How much water on the earth is made from hydrogen comes with solar wind?

Question

A great fraction of solar wind is protons or Hydrogen ions. Some part of it is captured by Earth. I am wondering how much water in my half a liter cup is made of hydrogen from the Sun.

Answer 1

Here is a way of getting a rough upper bound: the density $\rho$ of the solar wind is about 4 atoms per cm$^3$ (that we can assume are hydrogen), and it travels at $v=400-500$ km/s. That means that were the Earth to catch every atom directed at it it would gain $\pi R_\oplus^2 \rho v$ atoms per second, $2.0$ to $2.5\cdot10^{14}$. Which gives us a mass flow of $3.4$ to $4.3\cdot 10^{-13}$ kg/s. Earth has been around for 4.5 billion years, so over that time at most about 49 to 61 tons of hydrogen have arrived.

The total amount of liquid water on Earth is $V=1.386\cdot 10^9$ km$^3$ (which doesn't count the sizeable amount of water bound in rocks). About 11.19% of this mass is hydrogen, so the weight of hydrogen is $0.1119 V\rho_{water}=1.55\cdot 10^{20}$ kg. So the ratio between hydrogen from Earth and from the sun is about $2$ or $3\cdot 10^{15}$.

Still, in a half liter (about 27.7 mol water) there are $3.33\cdot 10^{25}$ hydrogen atoms, so there should be about 11 to 16 billion solar hydrogen atoms in the cup if this estimate is right.

Maybe Earth's magnetic field captures more atoms, which might increase the value. Likely most solar wind is instead pushed away and the true value is much lower. But I am pretty confident there are some solar atoms in the cup.

Answer 2

@anders answer below is not better. The solar-wind water accumulated primarily in the building blocks that became Earth, then to an extent the pre- and primordial-atmosphere’d Earth. The pebbles/boulders/planetesimals had, collectively, far greater surface area.

Still, we can’t literally answer the question, because much of this water is now hydrides, locked in the iron of the core, plus more locked in the minerals below the transition zone. We have poor constraints on these mineral compositions, much less the dissolved ions they carry. Even above the transition zone, we have only constrained the water level by its geoelectric implications, and its implications on tectonics (impurities will soften rock).

Figures for mantle water vary from, I’ve seen, one ocean (that is, 1x Earth’s visible, surface seas) to eleven ocean. A year or two ago, a paper posited that the evidence so far gives ~1 ocean in the core.