## 1 Introduction

[2] Approximate or semiempirical analytical expressions for the refraction of radiowaves from exoatmospheric sources—such as the Sun—are useful for many radar applications. For instance, radio signals from the Sun are used for monitoring of the antenna alignment and the receiving chain of weather radars [*Huuskonen and Holleman*, 2007; *Holleman et al.*, 2010a, 2010b]. As weather radars scan close to the horizon, the intercepted radio signals originate from a rising or setting Sun and thus atmospheric refraction is an issue.

[3] The energy paths of radiowaves in the atmosphere can be approximated by rays, and hence, their propagation can be obtained from the ray-tracing method [*Bean and Dutton*, 1966; *Doviak and Zrnić*, 1993]. The exact differential equation that specifies the ray path in a spherical stratified medium, like the Earth atmosphere, was first given by *Hartree et al.* [1946]. The ray propagation can be accurately calculated from this differential equation using, e.g., the Starlink positional astronomy library [*Disney and Wallace*, 1982; *Starlink*, 2011]. For many applications, however, modest precision is required, and easy-to-use analytical expressions are preferred. In *Huuskonen and Holleman* [2007], we presented empirical formulas for the refraction of radiowaves by fitting “Sonntag-like” equations [*Sonntag*, 1989; *Bennett*, 1982] to the Starlink calculations. Although these formulas are sufficiently accurate, we find it unsatisfactory that they do not originate from a physical model.

[4] The Effective Earth's Radius Model (*k*-Model) or 4/3-Model is commonly used to describe the propagation of radiowaves in a spherically stratified atmosphere [*Bean and Dutton*, 1966; *Doviak and Zrnić*, 1993]. The main assumption of this model is that the refractivity decreases linearly with increasing height. *Bean and Dutton* [1966] noted that “its success is due to the 4/3-Model being in essential agreement with the average refractivity structure near the Earth surface which largely controls the refraction of radio rays at small elevations.” Later, *Robertshaw* [1986] used the *k*-Model to approximate determinations of grazing angle, ground range, and slant range for higher altitude paths.

[5] In this paper, we present analytical formulas in closed form for the refraction angle of radiowaves from exoatmospheric sources as a function of elevation. These analytical formulas are derived from the *k*-Model and thus are fully compliant with this physical model. As this model is used in numerous radiowave applications, it is expected that these simple refraction formulas can be of wide use.