Comprehensive Study Guide for Artificial Terrestrial Satellites and Planetary Dynamics and Earth Movements

Introduction to Artificial Terrestrial Satellites

The movement of artificial satellites orbiting the Earth began historically on October 4, 1957, with the launch of the first artificial satellite of the Earth, Sputnik 1 (SAT). Since that historic event, approximately 8900 satellites have been launched into space by more than 40 different countries. These satellites serve various sectors, including commercial, government, civil, and military operators. The distribution of these satellites often focuses on Low Earth Orbit (LEO), which is defined by specific orbital parameters. According to data from the Union of Concerned Scientists covering the years 2000 to 2022, the perigee (the minimum height of a satellite's orbit) for these objects typically ranges from $0\,km$ up to $1800\,km$ . The launch mass of these satellites varies significantly, with some weighing as little as $1000\,kg$ and others reaching over $25000\,kg$ .

The Three Principal Cosmic Velocities

In the field of astronautics, three primary velocity values are considered essential for space flight and orbital mechanics. The first of these is the Primary Cosmic Velocity, also known as the circular velocity. This velocity is achieved when the gravitational force is equalized with the centripetal force. By performing the respective calculations where $F_g = F_c$ (gravitational force equals centripetal force) and $F_c = m \times a_{rad}$ , we can derive the formula for the orbital velocity:

$v = \sqrt{\frac{G \times m_E}{r}}$

Using the gravitational constant $G = 6.67 \times 10^{-11}\,N\cdot m^2/kg^2$ , the mass of the Earth $m_E = 5.97 \times 10^{24}\,kg$ , and an orbital radius $r = 6.68 \times 10^6\,m$ , the resulting primary cosmic velocity is approximately $7720\,m/s$ . This value represents the velocity a satellite possesses directly on the surface of the Earth. As a satellite moves further from the surface, its circular velocity decreases relative to the primary cosmic velocity. For example, the Hubble Space Telescope is located at an altitude of $593\,km$ above sea level. Using the formula $v_t = v_{sh} \times \frac{R}{R + h}$ where $R = 6680\,km$ , the tangential velocity calculated for the telescope is $7398\,m/s$ . The orbital period ( $T$ ) of an artificial satellite is determined by Kepler's Third Law, expressed as:

$T = 2\Pi \times \frac{R}{g} \times a^{3/2}$

In this context, $a$ represents the apogee of the orbit and $R$ is the radius of the Earth. It is noted that the orbit of an artificial satellite is subject to perturbations caused by the Earth's equatorial bulge (engrosamiento ecuatorial), atmospheric resistance, and the gravitational attraction of both the Moon and the Sun.

The Second Cosmic Velocity, also referred to as the parabolic velocity or escape velocity, is the minimum initial velocity that must be communicated to a body so that, starting its movement close to the Earth's surface, it overcomes the Earth's gravitational force entirely. This velocity depends on the height of the object and is expressed by the formula:

$v = R \times \sqrt{2 \times g \times (\frac{1}{R} - \frac{1}{r})}$

For these calculations, the radius of the Earth $R_t = 6378\,km$ and the acceleration of gravity $g = 0.00981\,km/s^2$ . For an object escaping the orbit, the radius $r$ is considered infinite ( $r = \infty$ ), making $\frac{1}{r} = 0$ . This simplifies the formula to:

$v = \sqrt{2 \times g \times R} = 11.2\,km/s$

The Third Cosmic Velocity, or hyperbolic velocity, is the minimum initial velocity under which a body, starting its movement near the terrestrial surface, first overcomes the Earth's gravitational pull, then overcomes that of the Sun, and finally leaves the Solar System. On the terrestrial surface, this third cosmic velocity is valued at $16.7\,km/s$ .

Dynamic Characteristics and Interior of the Earth

Earth is the third planet in the Solar System. Its specific mass and orbital situation make it a privileged planet, featuring an average temperature of approximately $15^\circ C$ , liquid water, and a dense atmosphere containing oxygen—conditions essential for the development of life. Unlike other planets, approximately $71\%$ of the Earth's surface is covered by water.

The interior composition of the Earth consists of several layers. The internal core is made of solid iron with a density of $13000\,kg/m^3$ . Surrounding this is the external core, which consists of liquid iron mixed with some nickel. The mantle contains substances rich in iron and magnesium; while primarily solid, its upper layers can move slowly and are considered plastic. This plastic region, known as the asthenosphere, is comprised of silicon-rich rock and is able to flow because its temperature is slightly higher than its melting point ( $1000^\circ C$ to $2000^\circ C$ ). The outermost layer is the crust, which is the thinnest layer with a depth ranging from $5\,km$ to $35\,km$ . The crust is solid because the temperature is lower than its melting point. The surface of the Earth is not static; it changes over time due to tectonic movement as tectonic plates shift.

The Terrestrial Atmosphere and Magnetosphere

Earth's atmosphere is composed of $78\%$ nitrogen, $21\%$ oxygen, and $1\%$ other gases (argon, carbon dioxide, and neon). It serves to regulate the planet's climate and protects it from solar radiation and meteoroids. The atmosphere is structured into five distinct layers:

Troposphere: Starting at the surface and rising to $12\,km$ . It contains $99\%$ of all water vapor and aerosols. It is the densest layer and is where most clouds exist.
Stratosphere: Located between $12\,km$ and $50\,km$ in height. This layer contains the ozone layer. Unlike the troposphere, temperature increases with altitude here. Some aircraft can reach this height.
Mesosphere: Between $50\,km$ and $80\,km$ . It contains a small amount of water vapor that forms noctilucent clouds (the highest clouds on Earth). This is where meteors typically burn up.
Thermosphere: Between $80\,km$ and $700\,km$ . Temperatures rise again due to low molecular density. The International Space Station orbits in this layer, and auroras are formed here.
Exosphere: The final layer, ranging from $700\,km$ to $10000\,km$ . Molecular density is extremely low, and particles escape into space. While there is no weather here, auroras sometimes form in its lowest parts. Most terrestrial satellites are located in the exosphere.

The Earth possesses a Magnetosphere (magnetic field) generated by the planet's rotation and the liquid outer core. This field protects the Earth from solar radiation. When charged particles from the sun fall into the atmosphere, they cause auroras. Although the polarity of the magnetic field can change, it is not predicted to occur within the next $1000$ years.

Ocean Tides and Gravitational Influence

Tides are the periodic rise and fall of oceanic waters, including the open sea, gulfs, and bays. They result from the gravitational attraction of the Moon and the Sun on the water and the Earth itself. Waters on the side closer to the Moon are pulled more intensely than the Earth itself, while water on the opposite side is pulled less intensely than the Earth, creating bulges on opposite sides. The Sun's gravitational attraction produces a similar effect, but its tide-generating force is $2.2$ times smaller than that of the Moon because the Sun is significantly further away.

Tide types include Flow (ascend), Ebb (descend), Spring tides (vivas), and Neap tides (muertas). During the New Moon (novilunio/conjunction) and Full Moon (plenilunio), the solar and lunar gravitational actions combine to create maximum flow tides. Conversely, during the first and third quarters of the Moon (cuadraturas), the solar action subtracts from the lunar action, resulting in minimum flow tides (neap tides). At specific points A and B on Earth, the lunar acceleration ( $W_A, W_B$ ) is greater than or less than the average Earth acceleration ( $W_T$ ), which weakens Earth's gravity and causes flow tides. At points F and D, the accelerations are directed at obtuse angles, increasing gravitational force and causing ebb tides.

Tide height varies by location. In internal seas like the Black Sea, flows are insignificant (only a few centimeters). In the open ocean, the magnitude does not exceed $1\,m$ , but in littoral regions like the Atlantic coast of Canada, tides can reach $18\,m$ . Atmospheric pressure also experiences tides, and the Earth's crust itself rises and descends by a few decimeters twice a day.

Rotational and Translational Movements of the Earth

The Earth moves in four primary ways. Translation is the movement of the Earth around the Sun, taking approximately $365$ days. However, the precise duration is $365.256363004$ days. To account for this, leap years add an extra day every four years. Further adjustments state that century years are only leap years if they are divisible by $400$ . The average orbital speed of Earth is $29702\,m/s$ .

Rotation is the movement of the Earth around its own axis, causing day and night. A complete revolution takes exactly $23\text{ hours, } 56\text{ minutes, and } 4\text{ seconds}$ . This is known as a Sidereal Day. The Solar Day, which is the time for the Sun to return to the midday position, lasts exactly $24\text{ hours}$ . The difference exists because the Earth must rotate slightly more to align with the Sun due to its concurrent translational movement.

Precession and Nutation

Because the Earth is not a perfectly solid sphere and is not homogeneous, its rotational axis undergoes complex movements. Precession is the slow movement where the Earth's axis describes a cone around the axis of the ecliptic. The cone has a radius of $23.26'$ and is tilted at an angle of $66.34'$ relative to the plane of the ecliptic. This movement has a period of approximately $26000$ years and is caused by the gravitational pull of the Sun and Moon on the Earth's equator.

Nutation refers to small oscillations or a "vibrating" back-and-forth movement occurring within the precession cycle, describing a small ellipse. This is caused by the Moon's gravitational pull and the fact that the Moon's orbit is not on the ecliptic, changing the Earth's axis angle by about $1/2^\circ$ every $18\text{ years}$ . The period of nutation is $18.6\text{ years}$ . Consequently, in one complete precession cycle, the Earth undergoes approximately $1385$ nutations.

Seasonal Cycles and Orbital Extremes

The combination of the Earth's axial tilt and its translation around the Sun creates the seasons. The Northern Hemisphere receives more heat from March 21 to September 23 (Spring and Summer) than from September 23 to March 21 (Autumn and Winter), while the Southern Hemisphere experiences the opposite. Key dates include:

Summer Solstice: June 21-23 (Summer in North, Winter in South).
Winter Solstice: December 21-22 (Winter in North, Summer in South).
Spring Equinox: March 21-22 (Spring in North, Autumn in South).
Autumn Equinox: September 22-23 (Autumn in North, Spring in South).

Earth's orbital path is elliptical, with the closest point to the Sun called perihelion and the furthest point called aphelion. At perihelion, the Earth receives $7\%$ more heat than at aphelion. This helps explain why the Northern Hemisphere has slightly less harsh winters and fresher summers compared to the Southern Hemisphere.