13.3 Orbital Mechanics and Satellites

Satellites

Satellites

A satellite is a mass that orbits (moves in circular motion) around a larger mass.

The Earth is a satellite orbiting the Sun and the Moon is a satellite orbiting the Earth.

How does a satellite stay in orbit?

• The gravitational force between the satellite and the mass which it orbits provides the centripetal force needed to keep the satellite moving in circular motion around the mass.

Gravitational Force as Centripetal Force

A satellite of mass m orbiting a mass of mass M

Danger

• When using gravitation force as centripetal force in a derivation, make sure to state that:

$$\text{Gravitational Force}(F_g)\text{provides}\text{Centripetal Force}(F_c)$$

• And then: $F_g=F_c$

Orbital Velocity

Derivation of Orbital Velocity ( $v$)

• Explicitly state that:

$$\text{Gravitational Force}(F_g)\text{provides}\text{Centripetal Force}(F_c)$$

• Equate Gravitational Force and Centripetal Force:

$$\begin{array}{rl}{F}_g& =F_c\\ \frac{GMm}{r^2}& =\frac{m{v}^2}{r}\end{array}$$

• Simplify and make $v$ subject:

$$\begin{array}{rl}\frac{GMm}{r^2}& =\frac{m{v}^2}{r}\left(\times \frac{r}{m}\right)\\ \frac{GM}{r}& =v^2\\ & \\ v& =\sqrt{\frac{GM}{r}}\end{array}$$

Orbital Period

Derivation of Orbital Period ( $T$)

• Explicitly state that:

$$\text{Gravitational Force}(F_g)\text{provides}\text{Centripetal Force}(F_c)$$

• Equate Gravitational Force and Centripetal Force:

$$\begin{array}{rl}{F}_g& =F_c\\ \frac{GMm}{r^2}& =mr{\omega }^2\end{array}$$

• Simplify:

$$\begin{array}{rl}\frac{GMm}{r^2}& =mr{\omega }^2\left(\times \frac{1}{mr}\right)\\ \frac{GM}{r^3}& ={\omega }^2\end{array}$$

• Substitute $\omega =\frac{2\pi }{T}$:

$$\begin{array}{rl}\frac{GM}{r^3}& ={\left(\frac{2\pi }{T}\right)}^2\end{array}$$

• Simplify and make $T$ subject:

$$\begin{array}{rl}\frac{GM}{r^3}& =\frac{4{\pi }^2}{T^2}\\ T^2& =\frac{4{\pi }^2{r}^3}{GM}\\ T& =\sqrt{\frac{4{\pi }^2{r}^3}{GM}}\\ & \\ T& =2\pi \sqrt{\frac{r^3}{GM}}\end{array}$$

Kepler's Third Law of Planetary Motion

$T^2\propto r^3$

Geostationary Orbit

Geostationary Orbit

A geostationary orbit is one in which the orbiting satellite is always above the same point on Earth.

A telecommunications satellite orbiting the Earth

Conditions for Geostationary Orbit

• Orbital period of exactly 24 hours.
• Orbits from West for East.
• Orbits directly above the Equator.

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