## 1. Introduction

[2] Knowledge of the aerosol optical thickness (AOT) throughout much of the shortwave spectral region (∼0.3–5 μm) is necessary to compute the shortwave aerosol radiative forcing at the surface and the top of the atmosphere. The AOT is easily measured in discrete spectral intervals with Sun photometers located at the surface, but gas and water vapor absorption prevent the measurement of AOT at all wavelengths of interest. This difficulty is easily circumvented because [*Angstrom*, 1929] noted that the spectral dependence of extinction by particles may be approximated as a power law relationship:

where τ(λ) is the aerosol optical thickness (AOT) at the wavelength λ, τ_{1} is the approximated AOT at a wavelength of 1 μm (sometimes called the turbidity coefficient, as per [*Angstrom*, 1964]), and α has come to be widely known as the Angstrom exponent.

[3] In addition to being a useful tool for extrapolating AOT throughout the shortwave spectral region, the value of the Angstrom exponent is also a qualitative indicator of aerosol particle size [*Angstrom*, 1929]; values of α ≲ 1 indicate size distributions dominated by coarse mode aerosols (radii ≳0.5 μm) that are typically associated with dust and sea salt, and values of α ≳ 2 indicate size distributions dominated by fine mode aerosols (radii ≲0.5 μm) that are usually associated with urban pollution and biomass burning [*Eck et al.*, 1999; *Westphal and Toon*, 1991]. *Kaufman et al.* [1994] demonstrated that the Angstrom exponent can be a good indicator of the fraction of small particles with radii *r* = 0.057–0.21 μm relative to larger particles with radii *r* = 1.8–4 μm for tropospheric aerosols.

[4] Since the Angstrom exponent is easily measured using automated surface Sun photometry [*Holben et al.*, 1998] and is becoming increasingly accessible to satellite retrievals [*Nakajima and Higurashi*, 1998; *Higurashi and Nakajima*, 1999; *Deuzé et al.*, 2000; *Ignatov and Stowe*, 2002; *Jeong et al.*, 2005], the true utility of this parameter lies in its empirical relationship to the aerosol size distribution. For instance, α has been used to characterize the maritime aerosol component at island sites [*Kaufman et al.*, 2001; *Smirnov et al.*, 2002, 2003], biomass burning aerosols in South America and Africa [*Dubovik et al.*, 1998; *Reid et al.*, 1999; *Eck et al.*, 2001b, 2003], and urban aerosols in Asia [*Eck et al.*, 2001a]. Measurements also indicate that the Angstrom exponent varies with wavelength, and that the spectral curvature of the Angstrom exponent contains useful information about the aerosol size distribution [*King and Byrne*, 1976; *King et al.*, 1978; *Eck et al.*, 1999, 2001a, 2001b, 2003; *Kaufman*, 1993; *O'Neill and Royer*, 1993; *O'Neill et al.*, 2001a, 2001b, 2003; *Villevalde et al.*, 1994].

[5] The focus of this paper is to explore the relationship between the spectral dependence of extinction and the size distribution of atmospheric aerosols. We begin by illustrating the sensitivity of α to the median radius of monomodal aerosol size distributions, using multiwavelength Mie calculations of τ(λ) for 38 monomodal lognormal aerosol size distributions. Then we explore the relationship between α and 45 bimodal lognormal aerosol size distributions, demonstrating that α is more sensitive to the fine mode volume fraction than the fine mode median radius. Next, we apply the same technique to explore the information content in the wavelength dependence of α (i.e., the curvature of α). Finally, we discuss application of the Angstrom exponent and the spectral curvature for setting limits on the possible size distributions associated with aerosol optical depth measurements.

[6] Before proceeding further, however, a word about the origination of the term “Angstrom exponent” is in order. The term “Angstrom exponent” originates from an early treatise by Anders Angstrom that provides the first documentation of equation (1) currently available in English [*Angstrom*, 1929]. However, that article cites an even earlier laboratory study whereby Lundholm documented the power law relationship for the absorption of thin powders at Uppsala in 1912, prompting at least one author to point out that we are honoring the wrong person [*Bohren*, 1989]. Unfortunately, the Lundholm citation of *Angstrom* [1929] is obscure and perhaps incorrect, as our library staff was unable to locate it. Compounding matters even further, it is not even clear whether Lundholm or Lindholm documented the power law in 1912, as *Angstrom* [1929] uses both forms of spelling. Our library staff did find references to a F. Lindholm at Uppsala in *Science Abstracts*, but none of the abstracts mention the spectral dependence of particulate extinction. Subsequent references to Lundholm and Lindholm on this topic have not appeared in the atmospheric literature, and Angstrom claimed full credit for introducing the methods of evaluating atmospheric turbidity parameters in later articles [*Angstrom*, 1930, 1961, 1964]. So who should get the credit for originating and promoting equation (1)? After presenting the discussion above, we maintain the standard nomenclature. Angstrom admittedly did not originate the empirical power law, but he did publish at least four articles documenting the relationship between α and particle size; meanwhile, we are unable to obtain any public documentation of equation (1) by Lundholm or Lindholm, making it impossible for us to fairly evaluate this person's contribution.