  The distribution of the inner orbital periods of solar-type main-sequence
(MS) triple star systems is known to be peaked at a few days, and this has been
attributed to tidal evolution combined with eccentricity excitation due to
Lidov-Kozai oscillations. Solar-type MS quadruple star systems also show peaks
in their inner orbital period distributions at a few days. Here, we investigate
the natural question whether tidal evolution combined with secular evolution
can explain the observed inner orbital period distributions in quadruple stars.
We carry out population synthesis simulations of solar-type MS quadruple star
systems in both the 2+2 (two binaries orbiting each other's barycentre) and 3+1
(triple orbited by a fourth star) configurations. We take into account secular
gravitational and tidal evolution, and the effects of passing stars. We assume
that no short-period systems are formed initially, and truncate the initial
orbital period distributions below 10 d accordingly. We find that, due to
secular and tidal evolution, the inner orbital period distributions develop
tails at short periods. Although qualitatively consistent with the
observations, we find that our simulated orbital period distributions only
quantitatively agree with the observations for the 3+1 systems. The observed
2+2 systems, on the other hand, show an enhancement of systems around 10 d,
which is not reproduced in the simulations. This suggests that the inner
orbital periods of 2+2 systems are not predominantly shaped by tidal and
secular evolution, but by other processes, most likely occurring during the
stellar formation and early evolution.
