Speeds and arrival times of solar transients approximated by self-similar expanding circular fronts

The NASA STEREO mission opened up the possibility to forecast the arrival times, speeds and directions of solar transients from outside the Sun-Earth line. In particular, we are interested in predicting potentially geo-effective Interplanetary Coronal Mass Ejections (ICMEs) from observations of density structures at large observation angles from the Sun (with the STEREO Heliospheric Imager instrument). We contribute to this endeavor by deriving analytical formulas concerning a geometric correction for the ICME speed and arrival time for the technique introduced by Davies et al. (2012, ApJ, in press) called Self-Similar Expansion Fitting (SSEF). This model assumes that a circle propagates outward, along a plane specified by a position angle (e.g. the ecliptic), with constant angular half width (lambda). This is an extension to earlier, more simple models: Fixed-Phi-Fitting (lambda = 0 degree) and Harmonic Mean Fitting (lambda = 90 degree). This approach has the advantage that it is possible to assess clearly, in contrast to previous models, if a particular location in the heliosphere, such as a planet or spacecraft, might be expected to be hit by the ICME front. Our correction formulas are especially significant for glancing hits, where small differences in the direction greatly influence the expected speeds (up to 100-200 km/s) and arrival times (up to two days later than the apex). For very wide ICMEs (2 lambda > 120 degree), the geometric correction becomes very similar to the one derived by M\"ostl et al. (2011, ApJ, 741, id. 34) for the Harmonic Mean model. These analytic expressions can also be used for empirical or analytical models to predict the 1 AU arrival time of an ICME by correcting for effects of hits by the flank rather than the apex, if the width and direction of the ICME in a plane are known and a circular geometry of the ICME front is assumed.