Length of a Comet's Plasma Tail
A comet's plasma tail points (very nearly) directly away from the Sun. Knowing the geometry of the comet's position, relative to the Sun and the Earth, as well as the angular extent of the tail as seen in the sky, it is possible to calculate the true physical length of the tail by a trigonometric consideration.
The following explanation was prepared by Knud Jepsen, physics teacher at Haslev Gymnasium (Denmark) and member of the Executive Council of the European Association for Astronomy Education (EAAE).
We have the triangle ESC (Earth-Sun-Comet) , for which the ephemeris (April 1, 1997; 0 UT) for Comet Hale-Bopp indicates EC = 1.351 AU; ES = 1.000 AU; SC = 0.914 AU, and also that the angle CES = 42.6 deg (Comet-Earth-Sun); the angle ECS = 47.7 deg (Earth-Comet-Sun) and thus the angle ESC = 89.7 deg (Earth-Sun-Comet).
The plasma-tail is positioned at the prolongation of SC (Sun-Comet) and extends to a point T (the tip of the tail).
The observation from the Earth of the angular extent of the plasma tail indicates that the angle TEC = 20 deg (Tip-Earth-Comet).
From triangle TEC (Tip-Earth-Comet) , we therefore find that the angle TEC = 20.0 deg (Tip-Earth-Comet) and that the angle ETC = angle ECS - angle TEC = 47.7 - 20.0 = 27.7 deg (Earth-Tip-Comet).
From the appropriate trigonometric formula, we then find the true length of the tail TC as:
TC = sin(20)*1.351/sin(27.7) = 0.994 AU = 148.2 million km.