ASTR 101Name:

Lab 2 Planetary Orbits In Terms of Kepler’s 3 Laws of Planetary Motion

Kepler found that the planets all have elliptical orbits. He was the first to discover this.

  1. What do Kepler’s three laws of planetary motion say?
    Write all three of Kepler’s laws of planetary motion down on a separate piece of paper, or in a digital document on a computer, that you submit prior to performing this lab. Keep in mind that:

(1) You must clearly cite your source(s) of information.

(2) If you quote, you must use quotation marks.

(3) Many web sites and some books state some of the laws incorrectly or imprecisely.

(4) Original writing by you is required. Therefore, if you quote the three laws from a source (using quotation marks, of course), you will at least need to follow each quoted law with your comment on what it means, using your own completely original composing.

An ellipse is shown below.

Semi- means half, so the semi-major axis is half of the major axis. By the same token, half of the minor axis is the semiminor axis. The semi-major axis of a planet’s elliptical orbit is its average distance from the Sun.

Draw an ellipse on the next page by putting the page on a piece of cardboard, putting pins or thumbtacks in each focus (big black dot), and tying a loop of thread or string so that the ellipse you draw stays within the left and right boundaries of the grid. Then answer the questions on that page.

Reminders: If units of measurement are involved, always write down the units. If numbers are involved, never put down more digits in your answer that can be justified by the input data.

Draw an ellipse on the grid above. Have the piece of paper on a piece of cardboard backing. Put thumbtacks or pushpins through the foci of your ellipse (the two black dots). Tie a loop of string that goes around the thumbtacks and, when stretched tight, does not go beyond the right and left-hand-sides of the grid. Keeping the string taut around the thumbtacks and making sure it does not slip, draw the ellipse with a pencil,.

  1. Use a ruler to measure the lengths of the axes on your ellipse in cm or mm.
    Write down the lengths (& units): a. minor axis:

b. semiminor axis:

c. major axis:

d. semi-major axis:

(Reviewwhat the previous page says about the importance of thesemi-major axis.)

3. What is the average distance from the ellipse to one of its foci?

Draw a straight line from the Sun (left focus, black dot) towards the letter A, stopping your line from the Sun where it hits the ellipse. Do the same with a line from the Sun towards the letter B (do not draw the line past the ellipse). The area in-between your two lines is a “swept area.” Lightly shade in your swept area (using a pencil, not a pen.)

4. What is the area of your swept area, in terms of grid squares? ______grid squares

Draw another line from the Sun (left-hand focus dot) to the aphelion, the point on the ellipse farthest from the Sun, on the right. Make that the beginning of a second swept area equal to the first swept area.

The area of the second swept area must equal the area of the first swept area. Figure out from where on the ellipse to draw the line back to the Sun to complete thesecond swept area.Assumethat the second swept area is, to a close approximation, a triangle.

As a triangle, its area = ½ x base x height, or, to symbolize each variable as a letter, A = ½bh.

You already know the value of area Aof the triangle (it equals the first swept area), and the value for base b(how many squares are spanned by the first line of the second swept area,from the Sun to the aphelion). The only unknown left in the equation is h, which you can now solve.

  1. First, take the equation A = ½bh. and solve it algebraically for h. Write out your complete algebraic solution in the space below, ending up with h = (something) .
  1. Now calculate your specific hnumber by writing your solution from the previous question in the space below, except now you should put your specific numbers in the equation, in place of A andb, and then perform the arithmetic to produce a number for h:

Re-state your result here: h = ______squares

Go upward from the aphelion that many squares (your h number) to find the point on the ellipse from which to draw a line back to the Sun. This will complete your second swept area, with an area equal to the first swept area. Lightly shade in this second swept area (with a pencil).

Get a protractor and measure the anglesof the two swept-out areas with the Sun at the vertex.

  1. What is the angle of the first swept area?
  1. What is the angle of the second swept area?
  1. What does a planet do to follow Kepler’s second law? (Describe what a planet must be doing to move much farther along its orbit when closer to the Sun, and much less of a distance along its orbit when farther from the Sun, in the same amount of time.)

The Earth is closest to the Sun in January, at perihelion, 146 million km (91 million miles) from the Sun, and farthest from the Sun in July, at aphelion, 152 million km (94.5 million miles) from the Sun.

When it is said that “the Earth is 150 million kilometers from the Sun,” or “the Earth is 93 million miles from the Sun,” that is not really a true statement (except by coincidence on two days each year).

  1. By the way, given that the distance from the Earth to the Sun varies from 91 to 94.5 million miles due to the elliptical shape of the Earth’s orbit, and given the fact that the Sun is at one focus of the ellipse, is that why summer is hotter, because the Earth is closer to the Sun in summertime in the northern hemisphere? Explain.
  1. What axis of the Earth’s elliptical orbit is 150 million km (93 million miles) in length?

Kepler’s 3rd Law relates the period of a planet to its distance from the Sun.

The orbital period (P) is the length of time it takes to complete one orbit around the Sun.

The distance from the Sun (a) is the semi-major axis of a planet’s elliptical orbit.

Using years for time units and AU for distance units, Kepler’s third law becomes, “The period of a planet (in years), squared, equals the semi-major axis of the planet (in AU), cubed,” or

P2 = a3

  1. Algebraically solve Kepler’s third law for the period of a planet, in the space below, expressing the answer as a raised to an exponent, without the square root sign:
  1. In the third column of the table below, the actual, measured orbital periods of the eight known planets, and thedwarf planetsCeres and Pluto, are already written down.
  2. Use your equation (derived above) to calculate the period of each planet or minor planet from its semi-major axis. Write your answers in the table below, in the fourth column.
    Only go to the same decimal place in writing each answer in column 4 as the number in the actual period in column 3.
    Calculate the discrepancy, the difference between the actual and the calculated orbital periods, and write the discrepancies in the last column. Leave off any negative signs.

Planet
or Minor Planet / Semi-major
Axis (AU) / Actual
Period (yr) / Calculated
Period (yr) / Discrepancy / Discrepancy %
Mercury / 0.3871 / 0.2408
Venus / 0.7233 / 0.61515
Earth / 1.0000 / 1.0000
Mars / 1.5237 / 1.8808
Ceres / 2.767 / 4.5998
Jupiter / 5.2028 / 11.86461
Saturn / 9.5388 / 29.4589
Uranus / 19.18 / 84.0013
Neptune / 30.0611 / 164.8095
Pluto / 39.44 / 247.68
  1. Which planet or minor planet has the greatest absolute discrepancy
    between the actual value and the calculated value of its period? ______
  2. What percentage of the actual period is this discrepancy? Divide the discrepancy into the actual period and multiply the result by 100 to get your answer. Show your complete calculation here:
  3. Which planet or dwarf planet has the greatest percentage discrepancy? ______
    a. What percentage is this discrepancy? Show your calculation and result in the following space:
  4. What may be the cause of some of the actual periods varying a tiny bit from the theoretical periods?

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