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The Three Laws of Planetary Motion


Using Slooh’s Online Telescope and integrated NGSS aligned Quest learning activities, you can create your own posters about Kepler’s laws while learning more about planetary motion. In the Footsteps of Kepler - Planetary Motions is one of 60+ curriculum-aligned STEM Quest learning activities on Slooh for students 4th grade to college.



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Kepler's Laws of Planetary Motion


Ellipses


To be able to understand Kepler’s laws, let’s first quickly look at the geometry of ellipses. Refer to the diagram below for the definitions of the various mathematical terms.


Ellipses are curves that can be described by two focus points, or focal points. A circle is a particular case of an ellipse with only one focus point at its center, where the distance between it and every point on the circle’s perimeter is the same. Ellipses look like a squashed circle.


The definitions of ellipticity, semi-major axis, semi-minor axis, periapsis, and apoapsis. Image by John H Boisvert for Slooh

The leftmost ellipse in the diagram has the locations of its center and two focal points marked. The foci are at unique locations inside an ellipse. The sum of the distances from each focus to any other point on the ellipse is the same for a given ellipse. The semi-major and semi-minor axes are the longest and shortest distances from the ellipse’s center to the perimeter. The apoapsis and periapsis are the longest and shortest distances from one focal point to the perimeter along the semi-major axis. An ellipse’s eccentricity measures how “circular” an ellipse is, as described in the diagram. The ellipses in the diagram have an eccentricity of 0.4.


There are many geometrical relationships between the eccentricity, semi-major axis, semi-minor axis, periapsis, and apoapsis; the diagram above contains a few equations of these relationships. The two foci points converge to the center when an ellipse loses ellipticity and becomes a circle. Not only that, notice how the semi-major axis, semi-minor axis, periapsis, and apoapsis all become the same length for circles.


Now let’s look at Kepler’s laws of planetary motion!


Kepler's First Law


Kepler’s first law states that the orbital path of a planet around the Sun is an ellipse with the Sun at one of the focal points. The early geocentric and heliocentric models of the Solar System break this law because they assumed the planets had circular orbits.


Kepler’s First Law. The orbital path of an object around a body is shaped like an ellipse with the body at one of the two foci. This orbital path has an eccentricity of 0.4. Image by John H Boisvert for Slooh

This law tells us that the distance between the planet and the Sun is not constant. A planet is at its closest approach at perihelion and its furthest point at aphelion. Note that these terms are similar to periapsis and apoapsis except with different suffixes. The suffix of peri-/apo- changes depending on the focal point (see table below), but the geometric meaning remains the same. Also note that, although the diagram shows a dramatic eccentricity of 0.4, most planets in the solar system have very low eccentricities. In other words, their orbital paths are nearly, but not precisely, circular.


You can observe Kepler’s first law in action when looking at the Earth-Moon system. The Moon orbits the Earth in an ellipse, with the Earth at a focal point. That means that the distance between the Earth and Moon is continuously changing. A supermoon occurs when a Full Moon is near perigee, and a micromoon happens when the Full Moon is near apogee.


Celestial Object

Suffix

Celestial Object

Suffix

Generic Object

-apsis

Jupiter

-jove

Mercury

-hermion

Saturn

-chron

Venus

-cythe

Sun

-helion

Earth

-gee

Star

-astron

Moon

-lune

Black Hole

-bothron

Mars

-areion

Galaxy

-galacticon

Suffixes for the periapsis and apoapsis depending on the kind of object the body is orbiting



Kepler's Second Law


Kepler’s second law states that a line from a planet to the Sun sweeps out equal areas during equal time intervals. See the diagram below for an illustration of this law.


The amount of time it takes for the planet to travel the perimeter of each shaded region is the same, and the area that those journeys carve out is also the same. This law means that the planet must be moving slower when traveling along the perimeter of the shaded region on the left than along the right region. In general, a planet slows down when it nears apoapsis, and it speeds up as it nears periapsis.


Kepler's Second Law. The area of both shaded regions, "A," and the length of time the orbiting object takes along each arc is the same. Image by John H. Boisvert for Slooh

Kepler's Third Law


Kepler’s third law states that the square of the orbital period is directly proportional to the cube of the semi-major axis of its orbit. Writing this in equation form gives the following:

In the equation, “p” is the orbital period (the time it takes the object to complete a full orbit), “a” is the semi-major axis, and “C” is the constant of proportionality. The constant is the same for every planet in the solar system. Graphically, this means that, when the square of the orbital period and the cube of the semi-major axis are plotted, they should form a linear graph, as shown below. In general, the further a planet is from the Sun, the longer it takes to complete its orbit.


A plot of the square of the orbital period vs. the cube of the semi-major axis for the Solar System planets. Image by John H. Boisvert for Slooh

 

More About Slooh's In the Footsteps of Kepler - Planetary Motions Quest


The wandering planets follow the same physical laws as all matter in the universe. In this quest, you will discover how humanity unveiled the extraordinary physical laws of orbital motion. You'll learn about Johannes Kepler and his three laws of planetary motion. You'll also learn about Sir Isaac Newton and his theory of gravitation. By the end of the quest, you will discover the similarities between Kepler's laws and Newton's gravity, and how these laws apply to our tiny corner of the galaxy—the Solar System.


Related Slooh Quests

  1. Solar System Explorer

  2. Stars Like Ours

  3. The Mystery of the Changing Moon


More About Slooh's Astronomy NGSS Aligned Learning Activities


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