Supplementary Methods

For a subsurface region as shown in Fig. 1 the mass deficit is given by

(2)

where e2=a2+ad+d2 and other quantities are defined in the main text. The quantity Rtis the radius at which diapir-induced uplift occurs, and is set to Rc (the core radius) for a diapir in the silicate core and to R for a diapir in the ice shell. A low-density region at depth will in general result in an interface uplift h and a corresponding mass excess of 2Rt2h(1-cos), where  is the density contrast between the material below and above the interface. In the isostatic case, the mass excess and mass deficit are equal, which implies that the isostatic topography h’ is given by

(3)

Note that this expression reduces to the usual (plane) isostatic case when a+d < Rt. In general, the subsurface mass deficit will not be completely compensated, owing to the rigidity of the lithosphere, and the interface topography will be given by h=Ch’, where C is the degree of compensation.

From Kaula (1968, p.67) the surface potential due to a degree-2 axisymmetric mass excess at the interface Rt is given by

(4)

where is a Legendre polynomial, is latitude and h<Rt, while the potential due to the internal mass deficit is given by

(5)

where terms in (a/Rt)3 and higher have been neglected. Making use of equations (2)-(5), for a diapir region of angular extent , the resulting degree-2 component of the potential is therefore given by

where the dependence on  is found using Blakely (1996, eq. 6.9). In the isostatic case C = 1 and the potential anomaly is positive, as expected.

Given the degree-2 component of the potential G20, the resulting degree-2 gravity anomaly g at a spacecraft altitude z is given by (e.g. McKenzie and Nimmo 1997)

where in this case l=2.

To calculate k2 we employed a multi-layered Maxwell viscoelastic code21 with either three or four layers. To obtain the long-term value of k2, after all viscous stresses have relaxed, we set the rigidities and viscosities of all layers except the elastic layer(s) to zero18. For the ice diapir case, we assumed that the silicate core was cold and elastic and adopted the following structure (thickness, density, rigidity and viscosity):

Silicate core161 km3500 kg m-3100 GPa1021 Pa s

Ice shellvariable950 kg m-30 GPa0 Pa s

Ice lithospherevariable (d)950 kg m-33 GPa1019 Pa s

The total ice layer thickness was 91 km and the thickness of the ice lithosphere d was varied as shown in Fig. 2. Because in general k2 < k2f, and also because of the form of equation (1), the details of the structure adopted were found to have very little effect on the reorientation at elastic thicknesses in excess of ~2 km. The main control on the degree of reorientation is the size of the potential anomaly, and not the resistance of the frozen-in bulge (which depends on k2f – k2).

For the silicate diapir case, it was assumed that the silicate core was warm and thus only the silicate lithosphere retained any elastic strength, giving the following structure:

Silicate corevariable3500 kg m-30 GPa0 Pa s

Silicate lithospherevariable (d)3500 kg m-3100 GPa1021 Pa s

Ice shell91 km950 kg m-30 GPa0 Pa s

The total silicate layer thickness was 161 km and the thickness of the silicate lithosphere d was varied as shown in Fig 2. Again, the details of the internal structure chosen had little effect on the degree of reorientation.

For a tidally-distorted satellite, the theory of 18is only applicable to reorientation about the tidal axis, which is energetically favoured and for which n=1 in equation (1). For completeness, we consider the more general case of a synchronous, tidally-distorted satellite in hydrostatic equilibrium. There is a simple relationship between the potential due to rotation and the potential due to tides (Murray and Dermott 1999); adding these two effects results in a triaxial ellipsoid, rather than the oblate spheroid generated by rotational effects alone. In the triaxial case, the extent of reorientation will depend on the longitude, as well as the latitude, of the starting load. Making use of Murray and Dermott (1999), Section 4.7, we may rewrite equations (27) of 18 such that the final constants in the expressions for I11L,L-D,ROT, I22L,L-D,ROT and I33L,L-D,ROT are -7,+2 and -5 rather than -1,-1 and -1, respectively. Diagonalization of the resulting matrix yields equation (1) with n=4. If reorientation occurs around the a (tidal) axis, the tidal deformation plays no role and we obtain equation (1) with n=1, as found by 18. This latter case is likely to be energetically favoured and is the one we adopt here.

Additional References

Blakely RJ, Potential theory in gravity and magnetic applications, Cambridge Univ. Press, 1996.

Kaula WM, An introduction to planetary physics: The terrestrial planets, John Wiley & Sons, 1968.

McKenzie D, Nimmo F, Elastic thickness estimates for Venus from line of sight accelerations, Icarus 130, 198-216, 1997.

Murray CD, Dermott SF, Solar system dynamics, Cambridge Univ. Press, 1999.