BeeTheory · Foundations · Technical Note XXIII
Modelling the Earth:
Visible Mass, Wave Mass, and Where Each Resides
This note applies BeeTheory to the Earth as a concrete, stratified spherical body. The Earth’s actual internal structure — inner core, outer core, mantle, crust — is fed into the BeeTheory framework with the kernel established in Note XXII. The result decomposes Earth’s gravitational mass into a “visible” (atomic) part and a “wave” part, and shows precisely where in space the wave mass resides.
1. The result first
Decomposition of the Earth’s mass
With $lambda = 0.098$ (Milky Way calibration, Note XX):
- Visible mass (atomic) : $5.97 \times 10^{24}$ kg (the value any local experiment measures)
- Wave mass (total, asymptotic) : $5.85 \times 10^{23}$ kg (delocalised over kpc)
- Visible fraction : $91.1\%$. Wave fraction : $8.9\%$
Of this wave mass, $99.997\%$ sits beyond the Solar System, between $\sim 100$ pc and a few kpc. Only $5 \times 10^{-3}$ kg of wave mass is enclosed within the Earth’s radius — utterly undetectable.
2. Internal structure of the Earth (standard model)
The Earth is a layered spherical body, with four major components defined by seismology and bulk-density measurements:
| Layer | Inner radius | Outer radius | Mean density | Mass |
|---|---|---|---|---|
| Inner core (solid Fe-Ni) | 0 km | 1 221 km | 12 950 kg/m³ | $9.87 \times 10^{22}$ kg |
| Outer core (liquid Fe-Ni) | 1 221 km | 3 480 km | 10 870 kg/m³ | $1.84 \times 10^{24}$ kg |
| Mantle (silicate rock) | 3 480 km | 6 346 km | 4 380 kg/m³ | $3.92 \times 10^{24}$ kg |
| Crust (light rock + ocean) | 6 346 km | 6 371 km | 2 700 kg/m³ | $3.43 \times 10^{22}$ kg |
| Total | — | $R_\oplus = 6 371$ km | $\rho_\text{avg} = 5513$ kg/m³ | $\mathbf{5.97 \times 10^{24}}$ kg |
For the BeeTheory wave-field calculation, all this internal structure is irrelevant as long as the total mass is correctly summed. The reason is the shell theorem combined with the coherence length: from a few hundred parsec away, the Earth is a point mass.
3. The BeeTheory calculation for Earth
Using the normalised kernel of Note XXII applied to a point mass $m = M_oplus = 5.97 times 10^{24}$ kg:
$$M_\text{wave}(
With $\lambda = 0.098$ and $\ell_0 = 1.59$ kpc, this gives the enclosed wave mass at any radius around the Earth. The values at key reference scales:
| Radius around Earth | $R/\ell_0$ | $M_\text{wave}(| Compared to $M_\oplus$ | |
|---|---|---|---|
| Cavendish lab (15 cm) | $3 \times 10^{-21}$ | $\sim 10^{-18}$ kg | $\sim 10^{-43}$ |
| Earth surface (6 371 km) | $1.3 \times 10^{-13}$ | $5 \times 10^{-3}$ kg = $5$ g | $8.3 \times 10^{-28}$ |
| Moon orbit (384 000 km) | $7.8 \times 10^{-12}$ | $18$ kg | $3.0 \times 10^{-24}$ |
| 1 AU (Earth–Sun) | $3.1 \times 10^{-9}$ | $2.7 \times 10^{6}$ kg | $4.6 \times 10^{-19}$ |
| 30 AU (Solar system edge) | $9.1 \times 10^{-8}$ | $2.4 \times 10^{9}$ kg | $4.1 \times 10^{-16}$ |
| $\ell_0$ (1.59 kpc) | $1.0$ | $1.5 \times 10^{23}$ kg | $0.0259$ |
| $5\,\ell_0$ ($\sim 8$ kpc) | $5.0$ | $5.6 \times 10^{23}$ kg | $0.094$ |
| $\infty$ | $\infty$ | $5.85 \times 10^{23}$ kg | $\lambda = 0.098$ |
A striking number
The wave mass enclosed within the Earth itself is just $5$ grams. The wave mass within Moon’s orbit is $18$ kg — about the mass of a child. Even out to Pluto’s orbit, only $\sim 2.4$ billion kg of wave mass exists — a number that sounds large but is $10^{16}$ times smaller than $M_\oplus$. The bulk of the wave mass — $99.99\%$ — sits beyond $100$ pc from the Earth, in the interstellar medium.
4. Where the wave mass actually sits
The total wave mass $\lambda M_\oplus = 5.85 \times 10^{23}$ kg is distributed in radial shells around the Earth. Most of it is far from the Earth itself:
| Spatial zone | Radial range | Wave mass | % of total |
|---|---|---|---|
| Inside Earth | 0 to $R_\oplus$ | $5 \times 10^{-3}$ kg | $\sim 10^{-27}\%$ |
| Cislunar (to Moon) | $R_\oplus$ to 384 000 km | $18$ kg | $\sim 10^{-23}\%$ |
| Solar System | to 30 AU | $2.4 \times 10^9$ kg | $\sim 10^{-15}\%$ |
| Solar System to $\ell_0/10$ | 30 AU to 160 pc | $2.7 \times 10^{21}$ kg | $0.47\%$ |
| $\ell_0/10$ to $\ell_0$ | 160 pc to 1.59 kpc | $1.5 \times 10^{23}$ kg | $\mathbf{26.0\%}$ |
| $\ell_0$ to $5\,\ell_0$ | 1.59 to 7.95 kpc | $4.1 \times 10^{23}$ kg | $\mathbf{69.5\%}$ |
| Beyond $5\,\ell_0$ | $> 7.95$ kpc | $2.4 \times 10^{22}$ kg | $4.0\%$ |
The wave mass of the Earth is overwhelmingly in the Milky Way disk, not on the Earth
$95.5\%$ of Earth’s total wave mass sits between $160$ parsec and $8$ kiloparsec from the Earth, deep in interstellar space. Only $0.47\%$ is closer than $160$ pc, and inside the Solar System, the wave mass contribution is essentially zero ($10^{-15}\%$ of the total). The Earth’s wave mass is therefore a member of the Galaxy’s overall wave field, not a localised “halo” around our planet.
5. Why the Earth’s orbit and dynamics are unaffected
5.1 Spherical symmetry preserves the orbit
The Earth is spherically symmetric (to excellent approximation). The wave field it generates is therefore also spherically symmetric. By the shell theorem, the gravitational influence of a spherically symmetric mass distribution on an external body depends only on the mass enclosed within that body’s radial distance. So the Moon, at $R = 3.8 \times 10^8$ m, sees only:
$$M_\text{effective}(\text{Moon}) \;=\; M_\oplus + M_\text{wave}(
The $18$ kg of wave mass enclosed within the Moon’s orbit is utterly negligible compared to the Earth’s $6 \times 10^{24}$ kg. The Moon’s orbital period is therefore set by the visible Earth mass alone, with a correction at the level of $10^{-23}$.
5.2 The Earth’s orbit around the Sun is similarly unaffected
Treating the Sun-Earth system reciprocally: the Sun also generates a wave field. By the same calculation:
| Body | Visible mass | Wave mass at $r = 1$ AU | Relative contribution |
|---|---|---|---|
| Earth | $5.97 \times 10^{24}$ kg | $2.7 \times 10^6$ kg | $5 \times 10^{-19}$ |
| Sun | $1.99 \times 10^{30}$ kg | $9.1 \times 10^{11}$ kg | $5 \times 10^{-19}$ |
The wave-mass contribution to Earth’s orbital dynamics is below $10^{-18}$ of the visible-mass contribution. The Earth’s orbit around the Sun is therefore identical to its Newtonian prediction, within experimental precision.
5.3 The Earth’s rotation around the galactic centre
This is where the wave mass does matter. The Earth (or rather the Sun) orbits the Milky Way centre at $R_odot = 8$ kpc with $V_odot approx 229$ km/s. The wave mass affecting this orbit is not Earth’s alone — it is the cumulative wave field of all the $10^{11}$ stars and the gas of the entire galactic disk, each contributing its own $\lambda M_i$ of wave mass spread over $\ell_0$ around it. The aggregate is sufficient to account for the observed rotation curve (see Notes XX–XXI).
The Earth’s wave mass is one drop in the Milky Way’s wave ocean
The full $\lambda M_\oplus = 5.85 \times 10^{23}$ kg of Earth’s wave mass is approximately $10^{-18}$ of the Milky Way’s total baryonic mass. The Sun’s wave mass is $\sim 10^{20}$ kg, also negligible at galactic scale. It is only the sum of $10^{11}$ stellar wave contributions, plus the gas, that produces the rotation curve we observe.
6. Two interpretations — both operationally equivalent
There are two consistent ways to read the Earth’s mass figures, both physically equivalent:
Interpretation A — “extended Earth”
The Earth’s atomic mass is $M_\text{vis} = 5.97 \times 10^{24}$ kg. The Earth’s total gravitational influence is $M_\text{vis}(1+\lambda) = 6.56 \times 10^{24}$ kg, but $\lambda M_\text{vis}$ of this is spread over the surrounding $\sim$kpc as wave mass. Locally, we measure only $M_\text{vis}$; the wave part is delocalised.
Interpretation B — “locally measured mass”
The Earth’s locally measurable mass is $5.97 \times 10^{24}$ kg. This includes both the atomic mass and the small enclosed wave mass (which is $\sim 10^{-27}$ of total — negligible). The atomic mass is therefore $5.97 \times 10^{24}$ kg to extreme precision, and the “extra” wave mass exists at kpc distances where it cannot be attributed cleanly to “Earth” alone.
Both interpretations agree on every observable: Cavendish reads $5.97 \times 10^{24}$ kg, the Moon’s orbit confirms it, and the wave mass becomes relevant only at galactic scales — where it is collectively responsible for rotation-curve anomalies, attributed to dark matter in the standard interpretation.
7. Summary
1. The Earth’s stratified internal structure — inner core, outer core, mantle, crust — is irrelevant to its wave-mass calculation at galactic scales. From kpc distances, only the total mass $M_\oplus = 5.97 \times 10^{24}$ kg matters.
2. With $\lambda = 0.098$, the total wave mass associated with the Earth is $5.85 \times 10^{23}$ kg ($8.9\%$ of the total gravitational influence).
3. This wave mass is spread over kiloparsec scales: $95\%$ of it sits between $\ell_0/10 = 160$ pc and $5\,\ell_0 = 8$ kpc from the Earth.
4. Within the Earth’s volume itself, only $5$ grams of wave mass exist. Within the Moon’s orbit, $18$ kg. Within the entire Solar System, $2.4 \times 10^9$ kg — all utterly negligible compared to $M_\oplus$.
5. Earth’s orbit around the Sun and the Moon’s orbit around the Earth are therefore unaffected by BeeTheory’s wave field — the modification is at the $10^{-18}$ level.
6. The Earth’s wave mass is a member of the Milky Way’s collective wave field, not a localised halo. It contributes — together with the wave masses of all other stars and gas — to the dynamics of the galactic rotation curve.
References. Dziewonski, A. M., Anderson, D. L. — Preliminary reference Earth model, Phys. Earth Planet. Inter. 25, 297 (1981). PREM, the standard Earth density profile. · Cavendish, H. — Experiments to determine the density of the Earth, Phil. Trans. R. Soc. London 88, 469 (1798). · Newton, I. — Philosophiae Naturalis Principia Mathematica (1687). Shell theorem. · Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, v2, BeeTheory.com (2023).
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