A&B's Astronomy Lab, Fall 2001, No. 5

Gravity and Orbit Simulations

What happened between Newton and an apple?
Why we are stuck on the ground? Why planets are orbiting around the Sun?
Why stars make groups? Let's feel the gravity by simulations.

Gravity

Ancient to Newton Einstein to Kip Thorne

The Ancients
Gravity is a force of attraction between all matter. It is the weakest known force in nature, but it still manages to hold galaxies and the solar system together. The ancient Greek philosophers thought that the motions of the stars and planets were totally unrelated to events on the earth. The heavens were the realm of the gods, where everything existed in perfection. One of these philosophers was called Aristotle. He thought that that the stars and planets followed a so called "natural" motion, and that the force that made them move made contact between them. This force is the force of gravity and today we know that it works without actually touching objects but the ideas of the ancient Greeks survived right up until the 16th century when our understanding of gravity was changed dramatically by scientists like Galileo and Newton.
Galileo (1564 - 1642)
Galileo was an Italian mathematician, astronomer and physicist. He was one of the first scientists to agree with the Copernicun view that the earth orbited around the sun. This was a revolutionary thought at the time because the established view held from ancient times was that the earth was the centre of the universe and the planets and stars revolved about it. Galileo's new ideas led to his trial at the Inquisition of Rome, where he was sentenced to house arrest. He remained under house arrest for 8 years until his death. Galileo is considered to be the founder of mechanics and experimental physics. He is also credited with the invention of the telescope, through which he discovered lunar craters, moons around Jupiter the rings of Saturn and sunspots amongst other things.
Brahe (1546 - 1601)
Tycho Brahe lived in the same period as Galileo, although he initially studied law. As a teenage boy Brahe witnessed an eclipse of the moon after which he decided to dedicate his life to astronomy. In 1572 he witnessed a nova (an old star exploding). This shook up the view of the sky at the time which was that the heavens were perfect and unchanged. He spent his life collecting incredibly accurate astronomical data without the aid of a telescope. Near the end of his life he taught a student by the name of Kepler who helped him collect his astronomical data.
Kepler (1571 - 1630)
Kepler was Brahe's at the time of his death. He subsequently took Brahe's data and formulated the laws of planetary motion known as Kepler's Laws. Kepler was able to show that the planets moved in eliptical paths around the sun and not in circular paths as previously thought. In a similar way to Brahe he was inspired by a supernova which occured in 1604, which was much larger than the one that Brahe witnessed. The stage was now set for Newton to describe a more general law that governs gravitational forces.
Newton (1642 - 1727)
Sir Isaac Newton was an English mathematician and physicist who laid the foundations for modern physics. In 1665 the plague had shut down Cambridge University where Newton had been working. He subsequently worked from home on circular motion and other ideas. When the university reopened two years later Newton used Kepler's laws and his own observations to derive the universal law of gravitation. His observations of the motion of the moon, the tides and even comets helped him derive the law. Newton made many other very significant contributions to science in his life that make him one of the most important figures in science. Albert Einstein (1879-1955)
When he was young Einstein did not show much promise as a student. He left school early after one of his teachers claimed that he "..will never amount to anything...". Later when receiving a higher education, Einstein attended few lectures but still managed to pass his course! In 1905 he earned his PhD. He wrote several scientific papers which eventually won the Nobel prize in Physics in 1921 for his work on the photoelectric effect. In 1915 Einstein published his General Theory of Relativity. This relates gravity to space and time. Among other astounding predictions, this theory predicted the bending of light by a massive object such as the sun. When this was confirmed in 1919 by scientists observing an eclipse, Einstein became an instant celebrity. To his last days Einstein worked on unifying gravitation and electromagnetism into one force, right up to his death bed where he died with the last few equations of his work resting alonside him.
Stephen Hawking (1942-present)
Stephen Hawking decided to be a physicist at the youthfull age of 15. He chose physics as he wanted to understand more about how the universe works. Upon entering Oxford University Hawking wanted to study Mathematics, which wasn't available at the time. As a result he studied physics. Upon graduating Hawking took up research into cosmology (the study of the universe as a whole) at Cambridge University. In 1974 Hawking found that quantum mechanics dictated that a black hole may evaporate. The radiation it emits in doing so is called "Hawking Radiation". Over the years Hawking has continued to make important contributions to cosmology, black hole physics and physics in general. He is one of the best know contemporary scientists even though he has never won the Nobel Prize and has been crippled with ALS (Amytrophic Lateral Sclerosis) for nearly 30 years.
Kip Thorne (-present)
Kip Thorne is the Professor of Theoretical Physics at Caltech (California Institue of Technology). He has done research in several areas including gravitational radiation, black holes, neutron stars and the nature of space, time and gravity. He obtained his Bachelor of Science at Caltech in 1962, and his PhD just three years later at Priceton University. He was appointed Professor of Theoretical Physics at Caltech in 1970, and Richard Feynman Professor of Theoretical Physics at Caltech in 1991. He research focuses on gravitational physics and astrophysics, which includes black holes and gravity waves. Currently, Thorne is looking at general relativity and quantum field theory in curved space-time.

Kepler's Three Laws of Planetary Motion

Kepler obtained Brahe's data after his death despite the attempts by Brahe's family to keep the data from him in the hope of monetary gain. There is some evidence that Kepler obtained the data by less than legal means; it is fortunate for the development of modern astronomy that he was successful. Utilizing the voluminous and precise data of Brahe, Kepler was eventually able to build on the realization that the orbits of the planets were ellipses to formulate his Three Laws of Planetary Motion.

I. The orbits of the planets are ellipses, with the Sun at one focus of the ellipse.
Kepler's First Law is illustrated in the image shown above. The Sun is not at the center of the ellipse, but is instead at one focus (generally there is nothing at the other focus of the ellipse). The planet then follows the ellipse in its orbit, which means that the Earth-Sun distance is constantly changing as the planet goes around its orbit. For purpose of illustration we have shown the orbit as rather eccentric; remember that the actual orbits are much less eccentric than this.

II. The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse.
Kepler's second law is illustrated in the figure. The line joining the Sun and planet sweeps out equal areas in equal times, so the planet moves faster when it is nearer the Sun. Thus, a planet executes elliptical motion with constantly changing angular speed as it moves about its orbit. The point of nearest approach of the planet to the Sun is termed perihelion; the point of greatest separation is termed aphelion. Hence, by Kepler's second law, the planet moves fastest when it is near perihelion and slowest when it is near aphelion.

III. The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semimajor axes:
In this equation P represents the period of revolution for a planet and "a" represents the length of its semimajor axis. The subscripts "1" and "2" distinguish quantities for planet 1 and 2 respectively. The periods for the two planets are assumed to be in the same time units and the lengths of the semimajor axes for the two planets are assumed to be in the same distance units. Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit. Thus, we find that Mercury, the innermost planet, takes only 88 days to orbit the Sun but the outermost planet (Pluto) requires 248 years to do the same.

Gravity Between the Galaxies - Galaxy Mergers

Gravity
Ancient to Newton	Einstein to Kip Thorne
The Ancients Gravity is a force of attraction between all matter. It is the weakest known force in nature, but it still manages to hold galaxies and the solar system together. The ancient Greek philosophers thought that the motions of the stars and planets were totally unrelated to events on the earth. The heavens were the realm of the gods, where everything existed in perfection. One of these philosophers was called Aristotle. He thought that that the stars and planets followed a so called "natural" motion, and that the force that made them move made contact between them. This force is the force of gravity and today we know that it works without actually touching objects but the ideas of the ancient Greeks survived right up until the 16th century when our understanding of gravity was changed dramatically by scientists like Galileo and Newton. Galileo (1564 - 1642) Galileo was an Italian mathematician, astronomer and physicist. He was one of the first scientists to agree with the Copernicun view that the earth orbited around the sun. This was a revolutionary thought at the time because the established view held from ancient times was that the earth was the centre of the universe and the planets and stars revolved about it. Galileo's new ideas led to his trial at the Inquisition of Rome, where he was sentenced to house arrest. He remained under house arrest for 8 years until his death. Galileo is considered to be the founder of mechanics and experimental physics. He is also credited with the invention of the telescope, through which he discovered lunar craters, moons around Jupiter the rings of Saturn and sunspots amongst other things. Brahe (1546 - 1601) Tycho Brahe lived in the same period as Galileo, although he initially studied law. As a teenage boy Brahe witnessed an eclipse of the moon after which he decided to dedicate his life to astronomy. In 1572 he witnessed a nova (an old star exploding). This shook up the view of the sky at the time which was that the heavens were perfect and unchanged. He spent his life collecting incredibly accurate astronomical data without the aid of a telescope. Near the end of his life he taught a student by the name of Kepler who helped him collect his astronomical data. Kepler (1571 - 1630) Kepler was Brahe's at the time of his death. He subsequently took Brahe's data and formulated the laws of planetary motion known as Kepler's Laws. Kepler was able to show that the planets moved in eliptical paths around the sun and not in circular paths as previously thought. In a similar way to Brahe he was inspired by a supernova which occured in 1604, which was much larger than the one that Brahe witnessed. The stage was now set for Newton to describe a more general law that governs gravitational forces. Newton (1642 - 1727) Sir Isaac Newton was an English mathematician and physicist who laid the foundations for modern physics. In 1665 the plague had shut down Cambridge University where Newton had been working. He subsequently worked from home on circular motion and other ideas. When the university reopened two years later Newton used Kepler's laws and his own observations to derive the universal law of gravitation. His observations of the motion of the moon, the tides and even comets helped him derive the law. Newton made many other very significant contributions to science in his life that make him one of the most important figures in science.	Albert Einstein (1879-1955) When he was young Einstein did not show much promise as a student. He left school early after one of his teachers claimed that he "..will never amount to anything...". Later when receiving a higher education, Einstein attended few lectures but still managed to pass his course! In 1905 he earned his PhD. He wrote several scientific papers which eventually won the Nobel prize in Physics in 1921 for his work on the photoelectric effect. In 1915 Einstein published his General Theory of Relativity. This relates gravity to space and time. Among other astounding predictions, this theory predicted the bending of light by a massive object such as the sun. When this was confirmed in 1919 by scientists observing an eclipse, Einstein became an instant celebrity. To his last days Einstein worked on unifying gravitation and electromagnetism into one force, right up to his death bed where he died with the last few equations of his work resting alonside him. Stephen Hawking (1942-present) Stephen Hawking decided to be a physicist at the youthfull age of 15. He chose physics as he wanted to understand more about how the universe works. Upon entering Oxford University Hawking wanted to study Mathematics, which wasn't available at the time. As a result he studied physics. Upon graduating Hawking took up research into cosmology (the study of the universe as a whole) at Cambridge University. In 1974 Hawking found that quantum mechanics dictated that a black hole may evaporate. The radiation it emits in doing so is called "Hawking Radiation". Over the years Hawking has continued to make important contributions to cosmology, black hole physics and physics in general. He is one of the best know contemporary scientists even though he has never won the Nobel Prize and has been crippled with ALS (Amytrophic Lateral Sclerosis) for nearly 30 years. Kip Thorne (-present) Kip Thorne is the Professor of Theoretical Physics at Caltech (California Institue of Technology). He has done research in several areas including gravitational radiation, black holes, neutron stars and the nature of space, time and gravity. He obtained his Bachelor of Science at Caltech in 1962, and his PhD just three years later at Priceton University. He was appointed Professor of Theoretical Physics at Caltech in 1970, and Richard Feynman Professor of Theoretical Physics at Caltech in 1991. He research focuses on gravitational physics and astrophysics, which includes black holes and gravity waves. Currently, Thorne is looking at general relativity and quantum field theory in curved space-time.

Kepler's Three Laws of Planetary Motion
Kepler obtained Brahe's data after his death despite the attempts by Brahe's family to keep the data from him in the hope of monetary gain. There is some evidence that Kepler obtained the data by less than legal means; it is fortunate for the development of modern astronomy that he was successful. Utilizing the voluminous and precise data of Brahe, Kepler was eventually able to build on the realization that the orbits of the planets were ellipses to formulate his Three Laws of Planetary Motion.
I. The orbits of the planets are ellipses, with the Sun at one focus of the ellipse. Kepler's First Law is illustrated in the image shown above. The Sun is not at the center of the ellipse, but is instead at one focus (generally there is nothing at the other focus of the ellipse). The planet then follows the ellipse in its orbit, which means that the Earth-Sun distance is constantly changing as the planet goes around its orbit. For purpose of illustration we have shown the orbit as rather eccentric; remember that the actual orbits are much less eccentric than this.
II. The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse. Kepler's second law is illustrated in the figure. The line joining the Sun and planet sweeps out equal areas in equal times, so the planet moves faster when it is nearer the Sun. Thus, a planet executes elliptical motion with constantly changing angular speed as it moves about its orbit. The point of nearest approach of the planet to the Sun is termed perihelion; the point of greatest separation is termed aphelion. Hence, by Kepler's second law, the planet moves fastest when it is near perihelion and slowest when it is near aphelion.
III. The ratio of the squares of the revolutionary periods for two planets is equal to the ratio of the cubes of their semimajor axes: In this equation P represents the period of revolution for a planet and "a" represents the length of its semimajor axis. The subscripts "1" and "2" distinguish quantities for planet 1 and 2 respectively. The periods for the two planets are assumed to be in the same time units and the lengths of the semimajor axes for the two planets are assumed to be in the same distance units. Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the radius of its orbit. Thus, we find that Mercury, the innermost planet, takes only 88 days to orbit the Sun but the outermost planet (Pluto) requires 248 years to do the same.