When is the appropriate time to email professors as a prospective PhD student? I think my research interests and experience are perfectly aligned with a professor. And I plan on applying to the department when the admissions open. 

But when is the right time to email the professor as a prospective PhD student? I don't want to shoot off an email and get ignored by being either too early or too late. 

In this university (in the US) the admissions are granted by committees and not individual professors. Also I speak as an international student.
The worst time to contact faculty is when they are deliberating the current round of applications and while they are recruiting the accepted students. In the USA, this would be December through April.

Conversely, best time would be May through October.