Critical surface density, convergence, and the Jacobian
The point of the critical surface density is not that \(\kappa\) has the units of a density. It does not. \(\kappa\) is dimensionless. The physical density is the projected surface mass density \(\Sigma\), and the definition
\[\kappa(\mathbf{\theta})
\equiv
\frac{\Sigma(D_l\mathbf{\theta})}{\Sigma_{\mathrm{crit}}}\]
turns that physical surface density into a dimensionless focusing strength. The connection to the lensing Jacobian is
\[\boxed{
\kappa_{\mathrm{Jacobian}}
=
\frac{1}{2}\nabla_\theta\cdot\mathbf{\alpha}
}
\qquad\text{and}\qquad
\boxed{
\nabla_\theta\cdot\mathbf{\alpha}
=
2\frac{\Sigma}{\Sigma_{\mathrm{crit}}}
}.\]
Putting them together gives
\[\boxed{
\kappa_{\mathrm{Jacobian}}
=
\frac{\Sigma}{\Sigma_{\mathrm{crit}}}
}.\]
1. From the Jacobian to a divergence
Start with the lens equation,
\[\mathbf{\beta}
=
\mathbf{\theta}
-
\mathbf{\alpha}(\mathbf{\theta}).\]
Differentiating with respect to the image-plane coordinate gives the Jacobian
\[A_{ij}
=
\frac{\partial\beta_i}{\partial\theta_j}
=
\delta_{ij}
-
\frac{\partial\alpha_i}{\partial\theta_j}.\]
Define the deformation matrix
\[U_{ij}
=
\frac{\partial\alpha_i}{\partial\theta_j},\qquad
A=I-U.\]
In lensing this matrix is decomposed as
\[U
=
\begin{pmatrix}
\kappa+\gamma_1 & \gamma_2\\
\gamma_2 & \kappa-\gamma_1
\end{pmatrix}.\]
The trace is therefore
\[\mathrm{Tr}(U)
=
\frac{\partial\alpha_1}{\partial\theta_1}
+
\frac{\partial\alpha_2}{\partial\theta_2}
=
2\kappa.\]
So the \(\kappa\) that appears in the Jacobian is half of the divergence of the reduced deflection field:
\[\boxed{
\kappa
=
\frac{1}{2}
\left(
\frac{\partial\alpha_1}{\partial\theta_1}
+
\frac{\partial\alpha_2}{\partial\theta_2}
\right)
=
\frac{1}{2}\nabla_\theta\cdot\mathbf{\alpha}
}.\]
2. From surface density to reduced deflection
The physical deflection angle caused by a projected mass distribution is
\[\hat{\mathbf{\alpha}}(\mathbf{\xi})
=
\frac{4G}{c^2}
\int \mathrm{d}^2\xi'\,
\Sigma(\mathbf{\xi}')
\frac{\mathbf{\xi}-\mathbf{\xi}'}
{|\mathbf{\xi}-\mathbf{\xi}'|^2}.\]
The lens equation uses the reduced deflection angle,
\[\mathbf{\alpha}(\mathbf{\theta})
=
\frac{D_{ls}}{D_s}
\hat{\mathbf{\alpha}}(D_l\mathbf{\theta}).\]
Use \(\mathbf{\xi}=D_l\mathbf{\theta}\), \(\mathbf{\xi}'=D_l\mathbf{\theta}'\), and \(\mathrm{d}^2\xi'=D_l^2\mathrm{d}^2\theta'\). Then
\[\mathbf{\alpha}(\mathbf{\theta})
=
\frac{4G}{c^2}
\frac{D_lD_{ls}}{D_s}
\int \mathrm{d}^2\theta'\,
\Sigma(D_l\mathbf{\theta}')
\frac{\mathbf{\theta}-\mathbf{\theta}'}
{|\mathbf{\theta}-\mathbf{\theta}'|^2}.\]
Now define the critical surface density
\[\boxed{
\Sigma_{\mathrm{crit}}
=
\frac{c^2}{4\pi G}
\frac{D_s}{D_lD_{ls}}
}.\]
This is chosen so that
\[\frac{4G}{c^2}
\frac{D_lD_{ls}}{D_s}
=
\frac{1}{\pi\Sigma_{\mathrm{crit}}}.\]
Therefore the reduced deflection can be written in the compact dimensionless form
\[\mathbf{\alpha}(\mathbf{\theta})
=
\frac{1}{\pi}
\int \mathrm{d}^2\theta'\,
\kappa(\mathbf{\theta}')
\frac{\mathbf{\theta}-\mathbf{\theta}'}
{|\mathbf{\theta}-\mathbf{\theta}'|^2},\]
where \(\kappa(\mathbf{\theta}')=\Sigma(D_l\mathbf{\theta}')/\Sigma_{\mathrm{crit}}\). This is the first bridge: \(\Sigma_{\mathrm{crit}}\) absorbs the gravitational constant, the factor \(c^2\), and the lensing geometry into the normalization of \(\Sigma\).
3. Taking the divergence gives \(\kappa\)
Take the two-dimensional divergence with respect to \(\mathbf{\theta}\). The only identity needed is
\[\nabla_\theta\cdot
\left[
\frac{\mathbf{\theta}-\mathbf{\theta}'}
{|\mathbf{\theta}-\mathbf{\theta}'|^2}
\right]
=
2\pi\delta^{(2)}(\mathbf{\theta}-\mathbf{\theta}').\]
Then
\[\begin{aligned}
\nabla_\theta\cdot\mathbf{\alpha}(\mathbf{\theta})
&=
\frac{1}{\pi}
\int \mathrm{d}^2\theta'\,
\kappa(\mathbf{\theta}')
2\pi\delta^{(2)}(\mathbf{\theta}-\mathbf{\theta}')\\
&=
2\kappa(\mathbf{\theta}).
\end{aligned}\]
Combining this with the Jacobian trace relation from section 1 gives
\[\kappa_{\mathrm{Jacobian}}
=
\frac{1}{2}\nabla_\theta\cdot\mathbf{\alpha}
=
\frac{\Sigma}{\Sigma_{\mathrm{crit}}}.\]
4. Why it is called critical
The Einstein-ring derivation gives a quick intuition for the word critical. For an axisymmetric lens, the physical deflection at impact parameter \(\xi\) is
\[\hat{\alpha}(\xi)
=
\frac{4GM(\lt \xi)}{c^2\xi}.\]
With \(\xi=D_l\theta\), the reduced deflection is
\[\alpha(\theta)
=
\frac{D_{ls}}{D_s}
\frac{4GM(\lt D_l\theta)}{c^2D_l\theta}.\]
An Einstein ring has \(\beta=0\), so the lens equation \(\beta=\theta-\alpha(\theta)\) gives \(\theta_E=\alpha(\theta_E)\). Thus
\[\theta_E^2
=
\frac{4G}{c^2}
\frac{D_{ls}}{D_lD_s}
M(\lt D_l\theta_E).\]
Write the enclosed mass in terms of the average surface density:
\[M(\lt D_l\theta_E)
=
\pi(D_l\theta_E)^2\bar{\Sigma}(\lt \theta_E).\]
Substituting and cancelling \(\theta_E^2\) gives
\[\bar{\Sigma}(\lt \theta_E)
=
\frac{c^2}{4\pi G}
\frac{D_s}{D_lD_{ls}}
=
\Sigma_{\mathrm{crit}}.\]
So \(\Sigma_{\mathrm{crit}}\) is the surface density scale at which the average density inside the Einstein radius is just high enough to make a ring. Locally, the same normalization says how much of the Jacobian trace is caused by the projected mass density.
5. The Poisson-equation route
The same result appears if one starts from the projected lensing potential. The three-dimensional Newtonian potential satisfies
\[\nabla^2\Phi=4\pi G\rho.\]
The two-dimensional lensing potential is
\[\psi(\mathbf{\theta})
=
\frac{2}{c^2}
\frac{D_{ls}}{D_lD_s}
\int \Phi(D_l\mathbf{\theta},z)\,\mathrm{d}z.\]
Because \(\mathbf{\xi}=D_l\mathbf{\theta}\),
\[\nabla_\theta^2=D_l^2\nabla_\xi^2.\]
Using the line-of-sight projection of the three-dimensional Poisson equation gives, for a localized lens,
\[\int \nabla_\perp^2\Phi\,\mathrm{d}z
=
4\pi G
\int \rho\,\mathrm{d}z
=
4\pi G\Sigma.\]
Therefore
\[\nabla_\theta^2\psi
=
\frac{8\pi G}{c^2}
\frac{D_lD_{ls}}{D_s}
\Sigma
=
2\frac{\Sigma}{\Sigma_{\mathrm{crit}}}.\]
Since lensing also defines
\[\kappa
=
\frac{1}{2}\nabla_\theta^2\psi,\]
one again obtains
\[\boxed{
\kappa
=
\frac{\Sigma}{\Sigma_{\mathrm{crit}}}
}.\]
That is the full bridge: physical mass density \(\rho\) projects to \(\Sigma\); \(\Sigma_{\mathrm{crit}}\) converts \(\Sigma\) into dimensionless convergence; and the convergence is exactly the trace part of the Jacobian deformation matrix.
临界面密度、会聚与 Jacobian
这里最容易混淆的是：\(\kappa\) 本身不是一个带有 density 单位的量。它是无量纲量。真正有单位的是投影表面质量密度 \(\Sigma\)，而定义
\[\kappa(\mathbf{\theta})
\equiv
\frac{\Sigma(D_l\mathbf{\theta})}{\Sigma_{\mathrm{crit}}}\]
就是把真实表面质量密度变成无量纲的局部聚焦强度。它和 lensing Jacobian 的连接是
\[\boxed{
\kappa_{\mathrm{Jacobian}}
=
\frac{1}{2}\nabla_\theta\cdot\mathbf{\alpha}
}
\qquad\text{以及}\qquad
\boxed{
\nabla_\theta\cdot\mathbf{\alpha}
=
2\frac{\Sigma}{\Sigma_{\mathrm{crit}}}
}.\]
所以立刻得到
\[\boxed{
\kappa_{\mathrm{Jacobian}}
=
\frac{\Sigma}{\Sigma_{\mathrm{crit}}}
}.\]
读这三个式子时，顺序很重要：Jacobian 先告诉我们 \(\kappa\) 是偏折场散度的 trace 部分；质量分布再告诉我们这个散度由 \(\Sigma\) 决定；\(\Sigma_{\mathrm{crit}}\) 则是把有单位的 \(\Sigma\) 变成无量纲 \(\kappa\) 的比例尺。也就是说，\(\Sigma_{\mathrm{crit}}\) 不是额外塞进 Jacobian 的物理量，而是从质量产生偏折、偏折产生局部形变这一整条链条里自然出现的归一化。
1. 从 Jacobian 到偏折场散度
从透镜方程开始：
\[\mathbf{\beta}
=
\mathbf{\theta}
-
\mathbf{\alpha}(\mathbf{\theta}).\]
对像平面角坐标求导，得到 Jacobian：
\[A_{ij}
=
\frac{\partial\beta_i}{\partial\theta_j}
=
\delta_{ij}
-
\frac{\partial\alpha_i}{\partial\theta_j}.\]
这里 \(A\) 描述的是“像平面小位移”到“源平面小位移”的局部线性映射。偏折角 \(\mathbf{\alpha}\) 本身只告诉你光线被偏到哪里；而 \(\partial\alpha_i/\partial\theta_j\) 告诉你偏折角在不同位置变化得多快。正是这个变化率造成小图像的局部放大、压缩和剪切。
定义 deformation matrix：
\[U_{ij}
=
\frac{\partial\alpha_i}{\partial\theta_j},\qquad
A=I-U.\]
在透镜里，我们把 \(U\) 分解成
\[U
=
\begin{pmatrix}
\kappa+\gamma_1 & \gamma_2\\
\gamma_2 & \kappa-\gamma_1
\end{pmatrix}.\]
所以它的 trace 是
\[\mathrm{Tr}(U)
=
\frac{\partial\alpha_1}{\partial\theta_1}
+
\frac{\partial\alpha_2}{\partial\theta_2}
=
2\kappa.\]
trace 的意思是两个坐标方向上的平均聚焦强度。因为 shear 是无迹的，它只负责一个方向增强、另一个方向减弱；真正同时作用在两个方向上的各向同性部分，就是 \(\kappa\)。所以从 Jacobian 的角度看，\(\kappa\) 必须等于 deformation matrix trace 的一半。
因此 Jacobian 里的 \(\kappa\) 就是 reduced deflection field 的一半散度：
\[\boxed{
\kappa
=
\frac{1}{2}
\left(
\frac{\partial\alpha_1}{\partial\theta_1}
+
\frac{\partial\alpha_2}{\partial\theta_2}
\right)
=
\frac{1}{2}\nabla_\theta\cdot\mathbf{\alpha}
}.\]
2. 从面密度到 reduced deflection
投影质量分布产生的物理偏折角是
\[\hat{\mathbf{\alpha}}(\mathbf{\xi})
=
\frac{4G}{c^2}
\int \mathrm{d}^2\xi'\,
\Sigma(\mathbf{\xi}')
\frac{\mathbf{\xi}-\mathbf{\xi}'}
{|\mathbf{\xi}-\mathbf{\xi}'|^2}.\]
这里 \(\mathbf{\xi}\) 是 lens plane 上的物理长度坐标，\(\mathbf{\xi}'\) 是质量面元的位置。积分里的每个质量面元都给光线一个朝向该面元的偏折贡献，所有面元加起来就是总偏折角 \(\hat{\mathbf{\alpha}}\)。
透镜方程里使用的是 reduced deflection：
\[\mathbf{\alpha}(\mathbf{\theta})
=
\frac{D_{ls}}{D_s}
\hat{\mathbf{\alpha}}(D_l\mathbf{\theta}).\]
这里的几何因子 \(D_{ls}/D_s\) 来自 source-lens-observer 的投影：同一个物理偏折角，对源平面位置的影响还取决于 lens 到 source 的距离相对于 observer 到 source 的距离。换句话说，\(\hat{\mathbf{\alpha}}\) 是光线实际被弯折的角度，而 \(\mathbf{\alpha}\) 是进入角坐标透镜方程的有效偏折。
令 \(\mathbf{\xi}=D_l\mathbf{\theta}\)、\(\mathbf{\xi}'=D_l\mathbf{\theta}'\)，并且 \(\mathrm{d}^2\xi'=D_l^2\mathrm{d}^2\theta'\)。代入后得到
\[\mathbf{\alpha}(\mathbf{\theta})
=
\frac{4G}{c^2}
\frac{D_lD_{ls}}{D_s}
\int \mathrm{d}^2\theta'\,
\Sigma(D_l\mathbf{\theta}')
\frac{\mathbf{\theta}-\mathbf{\theta}'}
{|\mathbf{\theta}-\mathbf{\theta}'|^2}.\]
现在定义临界面密度
\[\boxed{
\Sigma_{\mathrm{crit}}
=
\frac{c^2}{4\pi G}
\frac{D_s}{D_lD_{ls}}
}.\]
这个定义正好使得
\[\frac{4G}{c^2}
\frac{D_lD_{ls}}{D_s}
=
\frac{1}{\pi\Sigma_{\mathrm{crit}}}.\]
这一步就是 critical density 和偏折积分连起来的地方。原来的 prefactor 里有 \(G\)、\(c^2\)、\(D_l\)、\(D_s\)、\(D_{ls}\)，看起来像一串物理常数和几何因子；把它写成 \(1/(\pi\Sigma_{\mathrm{crit}})\) 后，积分里只剩下 \(\Sigma/\Sigma_{\mathrm{crit}}\)，也就是无量纲的 convergence。
于是 reduced deflection 可以写成无量纲的紧凑形式：
\[\mathbf{\alpha}(\mathbf{\theta})
=
\frac{1}{\pi}
\int \mathrm{d}^2\theta'\,
\kappa(\mathbf{\theta}')
\frac{\mathbf{\theta}-\mathbf{\theta}'}
{|\mathbf{\theta}-\mathbf{\theta}'|^2},\]
其中 \(\kappa(\mathbf{\theta}')=\Sigma(D_l\mathbf{\theta}')/\Sigma_{\mathrm{crit}}\)。这是第一层连接：\(\Sigma_{\mathrm{crit}}\) 把 \(G\)、\(c^2\) 和 lens-source-observer 的几何因子都吸收到 \(\Sigma\) 的归一化里。
3. 对 reduced deflection 取散度
现在对 \(\mathbf{\theta}\) 取二维散度。需要的关键恒等式是
\[\nabla_\theta\cdot
\left[
\frac{\mathbf{\theta}-\mathbf{\theta}'}
{|\mathbf{\theta}-\mathbf{\theta}'|^2}
\right]
=
2\pi\delta^{(2)}(\mathbf{\theta}-\mathbf{\theta}').\]
这个恒等式的含义是：核函数在普通位置的散度为零，但在 \(\mathbf{\theta}=\mathbf{\theta}'\) 的奇点处有一个 delta-function 贡献。直观地说，二维偏折核像一个从质量面元位置向外的径向场；只有当你把散度积分包住那个质量面元时，才会得到非零通量。
所以
\[\begin{aligned}
\nabla_\theta\cdot\mathbf{\alpha}(\mathbf{\theta})
&=
\frac{1}{\pi}
\int \mathrm{d}^2\theta'\,
\kappa(\mathbf{\theta}')
2\pi\delta^{(2)}(\mathbf{\theta}-\mathbf{\theta}')\\
&=
2\kappa(\mathbf{\theta}).
\end{aligned}\]
再和第 1 节的 Jacobian trace 关系合起来：
\[\kappa_{\mathrm{Jacobian}}
=
\frac{1}{2}\nabla_\theta\cdot\mathbf{\alpha}
=
\frac{\Sigma}{\Sigma_{\mathrm{crit}}}.\]
这一步很关键：delta function 把积分里的 \(\kappa(\mathbf{\theta}')\) 选到当前位置 \(\mathbf{\theta}\)，所以偏折场的散度不是某个抽象的新量，而是局部无量纲面密度的两倍。再除以 Jacobian trace 里的那个 2，就得到 Jacobian 里的 \(\kappa\)。
4. 为什么它叫 critical
Einstein ring 的推导可以很直观地解释 critical 这个词。对轴对称透镜，impact parameter 为 \(\xi\) 时的物理偏折角是
\[\hat{\alpha}(\xi)
=
\frac{4GM(\lt \xi)}{c^2\xi}.\]
这个式子可以看成点质量偏折角的推广：如果质量分布是轴对称的，那么半径 \(\xi\) 以内的总质量 \(M(\lt \xi)\) 像一个等效点质量一样决定该半径处的偏折强度。
用 \(\xi=D_l\theta\)，reduced deflection 是
\[\alpha(\theta)
=
\frac{D_{ls}}{D_s}
\frac{4GM(\lt D_l\theta)}{c^2D_l\theta}.\]
Einstein ring 对应源、透镜、观测者共线，也就是 \(\beta=0\)。透镜方程
\[\beta=\theta-\alpha(\theta)\]
因此给出
\[\theta_E=\alpha(\theta_E).\]
把上面的 reduced deflection 代入，得到
\[\theta_E^2
=
\frac{4G}{c^2}
\frac{D_{ls}}{D_lD_s}
M(\lt D_l\theta_E).\]
把包围质量写成平均表面密度：
\[M(\lt D_l\theta_E)
=
\pi(D_l\theta_E)^2\bar{\Sigma}(\lt \theta_E).\]
代入并消去 \(\theta_E^2\)，得到
\[\bar{\Sigma}(\lt \theta_E)
=
\frac{c^2}{4\pi G}
\frac{D_s}{D_lD_{ls}}
=
\Sigma_{\mathrm{crit}}.\]
所以 \(\Sigma_{\mathrm{crit}}\) 是这样一个面密度标尺：Einstein 半径以内的平均面密度刚好达到它时，就足以形成 Einstein ring。局部地看，同一个归一化告诉我们，投影质量密度贡献了 Jacobian trace 里的多少 \(\kappa\)。
这也解释了为什么 \(\kappa=1\) 有特殊意义。对轴对称情形，更准确地说是半径以内的平均 convergence 满足 \(\bar{\kappa}(\lt \theta_E)=1\) 时出现 Einstein ring；这就是 “critical surface density” 这个名字最直观的来源。
5. Poisson 方程路线
从投影透镜势出发也能得到同一个结果。三维牛顿势满足
\[\nabla^2\Phi=4\pi G\rho.\]
这个方程说的是：物质密度 \(\rho\) 决定三维引力势 \(\Phi\) 的曲率。引力透镜里我们关心的是沿视线投影后的二维效果，所以要把三维势沿 \(z\) 方向积分，变成二维透镜势。
二维透镜势为
\[\psi(\mathbf{\theta})
=
\frac{2}{c^2}
\frac{D_{ls}}{D_lD_s}
\int \Phi(D_l\mathbf{\theta},z)\,\mathrm{d}z.\]
因为 \(\mathbf{\xi}=D_l\mathbf{\theta}\)，所以
\[\nabla_\theta^2=D_l^2\nabla_\xi^2.\]
接下来把三维 Laplacian 拆成垂直于视线的二维部分和视线方向部分：
\[\nabla^2\Phi
=
\nabla_\perp^2\Phi
+
\frac{\partial^2\Phi}{\partial z^2}
=
4\pi G\rho.\]
沿视线积分：
\[\int \nabla_\perp^2\Phi\,\mathrm{d}z
+
\int \frac{\partial^2\Phi}{\partial z^2}\,\mathrm{d}z
=
4\pi G\int \rho\,\mathrm{d}z.\]
对一个局域透镜，边界处的势梯度项可以忽略，所以 \(\int \partial^2\Phi/\partial z^2\,\mathrm{d}z\simeq0\)。又因为
\[\Sigma(\mathbf{\xi})
=
\int \rho(\mathbf{\xi},z)\,\mathrm{d}z,\]
于是得到
\[\int \nabla_\perp^2\Phi\,\mathrm{d}z
=
4\pi G
\int \rho\,\mathrm{d}z
=
4\pi G\Sigma.\]
因此
\[\nabla_\theta^2\psi
=
\frac{8\pi G}{c^2}
\frac{D_lD_{ls}}{D_s}
\Sigma
=
2\frac{\Sigma}{\Sigma_{\mathrm{crit}}}.\]
而透镜里又定义
\[\kappa
=
\frac{1}{2}\nabla_\theta^2\psi,\]
所以再次得到
\[\boxed{
\kappa
=
\frac{\Sigma}{\Sigma_{\mathrm{crit}}}
}.\]
这就是完整桥梁：三维物质密度 \(\rho\) 投影成 \(\Sigma\)；\(\Sigma_{\mathrm{crit}}\) 把 \(\Sigma\) 归一化成无量纲 convergence；而 convergence 正是 Jacobian deformation matrix 的 trace 部分。