--- <iframe width="560" height="315" src="https://www.youtube.com/embed/YdOXS_9_P4U?si=1HwK-7KH9v2_zkJO" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe> ## On the Measure of All Things An introduction to a book on astronomy by Albert Einstein once referenced an idea he called one of "pure genius." This idea, which we will explore, concerns how Johannes Kepler deduced the shape of Earth's orbit—a feat of reasoning more clever than is commonly realized. This lecture, inspired by a conversation with the mathematician Terence Tao, will trace the historical and mathematical steps humanity took to measure our cosmos. This is the narrative of how humanity first measured the size of the Earth, then the solar system, and eventually the universe itself. It is a story where each measurement unlocks the path to the next, forming a proverbial ladder to the stars. The awe inspired by the unfathomable scales of the cosmos is matched, if not surpassed, by the cleverness of the reasoning applied at each step. > To measure an object $x$, one cannot simply observe $x$ in isolation; one must observe its impact on another object, $y$. This process requires clever ideation, precise data enabled by technology, and ultimately, the application of mathematics. > > `— Terence Tao` We begin, as we must, at home. ### The Measure of the Earth The foundational measurement of the distance ladder is the radius of the Earth. However, to even ask this question presumes a spherical Earth. How did ancient thinkers first deduce this shape? The most convincing early proof came not from observing the Earth itself, but from observing the Moon. We are stuck *on* the Earth, limited to a single vantage point. If we could step away and view it from multiple angles, its roundness would be obvious. A flat disc, for instance, would appear as an ellipse from any angle other than head-on. There exists a [geometric argument](https://mathoverflow.net/questions/39127/is-the-sphere-the-only-surface-with-circular-projections-or-can-we-deduce-a-sp) that a convex body whose every projection (or shadow) is a circle must be a sphere. While this may seem intuitive in three dimensions, it is notably false in two, where non-circular shapes can exist whose projections are all intervals of the same length. [[Aristotle - Ἀριστοτέλης|Aristotle (Classical Greece, 4th Century BCE)]] did not need a spacecraft; he used the Moon. During a lunar eclipse, the Moon passes into the Earth's shadow. By observing the shape of this shadow as it traverses the lunar surface, one can see it is consistently a circular arc. Composite photography of lunar eclipses today makes this visually explicit, providing direct evidence that the Earth is round. Such an observation requires no advanced technology, only careful attention to the sky. These same images also offer a visual clue to the relative sizes of the Earth and Moon, a point we will return to. ![[Wanderloots - Composite Lunar Eclipse - November 19, 2021.png]] With the Earth's sphericity established, its size could be measured. The first known successful attempt was by **Eratosthenes (Hellenistic Period, c. 276–194 BCE)**. The account begins with his knowledge of a particular well in the town of Syene (modern Aswan, Egypt). On the summer solstice, at high noon, the sun's rays shone directly to the bottom of this well, reflecting off the water. In his own city of Alexandria, Eratosthenes observed that this was not the case; his own well cast a shadow. [^EratosthenesWell] Recognizing that the Earth was round and that the Sun's rays arrive as effectively parallel lines, he understood the geometric implication.[^EratosthenesParallelRays] The phenomenon in Syene meant that the town lay on a specific line of latitude where the Sun is directly overhead at noon on the solstice[^SolsticeAxialRotation]—a line we now call the Tropic of Cancer (denoted as blue line).[^TwoTropics][^SyeneTropic] Alexandria, being further north, would see the Sun at an angle to the vertical. Using a gnomon (a type of ancient protractor), Eratosthenes measured this angle to be approximately 7 degrees off the vertical. ![[Grant Sanderson - Solstice Ray Illustration - 8 Feb 2025.png]] He deduced that this 7-degree angle $\theta$ at Alexandria corresponds to the angle subtended by the arc between Alexandria and Syene at the Earth's center. Therefore, the ratio of this angle to a full circle (360 degrees) must be equal to the ratio of the distance between the two cities to the Earth's total circumference.[^EarthSphere] $ \frac{7^{\circ}}{360^{\circ}} = \frac{\text{Distance}_{\text{Alexandria-Syene}}}{\text{Circumference}_{\text{Earth}}} $ The distance between the cities was recorded as about 5,000 stadia (approximately 500 miles). The historical record does not specify how this distance was measured. Theories suggest it could have been estimated from the travel times of merchant caravans along the Nile, or perhaps, as a recurring joke suggests, by a "graduate student" pacing the distance and counting the steps.[^EratosthenesRiverboatMerchants] This single direct measurement is the foundation upon which the entire cosmic distance ladder could, in principle, be built, though in practice the accumulation of errors would be prohibitive. With the angle $\theta \approx 7^{\circ}$ and the distance between the two cities $D_{AS} \approx 5000$ stadia, the circumference $C_{Earth}$ can be calculated directly from the proportion: $ C_{Earth} = D_{AS} \times \frac{360^{\circ}}{\theta} \approx 5000 \text{ stadia} \times \frac{360}{7} \approx 257,000 \text{ stadia} $ From this, the radius of the Earth, $R_{Earth}$, is found using the fundamental relationship $C = 2\pi R$: $ R_{Earth} = \frac{C_{Earth}}{2\pi} \approx \frac{257,000}{2\pi} \approx 40,900 \text{ stadia} $ Using the common conversion of 1 stadium ≈ 157.5 meters, this yields a radius of approximately 6,440 km, which is within 1.1% of the modern accepted mean value of 6,371 km. The precision of Eratosthenes's result is subject to the uncertainty in the conversion from stadia to modern units. Conventionally accepted conversions place his accuracy at around 10%, a remarkable achievement for its time. **Challenge:**[^Eratosthenes] With `θ = 7°`, `θ_rad ≈ 0.1222`, and arc `s = 5000` stadia (Alexandria→Syene), `R ≈ s/θ ≈ 5000/0.1222 ≈ 40,900` stadia; then `C = 2πR` as above. Redo the estimate with `θ = 6.5°` and `7.5°`. What percent change in `R` results from ±0.5°? [^Eratosthenes]: Ratios and proportions: arc/whole = angle/360°. Central angles and arc length `s = R·θ` (with `θ` in radians). Degrees ↔ radians: `θ_rad = (π/180)·θ_deg`. Similar triangles for parallel rays; circumference `C = 2πR`. Error sensitivity: how a small angle error changes `R`. ### The Measure of the Moon Once the Earth's radius, $R_{Earth}$, is known, the next logical step is to determine the distance to the Moon. Once again, eclipses provide the key. ![[Grant Sanderson - Lunar Eclipse Diagram - 8 Feb 2025.png]] During a lunar eclipse, the Earth's shadow in space is approximately the size of its diameter, or $2R_{Earth}$. Ancient observers, including **Aristarchus of Samos (Hellenistic Period, 3rd Century BCE)**, knew from extensive observation that the longest lunar eclipses last approximately 3.5 to 4 hours. They also knew that the Moon takes roughly one month (about 28 days[^LunarMonthLength]) to complete its orbit. By comparing the duration of the eclipse to the duration of the full orbit, one can determine the ratio of the Earth's diameter to the circumference of the Moon's orbit. A more precise calculation would account for the Moon's own diameter, as the total transit distance through the shadow is the Earth's diameter plus the Moon's diameter; ![[Grant Sanderson - Lunar Eclipse Radii - 8 Feb 2025.png]] Let $T_{eclipse}$ be the duration of the total lunar eclipse (~3.5 hours) and $P_{orbit}$ be the Moon's orbital period (~27.3 days, the sidereal month). The ratio of the distance the Moon travels during the eclipse (approximately the Earth's diameter, $2R_{Earth}$) to the circumference of its orbit ($2\pi D_{Moon}$) should be equal to the ratio of the durations: $ \frac{2R_{Earth}}{2\pi D_{Moon}} \approx \frac{T_{eclipse}}{P_{orbit}} $ Rearranging to solve for the distance to the Moon, $D_{Moon}$: $ D_{Moon} \approx \frac{R_{Earth}}{\pi} \left( \frac{P_{orbit}}{T_{eclipse}} \right) \approx \frac{R_{Earth}}{\pi} \left( \frac{27.3 \text{ days} \times 24 \text{ hr/day}}{3.5 \text{ hr}} \right) \approx \frac{R_{Earth}}{\pi} (187) \approx 59.5 R_{Earth} $ This geometric relationship reveals that the distance to the Moon is approximately 60 Earth radii. This is an exceptionally accurate figure, as the Moon's elliptical orbit varies between about 58 and 62 Earth radii. With the distance to the Moon established, its physical size can be calculated. While a lunar eclipse gives a rough visual sense that the Moon is about one-quarter the Earth's diameter, the Greeks had a more precise method. By measuring the time it takes for the full Moon to rise above the horizon—approximately two minutes—one is observing the Earth's 24-hour rotation[^MoonOrbit24] causing our line of sight to scan across the Moon's diameter. The ratio of these two minutes to 24 hours is equivalent to the ratio of the Moon's angular size to 360 degrees. Knowing the Moon's distance, this angular size can be converted into a physical radius. The angular diameter, $\alpha$, is determined by the ratio of the moonrise time ($t_{rise} \approx 2$ minutes) to the Earth's full rotation period ($T_{rot} \approx 24$ hours or 1440 minutes). In radians, this is: $ \alpha = 2\pi \left( \frac{t_{rise}}{T_{rot}} \right) = 2\pi \left( \frac{2 \text{ min}}{1440 \text{ min}} \right) \approx 0.0087 \text{ radians} \approx 0.5^{\circ} $ Using the small-angle approximation ($d \approx D\alpha$), the Moon's diameter $d_{Moon}$ is: $ d_{Moon} \approx D_{Moon} \times \alpha \approx (60 R_{Earth}) \times 0.0087 \approx 0.52 R_{Earth} $ This demonstrates the Moon's diameter to be roughly half that of the Earth (the more accurate value is ~0.27), with the discrepancy arising from the approximations used. These methods, reliant on simple geometry and naked-eye observation, provided a surprisingly accurate picture of the Earth-Moon system, with the understanding that the assumption of perfect circular orbits introduced some approximation. **Challenge:**[^Moon] From two minutes of moonrise over a 24-hour rotation, `α ≈ 2π(2/1440) ≈ 0.5°`; with `D_Moon ≈ 60·R_⊕`, estimate `d_Moon ≈ D·α`. If `t_rise = 2.2` minutes, how do `α` and the inferred `d_Moon` change? [^Moon]: Angular speed and periods: `ω = 2π/T`; relating times to angles. Small-angle approximation: angular size `α ≈ d/D` (radians). Radian measure and unit analysis; proportional reasoning. ### The Measure of the Sun Measuring the Sun's size and distance proved far more challenging. A crucial clue came from a remarkable coincidence: during a total solar eclipse, the Moon and the Sun appear to be almost exactly the same size in the sky. This implies that the ratio of their radii to their respective distances from Earth must be nearly equal:[^MoonSunAngularWidth] $ \frac{R_{Moon}}{D_{Moon}} \approx \frac{R_{Sun}}{D_{Sun}} $ The Greeks knew the ratio for the Moon, so if they could determine the distance to the Sun ($D_{Sun}$), they could immediately calculate its radius ($R_{Sun}$). But how to measure that distance? Aristarchus devised an ingenious method using the phases of the Moon.[^MoonCrescent] He reasoned that a perfect half-moon occurs not when the Moon is halfway between a new and full moon from our perspective, but when the angle formed by the Earth, Moon, and Sun is precisely 90 degrees. At this moment, our line of sight to the Moon is perpendicular to the Sun's illumination. ![[Grant Sanderson - Lunar Phase Diagram - 8 Feb 2025.png]] *The distances and moon positioned are greatly shortened in this diagram for illustrative purpose. In reality, the half moon is much closer to the mid point. If the sun's rays were perfectly parallel (infinite distance) it would be exactly the orbit's mid point.* This geometric configuration means that the time of the half-moon is slightly offset from the temporal midpoint (dotted line in diagram) between a new and full moon.[^HalfvsQuarterMoon] The magnitude of this offset, which manifests as a small angle $\theta$ at the Earth, is directly related to the Sun's distance. Simple trigonometry shows that the distance to the Sun is the distance to the Moon divided by the sine of this angle: [^SineFunctionDating] $D_{Sun} = D_{Moon} / \sin(\theta)$ ![[Grant Sanderson - Solar Theta Illustration - 8 Feb 2025.png]] Here, theory collided with the limits of technology. Aristarchus measured the time discrepancy to be six hours, when the true value is closer to thirty minutes.[^Aristarchus18mins] This error stemmed from the inability to precisely time the exact moment of a half-moon without telescopes and to keep accurate time, especially at night when their sundials were useless.[^AristarchusClocks] Aristarchus's method relies on measuring the angle $\phi$ (Sun-Earth-Moon) at the precise moment of a half-moon. The relationship is $\cos(\phi) = D_{Moon} / D_{Sun}$. The angle he needed to measure was the deviation of $\phi$ from a perfect right angle. His reported 6-hour time difference from the quarter-period of the Moon's orbit corresponds to an angular deviation of approximately $3^{\circ}$ from a right angle. He measured $\phi \approx 87^{\circ}$. $ \frac{D_{Sun}}{D_{Moon}} = \frac{1}{\cos(87^{\circ})} \approx \frac{1}{0.052} \approx 19.1 $ Consequently, his calculation was off by an order of magnitude. He concluded the Sun was 20 times more distant than the Moon (the true figure is closer to 370) and seven times larger in diameter than the Earth (the true figure is 109). ![[Grant Sanderson - Solar Theta Estimation - 8 Feb 2025.png]] The true time difference is closer to 30 minutes, corresponding to an angle $\phi \approx 89.85^{\circ}$. **Challenge:**[^SunAristarchus] Evaluate `D_⊙/D_Moon = 1/ cos φ` for `φ = 89.85°` and compare to `φ = 87°`. What angular precision in `φ` is needed to determine `D_⊙` within 10%? [^SunAristarchus]: Right-triangle trigonometry and the law of cosines at the half-moon. Inverse trigonometric functions and sensitivity near `90°`. Error propagation: small angle errors → large distance errors. #### Heliocentrism in Antiquity Despite its quantitative inaccuracy, Aristarchus's result led to a qualitatively revolutionary conclusion. By demonstrating that the Sun was significantly larger than the Earth, he argued it was illogical for the larger body to orbit the smaller one. He was thus the first known proponent of the heliocentric model, a fact later acknowledged by Copernicus in his own work. However, the Greek intellectual community dismissed his theory for sound mathematical reasons, again constrained by technology. They argued that if the Earth were orbiting the Sun, our vantage point would shift significantly over the course of a year. This motion should produce a visible stellar parallax—an apparent shift in the position of nearby stars relative to more distant ones.[^ParallaxSphereVs3D] The phenomenon is analogous to nearby trees appearing to move faster than distant mountains when viewed from a moving car. Since no such parallax was observed in the constellations between summer and winter, they concluded one of two things must be true: **either the Earth is stationary, or the stars are so astronomically far away that the parallax is too small to measure.** They opted for the more parsimonious explanation. Their logic was impeccable; their conclusion was wrong only because the universe is, in fact, unfathomably larger than they could have imagined. #### The Genius of Kepler We now jump forward nearly two millennia[^BetweenEratosthenes&Copernicus] to **Johannes Kepler[^Kepler’sPortrait] (Renaissance / Scientific Revolution, late 16th – early 17th Century CE)**. His work, an "idea of pure genius"[^Einstein’s“IdeaOfPureGenius”], is perhaps the most brilliant step on the entire ladder. ![[Grant Sanderson - Kepler Timelines - 8 Feb 2025.png]] Kepler inherited a crucial intellectual foundation from **Nicolaus Copernicus (Renaissance, 1473–1543 CE)**, who had revived the heliocentric model and, most importantly, had calculated the orbital periods[^TautologicalFullOrbit] of the planets with considerable accuracy (e.g., 687 days for Mars[^CopernicusMars687]) using centuries of observational data originating with the Babylonians. Kepler's initial ambition was to prove a speculative theory that the orbits of the six known planets were determined by the ratios of the five platonic solids nested within one another. To test this elegant hypothesis, he needed the most precise observational data available. This data belonged to **Tycho Brahe (Renaissance, 1546–1601 CE)**, an eccentric Danish aristocrat who had built an unparalleled observatory, Uraniborg, and compiled decades of meticulous planetary observations.[^BraheDataMars] ![[Tycho Brahe - Uraniborg Palace Observatory - 1598 CE.png]] *A painting of [Tycho Brahe](https://commons.wikimedia.org/wiki/Tycho_Brahe "Tycho Brahe")'s [Uraniborg](https://commons.wikimedia.org/w/index.php?title=Uraniborg&action=edit&redlink=1 "Uraniborg (page does not exist)") palace-observatory from his 1598 book "Astronomiae instauratae Mechanicae".* After a complex relationship, Kepler acquired Brahe's data[^KeplerStealingDataFromBrahe] and found that it disproved not only his own platonic solid model but also Copernicus's assumption of perfect circular orbits.[^MarsRetrograde] The data did not fit. With only a series of angular positions of planets in the sky, and with no knowledge of any of the absolute distances, how could one deduce the true shape of the orbits? His solution was a masterstroke of reasoning. He began with a simplified problem: imagine Mars were a fixed point in space. From Earth, at any given time, we can observe the direction to the Sun (inferred from the date) and the direction to this fixed Mars. The intersection of these two lines of sight would pinpoint Earth's location. By repeating this for many different dates, one could trace the shape of Earth's orbit relative to the Sun-Mars baseline.[^KeplerUseOfCelestialBodies] ![[Grant Sanderson - Mars Fixed Illustration - 8 Feb 2025.png]] Of course, Mars moves. But here, Kepler leveraged Copernicus's data: Mars returns to the exact same point in its orbit every 687 days. Kepler's insight was to treat Mars as a *temporarily* fixed reference point by analyzing Brahe's observations in time series spaced exactly 687 days apart. For each such series, he could triangulate several points on Earth's orbit. He then constructed what was essentially a massive jigsaw puzzle. Each piece of the puzzle was a set of points on Earth's orbit, conditional on Mars being at a specific, unknown location. By taking another time series starting just a day later, he created another puzzle piece corresponding to a slightly different position for Mars. Knowing that Mars's orbit must be a single, coherent path[^WhyPlanetsMoveOnAPlane], he realized all these pieces had to fit together perfectly. By painstakingly ensuring this consistency, he was able to simultaneously solve for the true shape of Earth's orbit and Mars's orbit.[^UniversalProblemSolving] ![[Grant Sanderson - Mars Periodic Illustration - 8 Feb 2025.png]] The result was the historic discovery that the orbits are not circles, but ellipses, and that planets sweep out equal areas in equal times (Kepler's First and Second Laws).[^KeplerCopernicusOrbits] This was an act of what we would now call sophisticated data analysis, built upon an intellectual inheritance stretching back through Copernicus to the astronomers of ancient Babylon.[^KeplerTriangulationEdgeCases][^EarthOrbitConsistancy] After Kepler, the *relative* structure of the solar system was understood with remarkable precision. Astronomers could draw a scale map of the planets' orbits. However, they still did not know the scale of the map itself. The absolute size of the solar system remained a mystery. The quest was now on to find a single, precise measurement of any interplanetary distance. With one such measurement, the entire scale of the solar system would lock into place, paving the way for the next rungs of the ladder: measuring the speed of light, the distance to the stars, and ultimately, the size of the observable universe. *** ### The Measure of Venus - Astronomical Unit (AU) <iframe width="560" height="315" src="https://www.youtube.com/embed/hFMaT9oRbs4?si=oLKm99zFDONCJgdG" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe> The foundational idea for measuring the distance to a planet like Venus involves the principle of **parallax**. By taking two simultaneous measurements of the planet's position from two widely separated points on Earth, one can triangulate its distance. This method gained prominence during the era of Captain Cook's voyages (mid-18th Century), which were undertaken not only for exploration but also for critical scientific missions.[^DidCaptainCookSetOutDiscoverAustralia] As an observer travels from the northern to the southern hemisphere, an object in the sky will appear to shift its position relative to the background constellations.[^TheRelativePositionOfHemispheres] This apparent shift is parallax, the same principle that underlies our binocular vision. The distance between our eyes provides a baseline that allows our brain to perceive depth for objects not significantly farther away than that baseline. To measure cosmic distances, we needed to create two "eyeballs" on opposite sides of the Earth. ![[Grant Sanderson - Jupiter Parallax Simple - 23 Feb 2025.png]] To transform this concept into a measurement, one must first know the baseline—the distance and orientation of the line connecting the two observation points. Since the size and geometry of the Earth were understood, this was achievable. The more formidable challenge was for each observer to measure their respective viewing angle to the target with extreme precision. With two angles of a triangle and the length of the side between them known, simple trigonometry allows for the calculation of the other side lengths, revealing the distance to the object. This process is deceptively simple in theory. In practice, the distances are immense. Even when Venus is at its closest approach to Earth (approximately 39 million kilometers), it is over 6,000 times farther away than the radius of the Earth. Consequently, the lines of sight from two observers on opposite sides of the planet are nearly parallel. The angular difference between these two lines is minuscule—about 1 arcminute, or one-sixtieth of a degree. This demands extraordinary precision. $\text{Angle deviation}=2\tan^{-1}(\frac{1}{6200})\approx1\text{ arc-minute}=\frac{1}{60}\cdot1\degree$ ![[Grant Sanderson - Jupiter Parallax Realistic - 23 Feb 2025.png]] A further complication was that the clocks of the era were not precise enough to perfectly synchronize observations. Moreover, there was no guarantee of clear viewing conditions at the prescribed moment. The solution to these challenges was an act of profound cleverness: using the **Transit of Venus**, a rare event where the planet passes directly in front of the Sun. Instead of trying to measure the planet's absolute position, observers could simply time the exact moment Venus first touched the edge of the Sun's disk and the moment it exited. From a northern hemisphere perspective, the path of Venus across the Sun might look a certain way. Due to parallax, an observer in the southern hemisphere would see Venus trace a slightly higher path. The goal is to measure precisely *how much* higher. Without photography, observers could not simply compare images. The Sun itself spans about 32 arcminutes of viewing angle, a known quantity. The parallax-induced deviation in Venus's path would be a fraction of a single arcminute. ![[Grant Sanderson - Jupiter Parallax Transit Arcminutes - 23 Feb 2025.png]] To measure this tiny fraction, each observer would record the total duration of the transit. Because astronomers knew the apparent speeds of both Venus and the Sun across the sky, they could calculate how long it *should* take Venus to traverse a distance equal to the Sun's diameter (approximately seven hours). By measuring the actual transit durations from two locations, they could determine the lengths of the two different chord paths across the Sun's disk. A bit of circle geometry then reveals the perpendicular distance between these two chords, which corresponds to that tiny fraction of an arcminute separating the two lines of sight. This, in turn, allows for the calculation of the absolute distance to Venus. This method, originally conceived by Edmund Halley (late 17th - early 18th Century), is a masterclass in indirect measurement. The pursuit of this data was an epic undertaking, exemplified by the story of French astronomer Guillaume Le Gentil (18th Century). Tasked with observing the 1761 transit, he was delayed by the Seven Years' War and was still at sea when the event occurred. Knowing the next transit was only eight years away, but the one after that was over a century later, he decided to wait. He set up in the Philippines for the 1769 transit, but on the day of the event, the sky was cloudy. Upon his eventual return to France, he discovered he had been declared legally dead, his wife had remarried, and his relatives had plundered his estate. The reason for such dedication was that this single measurement—the distance to Venus—was the key to establishing the scale of the entire solar system. Once known, it defined the **Astronomical Unit (AU)**—the distance from the Earth to the Sun—which became the most critical rung on the cosmic distance ladder.[^TransitOfVenus] Nearly every measurement beyond our solar system is expressed in terms of the AU. ![[Grant Sanderson - Jupiter Parallax Transit - 23 Feb 2025.png]] **Challenge:**[^VenusTransitAU] Two sites measure different transit durations `T1, T2`. From apparent solar speed, infer chord lengths `ℓ1, ℓ2` and deduce the parallax offset. Derive the relation between the chord offset and the parallax angle for a circular solar disk. [^VenusTransitAU]: Circle geometry: chords, offsets, and timing → path length. Similar triangles and parallax; spherical geometry for baselines on Earth. Converting duration differences to angular offsets; uncertainty from timing and seeing. ### The Measure of the Speed of Light - c Knowing the scale of the solar system enabled another monumental discovery. The Danish astronomer Ole Rømer (1676) was meticulously observing Jupiter and its moon, Io. Io orbits Jupiter with incredible speed, completing a revolution in just 42 hours, compared to our Moon's 28 days. Through a telescope, one can observe Io, a small white dot, repeatedly disappearing into Jupiter's massive shadow and then re-emerging. Rømer timed the precise moment of Io's reappearance from Jupiter's shadow over many cycles. He noted that while the orbits were regular, they were not perfectly clockwork. Over the course of months, he observed that Io was sometimes ahead of its predicted schedule and sometimes behind. The explanation, realized by Christiaan Huygens (late 17th Century), was that the discrepancy depended on the Earth's position relative to Jupiter. When Earth was on the same side of the Sun as Jupiter, the signal from Io's reappearance arrived earlier. When Earth was on the opposite side of its orbit, the signal was delayed by about 20 minutes.[^JupiterScheduleTIming] Huygens correctly inferred that this delay was the time it took for light to traverse the extra distance—the diameter of Earth's orbit, or two Astronomical Units. Thus, the speed of light could be calculated: $v_{light} \approx \frac{2 \text{ AU}}{20 \text{ minutes}}$.[^EllipticalOrbitStarObservation] Historically, these observations predated the precise Venus transit measurements, so the value of the AU was not yet accurately known, making their initial calculations for the speed of light appear unimpressive by modern standards. However, the discovery itself was revolutionary. At a time when light was widely assumed to travel instantaneously, this was the first concrete evidence of its finite speed. This astronomical observation laid the groundwork for more precise terrestrial experiments later on. Today, the process has come full circle: we now measure planetary distances with extreme accuracy using radar, a technique that relies on our precise knowledge of the speed of light $\textit{c}_{0}$. [^SpeedOfLight]: Time-of-flight reasoning; phase residuals vs. line-of-sight distance. Linear regression: slope of O–C (observed–calculated) times vs. Earth–Jupiter separation. Unit conversion: AU → meters; minutes → seconds; dimensional analysis. Worked problem: a $\Delta t \approx 20$ min delay across $\sim 2$ AU implies $c \approx (2\,\mathrm{AU})/(1200\,\mathrm{s}) \approx 2.5\times10^8\,\mathrm{m\,s^{-1}}$. **Challenge:** sketch how you would fit $c$ from a year-long time series of Io eclipses. ### The Measure of Nearby Stars With the AU established, the principle of parallax could be applied on a vastly larger scale to measure the distances to nearby stars. The method is identical in reasoning to the measurement of Venus, but the baseline is no longer the diameter of the Earth; it is the diameter of Earth's orbit. An astronomer measures a star's position relative to the distant, seemingly fixed background stars, and then waits six months for the Earth to travel to the opposite side of the Sun to make a second measurement. Over these six months, the line of sight to the star changes, causing its apparent position to drift subtly. This effect is observable, as seen in time-lapse imagery of our nearest stellar neighbor, Proxima Centauri. Yet, the true marvel is the subtlety of this effect and the fact that 19th-century astronomers could measure it. Let us consider the mathematics. For a star at distance $d$, the change in viewing angle is determined by the ratio of one AU to $d$. The total angular shift $\theta$ is approximately $2 \cdot \tan^{-1}\left(\frac{1 \text{ AU}}{d}\right)$. For Proxima Centauri, which is over 40 trillion kilometers (about 4.2 light-years) away, this distance is approximately 260,000 times the AU. Plugging this into the formula reveals an angular shift that is a tiny fraction of a degree. To place this in perspective: the full sky is 360 degrees. One degree is divided into 60 arcminutes. The Sun and Moon each span about 30 arcminutes. One arcminute is further divided into 60 arcseconds. The planet Jupiter, at its largest, spans about 50 arcseconds. The parallax shift for Proxima Centauri, our very closest neighboring star, is only about 1.5 arcseconds. This is the equivalent angular size of a dime viewed from 2.5 kilometers away. For all other stars, the effect is even smaller. Astronomers encode this tiny angle in a unit of distance: the parsec. By definition, a star at 1 parsec has a parallax of 1 arcsecond when observed six months apart, so in practice $ d_{\text{pc}} = \frac{1}{p(\text{arcsec})} = \frac{1000}{p(\text{mas})}, $ with the small-angle relation $p(\text{rad}) \approx \frac{1\,\text{AU}}{d}$. Useful conversions are $1\,\text{pc} \approx 206{,}265\,\text{AU}$ and $1\,\text{arcsec} = 4.8481\times10^{-6}\,\text{rad}$. Example: if $p = 10\,\text{mas}$, then $d = 100\,\text{pc}$; if $p = 1\,\text{mas}$, then $d = 1\,\text{kpc}$. The first successful measurement of stellar parallax was achieved in 1838 by Friedrich Bessel. This triumph sparked a monumental century-long effort to catalog the distances to as many stars as possible. However, this method is only effective for a tiny portion of our galaxy. [^StellarParallax]: Small-angle approximation and radians; parsecs and milliarcseconds. Separating proper motion from annual parallax; sinusoidal fits in RA/Dec. Uncertainty propagation: `σ_d` from `σ_p`; bias at low SNR. **Challenge:** a star has `p = 4.0 ± 0.2` mas. Estimate `d` and its fractional uncertainty. ### The Measure of the Milky Way To ascertain the full size of the Milky Way, a new idea was required. By the 19th century, parallax had provided distances for perhaps a thousand of the closest stars, within about 10 to 100 light-years. For these stars, astronomers knew two crucial things: their distance and their apparent brightness. This allowed them to calculate their *absolute* brightness using the **inverse-square law of light**. The law states that the apparent brightness of a light source is proportional to its true, absolute brightness, divided by the square of the distance to the observer:[^LogForumlaApparentMagnitude] $B_{apparent} \propto \frac{B_{absolute}}{d^2}$ If an observer moves twice as far away from a star, the light is spread over four times the area, and the star appears only one-quarter as bright. Astronomers recast this in magnitudes as a distance modulus relating apparent magnitude $m$, absolute magnitude $M$, and distance in parsecs: $ \mu \equiv m - M = 5\,\log_{10}\!\left( \frac{d}{10\,\text{pc}} \right). $ Equivalently, the flux–luminosity relation reads $F = \tfrac{L}{4\pi d_L^2}$, introducing the luminosity distance $d_L$; for nearby objects $d \approx d_L$. Example: for the Large Magellanic Cloud, $\mu_{\mathrm{LMC}} \approx 18.48\,\mathrm{mag}$ gives $d \approx 10^{(18.48+5)/5}\,\text{pc} \approx 49.8\,\text{kpc}$. Using this principle, astronomers could now begin to look for patterns. By plotting the known stars on a chart with color (spectral frequency) on the x-axis and absolute brightness on the y-axis, a distinct pattern emerged. This chart, known as the **Hertzsprung-Russell (H-R) Diagram**, was pioneered around 1911, built upon decades of painstaking data collection, much of it by a group of women at the Harvard College Observatory known as the "Harvard Computers." The diagram revealed that most stars fall along a distinct curve called the **main sequence**. This sequence represents stars, including our Sun, that are in the stable, hydrogen-burning phase of their life cycle. The diagram showed a powerful correlation: for main sequence stars, color is a reliable predictor of absolute brightness. Larger, brighter stars burn hotter and appear bluer, while smaller, dimmer stars burn cooler and appear redder. This discovery provided a new tool. If an astronomer could identify a very distant star as belonging to the main sequence, its color would reveal its absolute brightness. By then measuring its apparent brightness and applying the inverse-square law, its distance could be calculated. The key was classification. A star's spectrum—the pattern of light intensity across different wavelengths—contains absorption lines that reveal its chemical composition and physical state. Pioneering work by astronomers like Antonia Maury and Annie Cannon (late 19th - early 20th Century) created classification systems based on these spectra, allowing distant stars to be reliably placed within the H-R diagram's framework. This method allowed us to map our local neighborhood of the galaxy, but it too had its limits, as individual stars become too faint to resolve at galactic scales.[^MeasureEntireMilkyWay] **Challenge:** if a star is 100× fainter in flux, what is the magnitude difference (answer: 5 mag)? [^MilkyWayHR]: Inverse-square law → magnitudes and logarithms; color as a proxy for temperature. Linearizing relations on log axes; distinguishing populations (main sequence vs. giants). Selection effects and completeness: why nearby parallax samples bias trends. ### The Measure of Nearby Galaxies - Cepheid Variables To measure distances beyond the Milky Way, an even brighter light source was needed.[^WhyMilkyWayNotFullUniverse] Most stars in other galaxies blend into an unresolved haze. However, certain uniquely luminous stars, known as **Cepheid variable stars**, stand out. These are supergiants, thousands of times brighter than the Sun, whose brightness oscillates with a precise, regular period.[^CepheidsLightBloom] In the early 20th Century, the astronomer Henrietta Swan Leavitt, another of the Harvard Computers, was studying Cepheids.[^WhyLeavittStudyTheCepheidsOurOwnGalaxy] She discovered a fundamental relationship: a direct, linear law connecting a Cepheid's period of oscillation to its absolute brightness. The brighter the Cepheid, the longer its period.[^WhyCepheidsOscillate] In modern form, Leavitt’s law is written as a period–luminosity (PL) relation in a given band $X$, $ M_X = \alpha_X\,\log_{10} P + \beta_X + \gamma_X\,[\mathrm{Fe/H}], $ often combined into a reddening-free Wesenheit index, e.g. $W_I = I - R_{VI}(V - I)$, to mitigate extinction. Example: measure a Cepheid’s period $P$, read $M_X$ from the calibrated PL relation, then use $\mu = m_X - M_X$ to obtain $d$ from the distance modulus. This discovery provided the next crucial **standard candle**. If a galaxy was close enough to contain a resolvable Cepheid, an astronomer could observe it over time, measure its period, and use Leavitt's law to determine its absolute brightness.[^LeavittsLawLinear] Comparing that to its apparent brightness, the distance to the entire host galaxy could be calculated. This technique unlocked the distances to a few thousand nearby galaxies, revealing for the first time that the universe was filled with other "island universes" far beyond our own. But even this method fails for the most distant reaches of the cosmos. Other primary standard candles extend this reach. RR Lyrae stars obey metallicity-dependent relations such as $M_V = \alpha\,[\mathrm{Fe/H}] + \beta$, and in the near-infrared a PLZ form $M_{K_s} = a\,\log_{10} P + b\,[\mathrm{Fe/H}] + c$. The Tip of the Red Giant Branch (TRGB) provides a sharp standard candle in the $I$ band, commonly parameterized as $ M_{I,\,\mathrm{TRGB}} \approx -4.05 + 0.20\,\big[(V{-}I)_0 - 1.6\big]. $ Example: if a halo field shows $I_{\mathrm{TRGB}} = 24.1\,\mathrm{mag}$ and $M_{I,\mathrm{TRGB}} \approx -4.05$, then $\mu \approx 28.15$ and $d \approx 10^{(28.15+5)/5}\,\text{pc} \approx 4.3\,\text{Mpc}$. Because dust and bandpass effects bias magnitudes, practical distance work includes extinction and K-corrections via $ m_{\text{obs}} = M + \mu + A_\lambda + K_X(z), $ with $A_\lambda = R_\lambda\,E(B{-}V)$ and $K_X$ set by the source spectrum and filter $X$. Calibration anchors tie these rungs together: the Large Magellanic Cloud distance modulus $\mu_{\mathrm{LMC}} \approx 18.48\,\mathrm{mag}$ (from detached eclipsing binaries) sets a Cepheid zero-point; the geometric megamaser distance to NGC 4258, $D \approx 7.6\,\mathrm{Mpc}$, provides an independent check; and Gaia parallax measurements (with zero-point corrections) anchor nearby Cepheids and RR Lyrae directly. **Challenge:** show that a PL slope `∂M/∂log P = −3` implies a 10× longer period is 7.5 mag brighter. [^StandardCandles]: Regression on log-period vs. absolute magnitude; metallicity and color terms. Extinction and K-corrections; constructing reddening-free Wesenheit magnitudes. Hierarchical inference to combine parallaxes, anchors (LMC, NGC 4258), and SN calibrations. #### Galaxy-Scale Rungs Beyond individual stars, entire galaxies provide distance indicators. For spirals, the Tully–Fisher relation links luminosity and rotation width, $M = a\,\log_{10} W + b$ (equivalently $L \propto V_{\rm rot}^\alpha$). For ellipticals, the Faber–Jackson relation $L \propto \sigma^4$ generalizes to the Fundamental Plane, $ \log R_e = a\,\log \sigma + b\,\log \langle I \rangle_e + c, $ yielding distances from size, surface brightness, and velocity dispersion. Surface Brightness Fluctuations (SBF) convert pixel-to-pixel variance into a standard candle once calibrated versus color. Finally, water megamasers in circumnuclear Keplerian disks give geometric distances by combining proper motions $\mu$ and velocities $V_t$ with $v^2 = GM/r$. Example: a spiral’s HI width $W$ gives $M$ from the Tully–Fisher calibration; with observed $m$ one gets $\mu = m - M$ and hence $d$. [^GalaxyRungs]: Log–log scaling laws (TF, FJ, FP); fitting slopes and intercepts. Virial theorem links `M`, `σ`, `R_e`; rotation curves and inclination corrections. Fluctuation statistics for SBF; geometric modeling for maser disks. **Challenge:** if `L ∝ V^α`, show the Tully–Fisher slope in magnitudes is `−2.5·α` vs. `log W`. ### The Measure of the Expanding Universe The final rung of the ladder was placed by Edwin Hubble (1929). While measuring the spectra of distant galaxies, he noticed a systematic pattern. The characteristic absorption lines in their spectra—such as those from hydrogen—were not where they were expected to be. For most galaxies, these lines were shifted toward the red end of the spectrum. This phenomenon is known as **redshift**.[^HubbleDiscoverRedshiftGalaxies] Quantitatively, the redshift is $z = (\lambda_{\text{obs}} - \lambda_{\text{emit}})/\lambda_{\text{emit}}$. At low redshift, recessional velocity satisfies $v \approx cz$, where $c$ is the speed of light. Example: at $z = 0.01$, $v \approx cz \approx 3000\,\mathrm{km\,s^{-1}}$. Hubble plotted the measured redshift of each galaxy against its distance (as determined by Cepheids) and found another stunningly simple linear relationship, now known as **Hubble's Law**: the farther away a galaxy is, the greater its redshift.[^HubblesGraphTrends] In its local form, Hubble–Lemaître’s law reads $v = H_0 d$, so for small $z$ one may write $d \approx cz/H_0$. Scatter at very low $z$ is dominated by peculiar velocities $v_{\rm pec}$ (local motions) according to $v_{\rm obs} = H_0 d + v_{\rm pec}$. Example: with $H_0 = 70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ and $z=0.01$, $d \approx cz/H_0 \approx 43\,\mathrm{Mpc}$. At $z=0.003$, a $v_{\rm pec} \sim 300\,\mathrm{km\,s^{-1}}$ shifts an inferred distance by $\sim 4\,\mathrm{Mpc}$. This observation is now understood as direct evidence for the expansion of the universe, a prediction of Einstein's theory of general relativity.[^DoesGeneralRelativityPredictExpansion] As the fabric of spacetime expands, it stretches the wavelengths of light traveling through it. Light from a galaxy receding from us is therefore stretched, or redshifted. The farther away a galaxy is, the more space there is between it and us to expand, so it recedes faster, resulting in a greater redshift. This law provides the ultimate tool for cosmic measurement. To find the distance to a galaxy billions of light-years away, one simply measures its spectrum and its redshift. Hubble's Law then provides its distance.[^IsHubblesLawSufficient] This is the principle behind modern projects like the Sloan Digital Sky Survey, which has created 3D maps of our corner of the universe by measuring the redshifts of millions of galaxies.[^20%ObservableUniverse] These maps have revealed that galaxies are not randomly distributed but are organized into vast cosmic filaments and superstructures, a finding that aligns remarkably well with computer simulations of cosmic evolution under gravity. [^LowZExpansion]: Linear regression of `v` vs. `d`; propagating peculiar-velocity uncertainty. Series expansion: `d_L ≈ (cz/H0)[1 + ½(1−q0)z + …]` for intuition. Unit discipline and consistent frame corrections (heliocentric → CMB frame). **Challenge:** for `z = 0.02` and `H0 = 70`, estimate `d ≈ 86` Mpc; how much bias does a `300 km/s` peculiar velocity cause? #### Cosmological Distances On larger scales, cosmic expansion requires distance measures defined by the Friedmann–Lemaître model. The expansion rate evolves as $ H(z) = H_0\,\sqrt{\Omega_m(1{+}z)^3 + \Omega_k(1{+}z)^2 + \Omega_\Lambda + \Omega_r(1{+}z)^4}. $ The comoving distance is $ \chi(z) = c \int_0^z \! \frac{dz'}{H(z')}, $ and curvature enters via $S_k(\chi) = \{\sin(\sqrt{k}\,\chi)/\sqrt{k},\,\chi,\,\sinh(\sqrt{-k}\,\chi)/\sqrt{-k}\}$ for $k{>}0,=0,{<}0$. The angular diameter and luminosity distances are $ d_A = \frac{S_k(\chi)}{1{+}z}, \qquad d_L = (1{+}z)^2 d_A, $ the latter expressing Etherington reciprocity. The lookback time is $ t_L(z) = \int_0^z \! \frac{dz'}{(1{+}z')\,H(z')}. $ A useful density scale is the critical density $\rho_c = 3H_0^2/(8\pi G)$, with $\Omega_i = \rho_i/\rho_c$. Example: because $d_L = (1{+}z)^2 d_A$, at $z=1$ the luminosity distance is four times the angular-diameter distance. Modern geometric probes further constrain $H(z)$ and distances. Baryon Acoustic Oscillations (BAO) provide a standard ruler of length $r_d \approx 147\,\mathrm{Mpc}$ (comoving), yielding $D_A(z) = r_d/\theta$ transversely and $H(z)$ from the radial $\Delta z$; an isotropic combination is $ D_V(z) = \bigg[ (1{+}z)^2 D_A^2 \frac{cz}{H(z)} \bigg]^{1/3}. $ Gravitational lensing gives the Einstein radius $ \theta_E = \sqrt{\frac{4GM}{c^2}\,\frac{D_{ls}}{D_l D_s}}, $ and time-delay distances from multiple images via $\Delta t = (1{+}z_l)(D_{\Delta t}/c)\,\Delta\phi$ with $D_{\Delta t} = D_l D_s / D_{ls}$. [^CosmologicalDistances]: Calculus and differential equations: integrating `H(z)`; interpreting `Ω` parameters. Curvature via `S_k(χ)`; reciprocity `d_L = (1+z)^2 d_A` and lookback time `t_L`. Standard rulers (BAO) and lenses; combining angular and radial information. **Challenge:** in a flat, matter‑only universe (`Ω_m = 1`), derive `χ(z) = (2c/H0)[1 − 1/√(1+z)]`. ### The Measure of Mystery - Copernican principle Naturally, one must ask about the accuracy of these measurements. At the largest scales, redshift is our only tool, making it difficult to calibrate. However, a new method has recently emerged: the observation of gravitational waves from colliding black holes.[^WasItABlackHole] These events are known as **standard sirens**. Theory predicts the absolute amount of energy released in such a collision. By measuring the "loudness" of the gravitational wave signal received at our observatories, we can infer the distance to the event using an inverse-square law for gravity. For compact-binary “sirens”, the strain amplitude scales approximately as $ h \propto \frac{\mathcal{M}_c^{5/3}\,(\pi f)^{2/3}}{d_L}, $ where $\mathcal{M}_c$ is the chirp mass and $f$ is the observed frequency, yielding a direct measurement of the luminosity distance $d_L$ independent of the traditional ladder. [^StandardSirens]: Wave physics: amplitude–distance scaling; matched filtering and parameter inference. Joint constraints: combining `d_L` from GWs with host‑galaxy redshift to estimate `H0`. Model comparison and systematics: selection bias, inclination–distance degeneracy. **Challenge:** outline a Bayesian setup to infer `H0` from a set of siren events with redshift measurements. When these standard siren distances are cross-checked against redshift calculations for the same host galaxies, they match within about 10%. This is reassuring, confirming our cosmological model is broadly correct. However, that 10% discrepancy is itself a source of major controversy in modern cosmology, known as the Hubble tension.[^10%DiscrepancyInHubblesLaw] This same ~10% anomaly appears in other high-precision measurements, suggesting a potential flaw in our understanding. Is one of our laws of physics incomplete? This mystery strikes at a foundational assumption of modern science: the **Copernican principle**, which holds that the laws of physics are the same everywhere in the universe. This principle has been an article of faith since the Copernican Revolution, and it underpins our entire ability to extend the distance ladder.[^TypeIaSupernova] While this assumption has always been rewarded in the past, this persistent anomaly forces us to question it. Astronomy remains a living, evolving science, and our journey of measurement, which began with a simple walk in ancient Egypt, continues to the very edge of the observable universe.[^OmmitedTopics] --- [^EratosthenesWell]:  **Did Eratosthenes really check a local well in Alexandria?** [4:26](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=266s) This was a narrative embellishment on my part. Eratosthenes’s original work is lost to us. The most detailed contemperaneous account, by Cleomedes, gives a simplified version of the method, and makes reference only to sundials (gnomons) rather than wells. However, a secondary account of Pliny states (using [this English translation](https://www.attalus.org/translate/pliny_hn2c.html)), “Similarly it is reported that at the town of Syene, 5000 stades South of Alexandria, at noon in midsummer no shadow is cast, and that in a well made for the sake of testing this the light reaches to the bottom, clearly showing that the sun is vertically above that place at the time”. However, no mention is made of any well in Alexandria in either account. [^EratosthenesParallelRays]: **How did Eratosthenes know that the Sun was so far away that its light rays were close to parallel**? [4:50](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=290s) This was not made so clear in our discussions or in the video (other than a brief glimpse of the timeline at [18:27](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=1107s)), but Eratosthenes’s work actually came after Aristarchus, so it is very likely that Eratosthenes was aware of Aristarchus’s conclusions about how distant the Sun was from the Earth. Even if Aristarchus’s heliocentric model was disputed by the other Greeks, at least some of his other conclusions appear to have attracted some support. Also, after Eratosthenes’s time, there was further work by Greek, Indian, and Islamic astronomers (such as [Hipparchus](https://en.wikipedia.org/wiki/Hipparchus), [Ptolemy](https://en.wikipedia.org/wiki/Ptolemy), [Aryabhata](https://en.wikipedia.org/wiki/Aryabhata), and [Al-Battani](https://en.wikipedia.org/wiki/Al-Battani)) to measure the same distances that Aristarchus did, although these subsequent measurements for the Sun also were somewhat far from modern accepted values. [^SolsticeAxialRotation]: **Is it completely accurate to say that on the summer solstice, the Earth’s axis of rotation is tilted “directly towards the Sun”**? [5:17](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=317s)  Strictly speaking, “in the direction towards the Sun” is more accurate than “directly towards the Sun”; it tilts at about 23.5 degrees towards the Sun, but it is not a total 90-degree tilt towards the Sun. [^TwoTropics]: **Wait, aren’t there two tropics? The tropic of Cancer and the tropic of Capricorn?** [5:39](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=339s) Yes! This corresponds to the two summers Earth experiences, one in the Northern hemisphere and one in the Southern hemisphere. The [tropic of Cancer](https://en.wikipedia.org/wiki/Tropic_of_Cancer), at a latitude of about 23 degrees north, is where the Sun is directly overhead at noon during the Northern summer solstice (around June 21); the [tropic of Capricorn](https://en.wikipedia.org/wiki/Tropic_of_Capricorn), at a latitude of about 23 degrees south, is where the Sun is directly overhead at noon during the Southern summer solstice (around December 21). But Alexandria and Syene were both in the Northern Hemisphere, so it is the tropic of Cancer that is relevant to Eratosthenes’ calculations. [^SyeneTropic]:  **Isn’t it kind of a massive coincidence that Syene was on the tropic of Cancer?** [5:41](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=281s) Actually, Syene (now known as [Aswan](https://en.wikipedia.org/wiki/Aswan)) was about half a degree of latitude away from the tropic of Cancer, which was one of the sources of inaccuracy in Eratosthenes’ calculations.  But one should take the [“look-elsewhere effect](https://en.wikipedia.org/wiki/Look-elsewhere_effect)” into account: because the Nile cuts across the tropic of Cancer, it was quite likely to happen that the Nile would intersect the tropic near _some_ inhabited town.  It might not necessarily have been Syene, but that would just mean that Syene would have been substituted by this other town in Eratosthenes’s account. On the other hand, it was fortunate that the Nile ran from South to North, so that distances between towns were a good proxy for the differences in latitude.  Apparently, Eratosthenes actually had a more complicated argument that would also work if the two towns in question were not necessarily oriented along the North-South direction, and if neither town was on the tropic of Cancer; but unfortunately the original writings of Eratosthenes are lost to us, and we do not know the details of this more general argument. (But some variants of the method can be found in later work of [Posidonius](https://en.wikipedia.org/wiki/Posidonius), [Aryabhata](https://en.wikipedia.org/wiki/Aryabhata), and others.) Nowadays, the “[Eratosthenes experiment](https://eratosthenes.ea.gr/)” is run every year on the March equinox, in which schools at the same longitude are paired up to measure the elevation of the Sun at the same point in time, in order to obtain a measurement of the circumference of the Earth.  (The equinox is more convenient than the solstice when neither location is on a tropic, due to the simple motion of the Sun at that date.) With modern timekeeping, communications, surveying, and navigation, this is a far easier task to accomplish today than it was in Eratosthenes’ time. [^EarthSphere]: **I thought the Earth wasn’t a perfect sphere. Does this affect this calculation?** [6:30](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=390s) Yes, but only by a small amount. The centrifugal forces caused by the Earth’s rotation along its axis cause an [equatorial bulge](https://en.wikipedia.org/wiki/Equatorial_bulge) and a polar flattening so that the radius of the Earth fluctuates by about 20 kilometers from pole to equator. This sounds like a lot, but it is only about 0.3% of the mean Earth radius of 6371 km and is not the primary source of error in Eratosthenes’ calculations. [^EratosthenesRiverboatMerchants]: **Are the riverboat merchants and the “grad student” the leading theories for how Eratosthenes measured the distance from Alexandria to Syene?** [7:27](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=447s) There is some [recent research](https://doi.org/10.1007/978-3-642-18904-3) that suggests that Eratosthenes may have drawn on the work of professional [bematists](https://en.wikipedia.org/wiki/Bematist) (step measurers – a precursor to the modern profession of surveyor) for this calculation. This somewhat ruins the “grad student” joke, but perhaps should be disclosed for the sake of completeness. [^LunarMonthLength]: **How long is a “lunar month” in this context? Is it really 28 days?** [8:51](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=531s) In this context the correct notion of a lunar month is a “[synodic month](https://en.wikipedia.org/wiki/Lunar_month#Synodic_month)” – the length of a lunar cycle relative to the Sun – which is actually about 29 days and 12 hours. It differs from the “[sidereal month](https://en.wikipedia.org/wiki/Lunar_month#Sidereal_month)” – the length of a lunar cycle relative to the fixed stars – which is about 27 days and 8 hours – due to the motion of the Earth around the Sun (or the Sun around the Earth, in the geocentric model). A similar correction needs to be made around [14:59](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=899s), using the synodic month of 29 days and 12 hours rather than the “English lunar month” of 28 days (4 weeks). [^MoonOrbit24]: **Is the time taken for the Moon to complete an observed rotation around the Earth slightly less than 24 hours as claimed?** [10:47](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=647s) Actually, I made a sign error: the [lunar day](https://en.wikipedia.org/wiki/Lunar_day#Alternate_usage) (also known as a tidal day) is actually 24 hours and 50 minutes, because the Moon rotates in the same direction as the spinning of Earth around its axis. The animation therefore is also moving in the wrong direction as well (related to this, the line of sight is covering up the Moon in the wrong direction to the Moon rising at around [10:38](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=638s)). [^MoonSunAngularWidth]: **Is this really just a coincidence that the Moon and Sun have almost the same angular width?** [11:32](https://youtu.be/YdOXS_9_P4U?t=692) I believe so. First of all, the agreement is not _that_ good: due to the non-circular nature of the orbit of the Moon around the Earth, and Earth around the Sun, the angular width of the Moon actually fluctuates to be as much as 10% larger or smaller than the Sun at various times (cf. the “[supermoon](https://en.wikipedia.org/wiki/Supermoon)” phenomenon). All other known planets with known moons do not exhibit this sort of agreement, so there does not appear to be any universal law of nature that would enforce this coincidence. (This is in contrast with the empirical fact that the Moon always presents the same side to the Earth, which occurs in all other known large moons (as well as Pluto), and is well explained by the physical phenomenon of [tidal locking](https://en.wikipedia.org/wiki/Tidal_locking).) On the other hand, as the video hopefully demonstrates, the existence of the Moon was extremely helpful in allowing the ancients to understand the basic nature of the solar system. Without the Moon, their task would have been significantly more difficult; but in this hypothetical alternate universe, it is likely that modern cosmology would have still become possible once advanced technology such as telescopes, spaceflight, and computers became available, especially when combined with the modern mathematics of data science. Without giving away too many spoilers, a scenario similar to this was explored in the classic short story and novel “[Nightfall](https://en.wikipedia.org/wiki/Nightfall_\(Asimov_novelette_and_novel\))” by Isaac Asimov. [^MoonCrescent]: **Isn’t the illuminated portion of the Moon, as well as the visible portion of the Moon, slightly smaller than half of the entire Moon, because the Earth and Sun are not an infinite distance away from the Moon?** [12:58](https://youtu.be/YdOXS_9_P4U?t=778) Technically yes (and this is actually for a very similar reason to why half Moons don’t quite occur halfway between the new Moon and the full Moon); but this fact turns out to have only a very small effect on the calculations, and is not the major source of error. In reality, the Sun turns out to be about 86,000 Moon radii away from the Moon, so asserting that half of the Moon is illuminated by the Sun is actually a very good first approximation. (The Earth is “only” about 220 Moon radii away, so the visible portion of the Moon is a bit more noticeably less than half; but this doesn’t actually affect Aristarchus’s arguments much.) The angular diameter of the Sun also creates an additional thin band between the fully illuminated and fully non-illuminated portions of the Moon, in which the Sun is intersecting the lunar horizon and so only illuminates the Moon with a portion of its light, but this is also a relatively minor effect (and the midpoints of this band can still be used to define the terminator between illuminated and non-illuminated for the purposes of Aristarchus’s arguments). [^HalfvsQuarterMoon]: **What is the difference between a half Moon and a quarter Moon?** [13:27](https://youtu.be/YdOXS_9_P4U?t=807) If one divides the lunar month, starting and ending at a new Moon, into quarters (weeks), then half moons occur both near the end of the first quarter (a week after the new Moon, and a week before the full Moon), and near the end of the third quarter (a week after the full Moon, and a week before the new Moon). So, somewhat confusingly, half Moons come in two types, known as “first quarter Moons” and “third quarter Moons”. [^SineFunctionDating]: **I thought the sine function was introduced well after the ancient Greeks.** [14:49](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=889s) It’s true that the modern sine function only dates back to the [Indian](https://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%ADa%27s_sine_table) and [Islamic](https://en.wikipedia.org/wiki/Al-Khwarizmi#Trigonometry) mathematical traditions in the first millennium CE, several centuries after Aristarchus.  However, he still had Euclidean geometry at his disposal, which provided tools such as similar triangles that could be used to reach basically the same conclusions, albeit with significantly more effort than would be needed if one could use modern trigonometry. On the other hand, Aristarchus was somewhat hampered by not knowing an accurate value for $\pi$, which is also known as Archimedes’ constant: the [fundamental work of Archimedes](https://en.wikipedia.org/wiki/Pi#Polygon_approximation_era) on this constant actually took place a few decades after that of Aristarchus! [^Aristarchus18mins]: **I plugged in the modern values for the distances to the Sun and Moon and got 18 minutes for the discrepancy, instead of half an hour.** [15:17](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=917s) Yes; I quoted the wrong number here. In 1630, [Godfried Wendelen](https://en.wikipedia.org/wiki/Godfried_Wendelen) replicated Aristarchus’s experiment. With improved timekeeping and the then-recent invention of the telescope, Wendelen obtained a measurement of half an hour for the discrepancy, which is significantly better than Aristarchus’s calculation of six hours, but still a little bit off from the true value of 18 minutes. (As such, Wendelinus’s estimate for the distance to the Sun was 60% of the true value.) [^AristarchusClocks]: **Wouldn’t Aristarchus also have access to other timekeeping devices than sundials?** [15:27](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=927s Yes, for instance [clepsydrae](https://en.wikipedia.org/wiki/Water_clock) (water clocks) were available by that time; but they were of limited accuracy. It is also possible that Aristarchus could have used measurements of star elevations to also estimate time; it is not clear whether the [astrolabe](https://en.wikipedia.org/wiki/Astrolabe) or the [armillary sphere](https://en.wikipedia.org/wiki/Armillary_sphere) was available to him, but he would have had some other more primitive astronomical instruments such as the [dioptra](https://en.wikipedia.org/wiki/Dioptra) at his disposal. But again, the accuracy and calibration of these timekeeping tools would have been poor. However, most likely the more important limiting factor was the ability to determine the precise moment at which a perfect half Moon (or new Moon, or full Moon) occurs; this is extremely difficult to do with the naked eye. (The telescope would not be invented for almost two more millennia.) [^ParallaxSphereVs3D]: **Could the parallax problem be solved by assuming that the stars are not distributed in a three-dimensional space, but instead on a celestial sphere?** [17:37](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=1057s) Putting all the stars on a fixed sphere would make the parallax effects less visible, as the stars in a given portion of the sky would now all move together at the same apparent velocity – but there would still be visible large-scale distortions in the shape of the constellations because the Earth would be closer to some portions of the celestial sphere than others; there would also be variability in the brightness of the stars, and (if they were very close) the apparent angular diameter of the stars. (These problems would be solved if the celestial sphere was somehow centered around the moving Earth rather than the fixed Sun, but then this basically becomes the geocentric model with extra steps.) [^BetweenEratosthenes&Copernicus]: **Did nothing of note happen in astronomy between Eratosthenes and Copernicus?** [18:29](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=1109s) Not at all! There were significant mathematical, technological, theoretical, and observational advances by astronomers from many cultures (Greek, Islamic, Indian, Chinese, European, and others) during this time, for instance improving some of the previous measurements on the distance ladder, a better understanding of eclipses, axial tilt, and even axial precession, more sophisticated trigonometry, and the development of new astronomical tools such as the astrolabe. For instance [Al-Biruni](https://www.youtube.com/shorts/uT4-WVa-YfU) [^SineFunctionDating][^OmmitedTopics] his writings are all in Arabic, and he was nominally a subject of the Abbasid Caliphate whose rulers were Arab; but he was born in Khwarazm (in modern day Uzbekistan), and would have been a subject of either the Samanid empire or the Khrawazmian empire, both of which were largely self-governed and primarily Persian in culture and ethnic makeup, despite being technically vassals of the Caliphate. So he would have been part of what is sometimes called “Greater Persia” or “Greater Iran”. Another minor correction: while Al-Biruni was born in the tenth century, his work on the measurement of the Earth was published in the early eleventh century. **But the height of the mountain would be so small compared to the radius of the Earth! How could this method work?** Using the Taylor approximation $\cos \theta \approx 1 - \theta^2/2$, one can approximately write the relationship $R = \frac{h \cos \theta}{1-\cos \theta}$ between the mountain height $h$, the Earth radius $R$, and the dip angle  (in radians) as $R \approx 2 h / \theta^2$. The key point here is the inverse quadratic dependence on , which allows for even relatively small values of $h$ to still be realistically useful for computing $R$. Al-Biruni’s measurement of the dip angle $\theta$ was about $0.01$ radians, leading to an estimate of $R$ that is about four orders of magnitude larger than $h$, which is within ballpark at least of a typical height of a mountain (on the order of a kilometer) and the radius of the Earth (6400 kilometers). **Was the method really accurate to within a percentage point?** This is disputed, somewhat similarly to the previous calculations of Eratosthenes. Al-Biruni’s measurements were in cubits, but there were multiple incompatible types of cubit in use at the time. It has also been pointed out that atmospheric refraction effects would have created noticeable changes in the observed dip angle $\theta$. It is thus likely that the true accuracy of Al-Biruni’s method was poorer than 1%, but that this was somehow compensated for by choosing a favorable conversion between cubits and modern units. [^Kepler’sPortrait]: **Is that really Kepler’s portrait?** [18:30](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=1109s) We have [since learned](https://pubs.aip.org/physicstoday/article/74/9/10/928309/How-a-fake-Kepler-portrait-became-iconic) that this portrait was most likely painted in the 19th century, and may have been based more on Kepler’s mentor, [Michael Mästlin](https://en.wikipedia.org/wiki/Michael_Maestlin). A more commonly accepted portrait of Kepler may be found at his current [Wikipedia page](https://en.wikipedia.org/wiki/Johannes_Kepler). [^TautologicalFullOrbit]: **Isn’t it tautological to say that the Earth takes one year to perform a full orbit around the Sun?** [19:07](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=1147s) Technically yes, but this is an illustration of the philosophical concept of “[referential opacity](https://en.wikipedia.org/wiki/Opaque_context)“: the content of a sentence can change when substituting one term for another (e.g., “1 year” and “365 days”), even when both terms refer to the same object. Amusingly, the classic illustration of this, known as [Frege’s puzzles](https://en.wikipedia.org/wiki/Frege%27s_puzzles), also comes from astronomy: it is an informative statement that [Hesperus](https://en.wikipedia.org/wiki/Hesperus) (the evening star) and [Phosphorus](https://en.wikipedia.org/wiki/Phosphorus_\(morning_star\)) (the morning star, also known as Lucifer) are the same object (which nowadays we call Venus), but it is a mere tautology that Hesperus and Hesperus are the same object: changing the reference from Phosphorus to Hesperus changes the meaning. [^CopernicusMars687]: **How did Copernicus figure out the crucial fact that Mars takes 687 days to go around the Sun? Was it directly drawn from Babylonian data?** [19:10](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=1150s) Technically, Copernicus drew from tables by European astronomers that were largely based on earlier tables from the Islamic golden age, which in turn drew from earlier tables by Indian and Greek astronomers, the latter of which also incorporated data from the ancient Babylonians, so it is more accurate to say that Copernicus relied on centuries of data, at least some of which went all the way back to the Babylonians. Among all of this data was the times when Mars was in opposition to the Sun; if one imagines the Earth and Mars as being like runners going around a race track circling the Sun, with Earth on an inner track and Mars on an outer track, oppositions are analogous to when the Earth runner “laps” the Mars runner. From the centuries of observational data, such “laps” were known to occur about once every 780 days (this is known as the [synodic period](https://en.wikipedia.org/wiki/Orbital_period#Synodic_period) of Mars). Because the Earth takes roughly 365 days to perform a “lap”, it is possible to do a little math and conclude that Mars must therefore complete its own “lap” in 687 days (this is known as the [sidereal period](https://en.wikipedia.org/wiki/Orbital_period) of Mars). (See also this [post on the cosmic distance ladder Instagram](https://www.instagram.com/p/DF_fiYoogW2) for some further elaboration.) [^KeplerStealingDataFromBrahe]: **Did Kepler really steal data from Brahe?** [20:52](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=1252s) The situation is complex. When Kepler served as Brahe’s assistant, Brahe only provided Kepler with a limited amount of data, primarily involving Mars, in order to confirm Brahe’s [own geo-heliocentric model](https://en.wikipedia.org/wiki/Tychonic_system). After Brahe’s death, the data was inherited by Brahe’s son-in-law and other relatives, who intended to publish Brahe’s work separately; however, Kepler, who was appointed as Imperial Mathematician to succeed Brahe, had at least some partial access to the data, and many historians believe he secretly copied portions of this data to aid his own research before finally securing complete access to the data from Brahe’s heirs after several years of disputes. On the other hand, as intellectual property rights laws were not well developed at this time, Kepler’s actions were technically legal, if ethically questionable. [^MarsRetrograde]: **What is that funny loop in the orbit of Mars?** [21:39](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=1299s) This is known as [retrograde motion](https://en.wikipedia.org/wiki/Retrograde_and_prograde_motion). This arises because the orbital velocity of Earth (about 30 km/sec) is a little bit larger than that of Mars (about 24 km/sec). So, in opposition (when Mars is in the opposite position in the sky than the Sun), Earth will briefly overtake Mars, causing its observed position to move westward rather than eastward. But in most other times, the motion of Earth and Mars are at a sufficient angle that Mars will continue its apparent eastward motion despite the slightly faster speed of the Earth. [^KeplerUseOfCelestialBodies]: **Couldn’t one also work out the direction to other celestial objects in addition to the Sun and Mars, such as the stars, the Moon, or the other planets?  Would that have helped?** [21:59](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=1319s) Actually, the directions to the fixed stars were implicitly used in all of these observations to determine how the celestial sphere was positioned, and all the other directions were taken relative to that celestial sphere.  (Otherwise, all the calculations would be taken on a rotating frame of reference in which the unknown orbits of the planets were themselves rotating, which would have been an even more complex task.)  But the stars are too far away to be useful as one of the two landmarks to triangulate from, as they generate almost no parallax and so cannot distinguish one location from another. Measuring the direction to the Moon would tell you which portion of the lunar cycle one was in, and would determine the phase of the Moon, but this information would not help one triangulate, because the Moon’s position in the heliocentric model varies over time in a somewhat complicated fashion, and is too tied to the motion of the Earth to be a useful “landmark” to one to determine the Earth’s orbit around the Sun. In principle, using the measurements to all the planets at once could allow for some multidimensional analysis that would be more accurate than analyzing each of the planets separately, but this would require some sophisticated statistical analysis and modeling, as well as non-trivial amounts of compute – neither of which were available in Kepler’s time. [^WhyPlanetsMoveOnAPlane]: **Can you elaborate on how we know that the planets all move on a plane?** [22:57](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=1377s) The Earth’s orbit lies in a plane known as the [ecliptic](https://en.wikipedia.org/wiki/Ecliptic) (it is where the lunar and solar eclipses occur). Different cultures have divided up the ecliptic in various ways; in Western astrology, for instance, the twelve main constellations that cross the ecliptic are known as the Zodiac. The planets can be observed to only wander along the Zodiac, but not other constellations: for instance, Mars can be observed to be in Cancer or Libra, but never in Orion or Ursa Major. From this, one can conclude (as a first approximation, at least), that the planets all lie on the ecliptic. However, this isn’t perfectly true, and the planets will deviate from the ecliptic by a small angle known as the [ecliptic latitude](https://en.wikipedia.org/wiki/Ecliptic_coordinate_system#Spherical_coordinates). Tycho Brahe’s observations on these latitudes for Mars were an additional useful piece of data that helped Kepler complete his calculations (basically by suggesting how to join together the different “jigsaw pieces”), but the math here gets somewhat complicated, so the story here has been somewhat simplified to convey the main ideas. [^UniversalProblemSolving]: **What are the other universal problem solving tips?** [23:04](https://youtu.be/YdOXS_9_P4U?t=1384) Grant Sanderson has a list (in a somewhat different order) in [this previous video](https://www.youtube.com/watch?v=QvuQH4_05LI). [^KeplerTriangulationEdgeCases]: **Can one work out the position of Earth from fixed locations of the Sun and Mars when the Sun and Mars are in conjunction (the same location in the sky) or opposition (opposite locations in the sky)?** [23:28](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=1408s) Technically, these are two times when the technique of triangulation fails to be accurate; and also in the former case it is extremely difficult to observe Mars due to the proximity to the Sun. But again, following the Universal Problem Solving Tip from [23:07](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=1387s), one should initially ignore these difficulties to locate a viable method, and correct for these issues later. This [video](https://www.youtube.com/watch?v=Phscjl0u6TI) [series](https://www.youtube.com/watch?v=MprJN5teQxc) by Welch Labs goes into Kepler’s methods in more detail. [^KeplerCopernicusOrbits]: **So Kepler used Copernicus’s calculation of 687 days for the period of Mars. But didn’t Kepler discard Copernicus’s theory of circular orbits?** [24:04](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=1444s) Good question! It turns out that Copernicus’s calculations of orbital periods are quite robust (especially with centuries of data), and continue to work even when the orbits are not perfectly circular. But even if the calculations did depend on the circular orbit hypothesis, it would have been possible to use the Copernican model as a first approximation for the period, in order to get a better, but still approximate, description of the orbits of the planets. This in turn can be fed back into the Copernican calculations to give a second approximation to the period, which can then give a further refinement of the orbits. Thanks to the branch of mathematics known as [perturbation theory](https://en.wikipedia.org/wiki/Perturbation_theory), one can often make this type of iterative process converge to an exact answer, with the error in each successive approximation being smaller than the previous one. (But performing such an iteration would probably have been beyond the computational resources available in Kepler’s time; also, the foundations of perturbation theory require calculus, which only was developed several decades after Kepler.) [^BraheDataMars]: **Did Brahe have exactly 10 years of data on Mars’s positions?** [24:21](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=1461s) Actually, it was more like 17 years, but with many gaps, due both to inclement weather, as well as Brahe turning his attention to other astronomical objects than Mars in some years; also, in times of conjunction, Mars might only be visible in the daytime sky instead of the night sky, again complicating measurements. So the “jigsaw puzzle pieces” in [25:26](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=1526s) are in fact more complicated than always just five locations equally spaced in time; there are gaps and also observational errors to grapple with. But to understand the method one should ignore these complications; again, see “Universal Problem Solving Tip #1”. Even with his “idea of true genius”, it took many years of further painstaking calculation for Kepler to tease out his laws of planetary motion from Brahe’s messy and incomplete observational data. [^EarthOrbitConsistancy]: **Shouldn’t the Earth’s orbit be spread out at perihelion and clustered closer together at aphelion, to be consistent with Kepler’s laws?** [26:44](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=1604s) Yes, you are right; there was a coding error here. [^Einstein’s“IdeaOfPureGenius”]: **What is the reference for Einstein’s “idea of pure genius”?** [26:53](https://youtu.be/YdOXS_9_P4U?t=1613) Actually, the precise quote was “an idea of true genius”, and can be found in the introduction to Carola Baumgardt’s “[Life of Kepler](https://archive.org/details/johanneskeplerli0000caro/page/n5/mode/2up)“. [^DidCaptainCookSetOutDiscoverAustralia]: **Did Captain Cook set out to discover Australia?** [1:13](https://youtu.be/hFMaT9oRbs4?t=73) One of the objectives of Cook’s first voyage was to discover the hypothetical continent of [Terra Australis](https://en.wikipedia.org/wiki/Terra_Australis). This was considered to be distinct from Australia, which at the time was known as [New Holland](https://en.wikipedia.org/wiki/New_Holland_\(Australia\)). As this name might suggest, prior to Cook’s voyage, the northwest coastline of New Holland had been explored by the Dutch; Cook instead explored the eastern coastline, naming this portion New South Wales. The entire continent was later renamed to Australia by the British government, following a suggestion of Matthew Flinders; and the concept of Terra Australis was abandoned. [^TheRelativePositionOfHemispheres]: **The relative position of the Northern and Southern hemisphere observations is reversed from those earlier in the video.** [4:40](https://youtu.be/hFMaT9oRbs4?t=280) Yes, this was a slight error in the animation; the labels here should be swapped for consistency of orientation. [^TransitOfVenus]: **So, when did they finally manage to measure the transit of Venus, and use this to compute the astronomical unit?** [7:06](https://youtu.be/hFMaT9oRbs4?t=426) While Le Gentil had the misfortune to not be able to measure either the 1761 or 1769 transits, other expeditions of astronomers (led by Dixon-Mason, Chappe d’Auteroche, and Cook) did take measurements of one or both of these transits with varying degrees of success, with the measurements of Cook’s team of the 1769 transit in Tahiti being of particularly high quality. All of this data was assembled later by Lalande in 1771, leading to the most accurate measurement of the astronomical unit at the time (within 2.3% of modern values, which was about three times more accurate than any previous measurement). [^JupiterScheduleTIming]: **What does it mean for the transit of Io to be “twenty minutes ahead of schedule” when Jupiter is in opposition (Jupiter is opposite to the Sun when viewed from the Earth)?** [8:53](https://youtu.be/hFMaT9oRbs4?t=533) Actually, it should be halved to “ten minutes ahead of schedule”, with the transit being “ten minutes behind schedule” when Jupiter is in conjunction, with the net discrepancy being twenty minutes (or actually closer to 16 minutes when measured with modern technology). Both transits are being compared against an idealized periodic schedule in which the transits are occuring at a perfectly regular rate (about 42 hours), where the period is chosen to be the best fit to the actual data. This discrepancy is only noticeable after carefully comparing transit times over a period of months; at any given position of Jupiter, the Doppler effects of Earth moving towards or away from Jupiter would only affect shift each transit by just a few seconds compared to the previous transit, with the delays or accelerations only becoming cumulatively noticeable after many such transits. Also, the presentation here is oversimplified: at times of conjunction, Jupiter and Io are too close to the Sun for observation of the transit. Rømer actually observed the transits at other times than conjunction, and Huygens used more complicated trigonometry than what was presented here to infer a measurement for the speed of light in terms of the astronomical unit (which they had begun to measure a bit more accurately than in Aristarchus’s time; see the FAQ entry for [15:17](https://www.youtube.com/watch?v=YdOXS_9_P4U&t=917s) in the first video). [^EllipticalOrbitStarObservation]: **Shouldn’t one have to account for the elliptical orbit of the Earth, as well as the proper motion of the star being observed, or the effects of general relativity?** [10:34](https://en.wikipedia.org/wiki/Proper_motion) Yes; the presentation given here is a simplified one to convey the idea of the method, but in the most advanced parallax measurements, such as the ones taken by the [Hipparcos](https://en.wikipedia.org/wiki/Hipparcos) and [Gaia](https://en.wikipedia.org/wiki/Gaia_\(spacecraft\)) spacecraft, these factors are taken into account, basically by taking as many measurements (not just two) as possible of a single star, and locating the best fit of that data to a multi-parameter model that incorporates the (known) orbit of the Earth with the (unknown) distance and motion of the star, as well as additional gravitational effects from other celestial bodies, such as the Sun and other planets. [^LogForumlaApparentMagnitude]: **The formula I was taught for apparent magnitude of stars looks a bit different from the one here.** [14:53](https://youtu.be/hFMaT9oRbs4?t=893) This is because astronomers use a logarithmic scale to measure both [apparent magnitude](https://en.wikipedia.org/wiki/Apparent_magnitude) $m$ and [absolute magnitude](https://en.wikipedia.org/wiki/Absolute_magnitude) $M$. If one takes the logarithm of the inverse square law in the video, and performs the normalizations used by astronomers to define magnitude, one arrives at the standard relation $M = m - 5 \log_{10} d_{pc} + 5$ between absolute and apparent magnitude. But this is an oversimplification, most notably due to neglect of the effects of [extinction](https://en.wikipedia.org/wiki/Extinction_\(astronomy\)) effects caused by interstellar dust. This is not a major issue for the relatively short distances observable via parallax, but causes problems at larger scales of the ladder (see for instance the FAQ entry here for [18:08](https://youtu.be/hFMaT9oRbs4?t=1088)). To compensate for this, one can work in multiple frequencies of the spectrum (visible, x-ray, radio, etc.), as some frequencies are less susceptible to extinction than others. From the discrepancies between these frequencies one can infer the amount of extinction, leading to “dust maps” that can then be used to facilitate such corrections for subsequent measurements in the same area of the universe. (More generally, the trend in modern astronomy is towards “[multi-messenger astronomy](https://en.wikipedia.org/wiki/Multi-messenger_astronomy)” in which one combines together very different types of measurements of the same object to obtain a more accurate understanding of that object and its surroundings.) [^MeasureEntireMilkyWay]: **Can we really measure the entire Milky Way with this method?** [18:08](https://youtu.be/hFMaT9oRbs4?t=1088) Strictly speaking, there is a “[zone of avoidance](https://en.wikipedia.org/wiki/Zone_of_Avoidance)” on the far side of the Milky way that is very difficult to measure in the visible portion of the spectrum, due to the large amount of intervening stars, dust, and even a supermassive black hole in the galactic center. However, in recent years it has become possible to explore this zone to some extent using the radio, infrared, and x-ray portions of the spectrum, which are less affected by these factors. [^WhyMilkyWayNotFullUniverse]: **How did astronomers know that the Milky Way was only a small portion of the entire universe?** [18:19](https://youtu.be/hFMaT9oRbs4?t=1099) This issue was the topic of the “[Great Debate](https://en.wikipedia.org/wiki/Great_Debate_\(astronomy\))” in the early twentieth century. It was only with the work of Hubble using Leavitt’s law to measure distances to Magellanic clouds and “spiral nebulae” (that we now know to be other galaxies), building on earlier work of Leavitt and Hertzsprung, that it was conclusively established that these clouds and nebulae in fact were at much greater distances than the diameter of the Milky Way. [^CepheidsLightBloom]: **How can one compensate for light blending effects when measuring the apparent magnitude of Cepheids?** [18:45](https://youtu.be/hFMaT9oRbs4?t=1125) This is a non-trivial task, especially if one demands a high level of accuracy. Using the highest resolution telescopes available (such as HST or JWST) is of course helpful, as is switching to other frequencies, such as near-infrared, where Cepheids are even brighter relative to nearby non-Cepheid stars. One can also apply sophisticated statistical methods to fit to models of the point spread of light from unwanted sources, and use nearby measurements of the same galaxy without the Cepheid as a reference to help calibrate those models. Improving the accuracy of the Cepheid portion of the distance ladder is an ongoing research activity in modern astronomy. [^WhyCepheidsOscillate]: **What is the mechanism that causes Cepheids to oscillate?** [18:54](https://youtu.be/hFMaT9oRbs4?t=1134) For most stars, there is an equilibrium size: if the star’s radius collapses, then the reduced potential energy is converted to heat, creating pressure to pushing the star outward again; and conversely, if the star expands, then it cools, causing a reduction in pressure that no longer counteracts gravitational forces. But for Cepheids, there is an additional mechanism called the [kappa mechanism](https://en.wikipedia.org/wiki/Kappa%E2%80%93mechanism): the increased temperature caused by contraction increases ionization of helium, which drains energy from the star and accelerates the contraction; conversely, the cooling caused by expansion causes the ionized helium to recombine, with the energy released accelerating the expansion. If the parameters of the Cepheid are in a certain “instability strip”, then the interaction of the kappa mechanism with the other mechanisms of stellar dynamics create a periodic oscillation in the Cepheid’s radius, which increases with the mass and brightness of the Cepheid. For a recent re-analysis of Leavitt’s original Cepheid data, see [this paper](https://arxiv.org/abs/2502.17438). [^WhyLeavittStudyTheCepheidsOurOwnGalaxy]: **Did Leavitt mainly study the Cepheids in our own galaxy?** [19:10](https://youtu.be/hFMaT9oRbs4?t=1150) This was an inaccuracy in the presentation. Leavitt’s original breakthrough paper studied Cepheids in the [Small Magellanic Cloud](https://en.wikipedia.org/wiki/Small_Magellanic_Cloud). At the time, the distance to this cloud was not known; indeed, it was a matter of debate whether this cloud was in the Milky Way, or some distance away from it. However, Leavitt (correctly) assumed that all the Cepheids in this cloud were roughly the same distance away from our solar system, so that the apparent brightness was proportional to the absolute brightness. This gave an uncalibrated form of Leavitt’s law between absolute brightness and period, subject to the (then unknown) distance to the Small Magellanic Cloud. After Leavitt’s work, there were several efforts (by Hertzsprung, Russell, and Shapley) to calibrate the law by using the few Cepheids for which other distance methods were available, such as parallax. (Main sequence fitting to the Hertzsprung-Russell diagram was not directly usable, as Cepheids did not lie on the main sequence; but in some cases one could indirectly use this method if the Cepheid was in the same stellar cluster as a main sequence star.) Once the law was calibrated, it could be used to measure distances to other Cepheids, and in particular to compute distances to extragalactic objects such as the Magellanic clouds. [^LeavittsLawLinear]: **Was Leavitt’s law really a linear law between period and luminosity?** [19:15](https://youtu.be/hFMaT9oRbs4?t=1155) Strictly speaking, the [period-luminosity relation](https://en.wikipedia.org/wiki/Period-luminosity_relation) commonly known as Leavitt’s law was a linear relation between the absolute magnitude of the Cepheid and the _logarithm_ of the period; undoing the logarithms, this becomes a [power law](https://en.wikipedia.org/wiki/Power_law) between the luminosity and the period. [^HubbleDiscoverRedshiftGalaxies]: **Was Hubble the one to discover the redshift of galaxies?** [20:26](https://youtu.be/hFMaT9oRbs4?t=1226) This was an error on my part; Hubble was using earlier work of [Vesto Slipher](https://en.wikipedia.org/wiki/Vesto_Slipher) on these redshifts, and combining it with his own measurements of distances using Leavitt’s law to arrive at the law that now bears his name; he was also assisted in his observations by [Milton Humason](https://en.wikipedia.org/wiki/Milton_L._Humason). It should also be noted that Georges Lemaître had also independently arrived at essentially the same law a few years prior, but his work was published in a somewhat obscure journal and did not receive broad recognition until some time later. [^HubblesGraphTrends]: **Hubble’s original graph doesn’t look like a very good fit to a linear law.** [20:37](https://youtu.be/hFMaT9oRbs4?t=1237) Hubble’s original data was somewhat noisy and inaccurate by modern standards, and the redshifts were affected by the [peculiar velocities](https://en.wikipedia.org/wiki/Peculiar_velocity) of individual galaxies in addition to the expanding nature of the universe. However, as the data was extended to more galaxies, it became increasingly possible to compensate for these effects and obtain a much tighter fit, particularly at larger scales where the effects of peculiar velocity are less significant. See for instance [this article from 2015](https://www.pnas.org/doi/10.1073/pnas.1424299112) where Hubble’s original graph is compared with a more modern graph. This more recent graph also reveals a slight nonlinear correction to Hubble’s law at very large scales that has led to the remarkable discovery that the expansion of the universe is in fact accelerating over time, a phenomenon that is attributed to a positive [cosmological constant](https://en.wikipedia.org/wiki/Cosmological_constant) (or perhaps a more complex form of [dark energy](https://en.wikipedia.org/wiki/Dark_energy) in the universe). On the other hand, even with this nonlinear correction, there continues to be a roughly 10% discrepancy of this law with predictions based primarily on the cosmic microwave background radiation; see the FAQ entry for [23:49](https://youtu.be/hFMaT9oRbs4?t=1429). [^DoesGeneralRelativityPredictExpansion]: **Does general relativity alone predict an uniformly expanding universe?** [20:46](https://youtu.be/hFMaT9oRbs4?t=1246) This was an oversimplification. [Einstein’s equations of general relativity](https://en.wikipedia.org/wiki/Einstein_field_equations) contain a parameter $\Lambda$, known as the [cosmological constant](https://en.wikipedia.org/wiki/Cosmological_constant), which currently is only computable indirectly from fitting to experimental data. But even with this constant fixed, there are multiple solutions to these equations (basically because there are multiple possible initial conditions for the universe). For the purposes of cosmology, a particularly successful family of solutions are the solutions given by the [Lambda-CDM model](https://en.wikipedia.org/wiki/Lambda-CDM_model). This family of solutions contains additional parameters, such as the density of dark matter in the universe. Depending on the precise values of these parameters, the universe could be expanding or contracting, with the rate of expansion or contraction either increasing, decreasing, or staying roughly constant. But if one fits this model to all available data (including not just red shift measurements, but also measurements on the cosmic microwave background radiation and the spatial distribution of galaxies), one deduces a version of Hubble’s law which is nearly linear, but with an additional correction at very large scales; see the next item of this FAQ. [^IsHubblesLawSufficient]: **Is Hubble’s original law sufficiently accurate to allow for good measurements of distances at the scale of the observable universe?** [21:07](https://youtu.be/hFMaT9oRbs4?t=1267) Not really; as mentioned in the end of the video, there were additional efforts to cross-check and calibrate Hubble’s law at intermediate scales between the range of Cepheid methods (about 100 million light years) and observable universe scales (about 100 billion light years) by using further “[standard candles](https://en.wikipedia.org/wiki/Cosmic_distance_ladder#Standard_candles)” than Cepheids, most notably [Type Ia supernovae](https://en.wikipedia.org/wiki/Type_Ia_supernova) (which are bright enough and predictable enough to be usable out to about 10 billion light years), the [Tully-Fisher relation](https://en.wikipedia.org/wiki/Tully%E2%80%93Fisher_relation) between the luminosity of a galaxy and its rotational speed, and [gamma ray bursts](https://en.wikipedia.org/wiki/Gamma-ray_burst). It turns out that due to the accelerating nature of the universe’s expansion, Hubble’s law is not completely linear at these large scales; this important correction cannot be discerned purely from Cepheid data, but also requires the other standard candles, as well as fitting that data (as well as other observational data, such as the cosmic microwave background radiation) to the cosmological models provided by general relativity (with the best fitting models to date being some version of the [Lambda-CDM model](https://en.wikipedia.org/wiki/Lambda-CDM_model)). On the other hand, a naive linear extrapolation of Hubble’s original law to all larger scales does provide a very rough picture of the observable universe which, while too inaccurate for cutting edge research in astronomy, does give some general idea of its large-scale structure. [^20%ObservableUniverse]: **Where did this guess of the observable universe being about 20% of the full universe come from?** [21:15](https://youtu.be/hFMaT9oRbs4?t=1275) There are some ways to get a lower bound on the size of the entire universe that go beyond the edge of the observable universe. One is through analysis of the [cosmic microwave background radiation](https://en.wikipedia.org/wiki/Cosmic_microwave_background) (CMB), that has been carefully mapped out by several satellite observatories, most notably [WMAP](https://en.wikipedia.org/wiki/Wilkinson_Microwave_Anisotropy_Probe) and [Planck](https://en.wikipedia.org/wiki/Planck_\(spacecraft\)). Roughly speaking, a universe that was less than twice the size of the observable universe would create certain periodicities in the CMB data; such periodicities are not observed, so this provides a lower bound (see for instance [this paper](https://arxiv.org/abs/astro-ph/0604616) for an example of such a calculation). The 20% number was a guess based on my vague recollection of these works, but there is no consensus currently on what the ratio truly is; there are some proposals that the entire universe is in fact several orders of magnitude larger than the observable one, and possibly even infinite. The situation is somewhat analogous to Aristarchus’s measurement of the distance to the Sun, which was very sensitive to a small angle (the half-moon discrepancy). Here, the predicted size of the universe under the standard cosmological model is similarly dependent in a highly sensitive fashion on a measure $\Omega_k$ of the flatness of the universe which, for reasons still not fully understood (but likely caused by some sort of inflation mechanism), happens to be extremely close to zero. As such, predictions for the size of the universe remain highly volatile at the current level of measurement accuracy. [^WasItABlackHole]: **Was it a black hole collision that allowed for an independent measurement of Hubble’s law?** [23:44](https://youtu.be/hFMaT9oRbs4?t=1424) This was a slight error in the presentation. While the first gravitational wave observation by LIGO in 2015 was of a black hole collision, it did not come with an electromagnetic counterpart that allowed for a redshift calculation that would yield a Hubble’s law measurement. However, a later collision of neutron stars, observed in 2017, did come with an associated kilonova in which a redshift was calculated, and led to a Hubble measurement which was independent of most of the rungs of the distance ladder. [^10%DiscrepancyInHubblesLaw]: **Where can I learn more about this 10% discrepancy in Hubble’s law?** [23:49](https://youtu.be/hFMaT9oRbs4?t=1429) This is known as the [Hubble tension](https://youtu.be/hFMaT9oRbs4?t=1431) (or, in more sensational media, the “crisis in cosmology”): roughly speaking, the various measurements of Hubble’s constant (either from climbing the cosmic distance ladder, or by fitting various observational data to standard cosmological models) tend to arrive at one of two values, that are about 10% apart from each other. The values based on gravitational wave observations are currently consistent with both values, due to significant error bars in this extremely sensitive method; but other more mature methods are now of sufficient accuracy that they are basically only consistent with one of the two values. Currently there is no consensus on the origin of this tension: possibilities include systemic biases in the observational data, subtle statistical issues with the methodology used to interpret the data, a correction to the standard cosmological model, the influence of some previously undiscovered law of physics, or some partial breakdown of the Copernican principle. For an accessible recent summary of the situation, see [this video](https://youtu.be/I84skRsDO4M?t=649) by Becky Smethurst (“Dr. Becky”). [^TypeIaSupernova]: **So, what is a Type Ia supernova and why is it so useful in the distance ladder?** [24:49](https://youtu.be/hFMaT9oRbs4?t=1489) A [Type Ia supernova](https://en.wikipedia.org/wiki/Type_Ia_supernova) occurs when a [white dwarf](https://en.wikipedia.org/wiki/White_dwarf) in a binary system draws more and more mass from its companion star, until it reaches the [Chandrasekhar limit](https://en.wikipedia.org/wiki/Chandrasekhar_limit), at which point its gravitational forces are strong enough to cause a collapse that increases the pressure to the point where a supernova is triggered via a process known as [carbon detonation](https://en.wikipedia.org/wiki/Carbon_detonation). Because of the universal nature of the Chandrasekhar limit, all such supernovae have (as a first approximation) the same absolute brightness and can thus be used as standard candles in a similar fashion to Cepheids (but without the need to first measure any auxiliary observable, such as a period). But these supernovae are also far brighter than Cepheids, and can so this method can be used at significantly larger distances than the Cepheid method (roughly speaking it can handle distances of up to ~10 billion light years, whereas Cepheids are reliable out to ~100 million light years). Among other things, the supernovae measurements were the key to detecting an important nonlinear correction to Hubble’s law at these scales, leading to the remarkable conclusion that the expansion of the universe is in fact accelerating over time, which in the [Lambda-CDM model](https://en.wikipedia.org/wiki/Lambda-CDM_model) corresponds to a positive [cosmological constant](https://en.wikipedia.org/wiki/Cosmological_constant), though there are more complex “[dark energy](https://en.wikipedia.org/wiki/Dark_energy)” models that are also proposed to explain this acceleration. [^OmmitedTopics]: **Besides [Type Ia supernovae](https://en.wikipedia.org/wiki/Type_Ia_supernova), I felt that a lot of other topics relevant to the modern distance ladder (e.g., the [cosmic microwave background radiation](https://en.wikipedia.org/wiki/Cosmic_microwave_background), the [Lambda CDM model](https://en.wikipedia.org/wiki/Lambda-CDM_model), [dark matter](https://en.wikipedia.org/wiki/Dark_matter), [dark energy](https://en.wikipedia.org/wiki/Dark_energy), [inflation](https://en.wikipedia.org/wiki/Cosmic_inflation), [multi-messenger astronomy](https://en.wikipedia.org/wiki/Multi-messenger_astronomy), etc.) were omitted.** [24:54](https://youtu.be/hFMaT9oRbs4?t=1494) This is partly due to time constraints, and the need for editing to tighten the narrative, but was also a conscious decision on my part. Advanced classes on the distance ladder will naturally focus on the most modern, sophisticated, and precise ways to measure distances, backed up by the latest mathematics, physics, technology, observational data, and cosmological models. However, the focus in this video series was rather different; we sought to portray the cosmic distance ladder as evolving in a fully synergestic way, across many historical eras, with the evolution of mathematics, science, and technology, as opposed to being a mere byproduct of the current state of these other disciplines. As one specific consequence of this change of focus, we emphasized the _first_ time any rung of the distance ladder was achieved, at the expense of more accurate and sophisticated later measurements at that rung. For instance, refinements in the measurement of the radius of the Earth since Eratosthenes, improvements in the measurement of the astronomical unit between Aristarchus and Cook, or the refinements of Hubble’s law and the cosmological model of the universe in the twentieth and twenty-first centuries, were largely omitted (though some of the answers in this FAQ are intended to address these omissions). Many of the topics not covered here (or only given a simplified treatment) are discussed in depth in other expositions, including other Youtube videos. I would welcome suggestions from readers for links to such resources in the comments to this post. Here is a partial list: - “[Eratosthenes](https://www.youtube.com/watch?v=G8cbIWMv0rI)” – Cosmos (Carl Sagan), video posted Apr 24, 2009 (originally released Oct 1, 1980, as part of the episode “The Shores of the Cosmic Ocean”). - “[How Far Away Is It](https://www.youtube.com/watch?v=HgNJwg2GISs&list=PLpH1IDQEoE8QWWTnWG5cK4ePCqg9W2608)” – David Butler, a multi-part series beginning Aug 16 2013. - “[How the Bizarre Path of Mars Reshaped Astronomy [Kepler’s Laws Part 1]](https://www.youtube.com/watch?v=Phscjl0u6TI)“, Welch Labs, May 8, 2024. See also [Part 2](https://www.youtube.com/watch?v=MprJN5teQxc). - “[An ASTROPHYSICIST’S TOP 5 space news stories of 2024](https://youtu.be/I84skRsDO4M?t=649)“, Becky Smethurst (Dr. Becky), Dec 26, 2024 – covers the Hubble tension as one of the stories. - “[Measuring the Earth… from a vacation photo](https://www.youtube.com/watch?v=038AkmPvltA)“, George Lowther (Almost sure), Feb 22 2025. - “[How Did This Ancient Genius Measure The Sun?](https://www.youtube.com/watch?v=shkcVDHOvAI)“, Ben Syversen, Feb 28 2025.