Without the sun, we would have nothing that pulls us in with a gravitational force that has the earth orbit. The sun's mass is what draws earth near it, making it orbit, and without it, we would fly off in space. Something that seems off is how you readily admit "without the sun, our (your) day does not exist" yet are atill trying to argue the sun does not need to be for there to be a day...Do you not realize how conflicting that sounds on your behalf?
Sun light takes about 9 minutes to reach earth, meaning if there were no sun, we would still see it for another 9 mins before it dissappeared. After that it would be completely dark. We wouldn't be able to see the moon, either, as the sun light is what reflects on the moon to make it visible (not to mention the moon also wasn't created till day 4).
And a great example of how you must obscure science, or simply do not grasp it. See, there is already a mistake in your premise, which is very telling of your idea of science. The earth would still need a lot more than a mere rotation for there to be a day. The earth does not rotate around the sun (or in this case, a "light"). The earth orbits the sun, and would orbit the "light". The earth rotates (or spins) on its axis, and the earth revolves (or orbits) around the sun. Just to clarify that for you.
Especially since the plants and vegitation were created on day three, and still no sun. The "light" would have to provide the same amount of energy and light as the sun to produce photosynthesis.
My point here is that the "light" would have to be remarkably similar to the sun in literally every way. And since it is virtually no different from the sun, it very well could be the sun itself! And what's more, is if we look at Genesis 1:14 (to separate the day from the night) it describes the same event as Genesis 1:4-5 (and he separated the light from the darkness. God called the light “day,” and the darkness he called “night.”). Seems to me to be describing the same event twice. Literally, as you take it, word for word, it says the same event occuring twice. If there is a creating of the sun by that language in verse 14 and on, then there is doing so in verses 4-5 too.