How Did We Measure the Distance to the Sun?
– The Days the World Waited for Venus to Cross the Sun
"The Sun is about 150 million km away." This number sits quietly in textbooks, but you can't measure it with a ruler. The key was a day that comes only about twice a century, when Venus crosses the Sun as a tiny black dot. To avoid missing it, astronomers sailed to the far side of the Earth.
Stretch your arm straight out and raise your thumb. Look with only your right eye, then with only your left. Your thumb seems to jump left and right against the scenery behind it.
Now bend your arm, bring your thumb closer to your face, and do the same. The jump is bigger. The closer the object, the bigger the jump; the farther, the smaller. From that jump, you can work backward to the distance.
The distance to the Sun works on the same principle. But the Sun is so far away that, instead of two eyes, we needed two people, one in the north and one in the south of the Earth. And we needed Venus as a marker.
Why can Venus's transit give us the distance? Just two reasons
About 400 years ago, Kepler found the rules of planetary motion, and a "scale drawing" of the solar system became possible. Venus's path is about 0.72 times Earth's, and Mars's is about 1.5 times. What was missing was a single number: how many km one unit on the scale drawing is.
When a person in the north and a person in the south watch at the same time, Venus's track across the Sun looks slightly shifted up or down. If you know the size of the shift and the distance between the two people, the same logic as the thumb experiment gives the distance.
In short, the shape of the solar system was already known, and the one remaining job was to set the scale. Venus's transit was meant to do it. The method was proposed in 1716 by Edmond Halley, famous for his comet.
How much does Venus's path shift between north and south?
Look at the left of Figure 1. From an observing site in the northern part of the Earth, Venus appears slightly south of the Sun's centre. From a southern site, it appears slightly north. The two lines of sight cross at Venus, so on the Sun the up and down are swapped.
But this shift is surprisingly small. Even with sites 10,000 km apart north and south, the shift is only about 1/50 of the Sun's apparent diameter. Draw the Sun as large as a coin, and it is about the width of a single hair. Measuring that directly with a ruler was impossible with the telescopes of the day.
So Halley came up with measuring the shift by "time". As in the right of Figure 1, a path through the middle of the Sun takes longer to cross. A path nearer the edge is shorter. If you record with a clock the moments Venus enters and leaves the edge of the Sun, the difference in path length shows up as a time difference of a few minutes to over ten minutes.
What happened during the two chances in a hundred years?
Venus's transits come as a pair, eight years apart, only once in a little over a hundred years (Figure 2). When Halley made his proposal, the next chances were 45 years away, in 1761 and 1769. Halley himself died without seeing them.
In 1761 and 1769, the countries of Europe raced to send observing parties around the world. They needed places as far apart as possible, north and south, east and west. Britain's James Cook sailed to Tahiti in the South Pacific for the 1769 observation. This was the first great voyage that later made him famous as an explorer.
The unluckiest was probably France's Le Gentil. In 1761 he was on a ship bound for India, and the pitching deck made measurement impossible. Rather than go home, he waited eight years and made careful preparations in India for 1769. But on the day, the sky clouded over just for the length of the transit. When he finally returned to France after eleven years, he is said to have found that he had been declared dead and his property divided up.
So, did they get an answer?
They did, though not cleanly. When Venus touched the edge of the Sun, the black dot seemed to stick to the edge and stretch like a teardrop, and the moment it let go was unclear. Because of this look, called the "black drop", even observers at the same site recorded different times.
Even so, a 19th-century calculation combining the records from many places is said to have given about 153 million km. That is off from today's value by a little over 2%. For an age of sailing ships and pendulum clocks, with times compared between opposite sides of the Earth, that is a remarkable accuracy. Expeditions went out again in 1874 and 1882, and in 1874 French and American teams came to Nagasaki and Kobe. The place name "Venus Hill (Kinseidai)" on Suwayama in Kobe comes from that observation.
Today we can bounce radio waves off Venus or asteroids and measure the distance from the time the echo takes to return. In 2012, one astronomical unit was fixed at exactly "149,597,870,700 m", turning it from a quantity we measure into one we define.
For the 2004 transit, European astronomy institutions made a call, and schools and citizens recorded contact times in many places. When those records were combined exactly as Halley's method prescribes, the result is said to have differed from today's value by only about 0.01%. With accurate clocks and many pairs of eyes, a 300-year-old idea worked just as it was.
In 1761, Russia's Lomonosov recorded seeing a thin ring of light around the black disc when Venus was on the edge of the Sun. From this he is said to have concluded that "Venus has an atmosphere". It is an example of an observation made to measure distance producing a completely different discovery.
Summary
The "shape" of the solar system was known beforehand, thanks to Kepler's rules. All that was missing was one number: how many km one unit of that scale drawing is. To fix it, people watched Venus's transit from two distant places and measured the tiny shift in its path as a difference in time. For two chances in a hundred years, they crossed the seas at the risk of their lives.
The distance to the Sun was measured not with a ruler,
but by the "shift in Venus's shadow" seen from opposite ends of the Earth.
Rømer, who first measured the speed of light, did his calculation while this distance was still uncertain (How was the speed of light first measured?). Read alongside the story of measuring the Earth's size with the shadows of two sticks (How was the size of the Earth measured?), you can see the size of the universe being revealed one step at a time.
- Stretch out your arm, raise your thumb, and line it up with the corner of a distant wall or building. Close your right and left eyes in turn and see how far your thumb jumps against the background.
- Estimate how many metres that jump covers on the background (for example, how many windows wide). Your eyes are about 6 cm apart and your arm is about 60 cm long, a ratio of about 10 to 1.
- Multiply the estimated jump by 10 and you get the rough distance to the background. For example, if the jump is 3 m, the background is about 30 m away. Compare it with the real distance on a map app.
Put the north and south of the Earth in place of your two eyes, Venus in place of your thumb, and the Sun in place of the background, and you have the method of this article. *Never look directly at the Sun with the naked eye or binoculars; it will damage your eyes.
For those who want more – terms, formulas and links to textbooksWe state clearly which level each topic belongs to, from middle school science to university courses
- Middle schoolCovered in middle school science
- High schoolCovered in high school earth science or physics
- High school+Advanced high school material, or textbook sidebar content
- UniversityNot taught in high school; university courses (astronomy, celestial mechanics)
- ResearchNot yet taught as settled fact even at university; being studied by researchers now
Middle schoolTerms: this phenomenon has a name
- Transit of Venus: the phenomenon in which Venus is seen from Earth crossing in front of the Sun's disc.
- Parallax: the way a nearby object appears to shift against a distant background when you change your viewing position, or the angle of that shift.
- Astronomical unit: a unit of length based on the average distance between Earth and the Sun. It is now defined as 149,597,870,700 m.
Middle schoolHigh schoolChecking with a formula: from the angle of the shift to the distance to the Sun
Suppose two people 10,000 km apart, north and south, see Venus's path shifted by 36 arcseconds on the Sun (an arcsecond is a unit of angle, 1/3600 of a degree). Let's calculate the distance to the Sun from this.
| In symbols | AU ＝ ( B ÷ θ ) × ( a ÷ ( 1 − a ) ) |
| In words | Distance to the Sun = (distance between the two sites ÷ angle of Venus's shift on the Sun) × (a factor set by the size of Venus's orbit) |
| Where it comes from | It is the triangulation (parallax) rule "distance = baseline ÷ angle". Venus is closer than the Sun, so it shifts a lot, while the Sun is far, so it shifts only a little. The difference is the shift we see, and its proportion is set by the size of Venus's orbit (0.723 times Earth's), known from Kepler's third law. |
| Symbol | Meaning and unit |
| AU | Distance to the Sun (km). The astronomical unit |
| B | Separation of the two sites, perpendicular to the line of sight (km) |
| θ | Angle of the shift of Venus's path on the Sun (radians; 1 radian is about 206,265 arcseconds) |
| a | Size of Venus's orbit (taking Earth's orbit as 1) |
| Days for Venus to go once around the Sun | about 224.7 days |
| Days for Earth to go once around the Sun | about 365.25 days |
| Separation of the two sites B | 10,000 km (assumed) |
| Observed shift θ | 36 arcseconds (assumed) |
| 1 radian | about 206,265 arcseconds |
| Apparent diameter of the Sun | about 1,900 arcseconds (about 0.5 degrees) |
| Ratio of periods | 224.7 ÷ 365.25 ≒ 0.615 |
| Orbit size (third law) | 0.615 to the power 2/3 is about 0.723 |
| Closeness of Venus and Earth at transit | 1 − 0.723 ＝ 0.277 |
| Factor | 0.723 ÷ 0.277 ≒ 2.61 |
| Shift in radians | 36 ÷ 206,265 ≒ 0.0001745 |
| Separation ÷ angle | 10,000 ÷ 0.0001745 ≒ 57,310,000 km |
| Distance to the Sun | 57,310,000 × 2.61 ≒ 149,580,000 km |
| What fraction of the Sun's diameter is the shift | 1,900 ÷ 36 ≒ 53 |
The result is about 149.6 million km, which matches today's value (about 149.6 million km). But a shift off by just 1 arcsecond moves the answer by about 3%. A 36-arcsecond shift is only about 1/53 of the Sun's diameter, which shows how hard it was to measure directly. That is why Halley chose to measure by "time" rather than by angle.
High schoolHigh school+The trick of turning an angle into "time"
High schoolThe square of a planet's orbital period is proportional to the cube of the size of its orbit (Kepler's third law). Periods can be found accurately by watching for years, so the "shape" of the solar system was accurate early on. The annual parallax you learn in earth science (a way to measure the distance to stars) uses the same idea of triangulation.
High school+In Halley's method, you compare the time Venus takes to cross the Sun's disc, from entry to exit, at sites far apart north and south. Venus moves across the Sun at about 4 arcminutes per hour, so even a path shift of a few tens of arcseconds makes a difference of a few to over ten minutes in transit time (the number at the bottom right of Figure 1). The point is that even without the precision to measure angles, you can do the job if your clock is accurate. Later, France's Delisle proposed a method that uses only the entry or only the exit moment, which also allows sites far apart east and west.
UniversitySolar parallax and the definition of the astronomical unit
In astronomy, the angle subtended by Earth's equatorial radius as seen from the Sun is called the "solar parallax", and it was long used in place of the astronomical unit. Its value is about 8.794 arcseconds. Observations in the 18th century scattered between about 8.5 and 8.9 arcseconds, and reanalysis in the late 19th century is said to have settled it at around 8.80 arcseconds. In the 20th century, the lead role passed to triangulation of the asteroid Eros at close approach, and then to radar observations of Venus. A 2012 resolution of the International Astronomical Union defined the astronomical unit as a fixed value, independent of the Sun's mass and so on.
📖 Derivations and further reading: Astronomical unit (Wikipedia, in Japanese) ／ Transit of Venus (Wikipedia, in Japanese)
ResearchWhat we still don't clearly know
- Is the Earth–Sun distance slowly growing? An analysis of planetary observation records reported (in 2004) that it is growing by a dozen or so centimetres a year. This is said not to be explained by the Sun's mass loss through light and particles alone, and the debate continues.
- The air of planets crossing other stars. The faint dimming when a planet crosses in front of a star has revealed thousands of planets. Research on reading planetary atmospheres from that light is advancing, and the 2012 transit of Venus was also observed as a practice run for it.
- Did Lomonosov really see an atmosphere? Opinion is still divided on whether the ring of light from Venus's atmosphere could have been seen with the telescopes of the time. In 2012, some studies tried to recreate it with old-style telescopes.
So the content of this article, too, is "an explanation of what is understood so far". The distance itself is now fixed by definition, but how that distance changes over time is still being studied.
Links to textbooks (by level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Middle school | Science: the solar system and planets | How Venus looks, the order of the planets |
| High school | Earth science: planetary motion / distance to stars | Kepler's third law, triangulation as in annual parallax |
| High school+ | Advanced earth science, history of astronomy | Halley's method (comparing transit times), Delisle's method |
| University | Astronomy, celestial mechanics, positional astronomy | Solar parallax, radar ranging, definition of the astronomical unit |
| Research | Solar system dynamics / exoplanet observation | Change in the astronomical unit, observing atmospheres by the transit method |
| ― | Links to daily life | Calculations for sending probes to planets assume the astronomical unit is known accurately |
- Wikipedia, "Transit of Venus" (金星の太陽面通過, in Japanese)
- Wikipedia, "Astronomical unit" (天文単位, in Japanese)
- Wikipedia, "Kepler's laws" (ケプラーの法則, in Japanese)
- Halley, E. (1716). A New Method of Determining the Parallax of the Sun. Philosophical Transactions of the Royal Society, 29.
- Wulf, A. (2012). Chasing Venus: The Race to Measure the Heavens. Knopf.
- International Astronomical Union, 2012 General Assembly, Resolution B2 (redefinition of the astronomical unit)
*This article is a popular-science explanation. The figures given are guides and rough estimates to help you understand how things work. Anecdotes about historical figures follow what has been handed down. Always observe the Sun using safe methods, such as a proper solar filter.