射电天文学

reference: Essential Radio Astronomy by James J. Condon, and Scott M. Ransom

Radiation Fundamentals

Brightness and Flux Density

• Specific intensity / brightness $I_{\nu }$ or $I_{\lambda }$$$I_{\nu }\equiv \frac{\mathrm{d}P}{(\cos \theta \mathrm{d}\sigma )\mathrm{d}\nu \mathrm{d}\mathrm{\Omega }}$$$$I_{\lambda }\equiv \frac{\mathrm{d}P}{(\cos \theta \mathrm{d}\sigma )\mathrm{d}\lambda \mathrm{d}\mathrm{\Omega }}$$

units

• MKS units: metre, kilogram, second
• cgs units: centimetre, gram, second

• Total intensity $I$

  $$I\equiv {\int }_0^{\mathrm{\infty }}{I}_{\nu }(\nu )\mathrm{d}\nu ={\int }_0^{\mathrm{\infty }}{I}_{\lambda }(\lambda )\mathrm{d}\lambda$$

• Flux density $S_{\nu }$

  $$S_{\nu }={\int }_{\text{source}}{I}_{\nu }(\theta ,\varphi )\cos \theta \mathrm{d}\mathrm{\Omega }$$

  astronomers use smaller units

  $$1\text{jansky}=1\text{Jy}\equiv {10}^{-26}{\text{ W m}}^{-2}{\text{ Hz}}^{-1}$$

• Spectral luminosity $L_{\nu }$

  $$L_{\nu }=4\pi d^2{S}_{\nu }$$

• Luminosity $L$

  $$L\equiv {\int }_0^{\mathrm{\infty }}{L}_{\nu }\mathrm{d}\nu$$

Radiative Transfer

1. Absorption

   • the linear absorption coefficient $\kappa$

     $$\frac{\mathrm{d}{I}_{\nu }}{I_{\nu }}=-\kappa \mathrm{d}s$$

   • the optical depth or opacity $\tau$

     $$\tau \equiv -{\int }_{s_{\text{out}}}^{s_{\text{in}}}\kappa (s^{\prime })\mathrm{d}{s}^{\prime }$$

     thus

     $$\frac{I_{\nu }(s_{\text{out}})}{I_{\nu }(s_{\text{in}})}=\exp (-\tau )$$

     If $\tau \ll 1$, the absorber is said to be optically thin; if $\tau \gg 1$, it is optically thick.

2. Emission

   • the emission coefficient $j_v$$$j_{\nu }\equiv \frac{\mathrm{d}{I}_{\nu }}{\mathrm{d}s}$$
   • the equation of radiative transfer$$\frac{\mathrm{d}{I}_{\nu }}{\mathrm{d}s}=-\kappa I_{\nu }+j_{\nu }$$

In fact, the absorption coefficient and emission coefficient is not independent in full thermodynamic equilibrium (TE)

• Kirchhoff's law for a system in TE:

  $$\frac{j_{\nu }(T)}{\kappa (T)}=B_{\nu }(T)$$

  where $B_{\nu }(T)$ is the blackbody radiation. It applies in local thermodynamic equilibrium (LTE) as well as in TE.

Emission and Absorption of Radio Waves in the Earth's Atmosphere

• brightness temperature $T_b$$$T_b(\nu )\equiv \frac{I_{\nu }{c}^2}{2k{\nu }^2}$$

Using Kirchhoff's law to determine atmospheric absorption by measuring atmospheric emission as a function of zenith angle.

$$\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{I}_{\nu }}{\mathrm{d}s}}=-\kappa I_{\nu }+j_{\nu }\\ j_{\nu }=\kappa B_{\nu }(T_{\text{atm}})\end{array}\Rightarrow$$$$T_b=T_{\text{atm}}[1-\exp (-{\tau }_Z\sec z)]$$

Emission, Absorption, and Reflection from an Opaque Body

• absorption coefficient $a(\nu )$
• reflection coefficient $r(\nu )$
• opaque: $a(\nu )+r(\nu )=1$
• emission coefficient $e(\nu )$
• Kirchhoff's law for opaque bodies: $e(\nu )=a(\nu )=1-r(\nu )$

Blackbody Radiation

blackbody: $a(\nu )=1$

The Rayleigh–Jeans Approximation

1. standing waves

   $$\nu =\frac{c}{2a}\sqrt{n_x^2+n_y^2+n_z^2}$$

   modes' numbers

   $$N_{\nu }(\nu )\mathrm{d}\nu =\frac{4\pi {\rho }^2\mathrm{d}\rho }{8}\times 2=\pi {\left(\frac{2a\nu }{c}\right)}^2\frac{2a}{c}\mathrm{d}\nu$$

2. the average energy per mode

   $$⟨E⟩=\frac{{\int }_0^{\mathrm{\infty }}EP(E)\mathrm{d}E}{{\int }_0^{\mathrm{\infty }}P(E)\mathrm{d}E}=kT,P(E)\propto \exp (-\frac{E}{kT})$$

3. the spectral energy density

   $$u_{\nu }=\frac{1}{c}{\int }_{4\pi }{B}_{\nu }\mathrm{d}\mathrm{\Omega }=\frac{4\pi }{c}{B}_{\nu }$$

4. the Rayleigh-Jeans approximation

$$B_{\nu }=\frac{2kT{\nu }^2}{c^2}=\frac{2kT}{{\lambda }^2}$$

that is valid only in the low-frequency limit $h\nu \ll kT$

The Planck Radiation Law

1. obtain the Planck's law:

$$P(E)=P(nh\nu )\propto \exp (-\frac{nh\nu }{kT})$$$$⟨E⟩=\frac{\sum _{n=0}^{\mathrm{\infty }}nh\nu P(nh\nu )}{\sum _{n=0}^{\mathrm{\infty }}P(nh\nu )}=\frac{h\nu }{{\displaystyle \exp \left(\frac{h\nu }{kT}\right)-1}}$$$$B_{\nu }=\frac{2kT{\nu }^2}{c^2}\left[\frac{{\displaystyle \frac{h\nu }{kT}}}{{\displaystyle \exp \left(\frac{h\nu }{kT}\right)-1}}\right]=\frac{2h{\nu }^3}{c^2}\frac{1}{{\displaystyle \exp \left(\frac{h\nu }{kT}\right)-1}}$$

2. Stefan-Boltzmann law

$$B(T)\equiv {\int }_0^{\mathrm{\infty }}{B}_{\nu }(T)\mathrm{d}\nu =\frac{\sigma T^4}{\pi }$$

Stefan-Boltzmann constant

$$\sigma \equiv \frac{2{\pi }^5{k}^4}{15c^2{h}^3}\approx 5.67\times {10}^{-5}\frac{\text{erg}}{{\text{cm}}^2{\text{ s K}}^4\text{ (sr)}}$$

3. "Wien's displacement law"

$$\left(\frac{{\lambda }_{\text{max}}}{\text{cm}}\right)\approx 0.29{\left(\frac{T}{\text{K}}\right)}^{-1}$$

familiar form in optical one, but not the same, here use brightness.

Noise Generated by a Warm Resistor

• Nyquist formula

$$P_{\nu }=\frac{h\nu }{\exp \left({\displaystyle \frac{h\nu }{kT}}\right)-1}$$