{"id":4408,"date":"2021-03-25T13:19:50","date_gmt":"2021-03-25T20:19:50","guid":{"rendered":"http:\/\/universesandbox.com\/blog\/?p=4408"},"modified":"2021-05-20T11:26:30","modified_gmt":"2021-05-20T18:26:30","slug":"the-end-of-the-world-sciencelog-3","status":"publish","type":"post","link":"https:\/\/universesandbox.com\/blog\/2021\/03\/the-end-of-the-world-sciencelog-3\/","title":{"rendered":"The End of the World: Slower Than You Expected | ScienceLog #3"},"content":{"rendered":"\n
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How long do you think this would take?<\/em><\/figcaption><\/figure><\/div>\n\n\n\n
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Sure, the Sun\u2019s pretty useful, we guess. It feeds Earth\u2019s plant life, keeps us warm, and helps people see where they\u2019re going when they walk around outside. If the Sun suddenly disappeared from the Solar System (which you can do with the click of a button in Universe Sandbox!), we\u2019d be in big trouble.* In fact, right now you\u2019re probably imagining the desolate, frozen landscape that our planet would become without its Sun. But this apocalypse wouldn\u2019t happen quite as fast as you probably think:

If the Sun disappeared, it would take <\/span>over a century<\/span><\/i> for the Earth\u2019s oceans to completely freeze solid! <\/span><\/p>\n\n\n\n

Universe Sandbox lets you perform this kind of catastrophic experiment from the safety and comfort of your own home by simulating three phases of water (solid, liquid, and gas), and how they react to the changing environment. As a planet cools, its surface water will freeze into ice. Heat that planet back up with a laser, and the ice will melt and even vaporize into gas.<\/span><\/p>\n\n\n\n

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Using a laser to melt a hole in the ice on the frozen night side of a tidally locked Earth.<\/em><\/figcaption><\/figure><\/div>\n\n\n\n
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But you might have noticed that some of these phase changes take longer than you expect them to. If you\u2019ve found yourself wondering \u201cWhy is it taking so long for the oceans to freeze?\u201d or \u201cI\u2019ve been waiting for ages for the ice caps to melt, what\u2019s going on??\u201d, read on to learn more about the physics (and speed) of phase changes.<\/span><\/p>\n\n\n\n

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Energy Flow… Again<\/h3>\n\n\n\n
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In <\/span>ScienceLog #1<\/span><\/a>, we explained how the flow of energy into and out of a planet will affect that planet\u2019s temperature. In fact, the flow of energy also affects the phase of water.<\/span><\/p>\n\n\n\n

As you know if you\u2019ve ever boiled a pot of water, you need to <\/span>add<\/span><\/i> energy to turn water from a liquid to a gas. The opposite phase change\u2014 condensing water vapor into a liquid\u2014 involves the <\/span>release<\/span><\/i> of energy into the cooler environment surrounding the water. Similarly, energy needs to flow <\/span>into<\/span><\/i> a block of ice to melt it into water, but energy must flow <\/span>out of<\/span><\/i> a pool of water in order to turn it into ice. We can figure out how fast a phase change is occurring based on the speed at which energy is flowing into or out of the water.<\/span><\/p>\n\n\n\n

The key point here is that phase changes are not instantaneous. You\u2019ve probably already noticed that, if you pay attention to phase changes in your daily life: It can take a few days for snow to melt after a big blizzard, even if the temperature rises above freezing. Even ice doesn\u2019t melt instantly in your drink on a hot day. And of course, we all know that water never boils as fast as we want it to, even if we set it on high heat.<\/span><\/p>\n\n\n\n

The speed of a phase change of surface water in Universe Sandbox will depend on the temperature of the surface, the freezing or boiling point of water, and the mass of water that you\u2019re trying to change. This last factor, the mass of the water, is probably the source of most of the confusion about this issue in Universe Sandbox. Since we\u2019re all used to seeing phase changes in our everyday lives, we have some intuition for how fast we think they should happen. But the masses of the Earth\u2019s ice caps or oceans are much, much larger than an ice cube or a kettle of water, and this significantly slows down the rate of boiling, melting, and any other phase change.<\/span><\/p>\n\n\n\n

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This Earth is orbiting closer to the Sun than Mercury.<\/em><\/figcaption><\/figure><\/div>\n\n\n\n
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The heat from the too-close Sun is melting the Earth\u2019s ice quickly, as you can see in the Total Ice Mass graph on the left, but not instantly.<\/p>\n\n\n\n

That\u2019s why you might have to wait a while for your simulated planet\u2019s oceans to freeze or boil (depending on what you\u2019ve done to that poor planet). Of course, if you get impatient, you can always use the new Stabilize Phases button in the Surface tab to instantly change the surface water to the correct phase based on the local temperature. What a convenient apocalypse!<\/span><\/p>\n\n\n\n

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\u2026<\/span><\/p>\n\n\n\n

\u2026What\u2019s that? You still don\u2019t believe us that it would take a century to freeze the Earth\u2019s oceans?<\/span><\/p>\n\n\n\n

\u2026<\/span><\/p>\n\n\n\n

…You want some proof in the form of equations and hard numbers?<\/span><\/p>\n\n\n\n

\u2026<\/span><\/p>\n\n\n\n

\u2026All right, you asked for it. If you\u2019re still with us, read on for the juicy, math-y details:<\/span><\/p>\n\n\n\n

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Bonus Math: How Long Does It Take to Freeze the Earth\u2019s Oceans?<\/b><\/h3>\n\n\n\n
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We\u2019re going to put our money where our math is and walk through an example. Suppose we want to freeze all the water on Earth into ice. We could do this by deleting the Sun in the Solar System, although then we\u2019d have to wait for the Earth to slowly cool down. If we\u2019re impatient, we can skip ahead by just setting the Earth\u2019s Average Surface Temperature to the lowest possible temperature: -273\u00b0C, or zero Kelvin (also known as \u201cabsolute zero\u201d).<\/span><\/p>\n\n\n\n

If you try this in Universe Sandbox, you\u2019ll notice that after you change the temperature, the oceans are still made of liquid water. How long should we expect it to take to freeze all that water into ice: Days? Weeks? Months?<\/span><\/p>\n\n\n\n

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We\u2019ve just made the Earth as cold as it can be, but its oceans are still liquid!<\/em><\/figcaption><\/figure><\/div>\n\n\n\n
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Let\u2019s start by asking how much water we\u2019re trying to freeze. Earth\u2019s oceans have a mass of roughly 1.4 thousand billion billion kilograms. In scientific notation, that\u2019s 1.4 x 1021<\/sup> kg of water. To turn the liquid water into a solid, we need to remove energy from it. Since the water hasn\u2019t frozen yet, its temperature is sitting at the freezing point, around 273 Kelvin. Since the Earth itself is at zero Kelvin, the heat energy in the water will flow into the Earth (and then out into space). <\/span><\/p>\n\n\n\n

Our next question is: How <\/span>much <\/span><\/i>energy needs to flow out of the water in order to freeze it? To answer this question, we use a property of water called the <\/span>Heat of Fusion<\/span><\/a>. This property represents how much energy, in Joules, is required to melt one kilogram of ice into water, or, conversely, how much energy must be removed to freeze one kilogram of water into ice. You can look up the Heat of Fusion for many different materials online\u2014 For water, it\u2019s about 3.3 x 105<\/sup> Joules per kilogram.<\/span><\/p>\n\n\n\n

This means that the amount of energy that must be removed from Earth\u2019s oceans to freeze them entirely into water is:<\/span><\/p>\n\n\n\n

\\text{Energy} = \\text{Mass} \\times \\text{Heat of Fusion} = (1.4 \\times 10^{21} \\text{kg}) \\times (3.3 \\times 10^5 \\frac{\\text{J}}{\\text{kg}}) = 4.62 \\times 10^{26} \\text{J} <\/pre><\/div>\n\n\n\n
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That\u2019s roughly the amount of energy that would be released by two billion Tsar Bomba hydrogen bombs, the most powerful nuclear weapon ever created.<\/span><\/p>\n\n\n\n
Now we need to know the speed at which energy is flowing out of the water, and into its zero Kelvin environment. For this, we can use the <\/span>Stefan-Boltzman law<\/span><\/a>, which says that an object with temperature T<\/em> will lose energy through its surface at a rate of <\/span><\/p>\n\n\n\n
\\text{Rate} = \\sigma T^{4}A<\/pre><\/div>\n\n\n\n
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where \u03c3<\/em>, the Greek letter \u201csigma\u201d, represents the Stefan-Boltzmann constant, and the A<\/em> is the surface area of the object. <\/span><\/p>\n\n\n\n
The surface area of the Earth is about 5.1 x 1014<\/sup><\/span> m2<\/sup>, so the rate at which the oceans are losing energy is roughly<\/span><\/p>\n\n\n\n
\\text{Rate} = (5.7 \\times 10^{-8} \\frac{\\text{J}}{\\text{s m}^{2}~\\text{K}^{4}}) \\times (273~\\text{K})^{4} \\times (5.1 \\times 10^{14}~\\text{m}^{2}) = 1.61 \\times 10^{17}~\\text{J\/s}<\/pre><\/div>\n\n\n\n
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We can actually double-check this number in the game: First, put the Earth in an empty simulation. Then set Earth\u2019s Average Surface Temperature to 273 Kelvin and look at the Energy Radiation Rate property. As expected, it shows that this Earth is losing energy at a rate of 1.61 x 1017<\/sup> W (the Watts unit is equivalent to Joules per second). <\/span><\/p>\n\n\n\n
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This Earth\u2019s temperature is set to the freezing point of water, and the Energy Radiation Rate is exactly what we just calculated it should be with the Stefan-Boltzmann law.<\/em><\/figcaption><\/figure><\/div>\n\n\n\n
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Back to our zero-Kelvin Earth: you probably know that only about 70% of our planet\u2019s surface is covered in water. Since we\u2019re only interested in how fast the <\/span>oceans<\/span><\/i> are losing heat, we should use a reduced rate of<\/span><\/p>\n\n\n\n
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\\text{Rate} = 1.61 \\times 10^{17}~\\text{J\/s} * 0.70 = 1.13 \\times 10^{17}~\\text{J\/s}<\/pre><\/div>\n\n\n\n
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We now know how much energy we need the oceans to lose in order to freeze them all, and how fast they are losing energy to their surroundings. Now we can easily calculate the time it will take for the oceans to lose the required amount of energy:<\/span><\/p>\n\n\n\n
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\\text{Time} = \\text{Energy \/ Rate} = (4.62 \\times 10^{26}~\\text{J}) \/ (1.13 \\times 10^{17}~\\text{J\/s}) = 4.09 \\times 10^{9}~\\text{s}<\/pre><\/div>\n\n\n\n
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There are about 3.15 x 107<\/sup> seconds in a year, so that\u2019s<\/span><\/p>\n\n\n\n
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\\text{Time} = (4.09 \\times 10^{9}~\\text{s}) \/ (3.15 \\times 10^{7}~\\text{s\/yr}) = 132~\\text{yr}<\/pre><\/div>\n\n\n\n
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In other words, we estimate that it would take over 100 years(!) for the Earth\u2019s oceans to completely freeze if the Earth\u2019s temperature suddenly dropped to absolute zero. In real life, it would likely take even longer: The layer of ice that would form on top of the oceans would insulate the liquid water underneath, keeping it from freezing from much longer. Geothermal vents at the bottom of the oceans could also keep temperatures cozy for the microorganisms that live down there, possibly for billions of years.<\/span><\/p>\n\n\n\n
If you\u2019d rather go in the opposite direction and try to boil away Earth\u2019s oceans by heating up the planet, you might find that it takes even more energy! That\u2019s because the energy needed to change water from a liquid to a gas, known as the <\/span>Heat of Vaporization<\/span><\/a>, is almost ten times its Heat of Fusion. You can explore exactly this scenario in our Welcome | Part 2<\/strong> guide, which you can find in Home > Guides > Tutorials<\/strong>. You can also learn more about how Universe Sandbox simulates the surface temperatures of objects in the Surface Simulation<\/strong> or the Energy & Heating<\/strong> tutorials.<\/span><\/p>\n\n\n\n
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Assumptions Addendum<\/h3>\n\n\n\n
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Based on some comments we\u2019ve received about the assumptions we made for this calculation, we wanted to go into a bit more depth about what they are, and why they may (or may not) be important. You\u2019ll notice that because of these assumptions, the 132 years that we come up with really represents a minimum<\/em> amount of time it would take for the oceans to freeze solid.<\/p>\n\n\n\n