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As discussed in Section 2, when light impinges on a grain it is either scattered or absorbed. Let Io(lambda) be the intensity of the incident light at wavelength lambda, the intensity of light scattered into a direction defined by theta and phi (see Fig. 2) is

Equation 1 (1)

where 0° leq theta leq 180° is the scattering angle (the angle from the incident direction), 0° leq phi leq 360° is the azimuthal angle which uniquely determines the scattering plane defined by the incident direction and the scattering direction (see Fig. 2), 2 F(theta, phi) is the (dimensionless) angular scattering function, r >> lambda / 2pi is the distance from the scatterer, and k = 2pi / lambda is the wave number in vacuum. The scattering cross section Csca, defined as the area on which the incident wave falls with the same amount of energy as that scattered in all directions by the dust, may be obtained by integrating the angular scattering distribution F(theta, phi) / k2 over all solid angles

Equation 2 (2)

where F(theta, phi) / k2 (with a dimension of area), 3 after normalized by Csca, is known as the phase function or scattering diagram

Equation 3 (3)

The asymmetry parameter (or asymmetry factor) g is defined as the average cosine of the scattering angle theta

Equation 4 (4)

The asymmetry parameter g, specifying the degree of scattering in the forward direction (theta = 0°), varies from -1 (i.e. all radiation is backward scattered like a "mirror") to 1 (for pure forward scattering). If a grain scatters more light toward the forward direction, g > 0; g < 0 if the scattering is directed more toward the back direction; g = 0 if it scatters light isotropically (e.g. small grains in the Rayleigh regime) or if the scattering is symmetric with respect to theta = 90° (i.e. the scattered radiation is azimuthal independent and symmetric with respect to the plane perpendicular to the incident radiation).

Figure 2

Figure 2. Schematic scattering geometry of a dust grain in an incident radiation field of intensity Io which scatters radiation of intensity I(theta, phi) into a scattering angle theta (theta = 0°: forward scattering; theta = 180°: backward scattering), an azimuthal angle phi, and a distance r from the dust. In a Cartesian coordinate system, the incident direction defines the +z axis. The scattering direction and the incident direction define the scattering plane. In the far-field region (i.e. kr >> 1), I = Io F(theta, phi) / k2r2, where k = 2pi / lambda is the wave number in vacuum.

In radiative transfer modeling of dusty regions, astronomers often use the empirical Henyey-Greenstein phase function to represent the anisotropic scattering properties of dust [26]

Equation 5 (5)

Draine (2003) proposed a more general analytic form for the phase function

Equation 6 (6)

where eta is an adjustable parameter. For eta = 0 Eq. 6 reduces to the Henyey-Greenstein phase function. For g = 0 and eta = 1 Eq. 6 reduces to the phase function for Rayleigh scattering [12].

As discussed in Section 2, both scattering and absorption (the sum of which is called extinction) remove energy from the incident beam. The extinction cross section, defined as

Equation 7 (7)

is determined from the optical theorem which relates Cext to the real part of the complex scattering amplitude S(theta, phi) 4 in the forward direction alone [28]

Equation 8 (8)

The absorption cross section Cabs is the area on which the incident wave falls with the same amount of energy as that absorbed inside the dust; Cext, having a dimension of area, is the "effective" blocking area to the incident radiation (for grains much larger than the wavelength of the incident radiation, Cext is about twice the geometrical blocking area). For a grain (of size a and complex index of refraction m) in the Rayleigh limit (i.e. 2pi a / lambda << 1, 2pi a |m| / lambda << 1), the absorption cross section Cabs is much larger than the scattering cross section Csca and therefore Cext approx Cabs. Non-absorbing dust has Cext = Csca.

The albedo of a grain is defined as alpha ident Csca / Cext. For grains in the Rayleigh limit, alpha approx 0 since Csca << Cabs. For Non-absorbing dust, alpha = 1.

In addition to energy, light carries momentum of which the direction is that of propagation and the amount is hnu / c (where h is the Planck constant, c is the speed of light, and nu is the frequency of the light). Therefore, upon illuminated by an incident beam of light, dust will acquire momentum and a force called radiation pressure will be exerted on it in the direction of propagation of the incident light. The radiation pressure force is proportional to the net loss of the forward component of the momentum of the incident beam. While the momentum of the absorbed light (which is in the forward direction) will all be transfered to the dust, the forward component of the momentum of the scattered light will not be removed from the incident beam. Therefore, the radiation pressure force exerted on the dust is

Equation 9 (9)

where Cpr is the radiation pressure cross section, and Io is the intensity (irradiance) of the incident light.

In literature, one often encounters Qext, Qsca, Qabs, and Qpr - the extinction, scattering, absorption and radiation pressure efficiencies. They are defined as the extinction, scattering, absorption, and radiation pressure cross sections divided by the geometrical cross section of the dust Cgeo,

Equation 10 (10)

For spherical grains of radii a, Cgeo = pi a2. For non-spherical grains, there is no uniformity in choosing Cgeo. A reasonable choice is the geometrical cross section of an "equal volume sphere" Cgeo ident pi(3V / 4pi)2/3 approx 1.21 V2/3 where V is the volume of the non-spherical dust.

2 When the scattering is along the incident direction (theta = 0°, i.e. "forward scattering") or the scattering is on the opposite direction of the incident direction (theta = 180°, i.e. "backward scattering"), any plane containing the z axis is a suitable scattering plane. Back.

3 Also called the "differential scattering cross section" dCsca / dOmega ident F(theta, phi) / k2, it specifies the angular distribution of the scattered light [i.e. the amount of light (for unit incident irradiance) scattered into a unit solid angle about a given direction]. Back.

4 The angular scattering function F(theta, phi) is just the absolute square of the complex scattering amplitude S(theta, phi): F(theta, phi) = |S(theta, phi)|2. Back.

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