How did we figure out the Earth's mass?

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How did we figure out the Earth's mass?

Postby liveboy21 » Thu Aug 01, 2013 2:27 am UTC

The Earth is massive. How did we figure out its mass? How did we figure out that it has a core? Is in any way related to how we calculate mass for less than human sized objects?

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Re: How did we figure out the Earth's mass?

Postby brenok » Thu Aug 01, 2013 2:39 am UTC

I'm not sure if that's how it was first calculated, but you can do it by using Newton's Law of Gravitation:

F = G M m / R2

Where F is the weight of an object, M is the mass of Earth, m is the mass of the object and G is the gravitational constant.

To find the G you can use the Cavendish Experiment

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Re: How did we figure out the Earth's mass?

Postby Sizik » Thu Aug 01, 2013 2:43 am UTC

The entire purpose of the Cavendish experiment was to essentially calculate the mass of the Earth.
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Re: How did we figure out the Earth's mass?

Postby Deva » Thu Aug 01, 2013 2:58 am UTC

Earth's Core:
Source wrote:In 1896 when Emil Wiechert proposed his model of the Earth with an iron core and stony shell, scientists generally believed that the entire earth was a solid as rigid as steel. R. D. Oldham's identification of P and S waves in seismological records allowed him to detect a discontinuity corresponding to a boundary between core and shell (mantle) in 1906, and Beno Gutenberg established the depth of this boundary as 2900 km. But failure to detect propagation of S waves through the core was not sufficient evidence to persuade seismologists that it is fluid (contrary to modern textbook statements). Not until 1926 did Harold Jeffreys refute the arguments for solidity and establish that the core is liquid. In 1936 Inge Lehmann discovered the small inner core. K. E. Bullen argued, on the basis of plausible assumptions about compressibility and density, that the inner core is solid. Attempts to find seismic signals that have passed through the inner core as S waves have so far failed (with one possible exception), but analysis of free oscillations provided fairly convincing evidence for its solidity.

See Wikipedia for inner versus outer core. Mentions other facts too, such as pressure and temperature.
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Re: How did we figure out the Earth's mass?

Postby LaserGuy » Thu Aug 01, 2013 5:44 am UTC

liveboy21 wrote:Is in any way related to how we calculate mass for less than human sized objects?


Not at all. When we measure the mass of something here on Earth, what we're usually actually measuring is its weight. This is given by W = mg, where g is the strength of the local acceleration due to gravity. g can be measured easily enough, so we can get the mass from that. But if you're trying to measure the weight of something while you are, say, on a roller coaster, you'll find that your scale will give dramatically different readings depending on the direction of your current acceleration. Likewise, when you're on the Moon, your scale will read only a sixth as much as it does here on Earth. But your mass hasn't changed, of course--it's just that the calibration on the scale is set to work for Earth's gravity field. There are certain types of mass measurements we can do that are independent of the local gravity, of course, but such methods are less commonly used. This doesn't work for measuring the mass of the Earth though. You need to use the methods noted above.

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Re: How did we figure out the Earth's mass?

Postby Goemon » Thu Aug 01, 2013 5:56 am UTC

Kepler's third law is also used to measure the mass of planets and stars, though you have to have a value for G to use it:

M = (4pi R^3) / GT^2

Where R is the distance from the Earth to anything orbiting it (such as the moon or a satellite) and T is the time it takes the object to complete one orbit.
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Re: How did we figure out the Earth's mass?

Postby Diadem » Thu Aug 01, 2013 10:46 am UTC

Once you know G the measuring the mass of the earth becomes trivial. The hard part is figuring out G. As far as I know, there is no way to figure out the mass of the earth independent from G.
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Re: How did we figure out the Earth's mass?

Postby yurell » Thu Aug 01, 2013 11:10 am UTC

Most of the time when talking about celestial objects, we give GM, rather than M by itself, since that's the thing that we can infer 'directly' from the results of our measurements. The uncertainty on G is actually quite large, at E-4, which is equivalent of about two hundred Earths in the sun's mass.
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Re: How did we figure out the Earth's mass?

Postby skeptical scientist » Thu Aug 01, 2013 2:25 pm UTC

Diadem wrote:Once you know G the measuring the mass of the earth becomes trivial. The hard part is figuring out G. As far as I know, there is no way to figure out the mass of the earth independent from G.

We know the volume of the Earth and the approximate composition, which is enough to compute the mass. DId we use the mass to figure out the composition, or can that be done separately?
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Re: How did we figure out the Earth's mass?

Postby Copper Bezel » Thu Aug 01, 2013 2:49 pm UTC

Just from Wikipedia, Earth's mass came first experimentally, in the Cavendish Experiment mentioned earlier, but apparently for the purpose of determining the Earth's density. (The experimental setup would have determined the mass directly, since it was based on comparing the gravitational force of a weight on a small object against the force of the Earth's gravity on the same, so there weren't any volumes involved and the density figure had to have been calculated from there.) G was later calculated from the mass, and Wiechert's much later model of the Earth's structure that Deva cited was based on the density.
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Re: How did we figure out the Earth's mass?

Postby davidstarlingm » Thu Aug 01, 2013 3:09 pm UTC

So if we suddenly found ourselves on a strange world orbiting a different-sized star with a different day-length and a different year-length, and we were unable to immediately recall G, would we need to re-do the Cavendish experiment, or would there be another means of determining the mass of the planet we were on?

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Re: How did we figure out the Earth's mass?

Postby skeptical scientist » Thu Aug 01, 2013 3:23 pm UTC

Copper Bezel wrote:Just from Wikipedia, Earth's mass came first experimentally, in the Cavendish Experiment mentioned earlier, but apparently for the purpose of determining the Earth's density. (The experimental setup would have determined the mass directly, since it was based on comparing the gravitational force of a weight on a small object against the force of the Earth's gravity on the same, so there weren't any volumes involved and the density figure had to have been calculated from there.) G was later calculated from the mass, and Wiechert's much later model of the Earth's structure that Deva cited was based on the density.

Wikipedia wrote:In Cavendish's time, physicists used the same units for mass and weight, in effect taking g as a standard acceleration. Then, since Rearth was known, ρearth played the role of an inverse gravitational constant. ... For these reasons, physicists generally do credit Cavendish with the first measurement of the gravitational constant.
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Re: How did we figure out the Earth's mass?

Postby LaserGuy » Thu Aug 01, 2013 5:08 pm UTC

I think you can estimate the density somewhat accurately using earthquake reflections. Pretty much the same way they figured out the core composition.

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Re: How did we figure out the Earth's mass?

Postby Copper Bezel » Thu Aug 01, 2013 5:33 pm UTC

@ skeptical scientist,

Thank you for correcting my reading comprehension. I didn't understand that density was actually considered to be a preferable way of expressing the relationship, as opposed to using the mass, and I was reading back in modern assumptions about which constants and properties are the fundamental ones and couldn't understand why they'd add an extra step like that. I was also thinking in terms of the law of universal gravitation as it's normally stated and assuming that it was always expressed in masses, which is apparently not the case.

Some random translation of the Principia wrote:If to the several points of a sphere there tend equal centripetal forces decreasing in a duplicate ratio of the distances from those points ; and there be given both the density of the sphere and the ratio of the di-ameter of the sphere to the distance of the corpuscle from its centre ; I say, that the force with which the corpuscle is attracted is proportional to the semi-diameter of the sphere.


I like this - distance is a function of radius, so there's an r3 going one way and an r2 going the other, and gravitational force increases linearly with radius (changing distance, of course) just as it does with the density. I get it now. = )
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Re: How did we figure out the Earth's mass?

Postby LaserGuy » Thu Aug 01, 2013 6:32 pm UTC

davidstarlingm wrote:So if we suddenly found ourselves on a strange world orbiting a different-sized star with a different day-length and a different year-length, and we were unable to immediately recall G, would we need to re-do the Cavendish experiment, or would there be another means of determining the mass of the planet we were on?


Depends how accurately you need it, I guess. Using the density*volume method, you can get the right order of magnitude for the mass pretty easily.
Radius of the Earth is 6.4 x10^6 m, so volume is 1.1 x10^21 m^3.
If we assume the Earth is made of granite, which has a density of 2700 kg/m^3, we get the mass is 3 x10^24, which is within a factor of 2 of the correct value. If you guessed that the Earth was, say, 50% iron and 50% granite by volume, you'd end up with 5.8 x10^24, which is within a few percent of the right answer. This would work much better for a rocky planet, since it's mostly made of incompressible material and rocks have pretty similar densities. For a star or a gas giant, you could easily be off by a couple orders of magnitude unless you had a pretty good idea of what the interior density profile looked like.

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Re: How did we figure out the Earth's mass?

Postby davidstarlingm » Thu Aug 01, 2013 6:57 pm UTC

LaserGuy wrote:
davidstarlingm wrote:So if we suddenly found ourselves on a strange world orbiting a different-sized star with a different day-length and a different year-length, and we were unable to immediately recall G, would we need to re-do the Cavendish experiment, or would there be another means of determining the mass of the planet we were on?


Depends how accurately you need it, I guess. Using the density*volume method, you can get the right order of magnitude for the mass pretty easily.
Radius of the Earth is 6.4 x10^6 m, so volume is 1.1 x10^21 m^3.
If we assume the Earth is made of granite, which has a density of 2700 kg/m^3, we get the mass is 3 x10^24, which is within a factor of 2 of the correct value. If you guessed that the Earth was, say, 50% iron and 50% granite by volume, you'd end up with 5.8 x10^24, which is within a few percent of the right answer. This would work much better for a rocky planet, since it's mostly made of incompressible material and rocks have pretty similar densities. For a star or a gas giant, you could easily be off by a couple orders of magnitude unless you had a pretty good idea of what the interior density profile looked like.

So if you wake up in the morning on a strange planet with similar (though perhaps slightly higher or lower) gravity, a completely different-looking sky, and a nondescript landscape, you'd have to re-do the work of Eratosthenes to get the circumference and radius, then make a ballpark assumption of density to estimate mass?

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Re: How did we figure out the Earth's mass?

Postby Copper Bezel » Thu Aug 01, 2013 7:42 pm UTC

Well, that's still assuming that you've forgotten G, since if you haven't, you guess the composition and drop a kitten from something tall. (Although either way, it still depends on other available references.)
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Re: How did we figure out the Earth's mass?

Postby Goemon » Fri Aug 02, 2013 3:39 am UTC

davidstarlingm wrote:So if you wake up in the morning on a strange planet with similar (though perhaps slightly higher or lower) gravity, a completely different-looking sky, and a nondescript landscape, you'd have to re-do the work of Eratosthenes to get the circumference and radius, then make a ballpark assumption of density to estimate mass?


Well, if you're just going to assume that the composition is basically rock in order to estimate the mass anyway, (which works pretty well for the Earth by LaserGuy's calculations) then you can just weigh yourself to get an estimate of the planet's mass. Surface gravity of body with uniform density is proportional to radius; so if you weigh 10% more on the strange world, the radius should be about 10% greater, so the mass is 1.10^3 times greater than Earth.

F = mMG / r^2

g = (acceleration at surface) = F/m = MG / R^2

M = (density) * R^3 (for uniform density, such as mostly rocky planets) => g = (density) * GR
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Re: How did we figure out the Earth's mass?

Postby skeptical scientist » Fri Aug 02, 2013 3:52 am UTC

Eratosthenes experiment is pretty trivial to do. If you're on a new planet, exploring and mapping your new environs is going to be a pretty high priority. Once you've mapped a few hundred miles and computed latitudes, you're done.
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Re: How did we figure out the Earth's mass?

Postby davidstarlingm » Fri Aug 02, 2013 2:09 pm UTC

skeptical scientist wrote:Eratosthenes experiment is pretty trivial to do. If you're on a new planet, exploring and mapping your new environs is going to be a pretty high priority. Once you've mapped a few hundred miles and computed latitudes, you're done.

I wonder what the most unexpected possible sight would be if you were expecting a fairly Earthlike planet.

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Re: How did we figure out the Earth's mass?

Postby mfb » Fri Aug 02, 2013 5:02 pm UTC

davidstarlingm wrote:
skeptical scientist wrote:Eratosthenes experiment is pretty trivial to do. If you're on a new planet, exploring and mapping your new environs is going to be a pretty high priority. Once you've mapped a few hundred miles and computed latitudes, you're done.

I wonder what the most unexpected possible sight would be if you were expecting a fairly Earthlike planet.
The sight of skeptical scientist running around and making maps of hundreds of miles is certainly close to that.

Guessing the density and measuring the local g-value is easier.

yurell wrote:The uncertainty on G is actually quite large, at E-4, which is equivalent of about two hundred Earths in the sun's mass.
Thanks. I knew those values, but I never combined them in that way.
We should send a G experiment to space.

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Re: How did we figure out the Earth's mass?

Postby Copper Bezel » Fri Aug 02, 2013 5:52 pm UTC

Because it would help, or because things are cooler in space? What kind of experimental setup do you have in mind?
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Re: How did we figure out the Earth's mass?

Postby zenten » Sat Aug 03, 2013 3:55 am UTC

davidstarlingm wrote:
skeptical scientist wrote:Eratosthenes experiment is pretty trivial to do. If you're on a new planet, exploring and mapping your new environs is going to be a pretty high priority. Once you've mapped a few hundred miles and computed latitudes, you're done.

I wonder what the most unexpected possible sight would be if you were expecting a fairly Earthlike planetary .


A shape that is nothing like a sphere.

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Re: How did we figure out the Earth's mass?

Postby davidstarlingm » Sat Aug 03, 2013 4:17 am UTC

zenten wrote:
davidstarlingm wrote:I wonder what the most unexpected possible sight would be if you were expecting a fairly Earthlike planetary .


A shape that is nothing like a sphere.

Walking, walking....hmm, this is a steep hill....HOLY CRAP WHERE DID THE PLANET GO!?

Though weather patterns wouldn't be very stable, I don't think.

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Re: How did we figure out the Earth's mass?

Postby Dopefish » Sat Aug 03, 2013 4:46 am UTC

I want a klein bottle planet. It would be a thing.

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Re: How did we figure out the Earth's mass?

Postby brenok » Sat Aug 03, 2013 5:02 am UTC

A torus would suffice to me. Mainly, because it is possible with just three spatial dimensions.

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Re: How did we figure out the Earth's mass?

Postby addams » Sat Aug 03, 2013 11:08 pm UTC

davidstarlingm wrote:
LaserGuy wrote:
davidstarlingm wrote:So if we suddenly found ourselves on a strange world orbiting a different-sized star with a different day-length and a different year-length, and we were unable to immediately recall G, would we need to re-do the Cavendish experiment, or would there be another means of determining the mass of the planet we were on?


Depends how accurately you need it, I guess. Using the density*volume method, you can get the right order of magnitude for the mass pretty easily.
Radius of the Earth is 6.4 x10^6 m, so volume is 1.1 x10^21 m^3.
If we assume the Earth is made of granite, which has a density of 2700 kg/m^3, we get the mass is 3 x10^24, which is within a factor of 2 of the correct value. If you guessed that the Earth was, say, 50% iron and 50% granite by volume, you'd end up with 5.8 x10^24, which is within a few percent of the right answer. This would work much better for a rocky planet, since it's mostly made of incompressible material and rocks have pretty similar densities. For a star or a gas giant, you could easily be off by a couple orders of magnitude unless you had a pretty good idea of what the interior density profile looked like.

So if you wake up in the morning on a strange planet with similar (though perhaps slightly higher or lower) gravity, a completely different-looking sky, and a nondescript landscape, you'd have to re-do the work of Eratosthenes to get the circumference and radius, then make a ballpark assumption of density to estimate mass?


Sorry to interrupt, but
if you woke up on a different planet with different gravity and different looking sky;
The Mass of the whole might not be the first thing you may be needing math for.

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Re: How did we figure out the Earth's mass?

Postby gmalivuk » Sun Aug 04, 2013 6:10 am UTC

davidstarlingm wrote:
zenten wrote:A shape that is nothing like a sphere.

Walking, walking....hmm, this is a steep hill....HOLY CRAP WHERE DID THE PLANET GO!?
What exactly is "nothing like" a sphere?

In any case, anything that significantly deviates from having its whole surface at the same potential would be rather interesting to move around on. A cube, for example, would have clumps of atmosphere around the middle of each side, with edges and corners that seemed miles and miles high (though never particularly steep). And then when you reach the edge, you'd have just climbed up a 45 degree slope and you'd see the same slope going back down in front of you.

I can't begin to imagine what weather patterns would look like, though.
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Re: How did we figure out the Earth's mass?

Postby addams » Sun Aug 04, 2013 6:51 am UTC

gmalivuk wrote:
davidstarlingm wrote:
zenten wrote:A shape that is nothing like a sphere.

Walking, walking....hmm, this is a steep hill....HOLY CRAP WHERE DID THE PLANET GO!?
What exactly is "nothing like" a sphere?

In any case, anything that significantly deviates from having its whole surface at the same potential would be rather interesting to move around on. A cube, for example, would have clumps of atmosphere around the middle of each side, with edges and corners that seemed miles and miles high (though never particularly steep). And then when you reach the edge, you'd have just climbed up a 45 degree slope and you'd see the same slope going back down in front of you.

I can't begin to imagine what weather patterns would look like, though.

Really? A cube?
I don't know much about math.

But; Cubes are not stable.
Correct?

Weather? You are imagining a cube large enough for Weather?
Is this thing theoretically static? pfft. I can't do it.

No such thing as a Cube Like that.
Well: The Borg.

Weather. I like weather.
Cube Weather.

If in your made up world there was an atmosphere like Earths.
And; If it were Thick enough to remain Stable.

Then; It would be as you described.
The Air would be Thick, Thick Thick over the Flat Planes and Thin over the edges.

How are you going to keep it from spinning.
It could spin for a while. Then what?
Tunnels could be the Teams first project.

Great! Are there populations on the Cube.
That might be fun.

You know the doom and glooms, would have to put in their Two cents.
If we put in tunnels the tunnels will make the edges delicate and Poof!

The edges break away and you get a roundish cube.
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