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Astronomical Observing Techniques 2019 Lecture 13: a Summary Astronomical Observing Techniques 2019 Lecture 13: A summary Huub Röttgering [email protected] Questions related to the topics emphasized on the slides • What is the definition? • What is the principle? –How is it applied? Give example(s). • Why is this important/relevant? • Equation? –Do I really understand it, also intuitively? –(If we derived it,) How is it derived? –Why is it important? –What is its application? 2 Preparation • Be able to answer all these questions – If in doubt, read up using the reading material • Understand the questions and answers of the assignments and test exams Also: bigger questions: ask them now and the TAs will come back to you fully prepared for tomorrows session 3 Lecture 1: Black Bodies in Space 1. Blackbody Radiation 1. Planck Curve and Approximations 2. Effective Temperature 3. Brightness temperature 4. Lambert 2. Photometry 1. Flux and Intensity 2. Magnitudes 3. Photometric Systems 4. Color Indices 5. Surface Brightness 4 Blackbody Radiation • Cavity at fixed T, thermal equilibrium • Incoming radiation is continuously absorbed and re-emitted by cavity wall • Small hole → escaping radiation will approximate black-body radiation independent of properties of cavity or hole 5 Kirchhoff’s Law Gustav Kirchhoff stated in 1860 that “at thermal equilibrium, the power radiated by an object must be equal to the power absorbed.” • Blackbody cavity in thermal equilibrium with completely opaque sides • Opaque -> transmissivity τ = 0 • emissivity ε = amount of emitted radiation • Thermal equilibrium -> ε + ⍴ = 1 =1− + + =1 = = 0 Blackbody absorbs all radiation: ⍺=ε=1 Kirchhoff’s law applies to perfect black body at all wavelengths 6 Units: W Hz-1 m-2 7 Planck Curve: Emission, Power & Temperature Total radiated power per unit surface proportional to fourth power of temperature T: I T dd = M = T 4 ( ) σ = 5.67·10-8 W m-2 K-4 (Stefan-Boltzmann constant) Assuming BB radiation, astronomers often specify the emission from objects via their effective temperature. 8 Wien’s Law Relation between temperature and frequency/wavelength at which the BB attains its maximum: c −3 −3 T = 5.09610 mK or maxT = 2.9810 mK vmax Cooler BBs have peak emission at longer wavelengths and at lower intensities: T=293K 10000 1000 100 10 1 effective temperature [K] temperature effective 0,100 1,000 10,000 100,000 1000,000 wavelength [μm] 9 Planck Curve: Approximations Planck: 2h 3 1 I (T ) = c2 h exp −1 kT High frequencies (hv >> kT) ➔ Wien approximation: 2h 3 h I (T ) = exp− c2 kT Low frequencies (hv << kT) ➔ Rayleigh-Jeans approximation: 2 2 2kT I (T ) kT = c2 2 10 Effective Temperature The effective temperature of a body such as a star or planet is the temperature of a black body that would emit the same total amount of electromagnetic radiation. Effective temperature is often used as an estimate of a body's surface temperature 11 Brightness Temperature: Definition Brightness temperature: temperature of a perfect black body that reproduces the observed intensity of a grey body object at frequency ν. For low frequencies (hv << kT): Rayleigh- c2 Tb = ( )T = ( ) Iv Jeans 2kv2 Only for perfect BBs is Tb the same for all frequencies. 12 Lambert: Cosine Law* Lambert’s cosine law: radiant intensity from an ideal, diffusively reflecting surface is directly proportional to the cosine of the angle θ between the surface normal and the observer. Johann Heinrich Lambert (1728 – 1777) *Empirical law…. , mathematical derivation is complicated 13 Magnitudes: Apparent Magnitude Apparent magnitude is relative measure of monochromatic flux density Fλ of a source: F m − M 0 = −2.5log F0 M0 defines reference point (usually magnitude zero). In practice, measurements through transmission filter T(λ) that defines bandwidth: m − M = −2.5log T F d + 2.5log T d ( ) ( ) 0 0 14 Magnitudes: Absolute Magnitude • absolute magnitude = apparent magnitude of source if it were at distance D = 10 parsecs M = m + 5 − 5log D • m ≈ -27, M = 4.83 at visible wavelengths • MMilky Way = −20.5 → M- MMilky Way = 25.3 10 • Luminosity LMilky Way = 1.4×10 L 15 Magnitudes: Bolometric Magnitude Bolometric magnitude is luminosity expressed in magnitude units = integral of monochromatic flux over all wavelengths: F()d 0 −8 W M bol = −2.5log ; Fbol = 2.5210 2 Fbol m If source radiates isotropically: L 26 M bol = −0.25+ 5log D − 2.5log ; L = 3.82710 W L Bolometric magnitude can also be derived from visual magnitude plus a bolometric correction BC: M bol = MV + BC BC is large for stars that have a peak emission very different from the Sun’s. 16 Photometric Systems: AB and STMAG For given flux density Fv, AB magnitude is defined as: m(AB ) = −2.5log F − 48.60 • object with constant flux per unit frequency interval has zero color • zero point defined to match zero points of Johnson V-band • used by SDSS and GALEX -1 2 -1 • Fv in units of [erg s cm Hz ] STMAG system defined such that object with constant flux per unit wavelength interval has zero color. STMAGs are used by the HST photometry packages. 17 Color Index: Interstellar Extinction • Absolute magnitude definition: • Interstellar Mextinction= m + E5or− absorption5log D A affects the apparent magnitudes E(B −V ) = A(B)− A(V ) = (B −V )observed − (B −V )intrinsic • Need to include absorption to obtain correct absolute magnitude: M = m + 5 − 5log D − A 18 Photometric Systems: Conversions Fλ F 19 Surface Brightness: Point & Extended Sources Point sources = spatially unresolved Extended sources = well resolved Brightness ~ 1 / distance2 Surface brightness ~ const(distance) Size given by observing conditions Brightness ~ 1/d2 and area size ~ 1/d2 Surface brightness [mag/arcsec2] is constant with distance! 20 Lecture 2: Monsieur Fourier and his Elegant Transform 1. Introduction – Intuition, hearing, seeing 2. Fourier Series of Periodic Functions 3. Fourier Transforms – Properties – Examples – Numerical Fourier Transforms 4. Convolution, cross- and auto correlation – Parseval and Wiener-Khinchin Theorems 5. Sampling – Nyquist theorem – Aliasing 21 1. Introduction Fourier Transformation Functions f(x) and F(s) are Fourier pairs +¥ F(s) = ò f (x)×e-i2p xsdx -¥ +¥ f (x) = ò F(s)×ei2p xsds -¥ 22 23 Fourier Fourier Reminder: Reminder: Spatial Spatial decomposition frequency frequency 1D Fourier analysis Series of a step edge A discrete discrete sum of sines A x Fourier transform: just a change of basis 24 25 |F(u,v)| f(x,y) Example: Example: action original of filters filters on a ``Signal at the pupil pupil the at ``Signal (‘mirror’) a telescope’’ of real real l o w p image a s s high pass 2. Fourier Series of Periodic Functions Decomposition using sines and cosines as orthonormal basis set Periodic function: f (x) = f (x + P) a ¥ 2pnx 2pnx Fourier series: 0 é æ ö æ ö ù f (x) = + åêan cosç ÷ + bn sinç ÷ ú 2 n=1 ë è P ø è P ø û P/2 2 æ 2pnx ö a = f (x)cos dx n ò èç ø÷ Fourier coefficients: P - P/2 P P/2 2 æ 2pnx ö b = f (x)sin dx n ò èç ø÷ P - P/2 P Period: P Frequency: ν = 1/P Angular frequency: ω = 2π/P 26 Symmetries and Fourier Transforms symmetry properties of Fourier transforms have many applications • Define – even function: feven(-x) = feven(x) – odd function: fodd(-x) = -fodd(x) • Hence – even part of f(x): feven(x) = ½[f(x)+f(-x)] – odd part of f(x): fodd(x) = ½[f(x)-f(-x)] – arbitrary function: f(x) = feven(x) + fodd(x) 27 Fourier Transform Symmetries f (x) = feven (x) + fodd (x) feven (-x) = feven (x) fodd (-x) = - fodd (x) e-i2p xs = cos(2p xs) - isin(2p xs) +¥ Þ F s = 2 f x cos 2p xs dx ( ) ò even ( ) ( ) 0 +¥ - i 2 f x sin 2p xs dx ò odd ( ) ( ) 0 f(x) real: feven(x) transforms to (even) real part of F(s), fodd(x) transforms to (odd) imaginary part of F(s). 28 Fourier Transform Similarity 1 s Expansion of f(x) contracts F(s): f (x)→ f (ax) F a a 29 Fourier Transform Properties LINEARITY: a·f(x)+b·g(x) ⇔ a·F(s) + b·G(s) −i2as TRANSLATION: f (x − a) e F(s) n f (x) n DERIVATIVE: (i2s) F(s) xn -1 INTEGRAL: ò f (x)¶x Û (i2ps) F(s) + cd (s) 30 Important 1-D Fourier Pairs 1 2 2 f (x) = e-px F(s) = e-ps f (x) = const F(s) = d(s) 31 Important 1-D Fourier Pairs 2 f (x) = P(x) F(s) = sinc(s) f (x) = L(x) F(s) = sinc2 (s) 32 Important 1-D Fourier Pairs 3 1 f (x) = cos(px) F(s) = d (s ± ) 2 33 Convolution Convolution of two functions, ƒ∗g, is integral of product of functions after one is reversed and shifted: + h(x) = f (x) g(x) = f (u) g(x −u) du − f (x) Û F(s) g(x) Û G(s) h(x) = f (x)* g(x) Û F(s)×G(s) = H (s) 34 Cross-Correlation Cross-correlation (or covariance) is measure of similarity of two waveforms as function of time-lag between them. + k(x) = f (x) g(x) = f (u) g(x + u) du − Difference between cross-correlation and convolution: • Convolution reverses the signal (‘-’ sign) • Cross-correlation shifts the signal and multiplies it with another Interpretation: By how much (x) must g(u) be shifted to match f(u)? Answer given by maximum of k(x) 35 Cross-Correlation in Fourier Space f (x) Û F(s) g(x) Û G(s) h(x) = f (x) Ä g(x) Û F(s)×G* (s) = H (s) In contrast to convolution, in general f Ä g ¹ g Ä f 36 Auto-Correlation Theorem Auto-correlation is cross-correlation of function with itself: + k(x) = f (x) f (x) = f (u) f (x + u) du − + + f (x)Ä f (x) Û F(s)F* (s) = F(s) 2 37 Parseval’s Theorem Parseval’s theorem (or Rayleigh’s Energy Theorem): Sum of square of a function is the same as sum of square of the Fourier transform: + + f (x) 2dx = F(s) 2ds − − Interpretation: Total energy contained in signal f(x), summed over all x is equal to total energy of signal’s Fourier transform F(s) summed over all frequencies s.
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