Love number

Computation of the amplitude of tides for a realistic layered model is challenging. There is no simple formula available for evaluating the Love numbers h2 and k2. We have to rely on results from proprietary computer... 

Figure 7. (a) The values of the Love numbers for various assumptions regarding the... 

Figure 8 shows the results reported by Yoder and Sjogren, but in a way that elucidates the trends and is more accurately labeled than in the NASA report. First, we identify the important end-member cases that were not marked on Figure 7a. If all the layers were solid, the Love number values would lie at the spot marked by the small black spot at the lower left in Figure 8. If all the layers were fluid, then /r2 and k2 would be 2.05 and 1.05 respectively (Moore and Schubert 2000). Also, recall that if the same mass were uniformly distributed, rather than in layers, and were fluid and incompressible, then h<2 and 2 would be 2.5 and 1.5, respectively, according to the classical result by Love (1944).

We also have information about how the Love numbers would change if we started from any point along that line and increasingly softened the... 

While there would be a measurable difference in the Love numbers between the case of a 25 km ice shell and a 65 km ice shell, there is little change once the ice is thinner that about 25 km. This result was noted in the NASA report, and it (as well as most of these results) was corroborated by calculations by Moore and Schubert (2000). 

Hurford, T. A., and Greenberg, R. (2002). Tides on a Compressible Sphere Sensitivity of the h2 Love Number. In Lunar and Planetary Science Conference Abstracts, pages 1589-1590. 

Love s (1911) second case has a 3.3 and (3 2.1, yielding g/X = 1944/2041 (Poisson s ratio ps 1/4) and ggR/g = 1107/224. Again Love found that h is enhanced compared to the Love number of an incompressible case. The enhancement for these given parameter values is 22%. 

This remarkable result, that there exists Love number values that are infinite and/or negative, is not unique to this single set of parameters. Figure 5 shows the enhancement h/hinc0mpressibie for various values of ggR/g. If g and g are constrained to the reasonable values used by Love, the various values of ggR/g correspond to various radii. According to... 

From this understanding of how the interior deforms and changes density we get insight into how the cases with negative Love number can be physically plausible. For cases as h —> oo, the planet elongates toward the tide raiser (Figure 7a), while the mass density along this axis... 

At the surface nr(A) = 5R and the Love function h(R) matches the expression for the Love number as given in equation 4, (3). It has been shown that when C2 — 0, A2 and B2 — 00. Thus h(r) also becomes large under those conditions. 

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In addition, an incremental (deformation) potential 5w is produced by the re-distribution of masses which again is factorized by a number, the second Love number k, on the earth s surface... 

For realistic earth models the Love-Shida numbers prove to be dependent not only on the degree n of the tidal waves but also on the order and the frequency of the respective partial tides. Table 2 contains the values of the Love numbers calculated by Wahr (1981) for the earth model 1066 A of Gilbert and Dziewonski. 

Nevertheless, there is a large degree of uncertainty in fixing some numbers to the secular Love numbers ho, ko. 

Considering these problems, three concepts have been proposed with respect to the treatment of the permanent tidal effects. In the "non-tidal" case, recommended in a lAG Resolution of 1979, any direct and indirect effects, periodic and permanent, are reduced for. It is evident that this concept relies on the secular Love numbers ho, ko due to the disadvantages discussed above this approach is unsuitable from theoretical and practical points of view. 

The reduction to the "zero" case on the basis of the harmonic development (8) is achieved by applying the frequency-dependent Love numbers corresponding to the diurnal and semi-diurnal tidal waves. The permanent MqSq partial tide is correctly treated by formally setting ho=0, ko=0, 7o=l+ko-ho=l. 

If the reduction is to be based on Laplace s tidal representation it is impossible to choose frequency-dependent Love numbers. In this case average Love numbers, e. g. 

The reductions to the "mean" case can be derived analogously. If the tidal correction formula is based on the harmonic development (8) we simply have to apply the corresponding frequency-dependent Love numbers for the diurnal, semi-diurnal and long—periodic partial tides. The permanent MqSo term is correctly evaluated by formally setting ho O, ko=0, 7o=0. 

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