Earth Satellites

Earth Satellites

An Earth satellite is an object that orbits the Earth. This can be a natural satellite, like the Moon, or an artificial satellite launched by humans for various purposes. Artificial satellites are widely used for communication (like DTH TV, mobile networks), navigation (like GPS, India's NAVIC/IRNSS), weather forecasting, remote sensing, scientific research, and military applications.

A satellite stays in orbit because of the balance between its inertia (its tendency to move in a straight line at a constant speed) and the Earth's gravitational pull. As the satellite moves forward, the Earth's gravity constantly pulls it towards the centre, causing it to fall towards the Earth. However, because of its forward motion (velocity), it simultaneously moves horizontally such that the Earth's surface curves away beneath it at the same rate it falls, effectively keeping it in orbit around the Earth rather than hitting the ground.

Orbital Velocity

For a stable circular orbit at a constant altitude (and thus a constant distance from the Earth's centre), the gravitational force provides the necessary centripetal force. Let $M_E$ be the mass of the Earth, $m_s$ be the mass of the satellite, and $r$ be the distance of the satellite from the centre of the Earth ($r = R_E + h$, where $R_E$ is Earth's radius and $h$ is the altitude). The gravitational force is $F_g = G \frac{M_E m_s}{r^2}$. This force acts as the centripetal force $F_c = \frac{m_s v_o^2}{r}$, where $v_o$ is the orbital velocity.

$ F_g = F_c $

$ G \frac{M_E m_s}{r^2} = \frac{m_s v_o^2}{r} $

The mass of the satellite $m_s$ cancels out, indicating that the orbital velocity for a given orbit is independent of the satellite's mass. This is similar to how the acceleration due to gravity is independent of a falling object's mass.

$ \frac{GM_E}{r^2} = \frac{v_o^2}{r} $

$ v_o^2 = \frac{GM_E}{r} $

$ v_o = \sqrt{\frac{GM_E}{r}} $

where $r = R_E + h$. This formula gives the orbital velocity for a circular orbit at a distance $r$ from the Earth's centre. Note that as the orbital radius $r$ increases (higher altitude), the required orbital velocity decreases.

Orbital Period

The orbital period $T$ is the time it takes for the satellite to complete one full orbit. For a circular orbit, the distance travelled in one period is the circumference of the orbit, $2\pi r$. The period is distance divided by speed:

$ T = \frac{\text{Circumference}}{\text{Orbital Speed}} = \frac{2\pi r}{v_o} $

Substitute the expression for $v_o$:

$ T = \frac{2\pi r}{\sqrt{GM_E/r}} = 2\pi r \sqrt{\frac{r}{GM_E}} = 2\pi \sqrt{\frac{r^3}{GM_E}} $

$ T^2 = \frac{4\pi^2 r^3}{GM_E} $

This formula shows that the square of the orbital period is proportional to the cube of the orbital radius ($T^2 \propto r^3$). This is Kepler's Third Law, derived here for a circular orbit around the Earth. The constant of proportionality is $\frac{4\pi^2}{GM_E}$, which depends on the mass of the central body (Earth).

Answer:

Energy Of An Orbiting Satellite

An orbiting satellite possesses both kinetic energy (due to its motion) and gravitational potential energy (due to its position in Earth's gravitational field). The total mechanical energy of the satellite is the sum of these two energies.

Kinetic Energy of a Satellite

For a satellite of mass $m_s$ in a circular orbit with velocity $v_o$ at distance $r$ from the Earth's centre, the kinetic energy is:

$ KE = \frac{1}{2} m_s v_o^2 $

We know that for a circular orbit, $v_o^2 = \frac{GM_E}{r}$. Substituting this into the KE equation:

$ KE = \frac{1}{2} m_s \left(\frac{GM_E}{r}\right) = \frac{GM_E m_s}{2r} $

Gravitational Potential Energy of a Satellite

The gravitational potential energy of a satellite of mass $m_s$ at a distance $r$ from the Earth's centre, relative to infinity, is:

$ PE = -\frac{GM_E m_s}{r} $

Total Mechanical Energy of a Satellite

The total mechanical energy $E$ is the sum of the kinetic energy and potential energy:

$ E = KE + PE $

$ E = \frac{GM_E m_s}{2r} + \left(-\frac{GM_E m_s}{r}\right) $

$ E = \frac{GM_E m_s}{2r} - \frac{GM_E m_s}{r} $

$ E = \left(\frac{1}{2} - 1\right) \frac{GM_E m_s}{r} $

$ E = -\frac{GM_E m_s}{2r} $

The total mechanical energy of a satellite in a circular orbit is negative and is equal to half of the potential energy (and the negative of the kinetic energy, $E = -KE$).

The negative sign indicates that the satellite is bound to the Earth's gravitational field. Work must be done on the satellite by an external force to increase its energy (make it less negative or positive) to move it to a higher orbit or escape from orbit entirely.

For elliptical orbits, the kinetic and potential energy vary throughout the orbit, but the total mechanical energy $E$ is constant (if no external forces like air resistance are doing work). For an elliptical orbit with semi-major axis $a$, the total energy is given by the same formula:

$ E = -\frac{GM_E m_s}{2a} $

where $a$ is the semi-major axis of the elliptical orbit.

This formula is also consistent with the concept of escape speed. To escape from orbit, the satellite's total energy must become zero or positive ($E \ge 0$). If $E = -\frac{GM_E m_s}{2r}$ (for a circular orbit), making $E \ge 0$ requires adding energy $\frac{GM_E m_s}{2r}$ (or more) to the system. This energy is added by boosting the satellite's speed using rockets.

Answer:

Geostationary And Polar Satellites

Artificial satellites are placed in specific orbits depending on their intended purpose. Two important types of orbits are those used by Geostationary Satellites and Polar Satellites, which serve different applications due to their unique characteristics.

Geostationary Satellites

A geostationary satellite is a satellite that orbits the Earth at a specific altitude and path such that it appears to remain stationary relative to a fixed point on the Earth's surface. This makes them ideal for communication and broadcasting purposes, as ground antennas can be fixed to point at a single location in the sky.

Characteristics of Geostationary Orbit

For a satellite to be geostationary, it must meet the following conditions:

1. Orbital Period: Its orbital period must be exactly equal to the Earth's rotational period relative to the stars (sidereal period), which is approximately 23 hours, 56 minutes, and 4 seconds (or 86,164 seconds). For practical purposes, it's often rounded to 24 hours when considering its apparent position relative to a point on the rotating Earth.
2. Orbit Type: The orbit must be circular.
3. Orbital Plane: The orbit must be in the equatorial plane of the Earth.
4. Direction of Orbit: It must orbit in the same direction as the Earth's rotation (west to east).

A satellite meeting these conditions will appear fixed in the sky from any point on the Earth within its view. The altitude required for a geostationary orbit can be calculated using the orbital period formula $T = 2\pi \sqrt{\frac{r^3}{GM_E}}$, setting $T = 86164$ seconds.

$ T^2 = \frac{4\pi^2 r^3}{GM_E} $

$ r^3 = \frac{GM_E T^2}{4\pi^2} $

Using the values for $G, M_E$ and $T \approx 86400$ s (approx. 24 hours), the orbital radius $r$ comes out to be approximately $42,200$ km from the Earth's centre. Since $R_E \approx 6,400$ km, the altitude $h = r - R_E \approx 42200 - 6400 = 35800$ km above the Earth's surface. This is a very high orbit, known as a geosynchronous orbit (if the period is 24 hours but the orbit is not necessarily equatorial or circular) or specifically a geostationary orbit (if it satisfies all conditions, remaining fixed over a point on the equator).

Applications of Geostationary Satellites

• Satellite Television Broadcasting (DTH services).
• Telecommunications (telephone, internet).
• Weather monitoring over large areas (though they provide frequent images of the same region).

Polar Satellites

A polar satellite is a satellite that orbits the Earth in a north-south direction, passing over or near the Earth's geographical poles on each orbit. These satellites are typically placed in much lower orbits compared to geostationary satellites, usually at altitudes between 700 km and 800 km.

Characteristics of Polar Orbit

The key characteristic is that the orbital plane is inclined at a large angle (close to 90 degrees) relative to the Earth's equatorial plane. As the Earth rotates beneath the satellite's orbit, the satellite passes over different longitudinal strips on each pass. This allows the satellite to scan the entire Earth's surface over a period of time (e.g., a day or a week) as the Earth rotates below it.

Many polar satellites are placed in Sun-synchronous orbits. In a Sun-synchronous orbit, the orbital plane is tilted at just the right angle relative to the Sun's direction such that the satellite crosses the equator at the same local solar time on every pass. This provides consistent lighting conditions for Earth observation and imaging.

Applications of Polar Satellites

• Earth Observation (monitoring land use, vegetation, deforestation).
• Remote Sensing (collecting data about Earth's surface).
• Weather Forecasting (providing global weather data and images).
• Environmental Monitoring (tracking pollution, ice caps).
• Military reconnaissance.

India's Polar Satellite Launch Vehicle (PSLV) is renowned for launching polar and Sun-synchronous satellites.

Weightlessness

The term "weightlessness" is often used to describe the sensation experienced by astronauts in orbit, or by people briefly during free fall (like in a falling elevator or a parabolic flight). However, it is a slightly misleading term. Astronauts in orbit are not truly weightless; they are still under the influence of Earth's gravity. What they experience is apparent weightlessness.

Understanding Weight

Recall that weight is the force of gravity acting on an object ($W = mg$). This force is always present as long as the object is in a gravitational field. On the International Space Station (ISS), orbiting about 400 km above Earth, the acceleration due to gravity is still significant, about 90% of the value on the surface ($g_{ISS} \approx 0.9 \, g_{Earth}$). So, astronauts in orbit still have weight.

The sensation we normally associate with weight comes from the normal force exerted by a supporting surface (like the floor, a chair, or a scale) on our body, counteracting the force of gravity. For example, when standing on a scale, the scale reading is the force it exerts upwards on you (the normal force), which in equilibrium equals your weight. The scale measures this normal force, which is our "apparent weight".

Apparent Weightlessness in Orbit

Astronauts in orbit, along with their spacecraft, are in a continuous state of free fall around the Earth. The gravitational force is the only significant force acting on them (neglecting small effects like air resistance in the very thin atmosphere at orbital altitude). Because they are constantly accelerating towards the Earth due to gravity, just like the spacecraft is, there is no supporting surface pushing back against gravity. There is no normal force acting on them from the floor, walls, or objects around them within the spacecraft.

Since there is no normal force or reaction force to counteract gravity, the sensation of pressure and support that we associate with weight is absent. This lack of a supporting force, while still under the influence of gravity, leads to the feeling of apparent weightlessness.

It's the same sensation you briefly feel at the top of a roller coaster or in a falling elevator – the sensation of free fall where the floor is no longer supporting you with the usual force.

In orbit, because the forward velocity is high enough, this free fall motion causes the object to follow a curved path around the Earth instead of crashing into it.

Apparent Weightlessness in Other Scenarios

• Falling Elevator: Inside a freely falling elevator (if that were possible), the occupants would also be in free fall. The floor would not be supporting them, and they would experience apparent weightlessness.
• Parabolic Flight: Aircraft can fly in a parabolic trajectory to simulate weightlessness for short periods (e.g., for astronaut training). During the curved, downward part of the parabola, the aircraft and everything inside are in free fall.

True weightlessness would only occur in the absence of a significant gravitational field, such as in deep space, infinitely far from any massive body. In such a region, the force of gravity on an object would be zero, and therefore its weight would truly be zero.

So, while astronauts in orbit are often said to be "weightless", it's more accurate to say they are in a state of continuous free fall, experiencing apparent weightlessness due to the absence of supporting forces.