## Raindrops, stars, and photons

The bulk of this post was originally published on Prof. Andy Lawrence’s excellent blog, *The e-Astronomer, *which he wrote after reading some tweets I sent yesterday concerning the number of stars vs the number of raindrops that hit the UK in a year. I tweeted:

[1/4] Here’s a rain related #astronomy fact: Average rainfall in the UK last year (2012) was 1330mm or 1.33m Area of UK is 2.426×10^11 m^2

— Steve Owens (@darkskyman) July 28, 2013

[2/4] So approx. total amount of water that fell on the UK in 2012 is 3×10^11 m^3. The approx. average volume of a raindrop is 10^-9 m^3.

— Steve Owens (@darkskyman) July 28, 2013

[3/4] So the total number of raindrops that fell on the UK in 2012 is approx. 10^20 or 100,000,000,000,000,000,000 raindrops.

— Steve Owens (@darkskyman) July 28, 2013

[4/4] There are approx. 10^23 stars in the universe. So for every raindrop that hit the UK last year there are 1000 stars in the universe!

— Steve Owens (@darkskyman) July 28, 2013

Andy followed up:

Very nice, but I found myself thinking – how does the rate of raindrops compare to the rate of photons? The average raindrop flux is 13 raindrops/m^2/s. Lets compare to the Sun. The solar constant is 1360 W/m^2. If we take the typical photon as being at about 500nm with energy hc/λ = 4×10^-19J, we get roughly 3×10^21 photons/m^2/s.

So we get much much much more sunlight than rain! Woohee!

What about starlight? Well, as Mr Olbers pointed out, a Universe full of stars would make a sky as bright as the Sun in every direction. However, the Milky Way fades out, and the universe runs out too, in time and therefore space. So lets just get empirical. Cosmology types will often plot the ‘cosmic optical background’ at a level of about 10^-8 W/m^2/sr, about a factor of a thousand less than the CMB. However, that is the extragalactic background light; the summed emission from nearby stars is in fact much more. According to my Trusty Allen, star light from the whole sky is equivalent to 460 V=0 stars, or one star of V=-6.7. The apparent magnitude of the Sun is V=-26.7. So the scattered starlight is 20 magnitudes fainter or a factor of 10^8.

So in super-crude terms, starlight is giving us something like 3×10^13 photons/m^2/s. Still lots more than the 13 raindrops/m^2/s.

But what energy? Scaling down from the solar constant, starlight is giving us an energy flux of about 1.4×10^-5 W/m^2. What about raindrops? Each of those raindrops has mass 10^-6 kg. The terminal velocity of a raindrop depends on size, but at 1mm its about 4 m/s. So the KE per raindrop is about 8×10^-6 J and the energy flux is therefore 10^-5 W/m^2, about 6 times as much as starlight.

So… in particle count terms, the Sun wins hands down; starlight is down a factor of hundred million but still huge; and the raindrop count is pitiful, another factor of a trillion down.

In energy terms, the Sun still wins easily, with starlight a hundred millions times down; but the rain carries more energy than the starlight – just.