Orbiting Satellites (VCE SSCE Physics): Revision Notes

Orbiting Satellites

Introduction: Newton and orbital motion

In the 17th century, scientists struggled to explain why planets orbited the Sun. Nicolaus Copernicus had proposed in 1542 that planets moved in circular orbits around the Sun. Later, in 1609, Johannes Kepler used accurate astronomical data to discover that planetary orbits were actually elliptical, not circular.

Isaac Newton realised that both planetary motion and the Moon's orbit around Earth required some force acting toward the central body. His breakthrough was connecting "heavenly motion" (the Moon orbiting Earth) with everyday motion (an apple falling to Earth). This insight led to his universal law of gravitation and our understanding of how satellites orbit.

Newton's thought experiment

In his 1687 work System of the World, Newton described a thought experiment involving a cannonball fired horizontally from a high mountain, assuming no air resistance.

Newton's reasoning worked as follows:

• A cannonball fired with a certain velocity would follow a curved path and land at some point on Earth
• Cannonballs fired with progressively greater velocities would travel farther before landing
• At a sufficiently high velocity, the cannonball would travel all the way around Earth and return to its starting point
• The cannonball would then become a satellite of Earth, just like the Moon

The physics of satellite motion

Gravitational force as the centripetal force

When a satellite orbits a planet, the force due to gravity is the only force acting on it. For example, the Moon constantly falls toward Earth due to gravity, but it never hits Earth because it has a velocity perpendicular to the gravitational field that is large enough to maintain a circular orbit.

For any object moving in a circular path, the net force is the centripetal force, which always acts radially inward (toward the centre of the circle). For a satellite in orbit, the gravitational force provides this centripetal force.

Force due to gravity:

$F_g = G\frac{mM}{r^2}$

Where:

• $F_g$ = force due to gravity (N)
• $G$ = universal gravitational constant = $6.67 \times 10^{-11}$ Nm²kg⁻²
• $m$ = mass of satellite (kg)
• $M$ = mass of the planet or central body (kg)
• $r$ = distance between the centres of mass of $m$ and $M$ (m)

Centripetal force:

$F_c = \frac{mv^2}{r} = \frac{m4\pi^2r}{T^2}$

Where:

• $F_c$ = centripetal force (N)
• $m$ = mass of satellite (kg)
• $v$ = orbital speed (m s⁻¹)
• $r$ = orbital radius (m)
• $T$ = orbital period (s)

Deriving orbital motion formulas

By equating the gravitational force to the centripetal force, we can derive several useful formulas for satellite motion.

Orbital speed:

Starting with $G\frac{mM}{r^2} = \frac{mv^2}{r}$

Cancelling $m$ from both sides and rearranging:

$v = \sqrt{\frac{GM}{r}}$

Orbital period:

Starting with $G\frac{mM}{r^2} = \frac{m4\pi^2r}{T^2}$

Rearranging for $T$:

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

The orbital period is also independent of the satellite's mass.

Orbital radius:

We can also rearrange the period equation to find the orbital radius:

$r = \sqrt[3]{\frac{GMT^2}{4\pi^2}}$

Finding the orbital radius requires taking a cube root.

Calculating the mass of a planet

We can rearrange our equations to find the mass of a planet or central body if we know the orbital parameters of one of its satellites:

$M = \frac{4\pi^2r^3}{GT^2}$

This method allows us to calculate planetary masses by observing their moons.

Artificial satellites

An artificial satellite is any human-made structure placed in orbit around a planet or moon. Examples include Sputnik, the Hubble Space Telescope, the International Space Station, and the Mars Orbiter.

The space age began on 4 October 1957 when the Soviet Union launched Sputnik 1, the world's first artificial Earth satellite.

Sputnik 1 was launched into an elliptical low Earth orbit with:

• Mass: 84 kg
• Orbital period: 96.2 minutes
• Total orbits completed: 1440 before burning up on 4 January 1958

Geostationary satellites

A geostationary satellite is a special type of satellite whose orbit keeps it fixed relative to the same point on Earth's surface at all times.

Calculating the geostationary orbit:

Using the orbital radius formula with $T = 24$ hours $= 86400$ s:

$r = \sqrt[3]{\frac{GMT^2}{4\pi^2}}$

$r = \sqrt[3]{\frac{(6.67 \times 10^{-11})(5.98 \times 10^{24})(86400)^2}{4\pi^2}}$

$r = 4.2 \times 10^7 \text{ m}$

Since Earth's radius is approximately $6.4 \times 10^6$ m, the altitude above Earth's surface is:

Altitude $= 4.2 \times 10^7 - 6.4 \times 10^6 = 3.6 \times 10^7$ m $= 36000$ km

Normal force on astronauts in orbiting satellites

At the International Space Station, which orbits at an altitude of 408 km above Earth's surface, the gravitational field strength is 8.67 N kg⁻¹ (approximately 88% of Earth's surface value). This means both the astronauts and the ISS are accelerating toward Earth at 8.67 m s⁻².

However, orbiting astronauts don't feel gravity because:

• Gravity is the only force constantly acting on them
• They experience no normal force ($F_N = 0$)
• They are continually falling toward Earth under the influence of gravity

On Earth's surface, we feel gravity because the floor provides a normal force that prevents us from falling. This normal force creates the sensation of weight. In orbit, there is no floor pushing up, so there is no normal force and astronauts feel weightless.

Since the ISS accelerates toward Earth at exactly the same rate as the astronauts inside, objects appear to float relative to the space station. Both the station and everything inside it are in free fall together.

Remember!