Aberration of Starlight and Stellar Parallax

The Earth moves around the Sun in a planar orbit whose plane includes the Sun. (See Section 1.9.2.)
The orbit is approximately circular in shape, and has a radius
.
(See Sections 1.9.6, and Table 1.4.)
The plane that contains the Earth's orbit is known as the *ecliptic plane*. (See
Section 1.10.6.) Let us set up a Cartesian coordinate system in the ecliptic plane whose origin coincides with the
Sun, whose -axis is directed toward the northern ecliptic pole (i.e., the direction that is normal to the ecliptic plane in a northern sense), and whose -axis is directed toward
the *vernal equinox* (i.e., the point in the sky at which the Sun annually passes through the projection of the Earth's equatorial
plane in a northward sense). See Figure 3.3. The angle, , shown in the figure, is known as the Earth's *ecliptic
longitude*, and serves to locate the Earth on its orbit. Let be the displacement of the Earth from the Sun.
It is clear from simple geometry that the components of are

(3.13) |

(3.14) |

(3.15) |

Suppose that a light ray from a distant star is observed on the Earth. Let the phase velocity of the ray in the aether rest frame, in which the Sun is assumed to be stationary, be , where is the speed of light in vacuum. Because the Earth is actually moving with respect to the aether rest frame, the observed phase velocity of the light ray is

[See Equation (3.12).] Let be the angle subtended between and , and let be the angle subtended between and . See Figure 3.4. Thus, corresponds to the true angular location of the star (i.e., the location seen by an observer in the Sun's rest frame), whereas corresponds to the apparent location of the star seen on the moving Earth. Simple trigonometry reveals that(3.22) |

Let us write

(3.25) | ||

(3.26) |

(3.30) |

(3.35) | ||

(3.36) |

(3.37) | ||

(3.38) |

Let us now consider parallax. Let be the displacement of the Sun from a distant star, let be the corresponding displacement of the Earth, and let be the displacement of the Earth from the Sun. It is evident that

(3.39) |

(3.40) |

(3.41) |

(3.44) |

We can write and . Making use of very similar analysis to that used to calculate aberration, we obtain

Equations (3.47) and (3.48) can be combined to give Hence, Equations (3.49) and (3.50) yield(3.51) | ||

(3.52) |

(3.53) | ||

(3.54) |

Between 1725 and 1727, the astronomers James Bradley and Samuel Molyneux measured the position of the circumpolar star Draconis in an attempt to observe its parallax. They found that the angular position of the star underwent small annual variations, but that the deviation from the mean was greatest when the ecliptic longitude of the Sun matched that of the star, which is not the behavior expected from parallax. In 1728, Bradley correctly explained the observed variations in terms of the aberration of light. Note that this explanation depends crucially on the fact that the speed of light observed in a frame of reference that moves with respect to the rest frame of the aether is different to that observed in the rest frame. Incidentally, stellar parallax is so small an effect that it was not successfully measured until 1838, when Frederich Bessel measured the parallax of the star 61 Cygni.