Kepler's Third Law Calculator

This calculator computes the length of the semi-major axis of a planet's orbit around a central star using Kepler's Third Law, which provides a relationship between the orbital period and the semi-major axis. This relationship is fundamental in celestial mechanics, highlighting the harmonious patterns of planetary motion within our Solar System and beyond.
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CalculationInputs﻿﻿M
:1.989e+30kg​
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﻿﻿m
:5.97e+24kg​
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﻿﻿T
:1.00year​
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Outputs﻿﻿a
:1.496e+11m​
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﻿﻿a (AU)
:1.00​
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a=[T2G(M+m)​4π2]13​a=\left[\dfrac{T^2G(M+m)​}{4π^2}\right]^{\frac{1}{3}}​a=[4π2T2G(M+m)​​]31​​
Where:
﻿﻿a
 is the length of the semi-major axis of the planet's orbit, which is half the longest diameter of the elliptical orbit, measured in meters (﻿m
) or astronomical unit (﻿AU
) which is ﻿1AU=1.496×1011m
﻿
﻿﻿T
 is the orbital period, the time it takes for the planet to complete one full orbit around it's star, measured in years
﻿﻿M
 is the mass of the central star, measured in kilograms (﻿kg
),
﻿﻿m
 is the mass of the orbiting planet, also measured in kilograms (﻿kg
). Interestingly, Kepler's original formulation of the law considered only the central star's mass, whereas the inclusion of the planet's mass in modern versions acknowledges that both bodies exert gravitational forces on each other, albeit the planet's mass is often negligible compared to that of the central star.
﻿﻿G
 is the gravitational constant, which measures the intensity of gravitational attraction between masses, where ﻿G=6.67430×10−11m3kg−1s−2
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ExplanationKepler's Third Law, also known as the Law of Periods, establishes a mathematical relationship between the time it takes for a planet to complete an orbit around the Sun (its orbital period) and the size of its orbit, particularly the length of its semi-major axis. This law says that the square of the orbital period ﻿T
 of a planet is directly proportional to the cube of the semi-major axis ﻿a
 of its orbit, which is written as:
﻿
T2∝a3T^2 \propto a^3T2∝a3
📝 Equation for Kepler's Third Law
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Kepler's Third Law, aka The Law of Periods
Kepler's Third Law exemplifies the elegance and predictability of the cosmos, providing a mathematical framework that ties together the motion of celestial bodies across the universe. Its universal applicability underscores the fundamental role of gravity in shaping the cosmos, offering insights into the dynamics of solar systems and the intricate dance of stars and planets through space.
ApplicationsAstronomical Research: It aids in the study of planetary systems, enabling scientists to predict orbital periods based on distances from the central star.
Space Exploration: Essential for mission planning, especially in determining the paths of satellites and other spacecraft.
Educational Insight: Provides a fundamental grasp of planetary motion, enhancing our comprehension of the universe.
ReferencesChaisson, E., & McMillan, S. (2011). Astronomy: A Beginner's Guide to the Universe (7th ed.). Pearson.
Murray, C. D., & Dermott, S. F. (1999). Solar System Dynamics. Cambridge University Press.
Related ResourcesKepler's First Law Calculator
Y-coordinate of Keplerian Orbit
Orbital Mechanics Calculator
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