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Study Of Planetery Motion and Universal Gravitation

Historically, it appears that the Greeks gave us the word gravity, which translates as “heavy” according to some textbook authors.

During the 1600’s Galileo wanted to know what gravity is.  He studied the phenomenon and reasoned that all objects tend to fall at the same rate regardless of mass, if they are not affected by air.  (Air will cause relatively lightweight, aerodynamically shaped objects to fall more slowly than relatively heavy, non-aerodynamically shaped objects.) He never was able to explain what gravity was.

Tycho Brahe, acquiring an island astronomical observatory with the necessary astronomical observation tools of his day (Telescopes had not been invented yet.), spent 20 plus years observing the heavens. He and his assistants mapped the position and the change in position of every celestial object visible during this time.  He amassed a very large amount of data.

Johannes Kepler, acquiring Brahe’s data, spent a good 20 years analyzing the observations and came up with three laws of planetary motion.

Kepler’s Three Laws of Motion:

1. The paths (orbits) followed by the planets are elliptical in shape.

Ellipses have two axes.  The two axes stretching across the ellipse bisect each other at 90 degree angles.  Each one alone bisects the ellipse into two equal halves.  The longer of the two is called the Major Axis and divides the ellipse lengthwise.  The shorter of the two is called the Minor Axis and divided the ellipse widthwise.

Ellipses have a characteristic called eccentricity.  The symbol for eccentricity is the letter e.  The equation to calculate eccentricity is e = D. B. F. / M. A.  The letters DBF stand for the distance between foci (the two focus points of the ellipse).  The letters MA stand for the length of the major axis.

An Ellipse

Working with an ellipse
An ellipse has has a semi-major axis that is 4.125 cm long. It has a semi-minor axis of 3.65 cm. The distance of one of the two foci from the ellipse measured along the major axis is 2.205 cm.

a. Determine the length of the major-axis.
b. Determine the length of the minor-axis.
c. Determine the distance between the foci.
d. Determine the eccemtricity of this ellipse.

Part a:
E: Major-axis = 2 * semimajor axis
S: Major-axis = 2 * 4.125 cm
A: Major-axis = 8.25 cm

Part b:
E: Minor-axis = 2 * semiminor axis
S: Minor-axis = 2 * 3.65 cm
A: Minor-axis = 7.30 cm

Part c:
E: DBF = Major-axis - 2 * distance (ellipse --> focus)
S: DBF = 8.25 cm - (2 * 2.205 cm)
A: DBF = 3.84 cm

Part d:
E: e = DBF / Major-axis
S: e = 3.84 cm / 8.25 cm
A: e = 0.465

2. If an imaginary line were to be drawn from the sun to a planet, the line would sweep out equal areas of space in equal amounts of time.  This tells us that planets move faster when they are closest to the sun (perihelion), which is at one foci of the elliptical orbit, and mover slowest when they are farthest from the sun (aphelion).

Planetary Motion

Planet's velocity varies
At aphelion the planet's velocity is the slowest. Gravity is slowing it down as it moves away from the sun.
At Perihelion the planet's velocity is the fastest. Gravity accelerates it as it moves closer to the sun.

Calculating the average speed of a planet orbiting around the sun: There are two ways to look at this calculation. If you approach the calculation from the point of view of circular motion where you use the average orbital radius of a planet's orbit as the radius of a circular orbit, you will get a reasonable answer for the planet's average speed as it moves along its orbit. At perihelion we would expect the speed to be larger than this average value, and at aphelion we would expect the speed to be slower than this average value. On the other hand, you can approach the calculation from the point of view of Newton's Law of Universal Gravitation which uses the equation v = (G m / r)^(1/2). In this situation G is the proportionality constant found through the work of Henry Cavendish and equals 6.67 x 10^-11 N m^2 / kg^2, m is the mass of the object being orbited and r is the radius of the orbiting object's orbit.

Calculating the orbital speed of a planet
Using the earth as an example go to a planetary data table and find the radius of earth's orbit in meters (1.5 x 10^11 m). Also find or calculate the number of seconds in one year. If you calculate, you would use 365.25 days / year, 24 hours / day, 60 minutes / hour and 60 seconds per minute. (3.16 x 10^7 seconds / year)

Next find the circumference of the earth's orbit. Using a unit circle approach  assumes that a circle is an average of the actual elliptical orbit, and therefore the orbital radius of the earth's orbit is an average as well.

C = 2 p r = (2) (p) (1.5 x 10^11 m) = 9.425 x 10^11 m

And next find the number of seconds in the period of the earth. This is done by multiplying the period expressed in years by the number of seconds in ine earth year. Since we are doing earth which has a period of exactly one year we will be multiplying 1 year by 3.16 x 10^7 seconds / year.

T = (#years) (3.16 x 10^7 s / y) = (1 y) (3.16 x 10^7 s / y)
T = 3.16 x 10^7 s

Finally divide the circumference by the period.

v = C / T = 9.425 x 10^11 m / 3.16 x 10^7 s = 29,825 m/s

If you chose to use the equation based upon the G value in Newton's Law of Universal Gravity, you would do the following for the earth

v = (G * m / r)^(1/2)
v = [(6.67 x 10^-11) (1.99 x 10^30) / (1.5 x 10^11 m)]^(1/2)
v = 29,747 m/s

The difference in the two answers is due to differences caused by rounding and using a completely different approach. This difference is only 0.3 %. Both answers round off to between 2.97 and 2.98 x 10^4 m


3. The square of the ratio of the periods of any two planets revolving about the sun is equal to the cube of the ratio of their average distances (radii) from the sun.  The equation can be written as:

[T(of planet 1) / T(of planet 2)]^2 = [r(of planet 1's orbital radius) / r(of planet 2)'s orbital radius)]^3

Using Kepler's Third law to find the period of an orbiting object
Kepler's Third Law says that The squares of the periods of the planets are proportional to the cubes of their semimajor axes. As an example consider a simple solar system where two planets orbit about a star. One planet (planet a) orbits the star in 2 earth years and is 250 million kilometers from the star and the other planet (planet b), whose period is unknown is 500 million kilometers from the star. Find the period of the second planet.

Given: Ta = 2 y, Ra = 2.5 x 10^8 km, Tb = uknown, Rb = 5 x 10^8 km.
Find: The period of the second planet.
Equation: Ta^2 / Tb^2 = Ra^3 / Rb^3
         OR Ta^2 * Rb^3 = Tb^2 * Ra^3
        And Tb^2 = (Ta^2 * Rb^3) / Ra^3
Substitution: Tb^2 = {(2 y)^2 * (5 x 10^8 km)^3} / (2.5 * 10^8 km)^3
Then to find Tb you need to take the square root of the value for Tb^2.
Answer: Tb^2 = 32.0356 y^2, and so Tb = 5.66 years

This law also applies when comparing the orbits of two moons orbiting the same planet.

Copy this “Planetary Data Table” into your notes.

Planetary Data Table
Name of Object Pronunciation Average distance from the sun (m) Mass of the object (kg) Size of object Average radius (m)
Sun Sun 0 1.991 x 10^30 6.96 x 10^8
Mercury Mer qu ree 5.8 x 10^10 3.2 x 10^23 2.43 x 10^6
Venus Vee nus 1.081 x 10^11 4.88 x 10^24 6.073 x 10^6
Earth Earth 1.4957 x 10^11 5.979 x 10^24 6.3713 x 10^6
Mars Mars 2.278 x 10^11 6.42 x 10^23 3.38 x 10^6
Jupiter Joo pi ter (short i) 7.781 x 10^11 1.901 x 10^27 6.98 x 10^7
Saturn Sat ern 1.427 x 10^12 5.68 x 10^26 5.82 x 10^7
Uranus Yoo ray nes 2.870 x 10^12 8.68 x 10^25 2.35 x 10^7
Neptune Nep toon 4.5 x 10^12 1.03 x 10^26 2.27 x 10^7
Pluto Ploo toh 5.9 x 10^12 1.2 x 10^22 1.15 x 10^6

The earth’s orbit is elliptical in shape.  The sun is at one foci of the orbit.  The earth is closest to the sun (perihelion) around the winter solstice (usually on Dec. 21 or 22). It travels fastest as it moves closest to the sun.  The earth is farthest from the sun (aphelion) around the summer solstice (usually on June 21 or 22).  It travels slowest at it moves furthest from the sun.  Usually on March 21 or 22 the sun crosses the celestial equator from south to north.  This event is called the spring or vernal equinox.  The length of daylight and darkness are equal on this day.  Likewise, on September 21 or 22 the sun again crosses the celestial equator from north to south.  This event is called the fall or autumnal equinox. [The celestial equator is the projection of the Earth's equator onto the sky. During the summer months in the Northern hemisphere the sun appears above the celestial equator. During the winter months in the Northern hemisphere the sun appears below the celestial equator.]

Eccentricity of a planet's orbit is a measure of how much an orbit deviates from circular. A perfectly circular orbit has an eccentricity of zero; an eccentricit between 0 and 1 represents an elliptical orbit. A parabolic orbit has an eccentricity equal to 1; a hyperbolic orbic has an eccentricity greater than 1. Neptune, Venus, and Earth are the planets with the least eccentric orbits in our solar system. Pluto and Mercury are the planets with the most eccentric orbits in our solar system. [To learn more about ellipses, look up conic sections.]

Calculating Eccentricity
The equation for calculating the eccentricity of an ellipse, whether the ellipse is a geometric figure drawn on a sheet of paper or the eliptical orbit of a planet in a solar system is the distance between foci divided by the length of the major axis (e = DBF / Major Axis).

If an elipse has a major axis 100 cm in length and the DBF is 20 cm, what is the eccentricity of the ellipse?

Given: MA = 100 cm; DBF = 20 cm
Find: e, the eccentricity
Equation: e = DBF / MA
Substitution: e = 20 cm / 100 cm
Answer: e = 0.2

The Sun's Position in the Southern Sky

Ever since people began looking at the sky and what appears there, they have often thought of the earth as being inside of a very large sphere upon which the objects in the heavens can be viewed. In astronomy this concept of an imaginary sphere enclosing the earth has been adopted to explain and demonstrate the location and movement of objects seen in the heavens. This imaginary sphere is given a name. It is called the celestial sphere and is defined as the apparent surface of the imaginary sphere on which celestial bodies appear to be projected. This idea has lead to the building of domed ceiling rooms at planetariums where along with the aid of computers astronomers can project images of the night sky.

Note: That astronomical observatories are built for the purpose of housing telescopes, and that while planetariums may have one or more observatories, the definitions for a planetarium are as follows.

1. An apparatus or model representing the solar system.
2. An optical device for projecting images of celestial bodies and other astronomical phenomena onto the inner surface of a hemispherical dome.
3. A building or room containing a planetarium, with seats for an audience.

The path of the Sun across the celestial sphere is very close to that of the planets and the moon. With the invention of clocks, it was possible for astronomers to relate the path of the Sun in the daytime to the one of stars and planets at night. Because of its relation to eclipses, the path of the sun is known as the ecliptic. Because of the tilt of the earth's axis the sun does not rise and set exactly in the east and the west respectively. In fact it only rises and sets exactly in the east and the west respectively on the two days of the year called the equinoxes.

Rise and Set Positions of the Sun

The tilt of the earth’s axis is responsible for our seasons.  In winter the northern hemisphere is tilted away from the sun at an angle of 23.5 degrees.  In summer the northern hemisphere is tilted towards the sun at an angle of 23.5 degrees.  Only on two days of the year does the sun rise directly in the east and set directly in the west.  These two days are the day of the vernal (spring) equinox and the day of the Autumnal (fall) equinox. After the vernal equinox the sun will appear to rise further and further north of east each day until on the summer solstice it will rise 23.5 degrees north of east and set 23.5 degrees north of west.  Similarly, After the autumnal equinox the sun will appear to rise further and further south of east each day until on the winter solstice it will rise 23.5 degrees south of east and set 23.5 degrees south of west.

The Earth in its Orbit

The Astronomical Unit (AU): Astronomers deal in really large distances across space, so it is not surprising that they define unusually large units of lengths that are used to express large distances across space. One such unit is the Astronomical Unit, symbol AU. One AU equal the average distance between the sun and the earth, that is the average radius of earth's orbit. Thus 1 AU equals 1.5 x 10^11 m. To convert the orbital radii of other planets to AU's multiply the planet's orbit's radius expressed in meters by 1 AU / 1.5 x 10^11 m. Below is an example calculation finding Mars orbit's radius into AU's.

Converting  Mars orbit's radius to AU's
Mars orbit's radius is 2.278 x 10^11 m. To convert this to AU's multiply by the conversion factor defined above.

r = (2.278 x 10^11 m) (1 AU / 1.5 x 10^11 m)
r = 1.52 AU

Universal Gravitation:

When Isaac Newton was in his twenties an epidemic of the black plague ravaged London. He went to live at his family’s home in rural England to escape the plague.  While there he pursued the study of gravity and Kepler’s Laws.  He came up with two very important conclusions as a result of his efforts.  They are:

1. The force of gravity between two objects is proportional to the masses of the objects.  That is, if the mass is doubled the force of gravity is doubled.

2. The force of gravity between two objects is proportional to one over the distance between them squared.  That is, if the distance that separates the two objects is doubled the force of gravity is reduced to one fourth of its value before the distance between them was increased.  The distance is measured not from the outer edge of one object to the outer edge of the other object, but rather, it is measured from the center of mass of one object to the center of mass of the second object.

Law of Universal Gravitation Equation:

F(gravity) = G m(1) m(2) / d^2 where G is Newton's universal gravitational constant equaling 6.67 x 10^-11 N m^2 / kg^2.

Newton came up with this equation to mathematically express the relationships among the variables of gravitational force, mass, and distance.  Newton was not able to make the necessary measurements required to calculate the value of G.  Later, a man by the name of Henry Cavendish invented a device that would detect gravitational force and allow the calculation of G.  Henry Cavendish (1731-1810) was an English chemist and physicist. He was the first person to determine the value of G. His experiment has been called “the weighing of the earth.

Calculating the Force of Gravity between two objects
Two masses are brought near one another such that their gravitational force could be observed. Mass 1 is 100 g, while Mass 2 is 200 g. The distance separating them is 0.2 cm. Calculate the force of gravity pulling them together.

Given: Mass 1 = 100 g; Mass 2 = 200 g; d = 0.2 cm
Find: The force of gravity between them.
Equation: Fg = G m1 m2 / d^2
Substitution: Fg = 6.6 x 10^-11 N m^2 / kg^2 * 0.1 kg * 0.2 kg / (0.002 m)^2
Answer: 3.3 * 10^-7 N

Distances Between A Star and Its Planets:

When calculating the force of gravity between two planets, it is necessary to define the position of each of the planets and the radii of their orbits. The diagram belows shows an example of two planets aligned on the same side of the star they orbit.

Distances between Celestial Objects

Einstein's Perspective:

Albert Einstein was a German physicist who immigrated to the United States. He lived from 1879-1955. He changed our conception of the universe with his Theories of Special and General Relativity. His revolutionary ideas opened the door to the development of Quantun Physics during the 20th century. Important points concerning his theories are listed below.

1. Special relativity supplanted Newtonian mechanics, yielding different results for very fast-moving objects.
2. The Theory of Special Relativity is based on the idea that speed has an upper bound; nothing can pass the speed of light.
3. The theory also states that time and distance measurements are not absolute but are instead relative to the observer's frame of reference.
4. Space and time are viewed as aspects of a single phenomenon, called space-time. Energy and momentum are similarly linked.
5. Mass can be converted into huge amounts of energy, and vice versa, according to the formula E=mc2.
6. General Relativity expands the theory of special relativity to include acceleration and gravity, both of which are explained via the curvature of space-time.
7. In Astronomy his theories explained the perturbations in the orbits of Mercury.

Einstein descibed gravity in terms of a distortion of space and time by the presence of mass. Einstein likened the curve of Space-Time to the two dimensionsl model of a heavy mass placed on a rubber sheet. The distortion of the rubber sheet causes other objects in the vincinity of the heavy mass to "fall" into the depression made by the heavy mass sitting on the sheet. See the model below.

The Rubber Sheet Model


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