forcing-frequency.md

Reference (1)

Forcing Frequency

Given the fact that the tidal forcing magnitude decreases exponentially with harmonic order, we may limit the calculation to only the lowest harmonic degree n = 2. Generally, in the limit of R_p \ll a, it suffices to only consider the quadrupolar harmonic (n = 2). Nevertheless, Obliqua can also determine higher degree contributions. As of now, we are considering a coplanar geometry. Hence, terms with m = 1 associated with obliquity tides vanish. The Fourier modes of the second harmonic have frequencies given by

$$\sigma = m\Omega - k n_{\mathrm{orb}},$$

where \Omega is spin rate and n_{\mathrm{orb}} orbital mean motion, and for integer values of order -2 \leq m \leq 2 and harmonic \infty \leq k \leq \infty. In our formalism tides are occuring over a large time interval, a time step \Delta t. As such, we must account for tidal excitations that occur over a wide range of frequencies. We calculate the imaginary part of the nth harmonic degree (k_n) Love number (\Im[k_{n}(\sigma)]) for all relevant harmonnic frequencies for which the Hansen coefficient

$$X^{-(n+1), m}_k(e) = \frac{1}{2\pi} \int_0^{2\pi} \left(\frac{r}{a}\right)^n e^{im\Omega - ikn_{\mathrm{orb}}}\,dn_{\mathrm{orb}} \geq 0.01,$$

(i.e. ~1% corrections). This implies that we are considering the following set $K$ of harmonnic frequencies $k$:

$$\{k \in K \, \forall \, k : X^{-(n+1), m}_k \geq 0.01\ | k \in Z\}$$

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