Physics A Level
Chapter 17: Gravitational fields 17.7 Orbiting the Earth
Physics A Level
Chapter 17: Gravitational fields 17.7 Orbiting the Earth
The Earth has one natural satellite – the Moon – and many thousands of artificial satellites – some spacecraft and a lot of debris. Each of these satellites uses the Earth’s gravitational field to provide the centripetal force that keeps it in orbit. In order for a satellite to maintain a particular orbit, it must travel at the correct speed. This is given by the equation in topic 17.5 Orbiting under gravity:
${V^2} = \frac{{GM}}{r}$
It follows from this equation that, the closer the satellite is to the Earth, the faster it must travel. If it travels too slowly, it will fall down towards the Earth’s surface. If it travels too quickly, it will move out into a higher orbit.
Question
12) A satellite orbiting a few hundred kilometres above the Earth’s surface will experience a slight frictional drag from the Earth’s (very thin) atmosphere. Draw a diagram to show how you would expect the satellite’s orbit to change as a result. How can this problem be overcome if it is desired to keep a satellite at a particular height above the Earth?
Observing the Earth
Artificial satellites have a variety of uses. Many are used for making observations of the Earth’s surface for commercial, environmental, meteorological or military purposes. Others are used for astronomical observations, benefiting greatly from being above the Earth’s atmosphere. Still others are used for navigation, telecommunications and broadcasting.
Figure 17.11 shows two typical orbits. A satellite in a circular orbit close to the Earth’s surface, and passing over the poles, completes about 16 orbits in 24 hours. As the Earth turns below it, the satellite ‘sees’ a different strip of the Earth’s surface during each orbit. A satellite in an elliptical orbit has a more distant view of the Earth.
Geostationary orbits
A special type of orbit is one in which a satellite travels from west to east and is positioned so that, as it orbits, the Earth rotates below it with the same angular speed. The satellite remains above a fixed point on the Earth’s equator. This kind of orbit is called a geostationary orbit. There are over 300 satellites in such orbits. They are used for telecommunications (transmitting telephone messages around the world) and for satellite television transmissions. A base station on Earth sends the TV signal up to the satellite, where it is amplified and broadcast back to the ground. Satellite receiver dishes are a familiar sight; you will have observed how, in a neighbourhood, they all point towards the same point in the sky. Because the satellite is in a geostationary orbit, the dish can be fixed. Satellites in any other orbits move across the sky so that a tracking system is necessary to communicate with them. Such a system is complex and expensive, and too demanding for the domestic market.
Geostationary satellites have a lifetime of perhaps ten years. They gradually drift out of the correct orbit, so they need a fuel supply for the rocket motors that return them to their geostationary position, and that keep them pointing correctly towards the Earth. Eventually, they run out of fuel and need to be replaced.
We can determine the distance of a satellite in a geostationary orbit using the equation:
${T^2} = \left( {\frac{{4{\pi ^2}}}{{GM}}} \right){r^3}$
For a satellite to stay above a fixed point on the equator, it must take exactly 24 hours to complete one orbit (Figure 17.12).
We know:
$\begin{array}{l}
G = 6.67 \times {10^{ - 11}}\,N\,{m^2}\,k{g^{ - 2}}\\
T = 24\,hours = 86400\,s\\
M = 6.0 \times {10^{24}}\,kg\\
{T^2} = \left( {\frac{{4{\pi ^2}}}{{GM}}} \right){r^3}\\
{86400^2} = \left( {\frac{{4{\pi ^2}}}{{6.67 \times {{10}^{ - 11}} \times 6.0 \times {{10}^{24}}}}} \right){r^3}\\
{r^3} = \frac{{{{86400}^2}}}{{\left( {\frac{{4{\pi ^2}}}{{6.67 \times {{10}^{ - 11}} \times 6.0 \times {{10}^{24}}}}} \right)}}\\
{r^3} = 7.57 \times {10^{22}}{m^3}\\
r = \sqrt[3]{{7.57 \times {{10}^{22}}}}\\
\approx 4.23 \times {10^7}\,m
\end{array}$
So, for a satellite to occupy a geostationary orbit, it must be at a distance of $42300 km$ from the centre of the Earth and at a point directly above the equator. Note that the radius of the Earth is $6400 km$, so the orbital radius is 6.6 Earth radii from the centre of the Earth (or 5.6 Earth radii from its surface). Figure 17.12 has been drawn to give an impression of the size of the orbit.
perspective view; the Clarke belt is circular
Questions
13) For any future mission to Mars, it would be desirable to set up a system of three or four geostationary (or ‘martostationary’) satellites around Mars to allow communication between the planet and Earth.
Calculate the radius of a suitable orbit around Mars.
Mars has mass $6.4 \times {10^{23}}\,kg$ and a rotational period of 24.6 hours.
14) Although some international telephone signals are sent via satellites in geostationary orbits, most are sent along cables on the Earth’s surface. This reduces the time delay between sending and receiving the signal. Estimate this time delay for communication via a satellite, and explain why it is less significant when cables are used.
You will need the following:
- radius of geostationary orbit $= 42 300 km$
- radius of Earth $= 6400 km$
- speed of electromagnetic waves in free space $c = 3.0 \times {10^8}\,m\,{s^{ - 1}}$
EXAM-STYLE QUESTIONS
1) An astronaut is on a planet of mass $0.50{M_E}$ and radius $0.75{r_E}$, where ${M_E}$ is the mass of the Earth and rE is the radius of the Earth.
What is the gravitational field strength at the surface of the planet? [1]
A: $6.5\,N\,k{g^{ - 1}}$
B: $8.7\,N\,k{g^{ - 1}}$
C: $11\,N\,k{g^{ - 1}}$
D: $12\,N\,k{g^{ - 1}}$
2) Consider the dwarf planet Pluto to be an isolated sphere of radius $1.2 \times {10^6}m$ and mass of $1.27 \times {10^22}kg$.
What is the gravitational potential at the surface of Pluto? [1]
A: $ - 0.59\,J\,k{g^{ - 1}}$
B: $ - 7.1 \times {10^5}\,J\,k{g^{ - 1}}$
C: $ 0.59\,J\,k{g^{ - 1}}$
D: $ 7.1 \times {10^5}\,J\,k{g^{ - 1}}$
3) Two small spheres each of mass $20 g$ hang side by side with their centres $5.00mm$ apart. Calculate the gravitational attraction between the two spheres. [3]
4) It is suggested that the mass of a mountain could be measured by the deflection from the vertical of a suspended mass. This diagram shows the principle.
a: Copy the diagram and draw arrows to represent the forces acting on the mass. Label the arrows. [3]
b: The whole mass of the mountain, $3.8 \times {10^{12}}\,kg$, may be considered to act at its centre of mass. Calculate the horizontal force on the mass due to the mountain. [2]
c: Compare the force calculated in part b with the Earth’s gravitational force on the mass. [2]
[Total: 7]
5) This diagram shows the Earth’s gravitational field.
a: Copy the diagram and add arrows to show the direction of the field. [1]
b: Explain why the formula for potential energy gained ($mg\Delta h$) can be used to find the increase in potential energy when an aircraft climbs to a height of $10000 m$, but cannot be used to calculate the increase in potential energy when a spacecraft travels from the Earth’s surface to a height of $10000km$. [2]
[Total: 3]
6) Mercury, the smallest of the eight recognised planets, has a diameter of $4.88 \times {10^6}\,m$ and a mean density of $5.4 \times {10^3}\,kg\,{m^{ - 3}}$.
a: Calculate the gravitational field at its surface. [5]
b: A man has a weight of $900 N$ on the Earth’s surface. What would his weight be on the surface of Mercury? [2]
[Total: 7]
7) Calculate the potential energy of a spacecraft of mass $250 kg$ when it is $20000km$ from the planet Mars. (Mass of Mars $ = 6.4 \times {10^{23}}\,kg$, radius of Mars $ = 3.4 \times {10^6}\,m$.) [3]
8) Ganymede is the largest of Jupiter’s moons, with a mass of $1.48 \times {10^{23}}\,kg$. It orbits Jupiter with an orbital radius of $1.07 \times {10^6}\,km$ and it rotates on its own axis with a period of 7.15 days. It has been suggested that to monitor an unmanned landing craft on the surface of Ganymede a geostationary satellite should be placed in orbit around Ganymede.
a: Calculate the orbital radius of the proposed geostationary satellite. [2]
b: Suggest a difficulty that might be encountered in achieving a geostationary orbit for this moon. [1]
[Total: 3]
9) The Earth orbits the Sun with a period of 1 year at an orbital radius of $1.50 \times {10^{11}}\,m$. Calculate:
a: the orbital speed of the Earth [3]
b: the centripetal acceleration of the Earth [2]
c: the Sun’s gravitational field strength at the Earth. [1]
[Total: 6]
10) The planet Mars has a mass of $6.4 \times {10^{23}}\,kg$ and a diameter of $6790 km$.
a: i- Calculate the acceleration due to gravity at the planet’s surface. [2]
ii- Calculate the gravitational potential at the surface of the planet. [2]
b: A rocket is to return some samples of Martian material to Earth. Write down how much energy each kilogram of matter must be given to escape completely from Mars’ gravitational field. [1]
c: Use your answer to part b to show that the minimum speed that the rocket must reach to escape from the gravitational field is $5000\,m\,{s^{ - 1}}$. [2]
d: Suggest why it has been proposed that, for a successful mission to Mars, the craft that takes the astronauts to Mars will be assembled at a space station in Earth orbit and launched from there, rather than from the Earth’s surface. [2]
[Total: 9]
11) a: Explain what is meant by the gravitational potential at a point. [2]
b: This diagram shows the gravitational potential near a planet of mass M and radius R.
On a copy of the diagram, draw similar curves:
i- for a planet of the same radius but of mass $2M$–label this i. [2]
ii- for a planet of the same mass but of radius $2R$–label this ii. [2]
c: Use the graphs to explain from which of these three planets it would require the least energy to escape. [2]
d: Venus has a diameter of $12100 km$ and a mass of $4.87 \times {10^{24}}\,kg$.
Calculate the energy needed to lift one kilogram from the surface of Venus to a space station in orbit $900 km$ from the surface. [4]
[Total: 12]
12) a: Explain what is meant by the gravitational field strength at a point. [2]
This diagram shows the dwarf planet, Pluto, and its moon, Charon. These can be considered to be a double planetary system orbiting each other about their joint centre of mass.
b: Calculate the gravitational pull on Charon due to Pluto. [3]
c: Use your result to part b to calculate Charon’s orbital period. [3]
d: Explain why Pluto’s orbital period must be the same as Charon’s. [1]
[Total: 9]
13)This diagram shows the variation of the Earth’s gravitational field strength with distance from its centre.
a: Determine the gravitational field strength at a height equal to $2R$ above the Earth’s surface, where R is the radius of the Earth. [1]
b: A satellite is put into an orbit at this height. State the centripetal acceleration of the satellite. [1]
c: Calculate the speed at which the satellite must travel to remain in this orbit. [2]
d: i- Frictional forces mean that the satellite gradually slows down after it has achieved a circular orbit. Draw a diagram of the initial circular orbital path of the satellite, and show the resulting orbit as frictional forces slow the satellite down. [1]
ii- Suggest and explain why there is not a continuous bombardment of old
satellites colliding with the Earth. [2]
[Total: 7]
SELF-EVALUATION CHECKLIST
After studying the chapter, complete a table like this:
| I can | See topic… | Needs more work | Almost there | Ready to move on |
| understand the nature of the gravitational field | 17.1 | |||
| represent and interpret a gravitational field using field lines | 17.1 | |||
| recall and use Newton’s law of gravitation: $F = \frac{{G{m_1}{m_2}}}{{{r^2}}}$ | 17.1 | |||
| understand why g is approximately constant near the Earth’s surface | 17.2 | |||
| derive from Newton’s law of gravitation: | 17.2 | |||
| recall and use the equation: $g = \frac{{GM}}{{{r^2}}}$ | 17.2 | |||
| understand that the gravitational potential at infinity is zero | 17.3 | |||
| define gravitational potential at a point, $\varphi $, as the work done in bringing unit mass from infinity to that point | 17.4 | |||
| understand that the gravitational potential decreases, being more negative, as an object moves closer to a second object | 17.4 | |||
| recall and use the formula that the gravitational potential | 17.4 | |||
| use the formula: $\phi = GM\left( {\frac{1}{{{r^1}}} - \frac{1}{{{r^2}}}} \right)$ | 17.4 | |||
| understand that the potential energy of two point masses is equal to: $F = - \frac{{G{m_1}{m_2}}}{r}$ | 17.1 | |||
| solve problems involving circular orbits of satellites by relating the gravitational force to the centripetal acceleration of the satellite | 17.5 | |||
| understand that a satellite in a geostationary satellite remains above the same point on the Earth’s surface | 17.7 | |||
| understand that a geostationary satellite has an orbital period of 24 hours and travels from west to east. | 17.7 |