Lab 5_KeplersLawsONLINE
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Dec 6, 2023
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Lab 5: Kepler’s Laws of Planetary Motion Introduction:
Johannes Kepler was a German mathematician, astronomer, and astrologer in the 17th century. He used the astronomer Tycho Brahe's detailed observations of the planets develop a mathematical model to predict the positions on the sky of the planets. He published this model as three "laws" of planetary motion: 1st Law:
"The orbit of every planet is an ellipse with the Sun at one of the two foci." 2nd Law:
"A line joining a planet and the Sun sweeps out equal areas during equal intervals of time." 3rd Law:
"The square of the orbital period of a planet is directly proportional to the cube of the semimajor axis of its orbit." Ellipses are ovals with a long axis (the "major axis") and a short axis
(the "minor axis"). More technically, an ellipse is the set of points
where the sum of the distances to each of the foci (r
1
and r
2
) is always
the same. The separation of the foci (f) determines how oval-shaped or "squashed" the ellipse will be: if they are far apart, the major axis will
be much larger than the minor axis; but if they are close together, the major and minor axes are almost the same.
The term "eccentricity" refers to how oval-shaped an ellipse is. Eccentricity ranges from 0 for circular orbits to 1 for an orbit so oval-shaped that it is actually just a line. The larger the eccentricity the more oval-shaped (and less circular) the ellipse. The eccentricity, e, separation between foci, f, and the semi-major axis, a, are related to each other, ?
𝑒=
2𝑎 Objectives:
The scientific objective of this experiment is to develop an understanding of Kepler’s Law. Procedure:
Part 1: Ellipse and Eccentricity https://astro.unl.edu/classaction/animations/renaissance/kepler.html
Use
the above link to study the orbits of planets. Applet setup: Click the “start animation” button and then click on the tab labeled “Kepler’s 1
st
Law” in the lower left of the window. Click on “show empty focus,” “show center,” and “show semimajor axis.” You can adjust the orbit’s eccentricity by sliding the “eccentricity” bar on the right side of the window. 1.
Describe what happens to the shape of the ellipse when eccentricity gets larger. The larger the eccentricity the more oval and less circular the ellipse.
2.
Describe what happens when the eccentricity gets smaller. The more circular and the smaller the value or closer to zero is the eccentricity.
3.
Change the value of eccentricity until the orbit becomes circular in shape. What is the eccentricity of a circular orbit? A perfectly circular orbit would have an orbital eccentricity of 0.
Let’s examine the orbits of the planets. Be sure that the empty focus, center, and semi-major axis boxes are still turned on and then click “set parameters for” in the upper-right corner to select each planet in the Solar System. You must click “OK” to update the parameters. Complete the eccentricity and semi-major axis columns in the below. You will need to click each planet one at a time to view their orbits. Planets Eccentricity Mercury 0.2 Venus 0.007 Earth 0.01671
Mars 0.0934 Jupiter 0.048775 Saturn 0.0520 Uranus 0.0469 Neptune 0.0097 4.
Which planet has the most circular orbit? Venus 5.
Which planet has the most elliptical orbit? Mercury Part 2: Kepler’s Second Law Click Reset before starting this part. Click “Kepler’s 2nd Law.” Notice that you can control how long a sweep lasts by adjusting the fractional sweep bar. The default value is a sweep for 1/16 of the orbital period. Click “start animation” and observe how the speed of the planet changes. 1.
Describe how the orbital speed of the planet changes when it is near aphelion (which is the farthest point
from the Sun). The planet moves faster when it is nearer the Sun and slower when it is farther from the Sun. A planet moves with constantly changing speed as it moves about its orbit. The fastest a planet moves is at perihelion and the slowest is at aphelion. 2.
Describe how the speed changes when it is near perihelion (which is the closest point to the Sun). The planet's orbital velocity varies with distance from the Sun, at perihelion the planet is at maximum speed and at aphelion the planet is at its lowest space. Diagram below shows orbit of a satellite. Using this diagram and a ruler find the area between two sectors. You will need to print out the page. Measure the radius distance for each of the numbered positions on the diagram and record on the data table. Calculate the “Sector Area” of the ellipse according to the equation on the diagram.
Days Radius 1 (cm) Radius 2 (cm) Average radius
(cm) Angle (degrees) Sector Area
(cm
2
) 1-2 64 2-3 40 3-4 26 4-5 19 5-6 16 6-7 15 7-8 15 8-9 16 9-10 19 10-11 26 11-12 40 12-1 64
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Average Sector 3.
Are these areas the same? No they aren’t Part 3: Kepler’s Third law Kepler's third law allows you determine orbital size or orbital period for a planet orbiting the Sun. The equation relates the semi-major axis of a planet's orbit, a in AU, to how long it will take it to complete one orbit, P in Earth years, but this version of the equation is only valid for a planet orbiting our Sun: ?
2 =𝑎
3
Use Kepler’s third law to complete the table below. Planet Semi-major Axis (AU) Orbital Period (Earth years) Mercury 0.387 0.241 Venus 108.210 0.615 Earth 1.000 1 Mars 1.52 1.8 Jupiter 5.2028 11.9 Saturn 9.58 29.4 Uranus 19.2 84
Neptune 30.0611 165 1.
If you could move a planet so that it had a smaller orbit closer to the Sun (smaller a), what would happen
to its orbital period (get larger, stay the same, get smaller)? The orbital period would be shorter. 2.
Does eccentricity appear in the equation for Kepler's Third Law? Is the orbital period affected by whether an orbit is eccentric or not? the orbital period does not depend on eccentricity, a planet with a longer elliptical orbit will have the same orbital period as a planet with a circular orbit.