  Let $R$ be a unital $*$-ring. For any $a,w,b\in R$, we apply the defined
$w$-core inverse to define a new class of partial orders in $R$, called the
$w$-core partial order. Suppose $a,b\in R$ are $w$-core invertible. We say that
$a$ is below $b$ under the $w$-core partial order, denoted by
$a\overset{\tiny{\textcircled{\#}}}\leq_w b$, if $a_w^{\tiny{\textcircled{\#}}}
a=a_w^{\tiny{\textcircled{\#}}} b$ and $awa_w^{\tiny{\textcircled{\#}}}
=bwa_w^{\tiny{\textcircled{\#}}}$, where $a_w^{\tiny{\textcircled{\#}}}$
denotes the $w$-core inverse of $a$. Characterizations of the $w$-core partial
order are given. Also, the relationships with several types of partial orders
are considered. In particular, we show that the core partial order coincides
with the $a$-core partial order, and the star partial order coincides with the
$a^*$-core partial order.
