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<h2 class="hd hd-2 unit-title">A Historical Perspective</h2>
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<h2>4.3.1 A Historical Perspective</h2>
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<p>In our solar system, planets move around the Sun in orbits that are not circular, but elliptical. The orbital characteristics, such as shape, period, and velocity, can be described using three fundamental laws of planetary motion, published by the German mathematician and astronomer Johannes Kepler in the early 17th century. In fact, it is Kepler's work that helped pave the way for Isaac Newton's Law of Universal Gravitation, introduced in the late 17th century, which we discussed in Section 4.2.</p>
<p>Kepler's Laws of Planetary Motion are some of the most important concepts in orbital mechanics. In this section, we will go through each law in detail. To begin, however, we first take a brief trip back in time to understand the historical context and impact of Kepler's work.</p>
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<h3 class="hd hd-2">Video: A Historical Perspective of Orbital Mechanics</h3>
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<h2 class="hd hd-2 unit-title">Elliptical Orbits and Kepler's First Law</h2>
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<h2>4.3.2 Elliptical Orbits and Kepler's First Law</h2>
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<h3><strong>Kepler's First Law</strong></h3>
<p style="font-size: 16px; line-height: 25.6000003814697px;"><em><span style="color: #000000;"><strong>Celestial bodies move in orbits that are conic sections. Closed orbits, such as those of planets moving around the Sun, are ellipses, with the Sun at one of the foci.</strong></span></em></p>
<p style="font-size: 16px; line-height: 25.6000003814697px;">A conic section is a curve that is obtained by intersecting a cone with a plane. Examples of conic sections include circles, ellipses, parabolas, and hyperbolas. According to Kepler's First Law, planets in our solar system move around the Sun in elliptical orbits. Other types of orbits, such as circular and hyperbolic orbits, are also possible. The shape of an orbit is highly dependent on its eccentricity \(\epsilon\). For reference, the orbit of the Earth around the Sun has an eccentricity of 0.0167, which means that it is nearly circular.</p>
<p style="padding-left: 30px;"><img src="/assets/courseware/v1/1f19ec02728850e6388b8cfc284eb452/asset-v1:MITx+16.00x+2T2019+type@asset+block/ConicSections2_371.png" alt="Examples of conic sections" type="saveimage" target="[object Object]" /> <img src="/assets/courseware/v1/1524528dd97c14dd16ec397f20ed6460/asset-v1:MITx+16.00x+2T2019+type@asset+block/OrbitTypesFlipped3_281.png" alt="Examples of circular, elliptical, and hyperbolic orbits" type="saveimage" target="[object Object]" width="287" height="400" /></p>
<p style="padding-left: 60px;"><em>Left: Examples of conic sections (Image credit: <a href="http://commons.wikimedia.org/wiki/File:Conic_Sections.svg">Magister Mathematicae</a> via Wikimedia Commons)<br />Right: Eccentricity of circular, elliptical, parabolic, and hyperbolic orbits</em></p>
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<h2>4.3.3 Kepler's Second and Third Laws</h2>
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<h3><strong>Kepler's Second Law</strong></h3>
<p style="font-size: 16px; line-height: 25.6000003814697px;"><span style="color: #000000;"><em><strong>If two bodies revolve around each other under the influence of a central force, a line joining them sweeps out equal areas in the orbital plane in equal intervals of time.</strong></em></span></p>
<p style="font-size: 16px; line-height: 25.6000003814697px;">Kepler's Second Law implies that planets in the solar system move at variable speeds in their orbits --- faster when closer to the Sun, and slower when farther away. In the figure below, three segments of a planet's orbit are shown, which are each traversed in the same time \(t\) and sweep out equal area \(A\).</p>
<p style="text-align: center;"><img src="/assets/courseware/v1/759258bef492251af2086ade41ae080a/asset-v1:MITx+16.00x+2T2019+type@asset+block/Kepler2.gif" alt="Illustration of Kepler's Second Law" type="saveimage" target="[object Object]" /></p>
<p style="text-align: center;"><em><span>Illustration of Kepler's Second Law (Image credit: <a href="http://commons.wikimedia.org/wiki/File:Kepler2.gif">RJHall</a> via Wikimedia Commons)</span></em></p>
<h3><strong>Kepler's Third Law</strong></h3>
<p style="font-size: 16px; line-height: 25.6000003814697px;"><em><span style="color: #000000;"><strong>If two bodies revolve mutually about each other, the sum of their masses times the square of their period of mutual revolution is proportional to the cube of the semi-major axis of the relative orbit of one about the other.</strong></span></em></p>
<p style="font-size: 16px; line-height: 25.6000003814697px;">According to Kepler's Third Law, the square of a planet's orbital period \(P\) is proportional to the cube of the orbit's semi-major axis \(a \), such that \(P^2 \propto a^3\). The implication of this law is that the ratio of \(a^3/P^2\) is constant for all objects orbiting around the Sun. </p>
<p style="font-size: 16px; line-height: 25.6000003814697px;">The figure below plots \(a^3\) vs. \(P^2\) on logarithmic axes for the planets in our solar system (note that the orbits' semi-major axes are measured in astronomical units (AU), where 1 AU is equal to mean distance between the Earth and the Sun). The linear relationship between the data points verifies the relationship expressed by Kepler's Third Law.</p>
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<h2>4.3.4 Interactive Simulation: Kepler's Laws of Planetary Motion</h2>
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<p>Now that you are familiar with Kepler's Laws, use the simulation below to observe their implications for the motion of planets orbiting around a star. Try changing the presets to select various configurations of celestial bodies, and observe the resulting orbital paths, velocities, and periods.</p>
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<p style="text-align: center;"><em><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">Simulation credit: PhET Interactive Simulations, University of Colorado, <a href="http://phet.colorado.edu">http://phet.colorado.edu</a>.<br /> Used under a Creative Commons Attribution (CC BY) license.</span></em></p>
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<h2>4.3.5 Review of Concepts: Kepler's Laws</h2>
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Valid Orbits
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<p>Which of the following are valid orbits around the celestial body shown in red? Check all that apply. Note that two different views are shown for each orbit.</p>
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Instantaneous Orbital Velocity
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<p>Consider a planet in an elliptical orbit around the Sun. Rank the relative magnitude of the instantaneous orbital velocity of the planet at points A, B, and C.</p>
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Orbit of an Asteroid
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<label class="problem-group-label" for="input_7d8756be298944e59431128ec01022d7_2_1" id="label_7d8756be298944e59431128ec01022d7_2_1">Juno is a minor planet (asteroid) in the asteroid belt, which is located between the orbits of Mars and Jupiter. The orbital period of Juno around the Sun is 4.365 years. Using Kepler's Third Law, compute the average distance between Juno and the Sun (i.e., the semi-major axis of Juno's orbit). Provide your answer in AU with at least 2 decimal places. Hint: recall that 1 astronomical unit, or AU, is equal to average distance between the Earth and the Sun.</label>
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