Units and Calculations or 10 log 1010 x

Logarithmic Scaling • A logarithmic factor of 10 is a Bel, in honor of

Numerical Interlude To better than 1%: • • 0 d. B 3 d. B

What about the UNITS? • d. B always give ratios (pure numbers), e. g.

Useful Definitions • Power P = k. TB – K is Boltzman’s constant, 1.

Useful Definitions (cont’d) • Power Flux Density: – PFD is radiated power passing through

Useful Definitions (cont’d) • Isotropic Aperture (unity gain in all directions) at a wavelength

Example • Tsys: If you know a room temperature of 290 K is –

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Units and Calculations, or 10 log 10(10 x units) = x d. Bunits Mike Davis, SETI Institute Spectrum Management Summer School, Green Bank, 6/2002

Logarithmic Scaling • A logarithmic factor of 10 is a Bel, in honor of Alexander Graham Bell • Bels are rarely seen. However, 1/10 th of a Bel, 10 log 1010 x, the deci. Bel (d. B), is the lingua franca of Engineering • 1 0 d. B • 10 10 d. B 1/10 -10 d. B • 100 20 d. B 1/100 -20 d. B • 103 30 d. B 10 -3 -30 d. B etc.

Numerical Interlude To better than 1%: • • 0 d. B 3 d. B 6 d. B 9 d. B 10 d. B 7 d. B 4 d. B 1. 0 2. 0 4. 0 8. 0 10. 0 5. 0 2. 5 1. 25 • • 2 d. B 5 d. B 8 d. B 11 d. B /2 2 4 • 2 1. 5 d. B • 1 Stellar Magnitude is exactly -4 d. B

What about the UNITS? • d. B always give ratios (pure numbers), e. g. – Power: Pd. B = 10 log 10 P/P 0 – Several options for P 0 – watts, milliwatts, … – Append the unit to d. B: d. BW, d. Bm, … • Not limited to Power. For example – Bandwidth B: 10 MHz 70 d. BHz – Time : 2000 seconds 33 d. Bs • Seconds * Hz gives a pure number: – (B ): (70 d. BHz+33 d. Bs)/2 = 51. 5 d. B

Useful Definitions • Power P = k. TB – K is Boltzman’s constant, 1. 38 10 -23 Joules/Kelvin (-228. 6 d. BW/Hz/K) – T is absolute temperature in Kelvins – B is the bandwidth • At Room Temperature (290 K): k. T d. B = -204. 0 d. BW/Hz

Useful Definitions (cont’d) • Power Flux Density: – PFD is radiated power passing through a given area: W/m 2 • Spectral Power Flux Density: – PFD per unit bandwidth: W/m 2/Hz – 1 Jansky is 10 -26 W/m 2/Hz (sum of both polarizations) -260 d. BW/m 2/Hz

Useful Definitions (cont’d) • Isotropic Aperture (unity gain in all directions) at a wavelength : Ai = 2/4 [m 2] (This is the area of a circle with a circumference of . ) • The isotropic aperture drops off rapidly with : Wavelength 1 m 1 mm Isotropic Aperture -11 d. Bm 2 -71 d. Bm 2 • Effective Aperture with Gain G: Ae = G Ai =G 2/4 [m 2]

Example • Tsys: If you know a room temperature of 290 K is – 204 d. BW/Hz, what is Tsys = 29 K in these units? -214 d. BW/Hz • Ai: You are observing at 20 cm. What is your isotropic aperture in d. Bm 2? -25 d. Bm 2 • Radiometer Equation: You observe for 2000 seconds with a bandwidth of 10 MHz. What is your T/Tsys = 1/ (B )? -51. 5 d. B • What Spectral Power Flux Density arriving in an isotropic sidelobe equals this noise power? -240. 5 d. BW/m 2/Hz

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