Astronomy 10: Lecture 4

Lecture 4

Monday, June 3rd 2002. Reading: Cosmic Perspective Chapt. S1 and 3

Time Keeping

The Solar Day on Earth is in fact a Synodic Day. The Mean Solar Day is defined as the time it takes the Sun to transit the meridian twice. But again during that one day Earth has moved in its orbit (about 1°) and as seen in the Sidereal frame of reference it completes one spin before the Sun can again be in position to transit. The Synodic Day is defined as 24 hours. The Sidereal day, the actual time it takes Earth to complete one spin and the time between consecutive transits of any given star is about 23 hours and 56 minutes. It's 4 minutes shorter. This leads to there being 366 Sidereal days in a year. It also means that stars rise and set each day 4 minutes earlier than the previous.

Celestial Navigation: The Sky from different places on Earth

The Appearance of the Daily Motions of the Sky depends upon where you are on Earth.

First We examine the paths of Stars from several locations on Earth:
The North Pole
The Equator
40° N
30° S

The path of the Sun through the sky changes not only with position on Earth, but also with the time of the year. This is due to the 23.5° tilt of Earth's rotation axis to its orbital plane.

The North Pole
The Equator
The Tropic of Cancer
The Arctic Circle

The Shape and Size of Earth

  1. What is the Shape of the Earth?
    1. Looks Flat
      - The closer to Earth's Surface you get the more and more you cannot discern the curvature and it looks flat.
    2. Evidence for Curvature
      - As ships head out to sea they don't disappear all at once, but rather their masts disappear last.
      - The Sun's position in the sky is different from different locations on the same day.
      - As you move north or south you can see different stars that were previously hidden by the horizon. Not possible on a flat Earth.
    3. Evidence for Spherical
      - The Moon and the Sun are circular in projected appearance (the Moon's Phases give away its spherical shape), Why not Earth?
      - Earth's shadow on the Moon during a lunar eclipse is circular. Eclipses happen at all altitudes so Earth must be Spherical (Aristotle ~300 B.C.)
  2. What is the size of Earth?
    1. Eratosthenes figures it out.
      • Reads a report about the Sun being seen at the bottom of a well and no shadows being cast at noon on the Summer Solstice in Syene.
      • He never observed this to happen in Alexandria.
      • Must be due to Earth's curvature.
      • Meausure the angle of the shadow made by objects in Alexandria, on June 21st: 7°
      • Measure the distance between Alexandria and Syene: 500 Stadia = 800 km (hires a man to pace this out)
      • The ratio of the angle to that of a full circle would be equivalent to the ratio of the distance between Alexandria and Syene and the Circumference of the Earth: 40,000 km.
      • Real Answer: 40,030 km
      • Wow!!!

The Measurements of Aristarchus of Samos

Distance and Size of the Moon

In the 3rd Century B.C. (about 75 years before Eratosthenes), a great Greek thinker figured out a way to measure the relative sizes and distances of the Moon and Sun. His name was Aristarchus of Samos (circa 310 - 230 BC).

These calculations involve some straightforward logic and the application of simple geometry.

The first thing that Aristarchus noticed was that when the Moon was eclipsed by the Earth you could see the Earth's shadow creep across the face of the Moon. Earth's shadow is circular, and if you assume that the Sun's rays are roughly parallel to one another that implies that the size of Earth's shadow at the distance of the Moon is equal to the size of Earth. It's not exactly true, but it's very close. From the arc of Earth's shadow you can estimate the size of it's full circle, then you measure the size of the Moon's circle. This tells you the relative size of the Earth to the Moon. What you are measuring is the angular size of the Earth's shadow and the Moon. In the limit of small angles, physical size and angular size are directly proportional.

Aristarchus measured a ratio of Earth's size to the Moon's size to be:

D/DM = 2.86
(Where the symbol refers to Earth)
The real answer is 3.67. So the Moon is about one-third the size of Earth.

Using the angular size of the Moon we can estimate its distance as well using the relationship between angular size (), physical size (D), and distance (d):

= D/d
Where in this equation is in radians, the natural, unit-less measure of angles. The conversion from radians to degrees is simply:
2 radians = 360°
-> 1 radian = 57.3°
The angular size of the Moon is about 0.5° ( = 0.5°). Thus we have a distance to the Moon of about 30 Earth Diameters (dM = 30D)

Distance and Size of the Sun

Next Aristarchus reasoned that when the Moon was seen to be exactly in either 1st or 3rd Quarter Phase (exactly half lit up as seen from Earth), we can draw a right triangle between Earth, Moon, and Sun. The legs of the right triangle are the distance from the Earth to the Moon (dM) and the distance between the Moon and the Sun (dM). The hypotenuse is then the distance between the Earth and Sun (d). (The symbol refers to the Sun). We can measure the angle between dM and d (), it's the angular separation between the Sun and Moon in the sky. Once again a little simple trigonometry then gives us the distance to the Sun.

Aristarchus measured = 87°. This corresponds to a distance to the Sun of d = 573D. Now, the angular size of the Sun is also about 0.5° so by the same method as for the Moon Aristarchus estimated that the Sun was 5.33 times bigger than the Earth.

This measurement is a difficult one to make, and the real answer is = 89.9°. This corresponds to a distance to the Sun of d = 11,700D and a size of the Sun, D = 109D.

Either way, Aristarchus was able to establish that the Sun was bigger than Earth. Since this was so, he reasoned that it would make more sense for Earth to orbit the Sun rather than the other way around. But his contemporaries pointed to some arguments of Aristotle's that contradicted such an idea. One critical one being that the stars should exhibit parallax if Earth goes about the Sun. They do indeed exhibit this phenomenon but they are so far away that it cannot be measured without a telescope. It would be some 1900 years before Aristarchus would finally be proved correct by the work of Kepler and Galileo.

Motion of the Planets in the Sky

The History of Astronomy

Early Timeline

Ancient Astronomy

All cultures on Earth have made some understanding of the motions of the heavens. But the contribution by the Greeks of the use of a scientific reasoning was special and led to great discovery.

The Greek Contribution

Ptolemy (circa 140 AD)

Claudius Ptolemaeus, a.k.a Ptolemy, was also a director of the Library of Alexandria in his time. He is sometimes considered the last great natural philosopher of classical times. (I would argue however that this identification falls to the last director of the Library of Alexandria, Hypatia, who was brutally murdered by a Christian lynch mob in 415 AD.) He put together a model of the Universe that did not change much for 1500 years. Ptolemy used the collected observations of Hipparchus to build his model and to measure the parallax of the Moon, and thereby make another measurement of its distance.

His model was marked by using the geocentric model first developed by Eudoxus. He considered a heliocentric model, like that proposed by Aristarchus, but immediately rejected it based on Aristotelean physics. He made alterations to the geocentric model, added some epicycles within epicycles in order to make the model explain all of Hipparchus' data. He also offset some of the spheres so that they were not necessarily concentric with Earth. The most important thing about this model was that it was able to predict the positions of the planets with great accuracy (at least to the level that people were able to measure in those times). The model contained no explanation for how or why these motions occurred (it had no physics).

In subsequent centuries the alterations made to the model would be to add more epicycles into the model. This was done primarily by Arabic astronomers in medieval times. In the end some planets would have as many as 80 epicycles in order to be able to properly predict where they would be.

Ptolemy also codified the pseudoscience of astrology into the form which is still in use today. That just goes to show that great genius is no guarantee against being dead wrong.

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