Johannes Kepler formulated three fundamental laws describing the motion of planets orbiting the Sun based on astronomical data from Tycho Brahe. These laws explain the elliptical nature of planetary orbits, the variation of a planet's orbital speed at different positions, and the mathematical relationship between orbital period and orbital radius.
Illustrate the elliptical orbit with the Sun at one focus, equal swept areas over equal time intervals, and the derivation parameters of circular orbits.
All planets move in elliptical orbits with the Sun at one focus. An ellipse has two foci, \(F_1\) and \(F_2\), a major axis and a minor axis. Most planetary orbits have major and minor axes of almost equal length, so they are almost circular.
A line that connects a planet to the Sun sweeps out equal areas in equal times. If a planet takes the same time to travel from \(A\) to \(B\) as from \(C\) to \(D\), the swept area \(AFB\) equals area \(CFD\). The planet moves faster when closer to the Sun and slower when farther away.
The square of the orbital period of any planet is directly proportional to the cube of the radius of its orbit: \(T^2 \propto r^3\). A planet orbiting at a larger radius has a longer orbital period.
For a circular orbit, centripetal force is provided entirely by gravitational force. Equating \(\frac{mv^2}{r} = \frac{GMm}{r^2}\) gives \(v^2 = \frac{GM}{r}\). Substituting \(v = \frac{2\pi r}{T}\) gives \(T^2 = \left(\frac{4\pi^2}{GM}\right)r^3\).
Kepler's Third Law — General Form
Since \(G\) and \(M\) are constants, \(T^2 = kr^3\), where constant \(k = \frac{4\pi^2}{GM}\). For a planet-Sun system, \(M\) is the mass of the Sun. For a satellite-Earth system, \(M\) is the mass of the Earth.
Comparing Two Bodies Orbiting the Same Central Mass
The orbital periods \(T_1\) and \(T_2\) must be in the same unit. For satellites orbiting Earth, orbital radius \(r = R + h\), where \(R\) is the radius of the Earth and \(h\) is the altitude above the Earth's surface.
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What does Kepler's First Law state about the position of the Sun in a planet's orbit?
According to Kepler's Second Law, how does a planet's linear speed change as it moves closer to the Sun?
In the equation \(T^2 = \left(\frac{4\pi^2}{GM}\right)r^3\), what does \(M\) represent when calculating the orbital period of the Moon?
Satellite 1 orbits Earth at an altitude of \(600\ \mathrm{km}\) with a period of \(96\ \mathrm{min}\). Estimate the period of Satellite 2 at an altitude of \(20\,200\ \mathrm{km}\). Use Earth's radius \(R=6370\ \mathrm{km}\).
Planetary orbits are perfect circles.
Planetary orbits are ellipses; however, because their major and minor axes are very close in length, they appear nearly circular.
A planet moves at a constant speed throughout its entire orbit.
A planet's speed varies continuously; it moves faster when closer to the Sun and slower when further away to sweep out equal areas in equal times.
Objective: Test your understanding of Kepler's three laws and orbital parameters.
1 Based on Kepler's Second Law, which statement is correct regarding the motion of a planet?
2 What is the constant \(k\) equal to in the equation \(T^2 = kr^3\) for planets orbiting the Sun?
3 Kepler's First Law states that all planets move in orbits with the Sun at one focus.
4 In Kepler's third law formula, the square of the orbital period is directly proportional to the of the radius of orbit.
Drag each term to the correct meaning. If using a phone, tap the answer first, then tap the matching box.
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