  Ultracold quantum gases provide a unique setting for studying and
understanding the properties of interacting quantum systems. Here, we
investigate a multi-component system of $^{87}$Rb--$^{39}$K Bose-Einstein
condensates (BECs) with tunable interactions both theoretically and
experimentally. Such multi-component systems can be characterized by their
miscibility, where miscible components lead to a mixed ground state and
immiscible components form a phase-separated state. Here we perform the first
full simulation of the dynamical expansion of this system including both BECs
and thermal clouds, which allows for a detailed comparison with experimental
results. In particular we show that striking features emerge in time-of-flight
for BECs with strong interspecies repulsion, even for systems which were
separated in situ by a large gravitational sag. An analysis of the center of
mass positions of the BECs after expansion yields qualitative agreement with
the homogeneous criterion for phase-separation, but reveals no clear transition
point between the mixed and the separated phases. Instead one can identify a
transition region, for which the presence of a gravitational sag is found to be
advantageous. Moreover we analyze the situation where only one component is
condensed and show that the density distribution of the thermal component also
show some distinct features. Our work sheds new light on the analysis of
multi-component systems after time-of-flight and will guide future experiments
on the detection of miscibility in these systems.
