# Melting the Antarctic ice sheet

:: physics

How long might this take, in the worst case?

The Antarctic ice sheet has a volume of about $26.5\times 10^6\text{km}^3$, according to Bedmap21. This is $2.7\times 10^{16}\text{m}^3$ of ice. The density of ice is about $10^3 \text{kg}/\text{m}^3$ (about a tonne per cubic metre, which is approximately the same as water of course), so this is about $2.7\times 10^{19}\text{kg}$ of ice. The enthalpy of fusion of water is about $3.3\times10^5\text{J}/\text{kg}$ so, if we assume that the ice is all at freezing point2, then we require $8.9\times 10^{24}\text{J}$ to melt it all.

Let’s assume we use the Sun to do this. The solar constant is about $1.4\times 10^3 \text{W}/\text{m}^2$: this is the amount of power per square meter that the Sun provides at the top of the atmosphere. So, imagine we use all of the power that the Earth intercepts from the Sun to do this3. Well, the Earth’s radius is about $6.4\times 10^6\text{m}$ so the total power available is about $1.4\times 10^3 \times \pi \times (6.4\times 10^6)^2\text{W}\\ \approx 1.8\times 10^{17}\text{W}$, or about $1.8\times 10^{17}\text{J}/\text{s}$.

So to melt the Antarctic ice cap, using all of the power from the Sun that reaches the top of the atmosphere would take

Well, there are about $32\times 10^6$ seconds in a year, so this is about 1.5 years.

Of course we can’t use all the Sun’s power: even if we had the technology to do this (which we are not anywhere near doing!), doing this would cause an inconceivable catastophe for the rest of the planet: this would be a winter night which lasted for a year and a half. Everyone would die.

A plausible figure might be a tenth of one percent4: in this case the Antarctic ice sheet would melt in about $1500$ years.

1. The ice is, of course, far below freezing so the actual energy required will be much greater.

2. This is enormously more than the amount of power that we could plausibly use: see later.

3. This is just a number I have pulled out of thin air: one percent seems too high, so perhaps a tenth of one percent is plausible.