## FANDOM

159 Pages

Let us consider an object observed through an inhomogeneous medium. Let the distance to the source be $d$, and the velocity difference normal to the line of sight be $v$. Due to the passage of light through the medium, an observer sees more than a single image of the object. For simplicity, we assume that all these images lie on a single line parallel to $v$. The position of each image can be described by $\theta$, the angle relative to the line of sight. The farther an image is from the line of sight, the longer is takes for the light to reach the observer, so the time delay is

$\Delta t \approx \frac{d}{c} \theta^2$

where $c$ is the speed of light. Due to the Doppler shift, the frequency from each image will be slightly different

$\Delta \omega \approx \omega_0 \frac{v}{c} \theta$

where $\omega_0$ is the primary frequency at which the object is emitting. Eliminating the angle yields a quadratic relation between the time delay and the frequency shift

$\Delta t \approx \frac{d c}{v^2} \frac{\Delta \omega^2}{\omega_0^2}$

This parabola manifests itself in the secondary spectrum of the image. This is obtained in the following way. First, an observer observes the image for a certain period of time. Second, the observation period is divided into smaller segments, and the spectrum is calculated for each segment. This is called a dynamic spectrum, and it can be represented by a map where one axis is the time and the other axis is the frequency. Third, a Fourier transform of the dynamic spectrum yields the secondary spectrum. The secondary spectrum will sport the parabola discussed above.