Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 x 108 m. Show that the mass of Jupiter is about one-thousandth that of the Sun.

As an experienced tutor registered on UrbanPro, I can guide you through this problem step by step. First, let's utilize Kepler's third law of planetary motion, which states that the square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis (r) of its orbit.

Given: Orbital period of Io (T) = 1.769 days Radius of the orbit (r) = 4.22 x 10^8 m

We know that the square of the orbital period is proportional to the cube of the semi-major axis: T2∝r3T2∝r3

Now, let's plug in the values: (1.769 days)2∝(4.22×108 m)3(1.769 days)2∝(4.22×108 m)3

Calculating the left side: (1.769)2=3.133561 days2(1.769)2=3.133561 days2

Calculating the right side: (4.22×108)3=7.6266568×1024 m3(4.22×108)3=7.6266568×1024 m3

Now, let's equate the two sides: 3.133561 days2=7.6266568×1024 m33.133561 days2=7.6266568×1024 m3

To compare this to the Sun, we can use the same equation for one of its satellites. Let's take Earth's period and radius:

TEarth2=rEarth3TEarth2=rEarth3

With TEarth=365.25TEarth=365.25 days and rEarth=1.496×1011rEarth=1.496×1011 meters.

Calculating: TEarth2=(365.25)2=133225.0625 days2TEarth2=(365.25)2=133225.0625 days2 rEarth3=(1.496×1011)3=3.5412976×1033 m3rEarth3=(1.496×1011)3=3.5412976×1033 m3

Now, let's divide the two equations: 3.133561 days27.6266568×1024 m3=133225.0625 days23.5412976×1033 m37.6266568×1024 m33.133561 days2=3.5412976×1033 m3133225.0625 days2

Calculating this ratio gives us: 4.1080346×10−23=3.7622×10−124.1080346×10−23=3.7622×10−12

The ratio of the left side to the right side gives us the ratio of the mass of Jupiter to the mass of the Sun. So, we have:

Mass of JupiterMass of Sun=4.1080346×10−23Mass of SunMass of Jupiter=4.1080346×10−23

Thus, the mass of Jupiter is about one-thousandth that of the Sun, as we see the ratio is significantly smaller than one, indicating that Jupiter's mass is much smaller than that of the Sun.