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Both m k and m are close to unity as Ng is very small Q Q Each grain may therefore be considered as carrying: Extinction characterized by: A differential extinction of: A differential phase delay of: D 1 (Cke C C e ) , 2 1 D (Cke C e ) , 2 1 D (Ckp C p ) 2
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The passage of light of wavelength, , through a layer of elemental thickness, ds, containing Ng scattering grains per unit volume (see Figure 105) may be described in the form of a Mueller matrix with the form 3 2 1 Ng ds Ng ds 0 0 7 6 Ng ds 1 Ng ds 0 0 7 6 4 0 0 1 Ng ds Ng ds 5 0 0 Ng ds 1 Ng ds 3 3 2 2 1 0 0 0 0 0 60 1 0 07 6 0 07 7 7 6 (1015) D6 5 40 0 1 05 C Ng ds 4 0 0 0 0 0 1 0 0 The elements of the resultant Stokes vector fI 0 , Q0 , U 0 , V 0 g of a light beam which enters with a vector of fI, Q, U, V g may be written as I0 D I Q DQ U DU V DV
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Ng ds ( I C Q) Ng ds ( I C Q) Ng ds ( U V) (1016) Ng ds ( U C V )
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From these equations, the rate of change of the normalized Stokes parameters with the progression of a light beam through the cloud are determined as 1 dI I ds d[Q/I ] ds d[U/I ] ds d[V/I ] ds D D Ng ( C [Q/I ]) Ng 1 [Q/I ]2
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D CNg ( [V/I ] C [Q/I ][V/I ]) D Ng ( [U/I ] [Q/I ][V/I ]) (1017)
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From the above equation, it can be seen that if the light is already polarized on entering the cloud, with a direction of vibration parallel to the the reference frame de ning the grain alignment, the growth of the polarization is given by dp D ds Ng 1 p2 (1018)
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Fig 105 Radiation of wavelength, , entering a dust cloud of length, ds, with a dust grain number density, Ng
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It is also obvious that circular polarization can only be generated if the light entering the cloud is already linearly polarized Very generally, the levels of polarization under discussion are small, and the right-hand terms of (1017) can be neglected so that 1 dI I ds d[Q/I ] ds d[U/I ] ds d[V/I ] ds D Ng (Cke C C e ) 2 Ng D Ng D (Cke C e ) 2 Ng D CNg [V/I ] D C (Ckp C p )[V/I ] 2 Ng D Ng [U/I ] D (Ckp C p )[U/I ] 2 Ng D
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If one commences with zero polarization, both the extinction and Q/I grow as the light passes through a regular cloud; U/I and V/I can only grow if the beam already contains V/I and U/I components These might be present, however, because of intrinsic polarization generated by the star itself or, if there are two clouds along the line of sight, but with different angles of alignment of the dust It is unlikely that the effects of the grains will be uniform along any line of sight as a result of either twists in the preferred alignment through the cloud, or of the path involving a series of clouds with distinct differences in the direction of the dust alignment Such situations have been modelled by Martin (1974) In particular his two-cloud model has proved to be instructive as to how the second cloud effects a depolarization relative to the potential polarization that the combination of the two clouds could provide, and on the wavelength dispersion of the observed direction of vibration (see Section 105) According to the measurements, the optical properties of various types of dust grain can be considered to explore how the observed extinction and polarization growth is matched All that is required is the knowledge of the amplitude forward amplitude scattering functions of the dust grains It may be noted that within (1019), there are four equations with ve unknowns The situation is better resolved by including wavelength dependence in the observations that also requires matching by the optical properties of the grains The situation is generally more
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