I am not a 1st year engineering student, but will I do?
Distance of the Earth to the center of the galaxy: about 30,000 light years
Total distance traveled for one rotation = Pi x D = 188,400 light years
Speed of light = 299,792,458 meters per second
There are 86400 seconds in one day with approximately 365.25 days per year giving us approximately 31,557,600 seconds/year.
So the distance of one light year is around 9.46 x 1015meters.
This gives us a total distance travel for one rotation of 1.78 x 1021meters.
Rotational period of the galaxy: 2.25 x 108 years.
This gives us about 8 x 1012meters traveled per year.
8 x 1012 divided by 365.24 = approximately 2.2 x 1010 meters/day or 912,600 kilometers per hour.
This then is approximately 565,837 miles per hour or 157 miles per second.
Times 8.3 minutes = 78,186 miles (for some reason I had it in my head that the Sun traveled 270,000 miles in the time that it took Light to reach Earth from the Sun, bizarre).
So the question is whether the Earth rotates in an orbital plane around where the Sun was 8.3 minutes ago, which will always be 78,186 miles away from its current position (if your math is valid), or whether the Earth orbit is centered more closely to the actual position of the Sun at the present time.
And that answer will tell us the speed of Gravity.
Relative to me, the sun is motionless. It stays right where I want it, one AU from the earth. Nothing else matters. All your computations are for naught! Mrruuuhahahahahaha!