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1. Calculate the mass of the Sun based on data for Earth's orbit and compare the value obtained...

Question:

1. Calculate the mass of the Sun based on data for Earth's orbit and compare the value obtained with the Sun's actual mass.

2. Find the mass of Jupiter based on data for the orbit of one of its moons, and compare your result with its actual mass.

Kepler's Third Law:

When a celestial object is in orbital motion around the heavier object, the period of the object is related to the orbital radius and the mass of the heavier object. In detail, the square of the orbital period is proportional to the cube of the orbital radius.

Answer and Explanation:

(1)

According to Kepler's third law, the relationship between the orbital period T and the orbital radius r is given by

{eq}T^2=\dfrac{4\pi^2 r^3}{GM} \\ \rm Here:\\ \,\,\,\, \, \bullet \, G(=6.67\times10^{-11} N m^2/kg^2)\text{: gravitational constant}\\ \,\,\,\, \, \bullet \,M \text{: mass of the Sun}\\ \,\,\,\, \, \bullet \, T(=1\, yr) \text{: orbital period of the Earth}\\ \,\,\,\, \, \bullet \,r(=1.5\times 10^{11}\, m) \text{: distance from the Earth to the Sun} {/eq}

Please note, 1 year is equivalent to {eq}3.16\times 10^7 {/eq} seconds. So, the mass of the Sun is

{eq}\begin{align} M&=\dfrac{4\pi^2r^3}{GT^2}\\\\ &=\dfrac{4\pi^2\times (1.5\times 10^{11})^3}{6.67\times10^{-11}\times (1\times 3.16\times 10^7)^2}\\\\ &=\boxed{2\times 10^{30}\, kg} \end{align} {/eq}

The actual value of the Sun's mass is {eq}1.99\times 10^{30} {/eq} kg, so the calculation above is correct.

(2)

I will choose Io as one of Jupiter's moon.

According to Kepler's third law, the relationship between the orbital period T and the orbital radius r is given by

{eq}T^2=\dfrac{4\pi^2 r^3}{GM} \\ \rm Here:\\ \,\,\,\, \, \bullet \, G(=6.67\times10^{-11} N m^2/kg^2)\text{: gravitational constant}\\ \,\,\,\, \, \bullet \,M \text{: mass of Jupiter}\\ \,\,\,\, \, \bullet \, T(=1.769\, \rm days) \text{: orbital period of Io}\\ \,\,\,\, \, \bullet \,r(=4.2\times 10^5\, km) \text{: distance from Io to Jupiter} {/eq}

Please note, 1 km is 1000 meters, 1 day has 24 hours, and there are 3600 seconds in an hour. So, the mass of Jupiter is

{eq}\begin{align} M&=\dfrac{4\pi^2r^3}{GT^2}\\\\ &=\dfrac{4\pi^2\times (4.22\times 10^{8})^3}{6.67\times10^{-11}\times (1.769\times 24\times 3600)^2}\\\\ &=\boxed{1.9\times 10^{27}\, kg} \end{align} {/eq}

The actual value of Jupiter's mass is {eq}1.898\times 10^{27} {/eq} kg, so the calculation above is correct.


Learn more about this topic:

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Kepler's Three Laws of Planetary Motion

from Basics of Astronomy

Chapter 22 / Lesson 12
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