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11.1.12.1: Illustrations

  • Page ID
    34121
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    Illustration 1: Projectile and Satellite Orbits

    Newton, in his consideration of gravity, realized that any projectile launched from the surface of Earth is, in a sense, an Earth satellite (if only for a short time). For example, in this animation a ball thrown from a tall building sails in a modest orbit that soon intersects Earth not far from its point of launch.

    If the ball were fired with a greater initial velocity, it would travel farther. Further increasing the speed would result in ever larger, rounder elliptical paths (they would be elliptical if Earth was not in the way) and more distant impact points. Finally, at one particular launch speed the ball would glide out just above Earth's surface all the way around to the other side without ever striking the ground. At a greater speed the orbit of the ball will be a circle.

    At successively greater launch speeds the ball would move in an ever-increasing elliptical orbit until it moved so fast that it would sail off in an open parabolic or (with an even faster launch speed) into a still flatter, hyperbolic orbit, never to come back to its starting point.

    This Illustration lets you change the launch speed (but maintain the launch direction) by mouse click. Click "+" to increase launch speed and click "-" to decrease launch speed. Press "Start" to fire the ball. Press "Reset" to change parameters back to default values. The red arrow represents the velocity vector. Click the left mouse button near the tip of the arrow and drag the mouse to change the ball's initial velocity (both direction and speed). Click the right mouse button to stop the animation; press it again to resume. When will the ball start to go around without striking the ground? When will the motion become a circular motion? Try clicking the "full" checkbox and find out what happens.

    Illustration authored by Fu-Kwun Hwang and Mario Belloni.
    Applet authored by Fu-Kwun Hwang, National Taiwan Normal University.

    Illustration 2: Orbits and Planetary Mass

    When we consider the elliptical orbits of the planets (Kepler's first law), we assume that the Sun is stationary at one focus of the ellipse. Why does this happen? The mass of the Sun must be much, much greater than the mass of the planets in order for the motion of the Sun to be ignored. But how large does the mass of the Sun need to be in order to achieve this idealized planetary motion? For reference, the Sun is about \(1000\) times heavier than Jupiter (the most massive planet) and about \(100\) million (\(10^{8}\)) times more massive than the least massive planet, Pluto. Restart.

    As you vary the mass ratio in the animation, the mass of the system changes such that the product of the masses, \(m_{1}\ast m_{2}\), remains the same. Therefore as you change the mass ratio, the force will remain the same for the same separation between the masses.

    The 1000:1 Mass animation closely resembles the Sun and Jupiter system (the distance is given in astronomical units [A.U.] and the time is given in 108 seconds). The green circle is like the Sun, while the red circle is like Jupiter. The force of attraction due to gravity is shown by the blue arrows (not shown to scale), and the relative kinetic energies are shown as a function of time on the graph (note that for this animation the unit for the kinetic energy is not given since we are comparing the relative amount for each object). Also note that the eccentricity of the orbit \(e = 0.048\), the perihelion and aphelion distances, and the planet's period closely match those of Jupiter.

    In the 100:1 Mass animation does the "Sun" remain motionless? What about the 10:1 Mass animation? The 2:1 Mass animation? The 1:1 Mass animation? What do you think this means for planetary dynamics in our solar system?

    For elliptical orbits, the force due to gravity changes magnitude since the separation changes. But at every instant, the forces of gravitational attraction (the force of the green circle due to the red circle and the force of the red circle due to the green circle) are always the same. This is Newton's third law. It is not too surprising that the law of universal gravitation (described by Newton) contains the third law (also described by Newton).

    At the same time, what happens to the kinetic energy of the system as a function of time? It too changes. But why? As the separation between the "Sun" and the "planet" changes, the gravitational potential energy of the system changes too. While the kinetic energy of the system changes, the sum of the kinetic energy and the potential energy of the system must—and does—remain a constant throughout the motion of the objects.

    Illustration 3: Circular and Noncircular Motion

    A planet (green) orbits a star (orange) as shown in the two animations. Restart. One animation depicts the Uniform Circular Motion of a planet and the other one depicts the Noncircular Motion of a planet (position is given in \(10^{3}\) km and time is given in years). This Illustration will compare the two motions by focusing on the velocity and the acceleration of the planet in each of the animations.

    Start the Uniform Circular Motion animation of the planet and watch its motion. How would you describe the motion of the planet (consider velocity and acceleration)? The speed of the planet is certainly a constant since the motion of the planet is uniform. But using our usual xy coordinates, the velocity certainly changes with time. Recall that the term velocity refers to both the magnitude and direction. However, if we use the radial and tangential directions to describe the motion of the planet, the velocity can be described as tangential, and the acceleration is directed along the radius (the negative of the radial direction). Click here to view the velocity vector (blue) and a black line tangent to the path. Click here to view the acceleration vector (red), also. Notice that the acceleration vector points toward the center star.

    Start the Noncircular Motion animation of the planet and watch its motion. How would you describe the motion? How would you now describe the motion of the planet (consider velocity and acceleration)? The speed of the planet is certainly no longer a constant, as the motion of the planet is no longer uniform. Again using our usual \(xy\) coordinates, the velocity certainly changes with time; now both the direction and the magnitude change. However, if we use the radial and tangential directions to the path of the planet, the velocity can be described as tangential and the acceleration is directed along the radius. Click here to view the velocity vector (blue) and click here to view the acceleration vector (red), also. Notice that the velocity and the acceleration are no longer perpendicular for most of the orbit of the planet.

    Notice that between points A and C the planet is speeding up, and between points C and A the planet is slowing down. This means that at points A and C the tangential component of acceleration is zero. It turns out that for a planet orbiting a star, if there are no other planets or stars nearby, the acceleration of the planet is directed exactly toward the star whether the motion of the planet is uniform or not.

    Illustration authored by Aaron Titus and Mario Belloni.

    Illustration 4: Angular Momentum and Area

    In the absence of a net external torque acting on a system, a particle's angular momentum remains constant. For this discussion, the particle is free, so angular momentum should be conserved. Is there a different way to state the concept of angular momentum conservation? There may be. Consider the statement, Does a particle sweep out equal areas in equal times (with respect to any origin)? Specifically, in this Illustration, does a free particle moving in a straight line sweep out equal areas in equal times?

    Press "start" to begin the animation, and let the show begin: A black dot will move freely from left to right. Different colors show the area the particle sweeps out with respect to some fixed point (the origin). Do all the areas have the same size? Click within each area and see what will happen. Certainly from mathematical equations we know the area of a triangle \(=\) width \(\ast\) height\(/2\). All of the areas have the same height and the same width (\(= v_{x}\ast dt\)).

    Kepler's second law states, During equal time intervals, the radius vector from the sun to a planet sweeps out equal areas. What does this tell you about the angular momentum of the planets? What does this tell you about the motion of the planets?

    Illustration authored by Fu-Kwun Hwang and Mario Belloni.
    Applet authored by Fu-Kwun Hwang, National Taiwan Normal University.

    Illustration 5: Kepler's Second Law

    A planet orbits a star under the influence of gravity (distance is given in astronomical units [A.U.] and time is given in years; the total area swept out by the planet's orbit is given in A.U.\(^{2}\)). The animation begins from the point of aphelion, the point where the planet is farthest from the star. The planet's orbit is elliptical, and its trail is shown as it orbits the star. Kepler's second law states that the planets sweep out equal areas in their orbits in equal times. What does this mean for the planet's orbit? If the planet had a circular orbit, the planet would undergo uniform circular motion and Kepler's second law is just a statement of equal speed; it confirms the statement of uniform circular motion. For elliptical orbits, therefore, the planet's motion must not be uniform. Restart.

    Starting at \(t = 0\), run the animation for \(3\) years (not real time, animation time!). How much area has been swept out by the planet in this time interval? There is \(28.43\text{ A.U.}^{2}\) swept out. What about from \(3\) to \(6\) years? Again \(28.431\text{ A.U.}^{2}\) is swept out. Does it matter where you are in the orbit? No. Try it for yourself. While the planet is closer to the star, its speed increases; when the planet is farther away from the star, its speed decreases.

    So what does Kepler's second law really tell us? The sweeping out of equal areas is equivalent to telling us that angular momentum is conserved! We know that if angular momentum is conserved (see Chapter 11), there is no net torque. Here the only force on the planet is gravity, and gravity cannot create a torque because the radius arm and the force are always in the same direction.

    Illustration authored by Steve Mellema, Chuck Niederriter, and Mario Belloni.
    Script authored by Steve Mellema and Chuck Niederriter.

    Illustration 6: Heliocentric vs. Geocentric

    Does Earth orbit around the Sun or does the Sun orbit Earth? For a very long time people thought that Earth was stationary (as the argument goes, otherwise the birds would be ripped from their perches!) and the Sun orbited Earth. This belief is where we get the terms sunrise and sunset. But the Sun does not orbit Earth; it is the other way around. In addition, the motion of the planets, as seen from the reference frame of the Sun (the heliocentric reference frame) is rather simple. But from the perspective of each individual planet (the geocentric reference frames of the Inner Planet and the Outer Planet) the motions of the other planets are rather complicated. The geocentric view is exactly what we see on Earth when we look at the sun and the other planets of the Solar System.

    In this Illustration two planets (the red circle is the inner planet and green circle is the outer planet) orbit a central star (the black circle) as shown in the animation. Along with the animation from the star's reference frame, the heliocentric point of view, two other animations show the motion as seen from each of the planets' reference frames, the geocentric points of view.

    As you view the animation, keep in mind that in the Inner Planet animation, if Earth is the red planet, the green planet behaves like Mars as seen from Earth, while in the Outer Planet animation, if Earth is the green planet, the red planet behaves like Venus as seen from Earth.

    Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.


    This page titled 11.1.12.1: Illustrations is shared under a CC BY-NC-ND license and was authored, remixed, and/or curated by Wolfgang Christian, Mario Belloni, Anne Cox, Melissa H. Dancy, and Aaron Titus, & Thomas M. Colbert.