This is not really a research problem. The answer is in exercise 3.12.4(e) in Engelking's General topology text. For each linearly ordered topological space $X$ we have that the 
cellularity of $X$ equals the hereditarily Lindelof number of $X$ (and you ought to be able to prove this). By definition every Souslin line has countable cellularity, i.e. every disjoint family of non-empty open sets is at most countable. The hereditarily Lindelof number, $hL(X)$, is defined as $\sup\{L(Y): Y \subseteq X\}$ where $L(Y)$ is the Lindelof number of $Y$, i.e. every open cover of $Y$ has a subcover of cardinality at most $L(Y)$.