Vector space and 3D flow field

[Leo Liu]

Homework Statement

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Relevant Equations

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Could someone explain the green highlight to me, please?

 

[anuttarasammyak]

The sum emits light of constant speed c with constant power. The emitted photons becomes more sparse at distant places as $r^{-2}$ so the sun is observed more dimmer.

 

[Leo Liu]

The sum emits light of constant speed c with constant power. The emitted photons becomes more sparse at distant places as $r^{-2}$ so the sun is observed more dimmer.

Thanks for providing some physical intuition. But is it because the surface density of a mass flow is
$$\frac{\dot m}{A}=\frac{\dot m}{4\pi r^2}$$
?

 

[anuttarasammyak]

Yes. What's wrong with it ?

 

[Leo Liu]

Yes. What's wrong with it ?

Nothing. Just making sure I understand where this proportionality comes from.

 

[ergospherical]

Another perspective: imagine two concentric spherical surfaces of radii $\rho_1$ and $\rho_2$ ($\rho_2 > \rho_1$) which bound a region $R$. In steady state the mass contained in $R$ is constant, so the mass fluxes into the inner surface and out of the outer surface are equal: $4\pi {\rho_1}^2 \delta_1 v = 4\pi {\rho_2}^2 \delta_2 v \, \implies \, \delta \rho^2 = \mathrm{constant}$.