Astronomy 700: Radiation. 1 Basic Radiation Properties

Astronomy 700: Radiation. 1 Basic Radiation Properties 1.1 Basic definitions Fundamental importance to Astronomy: Almost exclusive carrier of information Radiation: Energy transport by electromagnetic fields Other forms of energy transport: cosmic rays • stochastic transport (micro: conduction, macro: convection) • gravitational waves • bulk transport (organized flows) • plasma waves • ... • Transport time variability (see section of E&M) → 1.1.1 The spectrum The most natural description of electromagnetic radiation is through Fourier decomposition into waves: f(~r, t) f(~k,ν) (1.1) ↔ where E is some variable describing the radiation field. Question: Why is this so natural? As we will shortly see, electromagnetic radiation naturally decomposes into waves with wavelength λ and frequency ν 1 Often, it is convenient to write the wave vector ~k =2πk/λˆ and angular frequency ω =2πν. In vacuum, group and phase velocity of those waves are equal: 10 1 λν = ∂ω/∂k c 2.99792... 10 cms− (1.2) ≡ ≡ × Fourier decomposition allows us to describe the local spectrum of the radiation at a fixed point in space as the Fourier transform ∞ f f(ν)= dtei2πνtf(t) (1.3) F ≡ Z−∞ and the inverse Fourier transform 1 ∞ i2πνt − f f(t)= dνe− f(ν) (1.4) F ≡ Z−∞ Without going into any details on Lebesque integration, it is worth pointing out the following identity: The inverse Fourier transform of a delta function in frequency is ∞ 1 i2πνt i2πν0t − δ(ν ν )= dνe− δ(ν ν )= e− (1.5) F − 0 − 0 Z−∞ i2πν0 t Thus, the Fourier transform of e− is ∞ i2πν0t i2π(ν ν0)t e− = dte − = δ(ν ν ) (1.6) F − 0 Z−∞ as one would expect for a decomposition into a spectrum of different exponentials. N.B.: Fourier transform conventions often differ in normalization and definition of frequency (ω vs. ν), so take care to make sure not to miss factors of 2π along the way. Broad classification into spectral regions: Astronomy has fallen historically into different bands, enforced by fundamentally different observ- ing techniques. The following figure shows the rough classification of spectral regions (largely anecdotal): 2 radio IR UV X-rays γ-rays visible 9 12 15 18 ν [Hz] Figure 1.1: Common classification into different spectral bands, dictated by observational tech- nique 1.1.2 Wave-particle duality and the radiative limit Since we measure radiation only through the interaction with matter, wave propagation in vacuum is not all that interesting. When radiation interacts with matter, quantum mechanics can become important. As Planck found (and as we will see later in the semester), photon phase space is granular, with a fundamental quantized phase space volume of h3. Photon energy: Eν = hν 28 where h is the Planck constant h 6.626075... 10− ergss ≡ × Photons are relativistic, thus their momentum is directly proportional to their energy: Photon momentum: Pν = hν/c N.B.: The classical description of electromagnetic waves is valid only in the statistical limit of many photons. The optical limit of radiation (“ray treatment”) is further only valid in the limit of large distances r: r λ ≫ This is the limit that we will concentrate on for the next sections. Finally, the wave field we detect from astronomical sources is virtually always in the far field limit: 3 r λc/u, where u is the characteristic velocity of the radiating particles. ≫ The far-field limit guarantees that the optical limit is safely satisfied for the travel of light between the source and us, though quantum mechanical effects come into play when the radiation interacts with the detector, where characteristic distances can be smaller than the wavelength of the radiation. 1.1.3 Observables and basic definitions of radiation quantities Astronomers want to measure radiation. Suppose you wanted to build a light capturing device, what quantities could you measure, and what properties of the device would be important? Question: What are the properties describing a “ray” of light? Observable: total energy of light captured dE Detector properties: Angular resolution/pixel scale: solid angle dΩ • Bandwidth: frequency/wavelength range dν • Exposure: time interval of measurement dt • Effective area: total sensitive area of detector dA • Independent variables: position ~r • ˆ direction ~k • frequency ν • time t • (polarization angle ψ) • A) Specific intensity or (surface) brightness: Iν ˆ Define a quantity that describes completely how the measured energy dE depends on ~r, ~k, ν, t, and describes the radiation field as completely as possible (neglecting polarization) given the information from the detector: dE I (~r, t, k,νˆ ) (1.7) ν ≡ dν dtdΩ dA 4 Figure 1.2: Sketch of wave direction, differential surface element, and solid angle. Note: for Iν, dA is taken as perpendicular to kˆ, while flux Fν is measured with respect to some fixed dA and integrates over all solid angles (i.e., kˆ does not have to be perpendicular to dA). where the surface dA is taken perpendicular to the direction of the ray kˆ, as shown in the figure. From Iν we can construct a number of important integrated quantities, or “moments” with respect to different observables. B) Mean intensity: Jν The zeroth moment of Iν with respect to the polar angle cos θ is the mean intensity: 1 J dΩI (1.8) ν ≡ 4π ν Z4π C) Specific flux: Fν Energy flux at frequency ν across a surface dA, integrated over all photon directions (coming from both sides of the surface). F dΩcosθI (1.9) ν ≡ ν Z4π where θ is measured relative to the normal of dA. Fν is the first moment of Iν with respect to cos θ. For an unresolved object, we can typically not determine the solid angle spanned by the object. 5 The flux is therefore the most general quantity we can derive directly from any measurements for the object. N.B.: Fν =0 for isotropic radiation. Sometimes it is useful to define a photon number flux. Since eν = hν, this is simply Φν = Fν/hν (1.10) D) Total flux: F Total energy flux across dA: ∞ F dνF (1.11) ≡ ν Z0 E) Radiation pressure: prad Total momentum flux across dA: ∞ I p = dν dΩcos2 θ ν (1.12) rad c Z0 Z4π 2 Question: prad is the second moment of Iν w.r.t. cosθ. Where does the cos θ come from? F) Radiative energy density: uν Amount of radiative energy contained per unit volume. Differential volume element: dV = dAds, where ds is measured along any given ray. Since light always travels at c, we have dt = ds/c and can integrate over all rays and write duν = dE/(dAdsdν)= dE/(cdtdAdν): I dE u = du = dΩ ν = (1.13) ν ν c dAdsdν Z Z4π which can easily be verified to have the proper dimensions, and ∞ u = dνuν (1.14) Z0 6 N.B.: For isotropic radiation, u =3p, which is an important analog to relativistic particles (verify this as a quick exercise in integration in spherical polar coordinates). G) Photon density: nν Number of photons per unit frequency interval per unit volume. Since eν = hν, we have u n = ν (1.15) ν hν H) Luminosity: Lν Specific (spectral) luminosity Lν = dAFν (1.16) I and total, “bolometric” luminosity: L = dAF (1.17) I measure the total radiative output of an astronomical object. This is mostly what is of astrophysical interest and it has to be extrapolated from a set of measurements. 33 1 Example: The sun has a luminosity of L =3.83 10 ergss− . ⊙ × Question: At a distance of r =1.5 1013 cm, what is the solar flux measured on earth (i.e., the solar constant)? × I) Spectral index: α It is often convenient to plot spectra on a log-log plot in frequency, log Fν vs. log ν. Many emission processes produce powerlaw-type spectra over some frequency range, α F ν− (1.18) ν ∝ which appear as straight lines in a log-log plot. It is then useful to define a local spectral index ∂log F α ν (1.19) ≡− ∂log ν 7 which is the local slope of the spectrum in a log-log plot and is well defined even for non-powerlaw spectra, with the implicit understanding that α depends on frequency. Sometimes it is also useful to define a photon index Γ ∂logΦ Γ ν = α +1 (1.20) ≡− ∂log ν N.B.: Sometimes the sign convention for the powerlaw index is switched, so make sure you understand how α is defined in a specific application J) ν λ conversions: − Given λν = c, We can express any specific quantity fν = df/dν with respect to wavelength instead: df df dλ λ2 c f = = f or f f (1.21) ν dν dλ dν ≡ λ c λ ≡ ν=c/λ λ2 Note the sign conversion implicit here such that both fν and fλ are always defined as positive. 1.1.4 Inverse square law From eq. (1.16) and energy conservation, it follows that the flux from an isotropic radiation source must obey L F = ν (1.22) ν 4πr2 For large distances (r Robj), the small angle approximation implies that radiation appears quasi-isotropic and the inverse≫ square law holds even for intrinsically non-isotropic sources. This approximation holds as long as dln I /dθ r/R. ν ≪ 1.1.5 Conservation of surface brightness Consider an object of surface area dS and an observer along a ray ~k at distance r. In the small 2 angle approximation, the solid angle subtended by the source is dΩS = dS/r , while the area dA 2 over which a given bundle of rays is spread is proportional to dA = dΩAr .

Load more