The Universe in Three Dimensions

An essay I submitted for I contest during graduate school.

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For much of human history, we have looked up at the night sky and tried to reach for the stars — and yet it seems the more we reach, the farther they are.

The night sky has kept humanity engrossed since the very beginning of our species. Ancient civilizations used the night sky for a multitude of purposes — to keep track of time, to orient their cities, to navigate, to predict the future, and, most importantly, to tell stories about the planets and stars that wandered around.

One of the first viral videos I saw was of a small girl reaching for the moon and being confused when she could not grab it. Even as we age and gain a better sense of depth perception, I do not think humans ever quite get rid of our tendency to reach for the stars. Our inability to climb up to the heavens and view the stars in three dimensions has not stopped us from inventing a metaphorical ladder to peer in and imagine the vastness that lies directly above us. Astronomers, both ancient and modern, use something called a cosmic distance ladder — the first rung of the ladder pinpoints the distance to something close-by (such as the moon) with high precision and accuracy and the successive rungs rest upon the previous rungs to reach further away from us.

Unfortunately, our eyes can only perceive depth for a hundred or so feet. A mountain a hundred miles away or a very realistic cardboard cutout of a mountain a thousand feet away would look no different. The moon, our closest celestial companion, alone is 250,000 miles away. The story of how we built the distance ladder initially with just our eyes and, later, with telescopes is a testament to the resilience, persistence, creativity, and stubbornness innate to each and every one of us.

A star born in our Galaxy observed by the Babylonian astronomers in 3500 B.C.E. has moved at most a couple of degrees on the sky today due to its motion around the Galaxy’s center. Humanity’s place among the stars, on the other hand, been radically transformed in those 5500 years. Astronomers know today that there are hundreds of billions of stars in hundreds of billions of galaxies, thousands of billions (of billions) miles away. In this essay, I will give an overview of how we went from a universe that Ptolemy generously estimated to be 80 million miles wide to a universe of seemingly infinite size. I will talk about the scientists who led the many paradigm shifts that brought us to today, the many failures along the way, and the lessons to be had about the flow of science from the pursuit to measure the cosmos.

The Center of the Universe

What makes measuring the distance to the planets and stars difficult is that our view of the universe is inherently two-dimensional from a fixed point on Earth. History has been witness to the many creative ways, both metaphysical and empirical, that humans have tried to measure the distance to the stars. However, up until only a few hundred years ago, the earth was thought to be at the center of the universe and the distant stars were thought a shorter distance than we now know the sun to be. How did the universe expand to be so unimaginably vast? The first step was placing the sun at the center of the cosmos.

Most of the early estimates of the vastness of the universe are indebted to Aristarchus of Samos, who is famous for being the first to propose that the earth revolves around the sun. Aristarchus came up with a technique to measure the absolute distance to the moon and the sun using nothing but his eyes and his command of Euclidean geometry in third century B.C.E. Aristarchus’ calculations would provide Ptolemy with the first rung of a geocentric cosmic distance ladder to explore the vastness of the universe.

Figure 1. The earth-moon-sun system at half-moon. Aristarchus correctly pointed out that this is a right-angled triangle.

By the time Aristarchus was contemplating the distance to the moon and sun, it had already been proposed that the moon received its light from the sun. Aristarchus correctly claimed that when the moon appears as a half-circle, the sun, the earth, and the moon must form a right-angled triangle (as shown in Figure 1). Aristarchus proposed that if one were to measure the angular separation between the sun and the moon exactly at the half-moon, all angles of this triangle would be known.

Aristarchus, born nearly two millennia before Galileo, had to rely only on his eyes to deduce the angular separation between the moon and the sun. He chose a value of 87 degrees, off by more than 2.5 degrees. Having determined all the angles of this triangle, Aristarchus was now ready to measure the ratio of the distance from the earth to the moon and the earth to the sun. He had to laboriously use a Euclidean toolkit that did not include trigonometry to show that this ratio has to be greater than 18 and less than 20 in more than 3 pages of dense mathematics .

Armed with the relative distance to the sun and the moon, Aristarchus set out to calculate the absolute distance to both. A total lunar eclipse — when the moon passes directly behind the earth and the sun, moon, and earth form a straight line — occurs on average twice every three years. In his eighty or so years of life, Aristarchus would have observed a lunar eclipse dozens of times and must have noticed that the radius of Earth’s shadow on the moon is about twice the total radius of the moon on a total lunar eclipse. He showed that the geometry of the total lunar eclipse and knowing this ratio would allow him to go from a fractional distance to an absolute distance.

At the heart of Aristarchus’ method is something we all learned in high school geometry: similar triangles. Armed with the approximation that the radius of Earth’s shadow on the moon is about twice the total radius of the moon and the fractional distance I talked about earlier, Aristarchus was able to calculate the distance to both the moon and the sun as ~20 earth radii and ~380 earth radii, respectively. Aristarchus’ methodology was correct but his eyes were simply not precise a measuring tool. Ultimately, his estimates for the distance to the moon were off by a factor of three and by a factor of sixty for the sun. Regardless, Aristarchus had opened the cosmos to be measured for the first time in recorded history.

Though he would later put forth a heliocentric universe, Aristarchus calculated the distances to the moon and sun while assuming that the earth was the center around which all the stars and planets moved, like a carousel ride at an amusement park (as shown in Figure 2). At the time, the geocentric worldview, born out of Aristotle’s theory of the heavens, was considered to be as true as evolution is today. One of Aristotle’s most impactful successors was Claudius Ptolemy, whose writings shaped all astronomical pursuit for the next millennium and a half.

If I were to ask any graduate student in our department what tool they use the most for their research, it would undoubtedly be Astropy — a software package that is the culmination of a decade of effort to collect core astronomical knowledge into a computer program. Astropy allows any astronomer to load the program, access state-of-the-art astronomical knowledge, and perform complex calculations with ease. Ptolemy’s Almagest and Planetary Hypothesis played Astropy’s role for almost 1400 years.

Astropy’s most used functionality is probably using the sun’s position in the Galaxy to figure out how far stars are. Similarly, Ptolemy’s tables for the distances to the planets and stars were similarly the standard astronomical toolkit for millennia. The Ptolemaic algorithms for the distance to the planets and stars were updated versions of the calculations that Aristarchus performed. Ptolemy’s models for the cosmos also included detailed instructions on how to predict the motions of the planets, as well as providing the ratio of the closest and furthest approach of a celestial body around the earth.

Ptolemy’s conception of the universe was based on the Aristotelian idea that the cosmos were made up of crystalline spheres made of aether. Overlapping like the layers of an onion, these crystalline spheres held the planets and the stars and moved around the earth. To calculate the distance to the different celestial spheres, Ptolemy invoked Aristotle’s postulate that natureabhors a vacuum and suggested that the celestial spheres must be connected in some way. Thus, he claimed that the maximum distance of a celestial body on a particular crystalline sphere and the minimum distance of a celestial body on the next crystalline sphere must be the same. Ptolemy had already computed the greatest distance to the moon and the sun using Aristarchus’s method. But with his theory of the overlapping crystalline spheres, Ptolemy was able to successively peel back different layers of the cosmos and provide the absolute distances to each — the first time in human history we had a cosmic distance ladder.

Figure 3 shows a table of distances derived by Ptolemy for the different planets and fixed stars. Notice that the moon, sun, and the planets are given minimum, maximum, and mean distances from the earth, while the fixed stars are only given a mean distance of 20,000 times the earth’s radius, which we now know to be slightly smaller than the distance from the earth to the sun. Even Ptolemy was hesitant to put an upper bound on the distance of the stars from Earth, unintentionally foreshadowing the difficulty with comprehending the vastness of the cosmos.

Ptolemy’s geocentric cosmic distance ladder stood for millennia because it was erected in a geocentric universe. The ancient Greeks were staunch geocentrists because the planets and the stars danced in the night sky in a predictable way. There were only two reasonable explanations for the harmony of the cosmos: the stars and planets were moving around the earth, or the earth was rotating. Eratosthenes (a few years after Aristarchus) had geometrically determined the diameter of a spherical Earth and deduced that to cover such tremendous distances, the earth would have to spin at unfathomable speeds. There is no motion under our feet that would indicate the earth spinning, and hence the geocentric universe quickly became irrefutable knowledge

But not everyone bought into geocentrism. It might come as a surprise to those who associate heliocentrism with Copernicus, but it was Aristarchus again who was the first to propose a heliocentric model. A full 1800 years before Copernicus was born, Aristarchus claimed that the sun was at the center of the universe and that the planets (including the earth) and stars all moved around the sun. Aristarchus’s contemporaries either ignored or rejected his heliocentrism because it was thoroughly incompatible with Aristotelian physics. Unfortunately, the concepts and motivations behind Aristarchus’ theory and any response he may have had to skeptics have been lost to time. In fact, we only know of Aristarchus’ universe from Archimedes’ (of bathtub Eureka fame) attempt to measure the volume of the universe.

The story goes that Archimedes wanted to impress King Hiero II’s successor, King Gelo, with his command of large numbers. So Archimedes set out to calculate how many sand grains it would take to fill the universe. He quickly realized that the traditional geocentric model, with its tightly nested concentric spheres, was too small to truly impress his royal patronage. Instead, Archimedes drew on Aristarchus’ heliocentric model, providing us with a glimpse of Aristarchus’ heliocentrism, in an effort to present a grander, more impressive effort. Without passing judgment on the merit of heliocentrism, Archimedes writes,

==His [Aristarchus’] hypotheses are that the fixed stars and the sun== ==remain unmoved, that the earth revolves about the sun on the== ==circumference of a circle, the sun lying in the middle of the orbit,== ==and that the sphere of fixed stars, situated about the same center as== ==the sun, is so great that the circle in which he supposes the earth to== ==revolve bears such a proportion to the distance of the fixed stars as== ==the center of the sphere bears to its surface==

Aristarchus’ heliocentric universe was significantly larger than the traditional geocentric universe and allowed Archimedes to, for the first recorded time in human history, comprehend the vastness of a universe where the sun was at the center. If it weren’t for Archimedes’ attempts to impress his new King, humanity(including potentially Copernicus)might not have ever heard of Aristarchus’ heliocentric universe.

Still, the geocentric universe went mostly unquestioned for the next few hundred years. Advancement in mathematical techniques during the Islamic golden age and detailed astronomical observation during the Renaissance would eventually point out enough flaws in the model, however, to finally warrant serious consideration of other ideas. Nicolaus Copernicus’ treatise, De revolutionibus orbium coelestium, Tycho Brahe’s careful data-collecting, Johannes Kepler’s even more careful data-crunching, and Isaac Newton’s theory of the invisible forces that govern planetary motion kicked off the heliocentric revolution — removing the earth from the center of the universe and forever changing the course of astronomy.

The Milky Way’s Yardstick

As Archimedes’ simple calculations had shown, the heliocentric universe had the potential to be far vaster than its geocentric counterpart. The acceptance of the heliocentric model of the universe had toppled the geocentric cosmic distance ladder and just made the universe a whole lot bigger, but no one quite knew by how much. It was time to begin measuring the universe anew — this time starting with the stars.

The first rung of the new cosmic distance ladder is something called the stellar parallax. Parallax is the apparent shift of an object when viewed from different vantage points. An easy way to see the effect of parallax is to hold your hand out and alternatingly look at it with just your left eye and your right eye. Perform the same exercise with your hand slightly closer and the shift will be larger! The magnitude of the shift (larger for nearby objects, smaller for further objects) against a static background can be used to infer the distance to the object being viewed.

On a cosmic scale, the stars are too far away to measure the parallax with our naked eyes. In fact, the stars are too far away to measure the parallax even from different ends of the earth. The largest baseline that nature affords us in a heliocentric universe is the extreme points in the earth’s orbit around the sun (as shown in Figure 4). In theory, one can observe a nearby star in June and in December, the “left” and “right” eyes of Earth’s orbit, against the backdrop of faraway stars and deduce the star’s distance from Earth. The ancient Greeks had already noticed that stars do not show a parallax by eye and used that to bolster their argument against Aristarchus’ heliocentric universe. But once geocentrism was dethroned, the race to measure the elusive stellar parallax had begun.

Tycho Brahe learned about both the Copernican universe and the Ptolemaic universe at school in the mid-1500s Denmark. Tycho grew up slightly partial to a universe with the earth at the center because it was grounded in Aristotelian physics, where the motions of the celestial bodies could be explained by physics. The physics governing the Copernican universe, on the other hand, would not be discovered for more than a hundred years. The marking point in Tycho’s life was when a new body almost as bright as Venus and noticeably brighter than Mars appeared in the Cassiopeia constellation overnight in 1572. Tycho’s faith in the static Aristotelian universe was shaken. The body, which we now know to be a giant stellar explosion called a type Ia supernova, violated the constancy of the cosmos that Aristotle held to be true. How could such a bright object spontaneously appear in the night sky?

Tycho meticulously measured the brightness of the new body over many months and noticed that it was fading — again irreconcilable with Aristotle’s physics. Before it faded away completely, Tycho, still a staunch geocentrist, wished to measure the parallax of the supernova to determine what celestial sphere the body belonged to. Only after many months of careful observation did Tycho conclude that no parallax was visible and, consequently, the new object must be a part of the celestial sphere holding the fixed stars. The stars in that celestial sphere were evidently no longer fixed.

Ask any observational astronomer in our department what the hardest part of observing is, and they will all tell you that it is the uncertainty associated with your observing instrument. Over the next few decades, Tycho essentially was running a one-man survey of the cosmos, carefully collecting data on the motion of the planets and stars. Tycho, one of the keenest observers in the history of astronomy, had an extraordinary handle on the precision of his tools. But the stellar parallax eluded him the thirty years or so after the 1572 supernova till his death.

Some back-of-the-envelope math showed Tycho that if the parallax of the closest stars just escaped detection, the nearest stars would have to be further than 700 times the distance of the sun to Saturn given the precision of his observations (the closest star is now known to be ~30,000 times the distance from the sun to Saturn). In a letter, Brahe writes, “Deduce these things geometrically if you like, and you will see how many absurdities (not to mention others) accompany this assumption [of the motion of the earth] by inference.”

Tycho’s faith in the Ptolemaic universe was shaken by the 1572 supernova but he also rejected the Copernican system because of his failure to detect the parallax of stars. Blending elements of both the Copernican worldview and the traditional geocentric worldview, Brahe proposed his own arrangement of the cosmos: the sun and the moon orbiting earth, and the other planets orbiting the sun — a compromise of sorts. Brahe’s geo-heliocentric system easily explained the non-detection of stellar parallax. However, Tycho’s key contribution here was his removal of Aristotle’s nested crystalline spheres holding the planets and stars, which paved the way for Isaac Newton to develop his theory of invisible gravity to explain planetary motion.

The next one-hundred years saw the writing of Johannes Kepler’s laws of planetary motion, Galileo’s invention of the telescope, Galileo’s discovery of Venus’ lunar-like phases, Galileo’s discovery of inertia, and Newton’s law of gravity. These radical scientific advancements were the final nail in the coffin of geocentrism and emboldened the astronomers of the day even further to pursue the measurement of the distance to a star. Moreover, Galileo’s invention of the telescope had convinced the next generation of astronomers that the measurement of stellar parallax must be right around the corner.

Imagine yourself an archer aiming for the bull’s eye only 12 cm in diameter from 70 meters away. A strong wind might deflect your arrow by many dozens of centimeters. The weight of the arrow might make a difference in the correction you have to make. The elasticity of the bow could also cause a deflection. James Bradley, born almost 150 years after Brahe, was the first to fully characterize similar extrinsic challenges that astronomers faced in measuring the stellar parallax.

Bradley and his contemporary, amateur astronomer Samuel Molyneux, were determined to measure the parallax of a star that passes directly overhead in London, Gamma Draconis. Bradley had calculated that Gamma Draconis would show a cyclical wobble from southernmost in December and northernmost in June if we were indeed orbiting the sun. Only the magnitude of the wobbles were uncertain and would depend on the distance to the star — large wobbles would mean the star is close and small wobbles that the star is far.

My advisor often mentions that a lot of interesting science starts with a confused, “what?” Bradley and Molyneux’s data on Gamma Draconis turned out to be a real head-scratcher — they found that the star further south and reached its southernmost point in March and moved upwards to reach its northernmost point in September. Bradley and Molyneux meticulously gathered eighty positions of Gamma Draconis over many months and found the same trend. So troubled was Bradley that he used most of his savings to buy a new telescope and make the same measurement for dozens of stars. The same trend persisted.

Imagine a rainy Boston day. If I were standing still, the raindrops will appear to drop in straight lines. But if I started walking or running, the raindrops would seem to tilt towards me. Prepared as I am, I would have to tilt my umbrella slightly to keep my feet dry. The apparent tilting of the raindrops led Bradley to develop the theory of aberration. The starlight is the rain, we are the moving Earth, and the umbrella is our telescope. It turned out that Bradley, and those before him, had not properly taken into account the motion of the orbiting Earth, much like your stroll in the rain, when measuring the parallax. Astronomers now had to slightly tip their telescopes to correct for Earth’s motion through the stars. To use the bow-arrow analogy again, aberration would cause our metaphorical arrow to deflect by more than three full meters if we were aiming for a 12 centimeter box.

Bradley serendipitously discovered another complication during his hunt for the stellar parallax: the wobbling of the earth’s rotation axis due to the gravitational tug of the moon (“nutation”). Nutation could shift positions of stars by 0.003 degrees (a meter and a half in the archery scenario). The list of things in the way of measuring the true stellar parallax was getting unwieldy: aberration, nutation, refraction of starlight in atmosphere, telescope wobbling, etc. Bradley spent the rest of his life correcting for all these different effects, measuring the positions of hundreds of stars to get down to a precision of less than 0.001 degrees, but passed away without ever measuring the stellar parallax. To his credit, Bradley had systematically characterized all the known extrinsic factors at play when measuring stellar parallax and had discovered two fundamental astrophysical phenomena in his pursuit.

William Herschel, a part of the next generation of astronomers, is most famous for his discovery of Uranus — a serendipitous discovery that happened when he was surveying double stars (stars very close together in the sky) to detect stellar parallax. Herschel chose stars close together on the sky intentionally such that one was bright and supposedly close to Earth and the other fainter and further away — he had realized that the impact of the atmosphere, aberration, and nutation will all have a very similar effect on both stars. Therefore, any wobble of the brighter relative to the dimmer star could definitively be attributed to the parallax.

In a few years, Herschel had systematically catalogued the positions of more than eight hundred double stars. Much like Bradley’s, “what?”, the more data that Herschel collected, the more perplexed he got. It took Herschel twenty five years of data collection and analysis to definitively show that the doubles were actually orbiting around each other! This discovery suggested that they were right next to each other in the cosmos and could not be used to measure stellar parallax. Herschel had just serendipitously proved the existence of binary stars — astronomers a few generations later go on to use binary stars to measure the mass of a star for the first time and germinate modern stellar astrophysics as we know it today.

The key piece of technological innovation came with Joseph von Fraunhofer’s new telescopes just a couple of decades after Herschel’s non-discovery of the parallax in 1820s. Fraunhofer built two instruments that had the required precision to measure stellar parallax: the heliometer and the Dorpat refractor telescope. These telescopes were so precise that they were able to make out the year a quarter was minted from a full 5 kilometers away. Friedrich Bessel, Friedrich Georg Wilhelm von Struve, and Thomas Henderson all published the first few reliable distances to three different stars within a few years of each other (1837-1839). John Herschel, William Herschel’s son, in congratulating Bessel said, “I congratulate and myself that we have lived to see the great and hitherto impassable barrier to our excursions into the sidereal universe, that barrier against which we have chafed so long and so vainly… almost simultaneously overleaped at three different points. It is the greatest and most glorious triumph which practical astronomy has ever witnessed.” There were many more triumphs to come for practical astronomy in the next ~180 years.

Henritta’s Standard Candles

The discovery of the stellar parallax transformed our view of the universe and had finally afforded us with an accurate three-dimensional view of the night sky beyond just our solar system. However, well into the 1940s, we only knew the distance to stars that were extremely close-by. If the whole Galaxy were the size of Cambridge, we had only surveyed the space a mini-refrigerator would occupy. The precision needed to go beyond our solar neighborhood was immense — it would require building telescopes that had the resolving power to read the headline of a newspaper in Cambridge all the way from Andover — a full twenty-three miles away. Astronomers realized that stellar parallax provided exquisite distances to nearby stars but technology was not advanced enough to survey our whole Galaxy. Around the same time, a great debate was raging about whether our galaxy was the universe or whether there existed many other galaxies like ours — little island universes. How did we go from surveying a tiny proportion of our galaxy to making claims about galaxies not our own?

One of the first astronomy colloquia I attended in my second year of graduate school was titled, “Constructing the Glass Universe”. Not having read the abstract for the talk and the mysterious title left me with no real idea on what to expect. I left the talk mostly the same, but slightly and certainly changed. Dava Sobel, the speaker, gave us a history lesson on the group of women working at the Harvard College Observatory who completely and utterly transformed astronomy. Most in the room knew who Annie Jump Cannon (classified stars with a scheme still used today) and Cecilia Payne-Gaposhkin (first to show that the sun was mostly hydrogen and helium, starting the field of stellar evolution) were but Sobel did a phenomenal job of highlighting their importance in the history of astronomy. But the magnitude of the discoveries of Henrietta Swan Leavitt were a shock to most in the audience.

My office at the department sits two floors and a hallway below one of the best telescopes of the mid-1800s and early 1900s — the great refractor. After 30 years of operation, Harvard College Observatory’s (HCO) fourth director, Edward C. Pickering, redirected the telescope to be used mostly for capturing the stars on photographic plates. The data collected by Pickering’s survey of the stars was a “big data” problem before the term existed. Consequently, Pickering hired dozens of women to be the computers — yes, they were really called computers (and derogatorily “Pickering’s Harem” by the scientific community) — for the HCO. These computers, Leavitt among them, were paid 25 cents an hour (10 cents more than a cotton mill worker then, and $7/hr in today’s dollars) to exclusively look at and record astronomical photographs.

Leavitt was assigned to look at stars that varied in brightness by a large amount for unknown reasons — much like how the brightness of a beacon from a lighthouse would vary if I were to observe it from a fixed location. Leavitt had taken enough astronomy and physics classes to speculate the cause of these modulations but her assigned job was to just note down the time at which a star was its brightest and when it was its dimmest. Leavitt was observing many variable stars in the Small Magellanic Cloud (SMC), preparing circulars, and Pickering was publishing her results (giving credit only in the body but not as authorship). An astronomer from Princeton wrote, “What a variable-star fiend Miss Leavitt is. One can’t keep up with the roll of the new discoveries.”

Leavitt’s greatest contribution to astronomy, and perhaps one of the greatest ever leaps in the study of astronomical distances, was her observation that the periods with which some stars varied in brightness was closely related to their apparent brightness in the sky. And since stars from the SMC were thought to be more or less the same distance away, Leavitt reasoned that this relationship between the period and the brightness must be an intrinsic property of the variable stars.

Leavitt’s variable stars, now called cepheids, could then be used as something astronomers call “standard candles”. Say you have two identical candles, one right next to you and the other across the room, and you know the exact distance to the one right next to you. Given this information and some understanding of how faint things appear when they are further away, you can measure the size of the room by relating to the apparent brightness of the candle across the room and the intrinsic brightness you expect from the one very close to you. This apparent brightness, distance, and intrinsic brightness relation is exactly what Leavitt derived for the cepheids in the SMC.

The article presenting Leavitt’s work, with the author credit wrongly given to Pickering again, says in relation to the cepheids, “It is to be hoped, also, that the parallaxes of some variables of this type may be measured.” Leavitt correctly predicted that the stellar parallax 8 would help us anchor the cepheid scale and measure the distance further than anyone else had ever before. In a year, Hertzsprung and Russell had already published the distances to 13 cepheids, calibrated Leavitt’s period-luminosity relation, and provided the distance to the SMC, the first to ever do so for an object outside the Milky Way. A mathematical error or a proofing mistake led to Russell’s distance being printed as 3000 light-years instead of 30,000 light-years. 3000 light-years is big, certainly, but not as eye-poppingly large as 30,000 light-years, which would have surely raised some eyebrows as that number would place the SMC outside the Galaxy (then the whole known universe).

There was a storm brewing in the broader community around the same time. The great between Harlow Shapley and Heber Curtis at the Smithsonian Museum of Natural History in 1920 came as close to a duel as an academic field could. Shapley was a strong believer in the Galaxy, our Galaxy, comprising the whole universe. Curtis strongly believed that the visible spiral nebulae in the sky (one such nebula is shown in Figure 5) were actually separate island universes much like our Galaxy. One of Shapley’s primary arguments against the island universes was the apparent brightness of the supernovae observed in these nebulae — if the nebulae were placed far away, that would imply that one supernova alone could outshine a whole galaxy comprising billions of stars (which we now know to be true).The duel thankfully had no casualties — only a few hurt egos.

Edwin Hubble, only four years after the great debate, presented results at the American Astronomical Society meeting that definitively proved Curtis’ island universes stance. Hubble had exclusive access to the world’s largest telescope of the time. For months on end, he observed the spiral nebula Andromeda and found a Leavitt lighthouse, V1 — he marked it “Var!” on his photographic plate. Carefully measuring the brightness of V1 over many months, Hubble was able to determine the period of the cepheid and, consequently, determine that the distance to Andromeda was ~900,000 light-years — much greater than Shapley’s estimates of the size of the Milky Way back then. Hubble broke the news in a gentle letter to Shapley, “the straws are all pointing in one direction and it will do no harm to begin considering the various possibilities involved.”

The various possibilities all pointed to the same inescapable conclusion: there were many galaxies like ours in the universe. Only twenty years after Leavitt’s introduction of a tool to measure great distances, the universe had responded in kind.

The Continual Expansion of the Universe

Leavitt’s stars and Hubble’s discovery showed astronomers of the time that there was a whole universe of galaxies outside our own to be studied and measured. Around the same time, Albert Einstein’s theories on gravity were becoming widely accepted in the physics community and astronomers were still debating about its implications on cosmic scales. How did go from a universe with many galaxies to where we are today?

Einstein imagined space differently from Newton and Aristotle. Instead of a static and unmoving canvas that the stars and galaxies lie upon, Einstein found that space is like a very stiff trampoline. Space, like a trampoline, bends slightly when objects move on it. The mathematics of Einstein’s theory of gravity led Einstein to the inescapable conclusion that space cannot stay still — it must either be expanding or contracting.

Space’s necessary expansion or contraction was a contradiction that bothered Einstein greatly. The idea of a universe that is forever expanding or forever contracting was directly at odds with what most scientists, including Einstein, believed in at the time: a static universe. Einstein decided to add a cosmological constant, essentially a fudge factor, to the final equations of gravity to balance out the expansion or contraction of space, in what he later called the biggest blunder of his scientific career.

Meanwhile, by the mid-late 1920s, Hubble had convinced the astronomical world that our galaxy was not unique by measuring the distances to other nebulae using Leavitt’s cepheids. However, the cepheids had more stories to tell. Hubble’s contemporary, Vesto Slipher, had concurrently measured the doppler shift to many nebulae, now called galaxies, and shown that some were curiously moving away from the Milky Way at enormous speeds. However, since the distance to these galaxies was not known, astronomers of the time were unable to make strong claims about why they were moving so fast. Hubble pointed the 100-inch mammoth telescope atop Mt. Wilson at many nebulae once more to find cepheids in as many galaxies as he could.

Hubble laboriously calculated the distances using Leavitt’s method and velocities using Slipher’s sample for his sample of twenty-four galaxies. The relation between the two was almost exactly linear and implied that the further away a galaxy is, the faster it is moving away from us — the fabric of space holding the galaxies together was expanding. In Brian Greene’s words, “much as poppy seeds in a muffin that’s baking move apart as the dough swells, galaxies move apart as the space in which they’re embedded expands.” Had Einstein not added the cosmological constant to placate his need to live in a static universe, he would have actually predicted the expansion of the universe a couple of decades prior. Hubble’s discovery led to the ominous conclusion that a ruler to measure the universe was not as powerful anymore since the very concept of the distance that the ruler was trying to measure was getting larger.

If you were to point Hubble’s 100-inch telescope to some of the faintest galaxies in the sky, all you would see is a blob and no individual stars. Even with today’s massive 10-meter telescopes, we can see stars only so far away. Astronomers could only measure the distances with Leavitt’s method if individual stars in a galaxy could be picked out and carefully observed. Cepheids allowed us to reach 10,000 times further away than the best geometric parallax measurement today could. But Hubble’s observations had shown astronomers of the time that the universe was likely significantly larger than what cepheids alone could probe. The next rung of the cosmic distance ladder came through the same type of bright star that appeared in the sky that shook Tycho’s faith in the Ptolemaic universe and the ones that Shapley used to argue against the existence of many different galaxies in the universe — the supernova.

In the mid-1950s, astronomers determined that the binaries that Herschel first discovered in the 1800s were actually responsible for the stellar explosions that Tycho observed in 1572. One of the binary stars had to be an object called a white dwarf, the final evolutionary state of stars born below a certain mass, and the other is still hotly under debate. These are unusual stars because they are approximately the mass of the sun but very tightly packed into the size of the earth — a teaspoon of a white dwarf would be equivalent to the weight of two full elephants!

If the stars in a binary move too close to each other, the white dwarf will start accreting stellar material from its companion. Put more air into a balloon than it can handle, and it will inevitably explode. Similarly, Chandrashekhar found that if a white dwarf accretes enough mass to hit a total mass of 1.44 times the mass of the sun, it will explode in one of the most energetic explosions in the known universe. These supernovae are so explosive that they have the ability to outshine a whole galaxy!

Since type Ia supernovae all happen at around the same white dwarf mass, they all also must output about the same amount of energy. Cepheids’ periods predicted how intrinsically luminous they were. Type Ia supernovae all get to about the same amount of intrinsic brightness because they explode at about the same mass. Moreover, these supernovae can be used as standard candles in the same way as cepheids — observe one in a nearby galaxy with cepheids and the ladder gets higher.

Looking at the night sky is akin to looking through time. The constant speed of light makes it such that the light from a star 200 light-years away would take 200 years to get here — that star observed today would appear to us as it was in 1819. However, 200 light-years is a tiny distance by cosmic standards. Two teams, one born in Harvard and one in Berkeley, contended that the distances to Type Ia supernovae could allow us to deduce whether the expansion of the universe was occuring at a constant rate, decelerating, or accelerating. An analogy from Robert Kirschner’s The Extravagant Universe , the initial lead of the Harvard supernova team, aptly describes this with an analogy.

==The impact of throwing snowballs at a receding bus, like sending photons across== ==an expanding universe, depends on whether the busis travelingat aconstantrate,== ==or slowing down for the stop sign up ahead. If you throw atabus thatis cruising== ==along, the snowball takes longer to get there and makes a less satisfying splat. If== ==you throw atbusthat is slowingdown, thesnowball travels asmallerdistance and== ==makes a resounding thud. (…) Looking at supernova in an expanding universe== ==means looking at photons that travel an extra distance, so the supernova would== ==look fainter (…). This is something like hurling a snowball at a bus that is== ==accelerating away after it drops youoff. If the snowballcatches upatall, it barely== ==sticks.==

Both teams were almost certain that the expansion of the universe’s expansion must be decelerating. Imagine throwing a ball up in the air. The ball’s velocity would initially increase but the earth’s gravity will eventually take over and slow the speed of the ball. Similarly, astronomers expected all the matter (in the form of the planets, stars, and galaxies) in the universe to play a similar role and cause the expansion of the universe to be decelerating, and the furthest supernovae to be brighter than they would be in a constantly expanding universe.

You might guess where this is headed. In 1998, the two teams published within days of each other and independently showed that the farthest supernovae actually appeared to be dimmer . The universe was expanding at an accelerating rate! It was as if someone took the ball and attached little thrusters such that its acceleration was always positive and it was moving further, and further, and further away from Earth as time went on. The cause of the accelerating expansion is unknown as of the writing of this essay and has been dubbed to be due to “dark energy”.

Future

We have come far in the last 5500 years. The universe started at a mere 80 million miles wide. The universe today is unknowably, immensely, and spectacularly large in every sense of the word. Moreso, we have found that the universe is expanding at an accelerating rate. Ptolemy’s omission of a maximum distance to the fixed stars was markedly omniscient of the difficulty that we face today in comprehending the vastness of the cosmos.

The difficulty in measuring the universe has not stopped us from trying. In this pursuit, we actually did some neat science along the way as well: Aristarchus’ heliocentric system, the success of Ptolemy’s universe, Tycho’s supernova, Kepler’s laws of planetary motion, Bradley’s discovery of aberration and nutation, Herschel’s discovery of Uranus and binary stars, Leavitt’s study of variable stars, Hubble’s discovery of other galaxies, study of type Ia supernovae, and the accelerating expansion of the universe.

More interesting to me though is how the pursuit to measure the cosmos is an excellent learning lesson for the actual flow of science. Science is not revolutionized by lone geniuses. Paradigm shifts take time, effort, and a lot of failure. While researching for this essay, I feel that I took valuable lessons from the many historical figures involved. Learning about Aristarchus’ bold ideas, Ptolemy and Tycho’s meticulous observations, Bradley’s many failures, Herschel’s serendipitous discoveries, Leavitt’s unsung contributions, and Hubble’s privilege of resources have all made me appreciate that science is messy and the flow of scientific knowledge is anything but linear.

In the words of Douglas Adams, “Space […] is big. Really big. You just won’t believe how vastly hugely mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist, but that’s just peanuts to space.” The universe keeps getting larger and yet we continue looking.