The Properties of the Earth

This page provides an explanation of the properties relating to the earth and its moon, which has been established using Isaac Newton's universal gravitational theories.

Newton's Law of Gravitational Force

Isaac Newton's gravitational force law is as follows:
F = G.m₁.m₂ / R²
where:
F is the gravitational force of attraction {N}
G is Isaac Newton's gravitational constant {m³ / kg.s²}
m₁ = mass of a force centre (e.g. the earth) {kg}
m₂ = mass of a satellite (e.g. the earth's moon) {kg}
R = distance between the centre of m₁ and the centre of m₂ {m}

Earth's Mass

If we remove m₂ (the mass of the earth's moon) from the above force law for the moon's orbit, we get the earth's gravitational acceleration at its surface (e.g. sea-level):
g = G . m₁ / R² = 9.80663139027614 {m/s²}
From which we can accurately calculate Earth's mass thus:
The volumetric radius of the earth at sea-level is:
R = ³√(6356752 + 6378137²) = 6371000.68502598m
m₁ = g . R² / G
    = 9.80663139027614 x 6371000.68502598² / 6.67359232004332E-11
    = 5.9645197677618E+24kg

CalQlata's UniQon calculator has an alternative calculation method taking into account the uneven nature of the earth's composition giving 'g' a value of 9.80616 at latitude 45° and an alternative mass of 5.96629861115861E+24kg

Therefore, we can confidently predict the actual mass of the earth as:
m₁ = 5.9654091894602E+24kg ±0.01491%
however, because of the ability to match the former value for mass with Newton's laws, CalQlata shall set the earth's mass as 5.9645197677618E+24kg for all of its internal calculations

Earth's Density

The earth's surface area = 4π x 6356752 x 6378137 = 5.094938849322E+14m²

The earth's volume = 4/3π x 6371000.685³ = 1.08320726625321E+21m³

So the density of the earth can be calculated thus:
ρ₁ = 5.9645197677618E+24 / 1.0832072662532E+21 = 5506.35132682682kg/m³

Earth's Geosynchronous Orbit

Centrifugal acceleration: a = v² / R

Gravitational acceleration: g = G.m₁ / R²

Where v is the velocity of the satellite at R and g ≡ a

From the above equations:
R = G.m₁ / v²
v = 2.π.R/86400 m/s⁽⁴⁾
v² = R².(2.π/86400)² = R² x 5.288496871297E-09 m²/s²
R = G x m₁ / (R² x 5.288496871297E-09)
R = (G.m₁ / 5.288496871297E-09)

The altitude of the earth's geosynchronous orbit is 'R' minus the radius of the earth: 4.22215624808981E+07m - 6378137m = 3.584342548E+07m

Our Moon

A few useful facts about the earth's moon are listed below.

Orbit

Given that our moon's orbit is susceptible to precession due to its relatively low mass and its proximity to the gravitational influences of our sun and Jupiter, distance measurements such as perigee and apogee are a little difficult to pin down. Current opinion appears to indicate that our moon's orbital eccentricity (e) is 0.0549, perigee distance (Rᴾ) is 363,300km, apogee distance (Rᴬ) is 405,504km and its orbital period (t) is 27.32166 days. But using the laws of orbital motion, these figures give us an Earthly mass (m₁) of 6.03223163E+24kg. And if both of these distances are correct, the moon's orbital eccentricity would be 0.054763948!.
Something must be wrong with our current assumptions.

So, what are our moon's actual orbital properties?
First we must define those properties that cannot be disputed, i.e. those that can be measured directly: Earth's mass (from surface gravitational acceleration) and the moon's orbital period.
Using these values and Isaac Newton's constant of proportionality (K = (2π)² / G.m₁), we can also establish an accurate value for the length of half its major axis; a = ³√[t²/K]
Therefore, we now know the following about our moon's orbit:
m₁ = 5.964519767713130E+24 kg (not 6.03223163E+24 kg)
t = 2360591.424 s
K = 9.91801091E-14 s²/m³
a = 383,006.0987 km (not 384,450 km)

Having already established that the eccentricity is incorrect (above), we can only assume that either one of the following may be correct; 'Rᴾ' or 'Rᴬ':
If Rᴾ is correct: e = 0.051200487 and Rᴬ = 402,616.1974 km
If Rᴬ is correct: e = 0.058740321 and Rᴾ = 360,508.1974 km

Mass

According to NASA, the average gravitational acceleration on the surface of the earth's moon is 1.624m/s², its radius is 1737494.51373m and the average distance between the centres of the earth and its moon is 3.83E+08m⁽²⁾.

By applying Newton's formula to the earth's moon: g = G . m₂ / r²
m₂ = g.r² / G
    = 1.624 x 1737494.51373² / 6.67359232004332E-11 = 7.34637741371792E+22kg

Density

The moon's surface area: A = 4π x 1737494.51373² = 3.79364552126886E+13m²

The moon's volume is: V = 4/3π x 1737494.51373³ = 2.19714609341368E+19E+19m³

The moon has a density of: ρ₂ = m₂ / V = 3343.59987974397kg/m³

Earth's Gravitational Force on its Moon

According to CalQlata's calculations, the moon's average orbital velocity is 1021.3712417m/s. According to NASA⁽⁴⁾, the mean velocity of the moon is actually 1022m/s, which may be why it is gradually receding

Earth's gravitational acceleration at the orbital radius of its moon is calculated as follows:
Irrespective of which orbital distances (above) are correct, the mean radius is the same; 383006098.7m
From; F = m₁.v² / R we get; F/m₁ = v²/R = gm = 0.00272371437m/s²
Alternatively, from; F = G.m₁.m₂ / R² we get;
a = F/m = G.m₂ / R² = 0.00271346220m/s²
Therefore, the mean gravitational attraction by the earth on its moon is:
gm = (0.00272371437 + 0.00271346220)/2 = 0.00271858828m/s²

The gravitational force between the moon and the earth is:
F = m₂ . gm = 7.34637741371792E+22 x 0.00271858828 = 1.9971776E+20N
which is an average value
See Laws of Motion for accurate values

Earth's Structure

The earth generates 2.877018E+28 J of frictional energy during every orbit, which relates to 9.1229642E+20 Watts of power.
Its internal structure is described below:

radius
[m]
density
[kg/m³]
mass
[kg]
pressure
[N/m²]
R₁ (core) 1 13000 54454 1.3964E+12
R₂ (inner core OD) 1215000 13000 9.76699E+22 2.80675E+11
R₃ (outer core OD) 3470000 9085.8 1.79228E+24 9.28986E+07
R₄ (mantle OD) 6363000 1105 4.06946E+24 152549.9
R₅ (crust OD) 6371000 1250 5.76668E+21 10332.29
Earth's internal structure (refer to core pressure theory; Fig 1 for definitions of radii R₁ to R₅)
Note: The above densities apply only at the radii concerned. The density between any two radii vary linearly between them;
e.g. the density of the mantle matter varies from 9085.8 kg/m³ at the "outer core OD" to a gaseous 1105 kg/m³ immediately beneath the earth's crust.

Earth's Composition

As can be seen from the above Table, the Earth's core matter is considerably more dense than iron (7870 kg/m³), therefore much of the earth's core must comprise a great deal of the heaviest elements.

Inner Core:
Volume is; V = ⁴/₃π.R₂³ = 7.51307E+18 m³
Average density is; ρᶜ = 13,000 kg/m³
which comprises the heaviest elements - iron, Tantalum, Tungsten, Rhenium, Osmium, Iridium, Platinum, lead, Gold, Protactinium, Uranium, etc.

Outer Core:
Volume is; V = ⁴/₃π.(R₃³-R₂³)³ = 1.67503E+20 m³
The average density is; ρ = 10700 kg/m³
which comprises various amounts of the heavier elements such as; iron, cobalt, copper, nickel, niobium, molybdenum, silver, etc., and the silicates.

Mantle:
Volume is; V = ⁴/₃π.(R₄³-R₃³) = 9.04116E+20 m³
The average density is; ρ = 4500 kg/m³
which comprises the lighter elements - Boron, carbon, Magnesium, Aluminium, Silicon, Phosphorus, Sulphur, Potassium, Calcium, Scandium, etc.

Crust:
Volume is; V = ⁴/₃π.(R₄³-R₃³) = 4.07539E+18 m³
The average density is; ρ = 1415 kg/m³
which comprises similar elements to the mantle, but much colder (≈273 K excluding the effects of stellar radiation)

Comment & Discussion

When equating the earth's internal energy, we can estimate the EME emitted by the planet's surface as follows:
The average surface temperature of the earth's crust (at night): Ṯ ≈ 273 K
The mass of an electron: mₑ = 9.1093897E-31 kg
The mass of a neutron: mₙ = 1.6735325768E-27 kg
The average neutronic ratio of the crust elements: ψ ≈ 1.12
The crust radii: R₄ = 6363000 m & R₅ = 6371000 m

The mass of the earth's crust is:
m = ⁴/₃π(R₅³ - R₄³).ρ = 5.1E+21 kg
The number of proton-electron pairs radiating heat from the earth's crust is:
N = m / mₙ.(ψ+1)
The energy radiated from the crust matter is:
E = N.ψ . ½.mₑ/X . Ṯ = 2.8832519E+28 J
By a remarkable coincidence, spin theory predicts the same internal friction energy (2.87709E+28 J)!

The frequency of the electro-magnetic energy (EME) radiated by the electron's in the crust matter on the dark-side of the earth can be found thus (@ Ṯ):
The minimum orbital radius of its electrons:
Rₑ = XR / Ṯ = 5.855E-09 m
The maximum orbital velocity of its electrons:
vₑ = √[Ṯ / X] = 2.079821E+05 m/s
The maximum frequency of the EME radiated by the proton-electron pairs:
ƒ = vₑ / 2πRₑ = 5.65364778E+12 Hz
Note: the lowest radiated EME frequency radiated by the outer orbiting electrons is <1E+11 Hz
Therefore the EME radiated from the dark side of our planet ranges between 1E+11Hz < ƒ < 5.6E12Hz

Notes

  1. Nature magazine discusses problems with the accuracy of this value
  2. NASA
  3. NASA moon facts
  4. There are ≈86400 seconds in one day {24 x 60 x 60}