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Referring to Plasma Astrophysics, Part 1, Eq 4.32,

The number density of electrons in the beam at a distance $z$ from the injection plane at $z = 0$ is defined as,

$$ n_b(z) = \int_{0}^{\infty} \int_{0}^{\pi} f(z,v, \theta) v^2 dv 2 \pi sin \theta d \theta $$

where $f(z,v, \theta)$ is the distribution function defined in terms of

$z$ - coordinate along the direction of the magnetic field

$v$ - velocity of electrons

$\theta$ - angle between $z$ and $v$

How did this definition come about?

honeste_vivere
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Refrigerator
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1 Answers1

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You are just showing the zeroth velocity moment (e.g., see answer at https://physics.stackexchange.com/a/218643/59023 for examples) of a velocity distribution function. The differential comes about because someone assumed azimuthal symmetry, i.e., you have: $$ d^{3}v \rightarrow v^{2} \ dv \ 2 \pi \ \sin{\theta} \ d\theta $$

The units of $f\left( z, v, \theta \right)$ are usually in number per unit volume per velocity cubed. So if you integrate over $d^{3}v$ you get units of number per unit volume, also known as number density.

honeste_vivere
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  • $d^3 v$ seems like volume element in spherical coordinates here, $r^2 sin \theta d \theta dv d \phi$ integrated over $\phi$. This would mean finding the number density in an infinite sphere about r = 0. But since $r$ is replaced by $v$, we are just integrating over the whole velocity space. This is to get the number density at $z$ accounting that electrons can have velocities in any direction wrt the magnetic field? – Refrigerator Feb 26 '24 at 16:37
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    I agree the differential in spherical coordinates. The velocity distribution function in its most general form is a function of time, spatial position, and velocity. Here the authors split velocity into a speed and angle but it's just velocity. They also ignored the time-dependence (presumably to assume that only spatial variations occur). The integral you show would need to be done for every position $z$. – honeste_vivere Feb 26 '24 at 16:44