Keplerian Astronomy and Gravitation

• The development of the law of universal gravitation by Isaac Newton was based primarily on mathematical laws developed by Johannes Kepler to fit astronomical data. The study of the historical development of the law of universal gravitation is a prime example of how ideas in physics evolve.

• The first credible attempt at the scientific explanation of events in the heavens was done by the classical Greeks. Two schools of thought developed: The first school of thought placed the spherical earth at the center of the known universe. The stars in fixed positions in their constellations rotating nightly around the Earth embedded in the celestial sphere with the planets attached inside on rotating clear crystalline spheres. The idea of the Earth being at the center of the cosmos, called the **geocentric model**, was first successfully developed by **Eudoxos** (~320BC). **Claudius Ptolemy** (~120AD) advanced this view to explain all the observed heavenly motions. This became known as the **Ptolemaic system** and agreed with the laws of physics as then understood. The other view substituting the Sun at the center of the cosmos, known as the **heliocentric model**, was championed by **Aristarchus** (~280BC) because then new evidence suggested that the Sun was actually larger than the earth. Because the Ptolemaic system was more consistent with observations, the accepted physics and theology, it was nearly universally accepted until the seventeenth century.

• Publishing from his deathbed in 1543, **Nicolas Copernicus** set about to correct what he believed was an imperfection in the Ptolemaic system. Ptolemy had planets move in off-centered circles at varying speeds to explain the observed variation in brightness of the planets. In order to restore perfect circular motion while still explaining the observations of the occasional retrograde motions of the planets, Copernicus amplified Aristarchus' heliocentric solar system with a series of nested circular epicycles for each planet. The academic community of the time generally saw no benefits and multiple evidence opposing adoption of the Copernican model. (Both theories could make equally good (?) predictions but Copernicus' model clashed with with both physics and theology, and even Copernicus admitted that the expected stellar parallax was not observable.)

• The Danish **Tycho Brahe** (~1600AD) was a creative, diligent, amateur astronomer who because convinced of the value of precise measurements to improve inaccurate predictions by all the models of the heavens. (Brahe understood that if you use sloppy measurements as the basis for calculating the parameters of your models, the predictions will be sloppy too.) Measuring at the accuracy and precision limits of human vision, Brahe observed many of the available astronomical events during his lifetime, including a few key locations of Mars, the most difficult planet. Brahe was a brilliant observer and proposed a third possible model of heavenly motions. But in an age when algebra had not been accepted as a valuable tool, he sought assistance to calculate the parameters of his geocentric model from his meticulous measurements.

• Rival astronomer **Johannnes Kepler** was hired as an assistant to Brahe and assigned to analyze a small portion of the volumes of data which Brahe had collected during his working life. Kepler expected to solve the orbit of Mars in 8 days, but took 8 years. Brahe's precise data did not fit within the tiny experimental error any of the models of Ptolemy, Copernicus, or Brahe! Only after considering what he called a *clean path* did Kepler find that Mars did not fit the circle that all theories demanded, or even several ovals, but rather travelled in an ellipse around the Sun. Over a lifetime, Kepler formulated four planetary laws, three of which are still in use today. The near madness of Kepler's methods of successfully matching accurate measurements to mathematical equations still often gives those who would emulate him reason to pause.

• **Kepler's first law** of planetary motion (discovered 3rd): The painstaking and exhaustive study of the best astronomical observations yielded the first law of planetary motion, also known as the law of ellipses. This law states that the path of any satellite is an **ellipse**, with the central body (the sun in the case of our solar system) located at one of the 2 foci.

• **Kepler's second law** of planetary motion (discovered 2nd): Kepler found that the speed of planets is fastest at perihelion and slowest at aphelion. Kepler suspected that this related an inverse speed law and came up with the law of equal areas to express this idea. The law of equal areas states that the **area swept out by an imaginary radius from the sun to a planet sweeps out equal areas in equal time intervals.** Newton used this idea to develop the inverse square aspect of his law of gravitation.

• **Kepler's third law** of planetary motion (discovered last): The law of periods established a relationship between the speeds of planets in different orbits. This law is sometimes called the harmonic law (because Kepler wrote the music for each planet in his discovery process). This law states that the **ratio of the cube of the average orbital radius divided by the square of the planetary period** (the length of the planet's year) is a **constant**. The constant is the same for all of the planets in the solar system. Unlike the first two laws, which govern and predict the orbits of individual bodies, the third law encompasses all of the known planets in the solar system as a group. (Most scientists would prefer to forget Kepler's first successful discovery explaining the spacings of the known planets, which at the time established his reputation as an astronomer.) Equation: R^{3}/T^{2} = constant where R is the average orbital radius and T is the period, the length of the planet's year.

• During the plague year of 1666, **Isaac Newton** stayed in the country at a relative's home because the schools were all closed and it was known that fewer people caught the plague and died in the country. Unlike Kepler who published almost diary-like accounts of his discoveries, we have almost no record of how Newton arrived at his theory of gravity. Based on the little he later told friends, it appears that during the plague years he pondered some of the big physics questions of the time such as the paths of the planets and the causes of their motions. Perhaps he wondered if the gravity that governs an apple's fall from the tree might not reach out far enough to govern the fall of the moon as it accelerates (changes direction) about the earth, or the planets orbit the Sun. It is likely he understood that Kepler had derived formulae that applied to the accelerations of the planets as they sampled different gravity at various distances away. Newton claimed he *stood on the shoulders of giants* to reach the conclusion that the strength of the gravity depends on the inverse square of the distance between the bodies. Years later when asked by Edmond Halley about the force needed to cause heavenly bodies to move in ellipses in accord with Kepler's laws, Newton claimed he had already solved the problem during the plague, but still took several years putting it to paper for Halley to publish. It is not clear when Newton realized the reciprocal but equal nature of the force on the two bodies, but that likely lead directly to his understanding that gravity is proportional to both their masses. Since Newton had no way of actually measuring the very weak gravitational forces between measurable masses on Earth, his law was stated only as proportions: F_{g} ∝ m_{1}m_{2}/R^{2}

• Finally **Henry Cavendish** performed a series of experiments a century later in 1798 in which he used one of the most sensitive devices known to physicists called a torsion balance (where the tiny force slightly twists a long, thin fiber) to measure the actual gravitational attraction between known masses. After careful repeated experiments, Cavendish added the constant to Newton's law, which enabled it to be used in the computation of the exact gravitational force, which acts between bodies: F_{g} = Gm_{1}m_{2}/R^{2} This constant used (G) in the System International has the value of 6.67 X 10^{-11} N•m^{2}/Kg^{2}. m_{1}m_{2} is the product of two attracting masses measured in Kg, and R is the distance between their centers of mass measured in meters. Note: time must be expressed in seconds if substituted into an equation using this value for G.

• We still do not know the fundamental nature of gravity. We know that the Gravitational constant G is fundamental and probably constant over time everywhere in the universe. We have no theory that predicts its precise value. Einstein proposed that a mass distorts the space around itself. This has been widely confirmed and provides a useful tool to study gravity far in the past and very distant from us. The familiar electric and magnetic fields are in some ways similar to gravity but differ in others. We are familiar with the quanta of radiation, the carrier of the electromagnetic force, the photon. We routinely detect and measure their propagation in space. But we have not yet detected the graviton, the carrier of gravity, although we have some evidence it exists from objects in distance space. We know in some detail how gravity works, but we have yet to confirm what causes it. It has been suggested that the mass which emits gravity is related to a yet undiscovered elementary particle called the Higgs boson. Finding the Higgs boson is the primary purpose for the Large Hadron Collider which has been built at CERN near Geneva Switzerland.

Name | Radius Of Body (x 10 ^{6} m) |
Mass (Kg) |
Average Orbital Radius (x 10 ^{11} m) |

Sun | 696 | 1.991 x 10^{30} |
Tiny; such tiny wobbles are the basis for detecting distant planets! |

Mercury | 2.43 | 3.2 x 10^{23} |
0.580 |

Venus | 6.073 | 4.88 x 10^{24} |
1.081 |

Earth | 6.3713 | 5.979 x 10^{24} |
1.4957 |

Mars | 3.38 | 6.42 x 10^{23} |
2.278 |

Jupiter | 69.8 | 1.901 x 10^{27} |
7.781 |

Saturn | 58.2 | 5.68 x 10^{26} |
14.27 |

Uranus | 23.5 | 8.68 x 10^{25} |
28.70 |

Neptune | 22.7 | 1.03 x 10^{26} |
45.00 |

Pluto (?) | 1.15 | 1.20 x 10^{22} |
59.00 |

- An asteroid orbits the sun with a mean orbital radius of 3.65 X 10
^{11}m. Using earth as a standard (T=365 days), calculate its ratio of R^{3}/T^{2}then use that ratio which is universal for everything which orbits the Sun to compute the period of the asteroid. - Using the data from the table and earth as a standard, compute the period of Neptune.
- The moon has a period of 27.3 days and an orbital radius of 3.90 X 10
^{5}Km. Compute the period of a research satellite which has an orbital radius of 7.1 X 10^{3}Km. (The ratio R^{3}/T^{2}for masses orbiting Earth is different from masses orbiting the Sun. But Since we already have a natural satellite, the moon, we can calculate the ratio for earth satellites.) - Geosynchronous orbits enable satellites to remain
fixed

over one spot on the equator. By definition, they must have a period of 1 day. Using the moon as a standard, compute the orbital radius of such a satellite. - Using the table, compute Jupiter's period.
- A satellite is placed into a very high orbit (1/2 of the moon's orbit). Compute the period of the satellite.

Note: time must be expressed in seconds when using G.

- A ball of lead has a mass of 12.0000Kg and is separated from another ball by a distance of 1.00230 m. If the mass of the second ball is 21.0000Kg, compute the gravitational attraction between them.
- Using the above table, compute the gravitational attraction between the sun and Saturn.
- Two identical balls are placed 2.566 m apart. If the gravitational attraction between them is 1.855 x 10
^{-12}N, compute the masses of the balls. - One ball has a mass, which is three times the other. If a distance of 1.450 m separates them and they experience a gravitational attraction of 2.650 x 10
^{-12}N, compute their masses. - Calculate the mass of the earth given the fact that the gravitational attraction between the earth and a 2.00 Kg mass is 19.60N and the radius of the earth is 6.40 X 10
^{6}m. - Compute the speed of a satellite, which is in orbit 255 Km above the surface of Mars.
- What is the period of a satellite, which has an altitude of 310 Km above earth?
- Consider the sun to be a satellite of the Milky Way galaxy. The orbital radius of the sun is 2.2 x 10
^{20}m. The period of one rotation is 2.5 x 10^{8}years. Find the mass of the galaxy. Assuming that the average star in the galaxy has the mass of the sun, compute the number of stars in the galaxy. Find the orbital speed of the sun around the center of the galaxy. - A geosynchronous orbit places a satellite over one spot on the earth. These orbits all have a radius of 4.23 X 10
^{7}m. Compute the speed of such a satellite. Compute the period of the satellite. - A moon of Saturn has an orbital radius of 1.87 X 10
^{8}m and an orbital period of about 23 hours. Use Newton's version of Kepler's third law to compute the mass of Saturn. Compare your result to the mass found in the table above.

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