12. Gravity and Topography 126 12 Gravity and Topography In the limit of perfect hydrostatic equilibrium, there is an exact relationship between gravity and topography.... and no new information emerges: The topography is directly determined from measuring the gravity and vice-versa. (The surface is an equipotential). In the limit where the only non-uniformity in the body is topography (i.e. a uniform density body that is “bumpy” with the bumps supported by finite strength), there is also an exact relationship between gravity and topography because the topography tells us what gravity anomalies to expect. No new information emerges (except the confirmation that the gravity anomalies are indeed merely the consequence of the topography alone and the density of the material is uniform). One could, however, determine the density of the near surface material (i.e., crust) because the gravity directly associated with the topography depends on this. In the real situation of a solid planet, there is new information available from studying gravity and topography together because the density anomalies responsible for the gravity are neither entirely hydrostatic nor entirely confined to the surface and expressed as topography. However, this real situation is also very non-unique in its interpretation because the depth of the density anomalies is undetermined (unless you can use yet another data set, e.g., seismic tomography). The non-uniqueness problem is reduced by appealing to physically plausible models: ideas of isostasy, the nature of the lithosphere, and ideas about the nature of the dynamics resulting from density anomalies in a viscous mantle. To some extent, these ideas are testable by looking at the spectra, the nature of gravity and topography (and their interrelationship) in spherical harmonic space. This is a big subject and this chapter only summarizes the main points. 12.1 Uncompensated Topography Suppose that there is a topographic harmonic Hm in the crust (density ρc ), so that the resulting mass anomaly per unit area is ρc HmPm (cosθ)cosmϕ. The contribution to the gravity potential external to the planet is then ΔVm(r,θ,ϕ) = GPm(cosθ)cos mϕdcos′θ−11∫d′ϕ02π∫ρc′r2RR + HmPm..∫.(′r/ r+1).Pm(cos′θ)cos m′ϕdr ' = GPm(cosθ)cos mϕ.4πR2.Hmρc.12 + 1.Rr+1 and this can be written in the form: 3GMρcρ(2 + 1).HmR.Rr+1 (12.1) where ρ is the mean planet density and the Legendre function , etc. has been dropped for simplicity of writing (i.e., it is always implied.)12. Gravity and Topography 127 12.2 Compensated Topography Suppose we wished to “compensate” for this topography at some depth D. Compensation in the isostatic sense means that the total mass at any location above depth D must be the same as it is in a location where there are no anomalies whatsoever. In other words, if there is a density anomaly Δρ at depth D then it must have a thickness Hm* such that (R − D)2Hm* Δρ= −R2Hlmρc (12.2) This is what we mean by isostatic compensation in a spherical planet. See the figure below. [But note that there are other kinds of definitions that one could construct for “isostasy”. Ultimately one wants a specific physical model]. The parameter D is called the depth of compensation. It is a theoretical construct and might or might not correctly describe what happens in a real planet. Of course, positive topography must be compensated by a negative density anomaly (i.e., a density deficit). This could, for example, be a crustal root, but it could also be a hotter part of the mantle. Figure 12.1 It should be evident by inspection of the integration above that the gravity potential anomaly associated with the density anomaly at depth D is:12. Gravity and Topography 128 ΔVm* = −3GMρcρ(2 + 1).HmR.(R − D)r+1 (12.3) from which it follows that the total gravity potential anomaly is now: ΔVm(total) =3GMρcρ(2 + 1).HmR.R− (R − D)r+1 (12.4) 12.3 Geoid and It’s Relationship with Topography The geoid height is defined to be the distance above a reference surface (e.g. mean radius of planet, including the effect of rotation) of a constant potential surface. So if we define the height for a given harmonic component G then by definition GMR + Gm...+ ΔVm... (12.5) is independent of position (i.e. is a constant).[ Remember that the .... after Gm etc means the usual Legendre function or spherical harmonic]. So it follows (using first order Taylor series expansion) that Gm=3ρcHmρ(2 + 1) [No compensation]Gm=3ρcHmρ(2 + 1).(1 −ζ) [Includes isostasy]ζ≡ 1-DR (12.6) For D/R<<1 and  not too large, 1-ζ~ D/R and G is only weakly dependent on  and proportional to D. By comparing this prediction with data, one might hope to deduce the depth of compensation D.12. Gravity and Topography 129 Figure 12.2 Shown above is an attempted application to Mars. As you can see, it does not work very well. However, it does show that compensation (in some sense) is occurring and it requires relatively small values of D, e.g. 200km or less. One reason why the data are fit rather poorly is mantle dynamics, which is important at small harmonic degree (see below). Another reason is that the lithospheric flexes in a way that depends on some other length scale (related to but not equal to the depth of compensation D). This flexural response provides compensation for long wavelengths but poor compensation at shorter wavelengths (high harmonics). Flexure is discussed in an appendix to this chapter. Another reason is that the planet does not have uniform properties from region to region (i.e., the parameter D is not constant with latitude and longitude). One way to see the extent of isostasy is to look at the power spectrum. This adds up the sum of the squares of harmonics for a particular  value. Below, this is plotted in such a way that the topography and geoid power spectra would exactly agree if there was no compensation. The much lower variance of the geoid (at least at low harmonic degree) attests to the extent of isostasy.12. Gravity and Topography 130 Figure 12.3 In the case of Mars, most of the topography is associated with variations in crustal thickness and is very well compensated. This is illustrated below, showing the results of the MGS mission (Zuber et al, Nature, 2001)