Calculating pressure and flux tensors in radiative astrophysics

Question

I need to calculate the components of the radiative flux tensor and the pressure tensor, but I am having a little trouble figuring out what the integral should be. The flux is defined as

$$F_v = 4{\pi}H_v$$ where $$H_v = 1/4{\pi} \int I_v {\hat k} d{\Omega}$$

and

$${\hat k_i} = (sin{\theta}cos{\phi}, sin{\theta}sin{\phi}, cos{\theta})$$

I know I need to integrate over the solid angle, and so the limits of integration are 0 to 2pi but I need to determine the components for an observer distance d away. Do I need a second integral that involves d in some way, or do I just integrate over the solid angle?

The same question holds for the pressure tensor, which is defined similarly, except

$$K_v = 1/4{\pi}\int I_v {\hat k_i}{\hat k_j}d{\Omega}$$

I'm pretty sure I need to something like

$$1/4{\pi} \int_{0}^{2{\pi}} d{\phi} \int {\hat k_i}d{\theta}$$

but I don't know what the limits of integration are for an observer distance d from the center of the sphere.

Answer 1

First I start with the component formulation of the (monochromatic) radiative flux tensor $\mathbf{F_\nu}$, i.e. $\left( \mathbf{F_\nu} \right)^{i} \equiv F_\nu^{\;i}$,

$$ F_\nu^{\;i} := \oint_\Omega I_\nu(\mathbf{k}) \, k^{i} \, d\Omega \quad. $$

If we express the direction vector $\mathbf{k}$ in cartesian coordiantes, i.e. $\mathbf{k} = (k^i)_{i \in \{x,y,z\}} = (k^x, k^y, k^z)$ we can re-write¹ the radiation flux from above, we get

$$ \left( F_\nu^{\;x},\, F_\nu^{\;y}, \, F_\nu^{\;z} \right) = \left( \oint_\Omega I_\nu(\mathbf{k}) \, k^{x} \, d\Omega, \, \oint_\Omega I_\nu(\mathbf{k}) \, k^{y} \, d\Omega, \, \oint_\Omega I_\nu(\mathbf{k}) \, k^{z} \, d\Omega \right) \quad,$$ with

$$ \mathbf{k} = \left( sin (\theta) cos(\phi),\, sin (\theta) sin(\phi),\, cos(\theta) \right) \quad. $$

A useful resource for orientation about the definition of the solid angle $\Omega$ and how to apply certain ranges of integration is the nrao.edu website which also reminds that the solid angle increment $d\Omega$ in spherical coordiantes is given by $d\Omega = sin\theta \, d\theta \, d\phi$. Furthermore, as you asked as second part of your question, how to tackle ways to take a certain distance $d$ from the source of emission to the observer into account, the solid angle $\Omega$ needs to be changed accordingly based on how large the detector will be, see also the nrao.edu website and slide no. 5 here.

For the (monochromatic) pressure tensor $\mathbf{P_\nu}$ you need to be careful not to confuse the former with the $K$ integral or angularmoment of order two $\mathbf{K_\nu}$. However, both of which are directly related through,

$$ P_\nu^{\; ij} \equiv \frac{4\pi}{c_0} K_\nu^{\; ij} = \frac{1}{c_0} \oint I_\nu(\mathbf{k}) \, k^i \, k^j \, d\Omega \quad. $$

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Addendum: (more background info, in case you need it)

For any star we know that the effective temperature $T_{\text{eff}}$ of the radiation source depends on the total, actual radiation flux, integrated over all frequences if we can assume the star's spectrum to be reasonably close (or identical) to a black body,

$$ F_{tot.} = \int_{0}^{\infty}d\nu \, \int_\Omega I_\nu(\mathbf{k}) \, k^{i} \, d\Omega \equiv \sigma \, T_{\text{eff}} \overset{!}{=} \frac{L}{4 \pi r^2} \quad, $$

which finally links the flux density to the luminosity of the source.

Although the source intensity $I_\nu$ is independent of the distance $d$ between the source and the observer, the flux density is not independent of source distance, and scales due to the solid angle $\Omega$ with

$$ \int_\Omega \ldots d\Omega \propto \frac{1}{d^2}, $$

see also the nrao.edu site.

I hope this synopsis helps as a starting point.

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Footnotes & References:

¹ Hubeny I. & Mihalas, D.: Theory of Stellar Atmospheres. Princeton University Press, 2015, p. 72ff.