How do I make her chat me on Facebook?

ok so theres this girl right? freakishly gorgeous, good sense of humour, amazing smile. the works. BUT...she's on the other side of the social spectrum. she's friends with the stupid hockey f*ggots at my school, im more of the "average joe". I don't talk to her very much, but we've been on the same softball team 2 years in a row so we have that in common i guess. (no softball is not a chick sport, more guys play than girls....) anyways i try to talk to her on fb every once in a while, and by that i mean once or twice a year (usually only around the time of the softball season), and it usually goes pretty well. she seems interested in the chat, you know, answers with full sentences, answers that generate more conversation rather than single word answers that make it hard for me to keep the conversation going. but just when i think i have a legitimate conversation going she ALWAYS says "well i g2g. ttyl:)" every damn time. and i usually don't talk to her later. and if we do talk via fb chat i always start the conversation, never the other way around. so my thoughts about the situation are sort of scattered. im usually pretty good at concocting scenarios in which "force" the girl i want to talk to me, well, talk to me. for example. this one girl i know, that one "girl-friend" that every guy has that has a super hot friend that you desperately want to get introduced to, kept asking me for a piece of gum, but every time i was like naaaw. but this one time her super hot friend came to our math class so i sat alone whilst my ugly friend hung out with the super hot girl. the ugly friend asked for a piece at the beginning of class before the hot one got there btw. anyways, at the end of class i decided to give her a piece, knowing that the hot friend would want one as well, seeing as gum is just that universal desire that everyone wants when they see someone with it. it obviously worked and me and the hot friend started chatting after that. but thats just a simpler one of like 10. but i digress. back to the issue. this girl i like i just simply CANNOT read and its really effing frustrating. is there anyway i could even go as far as subconsciously make her want to chat me up, via fb or otherwise? i always am the one to start a conversation but i bloody hate it. i always think i'm annoying the person, no matter how good of friends we are. i really don't want to start the next conversation with her if not completely necessary, simply for fear that i'm annoying her, and for fear that she will talk about me being "the most annoying kid ever" to her sister and friends. can't be having that happen. what am i supposed to do?