Astronomy Assignment #5: Newton's Law of Gravity Solutions

Answer the following Review Questions from Nick Strobel’s AstronomyNotes: Chapter 5: Newton's Law of Gravity

Review Questions

1. What basic fundamental assumption did Newton make about the laws of nature on the Earth and in space?

Newton assumed that the same laws applied on Earth that were true in space. He rejected the Aristotelian idea that there were two different kinds of matter, terrestrial and celestial, with different rules. The entire Universe, Newton assumed, operated under one set of physical laws.

2. What things does gravity depend on?

Gravity depends on three things. First, the strength of the gravitational attraction between two objects depends directly on the mass of each object, or equivalently on the product of the masses of each object. Second, the strength of the gravitational attraction between two objects depends inversely on the separation of the two objects. In particular, the strength of gravity follows an inverse square relation so that the strength of the gravitational attraction between two bodies depends on . Finally, the strength of the gravitational attraction depends on the intrinsic strength of gravity as expressed through the Universal Gravitational Constant G. G has a very small value of 6.6710-11 in mks (meters-kilogram-seconds) units and illustrates that gravity is an intrinsically weak force. So weak, that for gravity to be “felt” between two objects, at least one of the objects must be astronomical in mass. 3. How does gravity vary with distance between objects and with respect to what do you measure the distances? The strength of the gravitational attraction between two objects depends inversely on the separation of the two objects. In particular, the strength of gravity follows an inverse square relation so that the strength of the gravitational attraction between two bodies depends on . The distance d is measured from the center of one object to the center of the other object so long as they are spherical in shape. 4. What would happen to the Earth's orbit if the Sun suddenly turned into a black hole (of the same mass)? Why? If the Sun suddenly turned into a black hole with the same mass as the Sun, the Earth would not feel any difference in the gravitational attraction to the now-black-hole-Sun. The gravitational attraction between any two objects depends only on the mass of each object and the separatin distance from their centers – not on the radius of either object. The Earth would get quite cold since black holes do not emit any radiation (i.e. heat). However, the Earth’s orbit would remain unchanged. 5. What important laws of planet motion can be derived from Newton's law of gravity? All three of Kepler’s Laws of Planetary Motion can be derived from Newton’s Law of gravity. In class we derived Kepler’s 3rd Law in its general form . On a personal note, I recall the first day of my 2nd year physics mechanics class in 1973 where the instructor entered the room and wrote his name on the chalk board, turned to the class and said “Let’s begin”. In 50 minutes he derived all three of Keplers Laws: Ellipical Orbits, Variable Speeds, P2=a3. I didn’t believe up to that time that anyone could do so much math in so short a time. 6. What is the difference between mass and weight? Mass is an intrinsic property of an object. The mass of an object will not change simply doe to its location or orientation. Mass is a measure of what a physicist calls inertia – the resistance to changes in an objects motion. In some ways you could think of mass as a measure of the number of electrons, protons, and neutrons in an object. Weight, on the other hand, depends an objects mass and on the strength of the gravitational field the object is immersed in. The weight of an object can change due to different positions on the Earth or on anther astronomical body. A very sensitive weight scale would show that you would weight just a tiny bit less if you were standing on the equator of the Earth rather than in Syracuse since the equatorial radius of the Earth is just slightly larger than the radius of the Earth here in Syracuse. Of course, on other planets or other celestial objects where the strength of gravity is very different due to the different mass of the planet or its different radius, you weight could be larger or smaller, but your mass would remain unchanged. 7. If Joe Astronaut has a mass of 40 kilograms on the Earth, how much mass would he have on an asteroid with 10 times less surface gravity than the Earth's surface gravity? Explain your answer. Mass is an intrinsic property of an object. The mass of an object will not change simply doe to its location or orientation. Mass is a measure of what a physicist calls inertia – the resistance to changes in an objects motion. In some ways you could think of mass as a measure of the number of electrons, protons, and neutrons in an object. Astronaut Joe would have the same mass on the asteroid as he would have on Earth or Mars or Jupiter of an the Sun….The mass does not depend on where the object is. 8. Why is gravity called an ``inverse square law''? Gravity called an ``inverse square law'' because its strength depends inversely on the square of the distance between the two objects. In mathematical lingo . To illustrate, if the separation between two objects is increased by a factor of two, the strength of their mutual gravitational attraction will decrease by a factor of four. I the separation between two objects is decreased by a factor of five, the strength of their mutual gravitational attraction will increase by a factor of 25. 9. What is the difference between a simple inverse relation and an inverse square relation? The difference between a simple inverse relation and an inverse square relation is that a simple inverse relation decreases much more slowly than an inverse square relation. For example, in a simple inverse relation doubling the distance would simply halve the strength. However, with an inverse square relation doubling the distance would decrease the strength by a factor of four. Inverse square relations rapidly weaken with distance. 10. If the Earth was 3 A.U. from the Sun (instead of 1 A.U.), would the gravity force between the Earth and the Sun be less or more than it is now? By how many times? If the Earth was 3 A.U. from the Sun (instead of 1 A.U.), the gravity force between the Earth and the Sun be less than it is now by 32 = 9 times? 11. If Mercury was 0.2 A.U. from the Sun (instead of 0.4 A.U.), would the gravity force between Mercury and the Sun be less or more than it is now? By how many times? If Mercury was 0.2 A.U. from the Sun (instead of 0.4 A.U.), the gravity force between Mercury and the Sun be more than it is now by 4 times.

12. Why do astronauts in orbit around the Earth feel ``weightless'' even though the Earth's gravity is still very much present? Astronauts in orbit around the Earth are in free fall as they orbit the Earth. They feel weightless because they are falling towards the Earth. They don’t hit the Earth because their horizontal velocity (i.e. orbital velocity) is so large that as they fall they move over the “edge” of the Earth. 13. What keeps satellites orbiting the Earth moving along their curved paths? The combination of gravity pulling a satellite down towards the Earth and the satellites large horizontal velocity that prevents the satellite from falling straight down keep a satellite in orbit. Without gravity pulling downward, the satellite would continue on straight linear path into space. With the large orbital horizontal velocity, the satellite would plunge straight down into the Earth. 14. What two things must be determined first in order to calculate the mass of a planet or a star? In order to calculate the mass of a planet or a star using one of its satellites two things must be observed. You must be able to measure the satellite’s orbital radius in meters from the center of the planet and the satellite’s orbital period in seconds. Once these two quantities are known, then you can use the modified form of Kepler’s 3rd Law to calculate the mass of the planet of star. 15. Jupiter's moon Io has about the same mass as the Moon and orbits Jupiter at about the same distance that the Moon orbits the Earth (center to center). Then why does Io take only 1.8 days to orbit Jupiter but our Moon takes 27.3 days to orbit the Earth? Io takes only 1.8 days to orbit Jupiter even though it has nearly the same orbital radius as our Moon because the mass of Jupiter is so much larger that the mass of the Earth. Recall that the formula for the orbital velocity is . Since Jupiter is over 300 times more massive than the Earth, the orbital velocity of Io will be about times faster than our Moon. Since the orbital radii of Io and the Moon are about equal, Io ought to complete on orbit around Jupiter about 17 times quicker than our Moon does to orbit the Earth. (note: ) 16. Astronomers were able to accurately measure the orbital periods of the moons of Jupiter since the time of Galileo, so why was an accurate value for Jupiter's mass not found for over 300 years until the astronomical unit was measured accurately? In order to calculate the mass of a planet or a star using one of its satellites two things must be observed. You must be able to measure the satellite’s orbital radius in meters from the center of the planet and the satellite’s orbital period in seconds. Once these two quantities are known, then you can use the modified form of Kepler’s 3rd Law to calculate the mass of the planet of star. Galileo did not know the satellite’s orbital radius in meters from the center of the planet because he did not know how far away Jupiter was from the Earth in meters. This was because the exact size of the AU was unknown in Galileo’s time.

17. Which would have a shorter orbital period, a planet orbiting a massive star at 3 A.U. or a planet orbiting a low-mass star at 3 A.U.? Explain your answer. Recall that the formula for the orbital velocity is . So a planet orbiting a high mass star will have a higher velocity than a planet orbiting a low mass star, as long as the orbital radius of the planet is unchanged. The planet with the shortest orbital period will be the faster planet and that will be the planet orbiting the higher mass star. 18. If a planet orbiting a massive star has the same orbital period as a planet orbiting a low-mass star, which of the planets orbits at a greater distance from its star? Explain your answer. Recall that the formula for the orbital velocity is . The period of the planet’s orbit will be given by . A little algebra will lead to. So if both planets have the same orbital period, then the planet around the more massive star must have a larger orbital radius so that the term is the same for both planets.

19. What two things does the escape velocity depend on? The escape velocity is the minimum velocity an object must have to break free of the gravitational hold of a body. The old adage “what goes up must come down” is not true if the object going up has a velocity at least as great as the escape velocity. An object moving with at least the escape velocity will never fall back to the object it left, but will continue to move farther away to “infinity and beyond” (My apologies to Buck Light-year). The escape velocity of an object, like the orbital velocity, depends upon the mass of the body it is trying to escape from and the distance from the center of the object it is trying to escape from: . 20. Why does the planet Saturn with over 95 times the Earth's mass have a smaller escape velocity at its cloud tops than the Earth has at its cloud tops? Our text book author has made an error in this question. In fact the escape velocity from the cloud tops on Saturn is greater than the escape velocity from the cloud tops of the Earth. However, the difference in those two escape velocities is less than what the difference is mass would suggest. The escape velocity of an object depends on physical parameters: (1) the mass of the central object and (2) the radial distance from which the object is trying to escape. It is true that Saturn’s mass is about 95 times larger than the Earth’s mass and that this alone would suggest escape from Saturn would require a higher velocity. However, the cloud tops of Saturn are 9.5 times farther from the center of Saturn that the Cloud tops of Earth are from the Earth’s center. So this larger distance from the center tends to lessen the escape velocity. To quantitatively determine the ratio of escape velocities perform the following algebraic manipulations:

So the escape velocity from the cloud tops of Saturn should be about 3.16 times the escape velocity from the cloud tops of the Earth. Checking the actual escape velocities with the National Space Science Data Center in Maryland shows that the escape velocity from the cloud tops of Saturn is 35.5 km/s and the escape velocity from the cloud tops of the Earth is 11.2 km/s. The ratio of the two is 3.17 consistent with our argument. 21. How can you predict the orbital period of Jupiter's satellite Europa from observations of the other jovian moon Io? The ratio of follows from Kepler’s 3rd Law as derived by Newton. So the ratio must be a constant since the right-hand-side of the equation above is a constant. So we can equate that ratio for different satellites orbiting Jupiter as shown below:

Solving for the orbital period of Europa gives

or even more simply as

The problem does not ask us to solve this problem only to describe how to solve it – So we stop here. 22. Earth 220 kilometers above the surface and takes about 1.5 hours to complete one orbit. How can you find out how far up to put a communication satellite, so that it takes 24 hours to circle the Earth? (Such an orbit is called a ``geosychronous orbit'' because the satellite remains above a fixed point on the Earth.)

This is a complex problem that can be broken into several simple parts.

First express the period of the orbiting satellite in seconds

Second we remind ourselves of the two expressions for the orbital velocity of satellite around the Earth:

and

Combining these two expressions yields

This can be algebraically re-arranged to read

Notice that the right hand side of the equation above is a constant. So the ratio

is always a constant for any satellite in any orbit around the Earth. Thus…for any two satellites designated A and B the ratios shown below must be equal.

Now we just fill in the values given in the problem

Solve the equation for x So, this geosynchronous satellite must have an orbital radius of 41,895 km. Thus, the satellite must orbit above the surface of the Earth. 23. Find out how long it will take the Cassini spacecraft to travel to Saturn 9.5 A.U. from the Sun Earth. (Hint: Lookup the Hohmann Transfer Orbit on the internet.) Note: I changed this question a bit. See above. The preferred method to travel between the planets is through the Hohmann Transfer Orbit. In this orbit, if the destination is a superior planet, the perihelion distance of the orbit is 1 AU and the aphelion distance is the semi- major axis of the destination planet. See figure. For a trip to Saturn the aphelion distance will be 9.5 AU. So the semi-major axis of this Hohmann Transfer Orbit to Saturn is simply

Now use Kepler’s 3rd Law to solve for the period of this transfer orbit:

The Hohmann Transfer Orbit has a period of 12.o years. Since the trip to Saturn is only half the orbit, it will take the Cassini spacecraft 6.0 years to travel to Saturn from the Sun Earth.