Reading the Notation¶
Open almost any radio-astronomy paper and within a paragraph you will meet a dense thicket of Greek letters, decorated variables, and units that the authors assume you already speak. This page is a decoder ring. You spot an unfamiliar symbol or convention in a paper, you look it up here.
This is the syntax companion to two sibling references:
- Math Preliminaries — teaches the underlying mathematics (Fourier transforms, complex numbers, statistics). Come here for how to read a symbol; go there for how it works.
- Glossary — defines the words (e.g. "visibility", "system temperature", "column density"). Come here for the symbol; go there for the concept in prose.
The aim below is to help you parse an equation, not to re-teach the physics. Where a symbol first appears in the course, the relevant chapter is noted.
The golden rule of notation
Notation is local. A symbol means whatever the author defined it to mean, usually near its first appearance or in a table of symbols. The conventions below are strong priors, not laws. Always check the paper's own definitions first — especially for \(\alpha\), \(*\), and the sign of an exponent.
1. Greek letters in common use¶
Radio astronomy leans heavily on the Greek alphabet, and several letters carry more than one meaning. The right reading is almost always fixed by context (an angle on the sky vs. a statistical quantity, for instance).
| Symbol | Name | Usual meaning in radio astronomy | Watch out |
|---|---|---|---|
| \(\nu\) | nu | frequency (Hz) — the radio astronomer's default x-axis | looks like Latin v; \(\nu\) has the little tail |
| \(\lambda\) | lambda | wavelength (m); \(\lambda = c/\nu\) | also eigenvalue in linear algebra |
| \(\alpha\) | alpha | spectral index (\(S_\nu \propto \nu^\alpha\)) or right ascension | the big clash — see §5 |
| \(\delta\) | delta (lower) | declination (sky latitude); also a small increment | distinct from \(\Delta\) |
| \(\theta\) | theta | an angle — beamwidth, zenith angle, scattering angle | often the angular size \(\theta_\mathrm{FWHM}\) |
| \(\phi\), \(\varphi\) | phi | azimuth / phase angle; also Galactic-rotation azimuth | \(\Phi\) (capital) often a flux or potential |
| \(\Omega\) | omega (cap) | solid angle (steradians); beam solid angle \(\Omega_A\), \(\Omega_\mathrm{MB}\) | \(\omega\) (lower) = angular frequency \(2\pi\nu\) |
| \(\sigma\) | sigma (lower) | standard deviation / RMS noise; also a cross-section \(\sigma\) (cm\(^2\)) | rms vs. cross-section is pure context |
| \(\Sigma\) | sigma (cap) | summation \(\sum\); also a surface density \(\Sigma\) | the operator vs. the quantity |
| \(\tau\) | tau | optical depth (dimensionless) or integration time (s) | depth in radiative transfer; time in the radiometer eqn |
| \(\rho\) | rho | correlation coefficient; also a mass density \(\rho\) | |
| \(\Delta\) | delta (cap) | a difference or interval: \(\Delta\nu\) (bandwidth), \(\Delta T\) (noise) | "change in", not a value itself |
| \(\mu\) | mu | the prefix micro- (\(10^{-6}\), e.g. \(\mu\)Jy); also a mean \(\mu\) | \(\mu\)Jy is microjansky |
| \(\eta\) | eta | an efficiency (aperture \(\eta_A\), beam \(\eta_B\), main-beam) \(\in [0,1]\) | |
| \(\kappa\) | kappa | opacity / absorption coefficient (\(\kappa_\nu\), cm\(^2\) g\(^{-1}\) or cm\(^{-1}\)) | |
| \(\epsilon\), \(\varepsilon\) | epsilon | emissivity / emission coefficient \(\epsilon_\nu\); also a small quantity | |
| \(\Phi\) | phi (cap) | a flux (e.g. photon flux) or a phase / potential | |
| \(\chi\) | chi | \(\chi^2\) the goodness-of-fit statistic; also the polarisation (E-vector) angle \(\chi=\tfrac12\arctan(U/Q)\) | \(\chi^2\) vs polarisation angle is pure context (Ch 37) |
| \(\beta\) | beta | a velocity in units of \(c\), \(\beta = v/c\); also a power-law slope | |
| \(\gamma\) | gamma | Lorentz factor \(\gamma = (1-\beta^2)^{-1/2}\); also a power-law index | |
| \(\pi\) | pi | the constant \(3.14159\ldots\) (and \(2\pi\) everywhere in Fourier work) | |
| \(\psi\) | psi | a position angle or phase | |
| \(\zeta\), \(\xi\) | zeta, xi | generic dummy variables of integration |
Disambiguating multi-use letters
- \(\alpha\): if it sits in an exponent on \(\nu\) or appears with \(\delta\) as a coordinate pair \((\alpha,\delta)\), it is right ascension; otherwise it is the spectral index.
- \(\tau\): inside \(e^{-\tau}\) it is optical depth; under a square root with a bandwidth it is integration time.
- \(\sigma\): next to a measurement ("\(3\sigma\) detection") it is noise; next to a number density it is a cross-section.
2. Core radio-astronomy symbols¶
These are the workhorses. The subscript \(\nu\) (or \(_\lambda\)) almost always reads "per unit frequency" — i.e. a spectral density (Ch. 1–2).
| Symbol | Reads as | Typical units | First seen |
|---|---|---|---|
| \(S_\nu\) | flux density at frequency \(\nu\) | Jy (\(=10^{-26}\) W m\(^{-2}\) Hz\(^{-1}\)) | Ch. 1 |
| \(I_\nu\), \(B_\nu\) | specific intensity / brightness (\(B_\nu\) = Planck) | W m\(^{-2}\) Hz\(^{-1}\) sr\(^{-1}\) | Ch. 1–2 |
| \(T_b\) | brightness temperature (temp. of a blackbody giving that \(I_\nu\)) | K | Ch. 1 |
| \(T_\mathrm{sys}\) | system temperature (total noise power as a temperature) | K | Ch. 3 |
| \(T_A\) | antenna temperature (signal power as a temperature) | K | Ch. 3–4 |
| \(T_\mathrm{rx}\) | receiver temperature (noise added by the electronics) | K | Ch. 4 |
| \(A_\mathrm{eff}\), \(A_e\) | effective collecting area of the antenna | m\(^2\) | Ch. 4 |
| \(\Omega_A\) | (antenna) beam solid angle | sr | Ch. 4 |
| \(G\) | gain (telescope K/Jy, or amplifier gain) | K Jy\(^{-1}\) or dimensionless | Ch. 4 |
| \(\mathrm{SEFD}\) | system-equivalent flux density (\(= 2kT_\mathrm{sys}/A_e\)) | Jy | Ch. 3–4 |
| \(V(u,v)\), \(\mathcal{V}\) | visibility — the complex correlator output | Jy (complex) | Ch. 7–8 |
| \(u,v,w\) | baseline coordinates in wavelengths (\(uv\)-plane = sky FT plane) | \(\lambda\) (wavelengths) | Ch. 8 |
| \(\mathbf{b}\), \(B\) | baseline vector / length between two antennas | m (or \(\lambda\)) | Ch. 7 |
| \(l,m\) | direction cosines on the sky (image-plane coordinates) | dimensionless | Ch. 8 |
| \(b\), \(\ell\) | Galactic latitude / longitude (note: \(\ell\) also = longitude) | deg | Ch. 11 |
| \(z\) | redshift (\(1+z = \nu_\mathrm{emit}/\nu_\mathrm{obs}\)) | dimensionless | Ch. 14 |
| \(\mathrm{DM}\) | dispersion measure (integrated electron column to a pulsar) | pc cm\(^{-3}\) | Ch. 13 |
| \(\mathrm{RM}\) | rotation measure (\(\Delta\chi = \mathrm{RM}\,\lambda^2\); \(\propto\!\int n_e B_\parallel\,dl\)) | rad m\(^{-2}\) | Ch. 37 |
| \(I,Q,U,V\) | Stokes parameters (total, two linear, circular polarisation) | same as \(S_\nu\) | Ch. 37 |
| \(N_\mathrm{H}\), \(N_\mathrm{HI}\) | column density of (neutral) hydrogen | cm\(^{-2}\) | Ch. 11 |
| \(\mathrm{EM}\) | emission measure (\(\int n_e^2\,dl\)) | pc cm\(^{-6}\) | Ch. 2 |
| \(v_\mathrm{LSR}\) | velocity w.r.t. the Local Standard of Rest | km s\(^{-1}\) | Ch. 11 |
| \(k\), \(k_B\) | Boltzmann constant (\(1.38\times10^{-23}\) J K\(^{-1}\)) | — | Ch. 1 |
| \(h\) | Planck constant (or, lowercase in cosmology, \(H_0/100\)) | — | Ch. 2 |
3. Operators & decorations¶
What the marks on and around a variable do to it.
| Notation | Reads as | Notes |
|---|---|---|
| \(f * g\) | convolution | the beam smears the sky: (true sky) \(*\) (beam) |
| \(z^*\) | complex conjugate | same star, different job — see §5; conjugate flips the sign of the imaginary part |
| \(\langle x \rangle\) | ensemble / time average | the expectation of \(x\); in noise theory \(\langle x\rangle\) often \(=0\) |
| \(\hat{x}\) | unit vector, estimate, or Fourier transform | \(\hat{\mathbf{n}}\) = direction; \(\hat{\theta}\) = an estimator; \(\hat{f}(\nu)\) = the FT of \(f\) |
| \(\bar{x}\) | mean of \(x\) (sometimes complex conjugate) | overbar = average; in some texts \(\bar{z}\) = conjugate |
| \(\tilde{x}\) | a transformed / modified quantity | often "\(x\) in the Fourier domain" or a smoothed version |
| \(x_\mathrm{rms}\), \(\sigma_x\) | the root-mean-square / scatter of \(x\) | |
| \(\propto\) | is proportional to | drops all constants: \(S_\nu \propto \nu^\alpha\) |
| \(\sim\) | "of order" / "scales as" | an order-of-magnitude statement, looser than \(\approx\) |
| \(\approx\) | approximately equal | a genuine numerical near-equality |
| \(\equiv\) | is defined to be | introduces a definition, not a derived result |
| \(\nabla\) | gradient / vector derivative ("del") | \(\nabla\cdot\) divergence, \(\nabla\times\) curl |
| \(\partial\) | partial derivative | \(\partial I_\nu/\partial s\) along a ray |
| \(\int\), \(\iint\) | integral, double integral | \(\iint \cdots\, dl\, dm\) integrates over the sky |
| \(\sum\), \(\prod\) | sum, product over an index | |
| \(\lvert z\rvert\) | magnitude / modulus | amplitude of a complex visibility |
| \(\arg z\) | argument / phase | the phase angle of a complex number |
| \(\mathcal{F}\{\cdot\}\) | the Fourier-transform operator | \(\mathcal{F}^{-1}\) is the inverse; sky \(\leftrightarrow\) visibilities |
| \(\mathbf{b}\), \(\vec{b}\) | a vector | bold and \(\vec{\ }\) arrow are interchangeable conventions |
Subscripts & superscripts. A subscript usually specifies (which quantity: \(T_\mathrm{sys}\), \(S_\nu\), \(\Omega_A\)); a superscript usually modifies (a power \(\nu^\alpha\), or a label like \(T^\mathrm{atm}\)). Numerical subscripts often pin a reference value: \(\nu_0\), \(T_0\), \(H_0\) all mean "at the fiducial / present epoch".
The two stars
* is overloaded. As an operator between two functions, \(f*g\), it is
convolution. As a superscript on a single symbol, \(z^*\), it is the
complex conjugate. A correlator computes \(V \propto \langle E_1 E_2^*\rangle\)
— both stars in one expression. See §5.
4. Units & their conventions¶
| Unit | Reads as | Notes |
|---|---|---|
| Jy | jansky | \(1\,\mathrm{Jy} = 10^{-26}\,\mathrm{W\,m^{-2}\,Hz^{-1}}\); mJy, \(\mu\)Jy common |
| Jy/beam | jansky per beam | the unit of a map pixel before deconvolution — see below |
| K | kelvin | as a brightness temperature, not a physical temperature, in maps |
| K km s\(^{-1}\) | brightness \(\times\) velocity | the integrated-line unit (Ch. 11), \(\propto N_\mathrm{H}\) |
| pc cm\(^{-3}\) | parsec per cm\(^{3}\) | the DM unit — a column of electrons (Ch. 13) |
| cm\(^{-2}\) | per square cm | column density \(N_\mathrm{H}\) (Ch. 11) |
| dB | decibel | \(10\log_{10}\) of a power ratio (gain, dynamic range) |
| mag | magnitude | logarithmic, backwards: brighter = smaller (Ch. 14) |
| dex | "decimal exponent" | one dex = one factor of 10 (a unit of log-spacing) |
Frequency, wavelength, velocity are interchangeable. Spectral axes get labelled in any of three ways, related by \(\lambda=c/\nu\) and a Doppler shift. The radio community uses the radio velocity convention:
which differs from the optical convention \(v_\mathrm{opt} = c(\lambda-\lambda_0)/\lambda_0\). The two disagree at high \(z\) — always check which a paper uses (Ch. 11, 14).
cgs vs SI
Much of the classic literature (and Essential Radio Astronomy) works in
cgs-Gaussian units: ergs, cm, gauss, and intensities in
erg s\(^{-1}\) cm\(^{-2}\) Hz\(^{-1}\) sr\(^{-1}\). Modern code and this course
lean SI (W, m, tesla). Factors of \(10^{7}\), \(10^{4}\), and \(4\pi\) between
the systems are a classic source of silent errors. Check the units on \(k\),
\(\sigma_T\), and \(\kappa_\nu\) before plugging numbers in.
Common prefixes: k (\(10^3\)), M (\(10^6\)), G (\(10^9\), as in GHz), T (\(10^{12}\)); and down: m (\(10^{-3}\)), \(\mu\) (\(10^{-6}\)), n (\(10^{-9}\)), p (\(10^{-12}\)).
Why \"per beam\"?
A radio interferometer map is the true sky convolved with the instrument's point-spread function (the "dirty/clean beam"). A point source therefore spreads its flux over a beam-sized blob, so each pixel carries flux density per beam area, i.e. Jy/beam. To recover a source's total flux in Jy you integrate over the source and divide by the beam area (in pixels). This is why beam \(\ne\) pixel matters — see §5.
5. Conventions & gotchas¶
A short field guide to the traps that cost real time.
Spectral-index sign. The spectral index \(\alpha\) is defined by a power law, but the sign convention is not universal:
With the \(+\alpha\) convention, synchrotron sources have \(\alpha \approx -0.7\) (negative); with the \(-\alpha\) convention the same source has \(\alpha \approx +0.7\). A paper will state which it uses; if a "steep-spectrum" source is quoted with a positive index, you are in the \(-\alpha\) camp (Ch. 2).
The \(\alpha\) collision. \(\alpha\) is both the spectral index and right ascension. They essentially never appear in the same equation, and context separates them instantly: paired with \(\delta\) it is a coordinate; sitting on \(\nu\) it is a slope. (§1)
The \(*\) collision. Convolution (\(f*g\)) vs. complex conjugate (\(z^*\)). Position disambiguates: between functions it convolves; as a superscript it conjugates. (§3)
Fourier sign and \(2\pi\). Fourier transforms come in several conventions that differ by the sign in the exponent and where the \(2\pi\) lives:
Radio interferometry conventionally uses \(e^{-2\pi i(ul+vm)}\) for the sky \(\to\) visibility direction (Ch. 8). A flipped sign mirror-images your map; a misplaced \(2\pi\) rescales it. See Math Preliminaries.
RA in hms vs degrees. Right ascension is written either in time units
(\(\mathrm{h\,m\,s}\), e.g. 05h34m31s) or in degrees, with
\(1^\mathrm{h} = 15^\circ\). Declination is always degrees (+22d00m52s).
A common bug: feeding an hms RA to code expecting degrees, off by a factor of 15.
Epoch / equinox. Coordinates depend on the date of the reference frame. J2000 (equinox 2000.0, ICRS for practical purposes) is today's default; older catalogues use B1950. Mixing them shifts positions by tenths of a degree (Ch. 10, 12).
Beam vs. pixel. A pixel is a sampling grid cell; a beam is the instrument's resolution element, typically several pixels across. Photometry, noise statistics, and "Jy/beam \(\to\) Jy" conversions all depend on the beam, not the pixel (§4, Ch. 9, 12).
FWHM vs. \(\sigma\). A Gaussian beam is quoted either by its full width at half maximum or by its standard deviation \(\sigma\). They are not the same number:
Telescope beams are almost always given as FWHM; smoothing kernels in code often take \(\sigma\). Mixing them mis-sizes the beam by a factor of \(\sim\)2.4 (Ch. 4, 9).
6. Worked examples — parsing every symbol¶
The method: read left to right, name each symbol, note its units, then read the whole thing as a sentence.
The radiometer equation¶
| Symbol | What it is | Units |
|---|---|---|
| \(\Delta T\) | the noise: smallest temperature change detectable (the \(1\sigma\) sensitivity) | K |
| \(T_\mathrm{sys}\) | system temperature — total noise power expressed as a temperature | K |
| \(n_\mathrm{pol}\) | number of polarisations summed (1 or 2) | dimensionless |
| \(B\) | bandwidth (\(\Delta\nu\)) over which you integrate | Hz |
| \(\tau\) | integration time (here \(\tau\) is time, not optical depth) | s |
| \(\sqrt{\;\;}\) | square root — sensitivity improves as the root of \(B\tau\), not linearly | — |
Read as: "the noise floor equals the system temperature divided by the square root of (number of polarisations \(\times\) bandwidth \(\times\) integration time)." Doubling your integration time only improves sensitivity by \(\sqrt{2}\) — the heart of "integrating down" (Ch. 3).
The visibility integral¶
| Symbol | What it is | Notes |
|---|---|---|
| \(V(u,v)\) | the visibility — what the interferometer measures | complex: amplitude + phase |
| \((u,v)\) | a baseline in the Fourier plane, measured in wavelengths | one antenna pair = one \((u,v)\) point |
| \(\iint \cdots\, dl\, dm\) | integrate over the whole sky | \((l,m)\) = direction cosines |
| \(I(l,m)\) | the sky brightness in that direction | the thing we want to recover |
| \(e^{-2\pi i(\ldots)}\) | the Fourier kernel | the \(-\) sign and \(2\pi\) are the conventions of §5 |
| \(i\) | \(\sqrt{-1}\) | makes \(V\) complex; its phase encodes source position |
Read as: "the visibility at baseline \((u,v)\) is the two-dimensional Fourier transform of the sky brightness." Each baseline samples one Fourier component; filling the \(uv\)-plane (by using many antennas and Earth rotation) and inverting the transform reconstructs \(I(l,m)\) — the whole programme of aperture synthesis (Ch. 7–9). Note \(V^*(-u,-v) = V(u,v)\) because the sky is real, so each baseline secretly gives you two \(uv\) points (the conjugate \(*\) of §3 at work).
When in doubt
Find the paper's symbol table or the sentence "where \(x\) is…" near the first use. Notation is a dialect; this page is a phrasebook, but the author always has the final say.