Rick Wiseman said:
The conjunction of the moon with Venus earlier this month brought a question to mind. Since the moon and Venus were the same angular distance from the sun, but showed different phases, could this observation, or perhaps observations over a number of months, be used to determine the distance to Venus? This assumes a known distance to the moon, which was found using parallax. Perhaps one of the old time astronomers used this measurement technique. This makes me wonder about all the fuss over transits of Venus to determine its distance. Thanks for all comments.
Rick W
That’s an astute observation, Rick. However, for determining the distance to Venus, the illumination phase of the Moon is not very helpful. The same can be said regarding the illumination of Venus, since it is certain to be very close to dichotomy (50% illumination) around the times of its greatest elongations from the Sun. That’s when the angle Earth-Venus-Sun is 90°, and the distance to Venus in astronomical units can be easily calculated. Although Galileo found the illumination of Venus to be solid evidence that Venus orbits the Sun.
My two diagrams seen below may be useful for what follows. For simplicity let’s assume non-eccentric co-planar orbits. At the time of greatest elongation, the angle Venus-Earth-Sun can be easily measured from Earth based observations as about 46°. Since that is one angle of a right triangle, the Earth-Venus distance in astronomical units would be the Cosine of 46° degrees or 0.69 AU. The Venus-Sun distance would be the Sine of 46° or 0.72 AU. We can then deduce that at inferior conjunction the Earth-Venus distance is (1 – 0.72 = 0.28) AU and (1 + 0.72 = 1.72) AU at superior conjunction.
The problem is obtaining the length of an astronomical unit (Earth-Sun distance) in familiar measures such as kilometers or miles. More than two millennia ago Aristotle and his contemporaries determined the size of the Earth and distance and size of the Moon with decent accuracy through trigonometric means. They tried to determine the distance to the Sun by taking note of the fraction of the lunar disk that was illuminated at the time of quadrature, i.e. the angle Moon-Earth-Sun equal to 90° at first or last quarter position. The actual moment of Half Moon (50% illumination) occurs when the angle Earth-Moon-Sun is 90°. Half Moon and quadrature are separated in time by about 17 minutes. That is geocentrically. On the moving surface of the Earth the determination is more complicated. Also, the Moon’s terminator is not sharp since the Sun is much larger than a point. In any event, the ancient measurements were so imprecise that they came up with a figure for the distance to the Sun that was far short of the truth.
During the 18th and 19th centuries, transits of Venus were helpful in determining the length of an astronomical unit by noting the parallax effect. Timings of the events from different locations on Earth allowed for trigonometric calculations of good approximations for the distance to the Sun. More recently, radar settled the question.
