In this work we are most interested in dissipation within Mars and the factors which control it (primarily temperature and grain size). We will therefore adopt a relatively elaborate model of the dissipation process (see section 3.2 below). Conversely, our model will in other ways be significantly simplified, e.g., in its treatment of phase changes and its calculation of pressure and density (section 3.1). These simplifications are adopted for three reasons: first, our current knowledge of Mars' internal structure and composition is highly uncertain; second, the approximations involved can be shown a posteriori not to affect our main conclusions; and third, a simplified approach makes the sensitivity of our results to uncertainties more transparent.
 Even the dissipation model has its simplifications. In particular, it assumes that the dissipative behavior (but not the density/rigidity structure) of the Martian mantle can be adequately represented by the experimentally characterized response of dry, melt-free Fo90 olivine [Jackson and Faul, 2010]. In detail, this assumption is unlikely to be correct. The bulk of the Martian mantle is most likely olivine but of a more iron-rich kind than typical terrestrial values [Dreibus and Wanke, 1985]. The water content and extent of melting within the Martian mantle are not well known, but both can have significant effects on dissipation. The lower Martian mantle hosts higher-pressure phases (wadsleyite, ringwoodite, and possibly perovskite) [Longhi et al., 1992] which are treated in a simplified manner below.
 At one level, our decision to model the dissipative component of the Martian mantle using results from Fo90 is a purely pragmatic decision: other minerals and compositions have not been sufficiently well characterized to allow their rheological behavior to be described. However, it also represents a reasonable initial approach: dissipation is likely to be dominated by the most abundant mineral present, and a dry, melt-free mantle represents a useful end-member. We discuss the potential effects of melt and water further below (section 5.1). The simplified treatment of phase changes can be justified a posteriori as our best fit models result in most dissipation happening in the upper (low-pressure) part of the mantle (see Figure 1). The role of iron in controlling olivine dissipation is currently poorly understood but may be quite dramatic [Zhao et al., 2009]; thus, the approach adopted here probably underestimates the dissipation for a given set of conditions.
 For our simplified Martian structure, we will assume a five-layer model consisting of core, a mantle split into olivine (ol), wadsleyite (wa), and ringwoodite (ri) phase assemblages, and a crust. For the purposes of calculating the gravity and pressure within Mars, we will assume that the mantle and core have constant densities ρm and ρc, respectively. For a specified value of ρc we calculate the core radius and mantle density which satisfy the measured bulk density and moment of inertia of Mars, taking the crust into account (see Table 3). This approach is evidently not self-consistent, but the errors introduced are much smaller than other uncertainties (see below).
 Taking the core radius to be Rc, the gravitational acceleration within the mantle is given by
where Gg is the gravitational constant, and here we are neglecting the small density contrast between crust and mantle. In the mantle the pressure increment dP is given by dP=−ρmgdr which yields
where R is the radius of Mars and Δρ=ρc−ρm.
 In reality, the mantle density ρ will vary with depth because of compressibility, thermal expansion, and phase changes. For a particular phase assemblage x, we calculate ρx as follows:
where Kx is the bulk modulus, T is the temperature, TR is a reference temperature, and αx is the thermal expansivity (assumed constant). The quantity ρ0,x is a reference density. In what follows, we vary the reference density ρ0 of the olivine phase so that the bulk density of the mantle equals ρm. The other two reference densities are kept fixed. The bulk modulus is calculated as described in section 3.2.
 While this approach is not fully self-consistent, the errors introduced are small. The difference between the analytical values for g(r) (equation (2)) and those obtained by numerically integrating the calculated density profile (equation (4)) never exceeds 3%. Similarly, the difference between the analytical value for P(r) (equation (3)) and that obtained by numerical integration never exceeds 4%. Given other major uncertainties (notably the location of the core-mantle boundary), such errors are acceptable.
 The temperature structure of the Martian mantle likely includes a conductive stagnant lid overlying a convecting adiabatic mantle [e.g., Ogawa and Yanagisawa, 2011]. A simple parameterization approximating this behavior may be obtained as follows. The horizontally averaged potential temperature Tp is given by
 Here Ts is the surface temperature, Tm is the potential temperature of the convecting interior, and depth z=R−r. The near-surface heat flux in this model is k(Tm−Ts)/L, where k is the thermal conductivity. The quantity L is an indication of how thick the conductive lid is; roughly 90% of the total temperature drop occurs across a thickness 1.5L, so we may take the stagnant lid thickness to be ≈1.5L. Because most materials do not undergo significant creep below about 70% of their melting point [Frost and Ashby, 1982], the effective elastic thickness of the lithosphere is approximated by L.
 To convert from potential temperature to actual temperature T(z), we multiply Tp(z) by an adiabatic factor fad. The adiabatic temperature gradient dT/dz=αgT/Cp, where αand Cp are thermal expansivity and specific heat capacity, respectively. Taking these two quantities to be constant and using equation (2) for g, we find
 Equations (5) and (6) can thus be combined to derive the real temperature profile.
 Our simplified model does not treat the crust in detail. This is because Q, our main focus, will not be significantly affected by a cold, thin surficial layer. However, because the crust may be enhanced in radiogenic elements, it will affect the temperature structure; in particular, the conductive heat flux through the lid derived from equation (5) may be less than the actual surface heat flux.
 Local dissipation in the Martian mantle is calculated by using a model based on laboratory experiments to determine the complex rigidity as a function of depth. We generally adopt the same approach and parameters as in Nimmo et al. ; only a summary is given below.
 Laboratory experiments on melt-free, polycrystalline olivine show that the dissipation factor Q goes as ωα, where ω is the forcing angular frequency and α is a constant ≈0.3 [e.g., Gribb and Cooper, 1998; Jackson et al., 2004]. One advantage of investigating Mars rather than the Moon is that the tidal frequencies are higher (∼10−4s−1rather than ∼10−6s−1). This means that, unlike the Moon, essentially no extrapolation in frequency is required when applying the laboratory-derived parameters to Mars.
 Qualitatively, dissipation appears to arise from two different mechanisms: a background effect, inferred to be due to a diffusive process and modeled as a distribution of relaxation times, and a superimposed peak corresponding to elastically accommodated grain boundary sliding. Both these mechanisms are included. The result is that Q and the rigidity modulus G are dependent on temperature, pressure, and forcing period. The sensitivity of Q to temperature and frequency is much larger than that of G; for Mars-sized objects the pressure sensitivity is minor but not negligible. The results are also sensitive to the grain size d adopted; we examine the effects of varying this parameter below.
 In detail, we treat the shear modulus as a complex quantity G∗=1/J∗, where J∗=Jr+iJi is the complex compliance and . At a particular frequency the dissipation factor Q=Jr/Ji. For the extended Burgers model, the real and imaginary components of the complex compliance are as follows:
 Here GU=1/JU is the unrelaxed (infinite-frequency) shear modulus calculated using equation (9) below, τ is a dummy variable, τM=η/GU is the Maxwell time for a material of steady state viscosity η, Δ describes the strength of the relevant relaxation mechanism, ωis the angular frequency, and τL and τH are the integration limits corresponding to short and long periods, respectively. In the low-frequency limit, equation (8) reduces to Maxwellian behavior, in which Ji=(ωη)−1.
 As noted above, laboratory experiments suggest that there are two different relaxation mechanisms operating, with different distributions of relaxation times. We therefore include both a high-temperature background with strength ΔB and an additional peak with strength ΔP. Both distributions are a strong function of grain size and temperature. Further details may be found in Jackson and Faul  and Nimmo et al. . Unless noted otherwise below, all rheological parameters used are identical to those given in Nimmo et al. [2012, Table 1]; one exception is the reference grain size (13.4 μm) which was incorrectly stated to be 3.1 μm in that paper.
 The elastic (unrelaxed) rigidity modulus GU depends on pressure and temperature and is calculated as in Nimmo et al. :
where PR and TR are the reference temperature and pressure, and and are experimentally measured quantities (see below). Note that here we have neglected higher-order pressure derivatives, owing to the relatively modest pressures on Mars. We adopt different values of the reference rigidity GU(TR,PR) and the partial derivatives for the three different phases (see Table 1).
Table 1. Parameter Values for the Nominal Modela
|R||3400||km|| (3)||Rc||1700||km|| (2)|
|ρm||3526||kg m−3|| (2)||ρc||7000||kg m−3|| (2)|
|Ts||220||K|| (5)||Olivine assemblage (ol)|
|L||125||km|| (5)||ρ0||3300||kg m−3|| (4)|
|TR||1173||K|| (9)||GU(TR,PR)||62.7||GPa|| (9)|
|PR||0.2||GPa|| (9)||∂G/∂T||-13.1||MPa K−1|| (9)|
|Tm||1600||K|| (5)||α||2×10−5||K−1|| (6)|
|Cp||1200||J kg−1K−1|| (6)||K(TR,PR)||114||GPa|| (10)|
| || || || ||∂K/∂T||-18||MPa K−1|| (10)|
| || || || ||∂K/∂P||4.2||-|| (10)|
|Wadsleyite assemblage (wa)||Ringwoodite assemblage (ri)|
|ρ0||3580||kg m−3|| (4)||ρ0||3770||kg m−3|| (4)|
|GU(TR,PR)||88.8||GPa|| (9)||GU(TR,PR)||102||GPa|| (9)|
|∂G/∂T||-15||MPa K−1|| (9)||∂G/∂T||-15||MPa K−1|| (9)|
|∂G/∂P||1.4||-|| (9)||∂G/∂P||1.4||-|| (9)|
|α||2×10−5||K−1|| (6)||α||2×10−5||K−1|| (6)|
|K(TR,PR)||159||GPa|| (10)||K(TR,PR)||167||GPa|| (10)|
|∂K/∂T||-12||MPa K−1|| (10)||∂K/∂T||-29||MPa K−1|| (10)|
|∂K/∂P||4.3||-|| (10)||∂K/∂P||4.1||-|| (10)|
 We calculate the (unrelaxed) bulk modulus K in an analogous fashion:
The bulk modulus is assumed to not vary significantly with period, but is important in determining the density structure (equation (4)).
 The calculations detailed in Nimmo et al.  yield the complex shear modulus G at a specified frequency, where the ratio of the real to the imaginary parts of the modulus contains information on the local value of Q. At high frequencies, the mantle response is essentially elastic and G≈GU. At lower frequencies, dissipation becomes more important and G<GU. To calculate the global, frequency-dependent quantities Q and k2, the method of Roberts and Nimmo  is used. The mantle is discretized into layers 50 km thick, while the core is assumed to be fully liquid (zero shear modulus) and have other properties given in Table 1. The outermost 50 km thick layer is taken to be the crust, with an assumed density of 2900 kgm−3and the same elastic parameters as mantle material. We assume that Mars is spherically symmetric.
 The Martian mantle, like that of the Earth, is expected to undergo phase changes [Longhi et al., 1992; Bertka and Fei, 1997]. The olivine-wadsleyite and wadsleyite-ringwoodite transitions are expected to occur at roughly 1100 km and 1400 km depths, respectively [e.g., Mocquet and Menvielle, 2000; Khan and Connolly, 2008], depending on composition and temperature. The depth to the perovskite transition is comparable to the mantle thickness, so whether or not it occurs depends on details of the model adopted, such as the core radius and mantle temperature [Bertka and Fei, 1997]. For simplicity, we fix the transition depths zol−wa=1150 km and zwa−ri=1400 km and neglect the perovskite phase entirely. The anelastic behavior of these higher-pressure phases has not yet been fully characterized [Nishihara et al., 2008; Kawazoe et al., 2010]. However, seismological observations on Earth suggest that Q does not increase in a discontinuous fashion at the top of the transition zone. Rather, most models show a continuous increase through the upper mantle and the transition zone [Matas and Bukowinski, 2007]. Thus, to first order it appears that the dissipative nature of the mantle material is not changed by the ol-wa or wa-ri phase transitions. To calculate the anelastic behavior of the higher-pressure assemblages, we therefore adopt the same parameters describing dissipation as for the olivine assemblage. Note, however, that ρ0 and GU and Kand their derivatives are different (see Table 1).
 Except as noted below, the material properties of the mantle are taken from Jackson and Faul [2010, Table 2]. These authors were using Fo90 olivine; more iron-rich olivine (likely appropriate for Mars) will have slightly different elastic properties [Bass, 1995]. For all three phases we obtain GU(TR,PR) and K(TR,PR) and their derivatives from Stixrude and Lithgow-Bertelloni [2005, 2011] for an Mg# of 75; equations (9) and (10) are then used to determine GU and K at the conditions of interest. With this approach we are assuming that the higher iron content only affects the elastic moduli but not the grain boundary viscosity or diffusivity of olivine. The latter assumption can be justified by the observation that silicon is the slowest-diffusing species, controlling strain rates for both grain boundary and volume diffusion [e.g., Hirth and Kohlstedt, 2003; Dohmen and Milke, 2010]. On a more pragmatic level, there are currently no experimental data on dissipative behavior in iron-rich olivines.
 The three reference densities ρ0 were chosen to approximately satisfy the petrologically derived density curve of Bertka and Fei  (see Figure 1b) while yielding the observed moment of inertia and bulk density. Varying these values has no significant effect on the resulting Q (see section 5.1). The thermal expansivities were likewise chosen to approximately match the more sophisticated calculations of Sohl and Spohn [1997, Figure 4]; likely variations have only minor effect on our results (section 5.1).
 Our baseline core parameters were chosen to be consistent with the observed bulk density and moment of inertia. We examine the effect of varying these parameters below (section 5.1) and conclude that, although they do affect the model k2, they have almost no effect on the model Q value. We emphasize again that Q is sensitive to temperature and grain size but not rigidity or density. As a result, more sophisticated and self-consistent models of rigidity and density will not produce appreciably different dissipation results compared with the simple models we have adopted here.