# Harvard Astronomy 201b

## CHAPTER: Definitions of Temperature

In Book Chapter on March 7, 2013 at 3:27 am

(updated for 2013)

The term “temperature” describes several different quantities in the ISM, and in observational astronomy. Only under idealized conditions (i.e. thermodynamic equilibrium, the Rayleigh Jeans regime, etc.) are (some of) these temperatures equivalent. For example, in stellar interiors, where the plasma is very well-coupled, a single “temperature” defines each of the following: the velocity distribution, the ionization distribution, the spectrum, and the level populations. In the ISM each of these can be characterized by a different “temperature!”

#### Brightness Temperature $T_B =$ the temperature of a blackbody that reproduces a given flux density at a specific frequency, such that $B_\nu(T_B) = \frac{2 h \nu^3}{c^2} \frac{1}{{\rm exp}(h \nu / kT_B) - 1}$

Note: units for $B_{\nu}$ are ${\rm erg~cm^{-2}~s^{-1}~Hz^{-1}~ster^{-1}}$.

This is a fundamental concept in radio astronomy. Note that the above definition assumes that the index of refraction in the medium is exactly 1.

#### Effective Temperature $T_{\rm eff}$ (also called $T_{\rm rad}$, the radiation temperature) is defined by $\int_\nu B_\nu d\nu = \sigma T_{{\rm eff}}^4$,

which is the integrated intensity of a blackbody of temperature $T_{\rm eff}$. $\sigma = (2 \pi^5 k^4)/(15 c^2 h^3)=5.669 \times 10^{-5} {\rm erg~cm^{-2}~s^{-1}~K^{-4}}$ is the Stefan-Boltzmann constant.

#### Color Temperature $T_c$ is defined by the slope (in log-log space) of an SED. Thus $T_c$ is the temperature of a blackbody that has the same ratio of fluxes at two wavelengths as a given measurement. Note that $T_c = T_b = T_{\rm eff}$ for a perfect blackbody.

#### Kinetic Temperature $T_k$ is the temperature that a particle of gas would have if its Maxwell-Boltzmann velocity distribution reproduced the width of a given line profile. It characterizes the random velocity of particles. For a purely thermal gas, the line profile is given by $I(\nu) = I_0~e^{\frac{-(\nu-\nu_{jk})^2}{2\sigma^2}}$,

where $\sigma_{\nu}=\frac{\nu_{jk}}{c}\sqrt{\frac{kT_k}{\mu}}$ in frequency units, or $\sigma_v=\sqrt{\frac{kT_k}{\mu}}$ in velocity units.

In the “hot” ISM $T_k$ is characteristic, but when $\Delta v_{\rm non-thermal} > \Delta v_{\rm thermal}$ (where $\Delta v$ are the Doppler full widths at half-maxima [FWHM]) then $T_k$ does not represent the random velocity distribution. Examples include regions dominated by turbulence. $T_k$ can be different for neutrals, ions, and electrons because each can have a different Maxwellian distribution. For electrons, $T_k = T_e$, the electron temperature.

#### Ionization Temperature $T_I$ is the temperature which, when plugged into the Saha equation, gives the observed ratio of ionization states.

#### Excitation Temperature $T_{\rm ex}$ is the temperature which, when plugged into the Boltzmann distribution, gives the observed ratio of two energy states. Thus it is defined by $\frac{n_k}{n_j}=\frac{g_k}{g_j}~e^{-h\nu_{jk}/kT_{\rm ex}}$.

Note that in stellar interiors, $T_k = T_I = T_{\rm ex} = T_c$. In this room, $T_k = T_I = T_{\rm ex} \sim 300K$, but $T_c \sim 6000K$.

#### Spin Temperature $T_s$ is a special case of $T_{\rm ex}$ for spin-flip transitions. We’ll return to this when we discuss the important 21-cm line of neutral hydrogen.

#### Bolometric temperature $T_{\rm bol}$ is the temperature of a blackbody having the same mean frequency as the observed continuum spectrum. For a blackbody, $T_{\rm bol} = T_{\rm eff}$. This is a useful quantity for young stellar objects (YSOs), which are often heavily obscured in the optical and have infrared excesses due to the presence of a circumstellar disk.

#### Antenna temperature $T_A$ is a directly measured quantity (commonly used in radio astronomy) that incorporates radiative transfer and possible losses between the source emitting the radiation and the detector. In the simplest case, $T_A = \eta T_B( 1 - e^{-\tau})$,

where $\eta$ is the telescope efficiency (a numerical factor from 0 to 1) and $\tau$ is the optical depth.