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 $n$th harmonic degree ($k_n$) Love number ($\Im[k_{n}(\sigma)]$) for all relevant harmonnic frequencies for which the Hansen coefficient

\[|X^{-(n+1), m}_k(e) |= \bigg| \frac{1}{2\pi} \int_0^{2\pi} \left(\frac{r}{a}\right)^n e^{im\Omega - ikn_{\mathrm{orb}}}\,dn_{\mathrm{orb}} \bigg| \geq 0.001,\]

(i.e. ~0.1% corrections). This implies that we are considering the following set $K$ of harmonnic frequencies $k$:

\[K = \{k \in \mathbb{Z} \, \big| \, |X^{-(n+1), m}_k| \geq 0.001 \}\]

Function Documentation

get_k_range(e, n, m)

get_hansen(ecc)