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. No responsibility is taken for the correctness of the information on this page.

Overview

• Flux conversion
• H i absorption lines
• Conversion of velocity and frequency
• Rest frames
• Moment analysis of spectral lines
• The angular separation of two points on a shpere

Flux conversion

Flux densities can be converted into brightness temperatures via

where λ is the wavelength of the observed radiation, S is the flux density, Ω is the beam solid angle of the telescope, and k denotes the Boltzmann constant. The equation can be simplified to

with [T_B] = K, [λ] = cm, [S] = mJy / beam, and [a] = [b] = arcsec, where a and b are the beam major and minor axis. If we assume a spherical beam with HPBW θ = a = b and a wavelength of λ = 21.1 cm (H i line), we can further simplify the equation to

where θ is given in arcseconds. In the case of H i observations the integrated flux over a spectral line can be directly converted into the corresponding H i column density N_HI under the assumption that the gas is optically thin (τ ≪ 1):

with [N_HI] = cm⁻², [T_B] = K, and [v] = km/s.

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:

where T_B(ν) is the observed brightness temperature profile of the H i line, T_S is the spin temperature of the gas, T_C is the brightness temperature of any background continuum emission, and τ(ν) is the optical depth of the gas as a function of frequency, ν. If we assume that the optical depth of the gas is small, i.e. τ(ν) ≪ 1, the equation can be simplified to

If we now define the brightness temperature of the spectral line as the difference between the continuum level and the observed brightness temperature, T_L(ν) ≡ T_C − T_B(ν), we can rewrite the above equation as

On the other hand, the relation bewteen the observed column density, N_HI, and the optical depth is expressed by

where C is a constant. By inserting Eq. (2) into Eq. (3) and assuming that the background continuum source is very bright (T_S ≪ T_C) we get the following expression for the column density:

From this we can directly estimate the relative strength of the H i absorption line, T_L / T_C, for a particular column density and spin temperature of the gas. The constant, C, is the same as in Eq. (1) if we integrate over velocity instead of frequency.

Conversion of velocity and frequency

The exact relativistic equation for the conversion of the observed frequency, ν, into radial velocity, v, reads

where c denotes the speed of light, and ν₀ is the rest frequency of the observed spectral line. There are two commonly used approximations to this equation which are valid for low velocities. The so-called »optical definition« reads

and the so-called »radio definition« is

The advantage of the »radio definition« is that equal increments in frequency correspond to equal increments in radial velocity. However, the »radio definition« has been deprecated by the International Astronomical Union (IAU) and should not be used anymore.

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 erroneously 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.

For objects located in the Galaxy at larger distances from the Sun one usually takes the local standard-of-rest (LSR) frame as reference for radial velocities. The LSR frame accounts for the peculiar motion of the Sun of 16.5 km/s with respect to the regular rotation of the Galaxy. Radial velocities in the LSR frame can be calculated from the BSR velocities via

where l and b are the Galactic longitude and latitude. 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

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

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 of spectral lines

Let's assume that the spectrum is given in terms of intensity A(v) (e.g. brightness temperature T_B) as a function of radial velocity v with a bin width of Δv. The zeroth moment of the spectrum is simply the integrated flux over the spectral line:

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

The second moment is a measure for the velocity dispersion, σ, 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:

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_B, and spin temperature, T_S:

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:

Here, m_H = 1.674 · 10⁻²⁷ kg is the mass of a hydrogen atom, and Δv is the FWHM of the H i line.

The angular separation of two points on a shpere

One of the frequently needed parameters in astronomy is the angular distance between two objects in the sky. Let's assume a polar coordinate system with an azimuthal angle 0 ≤ φ < 2π, a polar angle 0 ≤ θ ≤ π and a radius of unity (r = 1). The figure on the right illustrates how the angles φ and θ are defined. If we have two points P₁ and P₂ on the sphere defined by the above coordinate system, we can easily determine their angular separation α via

with P₁ = |P₁| = 1 and P₂ = |P₂| = 1. Substituting the corresponding angles φ₁, φ₂, θ₁ and θ₂ into the above equation yields

When using the above formula to determine the angular distance of two positions in the sky we have to keep in mind that the polar angle in celestial coordinate systems (e.g. the declination, δ, or the Galactic latitude, b), is usually running from −π/2 to +π/2 so that a conversion into polar coordinates is necessary before calculating α.