A stronomy Chapter II History of Astronomy and Kepler`s Laws

1. Astronomy Chapter II History of Astronomy and Kepler`s Laws

2. Chapter 2 The History of Astronomy Astronomy is one of the oldest sciences. Human beings have been interested in the unique beauty of the universe and have wondered about the heavens since time immemorial. However, lack of sophisticated instruments prevented accurate studies on the nature of the universe. This led to many misunderstandings about the nature of the cosmos and solar system. Due to the intense interest in the nature of the universe, however, many discoveries were made even with these unsophisticated instruments. In this chapter we will examine the most prolific ideas from early astronomers about the nature of the universe.

3. 2.1 Ancient Astronomy Babylonians first recorded astronomical data upon thousands of stone tablets. They recognised the motion of the Sun, and they divided the path of the sun ecliptic into twelve parts. The length of the year was determined to within 4 minutes accuracy. However they thought the sky was a dome supported by mountains. The early Greeks also thought that the sky was a dome of the heavens and that the Earth was a disc-like structure floating on water. Pythagoras was the first person to realize that the Earth was round and that the paths of the heavens were circular. Anaxagoras discovered that the Moon did not emit its own light, instead it reflects the sunlight. Plato claimed that all objects were spherical and that they all had perfect spherical motions.

4. Aristotle claimed that the Earth was composed of four elements;Earth, wind, fire and water. He also said that the objects in the sky were made of a fifth element which he called aether. He thought that both the Earth and the universe were spherical and that the Earth was at the centre of the universe. The Earth was thought to be at the centre of the universe until about the 3rd century B.C. After this the Sun was thought to be at the centre of the universe. This was first noted by Aristarchus, who tried to calculate the relative sizes and distances of the Moon and the Sun, however, his measurements were not accurate enough. Aristarchus was unsuccessful in measuring the relative distances and sizes of the Sun and the Moon but Eratosthenes managed to determine the size of the Earth

5. Determining the Circumference of the Earth Eratosthenes (276 BC) was the head librarian in Alexandria, Egypt, the center of learning in the ancient world. He estimated the circumference of the Earth with the following method: He knew that on the summer solstice, the longest day of the year, the angle of the sun above Syene, Egypt, would be 0° ,in other words, the sun would be directly overhead. So on the summer solstice, he measured the angle of the sun above Alexandria by measuring the shadow cast by a pole and got a 7.2° angle. The following figure shows how Eratosthenes’s earth measurement worked. Eratosthenes divided 360° by 7.2° and got 50, which told him that the distance between Alexandria and Syene (500 miles) was 1/50 of the total distance around the Earth. So he multiplied 500 by 50 to arrive at his estimate of the Earth’s circumference: 25,000 miles. This estimate was only 100 miles off the actual circumference of 24,900 miles!

6. By the third century B.C. the idea of epicycle was developed to explain planetary motion. The idea of an epicycle model helped to explain some planetary motions. Hipparchus, who had an observatory on the island of Rhodes, laid down many of the foundations of astronomy. One of his most important inventions was the stellar magnitude system used to estimate starbrightness. Hipparchus advocated the idea of a geo-centric universe i.e. Earth centred Universe. Ptolemy tried to summarise all knowledge of astronomy up until that time. Chinese astronomical thought was not as complex as Greek astronomy. They simply thought that the universe was the sphere of the sky (celestial sphere), which rotates daily. The Mayas in America developed a very accurate calendar which was based on the idea of a universe which consisted of layers both above and below the Earth.

7. 2.2 Astronomy After the 15th Century New ideas concerning planetary motion were declared by NicolausCopernicus (MikolajKopernik) and TychoBrahe in the fifteenth century and by JohannesKepler at the beginning of the sixteenth century. Copernicus thought that the Sun was at the centre of planetary orbits. He deduced the relative distances of planets from the Sun, using a Sun-centred model. TychoBrahe claimed that the Earth was stationary; the Sun revolving around the Earth, but the other planets revolving around the Sun.

8. In 1572 the common belief that the heavens were eternal, suddenly changed with the observation, by Tycho Brahe, of a supernova explosion. This, revealed that all the heavenly bodies have a life story. Planetary motion was advanced significantly by JohannesKepler, who carried out observations on the planet Mars. Kepler recognised that the Sun was not exactly at the centre of the planetary orbit, furthermore he recognised that the orbit of Mars was not circular. The modern view of planetary motion was first declared by Johannes Kepler. His three Laws of planetary motion are still valid today, with small modifications made by Isaac Newton.

9. 2.2.a Kepler’s Laws of Planetary Motion Johannes Kepler defined planetary motion in three laws: 1) All the planets revolve around the Sun. The orbits of the planets are elliptical (not circular). The Sun is at one of the foci of the elliptical orbit.

10. 2) The speed of the planets is not the same during a full cycle. A planet is faster when it is closer to the Sun, its speed decreasing when it gets farther from the Sun. The increase and decrease in the speed of planets is random. The shaded area swept out by the line joining the Sun and the planet in a given time interval is constant independent of the initial position of the planet.

11. 3) The Square of the period of revolution of the planet around the Sun is proportional to the cube of the semi-major axis (half of the long axis of an ellipse). Where the unit of period,P, is one year and the period of the semi-major axis,a, is an astronomical unit (AU). (AU is the semi-major axis of the Earth.) = a

12. What is an Astronomical Unit? The astronomical unit (symbol: AU) is a unit of length, roughly the distance from Earth to the Sun. However, that distance varies as Earth orbits the Sun, from a maximum (aphelion) to a minimum (perihelion) and back again once a year. Originally conceived as the average of Earth's aphelion and perihelion, it is now defined as exactly 149597870700 metres (about 150 million kilometres, or 93 million miles). The astronomical unit is used primarily as a convenient measuring unit for the distances within the Solar System or around other stars.

13. Distances of planets in terms of AU.

14. Example 2.1The semi-major axis of Pluto is about 39.5 AU (39.5 times that of the Earth). Find the period of Pluto by using Kepler's third law of planetary motion. Solution: The equation between the period and the semi-major axis is: Substituting the values for Pluto into the equation, The period of Pluto is found to be slightly more than 248 years.

15. 2.2.b Newton's Corrections of Kepler's Laws Kepler was very successful in determining the laws of planetary motion, however, there were still some errors in Kepler's laws. In addition to this, Kepler could not understand or explain his laws using physics. Newton managed to understand and explain Kepler's laws using physics and, thus, he was able to modify Kepler's laws by developing his law of universal gravitation, stating that: Any two masses in the Universe attract each other with a force which can be expressed as, Where, F is the force of attraction between two objects. G is the universal constant of gravitation, with a value 6.67x-11 N.m2/kg2 m1 and m2 are the masses of two objects. d is the distance between the the two masses.

16. The law of universal gravitation helped Newton to modify Kepler's laws as follows: 1) The orbits of planets are elliptical and planets do not revolve around the Sun, instead, they both revolve around their common centre of mass. Thus, the centre of mass, not the Sun, is at the focus. The reason that Kepler could not recognise the motion of the Sun was that the centre of mass is very close to the centre of the Sun, since the mass of the Sun is so large compared to that of any planet (the mass of the Sun is more then 99% of the entire solar system). 2) The angular momentum of the planet and the Sun are constant during the revolution. Angular momentum is given as m.v.r. Mass is always constant, so change in distance between the revolving object and the centre of mass results in a change in velocity of the object in order to keep the product constant. When the planet is close to the centre of mass (not to the Sun) it speeds up and when it is far from the centre of mass, it slows down.

17. 3) The square of the period is proportional to the cube of the semi major axis but they are actually not equal. The equation can be expressed as follows: The term (M + m) was added by Newton. M and m are the mass of the Sun and the mass of the planet, respectively. The mass of the Sun should be taken as the unit for this term The unit of the period is one year, the unit of the semi-major axis is AU.

18. More on Newton's Calculations Newton formulated what is now known as his 2nd Law of Motion: This enabled him to formulate how objects are influenced (or attracted) in a gravitational field:

19. He was also the first to identify the acceleration on objects forced to move in circles as: And therefore the net force:

20. And finally what is perhaps the greatest intellectual discovery of all time—the Law of Universal Gravitation: This simple algebraic expression Mm/r says how everything in the universe is related to everything else—a far-reaching statement indeed!

21. Although the orbits of the planets are ellipses, they are very close to circles. The gravitational pull of the sun provides the force that causes the planet to go in its nearly circular orbit. The gravitational pull of the Sun provides the centripetal force of the satellite.

22. Gravity provides the centripetal force of the satellite.

23. Recall that the tangential velocity of the an object in circular path is simply the circumference divided by period. The same is true for satellites in circular orbits: We can square both sides:

24. Equating equivalent expressions for v2:

25. The ratio of two measurable quantities—radius and period—equals a constant.

26. The ratio of two measurable quantities—radius and period—equals a constant. If the distance of the planets to the sun are expressed in convenient units like astronomical units (1AU = the distance from the earth to the sun) and the period T is expressed in earth years, then the constant k equals 1!

27. Question 1 The magnitude of the gravitational force between two masses, m1 and m2, is F. If m2 is now doubled and the distance between the masses is also doubled, by what factor does the gravitational force increase/decrease? A. F decreases by a factor of 1/2 B. F remains the same C. F increases by a factor of 2 D. F decreases by a factor of 1/4 Question 2 In the figure, between which two positions will the planet have the fastest orbital speed about the Sun: A. A and B B. B and C C. C and D D. D and A