I have asked several big guys to write letters for me - most of them I did't know personally. The general rule is that the bigger the guys, the more likely they are willing to help - that's exactly why they become big at first place.
Here's how I write my letter:
I introduce myself first with one sentence ("My name is XXX and I'm a research associate at XXX Institute")
After that, it always helps to tell them how you know them or their work (even though they don't know you). In my case, I usually say "I have always enjoyed reading your articles on XXX", or "I have followed your publication closely for the last 5 years", etc. These are actually compliments that instantly shorten the distance between you and the big guys. No matter how big they are, they are always glad to know that someone appreciates their work. Compliments always work. They never failed. They work one hundred percent and they always work like magic.
Next, tell them that you are asking for a favor, namely, to write a recommendation to the director of USCIS service center on your behalf. You don't have to explain it in detail because most of these guys have written such recommendations before.
Next, you may explain WHY you ask THEM to write the letter. Here's something very very important: never try to say how desperately you need a letter, or how important such letters are to you or to your petition. Why? Because your immigration petition is, under any circumstances, unimportant to THEM. You will not motivate someone to do something by telling them whatever they do is important to you. Instead, I usually tell them that I ask them to write a letter because "you are a top national expert in our field and therefore best person to write such a letter for me". This is another compliment that absolutely works. What motivate them to write a letter for you? Writing such a letter for you gives them a sense of importance because you told them that they are "top national experts" in the field and "most qualified person" to write such a letter. People may not do certain things for money, but they are almost always willing to do something that gives them a sense of importance.
Offer them a draft letter. The hidden reason is that if you write the draft letter, you can tailor the letter to your application. The apparent reason, though, is that you realize that the big guys are busy and appreciate their time.
One more trick, treat it as if they have already agreed to help you. That is, don't forget to THANK them at the end of request. Why? Because in reality, very few people will refuse to help you with something you have already thanked them for.
If you do the above, I can guarantee that you'll recieve good letters from big shots in your field. I'm not saying that this approach will work with an assistant professor (some of them are loners or strange people with little personal skills). But if you ask a chair or head of department, it'll always work. A person who doens't know how to respond such a request with kindness usually doesn't have the leadership skill to become a chair of department.
