In this explainer, we will learn how to calculate the orbital speed of an object moving along a circular orbit given its orbital radius and the mass of the object it orbits.
To begin, let us recall some of the key properties of circular orbital motion.
Remember that, for circular orbits, any orbiting body has a velocity with a constant magnitude but an ever-changing direction. The diagram below shows Jupiterโs moon Europa orbiting Jupiter. The direction of the moonโs velocity always points along a tangent to its orbital path, indicated by the blue arrow. Jupiterโs gravity acts as a centripetal force, which we know must always point radially inward, indicated by the red arrow.
This relationship helps explain why the moonโs orbital speed is constant: the gravitational force has no component in the same direction as the moonโs velocity. Thus, the magnitude of the velocity does not change due to gravity, but the direction constantly changes because the gravitational force constantly redirects it along a circular path. At any point in the orbit, the directions of the two quantities always point at a right angle, or , to each other.
As illustrated above, the gravitational force from a massive object always points inward, toward its own center of mass. Remember that the acceleration due to gravity, , at a point near a massive object, such as a planet, is given by where is the universal gravitational constant, is the mass of the planet, and is the distance between the point and the planet.
In this case, the gravitational acceleration, which always points inward and keeps the moon in a circular orbit, acts as a centripetal acceleration. Thus, for an object, such as a moon, in circular orbit, the centripetal acceleration of the object is given by where is the linear speed of the moon and is the radius of the orbit.
Since gravity is providing the centripetal acceleration, we can equate the values of and :
On both sides of the equation, appears in the denominator, so we can simplify the equation by multiplying both sides by :
We can take the square root of both sides to make the subject of the equation:
Now, we have an equation for the orbital speed of an object moving along a circular orbit, given its orbital radius and the mass of the object it orbits.
Notice that the equation does not depend on , the mass of the object in orbit. This means that all objectsโregardless of their massesโthat move in circular orbits with the same orbital radius, around planets with the same mass, will have the same orbital speed. Remember that this relationship only applies in the case of circular orbits, where orbital speed is constant; noncircular orbits do not obey this equation.
Before we practice some calculations, it should be noted that the universal gravitational constant is found commonly throughout astronomy and other fields of physics and has an unchanging value of .
Definition: Orbital Speed EquationโCircular Orbit
In the special case of a circular orbit, an objectโs orbital speed, , is given by the equation where is the universal gravitational constant, is the mass of the large object at the center of the orbit, and is the orbital radius.
Now that we have an equation for orbital speed, let us put it to use in some examples.
Example 1: Calculating Orbital Speed
For a satellite to follow a circular orbit around Earth at a radius of 10โโโ000 km, what orbital speed must it have? Use a value of kg for the mass of Earth and m3/kgโ s2 for the value of the universal gravitational constant. Give your answer to the nearest metre per second.
Answer
Here, we are given values for , , and . Notice that is given in kilometres, so we must convert to metres before we can use it in the equation:
We are now ready to substitute all of our values into the orbital speed equation:
Rounding to the nearest metre per second, we have found that the satellite must orbit at a speed of 6โโโ310 m/s.
Notice that we did not need to know the mass of the satellite to find the answer. This is because the orbital speed equation does not depend on the mass of the object in orbit, .
Sometimes, we will be given information about an orbital system other than just and . Depending on what information is provided, we can rearrange the orbital speed equation to solve for another quantity, besides . For instance, in the following example, we will calculate the orbital radius for a system.
Example 2: Calculating Orbital Radius
A planet has a circular orbit around a star. It orbits the star at a speed of 17.9 km/s, and the star has a mass of kg. What is the radius of the planetโs orbit? Use a value of m3/kgโ s2 for the universal gravitational constant and m for the length of 1 AU. Give your answer to the nearest astronomical unit.
Answer
Here, we are given values for , , and and we must solve for . To do this, we can rearrange the orbital speed equation so that becomes
Before we can calculate, we must convert the value for into units of metres per second:
Now, substituting our values, we have
However, this is not our final answer, since is presently expressed in metres and we need to convert it into astronomical units:
Rounding to the nearest astronomical unit, we find that the planet has an orbital radius of 3 AU.
Let us now shift our focus from these computational examples to a couple examples that are more conceptual.
Example 3: Variable Dependence in the Orbital Speed Equation
Which line on the graph shows the relation between orbital speed and orbital radius for objects moving along circular orbits due to gravity?
Answer
We can begin by recalling the orbital speed equation, which relates orbital speed, , and orbital radius, :
From this, we will devise a proportion to help us describe the relationship between orbital speed and radius. Since and are both constants in this context, they will not appear in our proportion. Using a 1 as a placeholder in the numerator, we have
Thus, we can say that and are inversely proportional to each other, since an increase in one quantity indicates a decrease in the other. Because of this inverse relationship, we know that should decrease as increases, so the green line is incorrect.
Further, since , the independent variable, appears in the denominator, we know that the relationship between and cannot be linear. Therefore, the red line is also incorrect.
The blue and orange lines have similar shapes, but they are quite different at the -axis. The orange curve intersects the axis, where the blue curve has a vertical asymptote. To determine how the graph should look here, let us think about how the orbital speed equation behaves near .
As mentioned before, appears in the denominator, and we know that we cannot meaningfully divide by zero. Thus, as goes to zero, the function goes to infinity and is undefined at the -axis.
Therefore, the blue line correctly shows the relation between orbital speed and orbital radius for objects in circular orbit.
Example 4: Variable Dependence in the Orbital Speed Equation
A satellite follows a circular orbit around Earth at a radial distance and with an orbital speed . If the satellite were moved closer to Earth, so that it followed a circular orbit with a radius of , at what speed, in terms of , would it have to move in order to maintain its orbit?
Answer
In this example, we will consider how orbital speed and the radius relate to each other by devising a proportion. First, we need to identify an equation that includes both of these quantities, so we will use the orbital speed equation:
Next, we identify the constant values in the equation so that we can remove them to form our proportion, since proportions explore how variables change with respect to one another. We know that represents a constant, so it will not appear in the proportion. And although appears as a variable in the orbital speed equation, within the context of this problem, represents the mass of Earth, which is essentially constant.
Now, we can write a proportion to show how and depend on each other using the symbol for proportionality, . We can no longer use an equals sign to relate and , since the true equality of their values does depend on some constant quantities that are not relevant to our proportion. The number 1 will act as a placeholder in the numerator where the constant values were in the original equation:
Thus, it can be stated that is inversely proportional to the square root of . The relationship is described as inverse because an increase in one quantity causes a decrease in the otherโremember that increasing a denominator corresponds to a decrease in the overall value and decreasing a denominator corresponds to an increase in the overall value. If the satellite is moved closer to Earth, we know that the orbital radius will decrease, so we can predict that the orbital speed will increase.
We can find the exact factor to describe the change in by substituting in the factor that describes the change in โa proportion behaves similarly to an equation in this way. We want to change the satelliteโs orbital radius from to , so is being multiplied by a factor of , which we substitute into the proportion:
Simplifying the math,
Therefore, multiplying by a factor of means must be multiplied by a factor of 3.
Thus, moving the satellite to a new orbital radius of will cause the satellite to move at a new orbital speed of .
In the previous example, we used the orbital speed equation without having any real, exact values to substitute in, yet we were still able to find a meaningful result by examining the relationship between the relevant variables.
As demonstrated in the following example, we are not limited to only using the orbital speed equation in our calculations. We may be given other information about a system, such as an orbital period, . Remember that the orbital period equation is
We may need to use this in conjunction with the orbital speed equation to learn about a system.
Example 5: Calculating the Mass of a Body in Orbit
Io is one of the four Galilean moons of Jupiter. Io makes one complete orbit of Jupiter every 1.77 days. Assuming that Ioโs orbit is circular with a radius of 422โโโ000 km, calculate the mass of Jupiter. Use a value of m3/kgโ s2 for the universal gravitational constant. Give your answer in scientific notation to two decimal places.
Answer
We want to solve for mass, which appears in our equation for orbital speed,
Let us now solve the orbital speed equation for . We start by squaring both sides to undo the radical on the right side:
Now, we will multiply both sides of the equation by :
We now have an equation we can use to solve for the mass of Jupiter, but notice that we do not yet have a value for each of the variablesโwe know the values of and , but we do not know the orbital speed of Io, so we need a way to calculate . We are given the orbital period of Io, , and we know an equation that relates orbital period and orbital speed:
We can rewrite the equation to solve for :
We know and , but notice that we are given a value for expressed in days, which is not an SI unit, so we must convert into seconds, the SI unit for time. The conversion is as follows:
Applying this to our value of ,
We will substitute this value into the orbital period equation to solve for . But first, because we are given a value of expressed in kilometres, we need to convert into metres. Recall that , so , but this value is better expressed in scientific notation as . Let us now substitute these values into the orbital period equation to solve for :
Now that we have values for , , and , we can substitute them into the rearranged orbital speed equation to calculate , the mass of Jupiter:
Thus, we have found the mass of Jupiter to be kg.
Let us finish by summarizing a few important concepts.
Key Points
- For an object to move in a circular orbit around a large object of mass at a radial distance , it must have an orbital speed given by .
- Orbital speed does not depend on the mass of the object in orbit.
- We can use to solve for or , and .
