The non central t - distribution
Recall that if $Z \sim n(0, 1)$ and $ U \sim {\chi^2}_v$ are independent then
$$\frac{Z}{\sqrt{\frac{U}{v}}}\sim t_v$$
and it follows for a random sample $X_1 \ldots X_n \sim n(\mu, \sigma^2)$ independent; that
$$\frac{\bar {X} - \mu}{\frac{s} {\sqrt{n}}} = \frac{\frac{\bar {X} - \mu}{\frac{\sigma} {\sqrt{n}}}}{\sqrt{\frac{\sum ({X_i} -\bar {X})^2} { \frac {{\sigma}^2} {n-1}}}} \sim t_{n-1}$$
Details
On the other hand, if $W \sim n (\Delta,1) $ and $U \sim {\chi}^2_v $ are independent, then $\frac{W}{\sqrt{\frac{U}{v}}}$ has a non central t-distribution with $v$ degrees of freedom and non centrality parameter $\Delta$. This distribution arises, if $X_1 \ldots X_n \sim n(\mu, \sigma^2)$ independent and we want to consider the distribution of:
$$\frac{\bar {X} - \mu}{\frac{S} {\sqrt{n}}} = \frac{\frac{\bar {X} - \mu}{\frac{\sigma} {\sqrt{n}}} + \frac{\mu - \mu_0 }{\frac{\sigma} {\sqrt{n}}}} {\frac{S}{\sqrt{n}}} = \frac {Z + \frac{\mu - \mu_0 }{\frac{\sigma} {\sqrt{n}}}}{\sqrt{\frac{U}{v}}}$$
Where $\mu \neq \mu_0$ which is a non central t with non centrality parameters
$$ \Delta = \frac{\mu - \mu_0 }{\frac{\sigma} {\sqrt{n}}}$$
with $n-1$ df. Here $ v = n-1 df$ since $Z \sim n (0,1) $ and $U \sim {\chi}^2_{n-1} $ in this equation