1 Introduction

• Radio telescopes are less familiar than optical telescopes
• Fundamental differences and common properties and features

This lecture

• What is specific to radio astronomy ?
• What do radio telescopes measure and how ?
• General concept of radio telescope beam
• Going beyond the diffraction limit of single-dish telescopes

1.1 What is radio astronomy

• A spectral window: from ≈10~MHz (0.01 GHz) to 2 THz (2000 GHz)
• Radio sources: thermal, non-thermal (bremsstrahlung, Compton), atomic (hyperfine, recombination), and molecular lines
• Detection of e-m waves rather than photons: amplitude & phase ! not only |amplitude|²
• hν≪ kT: anything that has T≠0K will emit at (sufficiently) long wavelength: B_ν(T) = 2kTν²/c²+O(hν/kT); however, this implies that the source signal is lost in a thermal noise;

Atmospheric windows

• Absorption by atmospheric molecules lead to atmospheric windows
• High-frequency limit (~1 THz): O₂, H₂O, CO₂
• Rotational dipolar magnetic transitions of O₂ at 60 GHz and 119 GHz
• Water: 22 GHz, 183 GHz (precipitable water vapour, pwv, in length unit: e.g. 1 cm or 1 g cm^-2)

1.2 Radio sources

• Synchrotron emission [radiation from ultra relativistic particles accelerated by a magnetic field; see Rybicki & Lightman] due to high-energy electrons accelerated in supernova remnants: account for ~90% of the galactic emission at 1 GHz
• Remaining emission at 1GHz is thermal emission from HII regions, e.g. the Orion Nebula
• Note that short-lived stars thus dominate the radio sky; true for spiral galaxies in general (elliptical are rather radio-quiet)
• Galactic interstellar gas: lines + continuum "noise"; HI (21cm)
• Galactic discrete sources: supernova remnants (Cas A, Crab)
• Look into highly obscured regions in the visible (A_V>few mag)
• Look at the cold universe: low frequency correspond to low energy transitions, usually involving low lying atomic/molecular energy levels, and also high-level recombination lines from atoms
• Maser emission: coherent emission from particles moving with equal velocity vectors
• Measure plasma effects such as Faraday rotation, dispersion, or scattering, are ∝ ν^-2;
• This list is not exhaustive

From the Moon

Thermal emission from the surface of the Moon taken 6 days apart with the JCMT.

to super massive black holes

• Image of the Messier 87 black hole, a massive galaxy in the nearby Virgo galaxy cluster, d=55e6 ly; M_BH = 6.5e9 M_sun
• Event Horizon Telescope (EHT): planet-scale array of eight ground-based radio telescopes; synthesized beam size of the EHT array: 20 μas
• Observations at λ1.3mm; data (350TB/day) are recorded with extreme timing precision (hydrogen maser) and recombined afterwards: VLBI

Supernova remnants

• The Crab Nebula: HST, Very Large Array (VLA), composite
• Radio synchrotron emission (at 3 GHz, VLA), due to the pulsar wind, is recorded as a continuum (not lines)

Emission from HI gas in M81

• complementary view between visible and HI line showing intergalactic, neutral gas
• HI line at 21cm (hyperfine transition – hence intrinsically weak – associated with spin I=1/2 of the proton leading to 2×1/2+1=2 hyperfine levels separated by 1420 MHz; see R&L textbook op. cit.)

Atomic and molecular gas in the merging Antenna galaxies

• Left: visible; right: ALMA
• 22 Mpc NGC 4038 and NGC 4039; merger started ~ few 100 Myr ago
• Long stripes of HI gas and triggered burst of star formation in the merging region

Reionization epoch

• UV photons emitted by the first stars modify the level population of the hyperfine levels of the ground state of HI
• Redshifted HI emission at ~80 MHz (z∼18, age of the universe ∼220 Myr) is the primary messenger from the dark ages when the first stars form.

Reionization epoch

• UV photons emitted by the first stars modify the level population of the hyperfine levels of the ground state of HI
• Redshifted HI emission at ~80 MHz (z∼18, age of the universe ∼220 Myr) is the primary messenger from the dark ages when the first stars form.

• MWA, LOFAR, HERA, SKA (and MeerKAT interferometer)

1.3 Instruments and facilities

Historical perspective

• Extraterrestrial signals at long wavelength were discovered by Karl Jansky (left), engineer at Bell Laboratory Telephone companies. See here for some historical notes by J. Kraus. (Other Bell Labs Nobel Prize laureates: Penzias and Wilson, Davisson; see here)
• Grote Reber "amateur" 9.5m diameter telescope (right) in the backyard of his house. He was able to map the 160-MHz continuum emission from the galaxy and identify few discrete sources (see Reber 1944).

The HI line

• Reber's discovery led the Dutsch astronomer, Jan Oort (Leiden Observatory), to see the potential of spectral line observations to measure the galactic rotation curve;
• His student, Henk C. van de Hulst, was assigned the study of which spectral line could exist and be measurable; the original paper was translated in English in "Classical Radio Astronomy
• The first measurement was eventually performed by Ewen and Purcell (see here for details)

Single-dish antennas

• Primary: spherical, 305m diameter, effective 213m
• "Gregorian dome": secondary+tertiary: 900 tons, at 150m above the primary
• Covering 50 MHz (6m) to 10 GHz (3cm) frequency range; source within 20° of zenith

• Cable (13 cm diameter!) damage in August 2020; another one last Nov. 7th;

Single-dish antennas: FAST-300m

• FAST (Five-hundred meter Aperture Spherical Telescope), Guizhou, Southwest China
• Covering the range 70-3000 MHz; effective diameter is 300m

Single-dish antennas: IRAM-30m

• IRAM-30m located in Sierra Nevada close to Granada (Spain), ∼3000m altitude; website
• Operates in the 3, 2, 1mm, and 0.8mm millimeter windows

Single-dish antennas: LMT-50m

• Sierra Negra, Mexico, 4600m altitude; website
• Operates primarily in the 3mm window

1.4 Thermal noise vs photon noise

• Black-body radiation is radiation in thermal equilibrium: B_ν(T) = 2hν³/c² / [exp(hν/kT) -1]

• Bose-Einstein statistics describe the occupation number: mean number of photons per field mode (i.e. per frequency, per polarization):

  <n_ν> = [exp(hν/kT)-1]^-1

• Mean-square fluctuation in photon occupancy:

  (Δ <n_ν>)² = <n_ν> (1 + <n_ν>)

• Noise:
  • At high frequency: photon noise (or shot noise)
    • hν ≫ kT, n_ν ≪ 1, and the first term dominates
    • fluctuation of the arrival time of photons on the detector, which is a random (Poissonian) process: Δ <n_ν>∼(<n_ν>)^1/2
  • At low frequency: thermal noise;
    • hν ≪ kT: <n_ν> ∼ kT/hν ≫ 1: photon bunching; particles do not arrive independently
    • flucuations: Δ <n_ν> ∼ <n_ν> ∝ T
    • wave (interference) term

1.5 Radiotelescopes: overview

• Radiotelescope: measures electric fields (amplitude, phase)
  • Input signal: fW
  • Output signal: mW
  • Amplification (120 dB) with minimum extra noise
  • Converts input electric field into a current

2 Antenna beam

• There is no fundamental difference between an optical telescope and a radio telescope: same physical principles!
• However, in practice, there are important differences:
  • radio astronomy is capable of measuring the amplitude and phase of the incoming e-m wave
  • coherent detectors are most often single-pixels: a resolved source is convolved by the antenna beam
  • sidelobes are a problem because anything that is hotter than a few K emits radio waves that can blur the desired scientific signal
• Apodization, i.e. decreasing the amplitude of the sidelobes, is obtained through the coupling of the detector to the optics of the telescope

2.1 Antenna power pattern

• Diffraction theory (Huygens-Fresnel): emitted field is the sum of spherical waves emitted by the wavefront during the propagation of the wave
• Think of the telescope in emission: the spherical waves are emitted by the primary aperture; what is the radiation diagram of the antenna?
• Apply some electric field (or current) on the antenna: E_ant(x,y)
• At large distances from the antenna, Huyghens-Fresnel reduces to Fraunhoffer diffraction
  • E_ff(l,m) ∝ FT[E_ant(x,y)]
  • Note: Far-field approximation requires the source be at d≫ R_ff ≈ 2D²/λ
  • IRAM-30m: D=30m, λ=3mm, R_ff ≈ 6E-3*(30/3e-3)² = 600 km;
  • VLT: D=8m, λ=0.5mic, R_ff≈ 1E-6*(8/0.5E-6)²=256E3 km; the Moon (384E3 km) is barely into the far-field;

A simple example

• Distribution of electric field on the reflector: complexe aperture function
  • for example, a 1D sine wave E_ant(x,t)=Π(x)exp(-i2πν t), Π(x)=1 for |x|≤1/2, 0 outside
  • Note that the instantaneous field is the real part of this;
• Field radiation pattern at large distance (far-field approx.) in direction ℓ=sinθ (≈θ):
  • E_ff(ℓ) = FT(E_ant(x,t)] = ∫ Π(x) e^{-i 2π xℓ/λ} dx
  • FT[Π(u)] = sin(π ℓ)/(π ℓ) = sinc(ℓ)
  • E_ff(ℓ) = sinc(ℓ)
• Consider a line of length D:
  • General property of FT: FT[f(x/a)} = |a|× FT(f)(aθ);
  • Hence, for line of length D: E_ff(ℓ)= FT[Π(x/D)]= D sinc(Dℓ): the larger the antenna, the narrower the field pattern!

• Power emitted: P(θ_λ) ∝ |E_ff(θ_λ)|²

Simple example in one dimension

• Aperture complex function: here, uniform illumination
• Far-field pattern: Fourier Transform is a sinc function
• Far-field power pattern: square modulus of E_ff

General properties of the antenna beam

• E_ant(x,y): bounded on a finite domain hence E_ff(θ,φ) also concentrated on a finite domain
• Relation between Fourier pair coordinates: (D/λ)² Ω ∼ 1, or area × Ω ∼ λ²
• Sharp boundaries of the dish: oscillations in E_ff (side-lobes)
• Apodization or taper: decrease the level of the sidelobes, to the cost of increasing ΔΩ

Definitions

• Power pattern: P(θ,φ) normalized such that P(0,0)=1

• Beam solid angle:

  Ω_A = ∫_4π P(ω) dω, usually ≤ 4π

• Main-beam solid angle:

  Ω_mb = ∫_mb P(ω) dω, usually ≤ 4π

• Effective area: A_e(θ,φ) = A_e P(θ,φ), A_e ≤ A_geom
• Fundamental relation (étendue) (shown in the notes) for diffraction-limited telescopes: A_e Ω_A = λ²

Coherence étendue: diffraction limited resolution

• Diffraction by a limited aperture
• Consider plane waves propagating at angles θ from the optical axis
• Path difference λ accross the reflector obtained for direction angle θ=θ_min
  • Directions θ<θ_min can not be distinguished
  • θ_min∼λ/D
  • Solid angle: Ω=π θ_min² = π λ²/D²
• Coherence étendue: Ω S=π² λ² /4 ≈ λ²

Effective area and the directivity of a radio telescope

• Étendue relation for a diffraction-limited telescope: A_e Ω_A = λ²
• A_e is the effective area of the telescope

• The effective collecting area averaged over all directions is

  <A_e> = ∫_4π A_e P(ω)dω / ∫_4πdω = λ² 4π, A_e/<A_e> = 4πΩ_A

• The average collecting area increases with λ
• On the contrary, at short wavelength, <A_e> is small, and since A_e/<A_e> = 4π/Ω_A, having a large peak effective area implies having a small Ω_A that is a very directive antenna;

Optical vs radio telescopes

• Optical Transfer Function (OTF) and aperture function
  • Optical transfer function T(u)
  • Telescope = linear systems
  • Transfer function: T(u)=A(u)⊗ A(u), the autocorrelation product of the aperture complexe function
• PSF and antenna beam
  • PSF = antenna beam = power pattern
  • In practice, PSF of optical telescopes \(\approx\) antenna beam in sub-mm
  • PSF or power pattern ≡ response to an impulse
  • Phase is lost in the PSF (\(\ne\) Transfer function)
• Imaging
  • Imaging with single-beam radio telescopes requires scanning the source
  • Imaging with optical incoherent telescopes can be done w/ one integration
  • Heterodyne multi-beams on radio telescopes ≪ CCD
  • Key: optical detectors sample the PSF
  • Continuum multi-beams on radio telescopes \(\approx\) CCD

2.2 Diffraction-limited vs seeing-limited

• Kolmogorov 1941 theory for incompressible, homogeneous, fully developed turbulence can be applied to the atmosphere
• It follows that the statistics of the refractive index follow those of the velocity of the air; these statistical properties relate the fluctuations of n(λ) with scale: D_n(r) = <[n(x+r)-n(x)]>_x ∼ r^2/3, with D_n the second order structure function (longitudinal);
• The refractive index also depends on the temperature, pressure, water vapour partial pressure, and on the wavelength as ∼ λ^-2;
• It can be shown the the phase structure function is D_φ(r) = 6.88 (r/r₀)^5/3
• r₀ is the Fried (1965) parameter and we have that: r₀ ∝ λ^6/5
• Typical values: r₀ ∼ 10-30cm at 500nm, and ∼ 100-300m at 1mm;
• Radio telescopes are diffraction-limited; optical telescopes are seeing-limited;

Seeing-limited

• Space-based telescopes are diffraction-limited over a wide range of λ;
• Ground-based optical telescope are seeing-limited unless adaptive optics is used

2.2.1 Realistic beams

Figure 33: The measured beam pattern of the 31 GHz and 54 GHz channels of the DMR instrument on COBE. Each channel is sensitive to two polarizations (circular and linear for the 31 GHz and 54 GHz channels, respectively). Note the reduction of the minor lobes to below -50 dB.

Gaussian beam patterns

Figure 34: The measured beam pattern of the 3mm channel of EMIR at the IRAM-30m telescope (from Kramer & Greve 2013 IRAM Report. See also Greve et al 1998).

• Most radio telescopes have Gaussian beams
• The tapering is obtained by the horn
• Asumming the beam is axisymmetric: P = exp(-θ²/2σ²), with σ = HPBW/\(\sqrt{8\ln2}\)
• The beam solid angle is Ω_A = ∫₀^π sinθ dθ

2.3 Beam convolution

• Source with specific intensity I_nu(Ω) and solid angle Ω_s
• Source flux density is: F_ν = ∫_source I_ν(θ,φ)dΩ
• Define an arbitrary reference position \((\theta_0, \phi_0)\); angles will be measured wrt to this reference;
• Different parts of the source emit incoherent light: total intensity is the sum of individual intensities

• Looking towards the reference position: the observed flux density is:

  F_ν(0) = ∫_4π P(ω) I_ν(ω) dω = ∫_Ωsource P(ω) I_ν(ω) dω + contamination

Convolution of the sky by the power pattern

• Now, we would like to obtain a map of the source, that is, to know the spatial distribution of I_nu; in radio astronomy, this is obtained by scanning the source, contrary to optical telescopes where large matrix of detectors can be used covering a much wider FOV;
• Looking towards position 2 with coordinates \((\theta_2, \phi_2)\) wrt to the reference position, the observed flux density is now: F_ν(Ω₂) = ∫_4π P(Ω₂-ω) I_ν(ω) dω

• What we sill measure, while scanning the source is thus the convolution of the brightness distribution by the power pattern:

  F_ν(Ω) = (I_ν * P)(Ω) = ∫_4π P(ω)I_ν(Ω-ω) dω = ∫_Ωs P(ω)I_ν(Ω-ω) dω + contamination

Beam smearing

• Consider three cases:
  • unresolved point source
  • finite source and infinite diameter telescope
  • general case

• Convolution effect: what a (radio) telescope delivers is F_ν(Ω) = (I_ν * P)(Ω)
• A direct consequence of the convolution effect is beam smearing
• Particular case: power pattern and the source are Gaussian
  • Gaussian * Gaussian = Gaussian
  • θ²_obs = θ²_{main beam} + θ²_source
• To obtain the source size, the observed size must be corrected for the telescope beam

2.4 Sampling theorem

• Sampling theorem
  • Band-limited functions (i.e. having a bounded support) in the Fourier domain can be fully recovered by discrete sampling.
  • To recover all the information from the PSF, sampling the PSF at the critical sampling which is twice the cutoff frequency, is necessary and sufficient
• The pupil of a telescope is such a bounded function: telescopes act as low-pass filters in the spatial frequency domains
  • We have seen that T_A(x) = P(x) * I_s(x) that is, FT(T_A)(u) = FT(P)(u) × FT(I_s)(u), with u=2π/x. What does FT(P) look like?
  • Assume a uniform pupil (i.e. a uniform current distribution in radio jargon) over a diameter d, f(x)=f₀, |x|≤ d/2, we have seen that P(x)=sinc(Dθ/λ)²
  • The inverse Fourier Transform of the power pattern is the Telescope transfer function, and is a "triangle": low-pass filter, with cut-off spatial frequency ∼ 1/D
  • Whatever the details of the power patterns, their Fourier transform are zero beyond a frequency ∼ 1/D;
  • Higher frequencies are lost

Sampling theorem

• Function having a bounded support in the Fourier domain can be recovered by discrete sampling.
• To recover all the information from the PSF, sampling the PSF at twice the cutoff frequency is necessary and sufficient
• If \(u_c\) is the cutoff in the Fourier domain, then sampling must be done at \(u_s \ge 2u_c\)
• For the 1D \(\Pi\) antenna: \(u_c = D/\lambda\) corresponds to the zero of the transfer function \({\cal T}(u)\).
• Critical sampling: \(u_s \ge 2D/\lambda\), or \(\theta_s \le \lambda/2D\).
• The first null of the power pattern is at \(\theta_0=\lambda/D\). Thus \(\theta_s \le \theta_0/2\).
• optical telescope: detector size \(< {\rm PSF}/2\)
• radio telescope: mapping with spatial sampling \(< {\rm PSF}/2\)

2.4.1 Aliasing

Brightness is corrupted by undersampled high-frequency which get introduced (aliased) at lower frequency;

3 Superheterodyne radio telescopes

• Long wavelength: why is different than visible? energy of photons is too small
  • for the Sun, T_eff ≈ 5800K
  • at long wavelength: B_ν(T_eff)≈ 2kT/λ²=1.6E-15 W/m2/Hz/sr (λ/1cm)²
  • in the visible: λ=0.5μm, ν=6E14 Hz, B≈ 2.2E-8 W/m2/Hz/sr
• At long wavelength, the only way is to measure the electric field;
• Antenna: passive device converting a current into e-m radiation or vice versa; could be a dipole; oscillating incident wave raises oscillating motion of e- in a conductor; the oscillating current (amplitude and phase) can be measured;

3.1 Detectors

Overview

• Astronomical power: few 10^-16 W, ν ∼10¹¹ Hz
• Electronics, spectrometers: a few mW, ν∼10⁸ Hz
• Receiving system: amplification 120 dB (gain G_dB=log₁₀ (G)) + down conversion
• Limiting case: receiver noise ∼ telescope losses + atmosphere + celestial background (e.g. CMB)
• T_rec < 100K (10K for spaceborn telescopes)
• Fundamental limitation: quantum noise limit, hν/k (5K at 100 GHz)
• Coherent and incoherent detectors, linear and quadratic

Coherent detection chain

Two types of (superconducting) detectors

Overview

The feed horn

• Converter and impedance matching (backshort) to see only radiation from the sky
• Corrugated to convert the circular symmetry and pure linear polarization electric field from the reflector into TE10 mode which will propagate in the in 2x1 mm waveguide
• Feed horns are monomode (mode selective): power received from an unpolarized source is 1/2 the total power; not true for bolometer (total power)
• Linear polar can be separated, detected with two detectors, and added to recover the total power

3.2 Frequency down-conversion and sidebands

• E(z) = E₀ sin(kz-2πν_sky t + φ) propagating along z
• intensity ∝ E₀²
• coherent detection: measure E₀ and φ
• Issue: at ν_sky ∼ 10¹¹ GHz, response time τ ≪ 10^-11s
• No low-noise electronic devices working at high enough frequency

The mixer (or diplexer)

• Solution: mixing ν_sky with ν_LO = ν_sky - ε
  • Add & multiply
  • A beat frequency ν_S-ν_LO and a high frequency ν_sky+ν_LO
  • E²(t)=LO + Sky + f(ν_S+ν_LO) + g(ν_S-ν_LO)
  • Diode delivering voltage V_out ∝ V_in²

Non-linear mixer: general idea

• V₀ is the operating voltage
• Non-linear response of the mixer: Taylor series around V₀

A diode

• I = I₀ (e^qV_b/kT - 1) ≈ qV_B/kT + 1/2 (qV_B/kT)²
• Output current: I ∝ V² ∝ E² hence to the input power

Frequency conversion

• U_sky (t) = E_sky cos(2πν_sky t + φ_sky)
• U_LO (t) = E_LO cos(2πν_LO t + φ_LO)
• Non-linear diode output: \(E^2_{\rm IF}(t) = \sum_{k=0,n} a_k(U_{\rm sky} + U_{\rm LO})^k\)
• Second order term: I(t) = LO(2ν_LO) + SKY(2ν_sky) + f(ν_sky+ν_LO) + g(|ν_sky-ν_LO|)
• Low-pass filtering: suppress 2ν_LO, 2ν_sky, and ν_sky+ν_LO

• Intermediate frequency (IF)

  • ν_IF = |ν_sky-ν_LO|
  • beat frequency or difference; usually 4 to 8 GHz where low-noise elec. devices are available

  I(t) = a₂ E_LO E_sky × cos[2π(ν_sky-ν_LO) t + φ_sky-φ_LO]

• Phase is preserved: requires controlled and stable φ_LO

Sidebands

• cos(248-240) = cos(240-232)
• ν_sky=232GHz from the Lower Side Bande (LSB) and ν_sky=240GHz from the Upper Side Bande (USB) are superimposed

• Two bands in sky frequency: LSB, USB
• Receivers can be
  • SSB: only band (either LSB or USB, observatory dependent)
  • DSB: both bands superimposed
  • 2SB: LSB and USB simultaneously and separated
• Sky frequency: ν_S = ν_LO-ε and ν_S = ν_LO+ε have the same IF frequency
• EMIR at IRAM are 2SB: use the phase sign difference between LSB and USB; instantaneous bandwidth of 32GHz (almost a bolometer !)
• Figure of merit for SSB and 2SB: band rejection (usually in dB) or image gain (dimensionless)

The mixer (or first detector)

• The critical element of a receiver
• Classical diode (junction) but photodiodes have frequency response limited to less than 1 GHz; this limit is due to the recombination time of charge carriers that have crossed the junction
• Usual mixer devices: Quantum detecting device (quasiparticle tunneling junction)
  • IRAM/EMIR, ALMA, etc…: SIS (involving superconductors)
• Figure of merit: T_M, noise due to the mixer itself
• Other sources of noise: local oscillator, thermal background, Johnson noise, shot noise (additional currents in the junction, e.g. tunnel, or thermally activated)
• Fundamental limit: T_M > hν/k (zero point fluctuation of phase and amplitude of the electromagnetic field)

Technologies

• High electron mobility transistors: HEMT
• SIS junctions (IRAM, ALMA)
• Hot electron bolometer (HEB)
• Central frequency ν₀ = 80-2000 GHz
• Instantaneous bandwidth: Δν = 1-32 GHz

Junctions

• Based on junction of type-n and type-p extrinsic semiconductors
• High-gain, low-noise
• Energy from input photons can break Cooper pairs in the semiconductor leading to a current
• Superconductor gap (meV) ∼ 1/1000 semiconductor (eV)
• Insulator (1nm thick)

Low-noise mixer designs

Performances

• Sensitivity of Rx in radioastronomy: T_rec
• Coherent detectors: T_rec > hν/k_B
• T_rec = N/k_B, N: noise power

The beat or intermediate frequency

• I(t) = a₂ E_LO E_sky × cos[2π(ν_sky-ν_LO) t + φ_sky-φ_LO]

• However, there is another frequency difference in the signal:

  I(t) = a₂ E_LO E_sky × cos[2π(ν_LO-ν_sky) t + φ_LO-φ_sky]

• Intermediate frequency (IF): beat frequency or difference
  • ν_IF = |ν_sky-ν_LO|a
  • Important: ν_IF ∼ 4 to 8 GHz where low-noise electronic devices are available
• Preserved sky frequency: ν_sky = ν_LO ± ν_IF

4 Temperature scales

• Spectrum of NH₃ (Cheung et al 1968) toward a dense, cold cloud
• Note the x and y axis units:
  • T_A is the antenna temperature
  • v_LSR is the source velocity in the Local Standard of Rest
    • Note the large range of values! the source is located towards the Galactic Center, hence at ∼ 8 kpc
    • Furthermore, the line is broad (several tens of km/s), but the spectral resolution (15 km/s) does not allow to resolve the line;

Modern spectra

• Spectral resolution has increased by orders of magnitude
• Sensitivity
• Above: the hyperfine manifold of the rotational transition (1-0) of HCN, towards the cold starless core L1498

Why using a temperature scale ?

• Historically, radio astronomy was developed at wavelengths where the Rayleigh-Jeans approximation is fullfilled; the black-body radiation is thus given by:

  B_ν(T) ≈ 2kT/λ²

• Astronomical sources at radio wavelength include thermal emission which are directly characterized by their temperature which, in the Rayleigh-Jeans domain, also provides the natural scale for the received power;

• Fundamentally, any resistive circuit radiates a non-zero power in the form of heat. This noise is due to the motion of charges in the circuit. It was shown independently by Johnson (1928) and Nyquist (1928), that the monochromatic power emitted is

  p_ν = kT, and does not depend on the resistance itself, but only on the temperature;

• Consequently, dark currents of radio receivers are conventionally expressed in temperatures and not in e-/s; this makes the comparison between the measured power to the receiver noise straightforward;

Orders of magnitude

• 1 Jy = 10^-26 W Hz^-1 m^-2 = 10^-23 erg s^-1 Hz^-1 cm^-2

• Orders of magnitude: observed at a frequency of 10 GHz (λ=3cm), a 3K black-body radiates a specific intensity:

  B_ν ≈ 2kT/λ² ≈ 10^-19 W Hz^-1 m^-2 sr^-1 or 10 MJy

• Simpler unit: specific intensity expressed in K:

  1 K at 10 Ghz = 3 10^-20 W Hz^-1 m^-2 sr^-1 = 3 MJy sr^-1

• Solid angle: 20" resolution corresponds to Ω_mb ≈ 1.133θ²=10^-8 sr

4.1 The antenna temperature

• As we shall seem the antenna temperature is not a physical temperature but a fictitious one that nevertheless is representative of the brightness of a source when this is spatially resolved.
• Consider first an unpolarized source (unresolved)
  • For such a source, we measure the flux density F_ν (not its specific intensity)

  • Monochromatic (and monomode) power collected by an area A_e:

    p_ν = 1/2 A_e F_ν, W Hz^-1

  • Remember: units of flux density F_ν in astrophysics is the Jansky (1Jy = 10^-26 W m^-2 Hz^-1)
  • Power received (in W) within the bandwidth Δν: \[ p = \frac{1}{2}A_e \cdot F_\nu \cdot \Delta \nu \]

• We now make the most simplifying (but still physical) assumptions

• Assume source is a black-body: I_ν(T) = B_ν(T), occupying a solid angle Ω_s:

  F_ν = B_ν(T) Ω_s

• Futher assume Rayleigh-Jeans is a valid approximation:

  B_ν(T) ≈ 2kT/λ²

• Thus: p_ν = A_e Ω_s kT/λ²
  • We have thus shown that the received power can be expressed in terms of the temperature of the blackbody
• We are now going to relate this power to the noise power at thermodynamic equilibrium of a resistor equivalent to the entire telescope.

• the increase of antenna temperature when looking at the source is thus related to T;

Physical origin of the antenna temperature concept

• A resistance R in thermal equilibrium at a temperature T>0 radiates a noise power

• Johnson 1928, Nyquist 1928 have shown that the expression of the radiated noise per unit spectral range only depends on T and is given, at long wavelength (Rayleigh-Jeans approximation), by:

  p_ν = kT [W Hz^-1]

• When exposed to the electric field from the source, the antenna behaves as a resistor: the output voltage rms (remember that P=U²/R) can be interpreted as the monochromatic power radiated by a resistor at temperature T_A

  p_ν=kT_A

• this is the definition of T_A, the antenna temperature
• Important: p_ν does not depend on R.
• T_A: the equilibrium temperature of the equivalent resistive antenna
• This means that, if everything – telescope, source – was in thermal equilibrium, then T_A would have to be related to the temperature of the source. Let's see how.

Relation between T_A and the temperature of the source

• Equating the two expressions of p_ν, one obtains kT_A = A_e Ω_s kT/λ² or

  T_A = T×(A_e Ω_s/λ²)

• For diffraction-limited telescopes: A_e Ω_A = λ², hence

  T_A = T × Ω_s/Ω_A

• This relation expresses conservation of energy in free space:T_A Ω_A = TΩ_s
• The source temperature (hence its brightness) is known if Ω_s is known: it "becomes" resolved if we know Ω_s.

Point-source sensitivity

• The point-source sensitivity is a very important figure of merit which characterizes the ability of an instrument to detect a given source in a given amount of time.
• When pointing the antenna towards a point source, the antenna temperature is increased by p_ν = 1/2 A_e F_ν;

• The sensitivity of the antenna is usually expressed in terms of the so-called K/Jy factor:

  K/Jy = A_e ×10^-26/2k = 3.6(-4) A_e, with A_e in m²

• Thus, the point-source sensitivity is primarily related to the collecting area and, not surprisingly, it increases with A_e
• Numerically, a sensitivity of 1 K/Jy corresponds to an effective area A_e ≈ 2671 m², or a 60 m diameter dish; interferometers, with their larger collecting area, are therefore naturally more sensitive than single-dish;

4.2 Temperature scales and source temperature

Brightness temperature and source temperature

• We define the (unphysical) brightness temperature T_B: I_ν(Ω) = B_ν(T_B,Ω)
• We define the (unphysical) radiation temperature T_R:
  • I_ν(Ω) = B_ν(T_B, Ω) = 2kT_R(Ω)/λ²
  • In other words, this is the Rayleigh-Jeans approximation of I_ν(Ω)

Antenna temperature: general formulation

• When observing towards a direction Ω₀, the telescope collects

  \[ k T_A(\Omega_0) = \frac{1}{2} \int_{4\pi} P(\omega) I_s(\Omega_0-\omega) d\omega \]

• Introducing the radiation temperature T_R and simplifying leads to:

  \[ \boxed{T_A(\Omega_0) = \frac{1}{\Omega_A} \int_{4\pi} P(\omega) T_R(\Omega_0-\omega) d\omega} \]

Beam-filling factor

• We have obtained: \[ T_A(\Omega_0) = \frac{1}{\Omega_A} \int_{4\pi} P(\omega) T_R(\Omega_0-\omega) d\omega \]
• For an extended source, Ω_s ≫ Ω_A, this gives: T_A = T_R
• For an unresolved source, we recover Ω_A T_A = T_R Ω_s
• The dilution factor Ω_s/Ω_A is called the beam dilution factor; a 100K source filling 1% of the beam will produce an increase of 1K in the antenna temperature;
• It expresses, and quantifies, the fact that we do not measure I_ν for an unresolved source, unless one knows, from other measurements, the source solid angle.

4.3 Advanced details

4.3.1 Antenna temperature

The antenna temperature from a resolved source

• Monochromatic power delivered (in W/Hz) when looking towards (θ₀ , φ₀)≡Ω₀:

  p_ν(Ω₀) = kT_A = 1/2 A_e F_s(Ω₀)

• Fundamental relation: Ω_A A_e = λ²
• We define the (unphysical) brightness temperature T_B: I_ν(Ω) = B_ν(T_B,Ω)
• We define the (unphysical) radiation temperature T_R:
  • I_ν(Ω) = B_ν(T_B, Ω) = 2kT_R(Ω)/λ²
  • In other words, this is the Rayleigh-Jeans approximation of I_ν(Ω)

• We thus obtain (show it) the following expression for the antenna temperature, for a source of solid angle Ω_s

  \[ \boxed{T_A(\Omega_0) = \frac{1}{\Omega_A} \int_{\Omega_s} P(\Omega) T_R(\Omega_0-\Omega) d\Omega} \]

The antenna temperature: definition

• Thus far, we have considered an antenna pointing at a source; yet, in practice, the antenna looks at the sky… and also at the ground, or even itself because of sidelobes;

• Therefore, the antenna temperature should be calculated by integrated over the entire 4π sr:

  \[ T_A(\Omega_0) = \frac{1}{\Omega_A} \int_{4\pi}P(\omega) T_R(\Omega_0-\omega) d\omega \]

The corrected antenna temperature: T_A^*

• The antenna temperature introduced before is: \[ T_A(\Omega_0) = \frac{1}{\Omega_A} \int_{\Omega_s} P(\Omega) T_R(\Omega_0-\Omega) d\Omega \]
• This definition ignores two important limitations: 1) the absorption by the atmosphere, and 2) the sidelobes of the real beam pattern
  1. Correct for atmospheric absorption: multiply by exp(τ_atm), with τ_atm the opacity of the atmosphere; note that this opacity depends on the wavelength and on the elevation of the source
  2. Correct for rear-sidelobes: what we want is a measure of the power coming from the 2π sr in front of the telescope, excluding what comes from the 2π sr behind (ground, telecsope, etc). Thus, the integration is now performed over 2π sr;
• Energy conservation gives: Ω_A T_A exp(τ_atm) = Ω_2π T_A^*, with T_A^* the corrected antenna temperature;
• The factor Ω_2π / Ω_A is called the forward efficiency and describes the fraction of the antenna temperature which actually comes from the forward 2π sr; usually F_eff ≈ 90%, and increases as λ decreases;
• The largest correction between T_A and T_A^* is due to τ_atm

4.3.2 The main-beam temperature

• Consider now the power collected only in the main-beam instead of the forward 2π sr
• Same reasoning as before, with Ω_mb=∫_mb P(Ω)dΩ instead of 2π
• We thus have: Ω_A T_A e^τ_atm = Ω_2π T_A^* = Ω_mb T_mb
• We also define the beam efficiency: B_eff = Ω_mb / Ω_A
• Important relation: Beff T_mb = F_eff T_A^*

• Main-beam temperature:

  T_mb(Ω₀) = F_eff T_A^*/Beff = 1/Ω_mb ∫_Ωs P(ω)T_R(Ω₀-ω) dω

• Useful relation for a Gaussian beam: Ω_mb = 1.133 θ_mb²

4.3.3 Temperature scales and efficiencies

• Forward efficiency: F_eff = Ω_2π / Ω_A ≈ 80-95%
• Beam efficiency: B_eff = Ω_mb / Ω_A < 80%
• Relation:
  • Beff T_mb = F_eff T_A^*
  • Ω_mb T_mb = Ω_2π T_A^*

Notes

• What you measure is T_A^* or T_mb, which are not equal to T_R, the R-J temperature of the source

• Even if looking at a blackbody, T_R is not the temperature of the blackbody because T_R assumes that hν≪ kT; you must correct for the neglected factor

  T_bb = T_R hν/k/[exp(hν/kT

• Conversion from K to Jy is telescope dependent (depends on \(A_e\)). Given by telescope staff; of order 5-10 Jy/K.

Measuring efficiencies?

• F_eff: so-called skydips, measuring T_sky at several elevations (airmass) and use the elevation-dependent atmosphere emission T_atm (1-e^-τ(El)) with τ(El) the opacity at elevation El; it is related to the zenith opacity as τ(El)=τ_z/sin(El)=τ_z sec(z), with z the zenith angle (=π/2-El), and sec(z)=1/cos(z).
• B_eff: from known-flux objet (solar system, planets)
• Done by telescope staff; tabulated
• F_eff and B_eff are frequency dependent quantities; they decrease with λ; this, essentially, comes from the fact that surface irregularities are more diffractive as λ decreases, leading to increased phase fluctuations;

Which temperature scale ?

• Assume uniform source brightness: I_ν = T_R
• Ω_s ≫ Ω_mb:
  • T_mb =1/Ω_mb T_R ∫_Ωs P(ω) dω ≈ T_R Ω_2π/Ω_mb ≫ T_R
  • Similarly, T_A^* ≈ T_R Ω_2π/Ω_2π = T_R
• Ω_s ∼ Ω_mb: source "fills" the beam
  • T_mb =1/Ω_mb T_R ∫_Ωs P(ω) dω ≈ T_R Ω_s/Ω_mb ≈ T_R
  • Similarly, T_A^* ≈ T_R Ω_mb/Ω_2π ≪ T_R
• Ω_s ≪ Ω_mb: unresolved source
  • P(ω)≈ 1
  • T_mb ≈ Ω_s/Ω_mb T_R ≪ T_R
  • Beam dilution

  • if the source size is known, the true brightness must be corrected for beam dilution:

    T_mb ⇒ T_mb × (Ω_mb+Ω_s)/Ω_s ≈ T_mb × (θ_mb²+θ_s²)/θ_s²

5 What do radio telescope measure?

• Power pattern: P(θ,φ); a normalized function which describes the directivity of the antenna
• The effective aperture may be written A_e(θ,φ) = A_e P(θ,φ)

• When looking at a point-like (meaning, unresolved) source of flux density F_ν, the monochromatic power is

  p = 1/2 A_e S

• What is the power received by an antenna when looking at an extended source?

5.1 Noise-like measurements

• Output voltage a of radio telescope = sum of noise voltages from many independent random contributions
• Central Limit Theorem: output voltage has a Gaussian distribution
• Top: N=100 samples, separated by δ t = 0.5μs; total time is τ=Nδ t=50μs; corresponding bandwidth Δν=1/2δt=1MHz, from Shannon sampling theorem;
• If there is no astronomical source, the average is 0, and it has an rms V_rms

Square-law detector

• Output voltage V₀ from a square-law detector with Gaussian noise input
• The average <V₀> is <V²>, which is no longer 0
• The rms is 2^1/2<V₀>
• The non-linear detector has transformed the zero-mean noise voltage into a non-zero noise power;

Time integration: the radiometer equation

• The output of the square-law detector is averaged with a running window of width 50 samples
• The mean <V₀> is unchanged, but the rms is divided by N^1/2
• Here, N=50;

Time integration: the radiometer equation

• Same as before, but with N=200
• The rms is thus a factor 2 better

• Radiometer equation:

  σ_T ≈ T_s / \(\sqrt{\Delta\nu \tau}\)

• The rms of the output square voltage, expressed in temperature, is the noise of the input signal T_s divided by the number of independent samples
• The weakest signals that are detectable are a few times σ_T

5.2 The radiometer equation

• The noise contribution from the atmosphere + all components of the telescope = T_sys, the system temperature

• Observing a source with a spectral resolution δν, during a time τ, leads to a fluctuation rms of the antenna temperature:

  σ_Ta = κ 2^1/2 T_sys / (δν τ)^1/2

• κ: a factor which is usually 1 or 1.414, depending on the way the ON-OFF is performed
• The weakest signals that are detectable are a few times σ_Ta
• T_sys ≈ 100 to 1000K, highly dependent on frequency (atmospheric lines)