Sunday, 7 July 2013

Astrometry- The Cosmic Distance Ladder

Van Gogh's painting, 'The Starry Night', resembles not only the Whirlpool galaxy but it proves its astronomical worth in a number of ways. For starters, one can deduce it was painted in the predawn hours due to the inclination of the moon to the horizon and that the brightest of its 'stars' is in fact Venus, attesting to the fact that the planets are usually the first to emerge in the evening. But what makes this painting sacred is not what it shows but what it represents: astrometry. Our obsession with the heavens means that we can measure distances with greater rigour and precision, from gnomons and sundials to standard candles and gravitational lensing; such are the rungs of the cosmic distance ladder. The earth is the first rung of that ladder. Aristotle and others provided the first indirect arguments that the earth is round using the moon. He knew that lunar eclipses happned when the moon was directly opposite the sun (opposite constellation of the Zodiac), so eclipses happen because the moon falls into the earth's shadow. But in a lunar eclipse, the shadow of the earth on the moon is always a circular arc and since the only shape that produces such a shadow is the sphere he inferred the earth was round. If the earth was circular yet flat like a disk, the shadows would be elliptical. Similarly, Eratosthenes calculated the radius of the earth to 40000 stadia. Having read of a well in Syene that reflected the overhead sun at noon of the summer solstice (June 21) because of its location on the tropic of cancer; he used a gnomon (in Alexandria) to measure the deviation of the sun from the vertical as 7 degrees. Knowing the distance from Alexandria to Syene to some 5000 stadia, it was enough to compute the earth's radius. Aristotle also argued the moon was a sphere (rather than a flat disk) because the terminator (boundary of the sun's light on the moon) was always an elliptical arc. The only shape with such a property is the sphere, should the moon be a flat disk, no terminator wouldn't appear. Aristarchus determined the distance from the earth to the moon as 60 earth radii (57-63 earth radii in actuality). He also computed radius of the moon as 1/3 the radius of the earth. Aristarchus had knowledge of lunar eclipses being caused by the moon passing through the earth's shadow and since the earth's shadow is 2 earth radii wide (diameter) and the maximum lunar eclipse lasted for 3 hours, it meant that it took 3 hours for the moon to cross 2 earth radii. And it also takes around 28 days for the moon to go around earth, sufficient to compute the moon's radius. In addition, the radius of the moon in terms of distance to the moon was determined by the time it takes to set (2 minutes) and the time it take to make a full (apparent rotation) is roughly 24 hours. Next, the Sun's radius was measured by Aristarchus by  relying on the moon. Having computed the radius of the moon as 1/180 the distance to the moon, he knew that during a solar eclipse that the moon covered the sun almost perfectly, using similar triangles, he inferred that radius of the sun was also 1/180 the distance to the sun. But to determine the distance to the sun, he knew that half moons happened when the moon makes a right angle between the earth and sun, full moons occurred when the moon was directly opposite the sun and new moons occurred when the moon was between the earth and sun. This meant that half moons occur slightly closer to new moons than to full moons. Simple trigonometry could then be used to compute the distance to the sun at 20 times further than the moon, but a time discrepancy of 1/2 hour meant that the actual distance is 390 times the distance of the earth to moon. This also lead to the conclusion that the Sun was enormously larger than the earth and the first heliocentric proposal, later adapted by Copernicus.

Continuing our trek up the cosmic distance ladder, the rung of the planets and speed of light is quite a story. The ancient astrologers realised that all the planets lie on the ecliptic (a plane) due to the fact that they only moved via the Zodiac (the set of 12 constellations around the Earth. Ptolemy produced inaccurate results due to his geocentric model while Copernicus made highly accurate conclusions, initially poring over the annals of the ancient Babylonians who knew that the synodic period of mars repeated itself every 780 days. The heliocentric model allowed Copernicus to calculate the actual angular velocity as 1/170 and knowing that the earth took 1 year to go around the sun he would subtract implied angular velocities to find that the sideral period of mars was 687 days.Copernicus determined the distance of mars from the sun to 1.5 AU (astronomical units) by assuming circular orbits and using measurements of mars' location in the Zodiac across various dates. Brahe made similar predictions but they deviated from the Copernican regime, Kepler maintained that this was so because the orbits were elliptical and not perfect circles as Copernicus has assumed. Kepler would attempt to compute the orbits of the earth and mars simultaneously and since Brahe's data only gave the direction of mars from the earth and not the distance, he would need to figure out the orbit of the earth using mars. Working under the assumption that mars was fixed and the earth was moving in an orbit, Kepler used triangulation to use Brahe's 687 day interval to compute the earth's orbit relative to any position of mars. Such allowed the more precise calculation of the AU by parallax (measuring the same object from two different locations on earth), especially during the transit of Venus across the sun in multiple places (including Cook's voyage). But the anomaly of the precession of Mercury (where the points of aphelion and perihelion progressively wind around one another in a circular manner) could not be reconciled with Newtonian mechanics, so general relativity was invoked.The first attempts at accurately measuring the speed of light (c) was by Rømer who measured by observing Io, one of Jupiter's moons that made a complete orbit every 42.5 hours. Rømer noticed that when Jupiter was aligned with the Earth, the orbit advanced slightly but when it was opposed, it slowed and lagged by around 20 minutes. Huygens inferred that this was because of the extra distance (2 AU) that light had to travel from Jupiter, so light travels 2 AU in 20 minutes; hence the speed may be computed to 300,000 km/s.

No comments:

Post a Comment