1 Radiation fundamentals

In defining the magnitude, we mentionned physical quantities such as the flux and luminosity without going in the details of how these quantities are defined. We now go into more details. In doing so, we will introduce a less familiar quantity, the specific intensity (also called brightness although this is a more ambiguous word and we prefer to call it specific intensity). In the process, we will review our knowledge of the black-body radiation which is so essential in astrophysics.

Definitions

• Luminosity: L or L_ν
• Flux density: F or F_ν
• Specific intensity (or brightness): I_ν

1.1 Luminosity

• This is an intrinsic property of a source
• Spectral luminosity L_ν: total power per unit bandwidth emitted by the source, in W/Hz
• Energy conservation: L_ν,1 = L_ν,2
• Warning: Luminosity is rarely called luminosity, but more frequently flux

1.2 Specific intensity

• Total, or integrated intensity: I=∫ I_ν dν

1.3 Flux density

• Expose an area dA to some radiation during a time dt. The energy in bandwidth dν crossing dA from all directions is:

  F_ν dA dt dν = ∫_Ω dE_ν

• In terms of the specific intensity: F_ν = ∫_Ω I_ν cos(θ) dΩ
• Units: F_ν in W/m²/Hz

• Detector of area 1m² will intercept F_ν W/m²/Hz from the source; obviously, the amount of energy depends upon the distance d:

  F_ν = L_ν / [4 π d²], W m^-2 Hz^-1

• The flux density is not an intrinsic property of the source

Spatially resolved source: spectral flux density and specific intensity

• Spectral flux density detected from \(A_k(\theta_k,\phi_k)\) within \(d \Omega(\theta_k,\phi_k)\): \[ d F_{\nu,k} = I_\nu(\theta_k,\phi_k) \cos\theta_k d \Omega(\theta_k,\phi_k) \]
• Total spectral flux density detected: \[ F_\nu = \int_{\Omega_S} I_\nu(\theta,\phi) \cos\theta d\Omega(\theta,\phi)\]

Units

• I_ν is measured in W/m²/Hz/sr
• F_ν is measured in W/m²/Hz
• More convenient unit is the Jansky (see black-body radiation)
  • 1 Jy = 10^-26 W/m²/Hz
• In cgs units: 1 J = 10⁷ erg
  • W/m²/Hz = 10³ erg/cm²/Hz

1.4 Solid angles

• \(\Omega = S(d)/d^2 = \int\int d^2 S \vec{u}\cdot\vec{N}/d^2\)
• dΩ = dS cosα/d²
• Spherical coordinates: dΩ = sinθ dθ dφ
• Uniform sphere: Ω=4π
• Cylindrical cone of half-opening angle α: Ω=2π(1-cosα)≈πα² for α≪1

Application

• Isotropic radiation field:
  • F_ν = ∫_Ω I cos(θ) dΩ
  • dΩ = sin(θ)dθ dφ
  • F=0
• Integrating over one half:
  • \(F_\nu = \int_0^{\pi/2} d\theta \int_0^{2\pi} d\phi I_\nu \cos(\theta)\sin(\theta) d\theta d\phi = \pi I_\nu\)
• Flux emitted by a star within solid angle ω is: L_ν = ω d² F_ν, where F_ν is the flux density at a distance d;
  • The total luminosity L_ν is thus (assuming it is isotropic) 4π d² F_ν

1.5 Fundamental properties of the specific intensity

How is I_ν related to the radiation field emitted by the source?

• \(d\Omega_A = dA_\perp /d^2\): \(dA_\perp = d A \cos\theta_A\)
• \(d\Omega_B = dB_\perp/d^2\): \(dB_\perp = d B \cos\theta_B\)

• p_ν^A: power emitted by dA and crossing dB = I_ν(A) dA_⊥ dΩ_B
• p_ν^B: power flowing through dB looking at dA = I_ν(B) dB_⊥ dΩ_A
• Assuming propagation in empty space: p_ν^A=p_ν^B
• I_ν(A) = I_ν(B): the detector measures the specific intensity emitted by the source; intrinsic property of the source
• Important: \(I_\nu\) can only be measured for spatially resolved sources

Conservation of the specific intensity (2)

• \(I_{\nu,1}\): specific intensity measured on the detector at distance \(d_1\)
• \(I_{\nu,2}\): specific intensity measured on the detector at distance \(d_2\)
• Source area increases as \(d^2\) while dΩ_det decreases as \(d^{-2}\)
• Conservation of energy in free space: \(I_{\nu,1} = I_{\nu,2} = I_\nu\)
• The detector measures the source brightness: Intrinsic property of the source

Resolved vs unresolved source

• Flux dilution when source becomes unresolved: specific intensity can only be measured for a resolved source unless its angular size (solid angle) is known;
• In contrast, if we do not know the source size, what we measure is the flux F_ν;
• More generally, for an unresolved source, one measure the flux; if the distance is known, the luminosity is recovered; if in addition the source size is known, one obtains the specific intensity.

1.6 Spectral Energy Density (SED) and Reduced Brightness

• SED: flux, or brightness, over a broad spectral range; here: sketch of the SED of a normal spiral galaxy
• This plot gives the sensation that the galaxy emits much more energy in the X-ray domain than in the radio domain.
• This is because the wavelength are shown on a logarithmic scale. However, displaying it on a linear scale would completely wash out the spectral details at short wavelength.
• Very broad spectrum: distortion of the spectrum: 1μm in the radio is tiny, but huge at short visible wavelenghts; F_λ will under-represent the flux emitted at radio wavelength, and overrepresent the flux at short wavelength.

Reduced brightness

• A very practical way to deal w/ this is to keep plotting the wavelength on a logarithmic scale, while replacing the flux F_λ by λ F_λ;

• Using the reduced brightness also preserves the link between the area under the curve and the corresponding integrated energy:

  \[ \int I_\nu d\nu = \int I_\lambda d\lambda \] and \[ \int \nu I_\nu d(\log\nu) = \int I_\nu d\nu \] so that also \[ \int \nu I_\nu d(\log\nu) = \int \lambda I_\lambda d(\log \lambda) \]

• Sky background radiation (zodiacal light subtracted); notice the CMB black-body
• By plotting ν I_ν againt log(λ), it is possible to 'measure' directly, w/ a ruler, the area under the curve to obtain the energy radiated within a given spectral range.

2 Black-body radiation

• The Cosmic Microwave Background (CMB)
• Notice the 1σ error bars which were multiplied by 400

The CMB in other units

• 1 Jy = 10^-26 W/m²/Hz

2.1 Definition

• A black-body is an ideal body which absorbs 100% of the radiation regardless of the wavelength, incidence angle; the black body has zero reflectance and zero transmittance;
• Black-body radiation does not depend upon its geometry
• It is unpolarized
• Its specific intensity depends on a single parameter: the temperature

2.2 Mathematical expressions

• The spectral interval may be frequency or wavelength

• Conservation of energy ensures that B_ν dν = B_λ dλ

  B_λ = B_ν c/λ², or λ B_λ = ν B_ν

• Two equivalent formulations:

  \[B_\nu = \frac{2h\nu^3}{c^2}\,\frac{1}{e^{h\nu/k_BT}-1}\]

  and

  \[B_\lambda = \frac{2hc^2}{\lambda^5}\,\frac{1}{e^{hc/\lambda k_BT}-1}\]

• Units:
  • B_ν is expressed in W/m²/Hz^-1/sr
  • B_λ is expressed in W/m²/m/sr

Asymptotic laws

• Wien's law: at short wavelength, λ ≪ hc/kT (high frequency hν≫ kT)

  \(B_\lambda = \frac{2hc^2}{\lambda^5}\,e^{-hc/\lambda kT}\)

  \(B_\nu = \frac{2h\nu^3}{c^2}\,e^{-h\nu/kT}\)

• Rayeigh-Jeans law: at long wavelength, λ ≫ hc/kT (low frequency hν≪ kT)

  \(B_\nu = \frac{2kT\nu^2}{c^2} = \frac{2kT}{\lambda^2}\)

  \(B_\lambda = \frac{2kTc}{\lambda^4}\)

• Approx. to within 1% if λT<3.1 mm K for Wien's law, and if λT > 80mm K for R-J's law.

Wien's displacement law

• Black-body radiation has a maximum at a wavelength λ_max which depends solely on T;

• The wavelength of the maximum of the black-body radiation is given by:

  \(\partial B_\lambda(T)/\partial \lambda = 0\) which gives

  λ_max T = 2.898E-3 m K (≈ 3 mm K), B_λ(CMB) peaks at λ≈ 1mm

• We could have also searched for the maximum of the B_ν(T) curve:

  \(\partial B_\nu(T)/\partial \nu = 0\), giving

  ν_max = 58.8×T GHz/K (or ν/100 GHz ≈ T/2); B_ν(CMB) peaks at ν≈160GHz

• Warning λ_max ≠ c/ν_max! This reflects the non-linear relation between ν and λ; however, if we had computed the maximum of ν B_ν or λ B_λ, we would have find another displacement law of the form ν'_max = f(T) and λ'_max = g(T), where ν'_max = c/λ'_max; this is because λ B_λ = ν B_ν.

• Substituting Wien's displacement law into B_λ or B_ν gives:

  B_λ^max(T) = 4.1e-6 T⁵

  B_ν^max(T) = 1.9e-19 T³ ≈ T³ (in eV/m²/Hz/sr)

Plots

• Left: B_λ (Gnuplot script); Right: B_ν
• Check the peak values; check the unit conversion for B_ν (right scale of right plot)

Black-body: generalized curve

• Fractional brightness is a universal function of λ T:

  \[b(x) = \frac{B_\lambda}{B_\lambda^m} = \frac{c_3^5}{x^5}\frac{e^{c_2/c_3}-1}{e^{c_2/x}-1},\quad x=\lambda T\]

• Plot: b(x) together with the cumulative fraction of it on the upper axis.

2.3 Stefan-Boltzman law and the effective temperature

• Integrated Planck function: \[ B(T) = \int_0^\infty B_\nu(T) d\nu = \int_0^\infty B_\lambda(T) d\lambda = \frac{\sigma T^4}{\pi}\]
• Show this using \(\int_0^\infty x^3 dx/(e^x-1) =\pi^4/15\)
• Stefan's constant: \[\sigma = 2\pi^5 k^4/15c^2h^3 = 5.6696\times 10^{-8}\rm W mm K^{-4}\]

Effective temperature

• Total amount of flux radiated at the source: \[F_{\nu, \rm surface} = \int_{2\pi} \cos\theta I_\nu d \Omega = \pi I_\nu\] [show it]
• Integrated flux at the source is thus: \[F=\int_0^\infty F_\nu d\nu=\int_0^\infty \pi I_\nu d\nu = \pi I\]
• Assuming black-body radiation: I_ν ≡ B_ν, so I≡ B(T), hence \[F = \sigma T_{\rm eff}^4\quad\rm W/m^2\]
• Integrated luminosity: intrinsic property of the source \[\boxed{L = 4\pi R^2 \sigma T_{\rm eff}^4\quad\rm W}\]

Do stars really radiate as a blackbody ?

• Absorption by atoms in the photosphere (H, H\(^-\))
• High abundance of metals in Pop I stars: absorption (lower right)
• Different elements trace differents heights hence diff. T_eff
• Can you recover the straight line in this plot? (Hint: use Wien's approximation to write colours as U-B=a+b/T_eff.)

3 Extinction

• Light propagating through a medium suffers scattering (redistribution in phase and/or frequency) and absorption (opacity):

  scattering + absorption = extinction

  • Note: albedo ω=C_sca/(C_sca + C_abs)=C_sca/C_ext
  • Absorption: I_λ = I_λ,0 exp(-τ_λ); τ_λ is the opacity at λ
• There is also emission by the medium itself
• Radiative transfer: solve {emission + extinction} processes
• In this Section:
  • Atmospheric absorption
  • Interstellar extinction by dust

Reflection nebulae

M78

M45 (Pleiades)

Interstellar clouds near bright stars; blue nebulosity is light scattered by dust; dark lanes are due to absorption;

3.1 Atmospheric absorption

• Different processes:
  • scattering: light pollution
  • refraction: λ-dependent light deviation
  • turbulence: degrades the spatial resolution
  • thermal emission: significant contribution at IR and mm

Atmospheric windows

• high-energy astrophysics (UV, X-ray, γ-ray) requires airborne facilities
• radio observatory benefit from transparent atmosphere; at the upper frequency end of the radio domain, water, O₂ lines define radio windows;

Atmosphere absorption

Atmospheric windows

• Atmospheric windows: outside the windows, atm. absorption is very strong; associated with absorption lines (O₂, O₃, CO₂) or continuum (UV)
  • electronic (atomic, molecular) transitions: O, N, CH₄, CO, H₂O, O₂, O₃
  • ro-vibrational and pure rotation lines: CO₂, NO, CO, H₂O, O₃
  • CO₂: mid infrared absorption
• Chemical composition of the atm. is ≈ constant up to ≈ 90 km
• Decrease opacity (airmass): altitude
  • Hydrostatic scale height H≈8 km
  • Airmass: 1/cos(z) ≈ 1/sin(El) (approximation valid down to El≈20^o)

Why are observatories in altitude?

• quantity of precipitable water: the column density of water molecules above the observatory: \[ w(H2O) = \int_{z_0}^\infty n(H2O) dz \] with

  n(H2O) [cm^-3] = 4.3x10²⁵ P/P₀ T/T₀ r(z)

• r(z) is the mixing ratio
  • mass of water per vol. / mass of air per vol;
  • rapidly varying function of altitude z
• typical values of w(H_2O): 0 to a few mm;

Effect of pwv on atmospheric transparency at mm and sub-mm

Left: NOEMA site (2 km); Right: ALMA site (5 km)

Effect of altitude on pwv

• Left: NOEMA site (2 km); Right: ALMA site (5 km) from APEX weather page
• ∼65% of pwv<1.5mm at Chajnantor, ∼25% at Plateau de Bure

3.2 Interstellar extinction

Space absorption correction

• Extinction of light on the way from the source to the atmosphere

• Positive correction to the apparent magnitude: A

  M = m + 5 -5log₁₀ d_pc - A = m - DM - A

• Extinction or reddening (in magnitudes):
  • F_λ,0: the un-reddened flux
  • A_λ = 2.5 log₁₀ (F_λ,0/F_λ)
  • wavelength dependent: extinction curves
• A_λ is proportional to the opacity τ_λ:
  • Recall that I_λ = I_λ,0 exp(-τ_λ)
  • A_λ = 2.5log₁₀ [exp(τ_λ)] = 2.5/ln(10) τ_λ = 1.086 τ_λ
• In band V, extinction is essentially due to interstellar dust
• Extinction is usually measured by comparing observations of a reddened star to an unreddened one having identical intrinsic SED;

3.2.1 Reddening, colour excess, extinction curves

• Reddening: E(B-V) = A_B - A_V
  • A_λ: reddening at one wavelength
  • Other usual quantity: A_λ-A_V = E(λ-V)
• By definition: M=m-DM-A, or m=M+DM+A
• Color index: B-V = M_B - M_V + A_B - A_V
  • B-V = (B-V)₀ + E(B-V)
  • (B-V)₀ = M_B-M_V: intrinsic colour
  • E(B-V) = (B-V) - (B-V)₀ = A_B-A_V: colour excess
• Extinction curves: relative colour excess vs wavelength
  • Usual quantities in ordinate: A_λ/E(B-V) or A_λ/A_V, etc…
  • Remember that magnitude differences are flux ratios hence with less sensitivity to multiplicative noise sources (provided that …)

• Important relation that you must know (Bohlin+78):

  N_H [cm^-2] = 5.8 x10²¹ E(B-V)

3.2.2 Average galactic extinction curves: dust is small

• Dust grains must be small:
  • large grains would be in the geometric optics limit, hence λ-independent extinction cross-section
  • variations of A_λ down to λ=0.1 mic implies that there must be small grains: 2πa≤ λ
  • Implies that a significant population of grains with a≤0.015mic must be present
• Note: 1/λ > 3 mic^-1<1.25=NIR (Rieke & Lebofsky 1985, IRTF); 1/λ > 3 mic^-1 corresponds to UV (see Bless and Savage 1972, Orbiting Astronomical Observatory 2 OAO-2);

3.2.3 Interstellar extinction curves: common features

• Extinction curve: A_λ/E(B-V) or A_λ vs 1/λ
• Shown here is
  • left: A_λ divided by A_Ic, the extinction in the Cousins I band (8020\(\mathring{A}\)), for different lines of sight
  • right: (A_λ-A_V)/(A_B-A_V)

Common features

• Extinction increases with decreasing λ
• More extinction at blue wavelength: reddening
• Bump at 4.5μm^-1 (217.5 nm)

Variability

• We also see large differences between los at 1/λ>2 mu^-1
• Lines of sight are characterized by R_V, the total-to-selective absorption
  • R_V = A_V/E(B-V) = A_V/(A_B-A_V): depends on dust composition and size, hence on the line-of-sight
  • R_V ≈ 3.1, on average in the diffuse ISM in the MW
  • show the important relation A_λ = E(λ - V) + R_V E(B-V)

3.2.4 Reddening laws: the Cardelli+89 paper

• Trace A(λ)/A_V curve instead of E(λ-V)/E(B-V)

• Average A(λ)/A_V curve, for 0.125≤λ≤3.5 mic, parameterized bya single parameter R_V=A_V/E(B-V), applicable for dense and diffuse ISM:

  A_λ/A_V = f(λ;R_V) = a(x) + b(x)/R_V, x=1/λ[mic^-1]

• See also Fitzpatrick+07

3.2.5 Source variability: grain growth

• Cardelli+89
• A(λ)/A_V instead of E(λ-V)/E(B-V)
• Total-to-selective extinction: R_V = A_V/E(B-V)≈ 3.1 on average in the Galaxy
• Flattening of the curve:
  • signature of grain growth
  • large grains: A_λ ind. of λ, R_V → ∞
  • dense clouds: R_V=4-5

3.2.6 Digressions

The 2175 angstrom feature

• conspicuous feature at λ^-1 = 4.6 micr^-1 or 2175 angstrom
• strong feature: must be an abundant material, hence made of the most abundant elements
• variable width

• sp²-bond of carbon in carbon sheets (graphite): PAHs best candidates, but still elusive

Diffuse Interstellar Bands

• Carriers are unknown
• Free-flying (large) molecules or ultra-small dust grains
• fine-structure spectral features (molecular rotation ?)
• As of today, still an open question