Physics and statistics from zero从零开始的物理与统计

From photons to a fitted spectrum从光子到拟合谱

XMM-Newton EPIC MOS is a CCD: each incoming X-ray photon is converted to a charge packet read out in an energy channel, giving a spectrum of counts vs channel. The instrument response turns photon energy into observed channel: the RMF (redistribution matrix file) maps true energy to detected channel, and the ARF (ancillary response file) gives the effective area vs energy. A spectrum file also carries BACKSCAL, the extraction-area scaling that converts counts to surface brightness. Background has two parts: the quiescent particle background (QPB, from closed-filter or blank-sky data) and the astrophysical sky background.

XMM-Newton EPIC MOS 是一个 CCD：每个入射 X 射线光子转换为一个电荷包，按能量通道读出，得到通道 vs 计数的谱。仪器响应把光子能量转换为观测通道：RMF（重分布矩阵文件）把真实能量映射到探测通道，ARF（辅助响应文件）给出能量函数的有效面积。谱文件还携带 BACKSCAL，即提取面积缩放，用于把计数转为表面亮度。背景有两部分：静止粒子背景（QPB，来自闭滤镜或空白天空数据）和天体物理天空背景。

Worked example: one run6 MOS spectrum (ObsID 0109270301, MOS2, region a032_040) has EXPOSURE = 26307.5 s and BACKSCAL = 35339748 in its SPECTRUM header (read with astropy, command on the howto page). If the source region area is \(A\) arcmin\(^2\), the surface brightness is roughly counts / (EXPOSURE \(\times\) BACKSCAL \(\times\) A).

用数字例子：一条 run6 MOS 谱（ObsID 0109270301，MOS2，区域 a032_040）的 SPECTRUM 头中 EXPOSURE = 26307.5 s，BACKSCAL = 35339748（用 astropy 读取，命令在 howto 页面）。若源区域面积为 \(A\) arcmin\(^2\)，表面亮度约为 counts / (EXPOSURE \(\times\) BACKSCAL \(\times\) A)。

What kT and norm meankT 与 norm 的含义

A thermal plasma (APEC) spectrum is set by the temperature \(kT\) (in keV) and the emission measure. In XSPEC the APEC norm is \(10^{-14}/(4\pi D^2)\int n_e n_H\,dV\) (cm\(^{-5}\)), with \(D\) in cm. For \(kT=0.7\) keV, Z=1 (angr), unabsorbed APEC, the flux per unit norm is \(3.0247\times10^{-9}\) erg/s/cm\(^2\) in 0.5–2 keV and \(3.9497\times10^{-9}\) erg/s/cm\(^2\) in 0.1–10 keV (ratio 0.766). So a SrcApec norm of \(3.19\times10^{-7}\) (the a=20' annulus) corresponds to a 0.5–2 keV flux of \(3.19\times10^{-7}\times3.0247\times10^{-9} = 9.6\times10^{-16}\) erg/s/cm\(^2\) before area normalisation.

热等离子体（APEC）谱由温度 \(kT\)（单位 keV）和发射量度决定。在 XSPEC 中，APEC norm 是 \(10^{-14}/(4\pi D^2)\int n_e n_H\,dV\)（cm\(^{-5}\)），\(D\) 单位 cm。对 \(kT=0.7\) keV，Z=1（angr），未吸收 APEC，每单位 norm 的通量在 0.5–2 keV 为 \(3.0247\times10^{-9}\) erg/s/cm\(^2\)，在 0.1–10 keV 为 \(3.9497\times10^{-9}\) erg/s/cm\(^2\)（比值 0.766）。所以 a=20' 环带的 SrcApec norm \(3.19\times10^{-7}\) 对应未归一化前的 0.5–2 keV 通量 \(3.19\times10^{-7}\times3.0247\times10^{-9} = 9.6\times10^{-16}\) erg/s/cm\(^2\)。

Source: output/m31hotism_theoretical_expectation_20260910_run1/theoretical_expectation_budget.csv.来源：output/m31hotism_theoretical_expectation_20260910_run1/theoretical_expectation_budget.csv.

Animation: add a constant sky level to a stellar profile动画：给恒星剖面加上一个恒定天空水平

Drag the slider to change the pedestal level拖动滑块改变底座水平

Foreground–source degeneracy前景–源退化

A foreground (LB + MWhalo + CXB) and an M31 SrcApec component can trade flux: making the foreground brighter lets SrcApec's norm shrink, and vice versa, with almost the same \(\chi^2\). The errors from a fit with the foreground frozen (conditional \(\Delta\chi^2=1\)) are far too small — after R008, profile-likelihood errors with the foreground free are required. The figure shows why: profiling the foreground widens the norm error by more than a factor of two.

前景（LB + MWhalo + CXB）与 M31 SrcApec 成分可以互换通量：前景越亮，SrcApec 的 norm 越小，反之亦然，\(\chi^2\) 几乎不变。前景冻结拟合得到的误差（条件性 \(\Delta\chi^2=1\)）远小于真实误差——R008 后，要求用前景自由的 profile-likelihood 误差。下图说明原因：对前景进行 profile 会将 norm 误差扩大两倍以上。

Source: illustrative toy based on R008 (conditional errors must grow \(\times2.84\) linear / \(\times1.96\) power law to reach \(\chi^2/\mathrm{dof}=1\)).来源：illustrative toy based on R008 (conditional errors must grow \(\times2.84\) linear / \(\times1.96\) power law to reach \(\chi^2/\mathrm{dof}=1\)).

Animation: chi-square against the norm, foreground frozen vs refit动画：卡方 vs 标准化量，前景冻结 vs 重拟合

The stellar-floor prediction恒星底板预测

The unresolved-stellar floor is \(L_X\) per stellar mass times stellar mass: \(L_X = (L_X/M_*)\times M_*\). Ponti+26 give \(L_X/M_* = 1.05\times10^{28}\) erg/s/\(M_\odot\) (all stars) or \(6.9\times10^{27}\) (bright resolved point sources removed), 0.1–10 keV, from a fixed \(kT=0.7\) keV, \(Z=Z_\odot\) APEC fit to the western Galactic hemisphere. For M31, \(M_*\) comes from WISE \(W1\) via \(M/L_{W1}\approx 0.6\) (Telford+2020, 0.4 dex spread).

未解析恒星底板是每恒星质量的 \(L_X\) 乘以恒星质量：\(L_X = (L_X/M_*)\times M_*\)。Ponti+26 给出 \(L_X/M_* = 1.05\times10^{28}\) erg/s/\(M_\odot\)（所有恒星）或 \(6.9\times10^{27}\)（移除亮已解析点源后），0.1–10 keV，来自对西银河半球固定 \(kT=0.7\) keV、\(Z=Z_\odot\) APEC 的拟合。对 M31，\(M_*\) 由 WISE \(W1\) 通过 \(M/L_{W1}\approx 0.6\) 得到（Telford+2020，0.4 dex 离散）。

Worked example: if an annulus has stellar mass \(M_* = 10^9\,M_\odot\), the bright-removed stellar floor predicts \(L_X = 6.9\times10^{27}\times10^9 = 6.9\times10^{36}\) erg/s (0.1–10 keV). Compare with the observed diffuse luminosity to see whether a residual remains.

用数字例子：若一环带恒星质量 \(M_* = 10^9\,M_\odot\)，则亮源移除后的恒星底板预测 \(L_X = 6.9\times10^{27}\times10^9 = 6.9\times10^{36}\) erg/s（0.1–10 keV）。与观测弥散光度比较，看是否还留有残余。

Survey map to spatial template; the sky pedestal调查图到空间模板；天空底座

A WISE mosaic becomes a spatial template by projecting its pixel values onto each MOS window (annulus). But a raw mosaic carries an off-disk sky level (the pedestal): B026 measured W1 off-disk = 6.87 DN, while on-disk medians are 12.92 (a=32–48'), 9.71 (48–64'), 8.08 (64–80'), 7.38 (80–96'), 7.08 (96–112') — the pedestal is 53–97% of the W1 template, and W4dust is \(\approx\)99% pedestal.

Why the pedestal does not fake a young component: adding a constant to every pixel of a template only moves the constant term \(C\) in the fit — it cannot change the amplitudes \(A_\mathrm{old}\) or \(A_\mathrm{young}\). With a free, unbounded constant the fit is shift-invariant: \(\sigma(A_\mathrm{old})\) and \(\sigma(A_\mathrm{young})\) are identical for raw and net templates; only the correlation \(\mathrm{corr}(A_\mathrm{young}, C)\) moves (from −0.9996 to −0.02…−0.13, R022). So R006 “mixed old+young not identifiable” is caused by the narrow W4dust range across windows, not by the sky pedestal (R022 narrows R013). “Old beats young by \(\Delta\chi^2=770\)” compared W1+pedestal against an essentially flat template, so “constant \(\to\) 0 with W1” is not evidence against a flat component. V6 rebuilds with net (sky-subtracted) templates.

WISE 马赛克通过把像素值投影到每个 MOS 窗口（环带）成为空间模板。但原始马赛克带有一个场外天空水平（底座）：B026 测得 W1 场外 = 6.87 DN，而盘中中位数为 12.92（a=32–48'）、9.71（48–64'）、8.08（64–80'）、7.38（80–96'）、7.08（96–112'）——底座是 W1 模板的 53–97%，W4dust 约为 99% 底座。

为什么底座不会伪造一个年轻成分：给每个像素加上同一个常数，只会把拟合中的常数项 \(C\) 平移，而不能改变 \(A_\mathrm{old}\) 或 \(A_\mathrm{young}\) 的振幅。在带自由无界常数的情况下，拟合是平移不变的：\(\sigma(A_\mathrm{old})\) 和 \(\sigma(A_\mathrm{young})\) 对原始模板与净模板完全相同；只有 \(\mathrm{corr}(A_\mathrm{young}, C)\) 发生变化（从 −0.9996 到 −0.02…−0.13，R022）。因此 R006 “混合 old+young 不可辨识”的成因是 W4dust 窄范围，而非天空底座（R022 收窄了 R013）。“old 胜 young，\(\Delta\chi^2=770\)” 把 W1+底座与一个基本平坦的模板相比较，所以“constant \(\to\) 0 with W1”不是反对平坦成分的证据。V6 用净（减天空）模板重建。

Source: cognition-os/bug_impacts/B026_wise_template_sky_pedestal.md (raw-map medians, every 4th pixel); 07_REVISION_LOG.md R022 (RT-08).来源：cognition-os/bug_impacts/B026_wise_template_sky_pedestal.md (raw-map medians, every 4th pixel)；07_REVISION_LOG.md R022（RT-08）。

The same pedestal logic applies to the one-foreground story on the state page: a constant foreground level cannot be separated from a constant term in the fit, which is exactly why the disk soft floor and the halo-field 0.7 keV term can be confused.

同样的底座逻辑也适用于状态页上的单一前景故事：恒定前景水平在拟合中无法与恒定项分开，这正是为什么圆盘软底板和晕场 0.7 keV 项会被混淆。