As I explained earlier.Ã‚Â Parallax won't work because the distance of the background objects aren't known.Ã‚Â No reference points.
These distances are previously assumed upon a constant light speed such as the 1987a supernova... which the formula assumes they know the distance of the radius, therefore they think they can figure the formula using the radius as a reference point.
Again, the reference points are basically assumed, and never verified.
Scott don't be discouraged. I need someone to help me enlarge this diagram. I've emailed ikester to see how. I want someone to show me where this thing is wrong. The concept is hard to explain, but better seen.
I hope to show by the diagram that two objects can be placed (top view graph) on a straight line from earth in a nearly infinite number of positions (distances),and if in a proportional distance from one another--the perpendicular view from earth will produce the same dimensional arc measurement as seen in the night sky (first person view).
Remember, we are seeing space in a spherical plane, so our view of the star and background object is a perpendicular view no matter how much farther the BO is than the star. We are seeing in 2D in a single view.
So in the diagram, I can place 1987A theoretically at 40 light years or 2000 ly on straight line (in the May view, 1987A and the background object would be in line). I make the first line--"line may view" where earth, 1987A and the BO are all in line. I then make a right angle by drawing from "point earth--may view" to "point earth--november view. These two lines, the diameter of earth's orbit and the line of May view to 1987A (and BO) make a right angle. I then draw two lines from point November to intersect 1987A and BO--"line november, 1897A" and "line november,background object. I then make a right angle on "line November,1987A" from "point 1987A" to the "point of intersection" on line "November, background object." This is the corresponding arc distance that will appear in the night sky in a "first person view."
Now I can move 1987A closer to earth and make the same thing happen in the "first person view." By drawing a perpendicular line from point 1987A in relation to line november, 1987A I can find the intersection point on line november, background object. This is the actual corresponding distance that I got when the star was much farther. Bottom line is that if 1987A and the background are a proportional distance from each other, then from a perpendicular view (how we see them in the sky) they will look the same.