[Ethan Clive Osgoode]

[mrjesse] If the earth were turning at the rate of 180 degrees per 8.5 minutes, how far lagged would the sun's optical image be from its real position?

[LeGrande] 180 degrees off.

So, then, it would be daytime on the side facing away from the sun, and nighttime on the side facing toward the sun, and the Earth, in this case, would cast a shadow toward the sun, according to you.

We need not look far for illustrations of this. Neptune is 30 AU from the sun. Neptune's period of rotation is 16 hours. For an observer on Neptune, the LeGrandean optical lag of the sun would be 30*8.3 = 249 minutes (4.15 hours) or 360*4.15/16 = 93.3 degrees. Thus, Neptune casts a shadow which is nearly perpendicular to the line joining Neptune and the Sun.

[LeGrande]

We need not look far for illustrations of this. Neptune is 30 AU from the sun. Neptune's period of rotation is 16 hours. For an observer on Neptune, the LeGrandean optical lag of the sun would be 30*8.3 = 249 minutes (4.15 hours) or 360*4.15/16 = 93.3 degrees. Thus, Neptune casts a shadow which is nearly perpendicular to the line joining Neptune and the Sun.

Why don't you do yourself a favor and look up aberration of light in Wikipedia. I don't think that I can help you.