Consider a source of radiation containing some absorbers /scatterers. For simplicity, let us assume the source has a uniform density of absorbers / scatterers each with a cross section . If the typical width of the source (in the direction along the path towards the observer) is , then assuming the source is stationary, the optical depth to the observer is given by the familiar expression:
However, let us now consider that the source is moving relativisticaly towards the observer. In addition, let us assume also that the source has a jet (conical) geometry. In this case, as the photons are propagating through the medium, the medium expands and dilutes. As a result, the resulting optical depth may be lower than that obtained for a stationary medium.
As before, let us assume that the width of the medium (across the direction of propagation of the source which is also the direction along the line of sight towards the observer) is . In addition let us denote the typical size of the source in the directions orthogonal to the direction of propagation by . Notice, that this size increases as time passes (we can approximate the speed of the source to instead of without much loss of generality here). Due to the lateral expansion of the jet, the density decreases as: . Therefore, the optical depth becomes:There are two interesting limits to this expression. First, if then we obtain . This is the regular expression derived for the stationary source above. However, if then . In this case the optical depth can be much smaller than the expression for the stationary source.