In what cases do we have to put the definite article the before each of these words:
and in what cases do we not need to?
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For Sun, you always need the definite article when referring to the star itself. The only time you don't need it is when you're referring to the Sun's light/heat output...
"I like sun" is just about valid, but sun there just functions as shorthand for sunshine. Certainly that's what it means in the more common form "I like the sun" (note lack of capitalisation).
For Moon I can't come up with any context where you don't need the article...
"We put a man on the Moon", but you couldn't do anything on Moon.
Earth can take it or leave it (unlike world, which always needs the article)...
"The Earth is flat".
"Climate change threatens the Earth"
"The astronauts returned to Earth".
"It's like nothing on earth".
"Where on earth have you been?" (contrast with "What in the world was that?")
When the Earth is spoken of as a physical body, occupying space-time, it's normally preceded by the article, and often capitalised. As are Sun and Moon, but there's no universality about either convention, nor is capitalisation necessarily governed by whether the article is present or not. I've capitalised every usage after the article (as do most people), but there are exceptions.
When Earth is used more "metaphorically" to mean our whole environment (really, just the thin skin of biosphere on the surface of the planetary body, where nearly all things that concern us take place) it's more normal to omit the article, and I wouldn't normally capitalise either.
"Sun" and "moon" can be countable nouns. "Earth" is the name of this planet (a proper noun). For instance, you wouldn't say "The Jack and the Jill rolled down the hill".
Moon is demonstrated as a countable noun with this sentence: Jupiter has many moons.
It used to be common to refer to any star as a sun, and in this context it makes sense to say "the" before it to make sure we are talking about our sun.
However, "star" is now used much more commonly to denote stars other than the Sun, so it would make sense to drop the "the" before mention of the Sun. Maybe it will happen, maybe it won't.
"Earth" as a proper noun is no different than talking about Jupiter or Mars. We never say "The Jupiter" or "The Mars".
But earth is also used as an uncountable mass noun, and like sun, could also have been used to refer to other planets. It is in a nice situation where it can be referred to as "the Earth" OR "Earth". A lucky little planet it is.
OK, one more attempt to synthesize an answer (with particular thanks to Ham and pavium). The word 'sun' may refer to sunshine (a patch of sun) and 'earth' to dried mud; that's no more relevant than 'moon' meaning point your backside at. "Sun" and "moon" have astronomical meanings, in which they take neither capital letter nor article: Mars has two small moons. ('Earth' in this sense has been superseded by planet; 'sun' here is a synonym of 'star') All three words may also refer specifically to the body in our solar system, in which case, as names, they require "the". "Earth" is a special case; as well as the name of this planet (with article), it may refer to the place where humans live (without): coterminous but not synonymous. If an asteroid passed close, as in the H.G. Wells story, the Earth would not be affected from an astronomer's viewpoint, but Earth as we know it would be devastated.
I think you can see things roughly as follows:
(As other posters have mentioned, there is also a secondary use of "sun" to mean "heat/light from the sun", in which case it behaves more like a mass noun such as "water".)
According to the International Astronomical Union and famous physicist Johnny Wheeler, you should omit the definite article and capitalize Sun, Moon, and Earth.
From Wheeler's book Spacetime Physics on page 32:
Note: Neither astronomers nor newspapers say "the Venus" or "the Mars." All say simply "Venus" or "Mars." Astronomers follow the same snappy practice for Earth, Moon, and Sun. More and more of the rest of the world now follows -- as we do in this book -- the recommendations of the International Astronomical Union.
The ocean's rise and fall in a never-ending rhythmic cycle bears witness to the tide-driving power of Moon and Sun. In principle those influences are no different from those that cause relative motion of free particles in the vicinity of Earth.
For Sun, Moon, and Earth, a determiner is always needed, if the word is being used to denote single countable nouns.
That is: Sun, Moon, and Earth.
What about the expression "What on earth?"
"earth" in this case is being used in a different way:
the inhabitants of this planet, especially the human inhabitants:
Thus, "earth" in this expression is being used as a single uncountable noun, which means that it doesn't need the determiner "the".
I think you should use 'the' before any of those words when capitalised like that. There is only one of each.
I've been trying to think of an example where you wouldn't, but I can't think of one.
This is not the same as using, for example, 'moon' in a generic sense "Jupiter's moons" or "the sky was full of suns"
You can use a or an when you are...
Referring to representations of the real thing.
I added a sun and a moon to our logo.
Talking about a hypothetical situation.
I can't imagine a sun with no sunspots.
An Earth with no humans? It may happen.
There is at least a Sun in every galaxy.
Using these words as adjectives.
We have yet to find an Earth-like planet.
Moon is Earth's satellite. Titan is a satellite, not a moon. No one says the Titan. But everyone says Saturn has many moons.
The name of Earth's satellite is simply Moon. We should call it Moon as if it were someone's name. If you want to use the, use it as the satellite: Saturn has many satellites, not moons. Moon is our satellite's proper name. Same goes for our star, Sun, and our planet, Earth. All stars, planets, and satellites have their own name, so use them.
You may think you sound silly, but the correct way to say it is:
It takes 365 days for Earth to rotate around Sun, while it takes Moon only 24 hours to rotate along with Earth.
Listen to this using the:
It takes 365 days for the planet to rotate around the star, while it takes the satellite only 24 hours to rotate along with the planet.
Mix it all together:
It takes 365 days for the planet Earth to rotate around the star Sun, while it takes the satellite Moon only 24 hours to rotate along with the planet Earth.