[??]r is a unit vector that has the same direction as the one of the gravitation force which is directed to the center of the planet Earth. Those two forces are equal. Then we have:

[??] = G[M/r2] [??]r (1.9)

In magnitude,

a = G[M/r2]. (1.10)

If the radius of the planet Earth is Re, the acceleration on the surface of the planet Earth is [g = G[M/Re2]. Comparing the acceleration on the surface of the planet Earth and the one at the distance r from the center of the planet Earth, we have:

a/g = (Re/r)2 (1.11)

1.7. Planetary motion and Kepler's laws

For centuries, physicists have been trying to explain the motion of the planets around the sun in the solar system. Among the theories that held sway over years, was the geocentric theory associated with Claudius Ptolemy (C. A.D. 150). This theory was successful in explaining the planetary motion with a degree of accuracy related to the level of knowledge in those days. In this theory, the planet Earth was the center of the universe. In fact, the word geocentric can be split in two words: geo which means Earth and centric or center. This theory gives a model called the Ptolemaic model and was accepted by old civilizations such as the Greek civilization. It is then assumed that all the celestial bodies such as the sun, the moon, the stars and others, circled around the Earth.

Before Ptolemy, other geek philosophers, Aristotle and his student Plato wrote on the theory of geocentrism considering the planet Earth as spherical, stationary at the center of the universe and other celestial bodies turn around the earth on circular orbits. A century later, Aristarchus of Samos (310 -210 B.C.), propose his theory considering the sun fixed at the center of the universe. In his theory the earth revolve around the sun in a circular orbit. He also noticed that the start appears fixed in position because their distances from the sun were huge compare with the distance sun-earth. This theory called heliocentric theory was not well accepted in those days. From the second century to the sixteen century only the Ptolemy theory was accepted and taught. Later, Nicolas Copernicus (1472-1543) will revive again the heliocentric theory of Aristarchus of Samos which will trigger a scientific revolution in which other great physicist such as Kepler, Galileo and Newton would play a key role. In his heliocentric theory, Copernicus considers the sun at the center of the universe and all the planets revolve around the sun in circular orbits. The fixed stars were assumed to lie in spheres surrounding the solar system. The heliocentric theory met some opposition. However, a famous Danish astronomer, Tycho Brahe (15461601) made very careful and unprecedented accurate measurements of the motions of the planets that convinced him about Copernican hypothesis. Brahe careful and convincible observations and measurement were bequeathed to German astronomer Johannes Kepler (1571-1630) who after analysis of the data introduced three laws that describe the motions of the planets around the sun.

Kepler's three laws can be stated as follows:

First law

The path or orbit of each planet in it motion around the sun is elliptic with the sun at one focus of the ellipse.

Second law

As the planet moves in its orbit, a line drawn from the sun sweep out equal area in equal intervals of time

Third law

The square of the periods of the planets are proportional to the cubes of their mean distance from the sun.

We can obtain the mathematical expression of the third law by using the combine Newton' second law of motion and Newton law of universal gravitation. Thus considering the mass of the sun as M and the mass of the planet as m moving with a velocity [??] and a period of revolution T, the gravitational for applied to the planet will be:

F = G[mM/r2] (1.12)

And the centripetal acceleration of the planet in motion is

a = v2/r = rω2 = 4π2r/T2 (1.13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.14)

So,

T2/r3 = rπ2/GM = constant (1.15)

We can see that T2 is proportional to r3

Note that r is the average distance is the average distance between the planet and the sun.

Problem

1. State all Newton three laws of motion and Newton's law of Universal gravitation.

1. Calculate the altitude to which a rocket must be fired in order that the acceleration due to gravity is half of that of the surface of the planet Earth.

2. An airplane traveling at 400mi/hr makes a complete circle in 10min. What is the centripetal acceleration of the airplane? Calculate the centripetal force on the airplane.

3. Calculate the gravitational force between the moon and the planet earth. The mass of the moon is 7.34767309 × 1022kg and the mass of the planet earth is 5.972 × 1024kg.

4. Considering the motion of the moon around the planet earth, calculate the velocity of the moon, the centripetal acceleration of the moon toward the planet earth and the centripetal force from the earth on the moon. The distance earth – moon is about 238,900 miles (384,400 km).

The Planet earth's atmosphere

2.1. Description of the solar system
2.2. Planet Earth atmosphere composition.
2.3. Particles in the troposphere
2.3.1. Sea Salt particles
2.3.2. Particles from fires, rocks, soils and
volcanoes

2.1. Description of the solar system.

The planet Earth is in the solar system, a set of eight planet revolving about the sun. Four of them are rocky and terrestrial worlds such as Mercury, Venus, Earth and Mars. The four others are giant gases such as Jupiter, Saturn, Uranus and Neptune. Between the orbits of Mars and Jupiter lies the asteroid belt, which includes the dwarf planet Ceres. Beyond the orbit of Neptune one can observe the disk-shaped Kuiper belt, in which dwarf planet Pluto resides.

The sun itself is a huge mass of fire of light of mean radius about 6.96 x 108 m and mass 1.991x1030 kg. The planet Earth has a mean radius of 6.38x106 m and a mass of 5.98x1024 kg. The environment surrounding the planet earth is the atmosphere of the planet earth. It is composed of several gases. Planet earth seen from the above the northern hemisphere rotates counterclockwise about it axis of rotation once in 24hours. It also revolves around the sun in 365 1/4 days.

2.2. Planet earth atmosphere composition.

Planet's earth's atmosphere is composed of clouds unequally distributed in the lower 6 to 10 km. 99% of the atmosphere is confined in the lowest 30km while half of the atmosphere is confined in the 6-km layer and that's where most of the clouds are found. Temperature distribution is shown in figure 2.4. The lowest part of atmosphere that is from the surface of the planet Earth up to about 10km is called troposphere. In this region, the temperature decreases with altitude. The word troposphere comes from two words: tropo which stands for turning, and sphere. In other world it stands for changing or turning sphere. The troposphere contains about 80% of the total atmospheric mass. This means that the heaviest particles in the atmosphere are found in the troposphere. Complex atmospheric phenomena occur in the troposphere. Evaporation and heat conduction at the surface of the planet Earth are responsible of the horizontal and the vertical temperature gradient. In the troposphere, it is known that the rising of air cools adiabatically or pseudo-adiabatically resulting in temperature decrease. The size of the troposphere depends on the season and the altitude and the location in the space surrounding the atmosphere. In the tropical regions, it is between 16km and 18km; over the poles, around 8 to 10km in the summer and almost absent in the winter. The troposphere ends with a region called tropopause which is characterized by an increase of static stability and above which resides a statistically stable region called stratosphere which ranges from 20km to 50km. In the stratosphere, the temperature first increases very slowly up to about 20 and 30km, and then increases more rapidly up to 50km where it becomes close the earth's temperature.

The region right after the stratosphere where the temperature is almost constant is called stratopause. Lots of phenomena here are still poorly understood. Above the stratopause, the atmosphere layer is called mesosphere which stands for the middle sphere. In this region, temperature decreases with the height up to around 180K at the height of 85km. Wind speed can also be very height and reach about 150m/s. Temperature distribution in the region is similar to what it is in the troposphere. This can imply that similar atmospheric processes might be happening. Energy source in this region may come from the solar radiation absorption in the stratopause. Above the mesosphere between 80km and 90km, is a region called mesopause where notilucent clouds are sometimes observed. Above the mesopause, which is above 90km, we have a region called thermosphere where temperature increases from 600K to abut 2000K at the height of 500Km. Thermal processes occur in this region because of the sun's activities. Several other phenomena occur in this region such as the ionization, the dissociation of molecules and the diffusion which is more important than mixing; heaviest gases become less concentrated; molecules are dissociated in atoms. Around 100km, atomic oxygen [O] increases with the height while molecules such as nitrogen [N2] decrease with the height. However, at 100km, atomic oxygen dominates. Later around 500Km to 1000km Helium becomes the more dominant molecule species. Above 1500km, the dominant agent will be atomic hydrogen [H]. The region between 110km and 200km is called thermopause. The region above the thermopause is the exosphere which extends from about 300km to about 500km. In this region, some particles can escape the gravitational field

2.3. Particles in the troposphere

Particles in the atmosphere are also called aerosols. There are four majors groups in the mechanism of formation of aerosols. The first group is the condensation and sublimation of vapor and the formation of anthropogenic source smoke. The second other group concern chemical reaction in the atmosphere. The third is the mechanical disruption and dispersal of matter at the earth's surface such as ocean or mineral. The fourth group is the coagulation of particles which gives larger articles. There's a fifth group which concerns the influx of extraterrestrial particle falling into the earth's atmosphere.

The particles found in the troposphere are both natural and man-made. They are sometimes very difficult to distinguish.

2.3.1. Sea Salt particles

Since ocean makes up about the ¾ of the total surface of the planet earth, we have a huge number of sea particles usually called salt. It is made of sodium chloride (Na+, Cl-). There are about 100 sea salt particles per cm3 in the atmosphere. These particles are widely spread over the atmosphere and participate in some chemical reactions with other particles and trace gases in the atmosphere and extra-terrestrial particles. Pure sodium chloride crystals are hygroscopic and form droplet when the relative humidity is more than 75%. However when the related humidity drops below 75% water vapor in which sea salt is dissolved will evaporate from the sodium chloride until the vapor pressure of the droplet becomes equal to the partial pressure of water vapor in the air. Droplets are mainly form by breaking myriads of air bubbles at the sea surface. Bubbles are produced by breaking small waves, rain or snow falling in water. Near the sea salt, the concentration in salt is mostly high and decrease with distance from the coast and with altitude. But the particles size distribution remains the same. That's why the removal process does not depend on the particles size. The removal process is however less better by sedimentation than by precipitation. The chloride ions in the troposphere are associated with other positive ions to form some giant aerosols in the atmosphere at the vicinity of the oceans.