nuclear-density material). In this sense, we can view our magnetically-levitated droplet as a ‘toy model’ of an incompressible gravitating body (possessing surface tension), composed of a small-diameter, massive core (the fictitious mass), and a low density outer region (the water). Note that this result differs from that of a uniform-density self-gravitating body, σ$_{G}$ = [(8 / 3) πGρl ( l − 1) / (2 l +1)] 1 / 2 [32]. Bastrukov has recently obtained a similar spectrum for a self-gravitating liquid droplet with radially-varying density distribution ρ ( r < a ) ∝ 1 /r , having a singular density at the center [33]. By adjusting the current in the magnet (i.e. B$_{0}$ ) we can change the quadrupole ( c $^{′}$$_{2}$) component of the trap. Although we have sought to minimize this component in this paper, it would be interesting to investigate the effect of this quadrupole component on the oscillation frequencies experimentally and compare it with the result we derive above. Unlike the odd harmonics, the even order harmonics have a significant effect on the eigenfrequencies to first order. Using the analogy with a model ‘star’, adding a quadrupole component to the effective gravity Γ is equivalent to deforming the shape of the core mass from spherical to oblate, perhaps due to rotation, for example. This project is supported by a Basic Technology Grant from EPSRC, UK; Grant Nos. GR/S83005/01 and EP/G037647/1. VII. APPENDIX We consider the effect of a trap component c ′ j on an arbitrary shape oscillation The corresponding velocity potential is Inserting Eqn. 22 and Eqn. 23 into Eqn. 6, and equating the time-varying terms, we obtain The product Y m l P$_{j}$ = N m l e im $^{Φ}$P m l P 0 j can be expanded as N m l e im Φ ∑ j + l p = | j − l | Q (0 , j ; m, l ; p ) P m p , where the Q in this case are Gaunt coefficients, as defined in Ref. [34], P m l are associated aω 2 ∑ l ≥ 1 l ∑ m = − l b m l Y m l l = ∑ l ≥ 1 l ∑ m = − l [ ( c ′ 0 + c ′ $_{j}$P$_{j}$ ) + T ρa $_{2}$( l − 1)( l + 2)$^{]}$ b m l Y m l . (24) φ ( r, θ, Φ) = − ϵω cos ωt ∑ l ≥ 1 l ∑ m = − l b m l r $^{l}$l − $^{1}$a − l $^{+1}$Y m l ( θ, Φ) . (23) r = R ( θ, Φ , t ) = a + ϵ sin ωt ∑ l ≥ 1 l ∑ m = − l b m l Y m l ( θ, Φ) . (22) 16