Titan is the largest moon of Saturn. Assuming that Titan follows a circular orbit, with a radius of 1220000 kilometers and an orbital speed of 5.57 kilometers per second, calculate the mass of Saturn. Use a value of 6.67 times 10 to the negative 11 meters cubed per kilogram second squared for the universal gravitational constant. Give your answer in scientific notation to two decimal places.
Here, we’re considering Titan, a moon in the special case of circular orbit around Saturn. We’ve been told to calculate the mass of Saturn. And at first that might sound pretty complicated. But we can do it using some fairly simple math that we’re already familiar with.
Let’s take a closer look at what we already know about the system. We know Titan’s orbital radius, which we’ll call 𝑟, and its orbital speed, which we’ll call 𝑣, as well as the value for the universal gravitational constant represented by 𝐺. Let’s recall that these are three of the four terms in the orbital–speed formula 𝑣 equals the square root of 𝐺𝑀 divided by 𝑟. All that we’re missing is 𝑀, which represents the mass of the large body at the center of orbit or, in this case, the mass of Saturn.
So to calculate the mass of Saturn, let’s start by copying the formula down here and solve it for 𝑀. We’ll square both sides of the formula to undo the radical that 𝑀 appears under. Then we’ll multiply both sides of the formula by 𝑟 divided by 𝐺 to cancel those terms over here and get 𝑀 by itself. Now writing this a little more neatly, we found that 𝑀 equals 𝑟𝑣 squared divided by 𝐺.
But before we can substitute these terms into the formula, we need to make sure they’re all expressed in base SI units. Notice that orbital radius is currently written in kilometers, so we wanna convert it into plain meters. So let’s recall that one kilometer equals 1000 meters. Now we’ll multiply 𝑟 by this conversion factor, which itself is just equal to one. So we can cancel out units of kilometers. And we have 𝑟 equals 1220000000 meters or in scientific notation 1.22 times 10 to the nine meters.
Now looking at orbital speed, it’s written in kilometers per second, so let’s convert it to meters per second. Again, remember that one kilometer equals 1000 or 10 to the three meters. So let’s make this substitution in the numerator. And we have 𝑣 equals 5.57 times 10 to the three meters per second. Moving on, we can see that 𝐺 is already expressed in meters, kilograms, and seconds, which are all base SI units. Now all these values are ready to be substituted into the formula 𝑀 equals 𝑟 times 𝑣 squared divided by 𝐺.
Finally, calculating and rounding to two decimal places, we have found that the mass of Saturn is 5.67 times 10 to the 26 kilograms.