# The mass of Earth.

1. Nov 26, 2009

### God's Pen

Hello
I want to know the ideas and the formulas used to calculate the mass of Earth.
thanks.

2. Nov 26, 2009

### mgb_phys

The speed that an object orbits a much larger object only depends on the mass of the larger object.
So if you have something orbiting the Earth, like the moon, and know its distance and how long it takes to go around - you can find the mass of the earth.

A physicist called Cavendish measured this in about 1800.

3. Nov 26, 2009

### Bob S

Here is a quick way. The volume is (4/3) pi R3. Plug in the radius of the Earth, and the density (~5.515 grams/cc), and you will get the correct answer within a factor of two. Mgb-phys has a much more accurate way, but you need to know the separation distance.
Bob S

4. Nov 26, 2009

### mgb_phys

Knowing the density of the Earth is a bit of a cheat.
Interestingly thats what Cavendish was trying to work out when he measured G - they wanted to know if the Earth was hollow !

5. Nov 26, 2009

### D H

Staff Emeritus
Before I answer: Is this homework?

6. Nov 26, 2009

### Bob S

So is knowing the distance to a satellite, like Sputnik, the ISS, or the Moon.
Bob S

7. Nov 26, 2009

### D H

Staff Emeritus
The distance to a satellite or the Moon is a directly measurable quantity. The density of the Earth is not. It is inferred from the Earth's size and mass. Using density to derive mass is a circular derivation.

8. Nov 26, 2009

### ideasrule

The density of Earth might not have been well-known in Cavendish's day, but scientists could at least assume it had the same density as molten rock and get the approximate mass of the Earth from that. Now, in the 21st century, we have a decent understanding of Earth's density profile thanks to scientists' analysis of the refraction of seismic waves as they pass through the Earth.

9. Nov 26, 2009

### DaveC426913

BobS: note that, even in principle, the best you can get for such a heterogenous object as Earth is the average density.

And the only way to derive the average density of the Earth is to first determine the volume ... and the mass.

Sure, if you only need an answer within an order of magnitude or so. The Earth is quite heterogenous, and our tools for measurement are not very accurate. There's a lot of assumption in the method you propose.

10. Nov 26, 2009

### ideasrule

The mass of solid basalt is 3000 kg/m^3, so even if you take that and multiply it by Earth's volume, you'd be within a factor of 2 of the right answer. If you take into account all the latest sesmic data and computer models, you'd probably get within 10% of the actual mass. Of course that's less accurate than watching a satellite orbit and calculating Earth's mass from that, but it refutes DH's comment that the density of Earth is not directly measurable.

11. Nov 26, 2009

### DaveC426913

As you said yourself: you're relying on computer models. We make computer models to predict what's probably there when we can't directly measure it.

10% eh? So, you'd only be off by give-or-take one hundred thousand million million million kg or so...

Last edited: Nov 26, 2009
12. Nov 27, 2009

### mgb_phys

They tried that.
The equations for gravity have a GM term in them where M is the mass of the earth, so you can find M by measuring 'G' in Cavendish's famous experiment or you can find M from the density and use that to get G.
There was a long and careful attempt to measure the volume of a large conical mountain by a lot of surveying - and then measure it's mass by the effect it had on a small pendulum.
It didn't work very well because you don't know what shape the mountain is under the surface.

Cavendish got GM to within about 1%, assuming the Earth has the density as surface rocks gets you about half the correct value.

13. Nov 27, 2009

### ideasrule

And if you calculate the mass of the visible universe, you'd be off by at least a million million million million million million million million million million million million million kilograms with the most precise tools available. I'm not sure why you think the absolute value of the error is more important than the percentage error.

I feel that we're arguing only about semantics. Here's what I'm saying:

1) The density profile of the Earth can be inferred using the refraction of seismic waves.
2) This density profile can be used to arrive at a reasonable figure for Earth's mass.
3) Computer models can improve the accuracy of this result.
4) The most accurate calculations of Earth's mass involve measuring G and g at a known distance r from the Earth.

14. Nov 27, 2009

### D H

Staff Emeritus
Can it really? Or do siesmologists instead calibrate those inferred density profiles against the already known mass of the Earth?

The current limiting factor in the accuracy in the Earth's mass is the uncertainty in the measured value of G. Suppose seismologists could independently arrive at an estimate of the Earth's mass that is more accurate than the paltry 1 part in 104 to which we know G. That improved estimate of the Earth's mass we would directly lead to a better estimate for G because we know the product G*Me to an accuracy of 2 parts in 109.

15. Nov 27, 2009

### DaveC426913

Well played... 10% is 10%.

It is possible in principle. I'll grant you that. In practice, it is an exceedingly backwards and impractical way of doing it.

16. Nov 27, 2009

### D H

Staff Emeritus
This is the freshman physics version. The truth is a lot more complex. The Earth cannot be treated as a point mass for any distance r for which the acceleration due to gravity is observable.

17. Nov 27, 2009

### mgb_phys

Are tidal effects on the moons period significant - compared to the accuracy in 'G' ?

18. Nov 27, 2009

### D H

Staff Emeritus
To get that precision of 2 parts in 109 in GMe, yes, tidal effects are important, and in many ways. To assess the GMe to a precision of one part in 105 (an order of magnitude better than the precision in G), tidal effects are still important. They obviously are of reduced importance, and the subtle effects of tides can probably be ignored.

That two parts in a billion precision was based on five years of LAGEOS-1 data and was published in 2000 (the 2005 results increase the precision to one part in a billion). The kinds of errors that contributed to a few parts in a billion precision include:
• Uncertainties in the location in the Earth's center of mass. An uncertainty of 3 millimeters corresponds to about one part in a billion uncertainty at LAGEOS' altitude.
• Uncertainties in the Earth's and the Moon's tidal Love numbers. The Moon and creates tides in the Earth, which changes the Earth's gravity field, which in turn changes LAGEOS' orbit.
• Uncertainties in the oceanic tidal responses. The ocean tides also affect the orbits of satellites; the effect is about 1/10 that of the solid body tides.
• Uncertainties in the positions of the ground stations that measured the distance to the LAGEOS via laser retroreflector returns. A 2 mm error RMS corresponds to a part per billion error in GMe.
• Which means properly modeling the Earth tides are important.
• Even modeling the snow load in Siberia is important for this kind of accuracy.

19. Nov 28, 2009

### JANm

A Robinson Crusoe question, nice problem for a mathematician with low memory for numbers and high for principal formulae. We know gravitation at the surface pi^2=GM/r^2.
The deep pit method of measuring circumference of the earth gives the radius r=31900/5.
So indeed one wishes to know G. In the museum in Munchen I have seen a iron ball of radius 30 cm. This thing is used to measure G...
But I remember there must be an astronomical solution based on sun, earth and moon. I can't just remember which astronomical facts were known to Robinson Crusoe when he had all the time of the world to calculate its mass,
i'll be back,
greetings Janm

20. Nov 28, 2009

### mikelepore

Would we be able to treat the earth as a point mass if, instead of defining distance r to the geometric center, we defined r to the earth's center of mass?

If so, could we locate the earth's center of mass by the intersection of several plumb lines?