Question Video: Calculating the Mass of a Star Given a Planet's Orbital Period and Radius | Nagwa Question Video: Calculating the Mass of a Star Given a Planet's Orbital Period and Radius | Nagwa

# Question Video: Calculating the Mass of a Star Given a Planet's Orbital Period and Radius Physics • First Year of Secondary School

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The formula ๐ = 4๐ยฒ๐ยณ/๐บ๐ยฒ can be used to calculate the mass, ๐, of a planet or star given the orbital period, ๐, and orbital radius, ๐, of an object that is moving along a circular orbit around it. A planet is discovered orbiting a distant star with a period of 105 days and a radius of 0.480 AU. What is the mass of the star? Use a value of 6.67 ร 10โปยนยน mยณ/kg โ sยฒ for the universal gravitational constant and 1.50 ร 10ยนยน m for the length of 1 AU. Give your answer in scientific notation to two decimal places.

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### Video Transcript

The formula ๐ equals four ๐ squared ๐ cubed divided by ๐บ๐ squared can be used to calculate the mass, ๐, of a planet or star given the orbital period, ๐, and orbital radius, ๐, of an object that is moving along a circular orbit around it. A planet is discovered orbiting a distant star with a period of 105 days and a radius of 0.480 AU. What is the mass of the star? Use a value of 6.67 times 10 to the negative 11 meters cubed per kilogram second squared for the universal gravitational constant and 1.50 times 10 to the 11 meters for the length of one AU. Give your answer in scientific notation to two decimal places.

So we have some planet in circular orbit around a star. Itโs pretty cool that given our understanding of physics and some fairly basic math, we can use information about a distant planetโs orbit to learn the mass of such a large and far away object as a star. Letโs take a closer look at the formula weโll use. ๐ equals four ๐ squared ๐ cubed divided by ๐บ๐ squared. Now, we have been given values for all the terms in this formula. But before we can substitute them in, they should all be expressed in base SI units.

๐บ, the universal gravitational constant, is already written in meters, kilograms, and seconds. So itโs good to go. And now letโs look at orbital radius, ๐, which we know equals 0.480 AU. And while the astronomical unit is used frequently throughout astronomy, itโs not in SI unit. So letโs convert it into meters. Weโve been told that one AU equals 1.5 times 10 to the 11 meters. Knowing this, we can multiply ๐ by 1.50 times 10 to the 11 meters divided by one AU, which is just equal to one. So we can cancel out the AU. And thus, we have found that ๐ equals 7.200 times 10 to the 10 meters.

Next, weโll look at orbital period, ๐, which is equal to 105 days, and days is not the SI unit of time. For this, weโll need to convert to seconds. Recall that one day equals 24 hours, an hour equals 60 minutes, and a minute equals 60 seconds. We can use these three equalities to write three conversion factors, each of which being equal to one. Now we can cancel units of days, hours, and minutes, leaving only seconds. And now multiplying through 105 times 24 times 60 times 60 seconds gives us an orbital period value ๐ equals 9.072 times 10 to the six seconds.

So our values are all set to calculate. Substituting them in the formula, we have ๐ equals four ๐ squared times 7.200 times 10 to the 10 meters quantity cubed divided by 6.67 times 10 to the negative 11 meters cubed per kilogram second squared times 9.072 times 10 to the six seconds quantity squared. Now there are a lot of units here, so letโs make sure that theyโre all working out to reach a final mass value in units of kilograms.

First, for visual clarity, letโs group the units over here, making sure to distribute the proper exponents. Now, letโs cancel units of meters cubed as well as seconds squared in the denominator, leaving only one over kilograms in the denominator or plain kilograms in the numerator. Now calculating, we have ๐ equals 2.684 times 10 to the 30 kilograms. And finally, rounding to two decimal places, we have found that the mass of the star is 2.68 times 10 to the 30 kilograms.

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