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I'm calculating the Alfven velocity in space plasma.

B = 4.2 nT (nanoTesla)

mu_0 = 1.25663706212e-06 (vacuum permeability in Henry per meter)

n = 21.8906 (number density in 1/cm^3)

m_p = 1.67262192369e-27 (proton mass in kg)

The Alfven velocity equation is:

V_A = B/sqrt(mu_0*n*m_p)

I calculated it as

V_A = (4.2*1e-9)/sqrt(mu_0*n*1e-6*mp)

here obviously, 4.2*1e-9 for changing from nanoTesla to Tesla and n*1e-6 for 1/cm^3 to 1/m^3

The result is 18741529400.533485 m/s or 18741529.400533486 km/s

Which is too fast, even faster than the speed of light, which is 3*10^8 m/s.

So, am I doing something wrong here?

Daniel
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  • Your conversion from cm$^{-3}$ to m$^{-3}$ is incorrect, it should be $10^6$ instead – Zade Johnston Nov 17 '22 at 02:00
  • Can you elaborate more? – Daniel Nov 17 '22 at 02:01
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    $\textrm{cm}^{-3} = (10^{-2} \textrm{m})^{-3} = 10^{6} \textrm{m}^{-3}$ – Zade Johnston Nov 17 '22 at 02:02
  • Oh, you're right. Thank you very much. Too dumb to realize this, hehe. – Daniel Nov 17 '22 at 02:13
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    Note that $v_A$ is derived in a non-relativistic system, so that it blows up for certain values of $B$, $n$, etc should give pause, as in your case, to double check the math. – Kyle Kanos Nov 17 '22 at 04:03
  • Does this help? https://physics.stackexchange.com/a/179057/59023 – honeste_vivere Nov 17 '22 at 13:35
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    The approximation goes as $V_{A}$ ~ 21.81 B[nT]/$\sqrt{n [ cm^{-3} ]}$ km/s, where the units in []'s are meant to imply the input units for the magnetic field, B, and the number density, n. So for your input values I get something like 19.58 km/s. Side note: No particle instrument in space is accurate to better than ~5-10%, so reporting more than one decimal place is not necessary. – honeste_vivere Nov 17 '22 at 13:37

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