In this explainer, we will learn how to describe the velocities and accelerations of planets, moons, and man-made satellites that are moving along circular orbits.
What all of these objects have in common is that they move in orbits, which means that they follow cyclical paths around some larger body. Planets orbit stars, and in turn moons orbit planets. “Satellite” is a general term for anything that orbits a planet or star, so planets and moons are also satellites. When we use the word “satellite,” though, we usually mean artificial satellites, which are machines that are launched into space and orbit Earth or another body.
The great insight Sir Isaac Newton had is that the force causing these orbits is the same force that causes objects to fall downward on Earth: the force of gravity.
Any object that has mass exerts a gravitational force on any other object that has mass. The force acts along the line that connects the centers of mass of the two objects and it is attractive, meaning that it acts to pull the two objects together.
The magnitude of the gravitational force depends on the masses of the two objects and the distance between them. A larger mass exerts a greater gravitational force, and the force is greater the closer the objects are together.
For example, if we consider the solar system, the object with the most mass is the Sun, which is orbited by eight planets (as well as many other objects). Venus and Earth have almost the same mass, so if they were at the same distance from the Sun, they would experience approximately the same gravitational force. In reality, Earth is further away from the Sun than Venus and therefore experiences a lower gravitational force.
We can also consider the example of the Moon. Although the Sun has significantly more mass than Earth, the Moon is much closer to Earth than to the Sun, and so the gravitational force that the Moon experiences is dominated by Earth.
Example 1: Understanding Newton’s Law of Gravitation
The diagram shows four planets orbiting a star. All four planets have the same mass, and all have circular orbits.
Which planet experiences the greatest force of attraction to the star due to gravity?
Which planet experiences the weakest force of attraction to the star due to gravity?
Does planet 3 experience a greater or lesser force of attraction to the star due to gravity than planet 2 experiences?
In the diagram, we can see four planets, which we are told have equal mass, on circular orbits around a star.
The magnitude of gravitational force between two objects depends on the masses of the objects and the distance between them. The greater the mass and the closer a planet is to its star, the greater the gravitational force between them will be. Since all the planets’ masses in this example are equal, the crucial variable is their distance.
Although the star in yellow is shown to be the same size as the planets, the fact that it is referred to as a star implies it has significantly more mass. In addition, the question only asks about the gravitational attraction to the star. This means we can ignore any gravitational forces between the planets and only consider the gravitational force between each planet and the star. The only distance we need to consider is, therefore, the distance between each planet and the star.
For the first part of the question, we need to decide which planet experiences the greatest force of attraction to the star due to gravity. For the greatest force of attraction, we need the planet with the smallest distance from the star, so that is planet 1.
The second part of the question asks which planet experiences the weakest force of attraction to the star. This will be the planet located at the largest distance from the star, which is planet 4.
Finally, we need to decide whether planet 3 experiences a greater or lesser force of attraction to the star due to gravity than planet 2 does. Planet 3 is located at a greater distance from the star than planet 2, so it experiences a lesser force of attraction.
Now let’s consider a small object moving past something with a lot of mass, say an asteroid moving past a star. Although both the star and the asteroid experience the same gravitational force, the asteroid has lower mass, so the force will have more effect on its motion. What happens to its path depends on the strength of the gravitational force and how fast the asteroid is moving.
If the asteroid is moving slowly, the gravitational force of the star will pull the asteroid in a curved path that eventually crashes into the star.
If, on the other hand, the asteroid was moving very quickly, the gravitational force of the star will have a smaller effect on the asteroid’s motion. It will be pulled in the direction of the star, causing a deflection in its motion, but it will be moving fast enough to escape the star’s gravitational pull.
An orbit is what happens if the asteroid is moving neither too quickly nor too slowly. If its velocity is just right, the asteroid will move around the star in a regular orbit.
When an object is in orbit around a large body, there is a perfect balance between its motion and the force of gravity acting on it. The force of gravity always acts toward the large body, pulling the object toward it and changing its direction, but the object is moving quickly enough that it never falls inward. We can think of it as constantly falling toward the large body, but moving past it so quickly that it always misses.
It is easiest to see the directions of the force and velocity when the orbit is circular.
In a circular orbit, the large body is located at the center, and the object follows a circular path around it. The gravitational force experienced by the object is always in the direction of the center of the circle. The object’s velocity is a tangent to the circle and pointing in the direction of motion and is always perpendicular to the force.
As the object moves around the circular orbit, its distance from the central body does not change, and we can assume that the masses of the object and the large body do not change, so the magnitude of the gravitational force remains the same. The direction of the force changes so that it always points toward the center of the circle.
Similarly, if there are no other forces acting on the object, then the magnitude of the velocity (also known as the speed) remains the same. The direction of the velocity vector, though, changes so that it is always perpendicular to the direction of the force.
What is happening here is that the gravitational force is causing the object to accelerate toward the large body. Acceleration is defined as the rate of change of velocity. We are most accustomed to thinking of acceleration as a change in speed, or the magnitude of velocity, but in this case the speed remains constant and the acceleration causes a change in the direction of the velocity.
In the diagram below, we see the same object at two positions in its orbit: when it is at the first position, at the top of the diagram, it experiences force and is moving with velocity , and in the second position on the right of the diagram, it experiences force and moves with velocity . The magnitude of is equal to the magnitude of , and the magnitude of is equal to the magnitude of . However, in the time between the first and second positions, the direction of both the force and the velocity has changed by so that they are always perpendicular to one another.
Most orbits are not circular, but elliptical. An object in an elliptical orbit follows a path in the shape of an ellipse. The large body is not located at the center of the ellipse, but is offset to one side or the other (the mathematical term is that the large body is at one of the foci of the ellipse).
For now, the important thing to be aware of is that the distance between the large body and the object changes throughout an elliptical orbit. This means that the magnitude of the gravitational force changes; it is stronger when the object is closer to the large body and weaker when it is further away.
The changing magnitude of the gravitational force also has an impact on the object’s velocity: In an elliptical orbit, not only does the direction of the velocity change, but also the magnitude does. The object moves more quickly when it is closer to the large body and more slowly when it is furthest away.
The diagram above shows the same object in two positions in its orbit. In the first position, in the lower left part of the diagram, the gravitational force is indicated by and the velocity is shown as , and in the second position, the object experiences force and moves with velocity . The directions of both vectors have changed between the two positions, and they are not necessarily perpendicular to each other. In terms of magnitude, the object is closer to the large body in the first position, so and .
In reality, all stable orbits are elliptical, and a circle is just a special type of ellipse. We usually show Earth’s orbit around the Sun as a circle, though really it is slightly elliptical; Earth is very slightly closer to the Sun in January than it is in July. The difference is small enough that we can usually approximate the orbit as a circle. At the other end of the spectrum are comets, which have highly elliptical orbits. Comets tend to originate in the outer solar system, further out than Neptune, but come all the way into the inner solar system when they are at their closest point to the Sun.
let’s look at the types of orbit in an example.
Example 2: Identifying Types of Orbit
The diagram shows two different possible orbits of an object around a star.
Which of the following correctly describes the shape of orbit (a)?
- Highly elliptical
Which of the following correctly describes the shape of orbit (b)?
- Highly elliptical
For the first part of the question, we are focusing on the diagram labeled (a), and we are asked to choose which type of orbit is shown.
For an object in orbit around another object, there are two types of paths: elliptical and circular. We can therefore immediately eliminate options (B) and (C), as these are not valid types of orbit.
Next, we need to decide whether the orbit is circular or elliptical. An ellipse is a squashed circle, whereas this shape looks quite regular. The answer is therefore (D), circular.
For the next part, we need to look at diagram (b). Following the same reasoning as above, we can eliminate options (A), (C), and (D), which leaves (B), highly elliptical, and (E), circular. In this case the orbit does look like a squashed circle, so we can eliminate (E). This leaves the only remaining possible answer as (B), highly elliptical.
The amount of time it takes an object to complete one orbit is known as the object’s orbital period.
Definition: Orbital Period
An object’s orbital period is the amount of time it takes to complete one orbit. It has units of time.
For example, the orbital period of Earth is approximately 365 days.
The orbital period of an object depends on the shape of the orbit, which determines the distance the object has to cover and the speed at which it travels.
The most straightforward orbits are circular. In a circular orbit, the speed is constant throughout the orbit and depends only on the mass of the large body at the center of the orbit and the object’s distance from that large body. An object closer to the large body experiences greater gravitational force and moves faster; its orbital path is also smaller, as a small circle has a smaller circumference than a large circle. An object located further away from the large body both moves more slowly and has a longer path to complete, so it will have a longer orbital period.
If we consider our solar system, we can approximate the orbits of the eight planets as circular. Closest to the Sun, we have Mercury, which has an orbital period of approximately 88 days. We know Earth’s orbital period is one year, or approximately 365 days, and at the outer extreme we have Neptune with an orbital period of 165 years.
let’s look at a couple of examples: one within the solar system, and one around another star.
Example 3: Understanding Orbital Speed
Earth orbits the Sun at a distance of km. Venus orbits the Sun at a distance of km. Which planet is moving faster around the Sun?
Recall that, for planets in orbit around a star, how fast the planet moves in its orbit depends only on the mass of the star and the distance between the star and the planet. A more massive star, or a smaller distance, result in the planet moving faster in its orbit.
In this question, we are given two planets, Venus and Earth, and their respective distances from the Sun. Since both planets orbit the same star, the Sun, the mass of the star is the same in both cases. The only factor affecting how fast the planets move is therefore their distance from the Sun: planets closer to the Sun move faster.
Venus, at km, is closer to the Sun than Earth, at km. Therefore, Venus moves faster around the Sun than Earth.
Example 4: Understanding Orbital Speed
The diagram shows four planets orbiting a star. All of the planets have circular orbits.
Which planet is moving the fastest?
Which planet is moving the slowest?
Is planet 3 moving faster than, slower than, or at the same speed as planet 2?
In the diagram, we have four planets in orbit around a central star, and we are told that all of the orbits are circular.
Recall that, for planets in circular orbits, the orbital speed is determined only by the distance between the planet and the star: the closer the planet, the faster the orbital speed.
For the first part of the question, we are asked to determine which planet is moving the fastest. This will be the planet closest to the star, which in this case is planet 1.
Next, we need to decide which planet is moving the slowest. Following the same reasoning as above, this will be the planet located the furthest from the star, which is planet 4.
Finally, we are asked whether planet 3 is moving faster than, slower than, or at the same speed as planet 2. planet 3 is located further from the star than planet 2, so it is moving slower than planet 2.
We can apply the same logic to artificial satellites in orbit around Earth. Satellites are located at various different heights above Earth’s surface, which means they experience different magnitudes of gravitational force and have different orbital periods. For satellites in circular orbits, the higher they are, the slower they move and the longer their orbital period is.
The International Space Station is a crewed satellite that orbits Earth at a height of approximately 409 kilometres above Earth’s surface. Its orbital period is about 90 minutes, meaning that it takes about an hour and a half to travel all the way around Earth. The International Space Station is bright enough to see with the naked eye; when it passes over your location at night, you can see it moving rapidly from west to east.
There are some situations in which we wish satellites to remain above the same position on Earth’s surface at all times. For example, communications satellites, so that ground-based antennas can remain pointing at a fixed position. The navigation satellites that work with global positioning systems also need to be in fixed location over Earth’s surface.
In order to remain above a fixed point on Earth’s surface, a satellite must move in the same direction that Earth rotates (west to east) and must follow the rotation exactly so that it takes the same amount of time to complete one orbit that Earth takes to rotate: approximately 24 hours. This is much longer than the orbital period of the International Space Station, so these satellites must be much further away. To remain above the same point on Earth’s surface, the satellite must be over the equator; this is why, in the northern hemisphere, communications receiving dishes point south.
Satellites that are located above the equator and move from west to east with an orbital period of 24 hours are known as geostationary satellites and their orbits as geostationary orbits.
Definition: Geostationary satellites
A geostationary satellite is a satellite that remains over the same position on Earth’s surface throughout its orbit.
To be geostationary, a satellite must orbit over Earth’s equator from west to east with an orbital period of 24 hours.
Examples of geostationary satellites are communications and navigation satellites.
Geostationary orbits are the orbits followed by geostationary satellites. They are approximately 35 786 kilometres above Earth’s surface.
It is important to note that geostationary satellites are not stationary; they only appear to be so when viewed from Earth’s surface as it rotates.
let’s finish with two examples concerning satellites orbiting Earth.
Example 5: Relating Orbital Period and Orbital Radius
The table shows the heights at which three satellites orbit Earth. Each satellite follows a circular orbit.
|Satellite||Terra||ICESat-2||Eutelsat 113 West A|
|Height above Earth’s Surface||709 km||496 km||35 800 km|
Which satellite takes the longest time to orbit Earth?
Which satellite takes the shortest time to orbit Earth?
Eutelsat 113 West A is a geostationary satellite. How long does it take to orbit Earth?
In the table, we are given the names and the heights of the orbits of three satellites. For the first part of the question, we need to decide which of the three satellites takes the longest time to orbit Earth.
We are told that all of the satellites follow circular orbits, and they all orbit Earth. Therefore, the only variable affecting the time to orbit Earth, or the orbital period, is the distance between the satellite and Earth. The satellite the furthest from Earth has the longest orbital period. Looking at the table, the furthest satellite from Earth is Eutelsat 113 West A, at 35 800 km. Therefore, Eutelsat 113 West A takes the longest time to orbit Earth.
Next, we need to choose which satellite takes the shortest time to orbit Earth. Using the same reasoning as above, the satellite with the shortest orbital period will be the one closest to Earth. This is ICESat-2, at 496 km. Therefore, ICESat-2 takes the shortest time to orbit Earth.
Finally, we are told that Eutelsat 113 West A is a geostationary satellite. Recall that this means its orbital period is the same as the time it takes Earth to rotate, which is one day. Eutelsat 113 West A therefore takes approximately 24 hours to orbit Earth.
Example 6: Relating Orbital Period and Orbital Radius
The table shows the orbital periods of three satellites in orbit around Earth. Each satellite follows a circular orbit.
|Orbital Period||24 hours||101 minutes||113 minutes|
Which satellite is closest to Earth?
Which satellite is farthest away from Earth?
Which satellite is in geostationary orbit?
In this question, we are given the orbital periods of three satellites, and we need to first determine which is the closest to Earth. We are told that each satellite follows a circular orbit.
Recall that, for satellites in a circular orbit, the orbital period, or time taken to complete one orbit, is related to the distance between the satellite and Earth. The closer the satellite, the faster it moves and the shorter its orbital period.
In order to determine which satellite is closest to Earth, we need to find which one has the shortest orbital period. In order to compare the orbital periods, we should first convert them all to the same units. Two of them are given in minutes, so we should convert the orbital period of Americom-8 to minutes. Recalling that , then .
Therefore, we have Americom-8 with an orbital period of 1 440 minutes, NOAA-15 at 101 minutes, and Jason-2 at 113 minutes. The one with the shortest orbital period is therefore NOAA-15, so this is the satellite that is closest to Earth.
Next, we need to decide which satellite is furthest away from Earth. Here, we are looking for the opposite of the above: the furthest satellite will move the slowest and have the longest orbital period. In this case, that is Americom-8.
Finally, we need to choose which satellite is in geostationary orbit. Recall that this means it has an orbital period equal to the period of rotation of Earth, which is one day, or approximately 24 hours. The satellite with an orbital period of 24 hours is Americom-8, so this is the geostationary satellite.
- The gravitational force between two objects depends on the objects’ masses and the distance between them: the magnitude of the force is higher for objects with more mass and if they are closer together.
- When an object is in orbit, it can follow a circular or an elliptical path.
- In circular orbits, the direction of velocity of the object is always perpendicular to the gravitational force. The magnitude of both the velocity and the force remains constant.
- In elliptical orbits, both the magnitude and the direction of the velocity and the force change with time.
- Orbital period is the amount of time taken for the object to complete one orbit.
- If two objects are in circular orbits around the same body, the object closer to the large body moves faster and has a shorter orbital period.
- A satellite that stays over the same position on Earth’s surface is said to be geostationary.
- Geostationary satellites have orbital periods of 24 hours and are positioned over the equator.