this animation is zooming out by a factor of 10 every 2 seconds maybe you’ve seen things like this before conveying the mindboggling scale of our universe but here in this video you and I are going to continue the Saga through the many moments of delightful Ingenuity throughout human history that led us to First discover how far away objects in the cosmos really are appreciating how we know these distances is to me more amazing than the distances themselves this is part two of a collaboration with Terence Tao and it’s okay if you haven’t yet seen part one each video should be relatively self-contained but for context we left off with Kepler’s ingenious method for deducing the shapes of all of the orbits of the planets around the Sun so people knew what the solar system looked like but they still didn’t have an exact sense of scale and this left astronomers hungry to measure any distance that they could in this system like maybe how far away a given planet is from Earth at a given moment since that would be enough to lock everything else into place now I’ll admit while I had vaguely learned about how this was done before I definitely had not appreciated the cleverness of details.
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You could measure distances to the planets like Venus — take two measurements on different sides of the Earth around the time of Captain Cook when they were traveling to discover Australia and so forth. Part of the reason for this was the scientific mission: they wanted to know the distance to Venus and Mars and so forth. They wanted people to take precise measurements, one in Greenwich in the UK and one somewhere in the southern hemisphere at exactly the same time of the same object. The key idea here is that as you sail down to the Southern Hemisphere and you observe a given object up in the sky its position in the sky, say relative to the background constellations, will appear to shift up as the angle of your line of sight slowly changes with your position. We call this parallax. It is the same parallax that we use with binocular vision — our eyes are a certain distance apart and so we can determine depth or any distance that’s not too much larger than the distance between our eyes. Say that now we figured out how to make two eyeballs on different sides of the earth. Now if you want to turn this into a measurement you first need to understand this line right here connecting the two different observation points, both its distance and its direction — because people knew the size of the Earth and they could tell where they were on the earth that would be taken care of. Then the hard part is for the first observer to take a precise enough measurement of the viewing angle to this object — they need to deduce this angle here — and for the second observer to do likewise. If you can do that you have a triangle where you know all three of the angles and you also know one of the side lengths, which with a little trigonometry tells you any other side length.
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Even if both observers are looking at Venus at its absolute closest to Earth (about 39 million km, over 6,000 Earth radii), zooming back in the lines of sight would be almost parallel. The difference in angle works out to about one arc minute (1/60 of a degree). So the measurements have to be extremely precise, and both observers must really be looking at the same thing at the same moment. At the time clocks were not good enough for a “make this measurement at this time” protocol. The workaround was the transit of Venus — Venus travels across the face of the Sun, and you can time exactly when Venus hits the edge of the sun.
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Watching the transit from the northern hemisphere would look slightly different than from the southern hemisphere. The animation exaggerates the difference — in reality the two views would look much more similar. There was no photography, so they couldn’t take pictures and closely compare them. The sun spans about 32 arc minutes in viewing angle. The deviation in viewing angle to Venus would be a fraction of an arc minute. To measure exactly what that fraction was, each observer would measure the duration of the transit (how long it takes from the moment Venus’s silhouette first appears on the disc to the moment it leaves). Critically, they also would have known how fast Venus and the sun are moving through the sky at this time, which would let them calculate how long it should take Venus’s apparent position to traverse the distance of say one Sun diameter — which turns out to be around 7 hours.
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Thanks to Kepler, they knew the relative shapes of the orbit and how long it takes each planet to go around the Sun, so in principle they could calculate how long it should take for this line to scan over those 32 arc minutes. Measuring the durations of the transit from two different locations tells you the lengths of two lines drawn across the sun’s disc. These are very similar but measurably different, and with a little circle geometry the lengths of those lines will tell you how far apart they are. That fraction of an arc minute separating the two observations then lets you deduce how far away Venus is in terms of the distance between those observers.
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Edmund Halley originally came up with this idea but didn’t live long enough to see it come to fruition. Guillaume Le Gentil was another explorer tasked with making one of these measurements for the transit of Venus in 1761 but was delayed by the Seven Years’ War. The next transit was 8 years later, but after that wouldn’t be for another 105 years. He extended his journey, and by 1769 he was set up in the Philippines ready to go — but on that day it was cloudy. Once he finally got back to France he found he had been declared dead, his wife had remarried, and his relatives had plundered his estate.
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Having one measurement was enough to lock in place the scale of the whole solar system — every other distance involved would follow. This mattered because you need it to work out the distance to the Sun. The distance of the Sun is the most important rung of the ladder — it’s called the astronomical unit, and almost everything beyond the solar system is measured in terms of it.
There’s another clever deduction you can make. Roemer was measuring Jupiter, which has a small moon called Io. Io orbits Jupiter very fast (42 hours) because it’s close to Jupiter. If you observe Jupiter with a telescope you see Io go behind Jupiter’s shadow and out, in and out. Roemer was marking down the precise moment of each cycle. Io goes dark when it falls into Jupiter’s shadow. As it emerges it becomes bright again. You’d expect these to happen in 42-hour increments like clockwork — except that Roemer observed Io was 20 minutes ahead of schedule or 20 minutes behind schedule at different times of the year.
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Picture Earth going around the Sun, Jupiter also going around the Sun. When Earth is on the same side as Jupiter, the timing was 20 minutes earlier than when Earth was on the opposite side of Jupiter. Roemer realized the reason was that light was taking 20 minutes to traverse the extra distance — two astronomical units traversed in about 20 minutes. Historically this was before the measurement to Venus, so their estimates for the speed of light don’t look impressive, but at that time it wasn’t even obvious that light had a speed. At astronomical scales it’s actually really slow. This clever measurement laid the groundwork for more precise experiments down on earth.
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These days the more accurate way to measure distances to planets like Venus is with radar — which relies on knowing the speed of light.
Now leaving the solar system: you can use parallax for nearby stars too. Same reasoning, but the two observation points are opposite sides of Earth’s orbit. Measure the star at one time of year, wait six months for Earth to be on the other side of the Sun, and measure again. A time-lapse of Proxima Centauri over 6 months does show it gently drifting against the background stars.
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Working out the math: for Proxima Centauri (over 40 trillion km / 4+ light years / ~260,000 astronomical units), the change in angle is a tiny tiny portion of a degree.
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Parallax for Proxima Centauri is only about 1.5 arc seconds — roughly the angular size of a dime held 2.5 km away. That’s the closest star; it only gets worse from there. The first measurement of this kind was successfully done in 1838 by Friedrich Bessel. Over the next century there was a monumental effort for many more measurements — but parallax only really works for a tiny portion of our galaxy.
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By the 19th century there were maybe a thousand stars close enough that parallax gave distances. Knowing distance plus apparent brightness gives absolute brightness, by the inverse square law of light propagation.
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Organize stars onto a plot: color on the x-axis (higher frequency blue on left, lower frequency red on right) and absolute brightness on the y-axis. Most stars live on the main sequence. This is the Hertzsprung-Russell diagram, initially created around 1911, with decades of data from the Harvard computers (a group of women at the Harvard College Observatory).
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The main sequence represents one common type of star, including our sun during its hydrogen-burning phase. Bigger brighter stars burn hotter (blue); smaller cooler stars burn redder. So for a distant star you can deduce its absolute brightness from its color, then use apparent brightness and the inverse square law to get distance. Spectral lines tell you what atoms are in the star, which classifies the star. Antonia Maury and Annie Cannon came up with useful classification systems.
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This law stops working past the Galaxy — individual stars become too faint. Some very bright stars (Cepheids, super giants thousands of times brighter than the Sun) can still be measured in other galaxies. Cepheids are variable stars — their brightness oscillates with periods of 10-20 days. Henrietta Swan Leavitt measured all the Cepheids in her own galaxy, plotted period against intensity, and found a linear law — the brighter the Cepheid, the longer the period. This gave a standard candle: if a galaxy has a Cepheid, observe its period, deduce its absolute intensity, compare to apparent brightness, and get the distance. This is the next rung.
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Edwin Hubble was measuring many galaxies. Each has spectral lines (e.g., hydrogen absorption). For some galaxies the spectrum was shifted to the red. Hubble plotted red shift against distance and got a linear relationship — Hubble’s Law. Nowadays we know this is because general relativity predicts a uniformly expanding universe: further things recede faster, with more red shift. This is the largest rung of the ladder.
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Virtually every object in the universe emitting light or radiation can now have its distance measured, usually by spectral line shifts. The Sloan Digital Sky Survey visualizes about half a million galaxies out to ~1.6 billion light years. One discovery: galaxies form filaments, massive structures. Predicted by simulations of billions of virtual galaxies orbiting by gravity.
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To calibrate distances at very large scales you stack many rungs: sun distance, parallax to nearby stars, main sequence fitting for other galactic stars, Cepheids between that and Hubble’s law. Gravitational wave measurements (standard sirens from black hole collisions) cross-check Hubble’s red shift calculations. They match within about 10% — reassuring, but that 10% is controversial. Something seems wonky about Hubble’s law at very large scales.
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The cosmological principle — the laws of the universe are pretty much the same everywhere — is an article of faith since the Copernican revolution. It’s always been rewarded: self-consistent, leads to new laws. But there’s a 9-10% anomaly that’s an ongoing area of study. Astronomy is a living subject.
Tao has an FAQ with details and corrections, and is working on a book on the topic with collaborator Tanya Klowden.