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rob
rob
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Here's a quick-and-dirty estimate. The gravitational self-energy of a uniform-density sphere is $$ U = \frac35 \frac{GM^2}R $$ Let's assume Theia had the same mass and density as Mars, and that Gaia contained the rest of the mass of the Earth-Moon system. The binding energies for the four bodies are then

theia/mars 4.82e+30 joules
gaia       1.90e+32 joules
earth      2.24e+32 joules
moon       1.24e+29 joules

You can see that Theia and the Moon contribute to the binding energy starting the third significant figure: moving the 90% of Theia's mass to Gaia, leaving us with the Earth and the Moon, must have released something like $0.3\times10^{32}\rm\,J$ of gravitational binding energy as heat.

Kinetic energies due to Earth's rotation and the Moon's orbit are irrelevant compared to Earth's binding energy --- the biggest contributor there is $0.25\times10^{30}\rm\,J$ associated with Earth's daily rotation. It's probably safe to assume the same about the progenitors.

Apparently a teraton of TNT is $4\times10^{21}\rm\,J$, if you insist on that comparison.

Note that I've assumed nothing about the geometry of the collision, whether it was head-on or glancing. I'm only making assumptions about the (well-known) final state and the (poorly constrained) initial state.

I don't have a better handy place to keep notes on how I did this calculation, so I'll stick it here. Feel free to remove; it will live on in the edit history.

from scipy import constants

_mass = {'earth': 5.9722e+24,
 'gaia': 5.40663266e+24,
 'mars': 6.390254e+23,
 'moon': 7.345806000000001e+22,
 'theia': 6.390254e+23}

_radius = {'earth': 6371000.0,
 'gaia': 6163183.690874354,
 'mars': 3389000.0,
 'moon': 1737000.0,
 'theia': 3389000.0}

_period = {'earth': constants.day, 
  'mars': constants.day + 39*constants.minute, 
  'moon': constants.day * 27.3}
_period['theia'] = _period['gaia'] = float('inf')

mass = lambda s: _mass[s]
radius = lambda s: _radius[s]
period = lambda s: _period[s]

volume = lambda s: 4*pi*radius(s)**3 / 3
density = lambda s: mass(s) / volume(s)
self_energy = lambda s: 3*constants.G * mass(s)**2 / radius(s) /5

omega = lambda period: 2*pi / period
moment = lambda s: 2./5 * mass(s) * radius(s)**2
k_rot = lambda s: moment(s) * omega(period(s))**2 / 2

for b in 'theia gaia earth moon'.split(): 
    print '    {:10s} {:0.3g} joules'.format(b, self_energy(b))

Here's a quick-and-dirty estimate. The gravitational self-energy of a uniform-density sphere is $$ U = \frac35 \frac{GM^2}R $$ Let's assume Theia had the same mass and density as Mars, and that Gaia contained the rest of the mass of the Earth-Moon system. The binding energies for the four bodies are then

theia/mars 4.82e+30 joules
gaia       1.90e+32 joules
earth      2.24e+32 joules
moon       1.24e+29 joules

You can see that Theia and the Moon contribute to the binding energy starting the third significant figure: moving the 90% of Theia's mass to Gaia, leaving us with the Earth and the Moon, must have released something like $0.3\times10^{32}\rm\,J$ of gravitational binding energy as heat.

Kinetic energies due to Earth's rotation and the Moon's orbit are irrelevant compared to Earth's binding energy --- the biggest contributor there is $0.25\times10^{30}\rm\,J$ associated with Earth's daily rotation. It's probably safe to assume the same about the progenitors.

Apparently a teraton of TNT is $4\times10^{21}\rm\,J$, if you insist on that comparison.

Note that I've assumed nothing about the geometry of the collision, whether it was head-on or glancing. I'm only making assumptions about the (well-known) final state and the (poorly constrained) initial state.

I don't have a better handy place to keep notes on how I did this calculation, so I'll stick it here. Feel free to remove; it will live on in the edit history.

from scipy import constants

_mass = {'earth': 5.9722e+24,
 'gaia': 5.40663266e+24,
 'mars': 6.390254e+23,
 'moon': 7.345806000000001e+22,
 'theia': 6.390254e+23}

_radius = {'earth': 6371000.0,
 'gaia': 6163183.690874354,
 'mars': 3389000.0,
 'moon': 1737000.0,
 'theia': 3389000.0}

_period = {'earth': constants.day, 
  'mars': constants.day + 39*constants.minute, 
  'moon': constants.day * 27.3}
_period['theia'] = _period['gaia'] = float('inf')

mass = lambda s: _mass[s]
radius = lambda s: _radius[s]
period = lambda s: _period[s]

volume = lambda s: 4*pi*radius(s)**3 / 3
density = lambda s: mass(s) / volume(s)
self_energy = lambda s: 3*constants.G * mass(s)**2 / radius(s) /5

omega = lambda period: 2*pi / period
moment = lambda s: 2./5 * mass(s) * radius(s)**2
k_rot = lambda s: moment(s) * omega(period(s))**2 / 2

for b in 'theia gaia earth moon'.split(): 
    print '    {:10s} {:0.3g} joules'.format(b, self_energy(b))

Here's a quick-and-dirty estimate. The gravitational self-energy of a uniform-density sphere is $$ U = \frac35 \frac{GM^2}R $$ Let's assume Theia had the same mass and density as Mars, and that Gaia contained the rest of the mass of the Earth-Moon system. The binding energies for the four bodies are then

theia/mars 4.82e+30 joules
gaia       1.90e+32 joules
earth      2.24e+32 joules
moon       1.24e+29 joules

You can see that Theia and the Moon contribute to the binding energy starting the third significant figure: moving the 90% of Theia's mass to Gaia, leaving us with the Earth and the Moon, must have released something like $0.3\times10^{32}\rm\,J$ of gravitational binding energy as heat.

Kinetic energies due to Earth's rotation and the Moon's orbit are irrelevant compared to Earth's binding energy --- the biggest contributor there is $0.25\times10^{30}\rm\,J$ associated with Earth's daily rotation. It's probably safe to assume the same about the progenitors.

Apparently a teraton of TNT is $4\times10^{21}\rm\,J$, if you insist on that comparison.

Note that I've assumed nothing about the geometry of the collision, whether it was head-on or glancing. I'm only making assumptions about the (well-known) final state and the (poorly constrained) initial state.

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Source Link
rob
rob
  • 451
  • 4
  • 13

Here's a quick-and-dirty estimate. The gravitational self-energy of a uniform-density sphere is $$ U = \frac35 \frac{GM^2}R $$ Let's assume Theia had the same mass and density as Mars, and that Gaia contained the rest of the mass of the Earth-Moon system. The binding energies for the four bodies are then

theia/mars 4.82e+30 joules
gaia       1.90e+32 joules
earth      2.24e+32 joules
moon       1.24e+29 joules

You can see that Theia and the Moon contribute to the binding energy starting the third significant figure: moving the 90% of Theia's mass to Gaia, leaving us with the Earth and the Moon, must have released something like $0.3\times10^{32}\rm\,J$ of gravitational binding energy as heat.

Kinetic energies due to Earth's rotation and the Moon's orbit are irrelevant compared to Earth's binding energy --- the biggest contributor there is $0.25\times10^{30}\rm\,J$ associated with Earth's daily rotation. It's probably safe to assume the same about the progenitors.

Apparently a teraton of TNT is $4\times10^{21}\rm\,J$, if you insist on that comparison.

Note that I've assumed nothing about the geometry of the collision, whether it was head-on or glancing. I'm only making assumptions about the (well-known) final state and the (poorly constrained) initial state.

I don't have a better handy place to keep notes on how I did this calculation, so I'll stick it here. Feel free to remove; it will live on in the edit history.

from scipy import constants

_mass = {'earth': 5.9722e+24,
 'gaia': 5.40663266e+24,
 'mars': 6.390254e+23,
 'moon': 7.345806000000001e+22,
 'theia': 6.390254e+23}

_radius = {'earth': 6371000.0,
 'gaia': 6163183.690874354,
 'mars': 3389000.0,
 'moon': 1737000.0,
 'theia': 3389000.0}

_period = {'earth': constants.day, 
  'mars': constants.day + 39*constants.minute, 
  'moon': constants.day * 27.3}
_period['theia'] = _period['gaia'] = float('inf')

mass = lambda s: _mass[s]
radius = lambda s: _radius[s]
period = lambda s: _period[s]

volume = lambda s: 4*pi*radius(s)**3 / 3
density = lambda s: mass(s) / volume(s)
self_energy = lambda s: 3*constants.G * mass(s)**2 / radius(s) /5

omega = lambda period: 2*pi / period
moment = lambda s: 2./5 * mass(s) * radius(s)**2
k_rot = lambda s: moment(s) * omega(period(s))**2 / 2

for b in 'theia gaia earth moon'.split(): 
    print '    {:10s} {:0.3g} joules'.format(b, self_energy(b))

Here's a quick-and-dirty estimate. The gravitational self-energy of a uniform-density sphere is $$ U = \frac35 \frac{GM^2}R $$ Let's assume Theia had the same mass and density as Mars, and that Gaia contained the rest of the mass of the Earth-Moon system. The binding energies for the four bodies are then

theia/mars 4.82e+30 joules
gaia       1.90e+32 joules
earth      2.24e+32 joules
moon       1.24e+29 joules

You can see that Theia and the Moon contribute to the binding energy starting the third significant figure: moving the 90% of Theia's mass to Gaia, leaving us with the Earth and the Moon, must have released something like $0.3\times10^{32}\rm\,J$ of gravitational binding energy as heat.

Kinetic energies due to Earth's rotation and the Moon's orbit are irrelevant compared to Earth's binding energy --- the biggest contributor there is $0.25\times10^{30}\rm\,J$ associated with Earth's daily rotation. It's probably safe to assume the same about the progenitors.

Apparently a teraton of TNT is $4\times10^{21}\rm\,J$, if you insist on that comparison.

Note that I've assumed nothing about the geometry of the collision, whether it was head-on or glancing. I'm only making assumptions about the (well-known) final state and the (poorly constrained) initial state.

Here's a quick-and-dirty estimate. The gravitational self-energy of a uniform-density sphere is $$ U = \frac35 \frac{GM^2}R $$ Let's assume Theia had the same mass and density as Mars, and that Gaia contained the rest of the mass of the Earth-Moon system. The binding energies for the four bodies are then

theia/mars 4.82e+30 joules
gaia       1.90e+32 joules
earth      2.24e+32 joules
moon       1.24e+29 joules

You can see that Theia and the Moon contribute to the binding energy starting the third significant figure: moving the 90% of Theia's mass to Gaia, leaving us with the Earth and the Moon, must have released something like $0.3\times10^{32}\rm\,J$ of gravitational binding energy as heat.

Kinetic energies due to Earth's rotation and the Moon's orbit are irrelevant compared to Earth's binding energy --- the biggest contributor there is $0.25\times10^{30}\rm\,J$ associated with Earth's daily rotation. It's probably safe to assume the same about the progenitors.

Apparently a teraton of TNT is $4\times10^{21}\rm\,J$, if you insist on that comparison.

Note that I've assumed nothing about the geometry of the collision, whether it was head-on or glancing. I'm only making assumptions about the (well-known) final state and the (poorly constrained) initial state.

I don't have a better handy place to keep notes on how I did this calculation, so I'll stick it here. Feel free to remove; it will live on in the edit history.

from scipy import constants

_mass = {'earth': 5.9722e+24,
 'gaia': 5.40663266e+24,
 'mars': 6.390254e+23,
 'moon': 7.345806000000001e+22,
 'theia': 6.390254e+23}

_radius = {'earth': 6371000.0,
 'gaia': 6163183.690874354,
 'mars': 3389000.0,
 'moon': 1737000.0,
 'theia': 3389000.0}

_period = {'earth': constants.day, 
  'mars': constants.day + 39*constants.minute, 
  'moon': constants.day * 27.3}
_period['theia'] = _period['gaia'] = float('inf')

mass = lambda s: _mass[s]
radius = lambda s: _radius[s]
period = lambda s: _period[s]

volume = lambda s: 4*pi*radius(s)**3 / 3
density = lambda s: mass(s) / volume(s)
self_energy = lambda s: 3*constants.G * mass(s)**2 / radius(s) /5

omega = lambda period: 2*pi / period
moment = lambda s: 2./5 * mass(s) * radius(s)**2
k_rot = lambda s: moment(s) * omega(period(s))**2 / 2

for b in 'theia gaia earth moon'.split(): 
    print '    {:10s} {:0.3g} joules'.format(b, self_energy(b))
Source Link
rob
rob
  • 451
  • 4
  • 13

Here's a quick-and-dirty estimate. The gravitational self-energy of a uniform-density sphere is $$ U = \frac35 \frac{GM^2}R $$ Let's assume Theia had the same mass and density as Mars, and that Gaia contained the rest of the mass of the Earth-Moon system. The binding energies for the four bodies are then

theia/mars 4.82e+30 joules
gaia       1.90e+32 joules
earth      2.24e+32 joules
moon       1.24e+29 joules

You can see that Theia and the Moon contribute to the binding energy starting the third significant figure: moving the 90% of Theia's mass to Gaia, leaving us with the Earth and the Moon, must have released something like $0.3\times10^{32}\rm\,J$ of gravitational binding energy as heat.

Kinetic energies due to Earth's rotation and the Moon's orbit are irrelevant compared to Earth's binding energy --- the biggest contributor there is $0.25\times10^{30}\rm\,J$ associated with Earth's daily rotation. It's probably safe to assume the same about the progenitors.

Apparently a teraton of TNT is $4\times10^{21}\rm\,J$, if you insist on that comparison.

Note that I've assumed nothing about the geometry of the collision, whether it was head-on or glancing. I'm only making assumptions about the (well-known) final state and the (poorly constrained) initial state.