500 Words On Eratosthenes and Quasars

Two-thousand years ago, Eratosthenes put a rod in the ground and conducted one of the first known investigations in geodesy — the study of the Earth’s shape and orientation. 


There were reports that at noon on the summer solstice in the city of Syene, the columns cast no shadows and the Sun hung directly overhead. Yet, at the same time in Alexandria, Eratosthenes saw that columns did cast shadows and pondered. With a spark of genius, he realised he could exploit this phenomenon to try and calculate a value for the circumference of the Earth.


Eratosthenes took a rod of known length and placed it vertically in the ground in Alexandria, then at noon on the summer solstice he measured the length of the shadow cast. Using simple trigonometry, he measured the angle made between the vertical rod and the shadow cast to be “one-fiftieth of a circle”. Treating the Sun’s rays as parallel, Eratosthenes could then use alternate-angles to say that the angle between Alexandria and Syene must be the same. He then found out the distance between the cities by employing a man to walk the distance and count his paces. The value gained was 5000 stadia, approximately 800km, and from there it was a simple calculation to work out the circumference of the planet: if 5000 stadia is one-fiftieth of the circumference, then the whole distance is 50 · 5000 = 250 000 stadia, or 40 000km. 


The story of Eratosthenes encapsulates the fundamental role of a physicist — a problem solver, applying the elegance of mathematics to the seemingly unsolvable questions nature poses us. Eratosthenes also proved to be incredibly accurate — the accepted value for the Earth’s circumference today is 40 075km, a difference of just 0.9%. Moreover, the basic principles of Eratosthenes method are still what scientists use today to establish geodetic values, albeit with a more sophisticated tool for measuring distance.


Modern geodesists use Very-Long-Baseline Interferometry (VLBI), a technique that started as a means to observe quasars. By treating quasars as fixed-points in space — due to their enormous distances from Earth — geodesists can use them as reference points. Quasars emit characteristic radio bursts which we detect using an array of radio telescopes across Earth. We can then measure the delay between two telescopes receiving the signal using hydrogen-masers, essentially extremely precise clocks. These rely on hydrogen atoms emitting photons at a precise frequency, which can be exploited to keep time at least ten-times better than the standard, caesium. Knowing the time it takes for the signal to travel, the distance between the telescopes can be calculated with an accuracy of 2mm over a 600-second observation. 


VLBI is a beautiful example of how diverse a subject physics is, but also how vital all aspects are. Geodesy, radio astronomy and quantum mechanics — three seemingly disparate fields — combined let us answer a question we’ve been grappling with for two-thousand years.