Let’s imagine that the Artemis astronauts had a train-style toilet that dumped into space. Actually, we don’t have to imagine that; Apollo and Artemis both vented urine to vacuum. Videos show a coruscating rush of glittery ice crystals flying away from the craft.
ISTM that the mean temperature of such ice in sunlight in high Earth orbit will be significantly above whatever is needed to drive sublimation. So what would the lifespan of these icy shards be? Minutes? Hours? Days?
Suppose I dropped a cubic meter of water ice in high Earth orbit: would it last a year?
Scale is clearly a factor here; a cubic kilometer of ice would perhaps last quite some time, outgassing furiously. Does anyone have an inkling as to how to relate volume to longevity?
Well, at the extreme, comets are “a cubic kilometer or so of ice”, and they can generally survive several passes through the inner solar system (for a few months each time, but part of that is generally closer to the Sun than the Earth is).
True, and/but they have varying amounts of dust that probably affects the sublimation rates.
Focusing on the smaller end of the scale, how would I estimate the lifespan of a cubic meter of ice? There’s the solar flux on the surface (roughly a kilowatt per m^2), and Google informs me that the sublimation rate is exponentially dependent on the ice temperature, ranging from essentially zero at 173K to 9mm/hr at 273K.
I’m not sure how to meaningfully estimate what the temperature of ice in Earth orbit is, though. It does seem as though a standard refrigerator ice cube wouldn’t last a day, though.
Here in the Great White North, as spring is sprung the snow piles from the winter start to melt. Many of these accumulations are a result of plowing the streets over and over, and so include the sand spread during the winter to avoid skidding. At a certain point, the pile melts sufficiently that there is a moderate cover of sand on the pile. You would think the sand, being non-white, would heat the snowpile and melt quicker, but in fact it seems to insulate and this causes snow piles to persist far longer than expected.
If we just take solar heating into account then we can estimate the temperature via radiation balance. Assuming the object is roughly spherical and is rotating, then we can use the Stefan-Boltzmann law to equate the solar radiation it absorbs from the Sun with the solar radiation it radiates. At a distance d from the Sun, the solar flux is
I = \frac{R_\odot^2}{d^2} \sigma T_\odot^4
where R_\odot is the radius of the sun and T_\odot is its temperature. (This is the solar flux you found, though note that 1 kW/m2 is the value at Earth’s surface; at the top of the atmosphere it’s more like 1360 W/m2.)
When this radiation hits the ice, it will be absorbed on one side and then re-radiated over the entire surface. In equilibrium these rates are equal, so
(\pi r^2) (1-a) I = 4 \pi r^2 \sigma T_\text{ice}^4
where r is the radius of the ice sphere, a is its albedo, and T_\text{ice} is its equilibrium temperature. Whacking all of this in together and solving for T_\text{ice} yields
“Clean” ice has an albedo of about 0.6–0.7, so if you plug in the numbers (with d as the distance from the Earth to the Sun) you get somewhere between 205–220 K. If the ice was darker in color it would heat more and sublimate faster.