Roy's largest root is a common test statistic in multivariate analysis, statistical signal processing and allied fields. Despite its ubiquity, provision of accurate and tractable approximations to its distribution under the alternative has been a longstanding open problem. Assuming Gaussian observations and a rank-one alternative, or concentrated noncentrality, we derive simple yet accurate approximations for the most common low-dimensional settings. These include signal detection in noise, multiple response regression, multivariate analysis of variance and canonical correlation analysis. A small-noise perturbation approach, perhaps underused in statistics, leads to simple combinations of standard univariate distributions, such as central and noncentral χ2 and F. Our results allow approximate power and sample size calculations for Roy's test for rank-one effects, which is precisely where it is most powerful.