Student Exploration: Orbital Motion – Kepler’s Laws

University:
High School
Course:
High School Physics
Academic year:

2025
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8
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Name: citlali Date: Student Exploration: Orbital Motion – Kepler’s Laws Directions: Follow the instructions to go through the simulation. Respond to the questions and prompts in the orange boxes. Vocabulary: astronomical unit, eccentricity, ellipse, force, gravity, Kepler’s first law, Kepler’s second law, Kepler’s third law, orbit, orbital radius, period, vector, velocity Prior Knowledge Questions (Do these BEFORE using the Gizmo.) 1. The orbit of Halley’s Comet, shown at right, has an oval shape. In which part of its orbit do you think Halley’s Comet travels fastest? Slowest? Mark these points on the diagram at right. 2. How might a collision between Neptune and Halley’s Comet affect Neptune’s orbit? it probably wouldn't have a massive effect on neptune orbit because neptune is bigger than the comet Gizmo Warm-up The path of each planet around the Sun is determined by two factors: its current velocity (speed and direction) and the force of gravity on the planet. You can manipulate both of these factors as you investigate planetary orbits in the Orbital Motion – Kepler’s Laws Gizmo. On the CONTROLS pane of the Gizmo, turn on Show trails and check that Show vectors is on. Click Play ( 1. What is the shape of the planet’s orbit? ). oval 2. Watch the orbit over time. Does the orbit ever change, or is it stable? it gets slower over time 3. Click Reset ( ). Drag the tip of the purple arrow to shorten it and reduce the planet’s initial velocity. Click Play. How does this affect the shape of the orbit? it becomes smaller and more flat Reproduction for educational use only. Public sharing or posting prohibited. © 2020 ExploreLearning™ All rights reserved Activity A: Shape of orbits Get the Gizmo ready: ● Click Reset. ● Turn on Show grid. Introduction: The velocity of a planet is represented by an arrow called a vector. The vector is described by two components: the i component represents east-west speed and the j component represents north-south speed. The unit of speed is kilometers per second (km/s). Question: How do we describe the shape of an orbit? 1. Sketch: The distance unit used here is the astronomical unit (AU), equal to the average Earth-Sun distance. Place the planet on the i axis at r = –3.00i AU. Move the velocity vector so that v = -8.0j km/s (|v| = 8.00 km/s). The resulting vectors should look like the vectors in the image at right. (Vectors do not have to be exact.) Click Play, and then click Pause ( ) after one revolution. Draw the resulting orbit on the grid to the right. 2. Identify: The shape of the orbit is an ellipse, a type of flattened circle. An ellipse has a center (C) and two points called foci (F1 and F2). If you picked any point on the ellipse, the sum of the distances to the foci is constant. For example, in the ellipse at left: a1 + a2 = b1 + b2 Turn on Show foci and center. The center is represented by a red dot, and the foci are shown by two blue dots. What do you notice about the position of the Sun? it is the midpoint. then it can form a right angle at the planets orbit point 3. Experiment: Try several other combinations of initial position and velocity. A. What do you notice about the orbits? the orbits consist of two orbitals which are circular orbit ans elliptical orbit B. What do you notice about the position of the Sun? it is stable at midpoint You have just demonstrated Kepler’s first law, one of three laws discovered by the German astronomer Johannes Kepler (1571–1630). Kepler’s first law states that planets travel around the Sun in elliptical orbits with the Sun at one focus of the ellipse. Reproduction for educational use only. Public sharing or posting prohibited. © 2020 ExploreLearning™ All rights reserved 4. Observe: Use the Gizmo to create an orbit that is nearly circular. Then create an orbit that is flattened. Observe the foci in each ellipse. A. What do you notice about the spacing of the foci when the ellipse is very round? it is closer B. What do you notice about the spacing of the foci when the ellipse is very flat? it is farther away 5. Calculate: The eccentricity of an ellipse is a number that describes the flatness of the ellipse. Eccentricity is equal to the distance between foci divided by the total width of the ellipse. There are no units for eccentricity. Click Reset. Move the planet to r = –5.00i AU (does not have to be exact) and drag the velocity vector to set the velocity close to –8.0j km/s. Click Play, and then click Pause after one full revolution. A. What is the distance between the foci? -5.3i-15.3j B. What is the approximate width of the ellipse? infinity since it is no longer an ellipse C. What is the eccentricity of the ellipse? -7.6j D. Click Reset, and change the initial velocity to –4.0j km/s. Click Play. What is the eccentricity of this ellipse? Distance between foci: -1.0i -19.2j Width: infinity Eccentricity: 9.6j 6. Draw conclusions: Think about the eccentricity and shape of each ellipse. A. What is the relationship between the eccentricity of an ellipse and its shape? when the eccentric is too much then it won't be able to form an ellipse B. What is the eccentricity of a circle? Explain. eccentricity of a circle is the set of points in a place which are equidistant from a fixed point in the plane surface. eccentricity is 0 C. What is the eccentricity of a completely flat ellipse? Explain. a completely flat ellipse, also known as a degenerate ellipse, its eccentricity = 1. This is because the eccentricity of an ellipse is defined as the ratio of the distance between the two foci of the ellipse to the length of its major axis. D. The eccentricity of Earth’s orbit is 0.017. What can you infer about the shape of Earth’s orbit? Reproduction for educational use only. Public sharing or posting prohibited. © 2020 ExploreLearning™ All rights reserved The Earth's orbit eccentricity, 0.017, infers that its shape is largely circular, characterized by a minor ellipsis due to the slight deviation from zero. The eccentricity of an orbit defines its shape. Activity B: Velocity and area Get the Gizmo ready: ● Click Reset. ● Turn off Show foci and center. Introduction: After establishing that planetary orbits were ellipses, Kepler next looked at the speed of a planet as it traveled around the Sun. Question: How does the velocity of a planet vary as it travels in its orbit? 1. Observe: Place the planet at –5.00i AU and set the velocity to -4.0j km/s (does not have to be exact). Turn off Show vectors. Click Play. Observe the speed of the planet. A. At what point does the planet move fastest? its closest to the sun B. At what point does it move slowest? its further from the sun 2. Observe: Click Reset, and turn on Show vectors. Look at the green vector that represents the force of gravity pulling on the planet. Click Play. A. In which direction does the green vector always point? goes through the sun B. In which part of the orbit does the gravity vector point in almost the same direction as the velocity vector? its headed towards the sun C. In which part of the orbit does the gravity vector point in the opposite direction as the velocity vector? it goes away and leaves the sun 3. Explain: Based on your observations of the gravity vector, why does the planet accelerate as it approaches the Sun and slow down as it moves away from the Sun? The orbital speed varies depending on how far it is when orbiting a planet and how close it is to the host planet. If close, the planet will be accelerated by the impact of the host planet. The example here is the sun and the earth. If the earth gets closer to the sun, our planet will spin faster by the force of gravity. 4. Measure: Click Reset. Imagine a line connecting the planet to the Sun. As the planet moves around the Sun, the line will sweep out an area. Click Play, and then click Sweep area. Record the area below, and then press Sweep area four more times to complete the table. Reproduction for educational use only. Public sharing or posting prohibited. © 2020 ExploreLearning™ All rights reserved Trial 1 2 3 4 5 Area (km2) 18.768 18.768 18.768 18.768 18.768 5. Analyze: What pattern do you notice? the number repeated everytime i press the result button 6. Test: Try the same experiment with several different orbits and different numbers of days to sweep an area. Does the same rule hold true for each orbit? yes Kepler’s second law states that a planet accelerates as it approaches the Sun and decelerates as it moves farther from the Sun. As it orbits, the planet sweeps out equal areas in equal times. 7. Think and discuss: Why do you think the area swept out by a planet in a given period of time remains constant, even as the planet speeds up and slows down? (Hint: Think of each area swept out as a triangle. The height of the triangle is the distance between the planet and the Sun, while the base of the triangle is equal to the distance the planet travels in the given time period.) Since the planets are orbiting in an elliptical shape, sometimes they will be close, sometimes they will be far away. When the planets are close to creasing, the opponent's gravity will make them go faster and when they are far away they will go slowly because gravity is not enough to affect the planet going in orbit. So, the area that the planet sweeps is equal in both cases. because when accelerating, the coverage will be larger and vice versa when decelerating the coverage will be smaller Reproduction for educational use only. Public sharing or posting prohibited. © 2020 ExploreLearning™ All rights reserved Activity C: Radius and period Get the Gizmo ready: ● Click Reset. ● Set the Sun mass to medium, close to the mass of our Sun. Introduction: Kepler’s last law relates the period (T) of a planet, or the time it takes to complete an orbit, to the planet’s orbital radius (a), or average distance from the Sun. Question: How does the period of a planet relate to its average distance from the Sun? 1. Predict: How do you expect the period of a planet to change as its average distance from the Sun increases? it will become longer 2. Measure: Place the planet at -4.00i AU and set the velocity to -15.00j km/s (does not have to be exact). Click Play, and then click Pause when the planet completes one orbit. A. What is the approximate period of the planet in days? 2000 days B. Select the TABLE tab and click Record data. What is the actual period (T) of the planet in Earth days and years? 15 days 5 C. What is the orbital radius (a) of the planet? years 2.49 D. Click Reset. Place the planet at -2.00i AU and set the velocity to -20.0j km/s. Click Play and Record data. What is the period now? E. What is the new orbital radius? 188 days 1 years 1.11 F. How does the distance between the planet and the Sun affect its period? as the distance increases, the period decreases 3. Gather data: Experiment with the Gizmo to create a series of larger and larger orbits. Record each orbit’s radius (a) and period (T). Leave the last two columns blank for now. Reproduction for educational use only. Public sharing or posting prohibited. © 2020 ExploreLearning™ All rights reserved Orbit 1 2 3 4 5 a (AU) .47 1.11 1.76 2.49 3.31 T (years) .453 1.638 3.273 5.511 8.460 a^3 .1038 1.367 5.451 15.438 38.264 t^2 .2059 2.683 10.712 30.371 71.516 4. Analyze: How does the period change as the orbital radius increases? 5. Interpret: Select the GRAPH tab, and check that the graph of T vs. a is selected. Use the zoom controls to adjust the graph so you can see all your data and it fills the graph. A. What does the graph indicate? B. Does the graph of T vs. a form a straight line? 6. Investigate: Choose other options in the dropdown menus until you find a graph that makes a perfectly straight line. (If you cannot see all of your data, click the [–] controls to zoom out.) Which relationship makes a straight line? 📷) icon to take a screenshot. Right-click the image, and click When you have a line, click the camera ( Copy Image. Paste it here: 📷 7. Calculate: On the table on the previous page, label the third column “a 3” and the fourth column “T 2.” Use a calculator to cube each value of a and square each value of T. What do you notice? Kepler’s third law states that the cube of a planet’s orbital radius is proportional to the square of a planet’s period: a3 = kT 2 for some constant k. If the radius is measured in astronomical units, the period is measured in Earth years, and the mass of the star is equal to the mass of our Sun, the value of k is equal to 1 AU3/y2. 8. Challenge: The orbital radii of the planets are given in the table below. Calculate the period of each planet (in Earth years), and then check your values with your teacher. Planet a (AU) T (years) Planet a (AU) T (years) Reproduction for educational use only. Public sharing or posting prohibited. © 2020 ExploreLearning™ All rights reserved Mercury 0.39 0.24 Jupiter 5.20 11.86 Venus 0.72 0.61 Saturn 9.54 29.46 Earth 1.00 1.00 Uranus 19.18 84.01 Mars 1.52 1.88 Neptune 30.06 164.8 Reproduction for educational use only. Public sharing or posting prohibited. © 2020 ExploreLearning™ All rights reserved