Exploring Kepler’s Laws of Planetary Motion

[Pages:18]Exploring Kepler's Laws of Planetary Motion:

As Modified by Isaac Newton

Version: 1.0 Author: Sean S. Lindsay History: Written February 2020

Goal: To become familiar with Kepler's Laws of Planetary Motion in the light of Newton's Laws of Motion and Newtonian Gravity. Specifically, this lab addresses:

1. What are Kepler's Three Laws of Planetary Motion? 2. What are orbital velocity and escape velocity? 3. How orbital period relates to the semi-major axis of an ellipse-shaped orbit 4. How planetary speed connects Kelper's First and Second Laws through an exploration of

eccentricity, perihelion, and aphelion 5. How Newtonian physics modifies Kepler's Laws of Planetary Motion

Tools used in this lab: ? University of Colorado, Boulder's PhET simulation: "My Solar System" ? Microsoft Excel

1. Kepler's Three Laws of Planetary Motion

In 1609, the German mathematician Johannes Kepler (27 December 1571 ? 15 November 1630) published two of his three laws of planetary motion in his work Astronomia nova (A New Astronomy). Using the Tycho Brahe's incredibly comprehensive, accurate, and precise data set of planetary positions, Kepler was able to move past the limitation of circular orbits embedded in the Copernican model of the solar system. From precise measurements of Mars' position, Kepler deduced that planetary orbits are not circular in shape; they are instead elliptical in shape with the Sun located at one of the two foci of the elliptical orbit. Mars was ideal for this task because other than Mercury, it has the highest eccentricity of any of the naked-eye visible planets. The non-circular, elliptical orbits explained the motions of Mars and allowed him to extend his ideas to how planets are moving along their orbits. Formulating their motion in terms of geometry, he discovered that planets sweep out equal areas in equal times. Here, the area swept out is defined at a segment of the ellipse carved out by imagining the area a line between the Sun and planet sweeping out an area as it moves along its orbit. In equal time intervals, that swept out area will always be equivalent meaning that the planet is orbiting the Sun at different speeds in different parts of its orbit: faster when closer to the Sun, and slower when farther away.

In 1619, Kepler published his third law of planetary motion in Harmonices Mundi (The Harmony of the

World). Using orbital period and semi-major axis distance (the average distance away from the Sun a planet

is) data, he discovered what he called a "music of the spheres." He viewed the relationship between the

periods and semi-major axis distances as a harmony of the worlds, and therefore third law is often referred

to as the harmonic law. In Harmonices Mundi, he penned (translated from Latin), "I first believed I was

dreaming... But it is absolutely certain and exact that the ratio which exists between the period times of

any two planets is precisely the ratio of the 3/2th power of the mean distance." Putting this in the natural

Earth-based solar system units with orbital period, P, measured in Earth-years and semi-major axis distance,

a, measured in Earth-Sun distances, or rather astronomical units, AU, this becomes

P2 = a3;

Kepler's 3rd Law of planetary motion.

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1.1 ? Kepler's First Law of Planetary Motion

The orbit of a planet is an ellipse with the Sun at one of the two foci.

Figure 1 shows the geometry of an ellipse and a planet on an elliptical orbit according to Kepler's First Law. While mathematically, an ellipse is defined as a conic section, it is easiest to think of an ellipse a "squashed" circle. A circle can be defined by its center and radius. The radius is distance from the center of the circle to any point on the edge of the circle. As a circle is deformed into an ellipse, it will have a long cut through the center and a short cut through the center. The longest cut through the ellipse is called the major axis, and the short cut is called the minor axis. Half the major axis, from the center of the ellipse to the edge of the ellipse along the major axis, defines the semi-major axis, a. Half the minor axis, from the center of the ellipse to the edge of the ellipse along the minor axis, defines the semi-minor axis, b. Notice that the major axis goes through both foci, while the minor axis goes through neither. As the ellipse becomes less circular, the foci (singular: focus) of the ellipse move farther from the center of the ellipse and the ratio of the major axis length to minor axis length increases. This deviation from circular is quantified by the ellipse's eccentricity, e, such that

=

$( - )

(1)

Notice that for ellipses, 0 < 1, where e = 0 makes it so a = b, and you have circle; and e = 1 means b = 0, and the shape is no longer an ellipse. As e increases toward 1, the foci move farther from the center of the ellipse and the major axis become increasingly longer than the minor axis making the ellipse appear more and more "squashed."

Figure 1a: The geometry of an ellipse. The major, semimajor, minor, and semi-minor axes are labeled. The two foci of an ellipse are equidistant from the center of the ellipse and fall along the major axis. These points determine the shape of the ellipse including the ratio of the major to minor axis lengths and the eccentricity.

Figure 1b: An elliptical orbit according to Kepler's first Law of Planetary Motion. The green circle represents a planet moving on an elliptical orbit. The Sun is placed at one of the two foci, while the other one remains empty. The point the planet is closest to the Sun is called perihelion. The point the planet is farthest from the Sun is called aphelion.

Figure 1b shows what an elliptical orbit for a planet looks like according to Kepler's First Law of Planetary Motion. The Sun is at one of the two foci of the ellipse; the other focus is at point of empty space. A little bit of geometry and calculation can show that the average distance between the edge of the ellipse and one of the foci is equal to the semi-major axis length, a. This means that the semi-major axis length is the mean (average) distance a planet is away from the Sun. The Sun being placed at one of the two foci also creates two special points in the planet's elliptical orbit around the Sun:

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? Perihelion: The point where the planet is closest to the Sun. The distance away from the Sun is

defined as

, = ( - ),

(2)

where a is the semi-major axis distance and e is the eccentricity of the ellipse.

? Aphelion (Pronounced "Ap ? Helion"): The point where the planet is farthest from the Sun. The

distance away from the Sun is defined as

, = ( + ),

(3)

where a is the semi-major axis distance and e is the eccentricity of the ellipse.

Notice that as the eccentricity increase, the perihelion distance gets smaller and the aphelion distance gets larger.

1.2 ? Kepler's Second Law of Planetary Motion

A line segment connecting the planet and the Sun sweeps out equal areas of its elliptical orbit in equal intervals of time. The geometry phrasing of Kepler's Second Law of Planetary Motion makes it rather difficult to internalize what is communicating about how planets move on their orbits. Figure 2 shows two equal areas of an elliptical orbit that a planet as swept out and the length of the arcs around the elliptical orbit the planet is taking. Area 1 is equal to Area 2, and therefore the time for the planet to travel the distance of Arc 1 and Arc 2 must be the same. Arc 1 is a longer distance, so the planet must be orbiting faster than it is as it moves along Arc 2. This insight gives an alternative, and more intuitive way to state Kepler's Second Law. Alternative to Kepler's Second Law: A planet moves faster near perihelion and slower near aphelion. It orbital speed is fastest at perihelion and slowest at aphelion.

Figure 2. Kepler's 2nd Law states that a line segment connecting the Sun to a planet will sweep out equal areas in equal amounts of time. Area 1 and Area 2 are equal, therefore, the time for the planet to travel Arc 1 is the same as Arc 2. Arc 1 is longer, so the planet must be moving faster near perihelion. Arc 2 is shorter, so the planet must be moving slower near aphelion.

Exploring the alternative to Kepler's Second Law further, we can start comparing the orbital speed at aphelion and perihelion compared to what the orbital speed would be if it were a circular orbit. Again, some calculation can show that the speed a planet is moving at perihelion is faster than what it would be for the orbital velocity of a circular orbit with radius equal to the semi-major axis length. The planet will be moving slower than the circular orbital velocity at perihelion.

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1.3 ? Kepler's Third Law of Planetary Motion

The square of the orbital period is directly proportional to the cube of the semi-major axis of

its orbit. If the orbital period is measured in years and the semi-major axis is measured in

astronomical units (AU), then this becomes

=

(4)

Here the orbital MUST be in years and the semi-major axis MUST be in AU. This law is sometimes called the period-radius law or the harmonic law. It connects the orbital period, and therefore how fast the planet is traveling on average to the average distance it is away from the Sun. While this equation may at first glance look relatively unremarkable, its determination by Johannes Kepler fundamentally changed humanity's view of its place in the solar system and universe. For the first time, the distances to all of the known planets could be determined using their observed orbital periods. The distances to the planets were much greater than expected meaning that we live in a solar system and universe much larger than previously conceived.

1.4 ? The Limitation of Kepler's Three Laws of Planetary Motion

While Kepler's Three Laws of Planetary Motion fundamentally changed our understanding of our solar system, they were not without their flaws. Kepler provided a mathematical solution to the data set amassed by Tycho Brahe. This is what scientists call an empirical solution. It gives mathematics that can be highly predictive, but it does not give an explanation for how the natural phenomenon is operating. This means that Kepler's Three Laws only applied to the conditions of the data, which were objects in orbit around the Sun. As determined, Kepler's 3rd Law does not work for objects not in orbit around this Sun, including the: Earth-Moon system; Jupiter-Galilean Moons system; planetary orbits in exoplanetary systems; orbits of binary stars; rotations of spiral galaxies; etc. To extend the Kepler's Laws to any system, a generalized and universal understanding of how objects move and the development of a theory of gravity (the force making the celestial bodies move) need to be invented. For this, we need Isaac Newton.

2 ? Understanding Kepler's Laws Through Newtonian

Mechanics

Isaac Newton (25 December 1642 ? 20 March 1726) was a British mathematician, physicist, and astronomer. He was fascinated by the works of Johannes Kepler and Italian astronomer and physicist, Galileo Galilei. Specifically, Newton wanted to understand the findings that Galileo made about how objects fall and what makes the planets orbit and follow Kepler's Three Laws of Planetary Motion. To accomplish this, Newton developed a theory of universal motion, humanity's first conception of gravity, and a new branch of mathematics, calculus, which was required for his laws of motion and understanding of the force of gravity. These generalized physical explanations of how things move in the universe, and that all mass generates an attractive force called gravity allowed him to finally understand Kepler's Laws of Planetary Motion, the acceleration of falling objects observed by Galileo, and apply them to the entire universe. Newton's new physics finally unlocks what is making the planets orbit the Sun, and what makes all objects in the universe move in response to gravity. It was the dawn of a new understanding on the nature of how our universe works.

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2.1 ? Newton's Three Laws of Motion & Newton's Universal Law of Gravity

Before we get to what Newton's three laws of motion are, we need to define a few terms required to understand Newton's ideas. Since, we are talking about how things move, we will start with the difference between speed and velocity. Figure 3 shows the relationship between velocity and speed. Velocity is a speed plus information on the direction of travel. Speed is just the magnitude (how fast) the object is traveling. For example, a car driving at 60 mph is a speed, but a car driving at 60 mph heading due North is a velocity. In science, however, we do not use miles per hour (mph) as our unit for speed and velocity; we use meters per second (m/s) as our standard.

A force is any influence that changes the motion of an

object. Force is measured in Newtons (N), which in base

units

is

(1

EF G HI

=

1

N).

So,

1

Newton

of

force

is

required to move 1 kilogram of mass at an acceleration

of 1 m/s2. The change in motion in response to a force

could be a change in speed and/or a change in direction.

In both cases, we say that the object experienced

Figure 3. The relationship of speed and velocity. Velocity is a speed plus a direction.

acceleration. An acceleration is simply a change in motion, or more formally, the rate of change of the

object's velocity. That is, the acceleration is how the velocity is changing as a function of time, and that

change can be a change in direction and/or a change in speed. An object's resistance to change its motion,

i.e., accelerate, is called inertia. The higher the inertia, the more resistant to a change in motion a mass is.

The more massive the object, the higher the inertia, so inertia is really a concept saying that it requires

more force to accelerate more massive objects. Another way to put that is that given the same force acting

on two unequal masses, the more massive object will accelerate less because it has higher inertia.

With our basic concepts defined, we can now understand Newton's Three Laws of Motion.

Newton's First Law of Motion: The Inertial Law

An object at rest will remain at rest unless acted upon by and outside force. An object in motion with a constant velocity (same speed and direction) will continue to move at the same constant velocity unless acted upon by and outside force.

Newton's First Law of Motion essentially states that an object will not change its motion unless it is forced to.

Newton's Second Law of Motion: The Force Law

When a force F acts on a body of mass m, it produces an acceleration a equal to the force divided by the mass, i.e., a = F/m. Equivalently, the force that caused an acceleration a for mass m is equal to the mass times the acceleration, i.e., F = ma.

Newton's Second Law of Motion is the law that explains how an object will change its motion, i.e., accelerate, when an outside force acts on it. It gives a mathematical relationship for what the acceleration is given the amount of force and the amount of mass.

Newton's First Third of Motion: The Reaction Law

When one body exerts a force on a second body, the second body exerts a force equal in magnitude, but opposite in direction, on the first body.

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Newton's Third Law of Motion explains that forces come in equal and opposite pairs. Any force acting on a body will also have an accompanying force that has the same strength (magnitude), but opposite in direction. An example would be standing on a scale. The force of gravity between you and the Earth pulls you down on the scale. In response, the springs of the scale push back up on you with an equal, but opposite direction (upward) force. This depresses the springs, which can be translated into your weight. Another example is how rockets fly at launch. When the rocket fuel is ignited, it is directed out of nozzles toward the ground. The force created by this rapidly escaping gas acts downward. By Newton's Third Law, there is an equal and opposite, upward force acting on the rocket causing it to lift off the ground.

Newton's Universal Law of Gravity

Everything that has mass produces an attractive force acting on all other objects that have mass. The strength of this force is directly proportional to the product of the masses and inversely proportional to the square of the distance between the masses. This force is called the gravitational force, or gravity for short.

In equation form, the force of gravity, Fg between two masses, m1 and m2, separated by a distance r, is

=

(5)

, where G is the Universal Gravitational Constant equal to 6.67 * 10-11 m3/(kg s2).

Newton wanted to explain the results of experiments conducted by Galileo Galilei, as well as, understand

what was causing the motion of the planets. Galileo conducted experiments to determine that when objects fall, they fall with a constant acceleration equal to 9.8 m/s2 that is independent of the mass of the

object. By Newton's Three Laws of Motion, there must be some force that is casing this constant

acceleration. What is this force? Why, it is gravity, of course! Living on a massive planet, the Earth's mass of 5.97*1024 kg produces a large gravitational force acting on other masses. Applying Newton's

Second Law, this force of gravity must be equal to the mass times the acceleration. We can calculate the force of gravity between a mass m, the Earth with mass, ME = 5.97 * 1024 kg at a distance of the center of the Earth to the surface of the Earth, or rather one Earth radius, RE = 6,378 km = 6.378 * 106 m. We set

that equal to Newton's second law to see how much acceleration the object of mass m feels. Since we are

talking about the acceleration due to gravity, let's call that g instead of a. Now, we have

P = QRS TUV

YQ Q

=

=

(6)

Plugging in the values for G, ME, and RE, gives the acceleration due to gravity g = 9.8 m/s2. This value for g is called the surface gravity of Earth.

2.2 ? Connecting Newton's Laws to Kepler's Laws

Clearly, Newton's Laws of Motion and his Universal Force of Gravity are powerful tools to understand how things move in response to the force of gravity from massive objects. In the previous example of Earth's surface gravity, we found the acceleration all objects near the surface of Earth experience, which dictates the motion of objects moving around on Earth. What about how the planets move in response to the Sun's gravity? The Sun has about a million times more mass the Earth with a mass of 1 MSun = 1.99 * 1030 kg. That is going to generate a seriously strong gravitational force pulling the planets toward the Sun.

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What keeps the planets from falling into the Sun? The answer to this question is same answer to the question, "what causes the planets to orbit the Sun?". The graphic in Figure 4 demonstrates the following narrative explaining how Newton's laws and Gravity explain planetary orbits. Consider a planet moving with a velocity, v, in the presence of the Sun's gravitational pull. By Newton's First Law of Motion, this force of gravity acting on the planet will change the object's velocity. In fact, Newton's Second Law of Motion tells us that the planet will experience an acceleration towards the Sun, which will constantly change the motion of the planet trying to make it move toward the Sun.

Figure 4. A graphical depiction demonstrating how Newton's Laws of Motion and the Universal Law of Gravity explain planetary orbits. The leftmost image of the planet shows that according to Newton's 1st Law, the planet wants to move straight down at a speed of vorb. This speed is the orbital velocity necessary to create centripetal force, Fc to balance the force of gravity Fg. Gravity deflects the motion of the planet toward the Sun changing the velocity. At the orbital velocity, this deflection makes a circular path around the Sun causing the planet to forever fall towards the planet while always missing it. This is the secret to orbiting.

The light image of the planet at a second point in its orbit shows how the velocity, force of gravity, and centripetal force vectors change to create orbital motion.

Orbital Velocity

So, what is the secret to orbiting? In order to offset the pull of gravity toward the Sun, the planet needs to move at just the right velocity. To understand this, think about spinning a mass around on a string. The mass has circular motion, and you can feel the tension in the string as it spins around. This tension in the string is the string exerting a force on the mass pulling it toward the center point of motion. This force that you feel pulling toward the center of the circle is called centripetal force, or "center-seeking" force a that body experiences to have it move on a curved path The centripetal force is given by the equation Fc = (mv2)/r, where m is the mass, v is the velocity and r is the distance from the center of the circle, i.e., the radius.

For an object in orbit around the Sun, this centripetal force is gravity. This gives us a way to determine the velocity, v, to keep the object in orbit. To find it, we use the fact that here the centripetal force equal to the force of gravity. We call this velocity the orbital velocity, vorb. The value of vorb can be found by setting the force of gravity equal to the centripetal force. Consider a planet with mass, Mp, orbiting the Sun with mass, MSun, at a distance a. Then,

P = `

cdeR Q

=

chQij

deR

=

hQij

=

$

(7)

7

So, for a planet to orbit the Sun at a distance a away from the Sun, it simply needs to be doing so at the orbital velocity given in Equation 7. Notice that as the distance increases, the velocity decreases indicating that objects orbiting farther from the Sun orbit slower, just as Copernicus originally stated in his original heliocentric model.

Given that the total distance traveled for one complete orbit is the circumference of a circle with radius, a, and that to orbit, an object needs to move along this distance at the orbital velocity, you can calculate how long it would take the planet to complete exactly one orbit, i.e., the orbital period by dividing the circumference of the circle (2) by the orbital velocity. This is the connection Newton made to explain Kepler's Third Law of Planetary Motion. The orbital period of a planet can be determined using Newton's Laws of Motion and Gravity.

Escape Velocity

If there is a velocity required to have a planet, or other object, orbit, then what happens when that velocity is increased? Is it possible to get an object moving fast enough to escape the gravitational pull of the Sun altogether? The answer is yes, and the minimum velocity required to do so is called the escape velocity. The derivation of escape velocity requires calculus (remember Newton needed to invent calculus), and I won't belabor you with those details, but if you are curious and know the maths, you simply need to integrate the gravitational potential energy from a distance R away from the source of gravity to infinity. That gives you the total change in gravitational potential required to escape the gravity of the massive object (e.g., the Sun). Set that change in gravitational potential energy to the kenetic energy (1/2 mv2), and solve for the that velocity, and you get the minimum velocity to escape. Nicely done, us. That velocity comes out to simply be

= $ =

(8)

which is about 1.414 hij. So, 41.4% increase in the orbital velocity will cause an object to escape the gravity of the thing it is orbiting. If that object is the Sun, then it will leave our Solar System. If that object is the Earth, you have just determined the velocity required to send a spaceship away from the Earth to another part of our solar system, say Mars, Venus, Jupiter, or beyond.

Modifying Kepler's Laws

Newton's insights led to a scientific understanding of Kepler's Three Laws of Planetary Motion. The natural processes that govern the orbits of planets, and any other objects in orbit, were now accessible, understandable, and mathematically predictable. What changes and explanations needed to be added to turn Kepler's Laws, which only apply to objects in orbit around the Sun, into ones that apply for any orbital system of objects (i.e., The Moon orbiting the Earth, the Jupiter-Galilean moons system and moons of other planets in general, planets orbiting other stars, stars orbiting within a galaxy, etc.)

Newton's Modification to Kepler's First Law of Planetary Motion

Orbits are ellipses with the center mass of the system located at one focus that is common to both ellipses (the other focus is empty).

Newton explained orbital motion understanding how the force of gravity of the Sun acting on a planet requires a planet to move at the orbital velocity to remain in orbit. However, what does Newton's Third Law of Motion tell us? If the Sun is exerting a force of gravity on the planet causing it to orbit, then there must be an equal in magnitude, but opposite in direction, force of gravity of the planet acting on the Sun. If there is a force acting on the Sun, then by Newton's Second Law, it must change its motion. What keeps the Sun from heading directly toward the planet and them colliding? The answer is the same as before ? The Sun must have its own orbit where it moves at the correct orbital velocity! With the Sun

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