Question

phet online lab - keplers laws
i. introduction
after completing this lab you should be able to a) describe the orbit of a planet in terms of keplers three laws of planetary motion, b) describe the geometric properties of a planets elliptical orbit, c) explain how the motion of a planet changes during its orbit and d) explain how changing the powers of the length of the semi - major axis and the period can result in a linear relationship.
you will use one of the online simulations from phet interactive simulations. the simulation is called keplers laws html5. the link for the simulation is: http://tinyurl.com/56br3k52. the qr code to the right will also take you to the simulation website.
ii. keplers first law
on the left side of the homepage of the simulation click on the first law window. when the window opens you see a yellow dot that represents a central very massive object such as the sun and a smaller purple dot that represents an orbiting body or planet such as the earth.
a. orbital changes
spend a few minutes investigating the features of the first law window. before you begin answering the questions below, click the reset button and then do not change anything. click on the play button.
(1) in as much detail as possible, describe the motion of the planet as it orbits the sun. also, how would you describe the shape of the planets orbit?

(2) on the right hand side of the screen, click the gravity force box. how does the gravitational force change as the planet orbits the sun? where is it the strongest and where is it the weakest?

(3) pause the simulation. change the velocity of the planet by changing the length velocity vector. how does changing the velocity affect the orbit of the planet?

(4) what happens if the velocity becomes too large or too small?

© brian swarthout 2024

Explanation:

(1)

• Motion Description: The planet moves in an elliptical orbit around the Sun (the yellow dot). As it orbits, its speed changes: it moves faster when closer to the Sun (perihelion) and slower when farther away (aphelion).
• Orbit Shape: The orbit is an ellipse, with the Sun located at one of the two foci of the ellipse. In the default simulation (before changing velocity), it may appear nearly circular, but fundamentally, it follows Kepler’s first law as an ellipse (with low eccentricity in the default setup, making it look close to a circle).

(2)

• Gravitational Force Change: The gravitational force between the planet and the Sun follows Newton’s law of universal gravitation, \( F = G\frac{Mm}{r^2} \), where \( G \) is the gravitational constant, \( M \) is the Sun’s mass, \( m \) is the planet’s mass, and \( r \) is the distance between them. As the planet orbits, \( r \) (the distance from the Sun) changes. So, the gravitational force is strongest when the planet is closest to the Sun (smallest \( r \), at perihelion) and weakest when the planet is farthest from the Sun (largest \( r \), at aphelion).

(3)

• Effect of Changing Velocity on Orbit: Changing the velocity (magnitude of the velocity vector) affects the shape and size of the orbit. If the velocity is increased, the orbit becomes more elliptical (higher eccentricity) or may even become a hyperbola/parabola if the velocity exceeds the escape velocity. If the velocity is decreased, the orbit also becomes more elliptical (but in the opposite “direction” of elongation) or may collapse into a smaller orbit (or the planet could crash into the Sun if velocity is too low). Specifically, increasing the tangential velocity makes the orbit more stretched out (larger semi - major axis or more eccentric), while decreasing it can make the orbit more compact or change its orientation of elongation.

(4)

• Too Large Velocity: If the velocity becomes too large (greater than the escape velocity of the Sun - planet system), the planet will escape the Sun’s gravitational pull. Its orbit will change from a closed ellipse to an open curve (a parabola if velocity equals escape velocity, a hyperbola if velocity exceeds escape velocity), and it will move off into space, never to return to its original orbit.
• Too Small Velocity: If the velocity becomes too small, the gravitational force of the Sun will pull the planet inwards. The planet will not have enough tangential velocity to maintain its orbit, and it will spiral towards the Sun (or crash into the Sun) because the centripetal force required for orbital motion (\( F_c=\frac{mv^2}{r} \)) will be less than the gravitational force (\( F_g = G\frac{Mm}{r^2} \)) at that distance, causing the planet to fall towards the Sun.

Answer:

(1)

• Motion: The planet moves in an elliptical (or nearly circular) orbit, faster near the Sun (perihelion) and slower far from it (aphelion).
• Orbit Shape: Elliptical (Sun at one focus); appears nearly circular in default simulation.

(2)

Gravitational force is strongest at perihelion (closest to Sun, smallest \( r \)) and weakest at aphelion (farthest from Sun, largest \( r \)), following \( F \propto \frac{1}{r^2} \).

(3)

Changing velocity alters the orbit’s shape/size: higher velocity makes the orbit more elliptical (or open) and larger; lower velocity makes it more elliptical (or smaller) or causes collapse.

(4)

• Too large: Planet escapes (orbit becomes hyperbolic/parabolic).
• Too small: Planet spirals into the Sun (loses orbital stability).