附录 E 基本方程

辐射的比强度 $I_{\nu}$ 定义为

\boxed{I_{\nu} \equiv \frac{dP}{{({\cos{\theta d\sigma}})}d\nu d\Omega},} \tag{\href{02-chapter-2-radiation-fundamentals#E2}{2.2}}

其中 $$dP$$ 是探测器在投影面积 $({\cos{\theta d\sigma}})$ 内、在立体角 $d\Omega$ 内、以及在频率范围从 $\nu$ 到 $\nu + {d\nu}$ 内接收到的功率。同样地， $I_{\lambda}$ 是单位波长的亮度：

\boxed{I_{\lambda} \equiv \frac{dP}{{({\cos{\theta d\sigma}})}d\lambda d\Omega}.} \tag{\href{02-chapter-2-radiation-fundamentals#E3}{2.3}}

这两个量的关系为

\boxed{\left. \frac{I_{\lambda}}{I_{\nu}} = \middle| \frac{d\nu}{d\lambda} \middle| = \frac{c}{\lambda^{2}} = \frac{\nu^{2}}{c}. \right.} \tag{\href{02-chapter-2-radiation-fundamentals#E5}{2.5}}

一个源的通量密度 $S_{\nu}$ 是每单位探测器面积接收到的谱功率：

\boxed{S_{\nu} \equiv \int_{source}I_{\nu}{(\theta,\phi)}\cos\theta d\Omega.} \tag{\href{02-chapter-2-radiation-fundamentals#E9}{2.9}}

如果源足够紧凑，使得 ${\cos\theta} \approx 1$ ，则

\boxed{S_{\nu} \approx \int_{source}I_{\nu}{(\theta,\phi)}d\Omega.} \tag{\href{02-chapter-2-radiation-fundamentals#E10}{2.10}}

通量密度的 MKS 单位是 $Wm^{- 2}{Hz}^{- 1}$ ； ${1{jansky}{({Jy})}} \equiv {10^{- 26}Wm^{- 2}{Hz}^{- 1}}$ 。

光谱亮度 $L_{\nu}$ 的定义是光源在频率 $\nu$ 辐射的单位频率总功率；其 MKS 单位为 W Hz $^{- 1}$ 。在自由空间中，并且在距离 $$d$$ 远大于光源尺寸的情况下，反平方定律

\boxed{L_{\nu} = {4\pi d^{2}S_{\nu}}} \tag{\href{02-chapter-2-radiation-fundamentals#E15}{2.15}}

将各向同性光源的光谱亮度与其通量密度联系起来。

吸收体在频率 $\nu$ 处的线性吸收系数定义为光子在厚度为 $$ds$$ 的薄层中被吸收的概率 $dP{(\nu)}$ ：

\boxed{\kappa{(\nu)} \equiv \frac{dP{(\nu)}}{ds}.} \tag{\href{02-chapter-2-radiation-fundamentals#E18}{2.18}}

不透明度或光学厚度 $\tau$ 定义为从光源端开始通过吸收体的所有微小概率的总和：

\boxed{\tau \equiv \int_{s_{out}}^{s_{in}} - \kappa{(s')}ds'.} \tag{\href{02-chapter-2-radiation-fundamentals#E23}{2.23}}

在频率 $\nu$ 处的发射系数是每单位微小距离 $$ds$$ 的比强度 $dI_{\nu}$ 的微小增加：

\boxed{j_{\nu} \equiv \frac{dI_{\nu}}{ds}.} \tag{\href{02-chapter-2-radiation-fundamentals#E26}{2.26}}

辐射传输方程为

\boxed{\frac{dI_{\nu}}{ds} = - \kappa I_{\nu} + j_{\nu}.} \tag{\href{02-chapter-2-radiation-fundamentals#E27}{2.27}}

对于任何处于局部热力学平衡 (LTE) 的物质，基尔霍夫定律通过黑体辐射的比强度 $B_{\nu}$ 连接发射系数和吸收系数：

\boxed{\frac{j_{\nu}}{\kappa} = B_{\nu}{(T)}.} \tag{\href{02-chapter-2-radiation-fundamentals#E30}{2.30}}

具有任意比强度 $I_{\nu}$ 的辐射源的亮温度定义为

\boxed{T_{b}{(\nu)} \equiv \frac{I_{\nu}c^{2}}{2k\nu^{2}}.} \tag{\href{02-chapter-2-radiation-fundamentals#E33}{2.33}}

对于LTE中的不透明物体，基尔霍夫定律将发射系数 $e_{\nu}$ （物体每单位面积发射的光谱功率与黑体每单位面积发射的光谱功率之比）与吸收系数 $a_{\nu}$ （物体吸收辐射的比例）以及反射系数 $r_{\nu}$ （物体反射辐射的比例）联系起来：

\boxed{e_{\nu} = a_{\nu} = 1 - r_{\nu}.} \tag{\href{02-chapter-2-radiation-fundamentals#E47}{2.47}}

辐射的光谱能量密度为

\boxed{u_{\nu} = \frac{1}{c}\int I_{\nu}d\Omega.} \tag{\href{02-chapter-2-radiation-fundamentals#E76}{2.76}}

当 ${h\nu} \ll {kT}$ 时，黑体辐射的比辐亮度的瑞利--金斯近似为

\boxed{B_{\nu} = \frac{2kT\nu^{2}}{c^{2}} = \frac{2kT}{\lambda^{2}}.} \tag{\href{02-chapter-2-radiation-fundamentals#E79}{2.79}}

光子的能量为

\boxed{E = h\nu.} \tag{\href{02-chapter-2-radiation-fundamentals#E81}{2.81}}

在任意频率下黑体辐射比辐亮度的普朗克公式为

\boxed{B_{\nu} = \frac{2h\nu^{3}}{c^{2}}\frac{1}{{\exp\left( \frac{h\nu}{kT} \right)} - 1}.} \tag{\href{02-chapter-2-radiation-fundamentals#E86}{2.86}}

黑体辐射的总强度为

\boxed{B{(T)} \equiv \int_{0}^{\infty}B_{\nu}{(T)}d\nu = \frac{\sigma T^{4}}{\pi},} \tag{\href{02-chapter-2-radiation-fundamentals#E89}{2.89}}

其中斯特藩-玻尔兹曼常数 $\sigma$ 定义为

\boxed{\sigma \equiv \frac{2\pi^{5}k^{4}}{15c^{2}h^{3}} \approx 5.67 \times 10^{- 5}\frac{erg}{{cm}^{2}sK^{4}{sr}}.} \tag{\href{02-chapter-2-radiation-fundamentals#E90}{2.90}}

黑体辐射的总能量密度为

\boxed{u = \frac{4\sigma T^{4}}{c} = aT^{4},} \tag{\href{02-chapter-2-radiation-fundamentals#E93}{2.93}}

其中 $a \equiv {{4\sigma}/c} \approx {{7.56577 \times 10^{- 15}}{erg}{cm}^{- 3}K^{- 4}}$ 是辐射常数。

黑体辐射的光子数密度为

\boxed{\left( \frac{n_{\gamma}}{{cm}^{- 3}} \right) \approx 20.3\left( \frac{T}{K} \right)^{3}.} \tag{\href{02-chapter-2-radiation-fundamentals#E100}{2.100}}

黑体辐射的平均光子能量为

\boxed{{\langle E_{\gamma}\rangle} \approx \, 2.70kT.} \tag{\href{02-chapter-2-radiation-fundamentals#E101}{2.101}}

峰值黑体亮度的频率（每单位频率） $B_{\nu}$ 为

\boxed{\left( \frac{\nu_{\max}}{GHz} \right) \approx 59\left( \frac{T}{K} \right).} \tag{\href{02-chapter-2-radiation-fundamentals#E104}{2.104}}

峰值黑体亮度的波长（每单位波长） $B_{\lambda}$ 由维恩位移定律给出：

\boxed{\left( \frac{\lambda_{\max}}{cm} \right) \approx 0.29\left( \frac{T}{K} \right)^{- 1}.} \tag{\href{02-chapter-2-radiation-fundamentals#E106}{2.106}}

各向同性辐射的通量密度为

\boxed{S_{\nu} = \pi I_{\nu}.} \tag{\href{02-chapter-2-radiation-fundamentals#E109}{2.109}}

在极限 ${h\nu} \ll {kT}$ 下，温热电阻器产生的光谱功率的奈奎斯特近似为

\boxed{P_{\nu} = kT.} \tag{\href{02-chapter-2-radiation-fundamentals#E117}{2.117}}

在任何频率下，精确的奈奎斯特公式为

\boxed{P_{\nu} = \frac{h\nu}{{\exp\left( \frac{h\nu}{kT} \right)} - 1}.} \tag{\href{02-chapter-2-radiation-fundamentals#E119}{2.119}}

使宇宙闭合所需的临界密度为

\boxed{\rho_{c} = \frac{3H_{0}^{2}}{8\pi G} \approx 8.6 \times 10^{- 30}g{cm}^{- 3}.} \tag{\href{02-chapter-2-radiation-fundamentals#E126}{2.126}}

红移 $$z$$ 定义为

\boxed{z \equiv \frac{\lambda_{o} - \lambda_{e}}{\lambda_{e}} = \frac{\lambda_{o}}{\lambda_{e}} - 1 = \frac{\nu_{e}}{\nu_{o}} - 1,} \tag{\href{02-chapter-2-radiation-fundamentals#E127}{2.127}}

其中 $\lambda_{e}$ 和 $\nu_{e}$ 分别是红移为 $$z$$ 的源发射的波长和频率， $\lambda_{o}$ 和 $\nu_{o}$ 是在 $$z = 0$$ 处观测到的波长和频率。

红移 $$z$$ 与膨胀尺度因子 $$a$$ 的关系为

\boxed{{(1 + z)} = a^{- 1}.} \tag{\href{02-chapter-2-radiation-fundamentals#E128}{2.128}}

红移为 $$z$$ 的宇宙微波背景温度为

\boxed{T = T_{0}{(1 + z)}.} \tag{\href{02-chapter-2-radiation-fundamentals#E129}{2.129}}

距离 $$r$$ 处由电荷 $$q$$ 在加速度 $\overset{˙}{v}$ 方向偏 $\theta$ 角度处辐射的电场为

\boxed{E_{\bot} = \frac{q\overset{˙}{v}{\sin\theta}}{rc^{2}}.} \tag{\href{02-chapter-2-radiation-fundamentals#E136}{2.136}}

在真空中，波印廷通量或单位面积的功率为

\boxed{|\overset{\rightarrow}{S}| = \frac{c}{4\pi}E^{2}.} \tag{\href{02-chapter-2-radiation-fundamentals#E139}{2.139}}

由加速电荷辐射的总功率由拉莫尔公式给出

\boxed{P = \frac{2}{3}\frac{q^{2}{\overset{˙}{v}}^{2}}{c^{3}},} \tag{\href{02-chapter-2-radiation-fundamentals#E143}{2.143}}

该公式仅在 $v \ll c$ 时有效。

三角函数的指数表示法为

\boxed{e^{- {i\omega t}} = \cos{(\omega t)} - i\sin{(\omega t)}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E2}{3.2}}

电流定义为电荷对时间的导数：

\boxed{I \equiv \frac{dq}{dt}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E4}{3.4}}

短天线的功率分布为

\boxed{P \propto \sin^{2}\theta.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E14}{3.14}}

由电流 $I = {I_{0}e^{- {i\omega t}}}$ 驱动的短 ( $l \ll \lambda$ ) 偶极子的发射功率为

\boxed{{\langle P\rangle} = \frac{\pi^{2}}{3c}\left( \frac{I_{0}l}{\lambda} \right)^{2}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E17}{3.17}}

辐射电阻定义为

\boxed{R \equiv \frac{2{\langle P\rangle}}{I_{0}^{2}}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E25}{3.25}}

能量守恒意味着任何无损天线的平均功率增益为

\boxed{{\langle G\rangle} = 1} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E32}{3.32}}

以及

\boxed{\int_{sphere}Gd\Omega = 4\pi.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E33}{3.33}}

波束立体角定义为

\boxed{\Omega_{A} \equiv \frac{4\pi}{G_{\max}} = \frac{1}{G_{\max}}\int_{4\pi}G{(\theta,\phi)}d\Omega.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E34}{3.34}}

天线的有效面积定义为

\boxed{A_{e} \equiv 2P_{\nu}/S_{\nu},} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E35}{3.35}}

其中 $P_{\nu}$ 是由总通量密度为 $S_{\nu}$ 的非极化点源产生的输出功率密度。

任何无损天线的平均有效面积为

\boxed{{\langle A_{e}\rangle} = \frac{\lambda^{2}}{4\pi}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E41}{3.41}}

互惠性意味着

\boxed{G{(\theta,\phi)} \propto A_{e}{(\theta,\phi)}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E44}{3.44}}

互惠性和能量守恒意味着

\boxed{A_{e}{(\theta,\phi)} = \frac{\lambda^{2}G{(\theta,\phi)}}{4\pi}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E46}{3.46}}

天线温度定义为

\boxed{T_{A} \equiv \frac{P_{\nu}}{k}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E47}{3.47}}

由非极化点源产生的天线温度，其流量密度为 $$S$$ ，是

\boxed{T_{A} = \frac{A_{e}S}{2k}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E48}{3.48}}

如果 $A_{e} \approx {2761m^{2}}$ ，则点源灵敏度为 $1K{Jy}^{- 1}$ 。

对于覆盖固体角 $\Omega_{s}$ 的亮温为 $T_{b}$ 的均匀紧凑源，

\boxed{\frac{T_{A}}{T_{b}} = \frac{\Omega_{s}}{\Omega_{A}}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E56}{3.56}}

主波束固体角定义为对主波束积分到第一个零点为止：

\boxed{\Omega_{MB} \equiv {\frac{1}{G_{\max}}{\int_{MB}{G{(\theta,\phi)}{d\Omega}}}}} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E57}{3.57}}

并用于主波束效率的定义：

\boxed{\eta_{B} \equiv \frac{\Omega_{MB}}{\Omega_{A}}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E58}{3.58}}

在焦距为 $$f$$ 的抛物面反射器顶点上方轴向距离 $$r$$ 处的高度为 $$z$$

\boxed{z = \frac{r^{2}}{4f}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E60}{3.60}}

光阑的远场距离：直径为 $$D$$ 的光阑在波长 $\lambda$ 下的远场距离为

\boxed{R_{ff} \approx \frac{2D^{2}}{\lambda}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E64}{3.64}}

在远场中，光阑天线的电场分布是光阑照明的傅里叶变换：

{{l} \equiv {\sin\theta}},

{{u} \equiv \frac{x}{\lambda}},

{{f{(l)}} = {\int_{aperture}{g{(u)}e^{- {i2\pi lu}}{du}}}}.

( 3.69 ) ( 3.72 ) ( 3.73 )

均匀照明线性光阑的功率分布为

\boxed{P{(\theta)} \propto {sinc}^{2}\left( \frac{\theta D}{\lambda} \right),} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E79}{3.79}}

其中 ${{sinc}{(x)}} \equiv {{\sin{({\pi x})}}/{({\pi x})}}$ ，半功率波束宽度为

\boxed{\theta_{HPBW} \approx 0.89\frac{\lambda}{D}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E82}{3.82}}

具有渐缩照明的典型射电望远镜的半功率波束宽度(HPBW) 为

\boxed{\theta_{HPBW} \approx 1.2\frac{\lambda}{D}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E96}{3.96}}

二维光阑场分布为

\boxed{f{(l,m)} \propto \int_{- \infty}^{\infty}\int_{- \infty}^{\infty}g{(u,v)}e^{- {i2\pi{({{lu} + {mv}})}}}dudv,} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E97}{3.97}}

其中 $$m$$ 是 $$x$$ 轴上 $$l$$ 的 $$y$$ 轴类比，以及 $v \equiv {y/\lambda}$ 。二维光阑的电场分布是光阑场照明的二维傅里叶变换。

均匀照明矩形孔径的功率模式为

\boxed{G \approx \frac{4\pi D_{x}D_{y}}{\lambda^{2}}{sinc}^{2}\left( \frac{\theta_{x}D_{x}}{\lambda} \right){sinc}^{2}\left( \frac{\theta_{y}D_{y}}{\lambda} \right).} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E107}{3.107}}

孔径效率定义为

\boxed{\eta_{A} \equiv \frac{\max{(A_{e})}}{A_{geom}}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E111}{3.111}}

高斯波束的波束立体角为

\boxed{\Omega_{A} = \left( \frac{\pi}{4{\ln 2}} \right)\theta_{HPBW}^{2} \approx 1.133\theta_{HPBW}^{2}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E118}{3.118}}

表面误差均方根为 $\sigma$ 的反射器表面效率 $\eta_{s}$ 由Ruze 方程给出：

\boxed{\eta_{s} = \exp\left\lbrack - \left( \frac{4\pi\sigma}{\lambda} \right)^{2} \right\rbrack.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E129}{3.129}}

噪声温度定义为

\boxed{T_{N} \equiv \frac{P_{\nu}}{k}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E149}{3.149}}

系统噪声温度是来自所有源的噪声贡献之和：

\boxed{T_{s} = T_{cmb} + T_{rsb} + \Delta T_{source} + {\lbrack 1 - \exp{( - \tau_{A})}\rbrack}T_{atm} + T_{spill} + T_{r} + \cdots.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E150}{3.150}}

理想总功率辐射计方程为

\boxed{\sigma_{T} \approx T_{s}\left\lbrack \frac{1}{\Delta\nu\tau} \right\rbrack^{1/2}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E154}{3.154}}

实际总功率辐射计方程包括增益波动的影响：

\boxed{\sigma_{T} \approx T_{s}\left\lbrack \frac{1}{\Delta\nu\tau} + \left( \frac{\Delta G}{G} \right)^{2} \right\rbrack^{1/2}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E158}{3.158}}

Dicke 开关辐射计方程为

\boxed{\sigma_{T} \approx 2T_{s}\left\lbrack \frac{1}{\Delta\nu\tau} \right\rbrack^{1/2}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E162}{3.162}}

rms 混淆噪声由高斯波束中未解决的连续谱源引起，半高全宽 (HPBW) 为 $\theta$ ，频率为 $\nu$ ，值为：

\boxed{\left( \frac{\sigma_{c}}{{mJy}{beam}^{- 1}} \right) \approx \begin{cases} {\left. {0.2(}\frac{\nu}{GHz} \right)^{- 0.7}\left( \frac{\theta}{arcmin} \right)^{2}} & {{({\theta > {0.17{arcmin}}})},} \\ {\left. {2.2(}\frac{\nu}{GHz} \right)^{- 0.7}\left( \frac{\theta}{arcmin} \right)^{10/3}} & {{({\theta < {0.17{arcmin}}})}.} \end{cases}} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E163}{3.163}}

比混淆极限 $\approx {5\sigma_{c}}$ 更暗的单个源无法可靠检测。

通过 $$Y$$ 因子方法可以测量接收机输入噪声温度 $T_{r}$ ，其值为：

\boxed{T_{r} = \frac{T_{h} - {YT_{c}}}{Y - 1}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E168}{3.168}}

双元干涉仪对亮度分布为 $I_{\nu}{(\hat{s})}$ 的源的响应为复可见性

\boxed{\mathcal{V}_{\nu} = \int I_{\nu}{(\hat{s})}\exp{( - i2\pi\overset{\rightarrow}{b} \cdot \hat{s}/\lambda)}d\Omega.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E186}{3.186}}

为了在带宽 $\Delta\nu$ 下最小化带宽模糊，图像角半径 $\Delta\theta$ 应满足：

\boxed{\Delta\theta\Delta\nu \ll \theta_{s}\nu.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E192}{3.192}}

为了在角半径为 $\Delta\theta$ 的图像中最小化时间模糊，平均时间应满足：

\boxed{\Delta\theta\Delta t \ll \frac{\theta_{s}P}{2\pi} \approx \theta_{s} \cdot 1.37 \times 10^{4}s.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E194}{3.194}}

三维干涉仪的源亮度分布 $I_{\nu}{(l,m)}$ 与可见度 $\mathcal{V}_{\nu}{(u,v,w)}$ 之间的关系为

\boxed{\mathcal{V}_{\nu}{(u,v,w)} = \int\int\frac{I_{\nu}{(l,m)}}{{({1 - l^{2} - m^{2}})}^{1/2}}\exp{\lbrack - i2\pi{(ul + vm + wn)}\rbrack}dldm.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E197}{3.197}}

对于限制在 $$(u,v)$$ 平面的二维干涉仪，源亮度分布 $I_{\nu}{(l,m)}$ 是条纹可见度 $\mathcal{V}_{\nu}{(u,v)}$ 的傅里叶变换：

\boxed{\frac{I_{\nu}{(l,m)}}{{({1 - l^{2} - m^{2}})}^{1/2}} = \int\int\mathcal{V}_{\nu}{(u,v,0)}\exp{\lbrack + i2\pi{(ul + vm)}\rbrack}dudv.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E198}{3.198}}

具有 $$N$$ 个天线且每个天线有效面积为 $A_{e}$ 的干涉仪的点源灵敏度（或以每波束立体角的流量密度单位表示的亮度灵敏度）为

\boxed{\sigma_{S} = \frac{2kT_{s}}{A_{e}{\lbrack{N{({N - 1})}\Delta\nu\tau}\rbrack}^{1/2}}.} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E203}{3.203}}

对应于点源灵敏度 $\sigma_{S}$ 和波束立体角 $\Omega_{A}$ 的亮度灵敏度（K）为

\boxed{\sigma_{T} = \left( \frac{\sigma_{S}}{\Omega_{A}} \right)\frac{\lambda^{2}}{2k},} \tag{\href{03-chapter-3-radio-telescopes-and-radiometers#E204}{3.204}}

其中 $\Omega_{A} = {{\pi\theta_{HPBW}^{2}}/{({4{\ln 2}})}} \approx {1.133\theta_{0}^{2}}$ 对于半功率波束宽度为 $\theta_{HPBW}$ 的高斯波束。

(非相对论)麦克斯韦分布粒子速度 $$v$$ 为

\boxed{f{(v)} = \frac{4v^{2}}{\sqrt{\pi}}\left( \frac{m}{2kT} \right)^{3/2}\exp\left( - \frac{mv^{2}}{2kT} \right).} \tag{\href{04-chapter-4-freefree-radiation#E34}{4.34}}

自由-自由发射系数为

\boxed{j_{\nu} = \frac{\pi^{2}Z^{2}e^{6}n_{e}n_{i}}{4c^{3}m_{e}^{2}}\left( \frac{2m_{e}}{\pi kT} \right)^{1/2}\ln\left( \frac{b_{\max}}{b_{\min}} \right),} \tag{\href{04-chapter-4-freefree-radiation#E39}{4.39}}

其中

\boxed{b_{\min} \approx \frac{Ze^{2}}{m_{e}v^{2}}.} \tag{\href{04-chapter-4-freefree-radiation#E43}{4.43}}

自由-自由吸收系数为

\boxed{\kappa = \frac{1}{\nu^{2}T^{3/2}}\left\lbrack \frac{Z^{2}e^{6}}{c}n_{e}n_{i}\frac{1}{\sqrt{2\pi{({m_{e}k})}^{3}}} \right\rbrack\frac{\pi^{2}}{4}\ln\left( \frac{b_{\max}}{b_{\min}} \right).} \tag{\href{04-chapter-4-freefree-radiation#E52}{4.52}}

在频率低到 $\tau \gg 1$ 的情况下，Hii 区变得不透明，其光谱接近温度为 $T \sim 10^{4}$ K 的黑体，并且流量密度随 $S \propto \nu^{2}$ 变化。在非常高的频率下， $\tau \ll 1$ ，Hii 区几乎透明，并且

\boxed{S_{\nu} \propto \frac{2kT\nu^{2}}{c^{2}}\tau{(\nu)} \propto \nu^{- 0.1}.} \tag{\href{04-chapter-4-freefree-radiation#E54}{4.54}}

在对数-对数图上，均匀 Hii 区的整体光谱在 $\tau \approx 1$ 频率附近出现断点。

等离子体的发射量定义为

\boxed{\frac{EM}{{pc}{cm}^{- 6}} \equiv \int_{los}\left( \frac{n_{e}}{{cm}^{- 3}} \right)^{2}d\left( \frac{s}{pc} \right).} \tag{\href{04-chapter-4-freefree-radiation#E57}{4.57}}

等离子体的自由-自由光学深度为

\boxed{\tau \approx 3.28 \times 10^{- 7}\left( \frac{T}{10^{4}K} \right)^{- 1.35}\left( \frac{\nu}{GHz} \right)^{- 2.1}\left( \frac{EM}{{pc}{cm}^{- 6}} \right).} \tag{\href{04-chapter-4-freefree-radiation#E60}{4.60}}

赖曼连续谱光子产生的每秒电离率 $Q_{H}$ 以维持一个 Hii 区为

\boxed{\left( \frac{Q_{H}}{s^{- 1}} \right) \approx 6.3 \times 10^{52}\left( \frac{T}{10^{4}K} \right)^{- 0.45}\left( \frac{\nu}{GHz} \right)^{0.1}\left( \frac{L_{\nu}}{10^{20}W{Hz}^{- 1}} \right),} \tag{\href{04-chapter-4-freefree-radiation#E62}{4.62}}

其中 $L_{\nu}$ 是任意频率下的自由-自由亮度 $\nu$ ，足够高以至于 ${\tau{(\nu)}} \ll 1$ 。

运动电荷上的磁力为

\boxed{\overset{\rightarrow}{F} = \frac{q{({\overset{\rightarrow}{v} \times \overset{\rightarrow}{B}})}}{c}.} \tag{\href{05-chapter-5-synchrotron-radiation#E1}{5.1}}

回旋频率定义为

\boxed{\omega_{G} \equiv \frac{qB}{mc}.} \tag{\href{05-chapter-5-synchrotron-radiation#E4}{5.4}}

（非相对论）电子回旋频率（单位 MHz）为

\boxed{\left( \frac{\nu_{G}}{MHz} \right) = 2.8\left( \frac{B}{gauss} \right).} \tag{\href{05-chapter-5-synchrotron-radiation#E7}{5.7}}

洛伦兹变换为

\boxed{x = \gamma{(x' + vt')},y = y',z = z',t = \gamma{(t' + \beta x'/c)},} \tag{\href{05-chapter-5-synchrotron-radiation#E12}{5.12}}

\boxed{x' = \gamma{(x - vt)},y' = y,z' = z,t' = \gamma{(t - \beta x/c)},} \tag{\href{05-chapter-5-synchrotron-radiation#E13}{5.13}}

其中

\boxed{\beta \equiv {v/c}} \tag{\href{05-chapter-5-synchrotron-radiation#E14}{5.14}}

以及

\boxed{\gamma \equiv {({1 - \beta^{2}})}^{- {1/2}}} \tag{\href{05-chapter-5-synchrotron-radiation#E15}{5.15}}

称为洛伦兹因子。如果 $({\Delta x'},{\Delta y'},{\Delta z'},{\Delta t'})$ 和 $({\Delta x},{\Delta y},{\Delta z},{\Delta t})$ 是两个事件之间的坐标差，(线性) 洛伦兹变换的微分形式为

\boxed{\Delta x = \gamma{(\Delta x' + v\Delta t')},\Delta y = \Delta y',\Delta z = \Delta z',\Delta t = \gamma{(\Delta t' + \beta\Delta x'/c)},} \tag{\href{05-chapter-5-synchrotron-radiation#E16}{5.16}}

\boxed{\Delta x' = \gamma{(\Delta x - v\Delta t)},\Delta y' = \Delta y,\Delta z' = \Delta z,\Delta t' = \gamma{(\Delta t - \beta\Delta x/c)}.} \tag{\href{05-chapter-5-synchrotron-radiation#E17}{5.17}}

电子的汤姆孙截面定义为

\boxed{\sigma_{T} \equiv \frac{8\pi}{3}\left( \frac{e^{2}}{m_{e}c^{2}} \right)^{2}.} \tag{\href{05-chapter-5-synchrotron-radiation#E33}{5.33}}

磁能密度由下式给出

\boxed{U_{B} = \frac{B^{2}}{8\pi}.} \tag{\href{05-chapter-5-synchrotron-radiation#E35}{5.35}}

单个电子的同步辐射功率为

\boxed{P = 2\sigma_{T}\beta^{2}\gamma^{2}cU_{B}\sin^{2}\alpha.} \tag{\href{05-chapter-5-synchrotron-radiation#E37}{5.37}}

对所有俯仰角取平均的同步辐射功率 $\alpha$ 为

\boxed{{\langle P\rangle} = \frac{4}{3}\sigma_{T}\beta^{2}\gamma^{2}cU_{B}.} \tag{\href{05-chapter-5-synchrotron-radiation#E42}{5.42}}

单个电子的同步辐射谱为

\boxed{P{(\nu)} = \frac{\sqrt{3}e^{3}B{\sin\alpha}}{m_{e}c^{2}}\left( \frac{\nu}{\nu_{c}} \right)\int_{\nu/\nu_{c}}^{\infty}K_{5/3}{(\eta)}d\eta,} \tag{\href{05-chapter-5-synchrotron-radiation#E66}{5.66}}

其中 $K_{5/3}$ 是修正贝塞尔函数，临界频率为

\boxed{\nu_{c} = \frac{3}{2}\gamma^{2}\nu_{G}\sin\alpha \approx \gamma^{2}\nu_{G} \propto E^{2}B_{\bot}.} \tag{\href{05-chapter-5-synchrotron-radiation#E67}{5.67}}

我们银河系中观测到的宇宙射线电子的能量分布大致为幂律：

\boxed{n{(E)}dE \approx KE^{- \delta}dE,} \tag{\href{05-chapter-5-synchrotron-radiation#E70}{5.70}}

其中 $$n{(E)}dE$$ 是能量在 $$E$$ 到 $$E + {dE}$$ 之间的单位体积电子数，以及 $\delta \approx {5/2}$ 。相应的同步辐射发射系数为

\boxed{j_{\nu} \propto B^{{({\delta + 1})}/2}\nu^{{({1 - \delta})}/2}.} \tag{\href{05-chapter-5-synchrotron-radiation#E78}{5.78}}

（负号约定的）同步辐射和反康普顿辐射的谱指数为

\boxed{\alpha = \frac{\delta - 1}{2}.} \tag{\href{05-chapter-5-synchrotron-radiation#E79}{5.79}}

相对论电子在磁场 $$B$$ 中以频率 $\nu$ 辐射的有效温度为

\boxed{\left( \frac{T_{e}}{K} \right) \approx 1.18 \times 10^{6}\left( \frac{\nu}{Hz} \right)^{1/2}\left( \frac{B}{gauss} \right)^{- {1/2}}.} \tag{\href{05-chapter-5-synchrotron-radiation#E85}{5.85}}

在足够低的频率 $\nu$ 下，

\boxed{S_{\nu} \propto \nu^{- {5/2}}} \tag{\href{05-chapter-5-synchrotron-radiation#E89}{5.89}}

以及

\boxed{\left( \frac{B}{gauss} \right) \approx 1.4 \times 10^{12}\left( \frac{\nu}{Hz} \right)\left( \frac{T_{b}}{K} \right)^{- 2}.} \tag{\href{05-chapter-5-synchrotron-radiation#E91}{5.91}}

对于给定的同步加速光度，电子能量密度为

\boxed{U_{e} \propto B^{- {3/2}}.} \tag{\href{05-chapter-5-synchrotron-radiation#E98}{5.98}}

宇宙射线和磁场的总能量密度为

\boxed{U = {(1 + \eta)}U_{e} + U_{B},} \tag{\href{05-chapter-5-synchrotron-radiation#E100}{5.100}}

其中 $\eta$ 是离子/电子能量比。

在总能量最小时，粒子与场的能量比为 $\sim 1$ （能量均分）：

\boxed{\frac{{particle}{energy}}{{field}{energy}} = \frac{{({1 + \eta})}U_{e}}{U_{B}} = \frac{4}{3}.} \tag{\href{05-chapter-5-synchrotron-radiation#E107}{5.107}}

最小能量磁场为

\boxed{B_{\min} = {{\lbrack{4.5{({1 + \eta})}c_{12}L}\rbrack}^{2/7}R^{- {6/7}}{gauss}}} \tag{\href{05-chapter-5-synchrotron-radiation#E109}{5.109}}

相应的总能量为

\boxed{E_{\min}{({total})} = c_{13}{\lbrack{(1 + \eta)}L\rbrack}^{4/7}R^{9/7}{ergs}.} \tag{\href{05-chapter-5-synchrotron-radiation#E110}{5.110}}

同步加速寿命大约为

\boxed{\tau \approx c_{12}B_{\bot}^{- {3/2}},} \tag{\href{05-chapter-5-synchrotron-radiation#E112}{5.112}}

函数 $c_{12}$ 和 $c_{13}$ 在高斯 CGS 单位下绘制在图 5.10 和 5.11 中。频率限制 $\nu_{\min} = 10^{7}$ 赫兹和 $\nu_{\max} = 10^{11}$ 赫兹是常用的。

爱丁顿极限的光度为

\boxed{\left( \frac{L_{E}}{L_{\odot}} \right) \approx 3.3 \times 10^{4}\left( \frac{M}{M_{\odot}} \right).} \tag{\href{05-chapter-5-synchrotron-radiation#E117}{5.117}}

非相对论汤姆逊散射功率为

\boxed{P = \sigma_{T}cU_{rad}.} \tag{\href{05-chapter-5-synchrotron-radiation#E132}{5.132}}

相对论多普勒方程为

\boxed{\nu' = \nu{\lbrack\gamma{(1 + \beta\cos\theta)}\rbrack}.} \tag{\href{05-chapter-5-synchrotron-radiation#E142}{5.142}}

发射的净反康普顿功率为

\boxed{P_{IC} = \frac{4}{3}\sigma_{T}c\beta^{2}\gamma^{2}U_{rad}.} \tag{\href{05-chapter-5-synchrotron-radiation#E152}{5.152}}

IC/同步辐射功率比为

\boxed{\frac{P_{IC}}{P_{syn}} = \frac{U_{rad}}{U_{B}}.} \tag{\href{05-chapter-5-synchrotron-radiation#E154}{5.154}}

具有初始频率 $\nu_{0}$ 的上转散射光子的平均频率为 $\langle\nu\rangle$

\boxed{\frac{\langle\nu\rangle}{\nu_{0}} = \frac{4}{3}\gamma^{2}.} \tag{\href{05-chapter-5-synchrotron-radiation#E160}{5.160}}

非相干同步辐射源的最大静止系亮温度受反康普顿散射限制为

\boxed{T_{\max} \sim 10^{12}K.} \tag{\href{05-chapter-5-synchrotron-radiation#E163}{5.163}}

运动源分量的表观横向速度为

\boxed{\beta_{\bot}{({apparent})} = \frac{\beta{\sin\theta}}{1 - {\beta{\cos\theta}}}.} \tag{\href{05-chapter-5-synchrotron-radiation#E167}{5.167}}

对于任何 $\beta$ ，满足 $\beta_{\bot}{({apparent})}$ 最大化的角度 $\theta_{m}$ 为

\boxed{{\cos\theta_{m}} = \beta} \tag{\href{05-chapter-5-synchrotron-radiation#E170}{5.170}}

以及

\boxed{\sin\theta_{m} = \gamma^{- 1}.} \tag{\href{05-chapter-5-synchrotron-radiation#E171}{5.171}}

最大的表观横向速度为

\boxed{\max{\lbrack\beta_{\bot}{({apparent})}\rbrack} = \beta\gamma.} \tag{\href{05-chapter-5-synchrotron-radiation#E172}{5.172}}

横向多普勒频移（在 $\theta = {\pi/2}$ 处）为

\boxed{\frac{\nu}{\nu'} = \gamma^{- 1}.} \tag{\href{05-chapter-5-synchrotron-radiation#E180}{5.180}}

多普勒增强对于多普勒因子 $\delta \equiv {\nu/\nu'}$ 的范围为

\boxed{\delta^{2 + \alpha} < \frac{S}{S_{0}} < \delta^{3 + \alpha}.} \tag{\href{05-chapter-5-synchrotron-radiation#E183}{5.183}}

恒星形成星系的热射电和非热射电光度为

\boxed{\left( \frac{L_{T}}{W{Hz}^{- 1}} \right) \approx {{5.5 \times 10^{20}}\left( \frac{\nu}{GHz} \right)^{- 0.1}\left\lbrack \frac{{SFR}{(M > 5M_{\odot})}}{M_{\odot}{yr}^{- 1}} \right\rbrack}} \tag{\href{05-chapter-5-synchrotron-radiation#E184}{5.184}}

以及

\boxed{\left( \frac{L_{NT}}{W{Hz}^{- 1}} \right) \approx 5.3 \times 10^{21}\left( \frac{\nu}{GHz} \right)^{- 0.8}\left\lbrack \frac{{SFR}{(M > 5M_{\odot})}}{M_{\odot}{yr}^{- 1}} \right\rbrack.} \tag{\href{05-chapter-5-synchrotron-radiation#E185}{5.185}}

脉冲星周期为 $$P$$ 的最小平均密度为

\boxed{\rho > \frac{3\pi}{GP^{2}}.} \tag{\href{06-chapter-6-pulsars#E5}{6.5}}

旋转的磁偶极子辐射功率为

\boxed{P_{rad} = \frac{2}{3}\frac{{({\overset{¨}{m}}_{\bot})}^{2}}{c^{3}}.} \tag{\href{06-chapter-6-pulsars#E10}{6.10}}

脉冲星的能量损失光度为

\boxed{- \overset{˙}{E} \equiv - \frac{dE_{rot}}{dt} = \frac{- {4\pi^{2}I\overset{˙}{P}}}{P^{3}}.} \tag{\href{06-chapter-6-pulsars#E20}{6.20}}

脉冲星的最小磁场强度为

\boxed{\left( \frac{B}{gauss} \right) > 3.2 \times 10^{19}\left( \frac{P\overset{˙}{P}}{s} \right)^{1/2}.} \tag{\href{06-chapter-6-pulsars#E26}{6.26}}

脉冲星的特征年龄定义为

\boxed{\tau \equiv \frac{P}{2\overset{˙}{P}}.} \tag{\href{06-chapter-6-pulsars#E31}{6.31}}

脉冲星的制动指数用其可观测的周期 $$P$$ 以及一阶和二阶时间导数表示为

\boxed{n = 2 - \frac{P\overset{¨}{P}}{{\overset{˙}{P}}^{2}}.} \tag{\href{06-chapter-6-pulsars#E37}{6.37}}

在频率 $\nu$ 时，冷等离子体的折射率为

\boxed{\mu = \left\lbrack 1 - \left( \frac{\nu_{p}}{\nu} \right)^{2} \right\rbrack^{1/2},} \tag{\href{06-chapter-6-pulsars#E39}{6.39}}

其中 $\nu_{p}$ 是等离子体频率

\boxed{\nu_{p} = \left( \frac{e^{2}n_{e}}{\pi m_{e}} \right)^{1/2} \approx 8.97{kHz}\left( \frac{n_{e}}{{cm}^{- 3}} \right)^{1/2}.} \tag{\href{06-chapter-6-pulsars#E40}{6.40}}

脉冲的群速度为

\boxed{v_{g} \approx c\left( 1 - \frac{\nu_{p}^{2}}{2\nu^{2}} \right).} \tag{\href{06-chapter-6-pulsars#E42}{6.42}}

脉冲星的色散延迟为

\boxed{\left( \frac{t}{\sec} \right) \approx 4.149 \times 10^{3}\left( \frac{DM}{{pc}{cm}^{- 3}} \right)\left( \frac{\nu}{MHz} \right)^{- 2},} \tag{\href{06-chapter-6-pulsars#E45}{6.45}}

其中

\boxed{{DM} \equiv {\int_{0}^{d}{n_{e}{dl}}}} \tag{\href{06-chapter-6-pulsars#E46}{6.46}}

其单位为 pc cm $^{- 3}$ ， $$d$$ 距离处脉冲星的色散测量值为

氢原子的玻尔半径为

\boxed{a_{n} = \frac{n^{2}\hslash^{2}}{m_{e}e^{2}} \approx 0.53 \times 10^{- 8}{cm} \cdot n^{2}.} \tag{\href{07-chapter-7-spectral-lines#E6}{7.6}}

复合线的频率为

\boxed{\nu = R_{M}c\left\lbrack \frac{1}{n^{2}} - \frac{1}{{({n + {\Delta n}})}^{2}} \right\rbrack,{where}\operatorname{\quad\quad}R_{M} \equiv R_{\infty}\left( 1 + \frac{m_{e}}{M} \right)^{- 1}.} \tag{\href{07-chapter-7-spectral-lines#E12}{7.12}}

$n \gg 1$ 的近似复合线间隔频率为 ${\Delta\nu} \equiv {{\nu{(n)}} - {\nu{({n + 1})}}}$

\boxed{\frac{\Delta\nu}{\nu} \approx \frac{3}{n}.} \tag{\href{07-chapter-7-spectral-lines#E15}{7.15}}

自发发射率为

\boxed{A_{{n + 1},n} \approx \frac{64\pi^{6}m_{e}e^{10}}{3c^{3}h^{6}n^{5}} \approx 5.3 \times 10^{9}\left( \frac{1}{n^{5}} \right)s^{- 1}.} \tag{\href{07-chapter-7-spectral-lines#E23}{7.23}}

归一化的高斯线型为

\boxed{\phi{(\nu)} = \frac{c}{\nu_{0}}\left( \frac{M}{2\pi kT} \right)^{1/2}\exp\left\lbrack - \frac{Mc^{2}}{2kT}\frac{{({\nu - \nu_{0}})}^{2}}{\nu_{0}^{2}} \right\rbrack,} \tag{\href{07-chapter-7-spectral-lines#E32}{7.32}}

其中

\boxed{{\Delta\nu} = {\left( \frac{8{\ln{2k}}}{c^{2}} \right)^{1/2}\left( \frac{T}{M} \right)^{1/2}\nu_{0}}} \tag{\href{07-chapter-7-spectral-lines#E35}{7.35}}

以及

\boxed{\phi{(\nu_{0})} = \left( \frac{\ln 2}{\pi} \right)^{1/2}\frac{2}{\Delta\nu}.} \tag{\href{07-chapter-7-spectral-lines#E37}{7.37}}

速率平衡表示为

\boxed{n_{U}A_{UL} + n_{U}B_{UL}\overline{u} = n_{L}B_{LU}\overline{u}.} \tag{\href{07-chapter-7-spectral-lines#E42}{7.42}}

连接爱因斯坦系数的详细平衡方程是

{{\frac{g_{L}}{g_{U}}\frac{B_{LU}}{B_{UL}}} = 1},

{{\frac{A_{UL}}{B_{UL}}} = \frac{8\pi h\nu_{0}^{3}}{c^{3}}}.

( 7.50 ) ( 7.51 )

谱线辐射传输方程是

\boxed{\frac{dI_{\nu}}{ds} = - \left( \frac{h\nu_{0}}{c} \right){(n_{L}B_{LU} - n_{U}B_{UL})}\phi{(\nu)}I_{\nu} + \left( \frac{h\nu_{0}}{4\pi} \right)n_{U}A_{UL}\phi{(\nu)}.} \tag{\href{07-chapter-7-spectral-lines#E57}{7.57}}

两能级系统的玻尔兹曼方程是

\boxed{\frac{n_{U}}{n_{L}} = \frac{g_{U}}{g_{L}}\exp\left( - \frac{h\nu_{0}}{kT} \right).} \tag{\href{07-chapter-7-spectral-lines#E64}{7.64}}

LTE 下的谱线不透明系数是

\boxed{\kappa = \frac{c^{2}}{8\pi\nu_{0}^{2}}\frac{g_{U}}{g_{L}}n_{L}A_{UL}\left\lbrack 1 - \exp\left( - \frac{h\nu_{0}}{kT} \right) \right\rbrack\phi{(\nu)}.} \tag{\href{07-chapter-7-spectral-lines#E67}{7.67}}

激发温度 $T_{x}$ 定义为

\boxed{\frac{n_{U}}{n_{L}} \equiv \frac{g_{U}}{g_{L}}\exp\left( - \frac{h\nu_{0}}{kT_{x}} \right).} \tag{\href{07-chapter-7-spectral-lines#E70}{7.70}}

复合线不透明系数是

\boxed{{\kappa{(\nu_{0})}} \approx {\left( \frac{n_{e}^{2}}{T_{e}^{5/2}\Delta\nu} \right)\left( \frac{4\pi e^{6}h}{3m_{e}^{3/2}k^{5/2}c} \right)\left( \frac{\ln 2}{2} \right)^{1/2}}} \tag{\href{07-chapter-7-spectral-lines#E94}{7.94}}

并且复合线不透明度是

\boxed{\tau_{L} \approx 1.92 \times 10^{3}\left( \frac{T_{e}}{K} \right)^{- {5/2}}\left( \frac{EM}{{pc}{cm}^{- 6}} \right)\left( \frac{\Delta\nu}{kHz} \right)^{- 1}.} \tag{\href{07-chapter-7-spectral-lines#E96}{7.96}}

复合线亮温表示为

\boxed{T_{L} \approx T_{e}\tau_{L} \approx 1.92 \times 10^{3}\left( \frac{T_{e}}{K} \right)^{- {3/2}}\left( \frac{EM}{{pc}{cm}^{- 6}} \right)\left( \frac{\Delta\nu}{kHz} \right)^{- 1}.} \tag{\href{07-chapter-7-spectral-lines#E97}{7.97}}

复合线/连续谱比是

\boxed{\frac{T_{L}}{T_{C}} \approx 7.0 \times 10^{3}\left( \frac{\Delta v}{{km}s^{- 1}} \right)^{- 1}\left( \frac{\nu}{GHz} \right)^{1.1}\left( \frac{T_{e}}{K} \right)^{- 1.15}\left\lbrack 1 + \frac{N{({He}^{+})}}{N{(H^{+})}} \right\rbrack^{- 1},} \tag{\href{07-chapter-7-spectral-lines#E98}{7.98}}

其中 ${\lbrack{1 + {{{N{({He}^{+})}}/N}{(H^{+})}}}\rbrack} \approx 1.08$ 。

从线/连续比得到的电子温度是

\boxed{\left( \frac{T_{e}}{K} \right) \approx \left\lbrack 7.0 \times 10^{3}\left( \frac{\nu}{GHz} \right)^{1.1}\, 1.08^{- 1}\left( \frac{\Delta v}{{km}s^{- 1}} \right)^{- 1}\left( \frac{T_{C}}{T_{L}} \right) \right\rbrack^{0.87}.} \tag{\href{07-chapter-7-spectral-lines#E99}{7.99}}

角动量量子化表示为

\boxed{L = n\hslash.} \tag{\href{07-chapter-7-spectral-lines#E100}{7.100}}

双原子分子的角动量为

\boxed{L = mr_{e}^{2}\omega,} \tag{\href{07-chapter-7-spectral-lines#E104}{7.104}}

其中

\boxed{m \equiv \left( \frac{m_{A}m_{B}}{m_{A} + m_{B}} \right)} \tag{\href{07-chapter-7-spectral-lines#E105}{7.105}}

是约化质量，而 $r_{e}$ 是质量为 $m_{A}$ 和 $m_{B}$ 的原子的间距。

具有转动惯量 $$I$$ 的双原子分子的转动能级为

\boxed{E_{rot} = \frac{J{({J + 1})}\hslash^{2}}{2I},J = 0,1,2,\ldots.} \tag{\href{07-chapter-7-spectral-lines#E107}{7.107}}

对于满足选择定则的跃迁

\boxed{\Delta J = \pm 1,} \tag{\href{07-chapter-7-spectral-lines#E108}{7.108}}

谱线频率为

\boxed{\nu = \frac{hJ}{4\pi^{2}mr_{e}^{2}}.} \tag{\href{07-chapter-7-spectral-lines#E111}{7.111}}

激发频率为 $\nu$ 的 $\left. J\rightarrow{J - 1} \right.$ 跃迁所需的最低温度为

\boxed{T_{\min} \approx \frac{\nu h{({J + 1})}}{2k}.} \tag{\href{07-chapter-7-spectral-lines#E119}{7.119}}

自发发射系数为

\boxed{A_{UL} = \frac{64\pi^{4}}{3hc^{3}}\nu_{UL}^{3}|\mu_{UL}|^{2},} \tag{\href{07-chapter-7-spectral-lines#E131}{7.131}}

其中

\boxed{{|\mu_{J\rightarrow{J - 1}}|}^{2} = \frac{\mu^{2}J}{{2J} + 1}} \tag{\href{07-chapter-7-spectral-lines#E132}{7.132}}

且 $\mu$ 是分子的电偶极矩。

临界密度为

\boxed{n^{*} \approx \frac{A_{UL}}{\sigma v},} \tag{\href{07-chapter-7-spectral-lines#E135}{7.135}}

其中 $\sigma \sim {10^{- 15}{cm}^{- 2}}$ 是碰撞截面， $v \sim {10^{5}{cm}s^{- 1}}$ 是典型的 H $_{2}$ 分子速度。

我们银河系中的CO 到 H $_{2}$ 的转换因子 $X_{CO}$ 是

\boxed{X_{CO} = {(2 \pm 0.6)} \times 10^{20}{cm}^{- 2}{(K{km}s^{- 1})}^{- 1}.} \tag{\href{07-chapter-7-spectral-lines#E140}{7.140}}

Hi 超精细线频率是

\boxed{\nu_{10} = \frac{8}{3}g_{I}\left( \frac{m_{e}}{m_{p}} \right)\alpha^{2}{(R_{M}c)} \approx 1420.405751{MHz}.} \tag{\href{07-chapter-7-spectral-lines#E141}{7.141}}

Hi 超精细线发射系数是

\boxed{A_{10} \approx 2.85 \times 10^{- 15}s^{- 1}.} \tag{\href{07-chapter-7-spectral-lines#E146}{7.146}}

高旋转温度 $T_{s}$ 定义为

\boxed{\frac{n_{1}}{n_{0}} \equiv \frac{g_{1}}{g_{0}}\exp\left( - \frac{h\nu_{10}}{kT_{s}} \right),} \tag{\href{07-chapter-7-spectral-lines#E148}{7.148}}

其中 $g_{1}/g_{0}$ = 3。

Hi线不透明系数是

\boxed{\kappa{(\nu)} \approx \frac{3c^{2}}{32\pi}\frac{A_{10}n_{H}}{\nu_{10}}\frac{h}{kT_{s}}\phi{(\nu)}.} \tag{\href{07-chapter-7-spectral-lines#E153}{7.153}}

氢的柱密度 $\eta_{H}$ 定义为沿视线方向的密度积分：

\boxed{\eta_{H} \equiv \int_{los}n_{H}{(s)}ds.} \tag{\href{07-chapter-7-spectral-lines#E154}{7.154}}

如果 Hi 线是光学薄的 ( $\tau \ll 1$ )，那么Hi 柱密度为

\boxed{\left( \frac{\eta_{H}}{{cm}^{- 2}} \right) \approx 1.82 \times 10^{18}\int\left\lbrack \frac{T_{b}{(v)}}{K} \right\rbrack d\left( \frac{v}{{km}s^{- 1}} \right).} \tag{\href{07-chapter-7-spectral-lines#E155}{7.155}}

如果 $\tau \ll 1$ 一个星系的氢质量是

\boxed{\left( \frac{M_{H}}{M_{\odot}} \right) \approx 2.36 \times 10^{5}\left( \frac{D}{Mpc} \right)^{2}\int\left\lbrack \frac{S{(v)}}{Jy} \right\rbrack\left( \frac{dv}{{km}s^{- 1}} \right).} \tag{\href{07-chapter-7-spectral-lines#E166}{7.166}}

一个星系的总质量是

\boxed{\left( \frac{M}{M_{\odot}} \right) \approx 2.33 \times 10^{5}\left( \frac{v_{rot}}{{km}s^{- 1}} \right)^{2}\left( \frac{r}{kpc} \right).} \tag{\href{07-chapter-7-spectral-lines#E172}{7.172}}

附录 E 基本方程 ​

附录 E 基本方程