3Blue1Brown - Terence Tao on how we measure the cosmos | Part 1
The discussion begins with Terence Tao highlighting the genius of Kepler's deduction of Earth's orbit shape. The video then delves into the historical methods used to measure cosmic distances, starting with Eratosthenes' calculation of Earth's circumference using the angle of the Sun's rays in different locations. This method relied on basic geometry and the assumption of a spherical Earth. The video explains how lunar eclipses were used to estimate the distance to the Moon by observing the Earth's shadow on the Moon. Aristarchus's method for estimating the distance to the Sun involved measuring the angle during a half Moon, although his calculations were significantly off due to technological limitations. The narrative progresses to Kepler's work, which used Tycho Brahe's data to deduce the elliptical orbits of planets, marking a significant advancement in understanding planetary motion. Kepler's method involved observing Mars over time and using its periodic return to the same position to infer Earth's orbit. This approach, despite not providing absolute distances, allowed for the relative sizing of planetary orbits. The video concludes by noting that while Kepler's work laid the groundwork, the exact scale of the solar system remained unknown until later advancements.
Key Points:
- Eratosthenes calculated Earth's circumference using the Sun's angle at different locations, demonstrating early use of geometry in astronomy.
- Lunar eclipses helped estimate the Moon's distance by observing Earth's shadow, showcasing the use of natural phenomena in measurements.
- Aristarchus attempted to measure the Sun's distance using the Moon's phases, but was limited by the technology of his time.
- Kepler used Tycho Brahe's data to deduce planetary orbits as ellipses, a breakthrough in understanding celestial mechanics.
- Despite knowing orbital shapes, the absolute scale of the solar system was unknown until later precise measurements.
Details:
1. 🚀 Terence Tao and the Cosmic Journey
1.1. Introduction to Terence Tao and Einstein's Admiration
1.2. Tao's Fascination with Astronomy
1.3. Effective Science Communication
1.4. Understanding Cosmic Distances through Mathematics
2. 🌍 Measuring Earth's Size with Shadows
- Eratosthenes was the first known person to measure Earth's size using shadows and geometry, achieving an accuracy of approximately 10% without modern technology.
- He observed that in Syene, the Sun was directly overhead at noon on the summer solstice, while in Alexandria, the Sun was about 7 degrees off vertical.
- The distance between Syene and Alexandria was approximately 5000 stadia, equivalent to about 500 miles, though the exact conversion to modern units is uncertain.
- By calculating the angle of the Sun's shadow and knowing the distance between the two locations, Eratosthenes estimated Earth's circumference based on the geometric principle that the angle of the shadow corresponds to a segment of the Earth's 360-degree circumference.
- This method relies on the assumption that the Sun's rays are parallel due to its distance, making the angle of the shadow a reliable measure of Earth's curvature.
- Eratosthenes' calculation shows the innovative use of basic observational tools and understanding of geometry in ancient times to deduce large-scale geographic measurements.
- The accuracy of his findings is subject to conversion discrepancies between ancient stadia and modern miles, with estimates varying based on which conversion is used.
3. 🌕 Lunar Eclipses and Moon Measurements
- Lunar eclipses, lasting no longer than four hours, allow for calculations of the Earth-Moon distance using Earth's shadow, which spans approximately twice the Earth's radius.
- By comparing the duration of a lunar eclipse to the Moon's 28-day orbit, Aristarchus calculated the Moon's distance to be about 60 Earth radii, with variations between 58 and 62 Earth radii noted.
- The Moon's diameter is roughly a quarter of Earth's, a measurement challenged by the lack of photographic technology at the time.
- The rise time of the full Moon, approximately two minutes, aids in determining the size and distance ratio between Earth and the Moon.
- Ancient Greeks, despite limited technology, achieved reasonable estimates of the Moon's size and distance by acknowledging its elliptical orbit.
- Aristarchus used mathematical reasoning to relate the size of the Earth's shadow during an eclipse to the distance and size of the Moon, pioneering early astronomical measurement techniques.
4. ☀️ Solar Mysteries and Ancient Calculations
- Ancient Greeks developed methodologies to estimate the size and distance of celestial bodies like the Sun by comparing them to the Moon, leveraging observations of solar and lunar eclipses.
- The unique phenomenon where the Sun and Moon appear the same size during a solar eclipse enabled ancient astronomers to perform calculations regarding their relative sizes and distances.
- By understanding the Moon's phases and applying geometric principles, Greeks could calculate the Sun's distance. Aristarchus attempted to estimate this distance using the timing of the half Moon, although his calculations were off due to technological limitations.
- Despite inaccuracies, such as Aristarchus's estimate of the Sun being only 20 times the distance of the Moon, the mathematical approaches provided foundational understanding that the Sun is significantly farther than the Moon.
- The absence of precise instruments like clocks and telescopes limited the accuracy of these calculations but showcased impressive mathematical reasoning and ingenuity for the time.
5. 🔭 Aristarchus and the Heliocentric Leap
- Aristarchus was the first to propose the heliocentric model, suggesting the Earth revolves around the Sun, challenging the geocentric view.
- He estimated the Sun was 7 times larger than Earth, highlighting the significance of the Sun, although the actual size is 109 times larger.
- His model implied a universe far larger than previously thought, suggesting it was thousands of times larger, though it's actually billions and trillions of times larger.
- The Greeks dismissed the model due to the lack of observable parallax, which they argued would show a shift in star positions if Earth moved.
- This dismissal was based on the assumption that observable parallax was necessary, despite the model being mathematically sound.
- Aristarchus' work laid foundational ideas for future acceptance of the heliocentric model, even though it wasn't accepted until much later in history.
6. 🔍 Kepler's Genius: Unraveling Planetary Orbits
- Kepler built on Copernicus' model, which proposed that planets move around the Sun in circular orbits. Copernicus determined the orbital periods of planets, such as Mars taking 687 days to orbit the Sun.
- Kepler sought to determine the relative sizes of planetary orbits, initially proposing a theory linking these orbits to the five platonic solids, but found his theory did not fit the data.
- Kepler used observations from Tycho Brahe, who had decades of planetary data, to test his theories. Kepler discovered neither his theory nor Copernicus' could fit Brahe's data using circular orbits.
- Kepler deduced that planetary orbits are not circular but elliptical, by analyzing angles between Earth, Mars, and the Sun over multiple observations.
- Kepler used the known period of Mars' orbit (729 days) to develop a method of triangulating the orbits of planets relative to each other, using data series spaced by the Martian year.
- Kepler's analysis revealed that planets sweep out equal areas in equal times in their elliptical orbits, leading to the formulation of his second law of planetary motion.