User:Upthetrail/sandbox

In circumstellar environments it describes the difference in photons arriving at orbiting dust particles. Photons that originate from the limb that is rotating away from the particle are red-shifted, while photons emitted from the limb rotating toward the particle are blue-shifted. .

Robertson considered dust motion in a beam of radiation emanating from a point source. Guess also considered the problem but for a spherical source of radiation and found that for particles far from the source, the resultant forces are in agreement with those concluded by Robertson

The dimensionless dust parameter $$ \beta $$ is the ratio of the force due to radiation pressure to the force of gravity on the particle:



\beta = { F_r \over F_g } = { 3L  Q_{pr}  \over { 16 \pi GMc \rho s }  } $$

where $$Q_{pr} $$ is the Mie scattering coefficient, and $$ \rho $$ is the density and $$s$$ is the size of the dust grain.

The Equations of Motion for the dust grain are expressed by

m{ \operatorname{d^2}\vec{x}\over \operatorname{d}t^2 } = -GMm \Big( 1-\beta \Big) {\vec{x}\over r^3} +GMm \beta \Bigg \{ { -{ {\vec{x}\cdot \vec{v}} \over {cr}  } { \vec{x}\over r^3  } -{ \vec{v} \over {cr^2}  } + { R_s^2  \over {cr^4}  } \Big( \vec{\omega} \times \vec{x} \Big) } \Bigg \} $$ where $$ R_s$$ is the stellar radius.