## Useful equations for radio astronomy

On this page I have compiled a few equations which are regularly needed for the statistical analysis of spectra in radio astronomy, in particular for the 21-cm line of neutral hydrogen. Many of the equations presented on this page are generally only valid for sources at low redshift, and special, redshift-dependent corrections will have to be applied for objects at cosmological distances!

No responsibility is taken for the correctness of the information on this page. Please let me know if you find any errors or inaccurate information. The equations on this page are rendered by MathJax, and JavaScript will have to be enabled in your browser for them to be correctly displayed.

### Flux conversion

The concept of brightness temperature is based on the Rayleigh–Jeans approximation to Planck’s law. The brightness temperature, $T_{\rm B}$, of an astronomical source is defined as the temperature of a black body which emits the same spectral radiance, $B_{\nu}$, as the source, hence

\begin{equation} T_{\rm B} = \frac{\mathrm{c}^{2} B_{\nu}}{2 \mathrm{k_{B}} \nu^{2}} = \frac{\mathrm{\lambda}^{2} B_{\nu}}{2 \mathrm{k_{B}}} \end{equation}

where $\nu$ is the frequency of the emission and we have made use of the relation between wavelength and frequency of electromagnetic radiation, $\nu \lambda = \mathrm{c}$. The flux density of a source is simply defined as the integral of the spectral radiance over the solid angle of the source in the sky, leading to the relation between brightness temperature and flux density, $S_{\nu}$, of

\begin{equation} S_{\nu} = \frac{2 \mathrm{k_{B}}}{\lambda^{2}} \iint \limits_{\rm src} T_{\rm B} \mathrm{d} \Omega . \end{equation}

In the simple case where a source of constant brightness temperature fills the entire size of the telescope beam, we obtain the following relation between brightness temperature and flux density:

\begin{equation} \bbox[#F0F0F0, 10px, border:1px solid black]{T_{\rm B} = \frac{\lambda^{2} S_{\nu}}{2 \mathrm{k}_{\rm B} \Omega_{\rm bm}}} \end{equation}

where $\Omega_{\rm bm}$ is the beam solid angle of the telescope. Assuming a Gaussian beam of solid angle $\Omega_{\rm bm} = \pi \vartheta_{a} \vartheta_{b} \, / \, [4 \ln(2)]$, the equation can be simplified to

\begin{equation} \frac{T_{\rm B}}{\rm K} \approx 1.360 \, \left( \frac{\lambda}{\rm cm} \right)^{\!2} \, \frac{S_{\nu}}{\rm mJy} \, \left( \frac{\vartheta_{a} \vartheta_{b}}{\mathrm{arcsec}^{2}} \right)^{-1} \end{equation}

where $\vartheta_{a}$ and $\vartheta_{b}$ are the beam major and minor axis in arcseconds. If we assume a circular beam with FWHM $\vartheta = \vartheta_{a} = \vartheta_{b}$ and a wavelength of $\lambda = 21.1~\mathrm{cm}$ (H i line), we can further simplify the equation to

\begin{equation} \frac{T_{\rm B}}{\rm K} \approx 605.7 \, \frac{S_{\nu}}{\rm mJy} \, \left( \frac{\vartheta}{\mathrm{arcsec}} \right)^{-2} \end{equation}

where $\vartheta$ is again given in arcseconds. In the case of H i observations the integrated flux over a spectral line, $S$, can be directly converted into the corresponding H i column density, $N_{\rm H\,I}$, under the assumption that the gas is optically thin $(\tau \ll 1)$:

\begin{equation} \bbox[#F0F0F0, 10px, border:1px solid black]{\frac{N_{\rm H\,I}}{\rm cm^{-2}} \approx 2.33 \times 10^{20} \, (1 + z)^{4} \frac{S}{\rm Jy \, Hz} \, \left( \frac{\vartheta_{a} \vartheta_{b}}{\rm arcsec^{2}} \right)^{-1} .} \end{equation}

Note that the calculation of column density has a strong dependence on redshift, $z$. In the case of H i observations of the Milky Way at $z = 0$ this equation can be simplified and instead be expressed in terms of brightness temperature and velocity:

\begin{equation} {\frac{N_{\rm H\,I}}{\rm cm^{2}} \approx 1.823 \times 10^{18} \int \frac{T_{\rm B}}{\rm K} \, \frac{\mathrm{d}v}{\rm km \, s^{-1}} .} \end{equation}

Under the usual assumption of optically thin gas we can also derive the H i mass of a galaxy or gas cloud from the integrated H i flux,

\begin{equation} \bbox[#F0F0F0, 10px, border:1px solid black]{\frac{M_{\rm H\,I}}{M_{\odot}} \approx 49.7 \, \frac{S}{\mathrm{Jy \, Hz}} \, \left( \frac{D_{\rm L}}{\mathrm{Mpc}} \right)^{\!2}} \end{equation}

where $S$ is the integrated flux and $D_{\rm L}$ is the luminosity distance of the source.

### H i absorption lines

For the 21-cm line emission of neutral atomic hydrogen the equation of radiative transfer can be written in the following form:

\begin{equation} T_{\rm B}(\nu) = T_{\rm S} \left[ 1 - \mathrm{e}^{-\tau(\nu)} \right] + T_{\rm C} \, \mathrm{e}^{-\tau(\nu)} \end{equation}

where $T_{\rm B}(\nu)$ is the observed brightness temperature profile of the H i line, $T_{\rm S}$ is the spin temperature of the gas, $T_{\rm C}$ is the brightness temperature of any background continuum emission, and $\tau(\nu)$ is the optical depth of the gas as a function of frequency, $\nu$. If we assume that the optical depth of the gas is small, i.e. $\tau(\nu) \ll 1$, the equation can be simplified to

\begin{equation} T_{\rm B}(\nu) = T_{\rm S} \, \tau(\nu) + T_{\rm C} \, [1 - \tau(\nu)] \end{equation}

If we now define the brightness temperature of the spectral line as the difference between the continuum level and the observed brightness temperature, $T_{\rm L}(\nu) \equiv T_{\rm C} - T_{\rm B}(\nu)$, we can rewrite the above equation as

\begin{equation} \tau(\nu) = \frac{T_{\rm L}(\nu)}{T_{\rm C} - T_{\rm S}} \label{eqn_tau} \end{equation}

On the other hand, the relation bewteen the observed column density, $N_{\rm H\,I}$, and the optical depth is expressed by

\begin{equation} N_{\rm H\,I} = C \, T_{\rm S} \! \int \! \tau(\nu) \, \mathrm{d}\nu \label{eqn_nhiabs} \end{equation}

where $C$ is a constant. By inserting Eq. $\eqref{eqn_tau}$ into Eq. $\eqref{eqn_nhiabs}$ and assuming that the background continuum source is very bright $(T_{\rm S} \ll T_{\rm C})$ we get the following expression for the column density:

\begin{equation} N_{\rm H\,I} = C \, T_{\rm S} \! \int \! \frac{T_{\rm L}(\nu)}{T_{\rm C}} \mathrm{d}\nu \end{equation}

From this we can directly estimate the relative strength of the H i absorption line, $T_{\rm L} / T_{\rm C}$, for a particular column density and spin temperature of the gas. The constant, $C$, is the same as in Eq. $\eqref{eqn_nhi}$ if we integrate over velocity instead of frequency.

### Conversion of velocity and frequency

The special relativistic equation for the conversion of the observed frequency, $f$, of a source into radial velocity, $v$, under the assumption that the object is moving towards or away from the observer reads

\begin{equation} \frac{v}{\mathrm{c}} = \frac{f_{0}^{2} - f^{2}}{f_{0}^{2} + f^{2}} \label{eqn_relDoppler} \end{equation}

where $\mathrm{c}$ denotes the speed of light, and $f_{0}$ is the rest frequency of the observed line transition. This equation assumes that the observed redshift of a source is due to its relativistic velocity along the line of sight towards or away from the observer (relativistic Doppler effect). Note that the relativistic Doppler effect depends on the transversal velocity component of the object as well, and Eq. $\eqref{eqn_relDoppler}$ will therefore only be valid for pure line-of-sight motion.

Also note that the observed redshift of distant sources is largely due to the “cosmological expansion” of space, not due to their velocity with respect to the observer. Hence, the above relation will not yield sensible results for sources at higher redshift, and frequency (or redshift) rather than velocity should be used to characterise sources beyond redshift zero.

There are two commonly used approximations to this equation which are accurate for small velocities of up to a few hundred km/s. The so-called “optical definition” reads

\begin{equation} \bbox[#F0F0F0, 10px, border:1px solid black]{\frac{v_{\rm opt}}{\mathrm{c}} = \frac{f_{0}}{f} - 1 = z} \end{equation}

and the so-called “radio definition” is

\begin{equation} \frac{v_{\rm rad}}{\mathrm{c}} = 1 - \frac{f}{f_{0}} = \frac{z}{1 + z} \end{equation}

The advantage of the “radio definition” is that equal increments in frequency correspond to equal increments in radial velocity. However, the “radio definition” is deprecated by the International Astronomical Union (IAU) and should not be used any more, as the resulting velocity values are arbitrary and not physically motivated.

The “optical definition”, $v = \mathrm{c}z$, is also referred to as the recessional velocity and used as a convenient proxy for redshift when characterising distant objects, e.g. in redshift surveys. Despite having the dimension of a velocity, it must not be mistaken for a true velocity. Instead, it is simply the redshift of a source multiplied by a constant (in this case the speed of light) and entirely unrelated to the source’s peculiar velocity (with the exception of the nearest objects within the Local Group).

### Rest frames

The observed radial velocity of an astronomical object is subject to several projection effects such as the rotation and the orbital motion of the Earth, the motion of the Sun around the Galactic centre, the motion of our Galaxy within the Local Group, etc. To be able to interpret the observed radial velocity one must convert it into an appropriate rest frame.

A useful rest frame for objects in the solar neighbourhood is the so-called barycentric standard-of-rest (BSR) frame which uses the barycentre of the Solar System as reference point. Normally, the spectra observed with a radio telescope are already provided in the BSR frame. The BSR frame is often referred to as the heliocentric standard-of-rest (HSR) frame. The latter one, however, uses the barycentre of the Sun as reference point instead of the Solar System barycentre. The difference between barycentric and heliocentric velocities, however, is rather small and negligible in most cases.

For objects located in the Galaxy at larger distances from the Sun one usually uses the local standard-of-rest (LSR) frame as the reference for radial velocities. The LSR frame accounts for the peculiar motion of the Sun of about 16.55 km/s with respect to the regular rotation of the Galaxy. Radial velocities in the LSR frame can be calculated from barycentric velocities via

\begin{equation} v_{\rm LSR} = v_{\rm BSR} + 9 \cos(l) \cos(b) + 12 \sin(l) \cos(b) + 7 \sin(b) \end{equation}

where $l$ and $b$ are the Galactic longitude and latitude. This definition is the so-called “dynamical defintion” (also referred to as the LSRD) as specified by the IAU. There is an alternative “kinematical definition” (referred to as LSRK) which results in a slightly higher velocity of about 20 km/s in the direction of $(\alpha, \delta) = (270^{\circ},30^{\circ})$ in the B1900 system. However, the LSRD definition is the one most commonly used and usually referred to as the LSR.

For the description of circumgalactic objects it is useful to correct also for the rotation of our Milky Way of 220 km/s. The corresponding reference frame, the so-called Galactic standard-of-rest (GSR) frame, is derived from the LSR frame via

\begin{equation} v_{\rm GSR} = v_{\rm LSR} + 220 \sin(l) \cos(b) \end{equation}

For objects spread across the Local Group a reference frame accounting for the motion of our Milky Way of about 80 km/s with respect to the Local Group barycentre would be ideal. The corresponding radial velocities in the so-called Local Group standard-of-rest (LGSR) frame can be calculated from the GSR velocities via

\begin{equation} v_{\rm LGSR} = v_{\rm GSR} - 62 \cos(l) \cos(b) + 40 \sin(l) \cos(b) - 35 \sin(b) \end{equation}

In principal, one can correct the radial velocity for rest frames of even higher order in the hierarchy of the universe. The reference frames mentioned above, however, are the ones most frequently used.

### Moment analysis

#### Standard spectral moments

Let’s assume that the spectrum is given in terms of intensity $A(v)$ (e.g. brightness temperature $T_{\rm B}$) as a function of radial velocity $v$ with a bin width of $\Delta v$. The zeroth moment of the spectrum is simply the integrated flux over the spectral line:

\begin{equation} M_{0} = \Delta v \sum A(v) \end{equation}

The first moment defines the intensity-weighted velocity of the spectral line. It can be taken as a measure for the mean velocity of the gas. The first moment is defined by

\begin{equation} M_{1} = \frac{\sum v A(v)}{\sum A(v)} \end{equation}

The second moment is a measure for the velocity dispersion, $\sigma$, of the gas along the line of sight, i.e. the width of the spectral line. It is defined by the intensity-weighted square of the velocity:

\begin{equation} M_{2} = \sqrt{\frac{\sum (v - M_{1})^{2} A(v)}{\sum A(v)}} \end{equation}

#### Skewness and kurtosis

In addition to these spectral moments it is also occasionally useful to calculate higher-order moments of imaging data in order to determine the skewness and kurtosis of the distribution of flux density values. The skewness can be defined as

\begin{equation} S = \frac{\mathfrak{M}_{3}}{\mathfrak{M}_{2}^{3/2}} \end{equation}

while the kurtosis can likewise be written as

\begin{equation} K = \frac{\mathfrak{M}_{4}}{\mathfrak{M}_{2}^{2}} . \end{equation}

The higher-order moments, $\mathfrak{M}_{n}$, required for their calculation are defined as

\begin{equation} \mathfrak{M}_{n} = \frac{1}{N} \sum \limits_{i = 1}^{N} (y_{i} - \bar{y})^{n} \end{equation}

where $N$ is the number of data samples, $y_{i}$ are the flux density values, and $\bar{y} = \sum y_{i} \, / \, N$ is the mean flux density. Skewness is a measure for the level of asymmetry in the flux density distribution, while kurtosis can be used as a measure of how dominant the wings or tails of the distribution are. For pure Gaussian noise, i.e. for normally distributed values of $y_{i}$, we expect to measure $S = 0$ and $K = 3$.

### Temperature from H i lines

From the intensity and width of H i lines one can usually obtain a lower and upper limit of the kinetic temperature of the gas. The lower limit is given by the brightness temperature of the line. Due to its long life time, the 21-cm transition is usually collisionally excited, and the spin temperature of the gas is equal to the kinetic temperature. From the equation of radiative transfer we therefore get the following relation between brightness temperature, $T_{\rm B}$, and spin temperature, $T_{\rm S}$:

\begin{equation} T_{\rm B} = T_{\rm S} (1 - \mathrm{e}^{-\tau}) \le T_{\rm S} = T_{\rm kin} \end{equation}

An upper limit of the kinetic temperature can be derived from the line width. This is possible because the intrinsic line width of the H i line is very small due to the long life time of the transition. Hence, the observed line width is dominated by Doppler broadening due to effects such as the kinetic temperature of the gas, internal turbulence or rotation of the gas, or multiple clouds along the line of sight. From the Maxwell distribution we therefore get:

\begin{equation} T_{\rm kin} \le \frac{m_{\rm H} \Delta v^{2}}{8 \mathrm{k}_{\rm B} \ln(2)} \end{equation}

Here, $m_{\rm H} \approx 1.674 \times 10^{-27}~\mathrm{kg}$ is the mass of a hydrogen atom, and $\Delta v$ is the FWHM of the H i line.