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Solution to the minmus paradox


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1 hour ago, runner78 said:

Minmus is actually a hollow artificial satellite that uses ice as a camouflage and at the same time as a photovoltaic system by absorbing all heat energy that hits the surface and keeping the ice frozen.

But then you'll have to explain gravity, which is even harder. And before anyone suggests putting a chunk of neutron star or a tiny black hole in the center to generate the gravity, that would cause too much stress on the shell, causing it to collapse. So unless Ringworld Engineers have built Minmus out of scrith just to mess with Kerbals, I don't think that's going to work.

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2 hours ago, K^2 said:

But then you'll have to explain gravity, which is even harder. And before anyone suggests putting a chunk of neutron star or a tiny black hole in the center to generate the gravity, that would cause too much stress on the shell, causing it to collapse. So unless Ringworld Engineers have built Minmus out of scrith just to mess with Kerbals, I don't think that's going to work.

By means of a tractor beam that attracts everything that comes close to create the illusion of gravity.

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On 11/2/2021 at 4:23 AM, K^2 said:

At relevant time scales, more than a few meters of rock might as well be perfect insulation, so the surface can be considered as a infinite flat plane in equilibrium with stellar flux and background of the cosmos for purposes of thermal analysis. [. . .] day high is going to be about 70% above average equilibrium, which is way, way beyond the freezing temperatures.

I think that we imagine Kerbin-Minmus to see the same solar flux as does Earth outside the atmosphere, 1360 W/m², of which ice absorbs 700 W/m², 
so I figure the equilibrium is 700 W/m² = σ T4, so equilibrium-T = 330 K (57°C) at noon,
and 700 W/m² /π = σ T4, so average-T = 250 K (−23°C) averaging over the day.

But how much can the surface warm during the day?  700 W/m² coming in at noon, and at least σ (250K)4 = 200 W/m² radiating out.
So 500 W/m² net heating at noon, average over a full day is 1/π of that, and a Minmus day lasts 40 000 s, so 6 MJ/m² warming over the day.
Heat capacity of ice is around 2 MJ/m³/K, so 3 K×m of daily warming in some distribution, such as the top 100 mm heating 30 K.

Ice conducts only 2 W/m/K, so the 500 W/m² surface heating would cause a temperature gradient 250 K/m at the surface.  
So the 3-K·m daily warming is distributed roughly as 40 K daily warming of the surface, tapering in 160 mm depth to the average temperature.

So roughly a swing from 230 K to 270 K (-43° to −3°).  I have no trouble imaging a frozen Minmus, although I might be imagining a different situation than the KSP creators.

Spoiler

zYJsBh6.jpeg

Minmus rotates fast enough that only the top 20 cm changes temperature noticeably.

Sunlight would penetrate ice many centimetres before being absorbed as heat, but the case of surface heating and conduction is interesting, and it would apply to rock (which has similar thermal conductivity and capacity).

So here is imagining all the heat is absorbed at the surface.

import numpy as np
import matplotlib.pyplot as plt
# %%
day = 40_000  # seconds in a day
times = np.arange(0, 4*day, 10)  # seconds
 
solar_flux = (70  # mW/cm² at noon
              * np.maximum(0, np.cos(times * 2 * np.pi / day)))
net_surface_flux = np.empty_like(solar_flux)
plt.grid(True)
plt.plot(times / day, solar_flux)
 
# Store current temperature in each 1-cm layer, 1-meter depth
temperature_profile = np.full((100), 250.0)  # K
temperature_map = np.empty([temperature_profile.size, times.size])
K = 20  # mW/cm/K  thermal conductivity
C = 2000  # mJ/cm³/K heat capacity
sigma_SB = 5.67e-9  # mW/cm²/K⁴ Stefan-Botzmann constant
# %%
previous = times[0] + (times[0] - times[1])
for n, (time, flux) in enumerate(zip(times, solar_flux)):
    internal_flux = K * np.convolve(temperature_profile,
                                    [-1, 1], 'valid')  # mW/cm²
    dT_dt = np.convolve(internal_flux, [-1, 1], 'full') / C  # K/s
    net_surface_flux[n] = (flux
                           - sigma_SB * temperature_profile[0] ** 4)
    dT_dt[0] += net_surface_flux[n] / C
    temperature_profile += dT_dt * (time - previous)
    previous = time
    temperature_map[:, n] = temperature_profile
 
plt.imshow(temperature_map, cmap='hot', vmin=240, vmax=270, aspect=100)
plt.show()
plt.imsave('a.png',  temperature_map, cmap='hot', vmin=240, vmax=270)
plt.plot(times / day, net_surface_flux)
plt.grid(True)
plt.show()
plt.plot(times / day, temperature_map[0, :])
plt.grid(True)
plt.show()

But I agree the ice would evaporate away.  Evaporation rate at 0°C in vacuum is around 3 g/m²/s, so 10 mm/hour in terms of depth, evaporation slowing as evaporative cooling chills the surface.  The heat required is 3 MJ/kg, so 3000 MJ/m³ in terms of volume of ice.  A daily heating of 6 MJ/m² could evaporate 2 mm in a day.

Edited by OHara
inserted off-topic spoiler with numerical integration of heat on a fast-rotating planet
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1 hour ago, Opus_723 said:

Could someone clarify exactly why we are saying a frozen Minmus isn't possible? Kerbin is an Earth-analogue, and the "Snowball Earth" hypothesis is taken seriously enough by scientists. I don't see why Minmus couldn't have something similar going on?

Basically it's because Minmus is too warm for most ices to stay frozen all the time. Also the gravity is too low to hold any evaporated gases to form any type of atmosphere to allow them to refreeze. So over time, the ices will disappear and leave a barren rock that isn't shiny.

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19 hours ago, OHara said:

I think that we imagine Kerbin-Minmus to see the same solar flux as does Earth outside the atmosphere, 1360 W/m², of which ice absorbs 700 W/m², 

If it's very clean naturally formed ice, yes. Though, icy bodies tend to have higher albedos as their surface tends not to be that pristine. It's fine for an estimate, though.

19 hours ago, OHara said:

So roughly a swing from 230 K to 270 K (-43° to −3°).  I have no trouble imaging a frozen Minmus, although I might be imagining a different situation than the KSP creators.

You are playing very loose with heat conduction there. It takes time for a linear gradient to establish. As materials start to warm up, gradient starts out a lot steeper, meaning the surface temperature is going to be higher, and the skin layer is going to be very close to peak. The boundary condition here is nasty, so Mathematica told me to get lost, but I'm curious now, so I'll set up a quick simulation tomorrow. Looking at situation with real bodies at ~1AU, though, I don't expect it to be even close to 270K. But I owe you some numbers.

20 hours ago, OHara said:

But I agree the ice would evaporate away.  Evaporation rate at 0°C in vacuum is around 3 g/m²/s, so 10 mm/hour in terms of depth, evaporation slowing as evaporative cooling chills the surface.  The heat required is 3 MJ/kg, so 3000 MJ/m³ in terms of volume of ice.  A daily heating of 6 MJ/m² could evaporate 2 mm in a day.

Yeah, that's kind of the bigger problem here. The evaporation rates for ice I was able to find vary between about 1mm/hour and 10mm/hour in vacuum, but yeah, it's significant. And while temperature drop drastically reduces the rates, you have to have them go decidedly cryogenic if you want the ice to last for anything like geological time scale.

17 hours ago, Opus_723 said:

Could someone clarify exactly why we are saying a frozen Minmus isn't possible? Kerbin is an Earth-analogue, and the "Snowball Earth" hypothesis is taken seriously enough by scientists. I don't see why Minmus couldn't have something similar going on?

Between liquid water and atmosphere. Earth is pretty good at distributing the heat. Water can absorb a lot of day/night variation, and atmospheric circulation can actually help move heat from equator to poles, so that it can be radiated away from larger area. So under the right conditions, temperature on Earth can be prevented from rising much above average, and average can be well bellow freezing if the planet is covered with ice.

The second part is the aforementioned evaporation. If ice evaporates into an atmosphere, and your entire planet is frozen, that ice is going to get re-deposited somewhere else. Some amount of water will always be in atmosphere as moisture, but it's not a significant amount compared to ice on the surface. On the other hand, if you have no atmosphere, the vapor is going to be blown away by the solar wind. So anything that evaporates is effectively gone for good. Yes, some amount of re-deposition is still going to happen, but you are going to be losing most of the ice to the vacuum of space unless it's really, really cold and evaporation rate is negligible.

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17 minutes ago, K^2 said:

It takes time for a linear gradient to establish. As materials start to warm up, gradient starts out a lot steeper, meaning the surface temperature is going to be higher, and the skin layer is going to be very close to peak.

That's not how it works.  The heat flux causes the gradient. So there is a gradient with surface cooler than depth at dawn, the gradient following the net heat-flux on the surface as the sun rises.

There is well known solution, sometimes called "the wine cellar problem," of the one-dimensional heat equation that a lot of schools post online.  The boundary conditions for the Minmus question would be constant temperature at large depths, and radiative balance (plus evaporative cooling if you want to improve the model) determining the heat flux at the surface.

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I just assumed Minmus used to be an icy body, like Enceladus, that formed in the outer Kerbol system (a previous moon of Jool?) before being perturbed inward and captured by Kerbin-Mun at some point, and that the global ice and oceans boiled away and left behind the rocky core covered in salt flats:

Green Sandstone analysis suggests Minmus once had oceans.
Surface samples describe an "inedible crystalline substance," likely a salt of some kind.
Seismometer readings indicate the presence of subsurface liquid, which could be the remains of the saltwater ocean.
Salt flats can be very bright and resemble ice.

Seems like the simplest explanation for all the evidence. The volatile deposits that make Minmus useful as a fuel base in game are explainable as leftover pockets of saltwater or brines trapped beneath rock layers. 

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On 10/31/2021 at 4:59 PM, t_v said:

With our own moon, there are ice caps on the poles which don’t really melt even in summer, because they can reflect enough energy. Minmus, with such a high surface area to volume ratio, would be able to easily dissipate heat and stay frozen. Space is really really cold and even if you were in the sun, without insulation you would freeze. So minmus, which has no atmospheric insulation, would never get the chance to thaw out. 

No. The ice at the moons poles is in perpetual shadow, its not reflecting anything.

The equilibrium temperature being below zero is not enough. The thin surface layer of the ice would receive enough energy to directly sublimate.

 

To make an analogy that may be easier to understand: the way a puddle of water will evaporate even well below the boiling point of water is the sane way an icy body will sublimate even if it is below the freezing point of water.

If the gravity isn't strong enough to hold on to the vapor, it escapes.

On Earth, it goes into the atmosphere, but doesn't escape, so it can then rain or snow and come back later.

This wouldn't happen on minmus

Edited by KerikBalm
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On 11/1/2021 at 2:09 PM, OHara said:

That's also very true, and interesting to think about.

The equilibrium temperature of a bare Earth would be 255K = −18°C.   (If Earth was covered in snow and ice, it would reflect more of the mostly-visible light from the sun, but snow is not so white thermal infrared, so it would emit infrared reasonably well and make the equilibrium temperature even colder.)

Some ice would evaporate into water molecules with thermal velocities averaging 540 m/s, which is well below Earth's escape velocity (as well as Kerbin's, but enough to escape Minmus).

The evaporation would build up an atmosphere of water vapour, until the rate of re-condensing matches the evaporation rate, which happens the vapour pressure at the surface is about 100 Pa = 1 millibar.

Water vapour is itself a greenhouse gas, because the H-O-H bending vibrations absorb thermal infrared light (the same reason that snow emits significantly for radiative cooling).  So you might think that the atmosphere would become opaque to outgoing thermal radiation.  Then the planet would come to equilibrium where the upper atmosphere, at the average level visible from space in the thermal infrared, is at 255K.  The surface could be warmer, because the vapour is warmed by gravitational compression whenever convection moves it to lower altitudes.

But the small amount of water vapour in equilibrium over an icy earth isn't enough to make the atmosphere opaque, so the equilibrium surface temperature would remain around 255 K.  The prehistorical possibility of a snowball Earth (far, far prehistorical) seems plausible, and would seem difficult for Earth to have escaped. 

So assuming Kerbin-Minmus is imagined to receive the same solar flux as Earth (and I think that is the intention) Minmus would be cold enough for ice, but with insufficient gravity to hold on to what evaporates away.

Snowball earth make some sort of sense, the sun was dimmer back then but earth had lots of co2 probably as much as we have oxygen now or more and plenty of methane so it was pretty warm I believe.  But photosynthesis converted the CO2 to oxygen,  the methane was broken down and temperature fell and we got the snowball there most was covered in ice. This stopped most photosynthesis and blocked most other sort of CO2 absorption so CO2 from volcano piled up in the atmosphere making it heat up and earth became warmer again, indications this happened multiple time until stuff stabilized  after the cambrian explosion, I guess animals then stopped plants from growing unchecked consuming every last co2 molecule and the sun was brighter. 
Now if earth did not get photosynthesis I wonder if it would get an Venus style runaway greenhouse effect down the line with all that co2 as the sun getting hotter. 
Might be another great filter. 
 

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On 11/3/2021 at 10:52 AM, Timmon26 said:

I just assumed Minmus used to be an icy body, like Enceladus, that formed in the outer Kerbol system (a previous moon of Jool?) before being perturbed inward and captured by Kerbin-Mun at some point, and that the global ice and oceans boiled away and left behind the rocky core covered in salt flats:

The thing about that is if the surface IS made of ice, we would have to see a huge coma behind the moon, which we don't. I like the idea that most of the ice is already gone, leaving salt behind (as I previously stated).

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On 11/8/2021 at 10:49 AM, Nirgal said:

For what it's worth, I never considered Minmus to be icy, the flats remind me of Uyuni in Bolivia. A mysterious kind of salt, why can't that be the case?

Yea, I assumed they were salt flats...

Although that seemed a bit problematic for me... salt flats require relatively large bodies of water to form them... but at least with minmus you don't have to ask where all the water went.  You have to ask if it could have accumulated and held on to enough water to start with.

My other thought is that its like Pluto:

 mosaic-630x541.jpg

But its too close to the sun for that, so I move it to keep dres company...

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  • 3 weeks later...
  • 1 month later...

Minmus is made of mint-flavored ice cream. Period. At least it is in my headcanon... I mean look at it! It looks like a huge ice cream ball! In my heart, Minmus will always be ice cream and nothing- absolutely nothing- can change the fact that Minmus is made of mint-flavored ice cream.

However i'm ready to accept that it is coated with glass on the surface and that Minmus is still ice cream on the inside.

I just wanted to throw this in

 

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What about sugar? Simple organic molecules like amino acids have been observed in titanic quantities in nebulae, so the idea that a region rich in organics could form a small solid body with a crust rich in monosaccharides is at least tenuously believable. Long term exposure to sunlight and cold nights exposed to space would partially melt and refreeze the surface, perhaps until it resembles a glazed donut. Of course sunlight might also caramelize the surface until Minmus is a sickly brown, but maybe the green color reveals that there's some other chemical mixed in that counters that effect, or it's made of some form of sugar that caramelizes to more of a cyan color.

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  • 3 months later...
On 11/1/2021 at 11:47 PM, Kerb24 said:

Just say Minmus has upgraded from Ice cream to hard candy.

And she upgraded to netherite then?

(Nooo, from the color it would be a mix of diamonds and emeralds...)

But I still prefer ice cream. Because when in sunlight the ice can melt, but on the "night side" of Minmus the melted ice becomes ice again.

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19 minutes ago, Nazalassa said:

But I still prefer ice cream. Because when in sunlight the ice can melt, but on the "night side" of Minmus the melted ice becomes ice again.

The problem is not the temperature on average (put some water in space and it will freeze) but the fact that minmus doesn’t have enough gravity to hold any particles that do evaporate, so over time it would lose more and more mass. 

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15 hours ago, t_v said:

the fact that minmus doesn’t have enough gravity to hold any particles that do evaporate, so over time it would lose more and more mass. 

The Minmus Ice Cream is magnetive. Minmus too. So Minmus attracts its own Ice Cream gas.

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