I was hoping to launch the droplet then tell it to open a file but
when
the droplet is launched without a PDF being dropped on it, it
immediately
quits.
You might try something like this, Coj:
------------
set pdf_file to alias "path:to:your:new file.pdf"
launch application "name of your droplet"
tell application "name of your droplet" to open {pdf_file}