Reference (7)

Tidal potentials

Tidal Dissipation

To determine the total tidal heating, Obliqua loops over all $(n,m,k)$ pairs and calculates heating rates based on general eccentric expressions. While the model currently assumes co-planarity (precluding obliquity tides), it fully accounts for eccentricity through Hansen coefficients.

Tidal Potential and Normalization

For every triplet $(n, m, k)$, we calculate the Hansen coefficient $X_k^{-(n+1),m}(e)$ and the normalization factor $A_{n,m,k}$:

\[A_{n,m,k} = (2 - \delta_{m,0}\delta_{k,0}) (1 - \delta_{m,0}\delta_{k<0}) \sqrt{\frac{4\pi}{2n+1}\frac{(n-m)!}{(n+m)!}} P_n^m(0) X_k^{-(n+1),m}(e)\]

The associated tidal potential $U_{n,m,k}$ is defined as:

\[U_{n,m,k} = \frac{GM}{a} \left(\frac{R}{a}\right)^n A_{n,m,k}(e)\]

Heating Rates

The "black box" tidal models return an unnormalized heating profile $\tilde{H}(r, \sigma)$. To obtain the physical heating profile $H(r, \sigma)$ at a specific forcing frequency $\sigma$, we scale the output by the square of the tidal potential:

\[H(r, \sigma) = \tilde{H}(r, \sigma) \times |U_{n,m,k}|^2\]

The global heating rate $H(\sigma)$ for a given frequency is determined using the imaginary part of the tidal Love number $k_n(\sigma)$:

\[H(\sigma) = \text{prefactor} \times (-\text{Im}[k_n(\sigma)]) \times |U_{n,m,k}|^2\]

where the energy dissipation prefactor is:

\[\text{prefactor} = \frac{(2n + 1)R}{8\pi G} \sigma\]

Spectral Summation

Since the tidal forcing consists of a discrete set of frequencies, the total heating is determined by summing across the forcing frequency domain:

• Radial Heating Profile: $H(r) = \sum_{\sigma} H(r, \sigma)$
• Total Global Heating: $H = \sum_{\sigma} H(\sigma)$

Consistency Check: The volume integration of the radial profile $H(r)$ should match the global heating rate $H$, provided the surface load Love number $k'_n(\sigma)$ remains small compared to unity ($k'_n \ll 1$).

Outputs for Orbital Dynamics

In addition to the heating rates, Obliqua returns the complex tidal Love numbers $k_n(\sigma)$ and their corresponding forcing frequencies $\sigma$. these are required for calculating tidal torques and the long-term orbital evolution of the system.