Physics; Orbital Motion

Introduction

Even before they had telescopes, scientists were studying how planets orbited the Sun. Two men, Johannes Kepler and Tycho Brahe, are especially notable because their work on planets is still used today. Kepler discovered that the planets moved in elliptical orbits, and Brahe made the first accurate planetary observations. Kepler (1571-1630), born in Germany, established the three laws of planetary motion. Like Nicholas Copernicus (1473-1543), Kepler believed that Earth and the other planets revolve around the Sun. He wanted to find an explanation for the orbital paths of the planets. In 1596, Kepler published Mysterium Cosmographicum, in which he postulated that the orbits of the six known planets could be arranged in spheres nested around the five platonic solids. These solids are the octahedron, icosahedron, dodecahedron, tetrahedron, and cube. He knew he had to validate his model, so in 1600 he became the assistant of Tycho Brahe, a well-known Danish astronomer.

Stuff I supposedly need to remember

Remember that Kepler developed three laws to describe the motion of the planets across the sky. Kepler's first law (the law of orbits) states that all planets move in elliptical orbits with the Sun at one focus. Because the orbit is an ellipse, he also determined that the distance between the planet and the Sun must constantly change as a planet orbits around the Sun.

Orbital Paths

Remember these key points about uniform circular motion: An object travels at a constant speed in a circular path. The velocity is changing because the direction of the speed is changing, thus the object is accelerated. The magnitude of the velocity vector remains constant and always points in the direction of the motion.

Tycho Brahe

Brahe (1546-1601) established an observatory on an island where he spent 20 years recording the most precise astronomical observations the Western world had ever seen. Brahe was less competent in constructing models that would substantiate or explain his observations, so he hired Kepler. But Brahe feared that Kepler would surpass him as the best astronomer of his day. So Brahe restricted Kepler's access to his research and assigned Kepler to investigate the orbital path of Mars. After Brahe died just 18 months later, Kepler was able to use Brahe's research and his own work to formulate the correct model of the solar system and develop his three laws of planetary motion. His findings were remarkable because they were made before the invention of the telescope (which happened in 1608). Kepler's Laws still hold true: 1. All planets move in elliptical orbits with the Sun at one of the focal points. 2. The radius vector drawn from the Sun to a planet covers equal areas in equal time intervals. 3. The square of the orbital period of any planet is proportional to the cube of its average distance from the Sun.

This mess

Earth's gravitational force provides the centripetal force on a satellite. The satellite is always moving in a direction perpendicular to the centripetal force that acts upon it. This force is necessary to give the satellite centripetal acceleration. The magnitude of this force can be derived from Newton's second law (F = ma) and the law of universal gravitation.

Keplers law?

Kepler's second law (the law of areas) states that a line that connects a planet to the Sun covers equal areas in equal times as the planet travels around the ellipse. He discovered that the planets do not move around the Sun at a uniform speed. The planet's speed is greatest when it is closest to the Sun in its orbit —the perihelion. And it is slowest when it is farthest from the Sun —the aphelion. (Helios is the Greek word for sun.) For objects that orbit Earth, the closest point is the perigee and the farthest point is the apogee. (Geo is the Greek word for earth.) Kepler's first two laws were published in 1609 during his eight years in Prague. Kepler's third law was not published until 1619. We'll look at that law next.

Law of periods

Kepler's third law (the law of periods) states that the square of the period of any planet is proportional to the cube of the average radius of its orbit. Ks = T^2/r^2 where Ks = 2.97 × 10-19 s2/m3 In other words, Kepler's third law explains that the time it takes for one revolution of a planet around the Sun (the period) increases rapidly with the radius of the planet's orbit. Ks is independent of the mass of the planet. So Kepler's third law applies to any body that orbits the Sun. These bodies include planets, asteroids, comets, and other objects.

More stuff w/ this picture https://cdn.app.edmentum.com/EdAssets/b108bcd068a44751b5784e2a5f46306d?ts=636050288657230000

Let's look at centripetal force. It's that feeling you have of being forced to one side during the sharp turn of a car or to the back of a rollercoaster seat. Look again at Newton's law of universal gravitation, the force of gravity between two objects: Newton's first law says that a linear force is required to give an object a linear acceleration. His second law says that F = ma. In this image, note that the object is traveling in a circle. Look at the velocity and centripetal acceleration. We can conclude that any object moving in a circular path must have a net force acting on it that is directed toward the center of the circular path. Now let's see how Kepler's third law—the square of the orbital period of any planet is proportional to the cube of its average distance from the Sun—was derived. You aren't required to learn each step, but it's important to understand why certain variables and numbers show up in the final equation.

More Formulas

Let's review period and frequency for an object traveling in a circle. The image shows an object traveling at speed v around a circle of radius r. The period T is the time it takes for one complete revolution. The frequency f is the number of revolutions per second. This inverse relationship is shown in the formula T = 1/f. Again, for an object traveling in a circle of circumference 2πr at constant speed v, the relationship is v = 2(pi)r/t .

Imagining things isn't my favorite

Now imagine that you're an engineer responsible for placing a weather satellite in a circular orbit around Earth with an altitude of 120 kilometers. Let's calculate the velocity of the satellite and its period. Earth's mean radius is 6.37 × 106 meters, its mass is 5.98 × 1024 kilograms; the gravitational constant G is 6.67 × 10-11 N·m2/kg2, and the altitude is 1.2 × 105 m. Find the velocity rt = re + h, where rt = the total radial distance from the center of Earth to the center of the satellite, re = the mean radius of Earth, and h = the height above the surface of Earth.

Satelite?

Now, let's apply Newton's second law to a satellite: The critical step is to equate gravitational force to centripetal force. GMeM/r^2 = mv^2/r Then you can use algebra to get the velocity of a satellite. v = (sqrt)Gme/r Another one: T = (sqrt) 2(pi)r^3/Gms These equations represent a quantitative relationship for Kepler's third law. The law is not specific to the Sun and planets. It shows how a satellite in orbit at a constant height above Earth moves in uniform circular motion. It's important to know that a circular orbit is simply a special case of an elliptical orbit. It's also important to know that the closer the foci of an ellipse are to each other, the closer the ellipse is to being a circle.

Universal Gravitation

Sir Isaac Newton (1642-1727) built on the work of other scientists, including Brahe and Kepler. Newton's theory of gravity was based on the ideas of other scientists who tried to correctly understand planetary motion; describe planetary orbital paths; and develop rules to explain the orbital paths of planets. Newton formulated this equation for his law of universal gravitation, which states that the force of attraction between two planets is proportional to the product of the masses and inversely proportional to the square of the distance between their centers: F = Gm1m2/r^2 where G is the gravitational constant 6.67 × 10-11 N·m2/kg2; m1 is the mass of one object (for example, a planet); and m2 is the mass of a second object (for example, the Sun).

Not keplers law

The force of attraction between two planets is proportional to the product of the masses and inversely proportional to the square of the distance between their centers.