Good question. Lots of people don't care about this rule in real life, but the issue is that we need to say "as a result of [noun]." "The sun shining" is not a noun... "the sun" is a noun, but not "the sun shining." HOWEVER, if we tack on an 's, then we're okay: "as a result of the sun's shining." This has the effect of making "sun's" a possessive adjective modifying the gerund "shining" (and gerunds are nouns). Think of the phrase "Quit your whining"... we need to say "your" (possessive) because "whining" is being used as a noun; we couldn't say "Quit you whining." So you could grammatically say "as a result of the sun's shining" and be fine (although stylistically, there's a good argument to be made for going with "because the sun shines").