Man-made satellites are launched into space to orbit the Earth at specific altitudes to perform vital functions such as communication, weather forecasting, navigation, and Earth imaging. The motion of these satellites is governed by Newton's Universal Law of Gravitation and centripetal force principles. Understanding orbital motion also involves the concept of escape velocity, which is the minimum velocity needed for an object to overcome Earth's gravitational pull and escape completely into outer space.
Illustrate orbital mechanics, difference in satellite paths relative to Earth's rotation, and the energy balance needed for escape velocity.
Satellites move in circular orbits around the Earth where the centripetal force is provided by the gravitational force between the satellite and Earth.
\(F = \dfrac{GMm}{r^2}\) and \(F = \dfrac{mv^2}{r}\)
Equating forces gives \(v = \sqrt{\dfrac{GM}{r}}\).
Radius of orbit, \(r = R + h\), where \(R\) is the radius of the Earth and \(h\) is the altitude of the satellite above Earth's surface.
Therefore, \(v = \sqrt{\dfrac{GM}{R + h}}\).
The linear speed depends only on the radius of its orbit, \(r\). It does not depend on the mass of the satellite, \(m\).
Orbits in the Geostationary Earth Orbit. Its orbit is circular and in the equatorial plane, its direction of motion is the same as Earth's rotation, and its orbital period is \(T = 24\text{ hours}\). It therefore remains above the same geographical location. Example: MEASAT. Primary use: communication satellites.
Does not satisfy one or more conditions for a geostationary orbit. Its orbit may be lower, higher, inclined, or elliptical, and its direction need not match Earth's rotation. Its orbital period may be shorter than, longer than, or equal to 24 hours; for example, an inclined or elliptical geosynchronous satellite has the same period but appears to move in the sky. Other examples include TiungSAT, RazakSAT, Pipit, ISS, and GPS satellites. Primary uses include Earth imaging, GPS, and weather forecasting.
Escape Velocity
Minimum velocity needed by an object on the surface of the Earth to overcome gravitational force and escape to outer space. For Earth, \(v \approx 1.12 \times 10^4\text{ m s}^{-1} = 11.2\text{ km s}^{-1}\).
Earth's escape velocity is high (\(11\,200\text{ m s}^{-1}\)). Gas molecules in the atmosphere have an average speed of around \(500\text{ m s}^{-1}\), far below the escape velocity, so they cannot escape into outer space. This preserves the atmosphere. Commercial aircraft fly at around \(250\text{ m s}^{-1}\) and fighter jets at up to \(2\,200\text{ m s}^{-1}\), both well below escape velocity, allowing them to fly high without risk of escaping into outer space. Launching spacecraft into outer space requires massive quantities of fuel to produce enough thrust to achieve Earth's escape velocity.
Answer in your mind, then press “Check Answer”.
Does a heavier satellite require a higher linear speed than a lighter satellite to maintain the same orbit? Explain.
State two main differences between a geostationary satellite and a non-geostationary satellite.
Why is Earth able to retain its atmosphere while gases on Mars escape more easily?
Calculate the escape velocity from Earth's surface using \(G=6.67\times10^{-11}\ \mathrm{N\,m^2\,kg^{-2}}\), \(M=5.97\times10^{24}\ \mathrm{kg}\), and \(R=6.37\times10^6\ \mathrm{m}\).
Compare the escape velocities of the Moon and Mars using \(M_{Moon}=7.35\times10^{22}\ \mathrm{kg}\), \(R_{Moon}=1.74\times10^6\ \mathrm{m}\), \(M_{Mars}=6.42\times10^{23}\ \mathrm{kg}\), and \(R_{Mars}=3.39\times10^6\ \mathrm{m}\).
A rocket launched into orbit must achieve escape velocity.
A satellite placed into orbit only needs to reach orbital linear speed (\(v = \sqrt{\dfrac{GM}{r}}\)), which is lower than escape velocity (\(v = \sqrt{\dfrac{2GM}{r}}\)). Escape velocity is only required when a spacecraft needs to leave Earth's gravitational field completely.
Escape velocity depends on the mass of the spacecraft being launched.
Escape velocity depends strictly on the mass of the celestial body (\(M\)) and its radius (\(R\)). Both light and heavy spacecraft require the exact same escape velocity from Earth.
Objective: Identify orbital speed, geostationary and non-geostationary satellites, and escape velocity.
1 What happens to a satellite in orbit if its linear speed becomes less than the required orbital speed?
2 Which factor influences the escape velocity of an object from the surface of a planet?
3 A geostationary satellite has an orbital period of hours, which is identical to the period of rotation of the Earth.
4 The value of escape velocity from the surface of the Earth is approximately km s⁻¹.
Drag each term to the correct meaning. If using a phone, tap the answer first, then tap the matching box.
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