Video Transcript
A planet has a circular orbit around a star. It orbits the star at a speed of 17.9 kilometers per second, and the star has a mass of 2.18 times 10 to the 30 kilograms. What is the radius of the planet’s orbit? Use a value of 6.67 times 10 to the negative 11 meters cubed per kilogram second squared for the universal gravitational constant and 1.5 times 10 to the 11 meters for the length of one AU. Give your answer to the nearest astronomical unit.
Here, we have a planet orbiting around a star. And we’ve been told some of the properties of the system, such as the planet’s orbital speed, which we’ll call 𝑉, and the mass of the star, which we’ll call 𝑀. We’ve also been given the value of the universal gravitational constant represented by capital 𝐺 as well as the conversion factor for getting between meters and astronomical units, which we’ll use later.
We wanna know the orbital radius of the planet, which we’ll call 𝑟. So we’ll need a formula that relates these quantities. We’ll use the orbital speed formula 𝑉 equals the square root of 𝐺𝑀 divided by 𝑟, which only applies in the special case of circular orbit like we have here. Since we want to find the planet’s orbital radius, let’s copy the formula down here and solve it for 𝑟.
First, we’ll square both sides to undo the radical that 𝑟 appears under. Then, we’ll multiply both sides of the equation by 𝑟 over 𝑉 squared to cancel 𝑟 from the right-hand side and 𝑉 squared from the left-hand side so that we have 𝑟 by itself. But before we can substitute these terms in, they should all be expressed in base SI units. 𝑀 and 𝐺 are already good to go, but 𝑉 is expressed in kilometers per second, so we’ll need to convert it into meters per second.
Recall that one kilometer is equal to 10 to the three meters. So let’s make this substitution in the numerator. And we have 17.9 times 10 to the three meters per second. Or in scientific notation, that’s 1.79 times 10 to the four meters per second. Now, let’s substitute these values into the formula, and we have 𝑟 equals 6.67 times 10 to the negative 11 meters cubed per kilogram second squared times 2.18 times 10 to the 30 kilograms divided by 1.79 times 10 to the four meters per second quantity squared.
And before we calculate, let’s check on the units starting by distributing this exponent. And now let’s cancel units of per second squared as well as kilograms and two powers of meters. This leaves just one power of meters in the numerator, which is a good sign because we’re finding a distance. Now calculating, we found that 𝑟 equals 4.54 times 10 to the 11 meters.
But this isn’t our final answer because we’ve been asked to give our value to the nearest astronomical unit. We know that one AU equals 1.5 times 10 to the 11 meters. So let’s multiply 𝑟 by this conversion factor and cancel out units of meters. And now we have 𝑟 equals 3.03 astronomical units. So finally, rounding to the nearest astronomical unit, we have found that the radius of the planet’s orbit is three AU.
