This question was asked recently on a forum I lurk, but there is unquestionally no better place to ask this than The Dope.
As the title suggests, akin to the “grains of sand vs. stars” thought excercise, are there more stars in the universe than the total number of raindrops that have ever fallen in Earth’s existence?
Does the universe win simply by “infinity” alone?
If anyone is to differentiate between the observable universe, or the universe in all it’s entirety both are quite interesting to ponder, I have no specific stipulation.
For this I think you have to restrict the universe to the observable universe. If it is infinite then there is no question.
Remember that rain filled the oceans and that occurred over many millions of years. I can’t see that there are more stars than those raindrops (I’d be surprised if it was remotely close).
That said it is not uncommon to have our expectations on these things upended so it would be no surprise and cool to learn differently.
If there are 100 billion stars in our galaxy then you need that in raindrops to account for one galaxy. I’d be willing to bet we get 100 billion raindrops in a few good storms.
There are more trees on earth right now than there are stars in our galaxy. Even the small trees need a huge number of rain drops to grow, and they’ve been growing for a very long time.
Number of stars in the universe, (ESA): somewhere from 10^22 to 10^24 (1e+22 to 1e+24).
Average annual rainfall (wikipedia): 505,000 cubic km. Average rain drop size (here): 0.5-5 millimeters. A raindrop of 2mm diameter, would have a volume of 4 cubic mm.
Annual raidrops: 505,000 divided by 4 times 1e+18 (conversion from km to mm) = 1.26e+23.
So the yearly number of raindrops is about the same as the numbers of stars in the universe.
For C > A this comparison is pretty close, as the last detailed estimation of the number of galaxies revised the number up to 2 trillion. So approximately one galaxy for every 1.5 trees.
On the observable universe thing, it’s true it’s important to specify this. The OP suggests that the number of stars could be infinite; it indeed could be, but of course we can only talk about the observable part.
Earth’s surface area is 150,000,000 km^2, or 150,000,000,000,000 m^2 (150 trillion). Even at an average of one tree per square metre all over the land surface (some areas they don’t grow; some areas much denser) - that’s a lot of trees!
As an aside, you can’t get the number of stars in the (observable) Universe by multiplying the number of stars in our Galaxy by the number of galaxies in the (observable) Universe. Our Galaxy is significantly above-average in size. Most galaxies are dwarf ellipticals, much smaller than the spirals (in fact, there are about 50 of them orbiting ours), and the Milky Way is on the large end even among spirals (though the closest spiral to us, the Andromeda galaxy, is a bit larger yet).