Measuring tides and binary parameters from gravitational wave data and eclipsing timings of detached white dwarf binaries

The discovery of the most compact detached white dwarf (WD) binary SDSS J065133.33+284423.3 has been discussed in terms of probing the tidal effects in white dwarfs. This system is also a verification source for the space-based gravitational wave (GW) detector, evolved Laser Interferometer Space Antenna (eLISA) which will observe short-period compact Galactic binaries with $P_{orb}\lesssim 5$ hrs. We address the prospects of doing tidal studies using eLISA binaries by showing the fractional uncertainties in the orbital decay rate and the rate of that decay, $\dot{f}, \ddot{f}$ expected from both the GW and EM data for some of the high-$f$ binaries. We find that $\dot{f}$ and $\ddot{f}$ can be measured using GW data only for the most massive WD binaries observed at high-frequencies. Form timing the eclipses for $\sim 10$ years, we find that $\dot{f}$ can be known to $\sim 0.1$ for J0651. We find that from GW data alone, measuring the effects of tides in binaries is (almost) impossible. We also investigate the improvement in the knowledge of the binary parameters by combining GW amplitude and inclination with EM data with and without $\dot{f}$. In our previous work we found that EM data on distance constrained 2-$\sigma$ uncertainty in chirp mass to $15-25$ whereas adding $\dot{f}$ reduces it to $0.11$. EM data on $\dot{f}$ also constrains 2-$\sigma$ uncertainty in distance to $35$. EM data on primary mass constrains the secondary mass $m_2$ to factors of 2 to $\sim40$ whereas adding $\dot{f}$ reduces this to $25$. And finally using single-line spectroscopic constrains 2-$\sigma$ uncertainties in both the $m_2, d$ to factors of 2 to $\sim 40$. Adding EM data on $\dot{f}$ reduces these 2-$\sigma$ uncertainties to $\leq 25$ and $6$ respectively. Thus we find that EM measurements of $\dot{f}$ and radial velocity will be valuable in constraining binary parameters.