面向物理系本科一年级 · 可执行教程
教程目标：理解 Kaaret et al. (2020) HaloSat 的 empirical disk + adiabatic halo 模型如何从73个南天高银纬场拟合，再域外外推到M31方向，变成Figure 3上的一个 supplemental extrapolation 点。 Tutorial goal: Understand how Kaaret et al. (2020) HaloSat’s empirical disk plus adiabatic halo model, fitted to 73 southern high-latitude fields, is extrapolated outside its fit domain toward M31 to become a supplemental extrapolation point on Figure 3.
目标读者：物理系本科一年级，已学完普通物理（电磁学/光学），了解基本的原子物理概念（能级、跃迁），但不需要天文观测经验。 Target audience: First-year physics undergraduates who have completed general physics (electromagnetism/optics) and basic atomic physics, without requiring astronomical observing experience.
核心问题：HaloSat 是一颗 CubeSat，它用73个 b<-30° 的南天场拟合出银河系热气体的三维形态模型。M31 在 b≈-21.6°，全部十四场都在拟合域外。这个模型如何”外推”到M31方向？为什么它只是一个 supplemental extrapolation，而不是直接预测？
本教程只需要 Python 标准科学栈。所有数据文件已随教程提供。
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from pathlib import Path
plt.style.use('seaborn-v0_8-whitegrid')
plt.rcParams['font.size'] = 12
plt.rcParams['figure.dpi'] = 150
plt.rcParams['savefig.dpi'] = 300
# NOTE: do NOT add a CJK font fallback here — see SKILL.md
# All matplotlib text labels stay English-only.
DATA = Path("assets/data")
print("Environment ready!")Environment ready!HaloSat 是一颗专门观测软X射线（约0.2–2 keV）的 CubeSat——一种体积很小、成本较低的卫星。它不是成像望远镜，而是用 silicon-drift detectors（硅漂移探测器）和宽视场来测量大天区的X射线背景。它的科学目标是理解银河系周围的热气体——circumgalactic medium, CGM——是什么形状的。
银河系周围 \(10^6\)–\(10^7\) K 的热气体发出软X射线。但它的空间分布是盘状（disk-dominated）还是球状（spherical halo）？这关系到银河系如何吸积和释放气体。Kaaret et al. (2020) 用 HaloSat 的73个南天高银纬场（\(b<-30^\circ\)）的 emission measure（发射量度，EM）数据来回答这个问题。
Emission measure 的定义是沿视线积分 \(\int n_e n_H \, ds\)（单位 cm\(^{-6}\) pc），它正比于X射线发射强度。EM 越大，说明该方向上的热气体越多或路径越长。
Kaaret et al. (2020) 比较了 disk 和 halo 两种形态模型后，结论是：soft X-ray emitting CGM 在该南天样本中是 disk-dominated 的，而且有 patchiness（不均匀性）。最终采用的 composite model（empirical disk + adiabatic halo）的拟合优度是 \(\chi^2 \approx 71.7/70\)。
盘成分的密度模型是：
\[n_{\mathrm{disk}}(R, z) = n_0 \cdot \Sigma(R) \cdot \exp\left(-\frac{|z|}{z_0}\right)\]
其中： - \(n_0 = 0.0081\) cm\(^{-3}\) 是盘的密度归一化； - \(z_0 = 1.60\) kpc 是垂直标高； - \(\Sigma(R)\) 是一个 piecewise（分段）的径向函数，描述盘密度随银心距 \(R\) 的变化。
晕成分采用 Fang et al. (2013) 的 polytropic（多方）模型，描述在 NFW 势阱中处于流体静力学平衡的气体：
这是一个”绝热”气体晕——温度随密度按绝热关系变化，而不是等温的。
模型预测的 EM 通过沿视线积分 \((n_{\mathrm{disk}} + n_{\mathrm{halo}})^2\) 得到：
\[\mathrm{EM}_{\mathrm{model}} = \int (n_d + n_h)^2 \, ds\]
注意这里保留了 disk-halo cross term（交叉项） \(2 n_d n_h\)。这个 convention 是通过对73场 catalog 的 \(\chi^2\) closure 验证确定的——只有保留交叉项，才能复现论文的拟合统计量 \(\chi^2 = 71.665\)。
关键假设：论文没有说明拟合得到的 \(n\) 是 \(n_e\)、\(n_H\)、total-ion 还是 total-particle 密度。因此本项目不额外乘 \(n_e/n_H = 1.2\) 的 composition factor。这与 Ueda+2022 的处理不同。
graph TD
A["73 southern fields<br/>b < -30 deg"] --> B["Empirical disk<br/>n0=0.0081, z0=1.60 kpc"]
A --> C["Adiabatic halo<br/>Fang+2013 polytrope in NFW"]
B --> D["EM = ∫(n_d+n_h)² ds<br/>cross term retained"]
C --> D
D --> E["chi2 = 71.7/70<br/>catalog closure verified"]
E --> F["Extrapolate to M31<br/>b ≈ -21.6 deg (outside domain)"]graph TD
A["73 southern fields<br/>b < -30 deg"] --> B["Empirical disk<br/>n0=0.0081, z0=1.60 kpc"]
A --> C["Adiabatic halo<br/>Fang+2013 polytrope in NFW"]
B --> D["EM = ∫(n_d+n_h)² ds<br/>cross term retained"]
C --> D
D --> E["chi2 = 71.7/70<br/>catalog closure verified"]
E --> F["Extrapolate to M31<br/>b ≈ -21.6 deg (outside domain)"]
halosat = pd.read_csv(DATA / "m31_cgmsum_halosat_kaaret2020_disk_adiabatic_halo_m31_extrapolation.csv")
print(f"Loaded {len(halosat)} M31 sightlines")
print(f"Columns: {list(halosat.columns)}")
halosat.head(8)Loaded 14 M31 sightlines
Columns: ['obsid', 'galactic_l_deg', 'galactic_b_deg', 'nh_hi4pi_1e22_cm-2', 'outside_halosat_fit_domain', 'nominal_em_cm-6_pc', 'no_cross_term_em_cm-6_pc', 'absorbed_0p5_2p0_flux_per_em_fluxunit', 'nominal_absorbed_0p5_2p0_fluxunit', 'no_cross_term_absorbed_0p5_2p0_fluxunit', 'patchiness_sigma_absorbed_0p5_2p0_fluxunit', 'measurement_staterr_absorbed_0p5_2p0_fluxunit']| obsid | galactic_l_deg | galactic_b_deg | nh_hi4pi_1e22_cm-2 | outside_halosat_fit_domain | nominal_em_cm-6_pc | no_cross_term_em_cm-6_pc | absorbed_0p5_2p0_flux_per_em_fluxunit | nominal_absorbed_0p5_2p0_fluxunit | no_cross_term_absorbed_0p5_2p0_fluxunit | patchiness_sigma_absorbed_0p5_2p0_fluxunit | measurement_staterr_absorbed_0p5_2p0_fluxunit | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 800730201 | 119.831752 | -19.888005 | 0.057791 | True | 0.011788 | 0.009136 | 48.748698 | 0.574630 | 0.445350 | 0.165746 | 0.136534 |
| 1 | 800730301 | 119.508203 | -20.173004 | 0.057602 | True | 0.011785 | 0.009135 | 48.788971 | 0.574983 | 0.445679 | 0.165883 | 0.122502 |
| 2 | 800730501 | 119.959338 | -20.297171 | 0.055286 | True | 0.011722 | 0.009089 | 49.284947 | 0.577710 | 0.447959 | 0.167569 | 0.141052 |
| 3 | 800730601 | 120.221879 | -19.978270 | 0.057930 | True | 0.011735 | 0.009098 | 48.719218 | 0.571719 | 0.443229 | 0.165645 | 0.187279 |
| 4 | 800730701 | 120.355845 | -20.368759 | 0.057273 | True | 0.011672 | 0.009053 | 48.859083 | 0.570276 | 0.442322 | 0.166121 | 0.175246 |
| 5 | 800730801 | 120.090295 | -20.632593 | 0.055956 | True | 0.011666 | 0.009050 | 49.140924 | 0.573293 | 0.444720 | 0.167079 | 0.435051 |
| 6 | 800730901 | 119.868483 | -20.906513 | 0.055921 | True | 0.011655 | 0.009043 | 49.148343 | 0.572835 | 0.444437 | 0.167104 | 0.121189 |
| 7 | 800731101 | 120.520910 | -20.803429 | 0.056700 | True | 0.011602 | 0.009003 | 48.981375 | 0.568259 | 0.440995 | 0.166537 | 0.631616 |
数据包含14条M31方向的XMM视场。每条视场有：
| 列名 | 含义 |
|---|---|
obsid |
XMM-Newton 观测ID |
galactic_l_deg, galactic_b_deg |
场中心的银道坐标 |
nh_hi4pi_1e22_cm-2 |
HI4PI 中性氢柱密度（\(10^{22}\) cm\(^{-2}\)） |
outside_halosat_fit_domain |
是否在HaloSat拟合域外（全部为True） |
nominal_em_cm-6_pc |
nominal（含交叉项）EM（cm\(^{-6}\) pc） |
no_cross_term_em_cm-6_pc |
去交叉项的 EM（structural check） |
absorbed_0p5_2p0_flux_per_em_fluxunit |
单位 EM 对应的 absorbed 0.5–2.0 keV flux |
nominal_absorbed_0p5_2p0_fluxunit |
nominal absorbed 0.5–2.0 keV brightness（Figure 3 单位） |
no_cross_term_absorbed_0p5_2p0_fluxunit |
去交叉项的 absorbed brightness |
patchiness_sigma_absorbed_0p5_2p0_fluxunit |
patchiness scatter 转换后的 1σ |
measurement_staterr_absorbed_0p5_2p0_fluxunit |
观测统计误差（用于权重计算） |
HaloSat 的73个拟合场全部在 \(b < -30^\circ\) 的南天高银纬区域。M31 在 \(b \approx -21.6^\circ\)，十四场全部位于拟合域外。最近的一个拟合场中心距 M31 约 17.70°，超过了 HaloSat 约 7° 的零响应半径。
# Verify all 14 M31 fields are outside the fit domain
all_outside = halosat["outside_halosat_fit_domain"].all()
print(f"All 14 fields outside HaloSat fit domain: {all_outside}")
print(f"Galactic latitude range: [{halosat['galactic_b_deg'].min():.2f}, {halosat['galactic_b_deg'].max():.2f}] deg")
print(f"M31 is at b ≈ -21.6 deg, fit domain is b < -30 deg")
print(f"Nearest fit field center to M31: 17.70 deg (beyond ~7 deg zero-response radius)")All 14 fields outside HaloSat fit domain: True
Galactic latitude range: [-23.17, -19.89] deg
M31 is at b ≈ -21.6 deg, fit domain is b < -30 deg
Nearest fit field center to M31: 17.70 deg (beyond ~7 deg zero-response radius)# Load the 73 HaloSat southern fields
halosat_fields = pd.read_csv(DATA / "m31_cgmsum_halosat_kaaret2020_southern_fields.csv")
fig, ax = plt.subplots(figsize=(8, 5), subplot_kw={"projection": None})
# Plot HaloSat 73 field centers
ax.scatter(
halosat_fields["galactic_l_deg"],
halosat_fields["galactic_b_deg"],
s=18, c="#2ca6c4", alpha=0.7, edgecolor="none",
label=f"HaloSat 73 fit fields (b < -30 deg)"
)
# Plot selection envelope boundary at b = -30
ax.axhline(-30, color="#2ca6c4", linestyle="--", linewidth=1.0, alpha=0.5)
# Plot 14 M31 fields
ax.scatter(
halosat["galactic_l_deg"],
halosat["galactic_b_deg"],
s=60, c="#e07b3a", alpha=0.9, edgecolor="white", linewidth=0.5,
label=f"14 M31 XMM fields (b ≈ -21.6 deg)",
zorder=5
)
# M31 center
ax.scatter(121.17, -21.57, marker="*", s=200, c="#b74842", edgecolor="white",
linewidth=0.8, zorder=6, label="M31 center")
ax.set_xlabel("Galactic longitude l (deg)")
ax.set_ylabel("Galactic latitude b (deg)")
ax.set_title("HaloSat fit domain vs M31 footprint")
ax.legend(fontsize=9, loc="lower left")
ax.set_xlim(60, 130)
ax.set_ylim(-85, 5)
plt.show()
关键观察： - 73个青色点是 HaloSat 实际拟合的场中心，不是连续的天空覆盖。 - 橙色点是14个M31 XMM视场，全部在 \(b=-30^\circ\) 边界（青色虚线）之上——即拟合域外。 - 红色五角星是M31中心，最近的一个青色点离它约17.7°。
要把 HaloSat 的三维密度模型投影到 M31 方向，需要沿每条M31视线积分 \((n_d + n_h)^2\)：
# The nominal branch retains the cross term; the no-cross-term is a structural check
print("Emission measure comparison (cm^-6 pc):")
print(f" Nominal (with cross term): mean = {halosat['nominal_em_cm-6_pc'].mean():.6f}")
print(f" No cross term (structural chk): mean = {halosat['no_cross_term_em_cm-6_pc'].mean():.6f}")
print(f" Ratio (nominal / no-cross): mean = {(halosat['nominal_em_cm-6_pc'] / halosat['no_cross_term_em_cm-6_pc']).mean():.4f}")
print()
print("The cross term adds ~29% to the EM, confirming it is retained in the nominal branch.")Emission measure comparison (cm^-6 pc):
Nominal (with cross term): mean = 0.011560
No cross term (structural chk): mean = 0.008974
Ratio (nominal / no-cross): mean = 1.2881
The cross term adds ~29% to the EM, confirming it is retained in the nominal branch.HaloSat 的密度模型只告诉我们 \(n(r)\)，即气体在空间各处的密度。但要把 EM 转换成X射线 flux，还需要知道气体的温度 \(kT\)——因为温度决定了X射线谱的形状（哪些能量发射多少光子）。
问题在于：HaloSat 的 disk+halo 模型不预测M31方向的温度。它只在73个南天场各自拟合了温度，但M31在拟合域外，没有对应的拟合温度。
解决方案：用 HaloSat 73场的 sample median \(kT = 0.225\) keV 作为M31方向的补充温度。这是一个合理的”代表性”温度，但必须记住它是额外假设，不是模型自带的预测。
# Verify the median kT from the 73-field catalog
kt_values = halosat_fields["kt_keV"].dropna()
median_kt = kt_values.median()
print(f"HaloSat 73-field median kT = {median_kt:.3f} keV")
print(f"Adopted M31-direction kT = 0.225 keV (sample median)")
print(f"Note: This is an explicit augmentation — the density model has no M31-direction temperature prediction.")HaloSat 73-field median kT = 0.225 keV
Adopted M31-direction kT = 0.225 keV (sample median)
Note: This is an explicit augmentation — the density model has no M31-direction temperature prediction.有了 EM 和温度，还需要把 intrinsic（吸收前）的发射转换成 absorbed（吸收后）的观测值。转换链是：
# Each field has a per-EM conversion factor
fig, ax = plt.subplots(figsize=(8, 4.5))
sc = ax.scatter(
halosat["nh_hi4pi_1e22_cm-2"],
halosat["absorbed_0p5_2p0_flux_per_em_fluxunit"],
c=halosat["galactic_b_deg"],
cmap="viridis",
s=50, edgecolor="white", linewidth=0.5,
)
cbar = fig.colorbar(sc, ax=ax)
cbar.set_label("Galactic latitude b (deg)")
ax.set_xlabel("N_H (10^22 cm^-2, HI4PI)")
ax.set_ylabel("Absorbed flux per EM (flux unit / cm^-6 pc)")
ax.set_title("Per-field APEC+TBabs conversion factor")
plt.show()
print(f" N_H range: [{halosat['nh_hi4pi_1e22_cm-2'].min():.4f}, {halosat['nh_hi4pi_1e22_cm-2'].max():.4f}] x 10^22 cm^-2")
print(f" Flux/EM range: [{halosat['absorbed_0p5_2p0_flux_per_em_fluxunit'].min():.2f}, {halosat['absorbed_0p5_2p0_flux_per_em_fluxunit'].max():.2f}]")
print(" Higher N_H → more absorption → lower flux per EM (softer spectrum suppressed).")
N_H range: [0.0481, 0.0692] x 10^22 cm^-2
Flux/EM range: [46.39, 50.87]
Higher N_H → more absorption → lower flux per EM (softer spectrum suppressed).# Manual reconstruction: EM × (flux per EM) = absorbed flux
manual_nominal = halosat["nominal_em_cm-6_pc"] * halosat["absorbed_0p5_2p0_flux_per_em_fluxunit"]
csv_nominal = halosat["nominal_absorbed_0p5_2p0_fluxunit"]
print("Field-by-field reconstruction (first 5):")
for i in range(5):
print(f" Field {i+1}: manual={manual_nominal.iloc[i]:.6f}, CSV={csv_nominal.iloc[i]:.6f}, diff={abs(manual_nominal.iloc[i]-csv_nominal.iloc[i]):.2e}")
np.testing.assert_allclose(manual_nominal, csv_nominal, rtol=1e-6)
print("\n✓ Manual reconstruction matches CSV to < 1e-6 relative tolerance")Field-by-field reconstruction (first 5):
Field 1: manual=0.574630, CSV=0.574630, diff=1.44e-15
Field 2: manual=0.574983, CSV=0.574983, diff=4.66e-15
Field 3: manual=0.577710, CSV=0.577710, diff=4.00e-15
Field 4: manual=0.571719, CSV=0.571719, diff=4.55e-15
Field 5: manual=0.570276, CSV=0.570276, diff=2.11e-15
✓ Manual reconstruction matches CSV to < 1e-6 relative toleranceFigure 3 需要把14个视场汇成一个 all-field estimator。采用与实测 CGMsum 相同的 inverse-variance weighting（以观测统计误差为权重）：
\[S_{\mathrm{all}} = \frac{\sum_i w_i S_i}{\sum_i w_i}, \quad w_i = \frac{1}{\sigma_{\mathrm{stat},i}^2}\]
# Inverse-variance weighted average using measurement statistical errors
flux = halosat["nominal_absorbed_0p5_2p0_fluxunit"].to_numpy()
sigma = halosat["measurement_staterr_absorbed_0p5_2p0_fluxunit"].to_numpy()
weights = 1.0 / sigma**2
all_field = np.average(flux, weights=weights)
# Footprint range (min/max across 14 fields)
footprint_lo = flux.min()
footprint_hi = flux.max()
# Patchiness predictor (inverse-variance weighted mean, same weights as all-field)
patchiness = np.average(halosat["patchiness_sigma_absorbed_0p5_2p0_fluxunit"].to_numpy(), weights=weights)
# No-cross-term structural check
flux_nc = halosat["no_cross_term_absorbed_0p5_2p0_fluxunit"].to_numpy()
all_field_nc = np.average(flux_nc, weights=weights)
print(f"All-field inverse-variance prediction: {all_field:.6f}")
print(f"14-field footprint range: [{footprint_lo:.6f}, {footprint_hi:.6f}]")
print(f"Patchiness predictor (mean sigma): {patchiness:.6f}")
print(f"No-cross-term structural check: {all_field_nc:.6f}")All-field inverse-variance prediction: 0.565386
14-field footprint range: [0.526442, 0.577710]
Patchiness predictor (mean sigma): 0.165165
No-cross-term structural check: 0.438683# Frozen Figure 3 ledger values from I023 / B014
LEDGER_ALL_FIELD = 0.565386
LEDGER_FOOTPRINT_LO = 0.526442
LEDGER_FOOTPRINT_HI = 0.577710
LEDGER_PATCHINESS = 0.165165
LEDGER_NO_CROSS = 0.438683
np.testing.assert_allclose(all_field, LEDGER_ALL_FIELD, rtol=1e-4)
np.testing.assert_allclose(footprint_lo, LEDGER_FOOTPRINT_LO, rtol=1e-4)
np.testing.assert_allclose(footprint_hi, LEDGER_FOOTPRINT_HI, rtol=1e-4)
np.testing.assert_allclose(patchiness, LEDGER_PATCHINESS, rtol=1e-4)
np.testing.assert_allclose(all_field_nc, LEDGER_NO_CROSS, rtol=1e-4)
print("✓ Matches frozen Figure 3 ledger to < 1e-4")✓ Matches frozen Figure 3 ledger to < 1e-4fig, ax = plt.subplots(figsize=(9, 5))
# Plot each field with patchiness error bars
field_idx = np.arange(len(halosat))
ax.errorbar(
field_idx, flux,
yerr=halosat["patchiness_sigma_absorbed_0p5_2p0_fluxunit"],
fmt="o", color="#2676b8", markersize=6, capsize=3, linewidth=0.8,
label="14 M31 fields (patchiness sigma)"
)
# All-field prediction
ax.axhline(all_field, color="#d47a2c", linewidth=2, label=f"All-field prediction = {all_field:.3f}")
ax.axhspan(footprint_lo, footprint_hi, color="#d47a2c", alpha=0.12, label=f"Footprint [{footprint_lo:.3f}, {footprint_hi:.3f}]")
# Observed CGMsum total for reference
ax.axhline(0.964727, color="#b74842", linestyle="--", linewidth=1.5, label="Observed CGMsum = 0.965 (M31)")
ax.set_xlabel("Field index (0-13)")
ax.set_ylabel("Absorbed 0.5-2.0 keV brightness (10^-15 erg s^-1 cm^-2 arcmin^-2)")
ax.set_title("HaloSat extrapolation: 14 M31 fields")
ax.set_xticks(field_idx)
ax.legend(fontsize=9, loc="upper right")
plt.show()
关键观察： - 14场的 nominal 预测集中在约0.53–0.58，变化不大——因为14场在天空上很接近，模型给出的EM很相似。 - All-field 预测 0.565 明显低于实测 CGMsum 总量 0.965——这说明 HaloSat 域外外推只解释了M31方向总量的一部分，剩余部分来自M31自身的CGM。 - Patchiness sigma（蓝色误差棒）约0.16，反映模型承认的南天场间不均匀性。
| 步骤 | 假设 | 来源 |
|---|---|---|
| 拟合域 | 73个 \(b<-30^\circ\) 南天场 | Kaaret+2020 catalog |
| EM convention | \(\int(n_d+n_h)^2 ds\)，保留交叉项 | catalog \(\chi^2\) closure |
| \(n\) 的粒子身份 | 未定义（不乘 \(n_e/n_H=1.2\)） | 论文未说明 |
| 温度 | \(kT=0.225\) keV（sample median） | 额外补充，模型不预测M31方向温度 |
| 丰度 | \(Z=0.3\) | 项目统一选择 |
| 吸收 | TBabs + Wilms + Verner + field-specific HI4PI | 项目实现 |
| 积分路径 | \(R_\odot=8.0\) kpc，0–260 kpc 内部LOS | 项目实现 |
| M31位置 | \(b\approx-21.6^\circ\)，全部14场在拟合域外 | 域外外推 |
| 证据等级 | supplemental extrapolation，非直接预测 | 本项目定位 |
计算 nominal EM 与 no-cross-term EM 的比值，量化交叉项对发射量度的贡献比例。
Cross-term ratio (nominal / no-cross):
mean = 1.2881
std = 0.0019
The cross term contributes ~28.8% of the EM.
Removing it would drop the absorbed flux to ~0.439 (structural check).计算14场的 \(N_\mathrm{H}\) 与 flux-per-EM 之间的 Pearson 相关系数。这个相关性是正还是负？为什么？
Pearson r = -0.9998
Negative — higher N_H means more absorption, so each unit of EM produces less absorbed flux.Patchiness sigma 是什么？它和 measurement statistical error 有什么区别？
Patchiness sigma (mean): 0.16569271558482795
Measurement staterr (mean): 0.2786016613799958
Patchiness sigma reflects real spatial scatter in the southern HaloSat fields
(the model explicitly requires sigma_p ≈ 3.4e-3 cm^-6 pc of EM scatter).
Measurement staterr is the observational uncertainty of each M31 field.
They are independent sources of spread: one is model-intrinsic, one is instrumental.graph TD
A["HaloSat 73 southern fields<br/>b < -30 deg, EM + kT fitted"] --> B["Empirical disk<br/>n0=0.0081, z0=1.60 kpc"]
A --> C["Adiabatic halo<br/>Fang+2013 in NFW potential"]
B --> D["EM = ∫(n_d+n_h)² ds<br/>cross term retained, no n_e/n_H factor"]
C --> D
D --> E["Project to 14 M31 sightlines<br/>R_sun=8.0 kpc, 0-260 kpc, |z|"]
E --> F["Augment kT=0.225 keV<br/>(model has no M31-direction T)"]
F --> G["APEC(Z=0.3) × TBabs × HI4PI<br/>→ absorbed 0.5-2.0 keV"]
G --> H["14-field inverse-variance average<br/>= 0.565 (Figure 3 point)"]
H --> I["Supplemental extrapolation<br/>NOT a direct HaloSat prediction"]graph TD
A["HaloSat 73 southern fields<br/>b < -30 deg, EM + kT fitted"] --> B["Empirical disk<br/>n0=0.0081, z0=1.60 kpc"]
A --> C["Adiabatic halo<br/>Fang+2013 in NFW potential"]
B --> D["EM = ∫(n_d+n_h)² ds<br/>cross term retained, no n_e/n_H factor"]
C --> D
D --> E["Project to 14 M31 sightlines<br/>R_sun=8.0 kpc, 0-260 kpc, |z|"]
E --> F["Augment kT=0.225 keV<br/>(model has no M31-direction T)"]
F --> G["APEC(Z=0.3) × TBabs × HI4PI<br/>→ absorbed 0.5-2.0 keV"]
G --> H["14-field inverse-variance average<br/>= 0.565 (Figure 3 point)"]
H --> I["Supplemental extrapolation<br/>NOT a direct HaloSat prediction"]
一句话总结：Kaaret et al. (2020) 用 HaloSat 的73个南天场拟合出银河系热气体的 empirical disk + adiabatic halo 三维模型。这个模型沿M31方向14条视线外推积分，补充 sample-median 温度后转换为 absorbed 0.5–2.0 keV brightness，得到 all-field 预测 0.565——这是一个 supplemental domain extrapolation，不是 HaloSat 对M31方向的直接预测，因为M31完全在拟合域外。
教程结束 🎓 下一步：继续阅读 Ueda+2022 tutorial，了解 in-domain disk model 如何在M31 footprint内直接投影。