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Science Physics Kepler's Third Law

Solving for satellite orbit period.
orbit period
G is the universal gravitational constant
G = 6.6726 x 10-11N-m2/kg2

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satellite mean orbital radius (r)
planet mass (M)
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satellite mean orbital radius (r)
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planet mass (M)
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satellite orbit period (T)
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References - Books:
Tipler, Paul A.. 1995. Physics For Scientists and Engineers. Worth Publishers. 3rd ed.

Kepler's Third Law

Kepler's third law, the Law of Harmonies, is the final piece in the trio of laws governing celestial bodies' motion within our solar system. Conceived by the German mathematician and astronomer Johannes Kepler in the early 17th century, these laws revolutionized our understanding of planetary motion. In addition, they laid the groundwork for Isaac Newton's theory of gravitation. This article will provide an in-depth look into Kepler's third law, derivation, and significance in modern astronomy.

Background

Kepler's third law was published in 1619 in his book "Harmonices Mundi" or "The Harmony of the World." It built upon two earlier laws that Kepler had formulated based on his analysis of Tycho Brahe's meticulous astronomical observations:
  • Kepler's First Law (Law of Ellipses): With the Sun at one focus of the ellipse, each planet elliptically orbits the Sun.
  • Kepler's Second Law (Law of Equal Areas): A line connecting the Sun to a planet covers an equal region of space in equal amounts of time, meaning that planets move more quickly when closer to the Sun and more slowly when farther away.
Kepler's third law expanded upon these principles and provided a mathematical relationship between the orbital periods and average distances of planets from the Sun.

Statement of Kepler's Third Law

Kepler's third law declares that a planet's orbital period (T) squared is directly proportional to the semi-major axis (a) cubed of its elliptical orbit. Mathematically, it can be expressed as:
T² ∝ a³
To convert this proportionality into an equation, a constant of proportionality (k) is introduced:
T² = k * a³
This constant is the same for planets in our solar system, meaning the ratio of T² to a³ is consistent across all planetary orbits.

Derivation of Kepler's Third Law

Kepler derived the third law using empirical data and his profound understanding of geometry. However, Newton's laws of motion and the universal law of gravitation can obtain a more rigorous derivation.
Newton's Second Law of Motion declares that the total force (F) acting on an object is equal to its mass (m) times acceleration (a):
F = m*a
Newton's Universal Law of Gravitation defines the gravitational force between two bodies as:
F = (G*m1*m2) / r²
Where:
  • G - gravitational constant
  • m1 and m2 - two bodies masses
  • r - distance between their centers
By equating the centripetal force required to maintain the planet's circular motion with the gravitational force acting on the planet, we can derive the relationship between the orbital period and the semi-major axis:
T² = (4 * π² * a³) / (G * M)
Where M is the mass of the central body (in this case, the Sun).
For planets within the same solar system, the constants (4 * π²) / (G * M) remain the same, reducing the equation to the original form:
T² ∝ a³

Applications of Kepler's Third Law

Kepler's third law has played an essential role in understanding the solar system and beyond. Some of its applications include:
  • Determining astronomical distances: We can use Kepler's third law to calculate its average distance from the Sun by measuring a planet's orbital period.
  • Estimating the mass of celestial bodies: Combining Kepler's third law with Newton's laws of motion and universal gravitation allows astronomers to estimate the mass of celestial bodies, such as stars and exoplanets.
  • Analyzing binary star systems: Kepler's third law has been extended to study binary star systems, where two stars orbit each other due to their mutual gravitational attraction.
  • Galactic dynamics: Kepler's third law has also been applied to study the motion of stars within galaxies, providing valuable insights into the distribution of mass and the presence of dark matter.
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