  Given a graph $F$, the planar Tur\'an number of $F$, denoted
$\text{ex}_{\mathcal{P}}(n, F)$, is the maximum number of edges in an
$n$-vertex $F$-free planar graph. Such an extremal graph problem was initiated
by Dowden while determining sharp upper bound for
$\text{ex}_{\mathcal{P}}(n,C_4)$ and $\text{ex}_{\mathcal{P}}(n,C_5)$, where
$C_4$ and $C_5$ are cycles of length four and five respectively. In this paper
we determined an upper bound for $\text{ex}_{\mathcal{P}}(n,S_{2,2})$,
$\text{ex}_{\mathcal{P}}(n,S_{2,3})$, $\text{ex}_{\mathcal{P}}(n,S_{2,4})$,
$\text{ex}_{\mathcal{P}}(n,S_{2,5})$, $\text{ex}_{\mathcal{P}}(n,S_{3,3})$ and
$\text{ex}_{\mathcal{P}}(n,S_{3,4})$, where $S_{m,n}$ is a double star with $m$
and $n$ leafs. Moreover, the bounds for $\text{ex}_{\mathcal{P}}(n,S_{2,2})$
and $\text{ex}_{\mathcal{P}}(n,S_{2,3})$ are sharp.
