In my story people are traveling from somewhere beyond the opposite side of the galaxy to Earth. The distance traveled is approximately 66 million light-years. I would like the travelers to experience 2,000 years of time.
How close to the central black hole must the travelers pass and at what velocity must they travel to achieve this 2,000 period of time in the traveler's time frame?
Assume the starting point is beyond the opposite edge of the Milky Way galaxy exactly opposite Earth. My 66 million light-year reference is for convenience only. I recognize that the actual parabolic path will change the actual distance traveled.
Ignore the need to travel around stars, etc., especially near the galactic core. For the purposes of this question, assume the black hole and distance are the only relevant factors (e.g., there are no other gravity wells).
Assume the time to accelerate and decelerate are effectively instantaneous. How my travelers get up to speed and back down again is not part of the question.
Bonus points are awarded to the answer that results in a practical set of equations for estimating distance from the black hole and velocity given an arbitrary time experienced by the travelers. In other words, if someone else wants the same kind of results but for the travelers experiencing 8,278 years, those equations would get them to reasonable values for distance from the black hole and velocity. JBH has promised that 250 reputation points will be awarded as a bounty to the best answer that also achieves this goal. (He's about to move to Montana, so if he hasn't posted the bounty by Monday, he might need to be reminded.)
It is assumed that the gravitational effects of the black hole will compound the time dilation. If this is not the case, please explain why.
Image showing start and end points, distance between central black hole and the trajectory of the ship, etc.