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.

0.4–1.25 keV

这里所有主 surface brightness 都使用 absorbed v19 band convention。

All primary surface brightnesses use the absorbed v19 band convention.

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.

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，再经过80 eV Gaussian-smoothed emissivity closure、APEC band conversion 和每场 HI4PI absorption 得到的。

The orange band is not a broadband prior copied from a table. It comes from projecting an O VIII-constrained MW density geometry through 14 real sightlines, then applying an 80-eV Gaussian-smoothed emissivity closure, APEC band conversion, and 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_close=0.94987680 eV Gaussian-smoothed emissivity closure80-eV Gaussian-smoothed emissivity closure

phabs × APECabsorbed 0.4–1.25 keV surface brightnessabsorbed 0.4–1.25 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_close=ε_Loc/ε_{APEC,smoothed}=1.649/1.736=0.949876

raw target APEC narrow-band emissivity 为1.383×10⁻¹⁶，但 Figure 3 使用 Gaussian-smoothed 1.736×10⁻¹⁶ 做 closure。跳过 smoothing 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。

The raw target-APEC narrow-band emissivity is 1.383×10⁻¹⁶, but Figure 3 closes against the Gaussian-smoothed 1.736×10⁻¹⁶. Omitting the smoothing would change the line-to-broadband normalization incorrectly. No real MOS RMF/ARF is used here, 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_close × F[phabs(N_H,i) × APEC(kT=0.15,Z=0.1)]_{0.4–1.25 keV}

其中1 sr=(180×60/π)² arcmin²，最后除以10⁻¹⁵得到网页和 Figure 3 的 primary 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 primary units used on this page and in Figure 3. 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 0.4–1.25 keV
10.0000	2.400
20.0000	1.661
21.5733 (M31)	1.593
30.0000	1.334
45.0000	1.108
60.0000	1.023
90.0000	1.063

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.

1.5930M31中心，固定NHAt M31, fixed NH

1.5948+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 S(20 kpc)=0.9418。条件曲线在10/15/20/25/30 kpc约为1.818/1.267/0.942/0.737/0.599。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 S(20 kpc)=0.9418. The conditional curve is approximately 1.818/1.267/0.942/0.737/0.599 at 10/15/20/25/30 kpc. It is a population template, not an M31 profile recovered independently from the 14 XMM spectra.

07 · collapse into the contribution plane

空间信息如何被压缩进 draft Figure 3

How spatial information is collapsed into draft Figure 3

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。橙色竖向信息来自前面的14场模型投影。

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. The orange vertical information comes from the preceding 14-field model projection.

Step 1先只看非负 x=S_MW、y=S_M31 和 x+y=S_obs；这是贡献平面，不是空间坐标。Start with non-negative x=S_MW, y=S_M31, and x+y=S_obs. This is a contribution plane, not a spatial coordinate.
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水平线与横带加入 Zhang/Grayson 的 M31 conditional component templates；它们不是本项目直接测得的 M31 emission。Horizontal lines and bands add the Zhang/Grayson conditional M31 component templates; they are not directly measured M31 emission from this project.
Step 4橙色竖带加入14场 Locatelli footprint min–max；1.560是 side-balanced estimator，1.602是 all-field inverse-variance estimator，二者不可互换。Add the orange 14-field Locatelli footprint min–max. The 1.560 side-balanced estimator and 1.602 all-field inverse-variance estimator are not interchangeable.
Step 5Locatelli central 与 observed total 的交点落到负 M31 residual，只说明 named model+conversion 与总量有张力；它绝不是“负的 M31 辐射”。The Locatelli central value implies a negative residual against the observed total. That diagnoses tension in the named model+conversion, never negative physical M31 emission.

Draft Figure 3 MW-M31 contribution plane — 蓝/红斜带分别是 North/NW 与 South/SE 的 x+y constraints；黑色虚线是 all-field total。橙色 hatched band 是14个 response-matched field predictions 的 min–max，不是可信区间。水平线/带是 named M31 conditional templates。

Positive-quadrant log-log MW-M31 contribution plane — 这不是对原图的替代，而是扩大动态范围的补充诊断。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。Locatelli reference 并未被静默删除：底边三角明确表示它与 all-field total 没有非负 M31 交点。

1.431–1.66814-field min–max；不是interval14-field min–max; not an interval

1.608 / 1.511North / South model estimatorsNorth / South model estimators

1.560side-balanced central，图中竖线Side-balanced central, vertical line

1.602all-field inverse-variance，Table 5All-field inverse-variance, Table 5

Field-by-field decomposition of the Figure 3 MW reference prediction — fixed-NH geometry-only、逐场 HI4PI direct-EM conversion 和乘0.949876后的 response-matched values 分开显示。这个图解释橙色 footprint band 的宽度从哪里来。

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	1.081 ± 0.147	1.611	1.580
800731901	South/SE	11.17	-8.61	0.0648	0.931 ± 0.729	1.572	1.468
800731601	North/NW	13.07	+12.69	0.0630	0.936 ± 0.110	1.602	1.518
800732001	South/SE	13.21	-12.84	0.0692	0.963 ± 0.289	1.585	1.431
800731101	North/NW	13.45	+5.85	0.0567	1.005 ± 0.761	1.631	1.622
800732301	South/SE	16.73	-3.20	0.0529	0.842 ± 0.671	1.545	1.582
800730801	North/NW	18.92	+10.71	0.0560	0.890 ± 0.519	1.643	1.643
800730901	North/NW	19.01	+14.26	0.0559	1.060 ± 0.142	1.633	1.634
800730701	North/NW	19.54	+6.64	0.0573	0.898 ± 0.205	1.652	1.636
800730501	North/NW	23.40	+11.39	0.0553	1.324 ± 0.170	1.659	1.668
800730601	North/NW	25.03	+7.19	0.0579	1.522 ± 0.215	1.671	1.646
800732801	South/SE	26.10	-9.07	0.0481	0.798 ± 0.350	1.523	1.619
800730301	North/NW	28.69	+16.68	0.0576	1.121 ± 0.147	1.669	1.649
800730201	North/NW	28.79	+11.82	0.0578	1.272 ± 0.163	1.679	1.656

为什么1.560和1.602同时出现？Why do 1.560 and 1.602 both appear?

1.560先分别计算 North 与 South 的 inverse-variance mean，再给两个 sides 等权；它用于 Figure 3 的 side-balanced visual center。1.602直接对全部14场使用观测 inverse-variance weights；它与 all-field measured total 1.092 配对，用于 Table 5 conditional complement。它们是不同 estimator。

1.560 first computes inverse-variance means within North and South, then weights the two sides equally; it is the side-balanced visual center in Figure 3. 1.602 applies the observational inverse-variance weights directly to all 14 fields; it pairs with the all-field measured total 1.092 in Table 5. They are different estimators.

1.162–1.958 是 credible interval 吗？Is 1.162–1.958 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 envelope，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. It is a parameter-sensitivity envelope, not a posterior, confidence, or credible interval.

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.json6.5 KB data/m31_field_profile_decomposition.csv19.7 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.png100.9 KB figures/cgmsum_m31_radius_profile_2col.pdf30.8 KB figures/cgmsum_m31_radius_profile_2col.png174.7 KB figures/draft_figure3_contribution_plane.png356.6 KB figures/draft_figure3_contribution_plane_loglog_1col.pdf69.0 KB figures/draft_figure3_contribution_plane_loglog_1col.png221.6 KB figures/draft_figure3_contribution_plane_loglog_2col.pdf69.5 KB figures/draft_figure3_contribution_plane_loglog_2col.png310.7 KB figures/figure3_field_conversion_1col.pdf30.8 KB figures/figure3_field_conversion_1col.png101.9 KB figures/figure3_field_conversion_2col.pdf32.5 KB figures/figure3_field_conversion_2col.png187.9 KB figures/mw_latitude_profile_1col.pdf34.7 KB figures/mw_latitude_profile_1col.png124.7 KB figures/mw_latitude_profile_2col.pdf36.3 KB figures/mw_latitude_profile_2col.png258.2 KB figures/mw_signed_axis_profile_1col.pdf27.3 KB figures/mw_signed_axis_profile_1col.png110.3 KB figures/mw_signed_axis_profile_2col.pdf29.3 KB figures/mw_signed_axis_profile_2col.png215.9 KB source/README.md4.0 KB source/SITE_SPEC.md5.4 KB source/audit_apj_v19_cgmsum_conditional_priors.py19.4 KB source/build_explorer_site.py23.0 KB source/generate_explorer_products.py35.4 KB source/m31_cgmsum_conditional_prior_audit.json6.4 KB source/m31_cgmsum_conditional_prior_ledger.csv6.4 KB source/m31_cgmsum_locatelli2024_reference_beta0p5_m31_footprint_predictions.csv3.3 KB source/m31_cgmsum_v19_primary_measurements_public.csv11.9 KB source/make_apj_v19_cgmsum_conditional_figures.py32.1 KB source/pytest.ini166 B source/test_build_explorer_site.py10.1 KB source/test_generate_explorer_products.py11.8 KB source/test_verify_production.py4.6 KB source/verify_production.py5.3 KB

report_manifest.json

Build：2026-07-12 15:25 UTC。Canonical URL：m31cgm-mw-m31-profile-explorer.pages.dev。Build: 2026-07-12 15:25 UTC. Canonical URL: m31cgm-mw-m31-profile-explorer.pages.dev.