## 1 Introduction

[2] Thermochronometric dating of a wide assortment of minerals has become a standard tool in the analysis of tectonic, metamorphic, and even geomorphic problems [*Bernet et al.*, 2004; *House et al.*, 1998; *Hurford*, 1991; *Kamp et al.*, 1989; *Parrish*, 1983; *Reiners and Brandon*, 2006; *Reiners et al.*, 2003; *Wagner*, 1968; *Zeitler*, 1985]. Although a thermochronometric age, by definition is a cooling age, its utility is in the interpretation of that cooling age in terms of erosion, which is defined as surface removal of rock, driving exhumation or motion of the datable mineral toward the surface of the Earth. In either case, the conversion of a thermochronometric age to an erosion rate requires two components: a kinetic model for the dating system in order to calculate the temperature dependence of the closure process, and information about the motion of the dated mineral through the Earth's temperature field, generally through a thermal model.

[3] Thermochronometric systems are defined through a radiogenic parent-daughter relationship, where the daughter is either a radiogenic species or, in the case of fission-track dating, a crystal damage track. The kinetics of the processes vary, and there are many reviews of the various systems [*Reiners et al.*, 2005], but in nearly all cases the temperature dependence of the loss of the daughter product can be expressed through an Arrhenius expression with an activation energy controlling the rate and thus the effective temperature range of daughter loss. With an Arrhenius equation and the simplifying assumption of a constant rate of cooling, *Dodson* [1973] demonstrated that one can calculate a temperature corresponding to the measured age, which is commonly used as the effective closure temperature.

[4] A wide variety of approaches have been used to model the temperature field in the near surface. For direct interpretation of thermochronometric ages, some approaches include no heat transfer at all, instead inferring a temperature history without specifying an erosion function or explicitly including a heat transfer model [*Gallagher et al.*, 2005]. In other cases, ages are obtained over a range of elevations such that the gradient in age with elevation can be used to directly infer an erosion rate [*Brown*, 1991; *Fitzgerald et al.*, 1995; *Valla et al.*, 2010]; this method implicitly assumes that temperature is in a steady state. Analytical solutions have also been used to model the temperature field [*Brown and Summerfield*, 1997; *Mancktelow and Grasemann*, 1997; *Moore and England*, 2001], but many of these also use simplifying assumptions, such as steady state, to be practical [*Brandon et al.*, 1998; *Stuwe et al.*, 1994]. Numerical models of heat transfer in one [*Ehlers and Farley*, 2003; *Willett et al.*, 2003], two [*Batt et al.*, 2000; *Braun*, 2002; *Ehlers et al.*, 2003; *Fuller et al.*, 2006; *Herman et al.*, 2010], or even three dimensions [*Braun*, 2003; *Herman et al.*, 2009] have been used to include a range of tectonic kinematic models and to calculate heat transfer by conduction and advection.

[5] In spite of the many complex models available to convert ages to erosion rates [*Ehlers et al.*, 2005], in many cases, a simple analytical expression would be convenient to quickly convert ages to erosion rates. In this paper, we provide a set of such expressions based on analytical solutions for conductive or advective-conductive geotherms in the Earth. These include representations of the two most important physical properties in the system, an expression for closure of the mineral system through a first-order Arrhenius rate equation and upward advection of heat by the erosion process. Our approach is similar to that used by *Moore and England* [2001], but an important difference is that we include the closure behavior for our estimates of cooling ages.