Lecture 3: Astromony in History

PURDUE UNIVERSITY

EAS 105-THE PLANETS

Prof. Robert L. Nowack

Lecture 3

Aristotle (384-322 B. C.) was one of the first Greek philosophers to speculate that the Earth was round. The evidence he used was

(1) Shadow of Earth on Moon during a lunar eclipse (when Earth goes between the Sun and Moon).

(2) Stars like the North Star Polaris are higher in the sky as one goes north.

(3) Elephants are found east in India and west in Morocco from Greece. (Thus, these places must be nearby!)

Note: A Flat Earth Society exists even today that thinks the Earth is flat.

How do we measure the size of a spherical Earth? This was first done by Eratosthenes, an astronomer in Egypt (276-195 B. C.). His observation: the length of a shadow cast by the noonday Sun changed from one place on Earth to another.

Ex.) On the summer solstice, the spire in Alexandria still casts a shadow at noon. While in the town of Syene, ~ 5000 Stadia south (~ 500 miles), no shadow was cast at noon (i.e. Sun was at zenith there).

On the summer solstice, the shadow cast by the large tower in Alexandria was ~ 1/8 the height of the tower. Even at this time, philosophers speculated that Earth was spherical. Eratosthenes reasoned that the curvature of the Earth caused the variation of shadow length.

The distance between the two towns was: Distance ~ 5000 Stadia (500 miles). The ratio of shadow length to tower height was: (shadow length, s)/(h, tower height) = 1/8 (assume light rays are nearly parallel over a span of 500 miles). Now, use similar triangles.

r equals the Earth's radius, "d" equals the distance between the two towns.

The ratio of the distance between towns to the radius of Earth ~ 1/8 (since triangles 1 and 2 are similar)

Or, the radius of the Earth is about 8 times the distance between towns, or 4000 miles (the modern estimate for the radius of the Earth is 3958 miles)

Note: For any circle no matter what size: circumference/diameter = constant = . This constant is known as "pi", () = 3.1415 ... ~ 22/7. The circumference of the Earth is then (2 times the radius of the Earth) ~ 25,143 miles. (The modern estimate is 24,868 miles). Not bad for 200 B. C.

Aristarchus (310-230 B. C.) attempted to measure relative distances of the Sun and Moon. He investigated phases of the Moon. Phases of the Moon are based on light and shadows from the Sun.

The Moon revolves counterclockwise about the Earth in a plane ~ 5 degrees from the Ecliptic. The Moon's period of revolution about the Earth with respect to the fixed background stars, called the Sidereal Period, is 27.3 days. However, the period between successive phases of the Moon as seen from Earth, called the Synodic Period, is 29.5 days. Aristarchus attempted to measure the angle between the "new Moon" and "First Quarter" Moon.

He estimated an angle of 87°.

Note: Regardless of how big the triangle is, if the angles are known, the ratio of any two sides will be the same.

Using graph paper and a protractor, a small triangle can be drawn. From this, the ratio of the lengths of side 1 to side 2 can be found. For the angle equal to 87°, Aristarchus found: (side2)/(side1) ~ 19. Or for the large triangle in the sky that the Earth-Sun distance is 19 times the Earth-Moon distance. In fact, the correct angle is 89.85° and the correct ratio is 389.

Now, the Moon and Sun appear to be about the same angular size in the sky. From this relationship, we can use similar triangles again to show that: D_s/R_s ~ D_m/R_m = diameter/distance

Aristarchus concluded D_s/D_m ~ 19. In fact, the ratio of the Sun's diameter to the Moon’s diameter is about 0.389 or D_s/R_s ~ 389.

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Seen from the Earth, the Sun and Moon are about the same angular sizes. Angle 1 ~ Angle 2 ~ ½°. Therefore, they are similar triangles. The ratio of diameter to distance is the same.

These are relative distances and sizes. Can we get absolute sizes? Yes!

A Solar Eclipse occurs when the Moon moves in front of the Sun and casts a shadow on the Earth. A Lunar Eclipse occurs when the Earth moves in front of the Moon and casts a shadow on the Moon. If the Moon revolved exactly in the ecliptic, there would be a solar and lunar eclipse every month.

During a lunar eclipse, Aristachus found that the Earth's shadow on Moon (A-A') was twice the Moon's diameter.

Using a geometric construction, he then found the relative sizes of the Sun, Moon, and Earth. Thus, he found: D_e ~ 3.5 D_m or the diameter of the Earth, D_e is 3.5 times the diameter of the Moon D_m and D_s ~ 19 D_m. Modern estimate determined that D_s ~ 389 D_m.

Eratosthene’s estimate of the size of the Earth can be used to scale absolute sizes and distances of the Sun and Moon. Conclusion: Even at the time of the Greeks, rough distances and scale of Earth, Sun, and Moon could be determined.

Since even Aristachus realized that the Sun was much larger than Earth, he was one of the first to consider that the Sun was the center of the Solar System. However, Ptolemy (~ 140 A. D.) and others developed complicated models for Earth centered planetary motions which were used until the time of Copernicus (1473-1543).

What about the distances to the other planets? Venus is often called the Morning Star or Evening Star depending on whether it rises before the Sun or sets just after. In fact, Venus never gets more than ~ 47° from the Sun (4-6 fists at arms length). Mercury never gets more than 23° from the Sun. This can be understood if we assume Venus and Mercury have orbits about the Sun inside the Earth's orbit ("inferior" planets). "Superior" planets would have orbits outside the Earth's orbit.

For Venus, the maximum angle above the horizon at sunset is found to be 47°.

No matter how big the triangle, if the angles are known, the ratio of any two sides is known.

Thus, the ratio of Side 1 to Side 2 is the same as the ratio of the Venus-Sun distance to the Earth-Sun distance. Thus for an angle of 47°, the Venus-Sun distance is 0.73 times the Earth-Sun distance or Venus-Sun distance is 0.73 A.U.

For Mercury, the maximum angle from the Sun at sunset is 23°. This gives, from the ratio of two sides of a similar triangle, the Mercury-Sun distance as 0.39 times the Earth-Sun distance or the Mercury-Sun distance is 0.39 A.U.

Similar, but more complicated constructions can be used to find the relative distances to the superior planets. At the time of Copernicus, relative distances of known planets (known) from the Sun had been found.

Copernicus Modern

Mercury: 0.38 0.387

Venus: 0.72 0.723

Earth: 1.0 1.0 Au

Jupiter: 5.22 5.20

Saturn: 9.18 9.54

In Astronomical Units (Au)

These again were relative distances scaled by the Earth-Sun distance.

At the time of Copernicus, the Earth-Sun distance was not very accurate. (Recall: the Greek estimate R_E_-_S ~ 19 R_E_-_M ). In 1716, the Astronomer, Halley, showed that transits of an inferior planet across the face of the Sun could be used to scale the solar system. However, both Mercury and Venus have orbits slightly inclined to the ecliptic, and it was found that Venus makes a transit across the face of the Sun in pairs about 8 years apart once each century. Halley knew that two Venus transits occurred in 1631 and 1639, and the next would occur in 1761 and 1769.

Transit times that each observer saw would be proportional to the distance across circular disk of Sun. Now: (R_v-S)/(R₁) = (0.72 AU)/(0.28 AU) = 2.5 = (D)/(B).

Assuming B on Earth is 1000 km, then D on the Sun should be 2500 km. Using two or more chords across the Sun, the absolute diameter of the Sun can be found.

Using D_s, absolute distances of solar system can be determined.

It was found that 1 astronomical unit (Au) equals the Earth-Sun distance where this is 149.6 million kilometers or ~ 93 million miles.

This was one of the motivations of Captain Cook's voyage to the Pacific Ocean (as far from England as possible) to measure transit times it took Venus to go across the face of the Sun. This voyage also led to the discovery of Australia.