  This paper is concerned with ground states of two-component Bose gases
confined in a harmonic trap $V(x)=x_1^2+\Lambda^2 x_2^2$ rotating at the
velocity $\Omega >0$, where $\Lambda\ge 1$ and $(x_1, x_2)\in R^2$. We focus on
the case where the intraspecies interaction $(-a_1,-a_2)$ and the interspecies
interaction $-\beta$ are both attractive, i.e, $a_1, a_2$ and $\beta $ are all
positive. It is shown that for any $0<\Omega <\Omega ^*:=2$, ground states
exist if and only if $0<a_1,\, a_2<a^*:=\|w\|^2_2$ and
$0<\beta<\beta^*:=a^*+\sqrt{(a^*-a_1)(a^*-a_2)}$, where $w>0$ is the unique
positive solution of $-\Delta w+ w-w^3=0$ in $R^2$. By developing the argument
of refined expansions, we further prove the nonexistence of vortices for ground
states as $\beta\nearrow\beta^*$, where $0<\Omega <\Omega ^*$ and $0<a_1,\,
a_2<a^*$ are fixed.
