How quickly will the oceans freeze if Earth becomes a rogue planet?

Question

Assume Earth has been wrenched out of the Sun's orbit and has become a rogue planet (for the purposes of this question, assume that it happens near instantaneously, i.e. say the Sun just vanishes). The oceans will all be covered in ice in a matter of weeks, with the planet turning into Snowball Earth, and the atmosphere liquifying and falling as rain within a year or two before freezing solid.

According to this estimate, the equilibrium temperature will converge to around -240C and the oceans will freeze up entirely within 500,000 years.

Sunlight's absence forces Earth's surface temperature to become ~34 K – even during the postulated very harsh first-onset of a "Snowball Earth" period this planet's mean global temperature got to ~225 K – meaning the atmosphere will condense, forming an inconsistently thick draped stratum of solidified gases ~10 m-thick on the land and sea surface. Earth's atmosphere consists of 78% N2, 21% O2 and 1% other gases; the first substance to freeze would be H2O, followed by CO2, then N2 and finally O2. The pressure on the sub-aerial crust (1 atmosphere) will remain the same but the 10,329 kg/m2 will be in the form of solids, not gases. (Subsequently, Earth's world-ocean will commence to freeze, completing that phase transition in ~0.5 × 10^6 years. Without liquid water the tectonic plate motions would cease just as soon as all previously entrained water was lost, and without wet subducted tectonic plates, no active volcanism!)

Though there are also some projections which state that regions around hydrothermal vents will retain liquid water indefinitely, allowing extremophile life to subsist.

How long would it take for all or most of the oceans by volume to become ice? Is 500,000 years (as above) a plausible accurate estimate?

At what rate will the freeze progress - linearly, exponentially declining, or something else? What would be the depth of oceanic freeze after, say, 1 year, 10 years, 100 years, 1,000 years, and 10,000 years, respectively?

Thanks in advance!

Answer 1

The USGS estimates 10,633,450 km³ of water on earth (i'll call that 1e19 kg. i am going to go ahead and assign that a temperature of 10°C. With, roughly, energy of freezing 333kJ/kg and heat-capacity of 4kJ/(kg*K), that works out to 373kJ/kg * 1e19kg ... about 4e21kJ that need to leave earth for all the water to freeze. NASA has a nice graphic showing the current radiated energy as a colormap, which i am going to sum up (near-baselessly) as 200W/m². Water covers about 4e14m² (361132000km², Wikipedia) of earths surface, so that comes out as 8e16W. That means earth would only need to radiate for 10e5 seconds (about 25 hours) before it's a complete icicle.

so that's probably a bit fast.

Prolonging factors:

• radiated power goes with T^4, i.e. if the temperature (in Kelvin) is halved, only 1/16th of power will be radiated. I estimated the water as 10°C (283K), and assumed that all the water would simply cool at the same rate. This is obviously false. The outermost layer would cool most rapidly (reducing radiated power), with every meter of ice below acting as insulation.
• Earth itself is a heat source - conductive heat flux is about 1e-1W/m² (0.1 MW/km2, Wikipedia) - which pairs roughly with your 34K of a cold earth (going from currently 200W/m² to 1e-1W/m², which earth could sustain indefinitely, is a factor of 2e3 -> T^4 means going from currently 280°K to about 40°K ) - this heating from below the insulating layers will drastically slow cooling (if you could magic an insulation around the earth with less than 1e-1W/m² throughput, earth would actually warm up... (a vacuumplate of 10 meters thickness would make the surface a balmy 17°C!))
• Water ice acts as an insulator, with lambda=2.33W/(m·K), meaning an iceplate of , e.g., 5000 meter thickness would only let about 5e-4W/K pass through every m², so for a gradient of 40K outside and 273K inside, we would get about 1e-1W/m² ... remember earths thermal output? yay! below 5000 meters of ice, there might be actual near-freezing water, indefinitely!

NASA earth info

Answer 2

Oh boi, what a question. Since I am an oceanographer, this is more about what the process of freezing would potentially look like

To start, the oceans will not completely freeze. Rather the pure water freezes and the salt and other nutrients in the oceans will crystalize at around -21°C.
It is also worth while to note that bodies of water freeze from the top down once the entire body passes 4°C.

The oceans would freeze from the poles down (towards the equator) as the temperature decreases since the poles are already at or below 4°C. Due to Coriolis, the right side of the oceans (specifically Pacific and Atlantic) would freeze first

Factors such as starting temperature/salinity/density (different depending on geological location aka distance from the equator), depth (ie pressure), and current speed.

To simplify this, we will assume that the currents have slowed or almost stopped*, the depth/pressure is negligible, and will use temperature at 4°C.

*Thermohaline circulation would gradually slow until it came to a stop

Even with the assumptions, it gets gnarly very quick. It is not possible to calculate the time it would take to freeze.

I personally believe that the rate of freezing would decrease over time as the density of the water would increase thus the temperature needed for freezing would decrease. Aka the rate of freezing would be a logarithmic curve.

Edit: Remove oversimplified equation