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Lecture 13 • Today I plan to cover: – The jansky. – Photon noise at radio wavelengths? – Flux calibration. – A bit more about noise temperatures. – Polarized radio signals. – Radio spectroscopy. NASSP Masters 5003S - Computational Astronomy - 2009 The jansky • The standard unit of flux density in radio astronomy. • Symbol Jy. • Named after a pioneer in the field. • = 10-26 W m-2 Hz-1. Photon noise..? • High-energy astronomy usually is dominated by this. But radio υ hugely less! • Energy of a photon at 1.4 GHz is ~1e-24 J. • Hence 10 mJy over 1 MHz at this freq • ~ only 100 photons/sec. - a bit unexpected. NASSP Masters 5003S - Computational Astronomy - 2009 Flux calibration • • The bandwidth and gain of a radiometer tend not to be very stable. There are several methods of calibration. Eg: 1. Switching between the feed and a ‘load’ at a temperature similar to the antenna temperature. – To detector To detector “Warm load” at T “Warm load” at T But, the required physical temperature of the load resistor can be < 20 K... need to cool with He. NASSP Masters 5003F - Computational Astronomy - 2009 More widely used: 2. Periodic injection of a few % of noise into the feed. Noise sources can be made much more stable than noise detectors. • It is still good, while observing, to look occasionally at an astronomical source which has the following properties: – – • Known, stable flux density. Should also be unresolved (compact). These are difficult conditions to meet at the same time! Compact sources tend to vary with time. NASSP Masters 5003S - Computational Astronomy - 2009 Typical noise temperatures J D Kraus, “Radio Astronomy” 2nd ed., fig 8-6.(+ 7-25) NASSP Masters 5003F - Computational Astronomy - 2009 Polarized EM waves – conventions: y x Snapshot of a wave moving in the positive z direction. Left-hand circular polarization according to IEEE convention. (Physicists use the opposite convention.) z Direction of rotation of the field vector as seen by an observer. NASSP Masters 5003F - Computational Astronomy - 2009 Sources of polarized radio waves: • Thermal? No • Spectral line? No (unless in a strong B field) • Synchrotron? YES. – And this is the most common astrophysical emission process. • All jets emit synchrotron – and jets are everywhere. Magnetic field B Electron moving at speed close to c Linearly polarized emission. • Scattering off dust? YES. NASSP Masters 5003F - Computational Astronomy - 2009 How to describe a state of polarization? Stokes parameters I, Q, U and V. I = total intensity. Q = intensity of horizontal pol. U = intensity of pol. at 45° V = intensity of left circular pol. V axis U axis Q axis Therefore need 4 measurements to completely define the radiation. Polarization fraction d: Visualize with the “Poincaré sphere.” of radius I. Q2 U 2 V 2 d I NASSP Masters 5003F - Computational Astronomy - 2009 Antenna response, and coherency matrices. • The antenna response is different for different incoming polarization states. • This may be quantified by 4 ‘Stokes effective areas’ AI, AQ, AU, AV. • But it is more convenient to express both the radiation and the antenna response as coherency matrices: 1 S 2I I Q U iV U iV I Q and 1 A 2 Ae AI AQ A iA V U AU iAV AI AQ • Then the power spectral density detected is w = AeI×Tr(AS) (‘Tr’ = the ‘trace’ of the matrix, ie the sum of all diagonal terms.) NASSP Masters 5003F - Computational Astronomy - 2009 Depolarization due to finite resolution Arrows show the polarization direction. Half-power contour of the beam. Nett polarization observed. Waves from different areas of the source add incoherently. Result: some degree of depolarization. In general, the finer the resolution, the higher the polarization fraction. NASSP Masters 5003F - Computational Astronomy - 2009 Faraday rotation. • Any linear polarized wave can be decomposed into a sum of left and right circularly polarized waves. • In a magnetized plasma, the LH and RH components travel at slightly different speeds. • Result: – The plane of polarization rotates. – The amount of rotation θ is proportional to distance travelled x the field strength x the number density of electrons. – θ is also proportional to λ2. • Most due to Milky Way, but the Earth’s ionosphere also contributes – in a time-variable fashion. The ionosphere is a great nuisance and radio astronomers would abolish it if they could. NASSP Masters 5003F - Computational Astronomy - 2009 Faraday rotation J D Kraus, “Radio Astronomy” 2nd ed., fig 5-4 The slope of the line is called the rotation measure. Why is there progressive depolarization with increase in wavelength? NASSP Masters 5003F - Computational Astronomy - 2009 Faraday rotation – another cause of depolarization Because the rotation measure is not uniform and may vary within the beam. Eg: Half-power contour of the beam. NASSP Masters 5003F - Computational Astronomy - 2009 Radio spectroscopy • The variation of flux with wavelength contains a lot of information about the source. • We can pretty much divide sources into – Broad-band emitters, eg • Synchrotron emitters • HII regions (ie ionized hydrogen) • Thermal emitters – Narrow-band emitters (or absorbers), eg • HI (ie neutral hydrogen) • Masers • Neutral molecular clouds NASSP Masters 5003F - Computational Astronomy - 2009 Broad-band emitters • Most of these have spectra which, over large ranges of wavelength, can be described by a simple power law, ie S • For thermal sources, the Rayleigh-Jeans approximation to the black-body radiation law gives a spectral index α = -2. • Synchrotron sources have +ve α, averaging around +0.8. • HII regions exhibit a broken power law. NASSP Masters 5003F - Computational Astronomy - 2009 Wavelength or frequency? • Since λ=c/υ, a power-law spectrum S implies S • Ie, the sign of the spectral index depends whether you give the spectrum as a function of wavelength or frequency. • Make sure you know which convention is intended before using a spectral index (and always record this convention in your own reports). NASSP Masters 5003F - Computational Astronomy - 2009 Broad-band emitters J D Kraus, “Radio Astronomy” 2nd ed., fig 8-9(a) Mostly synchrotron. Steep-spectrum source Reversed-spectrum sources: mostly thermal, ie slope = 2. Log-log plot so all straight lines are power-laws. One spectrum, many spectra. Note too that nearly all broadband spectra are quite smooth. NASSP Masters 5003F - Computational Astronomy - 2009 HII regions • The gas here is ionized and hot (10,000 K is typical) – usually as a result of intense irradiation from a massive young star. • The radiation comes from electrons accelerated (diverted) as they come close to a positive ion. + -e • This radiation mechanism is called free-free, because the electron being accelerated is not bound to an atom either before its encounter or after. But it is basically a thermal process. • Otherwise known as bremsstrahlung (German for “braking radiation”. Yes spellt with 2 esses.) NASSP Masters 5003F - Computational Astronomy - 2009 Optical depth • Whenever you have a combination of radio waves and plasma, optical depth τ plays a role. – High τ = opaque – behaves like a solid body. – Low τ = transparent. • τ for a plasma is proportional to λ2. • Effective temperature Teff = T(1-e-τ). – Long λ - high τ - Teff ~ T – thus α = -2. – Short λ - low τ - Teff proportional to τ, thus λ2 means flux density S is constant, or α = 0. • Flat-spectrum source. – The troposphere is a good example of a thermal radiator which is optically thin at dm wavelengths. NASSP Masters 5003F - Computational Astronomy - 2009 Some more about synchrotron • Already covered the basics in slide 7. • Also subject to optical depth effects: J D Kraus, “Radio Astronomy” 2nd ed., fig 10-10 PKS 1934-63 – At low frequencies, opacity is high, the radiation is strongly self-absorbed: • α ~ -2.5. • Effective temperature limited to < 1012 K by inverse Compton scattering. NASSP Masters 5003F - Computational Astronomy - 2009 Narrow-band spectra • Molecular transitions: – Hundreds now known. – Interstellar chemistry. – Tracers of star-forming regions. – Doppler shift gives velocity information. • Masers: – Eg OH, H2O, NH3. – Like a laser – a molecular energy transition which happens more readily if another photon of the same frequency happens to be passing radiation is amplified, coherent. – Spatially localized, time-variable. • Recombination lines. NASSP Masters 5003F - Computational Astronomy - 2009 HI • The I indicates the degree of ionization. – Subtract 1 from this number to get the number of electrons stripped from the atom. • Eg FeVII, ‘iron seven’, means iron which has lost 6 electrons. Hence I means minus 0 electrons – just the neutral atom. Hydrogen has only 1 electron so the highest it can go is HII – which is just a bare proton. • The neutral H atom has a very weak (lifetime ~ 107 years!) transition between 2 closely spaced energy levels, giving a photon of wavelength 21 cm (1420 MHz). • But because there is so much hydrogen, the line is readily visible. NASSP Masters 5003F - Computational Astronomy - 2009 HI • Because the transition is so weak, and also because of Doppler broadening, hydrogen is practically always optically thin (ie completely transparent). • Thus the intensity of the radiation is directly proportional to the number of atoms. – For a cloud of mass M solar masses at a distance D megaparsecs, the measured flux φ (in W m-2) is ~ 6 10 23 M 2 Dc (I think.) NASSP Masters 5003F - Computational Astronomy - 2009 HI • Concept of column density in atoms per Distribution of atoms square cm. this way makes no difference. So many square cm Observer • Hydrogen will be seen in emission if it is warmer than the background, in absorption otherwise. NASSP Masters 5003F - Computational Astronomy - 2009 HI – Doppler information • Hubble relation between distance and recession velocity allows distance of far galaxies to be estimated. – Hence: 3D information about the large-scale structure of the universe. • Our Milky Way is transparent to HI – we can see galaxies behind it at 21 cm, whereas visible light is strongly absorbed. • Cosmic Doppler red shift z is given by obs true 1 v c v z 1 for v c 2 true c 1 v c NASSP Masters 5003F - Computational Astronomy - 2009 HI – Doppler information • Within galaxies: – Doppler broadening tells about the distribution of velocities within a cloud of hydrogen. – the Doppler shift of the HI line maps the rotation curve of the galaxy, eg: NGC 2403 Credit: F Walter et al (2008). (Courtesy Erwin de Blok.) NASSP Masters 5003F - Computational Astronomy - 2009 Model of galactic hydrogen • A good model for many galaxies is a disk of uniform surface density ρ within a radius R, rotating with a uniform speed |v|. – Such decoupling of speed from radius is not what is observed in eg planetary systems and is one of the indicators for large dominance of dark matter. • With distance D and inclination angle α this leads to a predicted spectral shape of R2 S f D 2 f 2 0 v c sin 2 2 f f centre 2 – Here I have used f instead of υ to avoid confusion with velocity v. (Caveat! This is my own derivation.) NASSP Masters 5003F - Computational Astronomy - 2009 Diagram of HI spectrum NASSP Masters 5003F - Computational Astronomy - 2009 An example exercise • You have an antenna of diameter 12 m, efficiency η=70%. The feed is linearly polarized and the detection chain has a noise temperature of 45 kelvin. How long do you have to observe to see an unresolved HI source of average flux density 2 Jy in a channel width of 15 kHz? – The relation between source flux density S and equivalent noise temperature T, for an unresolved source, is kT S pAeffective – where p is the fraction of signal the polarised detector is picking up. NASSP Masters 5003F - Computational Astronomy - 2009 An example exercise – HI emission is unpolarised, so p is 0.5 in the present case. – The effective area is the true area times the efficiency; = πr2η = in the present problem. – Hence 2 Tsource r S 2k • We need to observe long enough that the uncertainty in the total noise temperature (source plus instrumental) is equal to the source noise temperature. – A common trick: assume Tsource<<Tinst. – This is valid because we don’t need to know our observing time to more than a couple of significant figures! NASSP Masters 5003F - Computational Astronomy - 2009 An example exercise • We can therefore write, to a reasonable approximation: Ttotal Tinst T ~ t t • Setting this equal to the source temperature allows us to derive: 4 t Tinstk 2 r S 2 • = 41 s. (Don’t forget to convert Jy to MKS!) NASSP Masters 5003F - Computational Astronomy - 2009 Spectra - baselines • The spectrum of interest sits on top of a high ‘mesa’ due to system and background ‘temperature’ (ie noise). • We usually want to subtract the mesa and just leave the spectrum. • We could do that by alternating between onand off-source observations, and subtracting the two: – But this needs 4 times as much observing time to reach the same SNR! NASSP Masters 5003F - Computational Astronomy - 2009 Spectra - baselines • More commonly, the mesa is slowly varying compared to the spectrum, so one can fit some fairly smooth function to the mesa, then subtract it. • The examples (for which this has already been done) show it is not always so simple! • These show the infamous ‘Parkes ripple.’ NASSP Masters 5003F - Computational Astronomy - 2009 The ‘Parkes ripple’ A weak Fabry-Perot resonance occurs between the dish and the feed. 2D = nλ = (n+1)(λ-Δλ) => Δν ~ 5.5 MHz. D ~ 26 m NASSP Masters 5003F - Computational Astronomy - 2009