Final answer:
The question mixes two physics concepts: the Doppler shift and the relative orbital speeds in a binary star system. For Doppler shift, Star B moving directly away has the greatest shift, and Star A with perpendicular motion has the smallest. Orbital speed, following Kepler's laws, is greatest at points closest to the massive center in circular orbits.
Step-by-step explanation:
The ranking of orbital speeds for Star A, Star B, and the Sun from fastest to slowest depends on their relative positions and mass distribution in relation to their center of mass. According to the information provided, which outlines a scenario of Doppler shift and orbital mechanics, we are to assume that all stars are moving with the same velocity but in different directions. However, the question seems conflated with two distinct concepts: orbital speed around the center of mass in a binary system, and the Doppler shift due to motion relative to the observer on Earth. Therefore, it's not entirely clear how to rank the three without additional context about their masses and orbits.
Nonetheless, the orbital speed of an object is usually greatest when it is closest to the massive body around which it orbits due to conservation of angular momentum, as indicated by Kepler's laws of planetary motion. Hence, for objects in circular orbits, the one closer to the massive center will orbit faster. For Doppler shift, it is greatest when the star is moving directly away or towards the observer since the motion is entirely in the line of sight, which in this case would suggest Star B would show the greatest Doppler shift and Star A the smallest.
The concept of a binary system and its orientation to the observer is essential for determining the relative orbital speeds based on the mass distribution between two celestial objects. The more massive star has a smaller orbit and moves more slowly, while the less massive star moves faster over a larger orbit. As such, the orbital speed of the Sun in the Milky Way would depend on its distance from the galactic center, and in circular orbits, it moves fastest at the closest point to the massive center.
Regarding the rotation of stars, a rapidly rotating star like Altair would have a different rotational speed than the Sun which is slower in comparison.