Discuss three laws of Johannes Kepler that described the planets orbit around the Sun

Discuss three laws of Johannes Kepler that described the planets orbit around the Sun

1. Introduction
Johannes Kepler formulated three laws that described the planets orbit around the Sun.

His work paved the way for that of Sir Isaac Newton, who derived the underlying physical reasons why the planets behaved as Kepler had described. In this exercise, you will use computer simulations of orbital motions to experiment with the various aspects of Kepler’s three laws of motion. Kepler’s three laws of motion are as follows:

Kepler’s First Law states that a planet moves on an elliptical orbit around the Sun.
Kepler’s Second Law states that, for each planet, the arc swept out in space by an imaginary line connecting that planet to the Sun sweeps out equal areas of the ellipse in equal intervals of time.
Kepler’s Third Law states that planet’s orbital period squared = the average distance from the Sun cubed.
This means that planets orbit the sun more slowly than planets that are closer to the sun. In this lab, we will do some exercises to further our understanding of Kepler’s three laws.

Procedure:
Using a pencil copy the below a & b into a document and print it out or draw the dots on a piece of paper (you will later photo this sheet). Try to put one set of dots in the upper half of your page and the other set in the bottom half of the page. Place this sheet on a surface into which you can place push pins. Cut a piece of string about 4 – 5 inches long and fasten one end of the string to each dot with the tacks (a knot works best). Place your pencil in the string and trace out an ellipse. Note that you’ll have to draw one half of the ellipse at a time, flipping the string to the other side in order to do the other half. See this example: Kepler example.pdf.

a.

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b. Repeat the procedure for the dots below.

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Measure the distance between the dots for each ellipse and draw an x at a point halfway in between them. The two dots are the foci and the x is the center. Label the foci “C” and the center O.

Now we can calculate the eccentricity of our ellipses. Measure the distance from c to O, and also the distance along the center line from O to the ellipse. Divide the first distance by the second to get the eccentricity. Take a photo/scan of your ellipses and insert it into your lab Document. Also either type out your measurements/calculations or photo them as well to be in your lab document.

Assignment:
What are the eccentricities of the two ellipses you drew?
e1=
e2=
What do you think is the eccentricity of a perfect circle
Kepler’s 1’st Law states that the planets orbit the sun in an ellipse, with the sun at one focus.
Draw a picture of the Earth’s orbit around the Sun. Include, and label, the Earth and the Sun in your diagram.
What is the eccentricity of the Earth’s orbit? (You should find this information in your text – try the appendix.)
Should the orbit look more like your first or second drawing on the first page of this lab?
Take a photo/scan of your work and insert it into your lab Document.
Estimate the area in units of squares of the shaded region in the ellipse below. Area = _______ squares. Hint: the area of a triangle is (½base x height).
Now draw a pair of lines from the “sun” to the ellipse on the left side of the sun in such a way that they enclose the same area as the shaded area.

Using your string, to measure along the ellipse, measure the arc length between the two sides of the shaded area and the arc length between the two lines you drew. (The arc length is the “piece of crust” of your “pie slice”.)

Shaded length =
Your length =
Are the two arc lengths the same? If not, why not? Which is longer?
Take a photo/scan of your work and insert it into your lab Document.
Kepler’s second law says that a line drawn from the sun to a planet sweeps out equal areas in equal times.
What does this mean about the speed at which the planet orbits? Does the speed change during the completion of one orbit? Relate this to the drawing above.
Kepler’s Third Law states that p^2 ≈ a^3, where p = the period of the planet’s orbit around the sun, or how long it takes to orbit, (in Earth years) and a = the planet’s distance from the sun (in AU).
Use this law to answer the following and be sure to show all of your calculations and/or explain how you came to your answer:
Planet Q orbits a star of the same type and mass as our Sun. It orbits at the same distance as Earth, but has twice the mass of Earth. How long does it take planet Q to complete a single orbit?
Planet Z orbits the same star as Planet Q at a distance of 4 AU. Planet Z has the same mass as Earth. How long is Planet Z’s year?
Planet W is part of the same solar system. It’s year is 5 Earth years long. How far is Planet W from its star?
Did mass or eccentricity matter for the above questions based on the law stated above? Explain.
According to Kepler’s Third Law, planets closer to the sun have shorter years. We might ask the question, “Is this because they move faster, or because they don’t have as far to go?” Let’s investigate.

Speed is distance divided by time, so if we know how far a planet travels and how long it takes, we can calculate speed. Look up the period and orbital distance (in years and AU, respectively) of Mercury, Venus and Earth. Assuming circular orbits (the real orbits are pretty close), the distance traveled is 3.14 (= pi) times the orbital distance times 2. So in mathematical symbols, the planet’s speed is:

v = (2(pi)a)/(p) where v = speed

a = average distance from the sun in AU

p = orbital period in years

Fill out the table below, showing your calculation for the 3rd column. Be sure you’re using the correct units.

Planet

Average Distance (a) (AU)

Orbital Period (p) (years)

Average Speed (v) (AU/year) Show your full calculation on your sheet.

Mercury

Venus

Earth

d. So what is your conclusion? Do planets closer to the Sun actually move faster?

Now that we’ve investigated each of Kepler’s Laws, write out the laws below (providing any citations) and then explain in your own words what each law really tells us about planetary motion. Think about explaining the laws to your friend who hasn’t taken this class. Try to use simple explanations, talking about the general behavior of the planet would be helpful.

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