure3_field_conversion_1col.png111.4 KB figures/figure3_field_conversion_2col.pdf35.8 KB figures/figure3_field_conversion_2col.png182.2 KB figures/mw_latitude_profile_1col.pdf35.2 KB figures/mw_latitude_profile_1col.png123.3 KB figures/mw_latitude_profile_2col.pdf37.2 KB figures/mw_latitude_profile_2col.png268.1 KB figures/mw_signed_axis_profile_1col.pdf28.5 KB figures/mw_signed_axis_profile_1col.png120.5 KB figures/mw_signed_axis_profile_2col.pdf29.9 KB figures/mw_signed_axis_profile_2col.png222.1 KB source/README.md5.4 KB source/SITE_SPEC.md5.9 KB source/audit_apj_v19_cgmsum_bandpass.py4.4 KB source/audit_apj_v19_cgmsum_conditional_priors.py26.8 KB source/audit_external_mw_halo_comparators.py25.2 KB source/build_explorer_site.py24.5 KB source/generate_explorer_products.py37.1 KB source/m31_cgmsum_apec_bandpass_audit.json2.1 KB source/m31_cgmsum_apec_bandpass_matrix.csv11.8 KB source/m31_cgmsum_conditional_prior_audit.json10.2 KB source/m31_cgmsum_conditional_prior_ledger.csv13.2 KB source/m31_cgmsum_locatelli2024_reference_beta0p5_m31_footprint_predictions.csv4.3 KB source/m31_cgmsum_v19_primary_measurements_public.csv11.9 KB source/make_apj_v19_cgmsum_conditional_figures.py45.5 KB source/pytest.ini166 B source/test_build_explorer_site.py14.5 KB source/test_generate_explorer_products.py12.0 KB source/test_verify_production.py4.6 KB source/verify_production.py5.3 KB

report_manifest.json

Build：2026-07-14 16:39 UTC。Canonical URL：m31cgm-mw-m31-profile-explorer.pages.dev。Build: 2026-07-14 16:39 UTC. Canonical URL: m31cgm-mw-m31-profile-explorer.pages.dev.