## What-if 0067: "Expanding Earth"

What if there was a forum for discussing these?

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keithl
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### Re: What-if 0067: "Expanding Earth"

rmsgrey wrote:For 1cm/s, that's very roughly 1 atom added to a chunk of material for every 100 moles of that atom there were before, every second

M=(4/3)πρR3
dM/dt = 4πρR2(dR/dt)
(dM/dt)/M = 3(dR/dt)/R
(dR/dt)=0.01 m/s
R = 6,371,000 m (average radius for oblate earth)
(dM/dt)/M = 4.7e-9 /s
4.7e-9 * 6.02e23 --> 2.8e15 molecules per mole per second
If the extra mass is added with a relative velocity of 450 meters per second, the energy added per second to a gram of material is (dE/dt)/M = ½v²(dM/dt)/M = 4.8E-4 J/gm-sec = 40J/gm-day . The specific heat of granite is 0.8 J/gm-K, so the crust would heat by 50C/day .

I'll leave the computation of mechanical shear through the crust and against the mantle for others. Circulatory patterns in air demonstrate what Coriolis acceleration can do, and in this case Coriolis acceleration is shearing the plate, along with differential growth top to bottom. There's already a lot of built-up strain in the system; we can expect the heating and shear to release some of that.

My prescription: head for Antarctica, and assemble lots of rockets when you get there.
Last edited by keithl on Thu Oct 17, 2013 2:46 am UTC, edited 1 time in total.

cameroda
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### Re: What-if 0067: "Expanding Earth"

edo wrote:
Randall, in the title text of the last image wrote:If you liked it then you should have moved a mass inside its Roche limit.

Am I the only one to get this?
Spoiler:
"move a mass inside its Roche limit." = "Put a ring on it"

I guess there are not too many Beyonce fans on this thread...

gmalivuk
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### Re: What-if 0067: "Expanding Earth"

My guess would be that most of us got it but felt there was nothing to really comment about.
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rmsgrey
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### Re: What-if 0067: "Expanding Earth"

keithl wrote:
rmsgrey wrote:For 1cm/s, that's very roughly 1 atom added to a chunk of material for every 100 moles of that atom there were before, every second

[...]
--> 2.8e15 molecules per mole per second

Oops, yeah, I was off by a factor of R2/3 (and rounding errors) by cubing dR to find dV rather than cubing the linear scale factor 1+dR to give 1+dV...

Drecksau
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### Re: GPS messed up?

eran_rathan wrote:
syrinxsean wrote:So I don't get why GPS would be all messed up right off the bat. Yes, GPS is dependent upon very precise timing. But that's presumably timing for the satellites, not anything on the earth's surface. As GPS works in 3D (reporting altitude in addition to latitude and longitude), including in flight above the earth, presumably an increase of the earth's radius wouldn't affect the GPS satellites until they fell out of orbit due to increased gravity. What am I missing here?

Because GPS works by measuring distance to multiple satellites concurrently, and having precise ephemerides for each SV, thereby calculating position by solving the n-dimensional matrix for distance of each satellite. If the ephemerides are borked, so is GPS.

The GPS "configuration" gets updated and calibrated almost every day. A slightly change like in this situation would be covered by that handling IMO. Theoreticaly it would break the GPS, practicaly not.

Fooglmog
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### Re: What-if 0067: "Expanding Earth"

Out of curiosity, how fast will the effects of the change in gravity in this scenario percolate to distant bodies?

As Earth's mass increases, the change in gravity would have some impact on other solar (and galactic [and universal]) bodies. And presumably this change would take some time to spread, and wouldn't be instantaneous (ie. it would be possible to detect the change on Mars before Proxima Centauri). Would this spread at the speed of light? I can't think of any reason why it should, but I also can't think of any reason for it to spread at another specific speed, and "speed of light" is a nice default answer for these things.

Does anyone know?

rmsgrey
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### Re: What-if 0067: "Expanding Earth"

Fooglmog wrote:Out of curiosity, how fast will the effects of the change in gravity in this scenario percolate to distant bodies?

As Earth's mass increases, the change in gravity would have some impact on other solar (and galactic [and universal]) bodies. And presumably this change would take some time to spread, and wouldn't be instantaneous (ie. it would be possible to detect the change on Mars before Proxima Centauri). Would this spread at the speed of light? I can't think of any reason why it should, but I also can't think of any reason for it to spread at another specific speed, and "speed of light" is a nice default answer for these things.

Does anyone know?

Gravitational waves are believed to travel at the speed of light.

So, probably, yeah.

Pfhorrest
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### Re: What-if 0067: "Expanding Earth"

So by the time Earth has grown big enough to become a black hole, what's the closest object that still won't have noticed the supermassive body growing nearby?
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Copper Bezel
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### Re: What-if 0067: "Expanding Earth"

Too distant to care; c is still too fast for that to make a real difference. You saw jotun's math. And the increase in mass is steadily increasing, but there's no sudden inflection point.
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Milnoc
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### Re: What-if 0067: "Expanding Earth"

Just noticed the part where, after 5 years, your "weight" would increase from 70 to 88 "kg".

For shame, Randall! Using "weight" instead of "mass", and selecting a unit of measure that doesn't change even when gravity does change!

If you had chosen "pounds" as a unit of measure, you would have been in the clear!

gmalivuk
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### Re: What-if 0067: "Expanding Earth"

Isn't it fair to assume he's talking about weight in kilogram-force, since that's already what your bathroom scale is telling you? He's obviously well aware of the distinction, because he mentions that the measurement on a balance scale wouldn't change.
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scotty03
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### Re: What-if 0067: "Expanding Earth"

Breathing in this situation would be difficult. It's hard to suck in air against the weight of the water, which is why snorkels can only work when your lungs are near the surface.

Another reason snorkels only work near the surface is that if you are deeper, you require a longer tube, creating more 'dead space' ventilation.

Our bodies already contain a degree of dead space ventilation, it is the distance from the nose/mouth to the trachea, and also includes the trachea, right and left main bronchus, and further down until you get to the alveoli that actually perform the gas exchange process. By breathing through a pipe you massively increase the dead space ventilation, and your total lung volumes aren't large enough to expel the old air and get new air in. You instead just keep breathing the same air over and over again.

gmalivuk
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### Re: What-if 0067: "Expanding Earth"

Yeah, that's definitely another problem with breathing through a long tube, but I suspect the pressure becomes an issue before the dead space volume when you're under water. Inhaling through even a normal-sized snorkel is noticeably more difficult if you're vertical in the water instead of horizontal on the surface, and probably becomes near impossible for most people if they go down another meter or so, even if their lung capacity is large enough to continue getting at least some fresh air through an additional meter of tubing.
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rmsgrey
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### Re: What-if 0067: "Expanding Earth"

scotty03 wrote:
Breathing in this situation would be difficult. It's hard to suck in air against the weight of the water, which is why snorkels can only work when your lungs are near the surface.

Another reason snorkels only work near the surface is that if you are deeper, you require a longer tube, creating more 'dead space' ventilation.

Our bodies already contain a degree of dead space ventilation, it is the distance from the nose/mouth to the trachea, and also includes the trachea, right and left main bronchus, and further down until you get to the alveoli that actually perform the gas exchange process. By breathing through a pipe you massively increase the dead space ventilation, and your total lung volumes aren't large enough to expel the old air and get new air in. You instead just keep breathing the same air over and over again.

Dead space ventilation has an easy fix - breath in through the tube and exhale directly into the water around you (or use a second tube, and add valves...)

pickledish
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### Re: What-if 0067: "Expanding Earth"

The Earth would of expanded by 864 meters when t=1 day.This would be incorrect because the question says the radius of the world would be expanding by 1 cm per second

pickledish
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### Re: What-if 0067: "Expanding Earth"

Barstro wrote:
... Mt. Everest would ...would be taller—climbing would be more work.

Why would Mt. Everest be taller? Wouldn't it be the same height relative to the new sea level, since the earth is expanding at a constant rate?

It would be taller because we still have certain measurements of distance and the mountain is growing, too

gmalivuk
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### Re: What-if 0067: "Expanding Earth"

pickledish wrote:The Earth would of expanded by 864 meters when t=1 day.This would be incorrect because the question says the radius of the world would be expanding by 1 cm per second

Yes, and thus after one day that radius would have expanded by 864 meters.
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PinkShinyRose
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### Re: What-if 0067: "Expanding Earth"

gmalivuk wrote:Yeah, that's definitely another problem with breathing through a long tube, but I suspect the pressure becomes an issue before the dead space volume when you're under water. Inhaling through even a normal-sized snorkel is noticeably more difficult if you're vertical in the water instead of horizontal on the surface, and probably becomes near impossible for most people if they go down another meter or so, even if their lung capacity is large enough to continue getting at least some fresh air through an additional meter of tubing.

That could also depend on how well you've trained your intercostal muscles. You normally don't train them much beyond their normal use (this includes divers usually using pressurised air, but not vertical snorkellers I suppose), but I don't see why you couldn't...
rmsgrey wrote:
scotty03 wrote:
Breathing in this situation would be difficult. It's hard to suck in air against the weight of the water, which is why snorkels can only work when your lungs are near the surface.

Another reason snorkels only work near the surface is that if you are deeper, you require a longer tube, creating more 'dead space' ventilation.

Our bodies already contain a degree of dead space ventilation, it is the distance from the nose/mouth to the trachea, and also includes the trachea, right and left main bronchus, and further down until you get to the alveoli that actually perform the gas exchange process. By breathing through a pipe you massively increase the dead space ventilation, and your total lung volumes aren't large enough to expel the old air and get new air in. You instead just keep breathing the same air over and over again.

Dead space ventilation has an easy fix - breath in through the tube and exhale directly into the water around you (or use a second tube, and add valves...)

But normal snorkels don't contain valves, they are merely J-shaped tubes with soft mouthpieces... What would the advantage of the second tube be over dumping your respiratory waste into the water?

syrinxsean
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### Re: GPS messed up?

eran_rathan wrote:Because GPS works by measuring distance to multiple satellites concurrently, and having precise ephemerides for each SV, thereby calculating position by solving the n-dimensional matrix for distance of each satellite. If the ephemerides are borked, so is GPS.

Sure, but the satellite orbits aren't changing (at least, until they fall from orbit). The ephemerides shouldn't change just because the surface of the earth is getting closer. Unless the ephemerides are defined with respect to the surface, rather than the center of the earth, I don't see why an increase in the radius of the earth would map to anything other than just the appearance (GPS-wise) of increasing in altitude.

eran_rathan
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### Re: GPS messed up?

syrinxsean wrote:
eran_rathan wrote:Because GPS works by measuring distance to multiple satellites concurrently, and having precise ephemerides for each SV, thereby calculating position by solving the n-dimensional matrix for distance of each satellite. If the ephemerides are borked, so is GPS.

Sure, but the satellite orbits aren't changing (at least, until they fall from orbit). The ephemerides shouldn't change just because the surface of the earth is getting closer. Unless the ephemerides are defined with respect to the surface, rather than the center of the earth, I don't see why an increase in the radius of the earth would map to anything other than just the appearance (GPS-wise) of increasing in altitude.

No, the orbits would be changing, because the orbit is partially dependent on the mass of the body that is being orbited. Since the earth in this What-If is increasing in mass (expanding while maintaining the same density), the orbits are changing.
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gmalivuk
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### Re: What-if 0067: "Expanding Earth"

After a day, Earth's mass would increase by about 0.04%, which means the radius of a stable circular orbit at a particular speed would decrease by the same amount. For GPS satellites, this is a decrease of about 11km, plus the fact that the surface is itself almosy 1km higher. And because it takes less time to go around the shorter orbit, there's a horizontal error introduced as well.

I'd say a discrepancy of several km is absolutely a significant decrease in functionality.

Edit: And it's actually worse than that, because the horizontal error is cumulative. After 1 day, the period is 0.04% shorter, which at the satellite's distance and speed means 68km off after one revolution, or 16km along the ground. There are two orbits per day, so the error is 32km per day or 1.35km/h. This is a cumulative drift for each satellite, and admittedly they'd be drifting in different directions so some of the error might cancel out, but I think it's still reasonable to say the system as a whole would be essentially useless after a few hours.
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### Re: What-if 0067: "Expanding Earth"

The discussion here was predicated on "composition of rock" meaning "density of the Earth," while I took it to mean that the locations of the materials in the Earth remain what they are. In this scenario an expansion of the Earth's radius does not imply an increase in mass but rather a decrease in density. This would yield completely different results. It's also marginally less magical since there's no creation of mass or angular momentum like in the version discussed.

gmalivuk
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### Re: What-if 0067: "Expanding Earth"

Except, the original question asked about weight gain, which requires the increasing mass interpretation.
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