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If by some form of magic, the sun went out, but the earth was still warm a.k.a the Earth did not just freeze and kill us all, who would die first, and how long would it take for all terrestrial living things in the world to die?

I can hypothesize that all the plants would die out first before all the humans do, because we have canned food (hooray for canned food), and plants can only live around 6 months without light, so assuming everything was going normally and humans have had ~1 year to prepare for this event, a.k.a. governments spending all money on creating food, and none on their militaries, healthcare, etc, how long would it take for all the living things on the planet to die out? What about for each kind of living thing: bacteria, plants, animals, etc.?

EDIT: I don't know if this is fake news, but it says here that astronauts might be able to eat their own feces in the future. Could this prolong the life of humans? Could this machinery even be implemented without extra energy?

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    $\begingroup$ If the goal is to sustain humanity for as long as possible, the most workable thing for a government to do is stockpile as much food as possible, and then kill everyone except for a small population. This population would then remain small (but large enough to be a viable breeding population), for as long as the food lasted. I'm not sure how long you can reasonably keep food before it spoils though. $\endgroup$ – Gryphon - Reinstate Monica Dec 15 '17 at 1:52
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    $\begingroup$ That plan would also require a whole lot of people to willingly commit suicide, something that's rather unlikely to happen. $\endgroup$ – Gryphon - Reinstate Monica Dec 15 '17 at 1:53
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    $\begingroup$ @PipperChip, I'm not sure how feasible it is to manufacture food from hydrothermal vent-centered life, especially if you need enough to sustain a breeding population. $\endgroup$ – Gryphon - Reinstate Monica Dec 15 '17 at 2:07
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    $\begingroup$ I'm confused, the sun goes out but the earth doesn't lose heat? How is this question different from, "if all food-valuable plants died tomorrow and the world had a year to prepare, how long would the world survive?" $\endgroup$ – JBH Dec 15 '17 at 2:31
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    $\begingroup$ @Gryphon I never said the vent-based life would become our salvation. They just wouldn't die, making the "how long would it take for all the living things in the world to die" part of the question difficult. As long as volcanic activity exists for these vent communities to live around, living things will still exist. $\endgroup$ – PipperChip Dec 15 '17 at 2:57
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How fast will the Earth cool down?

This is going to be the dominant factor. If it takes too long to cool down, then maybe all the plants will die without sunlight and creatures will starve to death. But if it cools down fast enough, everyone will freeze.

We can apply the Stefan-Boltzman law to find out how fast we will cool down.

Earth's radiation

The simplest model of the Earth is that of radiative equilibrium. This sort of radiation is governed by the Stefan-Boltzmann law: $$j^* = \epsilon\sigma T^4$$ where $j^*$ is irradiance in W/m$^2$; $\epsilon$ is the emissivity of Earth; $\sigma$ is the Stefan-Boltzmann constant, $5.67\times10^{-8}$ W m$^{-2}$K$^{-4}$; and $T$ is temperature.

The effective emissivity of the Earth is often considered 0.612. The average temperature of the Earth's surface is 288 K.

Integrating over time

Now, here come the warnings. Earth's emissivity is powerfully controlled by cloud formation. Changing the temperature of the Earth will radically change the cloud makeup. In addition, if the Earth cools and snow and ice form, that will drastically change emissivity as well. However, I don't have the tools to model all those things, so I am going to consider a model where Earth's emissivity is constant as it cools down.

The heat loss of Earth is a function of its irradiance. But it is also a function of the Earth's heat resovoir. Let us assume (and this is very spherical cow) that the top 100m of the Earth's oceans along with the top 100m of its land area will freeze. The oceans cover $3.6\times10^{14}$ m$^2$; the volume at 100m depth is $3.6\times10^{16}$ m$^3$. The volumetric heat of seawater is $3.9\times10^6 \frac{\text{J}}{\text{m}^3\text{K}}$. Land covers $1.5\times10^{14}$ m$^2$;The specific heat of various types of rocks times the density of Earth's crust gives us a volumetric heat of $1.8\times10^6 \frac{\text{J}}{\text{m}^3\text{K}}$.

Overall, the heat capacity of the Earth is 1.7e23 J/K. Finally, the irradiance needs to be scaled by the surface area of the Earth ($5.1\times10^{14}$ m$^2$). This gives us what we need to define a differential equation for the cooling down of the Earth $$ \begin{align} \frac{dT}{dt}\ &= \frac{0.612\cdot5.67\times10^{-8}\text{ J m}^{-2}\text{s}^{-1}\text{K}^{-4}\cdot T^4 \text{ K}^4\cdot5.1\times10^{14}\text{ m}^2}{1.7\times10^{23}\text{ J/K}}\\ &=1.0\times10^{-16}\cdot T^4 \text{ K/s} \end{align}$$

How fast will the Earth cool?

At the above calculated rate, the Earth initially cools at 1 K in about 17 days.

We can also solve this differential equation for a temperature drop of 15 K, which will bring the Earth's top 100m to freezing conditions at 273 K.

$$\begin{align} \frac{dT}{dt} &= 1.0\times10^{-16}T^4\\ \int_{273\text{ K}}^{288\text{ K}}\frac{dT}{T^4}&=1.0\times10^{-16}\int dt\\ \frac{-1}{3}\cdot\left(\frac{1}{288^3}-\frac{1}{273^3}\right) &= 1.0\times10^{-16}t\\ t&=2.4\times10^{6}\text{ s} \end{align}$$ This number is in seconds, and is equivalent to about 281 days.

Conclusion

The Earth stores a lot of heat. The top 100m of soil and ocean store enough heat that radiating 1 K of it into space will take more than two weeks, and freezing it will take most of a year.

The dominant factor in killing all living things will be the death of plants by inability to photosynthesize. There are many plants on Earth, and they will take varying times to die, but it would be reasonable to assume that within 2 weeks, no new vegetation will grow. From that point, it will take as long as all herbivores will take to consume the remaining plant life to destroy the food chain. I would estimate that within a month, most animal life would be in extremis.

Some animals might survive longer, like vultures. Certain bacteria will survive indefinitely, such as chemotrophs. Bacteria living deep in the crust in deep sea vents might not be affected for hundreds of millions of years.

So in general, all life will not be extinguished for as long as billions of years, but surface life as we know it would collapse within about a month.

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    $\begingroup$ Shouldn't the temperatures be in Kelvin? The limits on integration would be initial temperature (say, 293K, 20 C) and the target temperature (278 K, 5 C) for a delta of 15 C. Using your constant, I get 109 years. I see where you were going with this, but you have to plug in the integration limits, not the delta. $\endgroup$ – asylumax Dec 15 '17 at 4:28
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    $\begingroup$ Also, the integration of T^(-4) is -1/(4*T^3), so that changes the answer some more. Looking at your original differential, if you put in T=300K, dT/dt is on the order of 0.07 K per day, so for 15 K, this is 200+ days. $\endgroup$ – asylumax Dec 15 '17 at 4:53
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    $\begingroup$ @asylumax Egads. Alright, I have 281 days to freeze now. $\endgroup$ – kingledion Dec 15 '17 at 6:48
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    $\begingroup$ The question says "If by some form of magic, the sun went out, but the earth was still warm a.k.a the Earth did not just freeze and kill us all" Assume this means heat but no light? $\endgroup$ – Slarty Dec 15 '17 at 16:34
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    $\begingroup$ @Slarty Well, perhaps I misinterpreted the intention of the question. I rather think that the OP didn't understand how fast the Earth would actually cool, so I think my answer provides some useful context. $\endgroup$ – kingledion Dec 15 '17 at 16:47
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If the Earth is somehow heated, then not everything is lost for the humanity.

Plants and animals will die out without the sunlight. First the diurnal ones, then plants and algae, then the nocturnal ones, then soil inhabitants and ocean bottom feeders. Warm Earth by itself can not provide any sustenance to the lifeforms, so after all atmospheric oxygen is used up, this will be the end. It will take thousands of years to get there, and some organisms can feed of volcanic chemicals and stay alive indefinitely, but this is a pretty bleak picture for the Earth. Without humans.

Humans can make things different - not for the entire Earth, but sufficiently to maintain limited biomes and some population. If we would have light, we can have food. To have light, we need energy. In this scenario, there still be enough of it. First, Moon will still be up there. Invisible, it would still cause tides. Tidal energy can be captured and used. Second, if the Earth is heated, we still going to have wind and ocean waves, maybe even hurricanes. That's even more energy.

Everything we need to do is to build a lot of tidal and wind power generators, and set up farms with artificial lighting (just like cannabis is grown at homes). Unfortunately, only a fraction of today's 7 billion population can be saved this way. So, some dramatic (and maybe violent) events would be inevitable.

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    $\begingroup$ Nuclear, coal and natural gas would also be used for power generation I think. Eventually fossil fuels would run out, but they'd be viable in the medium-term. $\endgroup$ – Ebonair Dec 16 '17 at 21:43
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The Solar System

The first thing that would happen to Earth, immediately after the sun disappeared... would be nothing. It takes a bit under ten minutes for light from the sun to hit Earth, so it would look like the sun was still there, for a while. Earth would even still be in the sun's gravitational field would also last some minutes, as the sun's gravity takes time to reach us.

Once we were hit by the most beautiful night sky in human history, the Earth would would be flung out into space, away from our solar system. From this clear view of the sky, we could easily see Jupiter and the other planets--sitting in the sky like nothing had happened. The sun's light is still reflecting off of them, and being further away they have not yet been "hit" by the lack of gravity.

The exception to this is Mercury and Venus, which being closer to the sun, have already gone through our dilemma. We probably won't be able to see them anymore, without the sun's light reflecting off of them (maybe we can with certain telescopes).

Oxygen

On the bright side, it'd actually take us forever to suffocate. Sure, all the plants die in a few days, and we have no way to recycle carbon to oxygen, but the atmosphere will stay mostly breathable for many centuries. Though, if we survived any length of time, the breathable air around cities could be pretty bad, without any parks or plants to regulate the local oxygen.

Many trees would survive for a year, before the growing cold killed them.

Freezing Cold

Within the first week, temperature would hit 0 Celsius, freezing, and you could walk across the frozen sea. That would be the good times. Many people would die in the following weeks from the sudden temperature shift, the poor in hot countries in particular would soon be freezing and sickening.

Storms

The Earth itself would become sick with the sudden temperature change, and terrible blizzards would rage across the world. The storms would eventually cease, maybe after the first month, and the weather would become much more stable across the globe.

In fact, when the last storm finished... there'd be no wind. Without the sun, wind would become a local phenomenon.

Snowy weather would become more and more rare as time passes. Without the sun to evaporate water, the only source of evaporation would be geothermal springs. And without the sun to heat the clouds and change temperature, they'd mostly stay up there in the cold sky; supercooling well past the freezing point.

A year later

After a year, it'd get real bad... Temperatures would hit -100 degrees Fahrenheit. The coldest temperature I've been able to find for human habitation was a research post, at -97 degrees Fahrenheit. A couple of cities have also survived winters of -90 Fahrenheit, though I don't know the details for how well they could keep that up.

While those equipped for it could survive, a huge portion of the human population would die. Every plant on Earth would die. Most of the animals across the world would die.

The animal world

For the first few months, the animals in the Night-half of the North Pole would notice no difference. They might get hit by the weeks of storms, but otherwise life would remain unchanged. By the time a year has passed, though, they would've reached a temperature on par with their coldest ever winters, and most of the animals would soon die. It is unlikely, but possible, that some animals will managed to survive for a while longer... only to die of starvation, as what little plantlife there is cannot be renewed.

Surprisingly, insects like the Red Flat Bark Beetle would be among those that survive the cold. They would eventually die out all the same, as their food source (wood-eating beetles) succumbed to the cold. Even if their food source survived, the dying trees would eventually be eaten away. The Upis beetle would outlive the Flat Bark, by freezing--going into cryogenic hibernation.

Cockroaches and rats would also survive in human compounds, though they might disappear entirely from some cities.

Animals who live near hotsprings may survive and adapt to this new world for a while... only to die when the death of plants and eternal winter causes them to starve.

Surviving

To survive the cold, you either need insulation and power, or hotsprings.

The easiest way to stay warm is to dig deep underground (which will get harder as the cold freezes the ground). Underground living will become quite comfortable, losing its chief disadvantages. Underground, you have to worry about rain, and sewage management. But in this scenario, all your sewage will be recycled, and there won't be a problem of rain anymore. Underground living will help you keep warm and safe, with minimal power consumption. You will, however, need to go to more effort to ventilate your air.

To survive starvation, you need either large food stores (which will run out), or a lot of power and a meticulous setup.

With power and insulation, it isn't too hard to keep warm, and even ventilate your compounds. The real drain on resources will be cultivating food. Mushrooms can be cultivated easily enough, and will begin to make up a large part of your diet as an economic food. But generating enough light to grow crops will be challenging. Without crops, most farm animals are useless.

So largely you'll be eating mushrooms, and vitamin tablets, and eventually lab-grown yeasts and the like to make up for the dietary deficiencies. The most obvious and important deficiency is vitamin D, as you no longer have the sun.

What experts survive this period on stockpiled foods will be working hard to genetically engineer better mushrooms and yeasts, while in the long term desperately searching for this new world's philosopher's stone: Fusion Power.

The Future of Power

If Fusion energy is developed, a small population of humans might live millennia is this cold night world.

Even if not, there is enough uranium for the current population of Earth to survive at its present level of consumption, for 200 years. If consumption of power becomes 1% of what it is now, nuclear power alone could keep humanity going for 2,000 years.

Clean Air

So, adding in coal and other fossil fuels, humanity should be able to survive with a reasonable population. Of course, without plants, the fear of carbon contaminating the atmosphere would become extreme. The fear would be much worse than the problem.

Even without a breathable atmosphere, it's quite practical to power a greenhouse and use it to recycle your air. This will be the main purpose of vegetables, to recycle air. The mushrooms are no good for this... they breathe oxygen too. So the more mushrooms you grow, the more vegetables you'll have to grow (like some kind of resource management sim!).

Mining and Scavenging

There will be a lot of scrambling in the early days to grab stuff from the stores (natural refrigeration) and steal supplies to survive. After people settle down into compounds, the main source of work will be scavengers braving the cold to get more supplies.

Eventually, when scavenging materials run short, we'll have to resume mining operations. In the extreme cold, it will be difficult to travel, prospect, and transport minerals. But the mining itself won't be too different. It's warm underground, and most of you will live underground anyway.

Conclusion

So, in the end, we might survive long enough to develop interstellar travel and adapt to this world - barring any unfortunate collisions with asteroids and meteors. Since the Earth is hurtling off into space all this time, this may even help us launch off towards the nearest star.

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A very spherical cow, indeed. When ice forms on the surface of the ocean, this will provide an insulation barrier. Would Earth wind up like Europa?

This question has been answered in The Straight Dope:

http://www.straightdope.com/columns/read/3293/if-the-sun-goes-out-how-long-till-the-earth-freezes/

The Earth may store a lot of heat, but we probably care about the surface temperatures, which will drop more quickly. As other have mentioned, at night, in desert, temperatures drop far more quickly.

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    $\begingroup$ There isn't much math in your link $\endgroup$ – kingledion Dec 15 '17 at 4:51

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