HW 7 KepleràLaws and Elliptical Orbits
Introduction: Tycho Brahe observed the regularly changing position of the planets relative to
the stars over several decades. Johannes Kepler used these observations to determine how objects
orbit the Sun and summarized his results in three Laws. Newton and Einstein later showed that
these Laws are only approximations, but that they are accurate enough for us to use in many
cases.
Homework7 Questions, Answer all questions below to get 100%:
Keplers First Law: An object which orbits the Sun moves along an ellipse with the Sun at one
focus.
An ellipse can be thought of as a stretched circle. An ellipse has two foci (the plural of focus)
which are located inside the ellipse on its major axis (the longest direction of the ellpse). The
center of the ellipse is exactly halfway between the foci.
An ellipseàeccentricity (e) is the distance between its foci divided by the length of its major
axis. An ellipse with e=0 is just a circle. An ellipse with e=1 is infinitely long and thin.
The semi-major axis (a) is half the length of the major axis. For a circle, a equals the radius ŠUse the above information to answer questions for the elliptical orbit in Figure 1.
Figure 1: An elliptical orbit. The cross shows the center of the ellipse. Each dotted square
measures 1 astronomical unit (AU) on a side.
1. One focus of the ellipse is shown by a black dot. Draw the Sun at the other focus.
2. How many AU across is the major axis of this ellipse?_______________
3. The semi-major axis of this ellipse equals how many AU?_________________
4. What is the eccentricity of this ellipse?______________________
Keplers Second Law: An object moves faster in its orbit when it is closer to the Sun, so that the
line connecting the object and the Sun sweeps out equal areas in equal times.
Note that KepleràThree Laws apply to ALL OBJECTS ORBITING THE SUN, not just planets.
The point in an objectàorbit where it is closest to the Sun, at radius rp, is called perihelion. The
point where it is farthest from the Sun, at radius ra, is called aphelion. To sweep out equal areas
??
in equal times, the object moves ???? times faster at perihelion than at aphelion. Perihelion and
??
aphelion occur on the major axis, so ???? + ???? = 2?? where a is the semi-major axis.
5. Mark perihelion of the orbit in Figure 1 with a P and aphelion with an A.
6. How much faster is the object moving at perihelion than at aphelion?___________
Keplers Third Law: For objects orbiting the Sun, the square of the objectàorbital period P,
measured in years, equals the cube of its semi-major axis a, measured in astronomical units.
??2 = ??3
The orbital period (P) of an object is how long it takes to move around its orbit once. The Earth
has P=1 Earth year and semi-major axis a=1 astronomical unit (AU) by definition. For Earth,
KepleràThird Law ??2 = ??3 gives 12=13 or (1×1)=(1x1x1), which is correct.
Figure 2: Circular and elliptical orbits of 5 asteroids (A through E) around the Sun (black dot).
The numbers show the distances in AU to the circular orbits of asteroids A, B, and C.
The orbits of five asteroids (labelled A through E) are shown in Figure 2. Some of the orbits are
circular and some are elliptical. Refer to Figure 2 when answering the following questions.
7. What is the perihelion of each orbit, in AU? Fill in these values in the table below.
8. What is the aphelion of each orbit, in AU? Fill in these values in the table below.
9. What is the semi-major axis of each orbit, in AU? Fill in these values in the table below.
10. Rank the orbital periods (P) of the asteroids from shortest to longest. Exact values are not
needed, just the relative rankings. Fill in these rankings in the table.
As an example of this kind of ranking, the eccentricities of each of the asteroidÊorbits have been ranked and entered in the table. Three of the orbits are circular and,
therefore, have zero eccentricity. Of the other two orbits, E is clearly more elliptical than D.
11. One of these asteroids is later discovered to be a comet. Based just on their orbits,
which asteroid is it most likely to be, and why?

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