According To Kepler's Third Law

Kepler'southward Three Laws

In the early 1600s, Johannes Kepler proposed 3 laws of planetary motion. Kepler was able to summarize the carefully collected information of his mentor - Tycho Brahe - with three statements that described the movement of planets in a sun-centered solar organisation. Kepler'south efforts to explain the underlying reasons for such motions are no longer accepted; even so, the actual laws themselves are yet considered an accurate description of the motion of any planet and any satellite.

Kepler's 3 laws of planetary motion tin be described every bit follows:

The path of the planets about the sun is elliptical in shape, with the center of the dominicus beingness located at i focus. (The Police force of Ellipses)
An imaginary line fatigued from the center of the lord's day to the centre of the planet will sweep out equal areas in equal intervals of time. (The Law of Equal Areas)
The ratio of the squares of the periods of any ii planets is equal to the ratio of the cubes of their average distances from the sun. (The Police of Harmonies)

The Law of Ellipses

Kepler'southward first law - sometimes referred to equally the police force of ellipses - explains that planets are orbiting the sun in a path described every bit an ellipse. An ellipse can easily be constructed using a pencil, two tacks, a string, a sheet of paper and a slice of cardboard. Tack the canvass of paper to the paper-thin using the two tacks. Then tie the cord into a loop and wrap the loop around the ii tacks. Have your pencil and pull the string until the pencil and two tacks brand a triangle (see diagram at the correct). Then begin to trace out a path with the pencil, keeping the string wrapped tightly effectually the tacks. The resulting shape will be an ellipse. An ellipse is a special curve in which the sum of the distances from every point on the curve to two other points is a constant. The two other points (represented here by the tack locations) are known equally the foci of the ellipse. The closer together that these points are, the more closely that the ellipse resembles the shape of a circumvolve. In fact, a circle is the special example of an ellipse in which the ii foci are at the same location. Kepler'southward first law is rather simple - all planets orbit the sun in a path that resembles an ellipse, with the sun beingness located at one of the foci of that ellipse.

The Law of Equal Areas

Kepler'southward second law - sometimes referred to as the law of equal areas - describes the speed at which whatsoever given planet volition move while orbiting the sun. The speed at which any planet moves through space is constantly changing. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Nonetheless, if an imaginary line were fatigued from the centre of the planet to the center of the sun, that line would sweep out the same area in equal periods of time. For instance, if an imaginary line were fatigued from the earth to the dominicus, then the expanse swept out by the line in every 31-day calendar month would be the same. This is depicted in the diagram below. Every bit can exist observed in the diagram, the areas formed when the earth is closest to the sun can exist approximated as a wide just short triangle; whereas the areas formed when the earth is uttermost from the sun can be approximated every bit a narrow but long triangle. These areas are the same size. Since the base of these triangles are shortest when the earth is uttermost from the sun, the earth would accept to be moving more slowly in gild for this imaginary surface area to be the same size as when the earth is closest to the sun.

The Police of Harmonies

Kepler'southward tertiary law - sometimes referred to equally the law of harmonies - compares the orbital period and radius of orbit of a planet to those of other planets. Unlike Kepler's first and second laws that depict the move characteristics of a unmarried planet, the third police force makes a comparing between the motion characteristics of different planets. The comparison being made is that the ratio of the squares of the periods to the cubes of their average distances from the dominicus is the same for every one of the planets. As an illustration, consider the orbital menstruum and average distance from sun (orbital radius) for World and mars as given in the table below.

Planet	Menstruation (southward)	Average Distance (thousand)	T^two/R³ (s^two/m³)
Earth	3.156 x 10⁷ due south	1.4957 ten ten¹¹	2.977 x 10^-19
Mars	5.93 10 10^vii south	2.278 x 10^eleven	2.975 x 10^-nineteen

Observe that the Tⁱⁱ/R^three ratio is the aforementioned for Globe every bit it is for mars. In fact, if the same T^two/R³ ratio is computed for the other planets, it tin be constitute that this ratio is nearly the same value for all the planets (see table below). Amazingly, every planet has the same T²/R^three ratio.

Planet	Menses (yr)	Average Distance (au)	T²/R³ (yrⁱⁱ/au^three)
Mercury	0.241	0.39	0.98
Venus	.615	0.72	one.01
Globe	1.00	ane.00	1.00
Mars	i.88	1.52	1.01
Jupiter	eleven.viii	5.20	0.99
Saturn	29.v	nine.54	ane.00
Uranus	84.0	19.xviii	1.00
Neptune	165	30.06	1.00
Pluto	248	39.44	one.00

( Notation : The average distance value is given in astronomical units where 1 a.u. is equal to the distance from the globe to the sun - i.4957 x 10¹¹ yard. The orbital flow is given in units of world-years where one world year is the fourth dimension required for the globe to orbit the sun - 3.156 x 10^seven seconds. )

Kepler's third law provides an accurate description of the menstruation and distance for a planet's orbits about the sun. Additionally, the same police force that describes the T²/R³ ratio for the planets' orbits nearly the sunday also accurately describes the T²/R³ ratio for whatever satellite (whether a moon or a human-made satellite) about any planet. In that location is something much deeper to be found in this T²/R^three ratio - something that must chronicle to basic primal principles of motion. In the next role of Lesson 4, these principles will be investigated as we describe a connection between the round movement principles discussed in Lesson i and the motion of a satellite.

How did Newton Extend His Notion of Gravity to Explicate Planetary Move?

Newton's comparison of the acceleration of the moon to the acceleration of objects on world allowed him to establish that the moon is held in a circular orbit by the strength of gravity - a forcefulness that is inversely dependent upon the altitude betwixt the two objects' centers. Establishing gravity every bit the crusade of the moon'southward orbit does not necessarily establish that gravity is the cause of the planet's orbits. How and then did Newton provide credible evidence that the force of gravity is meets the centripetal force requirement for the elliptical motion of planets?

Recall from earlier in Lesson 3 that Johannes Kepler proposed iii laws of planetary motion. His Police of Harmonies suggested that the ratio of the flow of orbit squared ( Tⁱⁱ ) to the mean radius of orbit cubed ( R³ ) is the same value thousand for all the planets that orbit the sun. Known data for the orbiting planets suggested the post-obit average ratio:

chiliad = 2.97 x 10^-nineteen s²/m³ = (T²)/(R³)

Newton was able to combine the police force of universal gravitation with circular motility principles to show that if the force of gravity provides the centripetal force for the planets' nearly circular orbits, so a value of 2.97 x ten^-19 southwardⁱⁱ/m³ could be predicted for the T^two/Rⁱⁱⁱ ratio. Here is the reasoning employed by Newton:

Consider a planet with mass Thou_planet to orbit in nearly circular motion virtually the dominicus of mass M_Dominicus. The net centripetal force acting upon this orbiting planet is given by the human relationship

F_cyberspace = (Thou_planet * v^two) / R

This cyberspace centripetal force is the result of the gravitational force that attracts the planet towards the sunday, and tin can be represented as

F_grav = (G* Thou_planet* M_Sun) / R^two

Since F_grav = F_net, the above expressions for centripetal force and gravitational forcefulness are equal. Thus,

(M_planet * vⁱⁱ) / R = (G* Yard_planet* M_Dominicus) / R²

Since the velocity of an object in near circular orbit can be approximated every bit five = (2*pi*R) / T,

v² = (4 * pi²* R²) / T^two

Substitution of the expression for 5ⁱⁱ into the equation higher up yields,

(M_planet * 4 * pi²* R²) / (R • T²) = (G* 1000_planet* K_Dominicus) / R^two

Past cross-multiplication and simplification, the equation tin be transformed into

T²/ R³= (M_planet * 4 * pi²) / (G* M_planet* M_Sun)

The mass of the planet tin can then exist canceled from the numerator and the denominator of the equation'due south correct-side, yielding

T²/ R³= (four * pi²) / (1000 * M_Sun)

The right side of the to a higher place equation will be the same value for every planet regardless of the planet'southward mass. Subsequently, it is reasonable that the T²/R³ ratio would be the aforementioned value for all planets if the force that holds the planets in their orbits is the force of gravity. Newton's universal law of gravitation predicts results that were consistent with known planetary information and provided a theoretical explanation for Kepler's Law of Harmonies.

Investigate!

Scientists know much more about the planets than they did in Kepler's days. Use The Planets widget bleow to explore what is known of the various planets.

Check Your Understanding

1. Our understanding of the elliptical motility of planets well-nigh the Sun spanned several years and included contributions from many scientists.

a. Which scientist is credited with the collection of the data necessary to support the planet's elliptical motion?

b. Which scientist is credited with the long and difficult task of analyzing the data?

c. Which scientist is credited with the accurate explanation of the data?

2. Galileo is often credited with the early discovery of iv of Jupiter's many moons. The moons orbiting Jupiter follow the same laws of motion as the planets orbiting the sun. One of the moons is called Io - its distance from Jupiter's center is 4.2 units and it orbits Jupiter in 1.8 Earth-days. Another moon is called Ganymede; it is x.7 units from Jupiter's center. Brand a prediction of the flow of Ganymede using Kepler'southward police of harmonies.

three. Suppose a modest planet is discovered that is 14 times every bit far from the sun as the Earth's distance is from the sun (i.5 x 10^xi m). Utilize Kepler's police force of harmonies to predict the orbital period of such a planet. GIVEN: T²/R³= 2.97 ten ten^-xix south²/chiliadⁱⁱⁱ

4. The average orbital distance of Mars is ane.52 times the average orbital distance of the Globe. Knowing that the Earth orbits the sun in approximately 365 days, utilise Kepler's law of harmonies to predict the time for Mars to orbit the sun.

Orbital radius and orbital period data for the 4 biggest moons of Jupiter are listed in the tabular array below. The mass of the planet Jupiter is 1.nine x 10²⁷ kg. Base your answers to the side by side five questions on this information.

Jupiter's Moon	Period (due south)	Radius (m)	T²/R³
Io	1.53 x 10⁵	4.2 x x^viii	a.
Europa	three.07 x ten⁵	vi.vii x x⁸	b.
Ganymede	half dozen.18 x 10^v	1.1 x x^ix	c.
Callisto	1.44 x 10^six	one.ix x x⁹	d.

5. Determine the T^two/R³ ratio (concluding cavalcade) for Jupiter's moons.

6. What blueprint practice you observe in the last cavalcade of data? Which police of Kepler'south does this seem to back up?

seven. Utilise the graphing capabilities of your TI calculator to plot T² vs. R³ (T² should exist plotted along the vertical axis) and to determine the equation of the line. Write the equation in slope-intercept form below.

Run across graph below.

8. How does the Tⁱⁱ/R³ ratio for Jupiter (as shown in the final column of the data table) compare to the T²/R^three ratio found in #7 (i.e., the slope of the line)?

9. How does the T²/R³ ratio for Jupiter (as shown in the final cavalcade of the data table) compare to the T²/R³ ratio found using the post-obit equation? (G=half-dozen.67x10^-11 North*m²/kg² and G_Jupiter = i.9 x x²⁷ kg)

T²/ R³ = (four * pi²) / (G * M_Jupiter) Graph for Question #vi

Render to Question #half dozen