Abstract
The speed of a wave travelling along a magnetic field line is set by the Alfven speed $v_A=B/\sqrt{\mu_0\rho}$. Because the name carries the word “magneto,” one is tempted to read the field as the deciding factor. This paper asks which of the two actually acts: the answer is the density. No new mathematical theorem and no new law is claimed.
First, we compute at five locations. In the solar corona $\mathbf{690.30}$ km/s; in the Earth’s outer core $\mathbf{8.92}$ mm/s (Section 2).
Second, this is the core of the paper. These two have the same field of $1$ mT, and still differ by a factor of $\mathbf{7.7382\times10^{7}}$ (Section 3).
Third, the ratio matches the square root of the density ratio exactly. The square root of the density ratio $5.9880\times10^{15}$ is $\mathbf{7.7382\times10^{7}}$: the field has cancelled out completely (Section 3).
Fourth, moving the field is not enough. Applying $\mathbf{1}$ T in the laboratory gives $6903.0$ km/s, only $10$ times the coronal value (Section 4).
Fifth, we make the $8.92$ mm/s of the Earth’s core concrete. It covers $\mathbf{281.51}$ km in a year, and crossing the $2260$ km outer core takes $\mathbf{8.0}$ years (Section 5).
Sixth, we take no credit for the $4\pi$ in $\mu_0$. As in Paper 267, it comes from the choice of units (Section 6).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed: the expression for the Alfven speed, the representative fields and densities of each region, and the comparison with the sound speed are all standard. We do not build magnetohydrodynamics: all we use is one square root and two ratios. We do not derive the Alfven speed: we merely quote $v_A=B/\sqrt{\mu_0\rho}$, and do not enter the argument that obtains it from magnetic tension. We do not treat magnetic reconnection: neither solar flares nor dynamos are treated at all. We do not discuss the origin of the Earth’s field: neither outer-core convection nor dynamo theory is treated. Density and field are merely placed as representative values. We claim no accuracy for the representative values: both fields and densities are values chosen to show orders of magnitude, and they move greatly with place and epoch. The coronal density and the Earth’s outer-core field in particular have a spread of one order. The sound speed carries premises: the $c_s$ of Section 4 assumes fully ionised hydrogen, $\gamma=5/3$, and the pressure of both electrons and protons. Changing the assumptions moves it by tens of percent. We do not call the $4\pi$ in $\mu_0$ a discovery: Paper 267 wrote that it comes from the choice of units. This paper treats it the same way, and its standing differs from the $4\pi$ of Paper 02. Relation to earlier papers: Paper 267 showed that the displacement current is not a current, and wrote that the $4\pi$ in $\mu_0$ comes from units. This paper uses the same $\mu_0$ and treats it the same way. Paper 256 treated the fact that a model cannot be built when there are two dimensionless numbers. This paper separates which of the variables inside one expression is acting. Paper 190 measured “rare” on a logarithmic scale. This paper likewise writes the ratio $7.74\times10^{7}$ in orders of magnitude. Paper 240 counted seven things called “mass”. This paper separates two variables inside one expression. What is added is computing and lining up the Alfven speed at five regions, seeing that the same field yields a factor of $7.7382\times10^{7}$, confirming that this ratio matches the square root of the density ratio exactly, and recasting the $8.92$ mm/s of the Earth’s outer core as “$8.0$ years to cross the outer core.”
1. Notation and Standing
Setting: In a conducting fluid (a plasma, or molten iron), a wave travels along the field lines. Its speed is
$$v_A=\frac{B}{\sqrt{\mu_0\rho}}$$where $B$ is the field, $\rho$ the mass density, and $\mu_0=4\pi\times10^{-7}$.
Question: Which of $B$ and $\rho$ sets this speed?
The answer is the density. Both appear in the expression, but in actual settings the range of the density is overwhelmingly wider.
Caution: The $4\pi$ inside $\mu_0$ comes from the choice of units: as in Paper 267, we take no credit for it.
2. Computing at Five Locations
Check 2.1: We insert representative fields and densities.
| Location | $B$ [T] | $\rho$ $[\mathrm{kg/m^3}]$ | $v_A$ |
|---|---|---|---|
| Laboratory plasma | $1.0$ | $1.67\times10^{-8}$ | $\mathbf{6903.0}$ km/s |
| Solar corona | $1.0\times10^{-3}$ | $1.67\times10^{-12}$ | $\mathbf{690.30}$ km/s |
| Solar wind ($1$ AU) | $5.0\times10^{-9}$ | $8.4\times10^{-21}$ | $48.67$ km/s |
| Interstellar medium | $5.0\times10^{-10}$ | $1.67\times10^{-21}$ | $10.91$ km/s |
| Earth’s outer core | $1.0\times10^{-3}$ | $1.0\times10^{4}$ | $\mathbf{8.92}$ mm/s |
Observation 2.2 Top and bottom are separated by a factor of $\mathbf{7.74\times10^{11}}$. The same single expression returns speeds spanning $11$ orders of magnitude.
Observation 2.3 A point to write honestly: every value is representative. Both the coronal density and the Earth’s outer-core field have a spread of one order. What this paper asserts is the structure of the orders, not the individual numbers.
3. The Core: With the Same Field, a Factor of $7.74\times10^{7}$
Setting: In the table above, the solar corona and the Earth’s outer core have the same field of $1$ mT. We take out only those two.
Check 3.1:
| Location | $B$ | $\rho$ | $v_A$ |
|---|---|---|---|
| Solar corona | $\mathbf{1.0\times10^{-3}}$ T | $1.67\times10^{-12}$ | $6.902983\times10^{5}$ m/s |
| Earth’s outer core | $\mathbf{1.0\times10^{-3}}$ T | $1.0\times10^{4}$ | $8.920621\times10^{-3}$ m/s |
| Ratio | $\mathbf{1.0000}$ | $5.9880\times10^{15}$ | $\mathbf{7.7382\times10^{7}}$ |
Observation 3.2 The ratio of the fields is exactly $1$. The speeds still differ by a factor of $\mathbf{7.74\times10^{7}}$: the field is not acting at all.
Check 3.3: We confirm that the ratio comes from the density.
| Quantity | Value |
|---|---|
| Density ratio $\rho_{\text{core}}/\rho_{\text{corona}}$ | $5.9880\times10^{15}$ |
| Its square root | $\mathbf{7.7382\times10^{7}}$ |
| Speed ratio (measured) | $\mathbf{7.7382\times10^{7}}$ |
| Difference | $\mathbf{0}$ |
Observation 3.4 They agree exactly. This is expected, since $v_A\propto\rho^{-1/2}$ when $B$ is the same; but that it is expected is the point of this paper: the field has cancelled out completely.
Observation 3.5 Separator. What must be moved for the speed to change? The density. Even though the name carries “magneto,” what makes the spread in real settings is the density.
4. Moving the Field Is Not Enough
Check 4.1: We move only the field, and see whether the density gap can be filled.
| Location | $B$ | $v_A$ | Relative to corona ($690.30$ km/s) |
|---|---|---|---|
| Laboratory plasma | $\mathbf{1.0}$ T ($1000\times$ corona) | $6903.0$ km/s | $\mathbf{10.0}\times$ |
| Solar corona | $1.0\times10^{-3}$ T | $690.30$ km/s | $1\times$ |
Observation 4.2 Raising the field by $1000$ raises the speed only by $10$. The density also differs by $10^{4}$, so the two effects cancel: $1000/\sqrt{10^{4}}=10$.
Observation 4.3 To bring the Earth’s core $8.92$ mm/s up to coronal values, the field must be raised by $7.74\times10^{7}$, that is to $\mathbf{7.7\times10^{4}}$ T. That is $\mathbf{1700}$ times the strongest steady field producible in a laboratory (around $45$ T): the field cannot fill the gap.
Check 4.4: We compare with the coronal sound speed. We assume fully ionised hydrogen, $\gamma=5/3$, $T=10^{6}$ K, counting the pressure of both electrons and protons.
| Quantity | Value |
|---|---|
| Sound speed $c_s=\sqrt{2\gamma kT/m_p}$ | $\mathbf{165.88}$ km/s |
| Alfven speed $v_A$ | $690.30$ km/s |
| Ratio $v_A/c_s$ | $\mathbf{4.1615}$ |
Observation 4.5 In the corona $v_A/c_s=4.16$, so the magnetic side is faster. This is why the field governs the dynamics, but this ratio too is set by the density.
Observation 4.6 A point to write honestly: the sound speed depends on the assumptions. Setting $\gamma$ to $1$ (isothermal) gives $128.5$ km/s and a ratio of $5.37$: we do not assert the number $4.16$ itself.
5. Making the $8.92$ mm/s of the Earth’s Core Concrete
Check 5.1: We recast $8.920621\times10^{-3}$ m/s into other units.
| Way of writing | Value |
|---|---|
| $v_A$ | $\mathbf{8.9206}$ mm/s |
| Distance covered in a year | $\mathbf{281.51}$ km |
| Time to cross the outer core ($2260$ km thick) | $\mathbf{8.0}$ years |
Observation 5.2 Inside the Earth, a magnetic disturbance takes $8$ years to cross the outer core. That is far slower than a person walking, and this is with a field of $1$ mT, the same strength as in the corona.
Observation 5.3 In the corona, meanwhile, the same $1$ mT gives $690.30$ km/s. Crossing the solar radius $7\times10^{5}$ km takes about $17$ minutes: with the same field: $8$ years, and $17$ minutes.
Observation 5.4 This is the content of the number $7.74\times10^{7}$. Written in orders of magnitude it is hard to feel; recast as a time it becomes clear: the reverse of Paper 190, which measured “rare” logarithmically: here the logarithm is undone to restore the feel.
6. Taking No Credit for the $4\pi$ in $\mu_0$
Observation 6.1 $\mu_0=4\pi\times10^{-7}=1.2566371\times10^{-6}$. A $4\pi$ sits in the denominator of the Alfven speed.
Observation 6.2 Paper 267 wrote that the $4\pi$ in $\mu_0$ comes from the choice of units. In Gaussian units $v_A=B/\sqrt{4\pi\rho}$, so $\mu_0$ disappears and the $4\pi$ is exposed: in either way of writing a $4\pi$ remains, but it is a $4\pi$ on the side of convention.
Observation 6.3 The conditions written by Paper 258 (point source, three dimensions, isotropy) are not met here. There is no source, and no inverse square: its standing differs from the $4\pi$ of Paper 02.
Observation 6.4 This paper therefore asserts nothing about this $4\pi$. Following Paper 267, it records only the point that the same symbol has a case where it arises from geometry and a case where it arises from units.
7. Roll Call and Closing
- Five locations (Check 2.1): From $6903$ km/s to $8.92$ mm/s, $11$ orders.
- The same $1$ mT, a factor of $7.74\times10^{7}$ (Check 3.1): The field ratio is exactly $1$.
- The ratio is the square root of the density ratio (Check 3.3): Difference $0$.
- The field cancels out (Observation 3.4): Against its name, it is not acting.
- $1000\times$ gives $10\times$ (Check 4.1): The density cancels it.
- Filling the gap needs $7.7\times10^{4}$ T (Observation 4.3): $1700$ times the strongest laboratory field.
- In the corona $v_A/c_s=4.16$ (Check 4.4): But it depends on assumptions.
- $8.0$ years to cross the outer core (Check 5.1): $281.51$ km a year.
- In the corona, $17$ minutes (Observation 5.3): With the same field.
Closing: The Alfven speed is called “the speed of magnetohydrodynamic waves,” but what acts is not the field. The solar corona and the Earth’s outer core have the same field of $1$ mT, and still their speeds differ by a factor of $\mathbf{7.7382\times10^{7}}$, a ratio that matches the square root of the density ratio exactly. The field has cancelled out completely. Moving the field is not enough: raising it by $1000$ in the laboratory raises the speed only by $10$, and bringing the Earth’s core up to coronal values would require $\mathbf{7.7\times10^{4}}$ T, $1700$ times the strongest laboratory field. Recast as a time, it becomes clear: with the same $1$ mT field, crossing the Earth’s outer core takes $8.0$ years, and crossing the Sun takes $17$ minutes. One thing separates them: writing out which variable inside the expression makes the spread in the actual setting. Write it out, and what the name points at separates from what acts. Fail to write it out, and one reads the difference between $8$ years and $17$ minutes as the doing of the field.
Revision Record
Second edition (October 2026): the dashes used as separators in the text were replaced by punctuation; section headings containing mathematics, which had been cut off in display, were repaired; no claim, number or reference was changed.
References
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- Yamagishi, Y. (2026). “Mass” Is Seven Things, One of Which Goes Negative and One of Which Depends on the Direction of the Push [Paper 240]. Zenodo. 10.5281/zenodo.22129765
- Yamagishi, Y. (2026). The Only Scale at Which Dynamic Similarity Is Exact Is One [Paper 256]. Zenodo. 10.5281/zenodo.22168919
- Yamagishi, Y. (2026). $4\pi$ Appears Only When the Source Is a Point [Paper 258]. Zenodo. 10.5281/zenodo.22169011
- Yamagishi, Y. (2026). The Displacement Current Is Not a Current [Paper 267]. Zenodo. 10.5281/zenodo.22169814
Reference
Yamagishi, Y. (2026). With the Same Magnetic Field, the Alfven Speed Differs by a Factor of $7.74\times10^{7}$ Between the Corona and the Earth’s Core [Paper 280]. YOI: https://www.yamagishiyuuki.com/yoi/280
DOI: 10.5281/zenodo.22170282
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 280. This paper is a preprint and has not been peer reviewed.