Is space cold? Part 2

part 1

In part one it was suggested that an object in the sun at distances of earth orbit, or less, will be hot.  What about the case for the shade in space, is it cold? 

Some say that when the Apollo astronauts moved from light to shadow on the surface of the moon, they went from a hot 107C or so to a cold -170C in a matter of seconds.

Neil Armstrong on the surface of the moon during Apollo 11

The Apollo astronauts had space suits with life support systems to keep them alive during extra vehicular activities (EVAs).  The cooling device called the porous plate sublimator was housed inside the backpack.

Kozloski's book U.S. Space Gear says that water froze in the pores of the porous plate sublimator because it was housed inside the space suit's backpack, was therefore in its shadow, was protected from the sunlight, and so was cold:

The astronauts' portable life support system, developed by Hamilton Standard, supplied oxygen and circulated it through the helmet and suit. It controlled relative humidity and held temperature to a comfortable 70 degrees F. A PLSS pump cooled water by sublimation* and recirculated it through the tubing of the liquid cooling garment.

* Hamilton Standard introduced a porous plate sublimator on the PLSS: Heated water would pass through the sublimator, freeze at pores of the nickel plate that was partially exposed to ambient space temperature, vaporize as heat was introduced through exchange fins, sublimate the ice film, and thereby free the vapor to be discharged.

Is it true, does space possess such a cold ambient temperature?  Well, it depends on the temperature of the other objects around you.

In deep space, with the warm sun far away, it would be cold.  But how quickly would you cool down?  Would it be a matter of seconds as is often portrayed?  And how fast would you cool down in a shadow on the moon?

To see how fast you'd cool down I used the Stefan-Boltzmann Law (idea I got from here reproduced here, formula lifted from here) to calculate the rate of temperature drop:

P = AεσT⁴........(1)

Where P (watts) is the radiated power from a body of area A (m²) at temperature T (K).

ε is emissivity, a dimensionless number between 0 and 1 that determines the efficiency of a body to radiate and absorb energy. A perfect radiator and absorber has an emissivity of 1. Soil, ice, rock, asphalt and human skin have emissivities slightly less than 1.

σ is the Stefan-Boltzmann constant, 5.67x10^-8 Wm²T⁴

T is the body temperature in Kelvin.

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P = Stefan Boltzmann constant*Area*emissivity*(T(astronaut)^4 - T(space)^4) ....(1)

Divided into the mass of the astronaut (assumed to be 150kg, heat capacity that of water, 4187 J/kg/K)

dT/dt = P/(mass*heat capacity)....(2)

Into (1),  P = 0.0000000567*3*0.9*(310^4 - 3^4) = 1414 watts radiated by astronaut.

1414 W output - 108 input from body heat = 1306 W.

Into (2), dT/dt = 1306/(150*4187) = 0.002 K per second drop in temperature.

From a starting temperature of human body temperature (37C), the following graph can be computed in Excel:

Rough estimate of an astronaut cooling down in deep space (parameters here).

Judge for yourself from the above graph.  It looks to me like for at least the first few minutes you wouldn't feel much of a difference in temperature.  You may feel a bit chilly by about the half hour mark. 

By the time 2 hours passed, the astronaut could be in pain due to the cold and would likely feel the need to exercise to keep warm.  If you were asleep you might die of hypothermia in several hours.  

If you were fit and awake you could perhaps survive the cold for a few hours by exercising.  Either way, awake or not, you'd be dead in a day or so.

In deep space heat can only be lost by radiation, so the rate of heat loss is less than it would be in cold areas filled with a medium such as air.  

For example, though the temperature is greater in Antarctica (-90C), it is less than deep space (-270C), and the heat capacity of the air is so much more it would extract heat from a human much quicker than deep space could. As a result Antarctica would feel way colder.  Instead of being dead in hours, you'd be dead in minutes, even with the space suit on.

At the slow rate of cooling shown in the above graph -- a matter of minutes or hours not seconds -- water couldn't freeze in the porous plate sublimator in a timely enough manner for the sublimator to be effective.  

But the real problem that would prevent water from freezing in the sublimator is the fact that the porous plate is not in deep space, but is surrounded by other warm objects.  This would keep the plate warm.

The sublimator, and everything else inside the backpack, is subject to mutual radiation, which equalises temperatures, keeping the various components inside the backpack warm.  The physical connections to the backpack would also keep the sublimator too warm for water to freeze.

This can be seen by considering two flat objects side by side in space, screening the sun's radiation.  These objects are opaque and block the transmission of light, and so are warmed.  They then emit radiation in all directions.

Two opaque black plates warmed in the sun

Now take one plate and put it behind the other.  The shaded plate represents the porous plate of the sublimator; the sunlit plate represents the case of the backpack.

For the first few seconds there's no noticeable change in temperature.  After a few minutes there's a moderate decrease in temperature of the shaded plate.  Because of the warmth provided by the sunlit plate the rate of temperature decrease is lower than for the case of the astronaut in deep space (graph above).

Shaded plate (representing contents of backpack) gets a little cooler in the shade

The flat surface of the sunlit plate emits EMR in equal measure on both sides.  If the shaded plate was close to the sunlit one, but not touching it, it could be as high as 84% of the temperature of the sunlit plate.  (Rough estimate I made assuming that the sunlit plate cuts the rays from the sun in half going through to the shaded plate.)

If you made a stack of plates not touching each other they would get progressively colder as you moved from the sunlit plate to the furthest plate from the sun.

Stack of plates that are progressively cooler away from the sun

The porous plate sublimator is joined by physical connections to the backpack, and by the liquid water in the cooling system.  In the two plate diagram above this can be represented by a simple physical connection:

Physical connections bring the two plate's temperatures closer together

This connection would have the effect of equalising the temperature of the two plates.

Ignoring this physical connection for the time being and starting from the two plate diagram and extending the front plate on all sides around the shaded plate to form a cube completely enclosing it.  

Cross section through six-sided cube enclosing the shaded plate, like the backpack that encloses the porous plate sublimator

When there were just two flat plates, the shaded plate had a warm surface on one side (the backside of the sunlit plate) and deep space at 3K on the other side.

Now with the shaded plate surrounded on all sides by the plate warmed in the sun, the inside plate radiates as much to the outside box as it receives from it. The shaded plate is therefore now fully part of the outside box, thermodynamically speaking, and will be at the same temperature.  Instead of 84%, it will now be at 100% of the sunlit plate's temperature.

Now we complete the picture by reintroducing the physical links, as is the case for the sublimator joined to the backpack, and you can see that the porous plate never had a chance to cool down; it would always be at, or very near, the temperature of the backpack/astronaut system.  

The porous plate sublimator was enclosed in the backpack:

Perhaps if the porous plate was strategically placed near an opening in the backpack, and was always in the shade it could work.  Remarkably NASA never made such a consideration, nor did they include such a cutaway in the shell of the backpack.  

Sublimator is enclosed by the backpack and has no outlet to the cool 3K background radiation of space

Even if there were to be such a cutaway hole in the backpack, and the plate was kept out of the sunlight in the shade, the backpack would still provide so much heat, it would never allow pores in the plate to get cold enough to freeze.

This heat from the backpack to the porous plate is in the form of the physical connections of the plate to the backpack, from the radiated heat of the backpack, and especially the coolant water that was in contact with the nickel porous plate.

The opening in the backpack that never was from NASA.  It could have allowed the porous plate to cool more in the shade and also stop water from recondensing on the inside of the backpack.  Such recondensation would have the unwelcome effect of retaining heat in the astronaut's system instead of being rejected to space.

Deep space, and shadows on the moon, are cold, but it takes time to cool down.  And when you're surrounded by warm objects it can take even longer.  

So no: space, or a shadow in space, is not necessarily cold.  In the case of the porous plate sublimator water can't freeze due to space's alleged cold "ambient temperature".