How to send an email to a well-known professor for a possible PhD student position? I am applying for graduate physics and I was wondering to work with a well-known professor in Caltech. He is a pioneer in the field that I am interested in. I have a kinda outstanding background, however, I am not sure if the professor is looking for exotic genius students who have completed their master at 18 or something! In this sense, I am very ordinary. How should I send an email to the professor and finally understand if I have any chance to do my PhD under his supervision or not?

P.S. I just want to know how one should contact a professor to make themselves sure that whether the professor is willing to accept them as a graduate student. I do not ask please decide on behalf of me.
Your email may run something like this (names and subject areas can be changed, of course):


  Dear Professor Carroll,
  
  Are you accepting PhD students at the     moment? I am planning to apply to CalTech this year for a PhD in physics and I am really interested in cosmology and in particular, your work on the arrow of time. I am currently a 3rd year undergraduate at University of Somewhere and have taken courses in differential geometry and astrophysics. 
  
  Best wishes,
  
  mathvc_