15.3 Satellites and Orbits | Gravitation | Physics 12

What is a Satellite?

A satellite is any object that orbits a larger body under the influence of gravity. Natural satellites include the Moon; artificial satellites are human-made objects placed in orbit for communication, navigation, weather monitoring, and scientific research.

Circular Orbital Motion

For a satellite of mass $m$ moving in a circular orbit of radius $r$ around Earth (mass $M_{E}$ ), the gravitational force provides the centripetal force:

$F_{g} = F_{c}$

$\frac{G M _{E} m}{r ^{2}} = \frac{m v ^{2}}{r}$

Solving for orbital velocity $v_{o}$ :

$v_{o} = \frac{G M _{E}}{r}$

where $r = R_{E} + h$ is the orbital radius (Earth's radius $R_{E}$ plus altitude $h$ ).

Since $g = \frac{G M _{E}}{r ^{2}}$ , this can also be written as:

$v_{o} = g r$

where $g$ is the gravitational field strength at the orbital radius.

Key Relationship: Speed vs. Orbital Radius

Since $v_{o} \propto \frac{1}{r}$ , higher satellites move slower. Doubling the orbital radius reduces orbital speed by a factor of $2$ .

Orbital Period

Using the circumference of the orbit $2 π r$ and orbital speed $v_{o}$ :

$T = \frac{2 π r}{v _{o}} = \frac{2 π r}{G M _{E} / r} = 2 π \frac{r ^{3}}{G M _{E}}$

This is consistent with Kepler's Third Law: $T^{2} \propto r^{3}$ .

Types of Satellite Orbits

Orbit Type	Description
Equatorial	Lies in the plane of Earth's equator
Inclined	At an angle to the equatorial plane
Polar	Passes over Earth's poles; useful for full-surface coverage
Geostationary	Equatorial orbit with $T = 24$ h; satellite appears stationary

Effect of Launch Speed

Too low ( $v < v_{o}$ ): Satellite falls back to Earth.
Just right ( $v = v_{o}$ ): Stable circular orbit — requires the least energy to maintain.
Too high ( $v_{o} < v < v_{esc}$ ): Satellite follows an elliptical orbit.
At or above escape speed ( $v \geq v_{esc}$ ): Satellite escapes Earth's gravity entirely.

Geostationary Satellites

A geostationary satellite orbits Earth with the same angular velocity as Earth's rotation, so it appears stationary above a fixed point on the equator.

Conditions for a Geostationary Orbit

Orbital period $T = 24$ hours (one sidereal day ≈ 23 h 56 min, but 24 h is used in FBISE)
Orbital plane must be in the equatorial plane (inclination = 0°)
Direction of rotation must be the same as Earth's rotation (west to east)

Calculating the Geostationary Orbital Radius

Setting $T = 24 h = 86, 400 s$ in the period formula:

$T = 2 π \frac{r ^{3}}{G M _{E}}$

$r^{3} = \frac{G M _{E} T ^{2}}{4 π ^{2}}$

$r = (\frac{G M _{E} T ^{2}}{4 π ^{2}})^{1/3}$

Substituting $G = 6.67 \times 1 0^{- 11} N m^{2} k g^{- 2}$ , $M_{E} = 5.97 \times 1 0^{24} kg$ :

$r \approx 4.23 \times 1 0^{7} m = 42, 300 km$

Since $R_{E} \approx 6, 400 km$ , the altitude above Earth's surface is:

$h = r - R_{E} \approx 42, 300 - 6, 400 = 35, 900 km \approx 36, 000 km$

Orbital Velocity of a Geostationary Satellite

$v_{o} = \frac{2 π r}{T} = \frac{2 π \times 4.23 \times 1 0 ^{7}}{86 , 400} \approx 3.07 km s^{- 1} \approx 3.1 km s^{- 1}$

Applications

Geostationary satellites are used for:

Television broadcasting and telecommunications
Weather monitoring (e.g., Meteosat)
GPS augmentation systems

Limitation

A geostationary satellite cannot be placed directly over the poles because the orbit must lie in the equatorial plane. Polar regions are served by satellites in inclined or polar orbits.

What is a Satellite?

Circular Orbital Motion

For a satellite of mass $m$ moving in a circular orbit of radius $r$ around Earth (mass $M_{E}$ ), the gravitational force provides the centripetal force:

$F_{g} = F_{c}$

$\frac{G M _{E} m}{r ^{2}} = \frac{m v ^{2}}{r}$

Solving for orbital velocity $v_{o}$ :

$v_{o} = \frac{G M _{E}}{r}$

where $r = R_{E} + h$ is the orbital radius (Earth's radius $R_{E}$ plus altitude $h$ ).

Since $g = \frac{G M _{E}}{r ^{2}}$ , this can also be written as:

$v_{o} = g r$

where $g$ is the gravitational field strength at the orbital radius.

Key Relationship: Speed vs. Orbital Radius

Since $v_{o} \propto \frac{1}{r}$ , higher satellites move slower. Doubling the orbital radius reduces orbital speed by a factor of $2$ .

Orbital Period

Using the circumference of the orbit $2 π r$ and orbital speed $v_{o}$ :

$T = \frac{2 π r}{v _{o}} = \frac{2 π r}{G M _{E} / r} = 2 π \frac{r ^{3}}{G M _{E}}$

This is consistent with Kepler's Third Law: $T^{2} \propto r^{3}$ .

Types of Satellite Orbits

Orbit Type	Description
Equatorial	Lies in the plane of Earth's equator
Inclined	At an angle to the equatorial plane
Polar	Passes over Earth's poles; useful for full-surface coverage
Geostationary	Equatorial orbit with $T = 24$ h; satellite appears stationary

Effect of Launch Speed

Too low ( $v < v_{o}$ ): Satellite falls back to Earth.
Just right ( $v = v_{o}$ ): Stable circular orbit — requires the least energy to maintain.
Too high ( $v_{o} < v < v_{esc}$ ): Satellite follows an elliptical orbit.
At or above escape speed ( $v \geq v_{esc}$ ): Satellite escapes Earth's gravity entirely.

Geostationary Satellites

A geostationary satellite orbits Earth with the same angular velocity as Earth's rotation, so it appears stationary above a fixed point on the equator.

Conditions for a Geostationary Orbit

Orbital period $T = 24$ hours (one sidereal day ≈ 23 h 56 min, but 24 h is used in FBISE)
Orbital plane must be in the equatorial plane (inclination = 0°)
Direction of rotation must be the same as Earth's rotation (west to east)

Calculating the Geostationary Orbital Radius

Setting $T = 24 h = 86, 400 s$ in the period formula:

$T = 2 π \frac{r ^{3}}{G M _{E}}$

$r^{3} = \frac{G M _{E} T ^{2}}{4 π ^{2}}$

$r = (\frac{G M _{E} T ^{2}}{4 π ^{2}})^{1/3}$

Substituting $G = 6.67 \times 1 0^{- 11} N m^{2} k g^{- 2}$ , $M_{E} = 5.97 \times 1 0^{24} kg$ :

$r \approx 4.23 \times 1 0^{7} m = 42, 300 km$

Since $R_{E} \approx 6, 400 km$ , the altitude above Earth's surface is:

$h = r - R_{E} \approx 42, 300 - 6, 400 = 35, 900 km \approx 36, 000 km$

Orbital Velocity of a Geostationary Satellite

$v_{o} = \frac{2 π r}{T} = \frac{2 π \times 4.23 \times 1 0 ^{7}}{86 , 400} \approx 3.07 km s^{- 1} \approx 3.1 km s^{- 1}$

Applications

Geostationary satellites are used for:

Television broadcasting and telecommunications
Weather monitoring (e.g., Meteosat)
GPS augmentation systems

Limitation

A geostationary satellite cannot be placed directly over the poles because the orbit must lie in the equatorial plane. Polar regions are served by satellites in inclined or polar orbits.