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## Gravitational mass

Posted by Chris on May 29, 2012 – 5:08 pm

Estimate the mass of the gravitational field of the Earth. What would the size of a cube of water with the same mass?

Use the following info (the real info is complex, but the following is close enough):
Relative density of inner core = 13, radius = 400 km
Relative density of outer core = 11, radius = 3400 km
Relative density of outer layers = 4.5, radius = 6400 km

This post is under “Fun Physics” and has 9 respond so far.

### 9 Responds so far- Add one»

1. 1. ragknot Said：

Are you saying that the “cube of water” has similiar densities like the example?

2. 2. Chris Said：

Hi Ragknot. No – the water has relative density 1. I only put it in for the sense of wonder the result should give.

3. 3. cazayoux Said：

I’ll give it a shot … The cube of water would have an edge of length 18,183km.

4. 4. cazayoux Said：

inner core volume = (4/3)pi(400^3)
outer core volume = (4/3)pi(3400^3 – 400^3)
inner core volume = (4/3)pi(6400^3 – 3400^3)

density is mass per volume.
volume * density = mass

inner core mass = 13 * (4/3)pi(400^3)
outer core mass = 11 * (4/3)pi(3400^3 – 400^3)
inner core mass = 4.5 * (4/3)pi(6400^3 – 3400^3)

Total mass ~= 6,011,969,519,000
Water density is 1, therefore, …
total volume = ~= 6,011,969,519,000

Cube root of total volume gives us the edge of the cube ~= 18,183 km

5. 5. Chris Said：

Hi cazayoux. You’ve misunderstood the question. That cube corresponds to the total mass of the Earth. The question is about the mass of the Earth’s gravitational field.

6. 6. Chris Said：

Here’s the starting point. I’ll post more in a day or so.

To do this problem, we need to show that a gravitational field stores energy. In fact the energy density u (J/m³) in a region where the field strength is g, is u = -g²/(8πG). Due to E = mc², we can define a mass density μ = u/c². Note that both u and μ are negative.

To see why, we’ll find the field associated with a large plate of areal mass density σ. Away from the edges of the plate, the g-field is perpendicular to the plate. Direct calculation using Newton’s law of graviation is tedious. A fairly easy way is to use the well known results for a thin spherical shell of of radius r, areal mass density σ and total mass M = 4πr²σ. Just outside the sphere, g = GM/r² = 4πGr²σ/r² = 4πGσ and just inside the sphere, g = 0. Consider a small section of the surface of the sphere. In isolation it must be making a field that points inwards and outwards equally. So the rest of the sphere must be adding a field that exactly cancels the inwardly pointing field, and it follows that it must be doubling the outwardly pointing field. We conclude that for a plate, g = 2πGσ.

Now split the plate into two equal plates. Note that for the pair of plates, the field strength is g. Let the area of each plate be A. Each plate has areal mass density σ/2 and produces a field g/2. Keep these two plates close to each other. Consider one of these plates to be fixed. The force acting on the second plate (due to the first) is (g/2)(σA/2) = gσA/4. If the plates are further separated by a small distance d, the work required to do that is gσAd/4. The volume of the universe outside the region defined by the plates is reduced by Ad. Where has that work gone to? It must have been used to remove the g-field in the region whose volume is Ad, and so the energy density of the g-field must be u = -(gσAd/4)/(Ad) = -gσ/4. But g = 2πGσ => σ = g/(2πG) and substituting that into the previous equation gives u = -g²/(8πG). From now on, I’ll drop the – sign.

The mass density of the g-field is μ = g²/(8πGc²).

G = 6.673*10-11, c² = 8.988*1016
For amusement, if g = 9.81, then μ = 6.385*10-7, => μ = 638.5 kg/(km)³

NB the 8πG factor also appears in Einstein’s field equations of general relativity: Gμν = 8πGTμν. Gμν is the Einstein (curvature) tensor, and Tμν is the stress-energy tensor. Note that it is quite common to choose physical units such that G = c = 1, and so it is common to see Gμν = 8πTμν.

7. 7. ragknot Said：

Wow, Chris, that’s a lot of info, but I have not read it yet. The relative density of water is 1.00, if the temperature is 4 deg C. One milliliter of it has a mass of exactly one gram.

8. 8. Chris Said：

For a spherically uniform distribution of mass, Newton’s law of gravitation has some rather nice consequences. First, for points outside the sphere, the g-field is identical to one the would be produced as if the entire mass of the sphere were concentrated at a point at the centre of the sphere. Second, the g-field inside a sphere at a distance r from the centre, is entirely due to the mass inside inside the sphere of radius r: the matter outside that radius doesn’t contribute to the internal g-field. Third, g-fields sum linearly.

In my last post I argued that μ = g2/(8πGc2) where μ is the mass density of the g-field. Hence the total mass of the g-field is M = ∫μ dV, where the integral is talken over all space and dV is a volume element. For a sphere, it is natural to use dV = 4πr2 dr. So M = ∫0 4πr2g2/(8πGc2) dr = 1/(2Gc2)∫0 g2r2 dr

We now need to find g everywhere. For a sphere of radius r and uniform density ρ, the mass is 4πρr3/3. For typing convenience, I’ll use k = 4π/3 ≈ 4.1888. So, the mass of the sphere is kρr3. The mass of a hollow sphere of inner radius ri and outer radius ro is kρ(ro3 – ri3).

In this problem we have four regions to consider: the inner core (radius ri, density ρi), the middle layer (aka outer core, radius rm, density ρm), the outer layer (radius ro, density ρo) and the region outside the Earth (radius ∞, density 0).

Let mi = kρiri3, mm = kρm(rm3 – ri3) and mo = kρo(ro3 – rm3)

For ri > r > 0: g = kGρir3/r2 = kGρir
For rm > r > ri: g = G(mi – kρmri3)/r2 + kGρmr
For ro > r > rm: g = G(mi + mm – kρorm3)/r2 + kGρor
For r > ro: g = G(mi + mm +mo)/r2

Using those results for g, we now find g2r2.
For ri > r > 0: g2r2 = k2G2ρi2r4
For rm > r > ri:
g2r2 = G2(mi – kρmri3)2/r2 + 2kG2ρm(mi – kρmri3)r2 + k2G2ρm2r4
For ro > r > rm:
g2r2 = G2(mi + mm – kρorm3)2/r2 + 2kG2ρo(mi + mm – kρorm3)r2 + k2G2ρo2r4
For r > ro: g2r2 = G2(mi + mm + mo)2/r2

To be continued…

9. 9. Chris Said：

…continued.

After some pretty boring integrations etc., we end up with the total mass of the g-field is around 3*1015 kg. As the density of water is 1000 kg m-3 that’d be a volume of 3*1012 m3 and so a cube of side around 14.4 km.

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