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3.1 The laws of planetary motion  (Page 5/17)

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Kepler’s three laws of planetary motion can be summarized as follows:

  • Kepler’s first law    : Each planet moves around the Sun in an orbit that is an ellipse, with the Sun at one focus of the ellipse.
  • Kepler’s second law    : The straight line joining a planet and the Sun sweeps out equal areas in space in equal intervals of time.
  • Kepler’s third law    : The square of a planet’s orbital period is directly proportional to the cube of the semimajor axis of its orbit.

Kepler’s three laws provide a precise geometric description of planetary motion within the framework of the Copernican system. With these tools, it was possible to calculate planetary positions with greatly improved precision. Still, Kepler’s laws are purely descriptive: they do not help us understand what forces of nature constrain the planets to follow this particular set of rules. That step was left to Isaac Newton.

Applying kepler’s third law

Using the orbital periods and semimajor axes for Venus and Earth that are provided here, calculate P 2 and a 3 , and verify that they obey Kepler’s third law    . Venus’ orbital period is 0.62 year, and its semimajor axis is 0.72 AU. Earth’s orbital period is 1.00 year, and its semimajor axis is 1.00 AU.

Solution

We can use the equation for Kepler’s third law, P 2 ∝ a 3 . For Venus, P 2 = 0.62 × 0.62 = 0.38 and a 3 = 0.72 × 0.72 × 0.72 = 0.37 (rounding numbers sometimes causes minor discrepancies like this). The square of the orbital period (0.38) approximates the cube of the semimajor axis (0.37). Therefore, Venus obeys Kepler’s third law. For Earth, P 2 = 1.00 × 1.00 = 1.00 and a 3 = 1.00 × 1.00 × 1.00 = 1.00. The square of the orbital period (1.00) approximates (in this case, equals) the cube of the semimajor axis (1.00). Therefore, Earth obeys Kepler’s third law.

Check your learning

Using the orbital periods and semimajor axes for Saturn and Jupiter that are provided here, calculate P 2 and a 3 , and verify that they obey Kepler’s third law. Saturn’s orbital period is 29.46 years, and its semimajor axis is 9.54 AU. Jupiter’s orbital period is 11.86 years, and its semimajor axis is 5.20 AU.

Answer:

For Saturn, P 2 = 29.46 × 29.46 = 867.9 and a 3 = 9.54 × 9.54 × 9.54 = 868.3. The square of the orbital period (867.9) approximates the cube of the semimajor axis (868.3). Therefore, Saturn obeys Kepler’s third law.

Got questions? Get instant answers now!

In honor of the scientist who first devised the laws that govern the motions of planets, the team that built the first spacecraft to search for planets orbiting other stars decided to name the probe “Kepler.” To learn more about Johannes Kepler’s life and his laws of planetary motion, as well as lots of information on the Kepler Mission, visit NASA’s Kepler website and follow the links that interest you.

Key concepts and summary

Tycho Brahe’s accurate observations of planetary positions provided the data used by Johannes Kepler to derive his three fundamental laws of planetary motion. Kepler’s laws describe the behavior of planets in their orbits as follows: (1) planetary orbits are ellipses with the Sun at one focus; (2) in equal intervals, a planet’s orbit sweeps out equal areas; and (3) the relationship between the orbital period ( P ) and the semimajor axis ( a ) of an orbit is given by P 2 = a 3 (when a is in units of AU and P is in units of Earth years).

Questions & Answers

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Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
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the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
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solar Univers
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no
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but thats just a theory
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GANGAIN Reply
these are Rocky substances between mars and jupiter
GANGAIN
Comets are cosmic snowballs of frozen gases , rock and dust that orbit the sun. They are mostly found between the orbits of Venus and Mercury.
Aarya
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John
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John
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sahil
how can we learn right and true ?
Govinda Reply
why the moon is always appear in an elliptical shape
Gatjuol Reply
Because when astroid hit the Earth then a piece of elliptical shape of the earth was separated which is now called moon.
Hemen
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lidiya Reply
Did you mean eye sight or sea level
Minal
oh sorry it's sea level
lidiya
according to the theory of astronomers why the moon is always appear in an elliptical orbit?
Gatjuol
hi !!! I am new in astronomy.... I have so many questions in mind .... all of scientists of the word they just give opinion only. but they never think true or false ... i respect all of them... I believes whole universe depending on true ...থিউরি
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Elyana
we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
Reply
w astronomy and cosmology!
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another theory of universe except big ban
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there many theory to born universe but what is the reality of big bang theory to born universe
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universal
there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
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Michele
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Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*). So it would just barely creep over the horizon.
Christopher
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Source:  OpenStax, Astronomy. OpenStax CNX. Apr 12, 2017 Download for free at http://cnx.org/content/col11992/1.13
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