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Lecture 11 (Was going to be –Time series –Fourier –Bayes but I haven’t finished these. So instead:) Radio astronomy fundamentals NASSP Masters 5003F -

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Presentation on theme: "Lecture 11 (Was going to be –Time series –Fourier –Bayes but I haven’t finished these. So instead:) Radio astronomy fundamentals NASSP Masters 5003F -"— Presentation transcript:

1 Lecture 11 (Was going to be –Time series –Fourier –Bayes but I haven’t finished these. So instead:) Radio astronomy fundamentals NASSP Masters 5003F - Computational Astronomy

2 Radio Astronomy Fundamentals Source (randomly accelerating electrons) Noisy electro- magnetic radiation (transfers energy) Antenna (simple dipole example) Load resistance R Types of electron acceleration: Thermal (random jiggling) Synchrotron (spiral) Spectral line (resonant sloshing)

3 NASSP Masters 5003F - Computational Astronomy Noise power spectrum Analyse the signal into Fourier components. jth component is: The Fourier coefficient V j is in general complex-valued. Power in this component is: Closely related to the ‘power spectrum’ we’ve already encountered in Fourier theory. V j (t) = V j exp(-i ω j t) = V j (cos[ω j t] + i sin[ω j t]) P j = V j *(t)V j (t)/R = V j *V j /R (cos 2 [ω j t] + sin 2 [ω j t]) = V j *V j /R

4 NASSP Masters 5003F - Computational Astronomy Averaging the power spectrum t = 1t = 16 t = 64t = 4

5 NASSP Masters 5003F - Computational Astronomy Total noise power output by the antenna. “Noise is noise”: signal from an astrophysical source is indistinguishable from contamination from –Background thermal radio noise. –Ditto from intervening atmosphere. –Noise generated in the receiver system. Each of these makes a contribution to the total. Thus the total noise power is P total = P source + P background + P atmosphere + P system

6 NASSP Masters 5003F - Computational Astronomy On-and-off source comparison The simplest way to determine the source contribution is to make 1 measurement pointing at the source, then a second pointing away from (but close to) the source, then subtract the two. Scanning over the source is also popular. Uncertainty in total power measurements is: A low-pass filter with a time constant t is another way of ‘averaging’.

7 NASSP Masters 5003F - Computational Astronomy Antenna detection efficiency The source radiates at S W m -2 Hz -1. The antenna has an effective area A e in the direction of the source. (Eg for a dish antenna pointed to the source, this is close to the actual area of the dish.) Thus the power per unit frequency interval picked up by the antenna is: However, antennas are only sensitive to one polarisation... w = A e S watts per herz.

8 NASSP Masters 5003F - Computational Astronomy Decomposition into polarised components The total power in the signal is the sum of the power in each polarization. An antenna can only pick up 1 polarization though.

9 NASSP Masters 5003F - Computational Astronomy Dependence on source polarisation If the source is unpolarized, the antenna will only pick up ½ the power, regardless of orientation. If the source is 100% polarized, the antenna will pick up between 0 and 100% of the power, depending on orientation (and type of detector – eg is detector sensitive to linear polarization, or circular). Obviously all values in between will be encountered. Thus measurement of source polarization is important.

10 NASSP Masters 5003F - Computational Astronomy Directionality of antennas. A radio telescope often (not always) incorporates a mirror. Parkes GMRT Ok as long as the roughness is << λ. An optically ‘smooth’ surface These are supposed to be smooth mirrors?

11 NASSP Masters 5003F - Computational Astronomy Directionality of antennas. Reflector Focal plane Point Spread Function Radio telescopes with a mirror can be analysed like any other reflecting telescope...

12 NASSP Masters 5003F - Computational Astronomy A more usual treatment: Beam width Side lobe It is often conceptually easier to imagine that the antenna is emitting radiation to the sky rather than absorbing it. Beam width ~ λ /D, same as for any other reflector. Eg Parkes 64m dish at 21 cm, beam width ~ 15’.

13 NASSP Masters 5003F - Computational Astronomy Going into a little more detail... Essential quantities: –The distribution of brightness B( θ, φ ) over the celestial sphere. (See next slide for definition of θ, φ.) The units of this are W m -2 Hz -1 sr -1 (watts per square metre per herz per steradian). –The effective area A e of the antenna, in m 2. (This is something which must be measured as part of the antenna calibration.) –The relative efficiency f( θ, φ ) of the antenna, which is normalized such that it has a maximum of 1. (This shape must also be calibrated.) –The received power spectrum w (units: W Hz -1 ).

14 NASSP Masters 5003F - Computational Astronomy Going into a little more detail... J D Kraus, “Radio Astronomy” 2 nd ed., Fig 3-2. Pointing direction of the antenna – NOT the zenith. Kraus uses P where I have f.

15 NASSP Masters 5003F - Computational Astronomy Going into a little more detail... The general relation between these quantities is: Remember that the ½ only applies where B is unpolarized. Further useful relations: It can be shown that Ω A = λ 2 /A e.

16 NASSP Masters 5003F - Computational Astronomy Going into a little more detail... Let’s consider two limiting cases: –B( θ, φ ) = B (ie, uniform over the sky); –B( θ, φ ) = S δ ( θ-0, φ-0 ) (ie, a point source of flux=S, located at beam centre). f f B B( θ,φ )=S δ w = ½ λ 2 B w = ½ A e S...the ½ still applies only for unpolarized B.

17 NASSP Masters 5003F - Computational Astronomy Conversion of everything to temperatures. Suppose our antenna is inside a cavity with the walls at temperature T (in kelvin). It can be shown that the power per unit frequency picked up by the antenna is Because of this linear relation between a white noise power spectrum and temperature, it is customary in radio astronomy to convert all power spectral densities to ‘temperatures’. Hence: w = kT watts per herz.

18 NASSP Masters 5003F - Computational Astronomy System temperature T source only says something about the real temperature of the source if –The source area is >> Ω A, and –The physical process producing the radio waves really is thermal. T atmosphere is a few kelvin at about 1 GHz. T background may be as much as 300 K if the antenna is seeing anything of the surroundings! Therefore avoid this. T system again says nothing about the real temperature of the receiver electronics. Rather it is a figure of merit – the lower the better. T total = T source + T background + T atmosphere + T system

19 NASSP Masters 5003F - Computational Astronomy The more usual way to write the measurement uncertainty: Thus the minimum detectable flux is and the minimum detectable brightness: Note: 1.B min not dependent on A e. 2.Factors of 2 only for unpolarized case.

20 NASSP Masters 5003F - Computational Astronomy A more realistic system: R M Price, “Radiometer Fundamentals”, Meth. Exp. Phys. 12B (1976), Fig 1, section “Back end” “Front end”

21 NASSP Masters 5003F - Computational Astronomy Jargon The ‘antenna’: –the reflecting surface (ie the dish). The ‘feed’: –usually a horn to focus the RF onto the detector. The ‘front end’: –electronics near the Rx (shorthand for receiver). The ‘back end’: –electronics near the data recorder. The LO: –local oscillator. A 38 GHz feed horn. The corrugations are good for wide bandwidth. RF: –Radio Frequency. IF: –Intermediate Frequency.


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