Energy - Earth, Moon and Man
I love simple physics, especially things that illustrate relative power or force. I'd been meaning to write something like the following, but it's hard to find the time to gather up the relevant data. Which makes it even cooler that someone else already did it. Enjoy!
-pmb
From http://crowlspace.blogdrive.com/archive/76.html.
Energy - Earth, Moon and Man
It's a worthwhile pursuit to
ponder the various energies of Earth, Moon and Sun.
Earth mass 5.9737E+24
Moon mass 7.34767E+22
mass sum 6.04718E+24
Gravity constant 6.67259E-11
Earth-Moon radius 384400000
angular velocity 2.66531E-06
dr/dt = 1.21048E-09
Earth/Moon system:
L12 momentum 2.85862E+34 kg.m^2.s^-1
E12 energy 3.80955E+28 J
dL/dt = 4.50092E+16
dE/dt 1.19964E+11
Earth alone:
Earth radius = 6378137 m
Earth I = 8.01889E+37 kg.m^2
Earth w = 7.29212E-05 rad/s
Le = 5.84747E+33 kg.m^2.s^-1
Ee = 2.13202E+29 J
(notice more energy is packed into Earth's rotation on its own
axis than in the Earth-Moon mutual motion)
dw/dt = (dL/dt)/Ie
= -5.6129E-22 rad/s
= 0.002092976s/day/Century
dEe/dt = 3.28213E+12 W
The above is the energy calculations for the Earth-Moon system. Earth slows
by ~ 0.002 seconds a day every Century, and this releases 3.3 terawatts as
frictional energy in the oceans.
What if humans got all their energy from the tides?
Energy consumption = 10000 W/person (in USA or Australia)
population = 6300000000
total 63 terawatt.year => compare current (= 15 TW.yr)
terawatt-year = 3.15576E+19 Joules
natural depletion time = 4.73364E+20 seconds
= 2058410434 years (2 billion years)
(actually it's an asymptotic approach to zero. Thus in about ~ 15 billion years the Earth/Moon will be synchronous with ~ 1/2000 th of its current energy)
@ 63 TW.yr, t = 107237508.4 (100 million years)
@ 2.6% growth, t = 570.803779 years
Amazing what exponential growth can do.
Of course only a tiny fraction could ever be used per year, but with enough tide machines we might tap ~ 100 TW.yr/yr, slowing Earth down in a mere ~ 66 million years. The synchronous radius is at 553757 km, and the length of day/month will be the synchronous
period 47.176 days.
Famed early 20th Century biologist J.B.S.Haldane once predicted fictionally that the desire for central heating would cause the tides to be excessively tapped and the Moon to spiral out then reverse and break-up into a ring, bombarding the Earth - in about 3,000,000 AD.
So what does our mere present day 15 TWyr mean in terms of barrels of oil?
oil barrel energy = 6100000000 Joules
million barrels/annum = 77,600.65574 (call it 80 billion barrels)
daily = 212.4590164
As for carbon dioxide produced here's some relative figures...
coal 32800000 J/kg 3.666666667 kg (CO2 per kg fuel burnt)
8945455 J/kg CO2
oil 44880000 J/kg 3.087719298 kg
14535000 J/kg CO2
wood/biomass 16400000 J/kg 1.833333333 kg
8945455 J/kg CO2
Current usage...
5% energy non-carbon fuels (nukes, hydro, wind, solar)
total carbon-fuel energy... 1.425E+13 J/s
coal = 0.5 7.125E+12 J/s 796493.9024 kg(CO2)/s
2.51354E+13 kg(CO2)/annum
oil = 0.4 5.7E+12 J/s 392156.8627 kg(CO2)/s
1.23755E+13 kg(CO2)/annum
biomass = 0.1 1.425E+12 J/s 159298.7805 kg(CO2)/s
5.02709E+12 kg(CO2)/annum
total (tons) = 42538052582 (42.5 billion tons)
(at present only about 3 billion tons/annum stays in the air - the rest is taken up by the oceans and plant-life. However we may hit the limit of that natural regulation before too long - last year's uptake was lower than previous years, causing fears that the limit has finally been reached)
And how much is in the air already?
CO2 mass = 380 ppm = 3.92654 kg/m^2,
thus total = 2.00056E+15 kg (2 trillion
tons) = 47.03 years @ current production levels
At 5% of the atmosphere CO2 is toxic for many people. That's 6,192 years of
current burning. But how much is there in the ground?
Fossil fuels:
commercial coal/oil: 9000 TWyr = 8.2 trillion tons of carbon
(mostly coal - oil is rarer)
all sources : 18,000 TWyr = 16.4 trillion tons carbon
- much is inaccessible commercially by current methods
Hence, at most, CO2 can only rise to ~ 33 times the present value. That's 1.25% so there is no danger of toxicity, but plenty of trouble from the greenhouse effect if we burn it all.
time to depletion:
@ current levels: 600 years & 1200 years respectively
@ 2.6% growth: 246 years & 273 years...
...exponential growth again. Theoretically coal can be used to form hydrogen
and this can be used in fuel cells @ 50-70% efficiency (x 2 current levels),
which buys just an extra 27 years in both cases. That's the doubling time at just 2.6% growth.
Conclusion: in just 300 years all fossil fuels will be gone. CO2 will be ~
12,500 ppm, unless we sequester it as carbonates and/or liquefied CO2 under
high-pressure on the ocean floors. Alternatively if someone can find a way of making stable solid carbon dioxide (called "carba" - analogous to solid SiO2, silica) then all problems will be solved.
Another handy unit of energy is the "megaton of TNT" which is exactly 10^15
gram-calories, a gram calorie being the energy needed to raise the
temperature of a gram of water by one degree Celsius. According to
relativity one kilogram of rest mass is equivalent to m.c^2 in energy, and
since c is 299,792,458 m/s that's almost 90,000 trillion joules per
kilogram, or exactly 21.47 megatons of TNT.
Some catastrophes for comparison...
Boiling an ocean takes ~ 2.7 megajoules/kg and there's 1.4E+21 kg of ocean.
That's 3.78 E+27 Joules = 900 billion megatons.
To melt the Earth's average 17 km of crust (= 17,000*3,000 kg/m^2 = 51E+7
kg/m^2) means raising the rock by ~ 1100 degrees @ 500 J/kg. That's 550,000
J* 51 million kg per sq. metre = 1.43 E+28 J = 3.4 trillion megatons TNT.
Finally to disrupt the Earth utterly requires sending its entire mass past
its own escape velocity. For a uniform density sphere that's E =
3/5.G.M^2/R, or 2.24E+32 J for the Earth - just a bit more than x1000 its
rotational energy. It's roughly 50,000 TRILLION megatons TNT, or 1 trillion
tons of antimatter explosive placed at the Earth's core... quite a trick.
In Greg Bear's "The Forge of God" some aliens drop two chunks of neutronium on
opposite orbits inside the Earth - one matter, the other antimatter.
Neutronium is 100 billion times denser than water and flies through solid
rock like it was rarefied air. Eventually the two chunks meet and detonate
the Earth.
Some advocate detonating Venus because it has such a rotten atmosphere, but the energy disparity...
removing the atmosphere @ 10^6 kg/sq.m, needs 54 MJ/kg, 4.6 x 10^14 sq.m... total = 2.5 x 10^28 J
disrupting Venus ~ M^2/R = (0.815)^2/0.95 (in Earth units) = 0.7 Earth disruption energy = 1.57 x 10^32 J
Ratio: disruption/removal = 6300 times...
thus removal is far, far cheaper energy-wise. If we import hydrogen from Uranus to make carbon and water out of the CO2 and the energy needed is even lower. By the end of that process Venus will have a 4 bar nitrogen/helium atmosphere and about a kilometre equivalent of water in its lowlands, most of which will soak away into the parched soil. Shallow salty seas and the high pressure will mean water vapour will be diluted and unlikely to escape to space like it did when Venus first dried up. The highlands should be pleasant.
Earth's orbital energy around the Sun is 2.65 E+33 J, or some ~ 12 times
higher than its disruption energy. Stopped in its tracks, say by a counter-rotating planet, and the Earth would fly apart as a cloud of hot gas at 30 km/s. This isn't so crazy
a scenario. According to Michael Woolfson, a planetary physics professor in
the UK, Earth and Venus were once cores of two gas planets that collided.
The resulting blast was hot enough (> 5,000,000 K) to cause nuclear
reactions in the planets' gases and form odd isotopes of otherwise common
elements. The collisional debris heavily eroded one of the planets' moons
almost stripping it to its iron core - that moon is now Mercury. Another
moon became Mars, while another Mars-sized moon smashed into Earth to form
the Moon - it's iron core sank into the Earth to join our core.
-pmb
From http://crowlspace.blogdrive.com/archive/76.html.
Energy - Earth, Moon and Man
It's a worthwhile pursuit to
ponder the various energies of Earth, Moon and Sun.
Earth mass 5.9737E+24
Moon mass 7.34767E+22
mass sum 6.04718E+24
Gravity constant 6.67259E-11
Earth-Moon radius 384400000
angular velocity 2.66531E-06
dr/dt = 1.21048E-09
Earth/Moon system:
L12 momentum 2.85862E+34 kg.m^2.s^-1
E12 energy 3.80955E+28 J
dL/dt = 4.50092E+16
dE/dt 1.19964E+11
Earth alone:
Earth radius = 6378137 m
Earth I = 8.01889E+37 kg.m^2
Earth w = 7.29212E-05 rad/s
Le = 5.84747E+33 kg.m^2.s^-1
Ee = 2.13202E+29 J
(notice more energy is packed into Earth's rotation on its own
axis than in the Earth-Moon mutual motion)
dw/dt = (dL/dt)/Ie
= -5.6129E-22 rad/s
= 0.002092976s/day/Century
dEe/dt = 3.28213E+12 W
The above is the energy calculations for the Earth-Moon system. Earth slows
by ~ 0.002 seconds a day every Century, and this releases 3.3 terawatts as
frictional energy in the oceans.
What if humans got all their energy from the tides?
Energy consumption = 10000 W/person (in USA or Australia)
population = 6300000000
total 63 terawatt.year => compare current (= 15 TW.yr)
terawatt-year = 3.15576E+19 Joules
natural depletion time = 4.73364E+20 seconds
= 2058410434 years (2 billion years)
(actually it's an asymptotic approach to zero. Thus in about ~ 15 billion years the Earth/Moon will be synchronous with ~ 1/2000 th of its current energy)
@ 63 TW.yr, t = 107237508.4 (100 million years)
@ 2.6% growth, t = 570.803779 years
Amazing what exponential growth can do.
Of course only a tiny fraction could ever be used per year, but with enough tide machines we might tap ~ 100 TW.yr/yr, slowing Earth down in a mere ~ 66 million years. The synchronous radius is at 553757 km, and the length of day/month will be the synchronous
period 47.176 days.
Famed early 20th Century biologist J.B.S.Haldane once predicted fictionally that the desire for central heating would cause the tides to be excessively tapped and the Moon to spiral out then reverse and break-up into a ring, bombarding the Earth - in about 3,000,000 AD.
So what does our mere present day 15 TWyr mean in terms of barrels of oil?
oil barrel energy = 6100000000 Joules
million barrels/annum = 77,600.65574 (call it 80 billion barrels)
daily = 212.4590164
As for carbon dioxide produced here's some relative figures...
coal 32800000 J/kg 3.666666667 kg (CO2 per kg fuel burnt)
8945455 J/kg CO2
oil 44880000 J/kg 3.087719298 kg
14535000 J/kg CO2
wood/biomass 16400000 J/kg 1.833333333 kg
8945455 J/kg CO2
Current usage...
5% energy non-carbon fuels (nukes, hydro, wind, solar)
total carbon-fuel energy... 1.425E+13 J/s
coal = 0.5 7.125E+12 J/s 796493.9024 kg(CO2)/s
2.51354E+13 kg(CO2)/annum
oil = 0.4 5.7E+12 J/s 392156.8627 kg(CO2)/s
1.23755E+13 kg(CO2)/annum
biomass = 0.1 1.425E+12 J/s 159298.7805 kg(CO2)/s
5.02709E+12 kg(CO2)/annum
total (tons) = 42538052582 (42.5 billion tons)
(at present only about 3 billion tons/annum stays in the air - the rest is taken up by the oceans and plant-life. However we may hit the limit of that natural regulation before too long - last year's uptake was lower than previous years, causing fears that the limit has finally been reached)
And how much is in the air already?
CO2 mass = 380 ppm = 3.92654 kg/m^2,
thus total = 2.00056E+15 kg (2 trillion
tons) = 47.03 years @ current production levels
At 5% of the atmosphere CO2 is toxic for many people. That's 6,192 years of
current burning. But how much is there in the ground?
Fossil fuels:
commercial coal/oil: 9000 TWyr = 8.2 trillion tons of carbon
(mostly coal - oil is rarer)
all sources : 18,000 TWyr = 16.4 trillion tons carbon
- much is inaccessible commercially by current methods
Hence, at most, CO2 can only rise to ~ 33 times the present value. That's 1.25% so there is no danger of toxicity, but plenty of trouble from the greenhouse effect if we burn it all.
time to depletion:
@ current levels: 600 years & 1200 years respectively
@ 2.6% growth: 246 years & 273 years...
...exponential growth again. Theoretically coal can be used to form hydrogen
and this can be used in fuel cells @ 50-70% efficiency (x 2 current levels),
which buys just an extra 27 years in both cases. That's the doubling time at just 2.6% growth.
Conclusion: in just 300 years all fossil fuels will be gone. CO2 will be ~
12,500 ppm, unless we sequester it as carbonates and/or liquefied CO2 under
high-pressure on the ocean floors. Alternatively if someone can find a way of making stable solid carbon dioxide (called "carba" - analogous to solid SiO2, silica) then all problems will be solved.
Another handy unit of energy is the "megaton of TNT" which is exactly 10^15
gram-calories, a gram calorie being the energy needed to raise the
temperature of a gram of water by one degree Celsius. According to
relativity one kilogram of rest mass is equivalent to m.c^2 in energy, and
since c is 299,792,458 m/s that's almost 90,000 trillion joules per
kilogram, or exactly 21.47 megatons of TNT.
Some catastrophes for comparison...
Boiling an ocean takes ~ 2.7 megajoules/kg and there's 1.4E+21 kg of ocean.
That's 3.78 E+27 Joules = 900 billion megatons.
To melt the Earth's average 17 km of crust (= 17,000*3,000 kg/m^2 = 51E+7
kg/m^2) means raising the rock by ~ 1100 degrees @ 500 J/kg. That's 550,000
J* 51 million kg per sq. metre = 1.43 E+28 J = 3.4 trillion megatons TNT.
Finally to disrupt the Earth utterly requires sending its entire mass past
its own escape velocity. For a uniform density sphere that's E =
3/5.G.M^2/R, or 2.24E+32 J for the Earth - just a bit more than x1000 its
rotational energy. It's roughly 50,000 TRILLION megatons TNT, or 1 trillion
tons of antimatter explosive placed at the Earth's core... quite a trick.
In Greg Bear's "The Forge of God" some aliens drop two chunks of neutronium on
opposite orbits inside the Earth - one matter, the other antimatter.
Neutronium is 100 billion times denser than water and flies through solid
rock like it was rarefied air. Eventually the two chunks meet and detonate
the Earth.
Some advocate detonating Venus because it has such a rotten atmosphere, but the energy disparity...
removing the atmosphere @ 10^6 kg/sq.m, needs 54 MJ/kg, 4.6 x 10^14 sq.m... total = 2.5 x 10^28 J
disrupting Venus ~ M^2/R = (0.815)^2/0.95 (in Earth units) = 0.7 Earth disruption energy = 1.57 x 10^32 J
Ratio: disruption/removal = 6300 times...
thus removal is far, far cheaper energy-wise. If we import hydrogen from Uranus to make carbon and water out of the CO2 and the energy needed is even lower. By the end of that process Venus will have a 4 bar nitrogen/helium atmosphere and about a kilometre equivalent of water in its lowlands, most of which will soak away into the parched soil. Shallow salty seas and the high pressure will mean water vapour will be diluted and unlikely to escape to space like it did when Venus first dried up. The highlands should be pleasant.
Earth's orbital energy around the Sun is 2.65 E+33 J, or some ~ 12 times
higher than its disruption energy. Stopped in its tracks, say by a counter-rotating planet, and the Earth would fly apart as a cloud of hot gas at 30 km/s. This isn't so crazy
a scenario. According to Michael Woolfson, a planetary physics professor in
the UK, Earth and Venus were once cores of two gas planets that collided.
The resulting blast was hot enough (> 5,000,000 K) to cause nuclear
reactions in the planets' gases and form odd isotopes of otherwise common
elements. The collisional debris heavily eroded one of the planets' moons
almost stripping it to its iron core - that moon is now Mercury. Another
moon became Mars, while another Mars-sized moon smashed into Earth to form
the Moon - it's iron core sank into the Earth to join our core.