Effect of solar radiation on interplanetary dust grains

Let , , be cylindrical polar coordinates in a frame of reference, centered on the Sun, that is aligned with the orbital plane of the dust grain, as described in Section I.1. Let and be the (relative) position and velocity of the grain, respectively. Consider a photon emitted by the Sun. Let and be the photon's energy and momentum, respectively, in the heliocentric frame. Let and be the corresponding quantities in the dust grain's instantaneous rest frame. According to standard relativistic theory (Rindler 1977),

(10.160) | ||

(10.161) |

(10.162) | ||

(10.163) |

(10.164) | ||

(10.165) |

Let and be the electromagnetic energy and momentum, respectively, absorbed by the grain per unit time in its instantaneous rest frame. Let and be the corresponding quantities emitted by the grain. Now, we are assuming that

(10.166) |

(10.167) |

(10.170) |

(10.171) |

(10.172) | ||

(10.173) |

(10.174) |

In the heliocentric frame, the net force per unit mass acting on the grain is

(10.175) |

(10.177) |

(10.178) |

Assuming that the total radiation pressure force, (10.176), is small compared to the force of gravitational attraction between the Sun and the dust grain—and can, thus, be treated as a perturbation—the grain's orbit can be modeled as Keplerian ellipse whose six elements evolve slowly in time under the influence of the pressure. The six elements in question are chosen to be the major radius, , the mean anomaly at epoch, , the eccentricity, , the argument of the perigee, , the inclination (to the ecliptic plane), , and the longitude of the ascending node (measured with respect to the vernal equinox), . (See Section 4.12.) The evolution of these elements is governed by the Gauss planetary equations, (I.53)–(I.58). Now, in the perturbative limit, the evolution of the dust grain's orbital elements takes place on a timescale that is much longer than its orbital period. We can concentrate on this evolution, and filter out any relatively short-term oscillations in the elements, by averaging the Gauss planetary equations over an orbital period. A suitable orbit-average operator is

(10.182) |

In order to be in the perturbative limit, the relative changes in the dust grain's orbital elements induced by the radiation pressure force in an orbital period must all be small. Because [given that , which follows because the grain is moving non-relativistically], this requirement yields , or

(10.189) |

(10.190) |

(10.191) |

For the case of a large dust grain in a circular orbit around the Sun, Equation (10.183) gives

(10.192) |

(10.193) |

(10.194) |