9.2.C - Orbits


In Physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the centre of a star system, such as the Solar System. While the orbits of planets are typically elliptical, we approximate them to circular orbits in this section of the course.

Circular motion is rotation along a circle: a circular path or a circular orbit. It can be uniform, that is, with constant angular rate of rotation, or non-uniform, that is, with a changing rate of rotation. Satellites orbiting the Earth and planets orbiting the Sun are examples of circular motion whereby the gravitational force keeps the mass orbiting at a certain velocity and radius.


In this unit you will learn about

  • Uniform circular motion and its relationship to orbits and gravitational force
  • Orbital speed and those factors that affect it
  • Orbital shapes (circular, elliptical and hyperbolic)
  • Kepler's laws (first, second and third)
  • Satellite orbits (LEO and geostationary)
  • Orbital decay and associated energy transformations


Uniform Circular Motion

Variables involved in keeping an object
in unifrom circular motion.

Uniform circular motion is circular motion with a uniform orbital speed. As an example of circular motion, imagine you have a rock tied to a string and are whirling it around you in a horizontal plane. Because the path of the rock is in a horizontal plane gravity plays no part in its motion.

If you were to let go of the string, the rock would fly off at a tangent to the circle - a demonstration of Newton's First Law of Motion. He said that an object would continue in uniform motion in a straight line unless acted upon by a net force. In the case of our rock, the force keeping it within a circular path is the tension in the string and it is always directed back towards the hand at the centre of the circle. Without that force the rock would travel in a straight line.

Vector diagrams showing the
magnitude and direction of the
change in velocity, acceleration
and centripetal force (red vectors).

So, if the magnitude of the velocity is constant but its direction is changing, the velocity must be changing and it is, therefore, accelerating. A net force is required to accelerate the rock and as stated above, this force is the tension in the string. The force needed to keep an object moving in a circle is also called centripetal force.

The same is true of a spacecraft in orbit around the Earth or any object in circular motion - some force is needed to keep it moving in a circle or accelerate it and that force is directed towards the centre of the circle. In the case of the spacecraft, it is the gravitational attraction between the Earth and the spacecraft that acts to maintain the circular motion and keep it in orbit.

The diagram on the left allows us to work out the direction of the centripetal acceleration - which must also be in the direction of the centripetal force. The object is shown moving between two points A and B on a horizontal circle. Its velocity has changed from v1 to v2. The magnitude of the velocity is always the same, but the direction has changed. Since velocities are vector quantities we need to use vector mathematics to work out the average change in velocity. In this example, the direction of the average change in velocity is towards the centre of the circle. This is always the case and thus true for instantaneous acceleration.

Equation - Centripetal Force

F

m

v

r

centripetal force

mass or object moving in a circle

velocity

radius of circle

newtons (N)

kilograms (kg)

metres per second (ms-1)

metres (s)

You should use this equation when you are determining the force, velocity, mass or radius of any object moving in uniform circular motion (at a constant speed in a circle).


Although the planets have elliptical orbits around the Sun, in
this course we approximate them to being circular.

Applying this equation to the Space module, let's consider the planets in their orbit around the Sun. The centripetal force in this case is provided by the force of gravity between the planet and the sun. This is different for each planet because they all have different masses. Each planet orbits the Sun such that the mathematical combination of mass, velocity and radius (according to the euqation above) provides the precise value for gravitational force (or centripetal force) needed to keep it in a stable orbit. If one of the variables changed for some reason, the planet's orbit would be compromised and it would no longer be able to stay in a stable orbit under the same conditions.

In summary, each variable in the equation for centripetal force has a unique and precise value to maintain a stable orbit. If one of them is changed, they all need to change and adjust to reach another stable orbit.

 

Orbital Speed

Newton's idea of escape velocity shows
that objects in orbit are actually falling
towards the Earth in projectile motion.
The Earth's surface curves away at the
same rate so they never quite make it
to the ground!

Isaac Newton surmised that if you climb to the top of a mountain and fire a cannon, the cannon ball will travel a certain distance and then hit the ground (A). If you fired the cannon so that the cannon ball had a larger velocity it would travel even further (B) and if you gave it even more velocity it would travel further still (B).

If you kept increasing the velocity of the cannon ball and there was no air resistance, a point would come when the cannon ball would be travelling part-way around the world. If the ball could be fired at just the right velocity, it would travel completely around the Earth and hit you in the back of the head (C). In this case it would fall at exactly the same rate as the Earth curves. If it was fired much faster than that, the canon ball would go into an elliptical orbit (D) or even faster it would travel off into space and never return (E).

To get a spacecraft into a stable orbit at a certain radius around the Earth, it must be launched so that when it reaches the desired radius it will have the velocity needed to maintain a stable orbit.

To place a space shuttle into a stable Earth orbit at a particular radius, the rocket engines must do two things. First, they must do the work needed (provide the energy) to overcome gravity and increase the gravitational potential energy of the space shuttle to that needed for the radius of the stable orbit. Secondly, the rocket engines must give the rocket enough velocity so that when the space shuttle gets to the required radius (altitude), it will have the orbital velocity needed to maintain a stable orbit.

This view shows the path taken by a space shuttle being launched
from the equator into a stable orbit of radius, r. The view shown is
looking fown towards the Earth from the north pole.

The space shuttle is launced vertically but will eventually turn so that it is travelling horizontal to the Earth's surface. At this radius, the force of gravity provides the acceleration needed to keep the space shuttle in a stable orbit with its particular value for orbital velocity.

There is clearly a relationship between orbital velocity and radius of a satellite in orbit and it is to this that we turn next. In a stable orbit, the gravitational force is the same as the centripetal force. Making the equations for these two quantities equal and solving for velocity will give us an expression linking orbital velocity and radius.

 

Equation - Orbital Velocity

v

G

m1

r

orbital velocity

universal gravitation constant (6.673 x 10-11)

mass of central body (not orbiting body)

distance between centres

metres per second (ms-1)

N m2 kg-2

kilograms (kg)

metres (m)

Use this equation when determining the orbital velocity required for a stable orbit at a certain radius.

 

Orbital Shapes

Shapes of circular, elliptical and hyperbolic orbits
and their eccentricity values.

The shapes or pathways taken by satellites come in many shapes and sizes. At the highest level, orbits can be classified as open or closed.

Closed orbits are those that are periodic whereby a satellite orbits a central body in a regular and consistent way. The planets around the Sun, the moon around the Earth and some comets are all examples of periodic and, therefore, closed orbits.

Open (escape) orbits are those where an object passes close by another and is affected by its gravity but will never return in any periodic way. Some comets are in open orbits where they pass by the Earth and have their velocity changed, but not by enough to put them into a closed orbit. They pass on into space and just keep going!

Closed orbits can be further classified into circular and elliptical orbits. Circular orbits are said to have an eccentricity of zero and trace out the path of a circle. Elliptical orbits have an eccentricity which is greater than zero and less than one and trace out a path rather like a squashed circle (ellipse). The higher the value for eccentricity, the less like a circle it is.

Open orbits are classified into parabolic and hyperbolic orbits. Parabolic orbits have an eccentricity of exactly one and are so defined because they have an orbital velocity exactly equal to the escape velocity (more on escape velocity in the next section, 9.2.D - Launches). Hyperbolic orbits have an eccentricity greater than one such that the orbital velocity is greater than the escape velocity.

 

Kepler's Laws

While only the third law is referred to specifically in the syllabus, a brief description of the first and second is supplied here for completeness.

First Law

Kepler's first law placing the Sun and the
focus of an elliptical orbit.

The path of the planets about the sun is elliptical in shape, with the center of the sun being located at one focus.

Kepler's first law is sometimes referred to as the law of ellipses and explains that planets are orbiting the Sun in a path described as an ellipse. An ellipse is a special curve in which the sum of the distances from every point on the curve to two other points is a constant. The two other points are known as the foci of the ellipse. The closer together that these points are, the more closely that the ellipse resembles the shape of a circle. In fact, a circle is the special case of an ellipse in which the two foci are at the same location. Kepler's first law is rather simple saying that all planets orbit the Sun in a path that resembles an ellipse, with the Sun being located at one of the foci of that ellipse.

Second Law

Kepler's second law. The planet moves
faster near the Sun, so the same area
is swept out in a given time as at larger
distances, where the planet moves
more slowly. The green arrow shows
the planet's velocity, and the purple
arrows represent the force on the
planet.

An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time.

Kepler's second law is sometimes referred to as the law of equal areas and describes the speed at which any given planet will move while orbiting the sun. The speed at which any planet moves through space is constantly changing. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun. Yet, if an imaginary line were drawn from the center of the planet to the center of the sun, that line would sweep out the same area in equal periods of time.

Third Law

The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun.

Kepler's third law is sometimes referred to as the law of harmonies and compares the orbital period and radius of orbit of a planet to those of other planets. Unlike Kepler's first and second laws that describe the motion of a single planet, the third law makes a comparison between the motion of different planets. The comparison being made is that the ratio of the squares of the periods to the cubes of their average distances from the sun is the same for every one of the planets.

Said another way, the radius cubed is directly proptional to the period squares for planets orbitting the Sun. Mathematically we express this lawin the equation shown below.

Equation - Kepler's Third Law

r

T

G

m1

radius of the orbit

period of the orbiting body

universal gravitation constant (6.673 x 10-11)

mass of central object

metres (m)

seconds (s)

N m2 kg-2

kilograms (kg)

Use this equation when calculating the period or radius orbit for a body that is orbiting around a larger central body.

 

A graph showing the mathematics of Kepler's third law for planets
orbiting the Sun.

The values on the right hand side of the equation are all constant for any group of objects orbiting a central body with mass m1, (such as the planets all orbiting the sun). This means that the ratio of the radius cubed to the period squared for each orbiting object will be the same or will be equal to a constant.

Graphically, if we plotted the radius cubed against the period squared for each planet in the solar system, we would expect each planet to be a data point on a straight line as shown in the graph on the right.

If we were to do the same thing for the moons of Jupiter instead of the planets, the graph would still be a straight line that passed through the origin, however, it would have a different gradient because the central mass m1 has changed.

Deriving Kepler's Third Law

A simple derivation of Kepler's third law using
equations for gravitational force, centripetal
force and orbital velocity.

This law can be derived very simply by:

  1. Equating the formulae for centripetal force and gravitational force

  2. Substituting for orbital velocity (v)

  3. Changing the subject so that all constants (G, m1) are on one side and the variables (T, r) are on the other side.

 

Satellite Orbits

Four common orbit types for satellites orbitting the Earth.

In the context of spaceflight, a satellite is an object which has been placed into orbit by human endeavour. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon.

The world's first artificial satellite, the Sputnik 1, was launched by the Soviet Union in 1957. Since then, thousands of satellites have been launched into orbit around the Earth.

A few hundred satellites are currently operational, whereas thousands of unused satellites and satellite fragments orbit the Earth as space debris.

Satellites are placed in one of several different types of orbit depending on the nature of their mission. Four common orbit types are Low Earth Orbits (LEOs), Medium Earth Orbits (MEOs), Highly Elliptical Orbits (HEOs) and Geostationary Orbits (GEOs). For the HSC Physics course you only need know about LEOs and GEOs.

Low Earth Orbits

LEOs occur at a radius of between 100 km and 1000 km above the Earth's surface with periods varying from 60 to 90 minutes. The space shuttle uses this type of orbit at an altitude of 200 - 250 km. LEOs have the smallest field of view but frequent coverage of specific or varied locations. Orbits at less than 400 km are difficult to maintain due to atmospheric drag and subsequent orbital decay (discussed below).

They are used mainly for military applications, Earth observation, weather monitoring and shuttle missions.

Geostationary Orbits

A geostationary satellite in orbit above the equator shown exactly
12 hours apart.

A geostationary orbit is a circular orbit in the Earth's equatorial plane (above the equator) at any point which orbits the Earth in the same direction and with the same period as the Earth's rotation (24 hours).

Geostationary orbits are useful because they cause a satellite to appear stationary with respect to a fixed point on the rotating earth. As a result an antenna can point in a fixed direction and maintain a link with the satellite. These satellite orbits occur at an altitude of approximately 35,786 km above ground. This altitude is significant because it produces an orbital period equal to the Earth's period of rotation (24 hours), which is known as the sidereal day.

 


Geostationary satellites (top view).
To an observer on the rotating
Earth, both satellites (red and yellow)
appear stationary in the sky at their
respective locations.

Geostationary satellites (Side view).

These orbits allow for the tracking of a stationary point on Earth and have the largest field of view. Some common applications for geostationaty satellites are outlined below.

GPS

GPS, or global positioning systems, enable an individual with a GPS device to know exactly where he is on earth at any given point. The earth-bound part of the satellite reads signals from the geostationary orbit satellites and determines the position of these satellites in comparison to the device's reader. This provides the user with a reliable reading as to where he is in the world.

Communication

Communication satellites are some of the most common geostationary orbit satellites orbiting the Earth. They receive information transferred to antennas from communication devices, such as cell phones, and bounce that information to other antennas, providing a reliable and quick relay from one communication device to another.

Military Applications

Military forces around the world employ geostationary orbit satellites to provide them with up-to-date pictures of their allies and enemies. These satellites are often referred to as spy satellites. Other military satellites include military communication satellites.

Weather

Many weather agencies have satellites that provide them with the most up-to-date information on what the weather is doing, including data on temperatures, cloud coverage, wind speeds and pressures. These geostationary orbit satellites have greatly improved the weather prediction process and have provided people with early warnings of dangerous weather conditions.

 

Orbital Decay

Orbital decay is the reduction in the height of an object's orbit over time due to the drag of the atmosphere on the object. All satellites in low Earth orbits are subject to some degree of atmospheric drag that will eventually decay their orbit and limit their lifetimes. Even at 1000 km, as thin as the atmosphere is, it is still sufficiently dense to slow the satellite down over a period of time. Atmospheric drag is seasonal and predictable for the most part, however, it increases during periods of high solar activity (sunspots), due to more frequent collisions between the object and surrounding air molecules. High solar activity brings an increase in solar radiation which heats up the atmosphere, making it expand. So the more solar activity, the more drag.

The total energy possessed by a satellite will be the sum of its kinetic and potential energy.

Etotal = Epotential + Ekinetic

Work must be done by the satellite to overcome the frictional drag. This means it loses GPE which results in a loss of altitude. The velocity must also increase if the radius is decreasing so some GPE is converted to kinetic energy (as the orbital velocity of the satellite increases) and the rest is lost as thermal energy due to friction.