Radiative Processes in Astrophysics

1. Radiation Lecture 01

• 1 Units
• 2 Proof that Specific Intensity is conserved along a ray
• 3 The Blackbody

Units

Here are some terms pertaining to telescope observations:

aperture area ($\Delta A$), solid angle on sky ($\Delta \Omega$), exposure time ($\Delta t$), collects energy ($\Delta E$), over waveband ($\Delta \nu$), but $\Delta \lambda \neq {c \over \Delta \nu }$.

${\Delta E \over \Delta t\Delta \nu \Delta A\Delta \Omega }\iff {dE \over dtd\nu dAd\Omega }\equiv I_{\nu }\,\!$

$I_{\nu }$ is the specific intensity per unit frequency.

Flux density is power per unit frequency passing through a differential area whose normal is ${\hat {n}}$. Thus, flux density is:

${F_{\nu }\equiv \int I_{\nu }\cos \theta d\Omega }\,\!$

Proof that Specific Intensity is conserved along a ray

The power received by the telescope is:

$P_{rec}=I_{\nu }d\Omega dA\,\!$

where $I_{\nu }(\alpha ,\delta )$ is the intensity as a function of right-ascension ($\alpha$) and declination ($\delta$). Say that $\Sigma _{\nu }(\alpha ,\delta )$ is the surface luminosity of a patch of sky (that is, the emitted intensity). Then power emitted by patch of sky is:

$P_{emit}=\Sigma _{\nu }{dA \over r^{2}}d{\tilde {A}}\,\!$

Recognizing that $d{\tilde {A}}=d\Omega r^{2}$:

$\Sigma _{\nu }=I_{\nu }\,\!$

This derivation assumes that we are in a vacuum and that the frequencies of photons are constant. If frequencies change, then though specific intensity $I_{\nu }$ is not conserved, ${I_{\nu } \over \nu ^{3}}$ is. Also, for redshift $z$,

$I_{\nu }\propto \nu ^{3}\propto {\frac {1}{(1+z)^{3}}}\,\!$

so intensity decreases with redshift. Finally:

${I_{\nu } \over \eta ^{2}}\,\!$

is conserved along a ray, where $\eta$ is the index of refraction.

The Blackbody

A blackbody is the simplest source: it absorbs and reemits radiation with 100% efficiency. The frequency content of blackbody radiation is given by the Planck Function:

$B_{\nu }={h\nu \over \lambda ^{2}}{2 \over (e^{h\nu \over kT}-1)}\,\!$ ${B_{\nu }={2h\nu ^{3} \over c^{2}(e^{h\nu \over kT}-1)}\neq B_{\lambda }}\,\!$

(The Planck Function for Black Body Radiation)

Derivation:

The # density of photons having frequency between $\nu$ and $\nu +d\nu$ has to equal the # density of phase-space cells in that region, multiplied by the occupation # per cell. Thus:

$n_{\nu }d\nu ={4\pi \nu ^{2}d\nu \over c^{3}}{2 \over e^{h\nu \over kT}-1}\,\!$

However,

$h\nu {n_{\nu }c \over 4\pi }=I_{\nu }=B_{\nu }\,\!$

so we have it. In the limit that $h\nu \gg kT$:

$B_{\nu }\approx {2h\nu ^{3} \over c^{2}}e^{-{h\nu \over kT}}\,\!$

Wein tail

If $h\nu \ll kT$:

$B_{\nu }\approx {2kT \over \lambda ^{2}}\,\!$

Rayleigh-Jeans tail Note that this tail peaks at $\sim {3kT \over h}$. Also,

$\nu B_{\nu }=\lambda B_{\lambda }\,\!$

1. Radiation Lecture 02
2. Radiation Lecture 03
3. Radiation Lecture 04
4. Radiation Lecture 05
5. Radiation Lecture 06
6. Radiation Lecture 07
7. Radiation Lecture 08
8. Radiation Lecture 09
9. Radiation Lecture 10
10. Radiation Lecture 11
11. Radiation Lecture 12
12. Radiation Lecture 13
13. Radiation Lecture 14
14. Radiation Lecture 15
15. Radiation Lecture 16
16. Radiation Lecture 17
17. Radiation Lecture 18
18. Radiation Lecture 19
19. Radiation Lecture 20
20. Radiation Lecture 21
21. Radiation Lecture 22
22. Radiation Lecture 23
23. Radiation Lecture 24