Re: "... and the months and years tick by"
"I'm not a rocket scientist but I'm fairly sure without any orbital speed it'll just fall into the sun,"
Perhaps, but I'm thinking that the vast difference in mass would produce a minimal amount of force given that the Earth's mass ratio of 1 : 330,000 with the mass of the Sun versus a space station (going by the mass of the ISS) would have a mass ratio of 1 : 4.75 x 10^24. Given that the distances are equal, I figure that the gravitation force exerted on the station would be 1/14.4 x 10^18 of that of the force exerted upon the earth.
I am going off of knowledge gained from my physics 101 class, I am by no means an expert on astrophysics, so my math may be off...