Astronomy 6570

Physics of the Planets

Light Scattering and Photometry

Summary: Photometry vs Spectroscopy• Photometry – low resolution, absolute flux

measurements (visible and near-IR)– Albedo at zero phase angle

• Constraint on “surface” composition and texture

– Brightness variation with• Phase angle : surface texture/particle sizes• Position on disk (limb darkening)

– Atmospheric cloud properties

• Polarization direction: cloud particle properties

– Geometric albedo + phase function à Bond albedo à net solar energy input

Summary (cont’d.)

• Spectroscopy – flux vs frequency or wavelength (UV, VIS, near-IR, mid-IR, radio region) – Reflected sunlight

• Identify atmos. constituents by selective absorption or emission lines & bands

• Surface composition (“active” species only)• Doppler shifts: atmos. rotation rates

– Thermal emission• Retrieve temperature profiles (atmos. and subsurface)• Identification of atmos. constituents• Infer wind velocities (geostrophic flows)• Measure overall energy balance of planet• Measure surface thermal inertia.

IntroductionRadiation emitted and reflected by a planet

wavenumber = 1/l = f/cso10,000 cm-1

è l = 10-4 cm = 1 µm.

Reflection and Emission

çπF0

I(θ)

) θ

Whole Sphere Observations

Some examples:

Sphere with uniform surface brightness independent of phase angle:

(Metallic sphere with perfect reflection:

Lambert sphere:

Some Phase Functions:

Geometric albedo spectra of outer planets in the visible & near-IR.

Note strong absorption bands due to CH4 on Titan & Saturn, and pressure-induced absorption at 2.1 µm due to H2 on Saturn

Scattering of light by atmospheric aerosols.

A fraction of the sunlight that is incident on a planet is scattered by atmospheric particles:

The power intercepted by a particle is:

The scattered power in all directions is:

Thus:

Pin = πr2 QextπF0 (90)

Pout = πr2 I Θ( )dΩ (91)

4π∫

I Θ( )dΩ = I4π∫ I Θ( )dΩ = 4π I ≡ω0QextπF04π∫ (92)

r the radius of the particleQext the extinction efficiency

Θ the scattering angleω0 the single scattering albedo

φ the phase function

Typo: I(q) à f(q)

It follows that:

Note that the phase function for small particles is normalized so that:

depend on:• The wavelength λ• The particle size, shape, composition, and refractive index

Note that Qext = Qsc + Qabs … extinction = scattering + absorptionwhere Qsc = w0 Qext and w0 is the fraction of the light actually intercepted by the particle which is scattered. It is equivalent to the Bond albedo for planets.

I Θ( )F0

=ω0Qext

4φ Θ( ) (93)

φ θ( )4π4π∫ dΩ = 1 (94)

ω0 ,Qext , and φ

Note that this differs from the convention for whole planets, where f(a) is normalized at a=0o. It is more commonly designated P(q).

Size parameter χ = 2π λ :

χ 1: Rayleigh scatteringχ~1: Mie-scattering for spherical particlesχ 1: geometrical optics (ray tracing)

For χ 1 (the Rayleigh scattering limit).

ϖ 0Qext ∝ χ 4 = 2π λ( )4

and

Qabs ∝ χ i Im n( ) .

For χ 1 (macroscopic particles, Qext → 2

(50% of the light removed is confined to a forward-scattering diffraction cone of angular width λ 2r).If we neglect the diffracted light Qext 1 and we have

I 0( ) F ≡ p = 14ϖ 0φ 0( ) .

Typo: x = 2pa/l

Recall that p = A/4 for an isotropically scattering planet, where f(q) = 1.

(Relevant to scattering by molecules and very small aerosols)

The amplitude of Rayleigh scattering depends primarily on x = 2pa / l, while the absorption depends on x and the absorption coefficient.

The scattered light is linearly polarized, especially at scattering angles near 90o.

Hansen & Travis (1974)

Note that Qsc = w0 Qext

Recall w0 is the fraction of the light actually intercepted by the particle which is scattered.

The gap between the primary and secondary rainbows.

Diffraction spike.

Mie scattering:

Light Scattering from a Particle Layer

Define the OPTICAL DEPTH of the layer by

τ λ = n z( )0

d∫ πr2Qe r,λ( )dz

where n(z) = # of particles/vol. For a particlesize distribution n(r, z) we have

τ λ = πr2Qe r,λ( )n r, z( )dr dzrmin

rmax∫0

d∫

In terms of the individual particle phase function P α( ) and single-scattering albedo ϖ 0, the reflected intensity due to the SINGLE SCATTERING only is given by:

I i,e,α( )

F= ϖ 0µ0

4 µ + µ0( )P α( ) 1− exp−τ µ + µ0( )

µµ0

⎡

⎣⎢

⎤

⎦⎥

⎛

⎝⎜⎞

⎠⎟

where µ0 ≡ cos i and µ = cose.For contributions from MULTIPLY-SCATTERED light see Chandrasekhar, S (1960) "Radiative Transfer".Limits: τ → 0 : I F→ 1

4ϖ 0τµP α( ) (e.g., Jovian rings)

τ → ∞ : I F→ 14ϖ 0µ0

µ + µ0

P α( ) (e.g., thick cloud)

πF

i e I(θ,ϕ)

0

dz

particle, r

Light Scattering from Airless BodiesLight scattering from solid surfaces is complicated further by surface microstructure ( èshadowing of soil particles) and by macroscopic (topographic) shadowing. Several quasi-empirical relations have been developed, such as:

Minnaert's "Law": I F = B0 α( )µ0kµ k−1

- developed for the Moon, where k 0.5@α 0 (1) - at zero phase, µ = µ0 ⇒ I F ~ µ2k−1

⇒ the Moon shows no limb darkening!Lambert "Law": I F = B0 α( )µ0

- I is independent of µ - corresponds to Minnaert k = 1

Buratti's relation: I F = Aµ0

µ + µ0

f α( ) + 1− A( )µ0

- linear combination of SS + Lambert; usually A ≥ 0.6 for icy sats.

Hapke's theoretical formulation:

I F = 14

ϖ 0 ′µ0

′µ + ′µ0

1+ B α( )⎡⎣ ⎤⎦P α( ) + H µ,ϖ( )H µ,ϖ( )−1{ }S i,e,d( )

B α ,h( ) = B0 1+ tan α 2( ) h⎡⎣ ⎤⎦and P α( ) modelled by Henyey-Greenstein expression

P α ,g( ) = 1− g2

1+ g2 + 2gcosα( )32

where g = cosα and −1≤ g ≤1.

(1) for real bodies k α( )

!µ , !µ0 = "effective" µs oppos'n effect Multiple scattering terms a la Chandrasekhar

Shadowing func’n

α = phase angleμ0 = cos iμ = cos e

The “Opposition Effect” is a pronounced brightening near a = 0o.

It is .seen commonly for bright, airless surfaces (including Saturn’s rings) & is due either to the absence of shadows at zero phase or to a coherent backscattering phenomenon in transparent surface materials (e.g., water ice).

VIMS 0 PHASE observation: A ring mosaic