As discussed in Section 2, when light impinges
on a grain it is either scattered or absorbed. Let
*I*_{o}()
be the intensity of the incident
light at wavelength ,
the intensity of light scattered into a direction
defined by and
(see Fig. 2) is

(1) |

where 0°
180° is the
*scattering angle* (the angle from the incident direction),
0°
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*(,
) is the
(dimensionless) angular scattering function,
*r* >> /
2 is the distance from the
scatterer, and *k* = 2 /
is the wave number in
vacuum. The *scattering cross section* *C*_{sca},
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*(,
) /
*k*^{2} over all solid angles

(2) |

where *F*(,
)
/ *k*^{2} (with a dimension of area),
^{3}
after normalized by *C*_{sca}, is known as the *phase
function* or *scattering diagram*

(3) |

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

(4) |

The *asymmetry parameter* *g*,
specifying the degree of scattering in the forward direction
( = 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
= 90°
(i.e. the scattered radiation is azimuthal
independent and symmetric with respect to
the plane perpendicular to the incident radiation).

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]

(5) |

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

(6) |

where is an
adjustable parameter. For
= 0 Eq. 6
reduces to the Henyey-Greenstein phase function. For *g* = 0 and
= 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

(7) |

is determined from the *optical theorem*
which relates *C*_{ext} to the real part of
the complex scattering amplitude
*S*(,
)
^{4}
in the forward direction alone
[28]

(8) |

The *absorption cross section* *C*_{abs} is
the area on which the incident wave falls with the same
amount of energy as that absorbed inside the dust;
*C*_{ext}, 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, *C*_{ext} 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. 2 *a*
/ << 1,
2 *a* |*m*|
/ << 1),
the absorption cross section *C*_{abs} is much larger than
the scattering cross section *C*_{sca}
and therefore *C*_{ext}
*C*_{abs}.
Non-absorbing dust has *C*_{ext} = *C*_{sca}.

The *albedo* of a grain is defined as
*C*_{sca}
/ *C*_{ext}. For grains in the Rayleigh limit,
0 since
*C*_{sca} << *C*_{abs}.
For Non-absorbing dust,
= 1.

In addition to energy, light carries momentum
of which the direction is that of propagation
and the amount is *h*
/ *c* (where *h* is the Planck
constant, *c* is the speed of light,
and 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

(9) |

where *C*_{pr} is the radiation pressure cross section,
and *I*_{o} is the intensity (irradiance) of the incident
light.

In literature, one often encounters
*Q*_{ext}, *Q*_{sca}, *Q*_{abs},
and *Q*_{pr} - 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 *C*_{geo},

(10) |

For spherical grains of radii *a*, *C*_{geo} =
*a*^{2}.
For non-spherical grains, there is no uniformity
in choosing *C*_{geo}.
A reasonable choice is the geometrical cross section
of an "equal volume sphere"
*C*_{geo}
(3*V* /
4)^{2/3}
1.21
*V*^{2/3} where *V* is the volume
of the non-spherical dust.

^{2} When the scattering is along
the incident direction (
= 0°, i.e. "*forward scattering*")
or the scattering is on the opposite direction of
the incident direction (
= 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*"
*d**C*_{sca} /
*d*
*F*(,
)
/ *k*^{2},
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*(,
)
is just the absolute square of the complex scattering amplitude
*S*(,
):
*F*(,
) =
|*S*(,
)|^{2}.
Back.