Here we derive the dispersion relation for radio waves in a cold, ionized medium, and from it the dispersive time delay.
Electromagnetic propagation through a cold plasma
We consider the oscillation of cold (the electron thermal speeds are negligible), free electrons in a propagating electric field. For a single electron, we have
\[m_e \ddot{\mathbf{r}} = e\mathbf{E}.\]The current density is
\[\mathbf{J} = \rho\mathbf{v} = n_e e \dot{\mathbf{r}}\]and its time derivative is
\[\frac{\partial\mathbf{J}}{\partial t} = \frac{\partial}{\partial t} \left(n_e e \dot{\mathbf{r}}\right) = n_e e \ddot{\mathbf{r}} = n_e e \left(\frac{e\mathbf{E}}{m_e}\right) = \frac{n_e e^2}{m_e}\mathbf{E}.\]Let’s now assume we have a plane wave propagating in the \(z\)-direction, linearly polarized along \(x\). Therefore, we can write the electric field as
\[\mathbf{E} = E_0 \exp\left[i(kz-\omega t)\right] \hat{\mathbf{x}}.\]This equation satisfies the wave equation, so it will be useful later to determine the two quantities:
\[\begin{aligned} \nabla^2\mathbf{E} &= \frac{\partial^2}{\partial z^2} E_0 \exp\left[i(kz-\omega t)\right] \hat{\mathbf{x}} = -k^2 E_0 \exp\left[i(kz-\omega t)\right] \hat{\mathbf{x}} = -k^2 \mathbf{E} \\ \frac{\partial^2\mathbf{E}}{\partial t^2} &= \frac{\partial^2}{\partial t^2} E_0 \exp\left[i(kz-\omega t)\right] \hat{\mathbf{x}} = -\omega^2 E_0 \exp\left[i(kz-\omega t)\right] \hat{\mathbf{x}} = -\omega^2 \mathbf{E}. \end{aligned}\]We will also use three of Maxwell’s equations in Gaussian (cgs) units:
\[\begin{aligned} \nabla \cdot \mathbf{E} &= 4\pi\rho \\ \nabla \times \mathbf{E} &= -\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t} \\ \nabla \times \mathbf{B} &= \frac{1}{c}\left(4\pi\mathbf{J} + \frac{\partial\mathbf{E}}{\partial t}\right). \end{aligned}\]We start by taking the curl of both sides of the second equation (Faraday’s law) and then plug in values appropriately
\[\begin{aligned} \nabla \times \left(\nabla \times \mathbf{E}\right) &= \nabla \times \left(-\frac{1}{c}\frac{\partial\mathbf{B}}{\partial t}\right) \\ \rightarrow \nabla \left(\nabla \cdot \mathbf{E}\right) - \nabla^2 \mathbf{E} &= -\frac{1}{c}\frac{\partial}{\partial t}\left(\nabla \times \mathbf{B}\right) \\ \rightarrow \nabla \left(4\pi\rho\right) - \left(-k^2\mathbf{E}\right) &= -\frac{1}{c}\frac{\partial}{\partial t}\left[\frac{1}{c}\left(4\pi\mathbf{J} + \frac{\partial\mathbf{E}}{\partial t}\right)\right] \\ \rightarrow 4\pi\cancel{\nabla \rho} + k^2\mathbf{E} &= -\frac{4\pi}{c^2}\frac{\partial\mathbf{J}}{\partial t} - \frac{1}{c^2}\frac{\partial^2 \mathbf{E}}{\partial t^2} \\ \rightarrow k^2\mathbf{E} &= -\frac{4\pi}{c^2}\frac{n_e e^2}{m_e}\mathbf{E} + \frac{\omega^2}{c^2}\mathbf{E} \\ \implies k^2c^2 &= -\frac{4\pi n_e e^2}{m_e} + \omega^2. \end{aligned}\]We can define the (angular) plasma frequency as
\[\omega_p^2 \equiv \frac{4 \pi n_e e^2}{m_e}\]and therefore we arrive at the dispersion relation
\[\omega^2 = k^2 c^2 + \omega_p^2.\]The plasma frequency is related to the electron density as
\[\nu_p = \frac{\omega_p}{2\pi} = \sqrt{\frac{n_e e^2}{\pi m_e}} \approx 8.979~\mathrm{kHz}~\left(\frac{n_e}{\mathrm{cm^{-3}}}\right)^{1/2}.\]The propagation speed is given by the group velocity
\[\begin{aligned} v_g \equiv \frac{\partial\omega}{\partial k} &= \frac{\partial}{\partial k} \omega \\ &= \frac{\partial}{\partial k} \left(k^2 c^2 + \omega_p^2\right)^{1/2} \\ &= \frac{\cancel{2}kc^2}{\cancel{2}\left(k^2 c^2 + \omega_p^2\right)^{1/2}} \\ &= \frac{\cancel{\frac{1}{c}}\left(\omega^2 - \omega_p^2\right)^{1/2} c^{\cancel{2}}}{\omega} \\ &= c \left(1 - \frac{\omega_p^2}{\omega^2}\right)^{1/2} \\ &= c \left(1 - \frac{\nu_p^2}{\nu^2}\right)^{1/2} \\ &\equiv c\mu, \end{aligned}\]where \(\mu \le 1\) is the index of refraction. Below the plasma frequency, \(\mu\) is imaginary and the waves cannot propagate. For the ionosphere, the electron density peaks at about \(10^6~\mathrm{cm}^{-3}\) and so the plasma frequency is about 9 MHz. In the ISM, for \(n_e \sim 0.1~\mathrm{cm}^{-3}\), the plasma frequency is about 3 kHz.
Dispersive time delay
The total propagation time as a function of path length through the medium is
\[\begin{aligned} t_\mathrm{total} &= \int_0^D \frac{dl}{v_g} \\ &= \int_0^D \frac{dl}{c}\left(1-\frac{\nu_p^2}{\nu^2}\right)^{-1/2} \\ &\approx \int_0^D \frac{dl}{c}\left(1+\frac{\nu_p^2}{2\nu^2}\right) \\ &= \int_0^D \frac{dl}{c} + \int_0^D \frac{dl}{c}\frac{\nu_p^2}{2\nu^2} \\ &= \frac{D}{c} + \frac{e^2}{2\pi m_e c}\frac{\int_0^D n_e(l)\,dl}{\nu^2} \\ &= t_\mathrm{geometric} + t_\mathrm{dispersive}, \end{aligned}\]where in the last step we break up the total time into the geometric travel time and the dispersive delay. Therefore,
\[\begin{aligned} t_\mathrm{delay} &= \frac{e^2}{2\pi m_e c}\frac{\int_0^D n_e(l)\,dl}{\nu^2} \\ &\equiv K\frac{\mathrm{DM}}{\nu^2} \\ &\approx 4.149~\mathrm{ms}~\left(\frac{\mathrm{DM}}{\mathrm{pc~cm^{-3}}}\right)\left(\frac{\nu}{\mathrm{GHz}}\right)^{-2}, \end{aligned}\]where \(\mathrm{DM} \equiv \int_0^D n_e(l)\,dl\) is the dispersion measure and \(K \equiv e^2/(2\pi m_e c) \approx 4.149~\mathrm{ms~GHz^2~pc^{-1}~cm^3}\) is the dispersion constant.