Get in touch with potential PhD supervisor without a topic? How specific should I be? I would like to work with a very specific person. That person has an interest in the overall topic I want to write my PhD in, but I have no idea what exactly that topic is going to be. But I am sure that I want to work with this person, and that this is the general area I want to tackle.

How do I get in touch? Should I better have some proposal ready? How specific would I need to be? Or do you think it's ok to just say it how it is?

I am worried to make a bad impression if I just contact him and say "I would like to do my PhD with you, but I have no idea what specific topic yet". I am in the political sciences and the person is in very high demand, which complicates things a bit further I guess. I am in the UK by the way.
From your question, it sounds like you have an area you are interested in, but are not sure what your exact hypothesis or research question should be. For a pre- or early program PhD student, this is normal. 

From my experience with PhD programs, students are often pushed to have their exact research question from the first day, but this is not usually not ideal. For most students (as it should be), your research question will change as you learn more about the topic. So, theoretically, how can you be so sure that a research question is the right one from the start without a serious review of the literature (which usually happens during your studies). 

That being said, one way to approach this is to ask the professor to coffee or a brief meeting and tell him or her, "I am very interested in your area of research. I do not know enough about it to have a specific research question in mind, but I am very interested in working with you so I can learn more about it and develop my own area of study."

Your question reminds me of one graduate student who was placed with me and was interested in criminology and health, but not sure how the two should intersect. He was very interested in the topic of HIV/AIDS. I'm a gerontologist and he was placed with me because a more suitable advisor was on sabbatical. He learned more about gerontology and ultimately changed his research question to be about aging prisoners, which is a really hot topic. He is doing well. You will figure this out, too. Good luck.