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ASTR Spring 2008 Joel E. Tohline, Alumni Professor 247 Nicholson Hall [Slides from Lecture17]

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Nicolaus Copernicus (1473 – 1543)

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Johannes Kepler (1571 – 1630)

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Galileo Galilei (1564 – 1642)

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Isaac Newton ( )

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Algebraic Relations understood by Isaac Newton Kepler’s 3 rd Law of planetary motion: Relating distance, time, and speed of travel for circular motion: Acceleration (A) required to achieve circular motion:

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A

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Algebraic Relations understood by Isaac Newton Kepler’s 3 rd Law of planetary motion: Relating distance, time, and speed of travel for circular motion: Acceleration (A) required to achieve circular motion:

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Gravitational Acceleration Isaac Newton hypothesized that the (nearly) circular motion of the planets is caused by a “gravitational force” that is exerted by the Sun on each of the planets. As a result of this radially directed force, each planet “feels” a constant acceleration (A) whose magnitude only depends on the planet’s distance from the Sun as follows:

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Gravitational Acceleration Isaac Newton hypothesized that the (nearly) circular motion of the planets is caused by a “gravitational force” that is exerted by the Sun on each of the planets. As a result of this radially directed force, each planet “feels” a constant acceleration (A) whose magnitude only depends on the planet’s distance from the Sun as follows:

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Gravitational Acceleration In summary, Newton hypothesized that a planet orbiting a distance “r” from the Sun feels a gravitational acceleration If this is true, then Kepler’s 3 rd Law is actually predictable!

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Algebraic Relations understood by Isaac Newton Newton’s Proposed Law of Gravitation: Relating distance, time, and speed of travel for circular motion: Acceleration (A) required to achieve circular motion:

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Consequence of Newton’s Law If the acceleration “A” causing circular motion of planetary orbits is due to Newton’s hypothesized “A grav ”, then … But, since, it must also be true that, In this way, Newton was able to derive Kepler’s 3 rd Law!

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Consequence of Newton’s Law If the acceleration “A” causing circular motion of planetary orbits is due to Newton’s hypothesized “A grav ”, then … But, since, it must also be true that, In this way, Newton was able to derive Kepler’s 3 rd Law!

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Consequence of Newton’s Law If the acceleration “A” causing circular motion of planetary orbits is due to Newton’s hypothesized “A grav ”, then … But, since, it must also be true that, In this way, Newton was able to derive Kepler’s 3 rd Law!

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Consequence of Newton’s Law If the acceleration “A” causing circular motion of planetary orbits is due to Newton’s hypothesized “A grav ”, then … But, since, it must also be true that, In this way, Newton was able to derive Kepler’s 3 rd Law!

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Consequence of Newton’s Law If the acceleration “A” causing circular motion of planetary orbits is due to Newton’s hypothesized “A grav ”, then … But, since, it must also be true that, In this way, Newton was able to derive Kepler’s 3 rd Law!

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Consequence of Newton’s Law Okay, I admit that the derivation of Kepler’s 3 rd Law is no big deal because Newton formulated his Law of Gravitation based on Kepler’s description of planetary motions. Are there other consequences of Newton’s Law? Newton asked whether his Law of Gravitation might also apply to the Earth-Moon system. In particular, he wondered … Is the “gravitational” acceleration that holds us onto the surface of the Earth also responsible for keeping the Moon in its orbit?

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Consequence of Newton’s Law Okay, I admit that the derivation of Kepler’s 3 rd Law is no big deal because Newton formulated his Law of Gravitation based on Kepler’s description of planetary motions. Are there other consequences of Newton’s Law? Newton asked whether his Law of Gravitation might also apply to the Earth-Moon system. In particular, he wondered … Is the “gravitational” acceleration that holds us onto the surface of the Earth also responsible for keeping the Moon in its orbit?

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Consequence of Newton’s Law Okay, I admit that the derivation of Kepler’s 3 rd Law is no big deal because Newton formulated his Law of Gravitation based on Kepler’s description of planetary motions. Are there other consequences of Newton’s Law? Newton asked whether his Law of Gravitation might also apply to the Earth-Moon system. In particular, he wondered … Is the “gravitational” acceleration that holds us onto the surface of the Earth also responsible for keeping the Moon in its orbit?

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Algebraic Relations understood by Isaac Newton Newton’s Proposed Law of Gravity (applied to Earth): Relating distance, time, and speed of travel for circular motion: Acceleration (A) required to achieve circular motion:

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Consequence of Newton’s Law Following the same steps as before, Newton deduced that objects orbiting the Earth should obey the relation, But what is the value of K Earth ? At the surface of the Earth (r = R = 6.4 x 10 6 m), an apple that falls from a tree experiences a downward acceleration of 9.8 m/s 2. Hence, from his Law of Gravitation, Newton obtained …

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Consequence of Newton’s Law Following the same steps as before, Newton deduced that objects orbiting the Earth should obey the relation, But what is the value of K Earth ? At the surface of the Earth (r = R = 6.4 x 10 6 m), an apple that falls from a tree experiences a downward acceleration of 9.8 m/s 2. Hence, from his Law of Gravitation, Newton obtained …

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Consequence of Newton’s Law Following the same steps as before, Newton deduced that objects orbiting the Earth should obey the relation, But what is the value of K Earth ? At the surface of the Earth (r = R = 6.4 x 10 6 m), an apple that falls from a tree experiences a downward acceleration of 9.8 m/s 2. Hence, from his Law of Gravitation, Newton obtained …

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Consequence of Newton’s Law Following the same steps as before, Newton deduced that objects orbiting the Earth should obey the relation, But what is the value of K Earth ? At the surface of the Earth (r = R = 6.4 x 10 6 m), an apple that falls from a tree experiences a downward acceleration of 9.8 m/s 2. Hence, from his Law of Gravitation, Newton obtained …

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Consequence of Newton’s Law Following the same steps as before, Newton deduced that objects orbiting the Earth should obey the relation, But what is the value of K Earth ? At the surface of the Earth (r = R = 6.4 x 10 6 m), an apple that falls from a tree experiences a downward acceleration of 9.8 m/s 2. Hence, from his Law of Gravitation, Newton obtained …

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Consequence of Newton’s Law So, knowing that the Moon orbits the Earth in a (nearly) circular orbit with radius r = 60 R Newton realized that he could predict the orbital period of the Moon …

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Consequence of Newton’s Law So, knowing that the Moon orbits the Earth in a (nearly) circular orbit with radius r = 60 R Newton realized that he could predict the orbital period of the Moon …

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Consequence of Newton’s Law So, knowing that the Moon orbits the Earth in a (nearly) circular orbit with radius r = 60 R Newton realized that he could predict the orbital period of the Moon …

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Consequence of Newton’s Law So, knowing that the Moon orbits the Earth in a (nearly) circular orbit with radius r = 60 R Newton realized that he could predict the orbital period of the Moon …

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Consequence of Newton’s Law So, knowing that the Moon orbits the Earth in a (nearly) circular orbit with radius r = 60 R Newton realized that he could predict the orbital period of the Moon …

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Amazing! Through his hypothesized Law of Gravitation, Isaac Newton was able to combine his knowledge of the distance between the Earth and the Moon with a measurement of the acceleration due to gravity at the surface of the Earth to predict (or explain) the orbital period of the Moon! This ranks as one of the greatest discoveries in human history, as it provided for the first time a quantitative description of the behavior of the physical universe.

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Amazing! Through his hypothesized Law of Gravitation, Isaac Newton was able to combine his knowledge of the distance between the Earth and the Moon with a measurement of the acceleration due to gravity at the surface of the Earth to predict (or explain) the orbital period of the Moon! This ranks as one of the greatest discoveries in human history, as it provided for the first time a quantitative description of the predictable behavior of the physical universe.

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Related Issues By convention, the constant “K” – that appears in Kepler’s 3 rd Law as well as in Newton’s Law of Gravitation – is now written in terms of the mass of the (dominant) object that is exerting the acceleration. Specifically, Hence, Newton’s Law of Gravitation is written in the form …

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Related Issues Newton’s Law also naturally explains other, noncircular “orbital” trajectories (for example, elliptical orbits, and parabolic & hyperbolic trajectories) – all trajectories are “conic sections” In other words, Newton’s Law of Gravitation also explains Kepler’s 1 st Law of planetary motion!

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Predictability … Weather satellites are launched into “geosynchronous” orbits where they appear to hover over a fixed location on the Earth’s surface. What is the radius of all geosynchronous orbits? The space shuttle is usually placed in a very low Earth orbit, that is, the radius of its orbit is essentially 1 R . What is the shuttle’s orbital period? The cheapest way to travel from Earth to Mars is to coast along a heliocentric orbit whose perhelion is the Earth-Sun distance (1 AU) and whose aphelion is the Mars-Sun distance (1.88 AU). What is the length of the semi-major axis of such an orbit? How long will it take to coast from the Earth to Mars along this orbit?

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Predictability … Weather satellites are launched into “geosynchronous” orbits where they appear to hover over a fixed location on the Earth’s surface. What is the radius of all geosynchronous orbits? The space shuttle is usually placed in a very low Earth orbit, that is, the radius of its orbit is essentially 1 R . What is the shuttle’s orbital period? The cheapest way to travel from Earth to Mars is to coast along a heliocentric orbit whose perhelion is the Earth-Sun distance (1 AU) and whose aphelion is the Mars-Sun distance (1.88 AU). What is the length of the semi-major axis of such an orbit? How long will it take to coast from the Earth to Mars along this orbit?

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Predictability … Weather satellites are launched into “geosynchronous” orbits where they appear to hover over a fixed location on the Earth’s surface. What is the radius of all geosynchronous orbits? The space shuttle is usually placed in a very low Earth orbit, that is, the radius of its orbit is essentially 1 R . What is the shuttle’s orbital period? The cheapest way to travel from Earth to Mars is to coast along a heliocentric orbit whose perihelion is the Earth-Sun distance (1 AU) and whose aphelion is the Mars-Sun distance (1.524 AU). What is the length of the semi-major axis of such an orbit? How long will it take to coast from the Earth to Mars along this orbit?

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