91影视

The expression for the Kepler鈥檚 third law is given by,

\({r_1} = {r_2}{\left( {\frac{{{T_1}}}{{{T_2}}}} \right)^{\frac{2}{3}}}\)

Kepler's second law: This law shows the relation between the distance between a planet and its principal moons and the period of the moon.

Refer to table 5-3, 鈥淧rincipal Moons of Jupiter鈥� for the mean distance and period.

The mean distance forIois\(r{'_I} = 422 \times {10^3}\;{\rm{km}}\).

The mean distance for Europa is\(r{'_E} = 671 \times {10^3}\;{\rm{km}}\).

The mean distance for Ganymede is\(r{'_G} = 1070 \times {10^3}\;{\rm{km}}\).

The mean distance for Callisto is\(r{'_C} = 1883 \times {10^3}\;{\rm{km}}\).

The period forIois\({T_I} = 1.77\;{\rm{Ed}}\).

The period for Europa is\({T_E} = 3.55\;{\rm{Ed}}\).

The period for Ganymede is\({T_G} = 7.16\;{\rm{Ed}}\).

The period for Callisto is\({T_C} = 16.7\;{\rm{Ed}}\).

The mean distance of Europa using the Kepler鈥檚 law can be calculated as,

\(\begin{aligned}{r_E} &= r{'_I}{\left( {\frac{{{T_E}}}{{{T_I}}}} \right)^{\frac{2}{3}}}\\{r_E} &= 422 \times {10^3}\;{\rm{km}}{\left( {\frac{{16.7\;{\rm{Ed}}}}{{1.77\;{\rm{Ed}}}}} \right)^{\frac{2}{3}}}\\{r_E} &= 1884 \times {10^3}\;{\rm{km}}\end{aligned}\)

The mean distance of Ganymedeusing the Kepler鈥檚 law can be calculated as,

\(\begin{aligned}{r_G} &= r{'_I}{\left( {\frac{{{T_G}}}{{{T_I}}}} \right)^{\frac{2}{3}}}\\{r_G} &= 422 \times {10^3}\;{\rm{km}}{\left( {\frac{{7.16\;{\rm{Ed}}}}{{1.77\;{\rm{Ed}}}}} \right)^{\frac{2}{3}}}\\{r_G} &= 1071 \times {10^3}\;{\rm{km}}\end{aligned}\)

The mean distance of Callistousing the Kepler鈥檚 law can be calculated as,

\(\begin{aligned}{r_C} &= r{'_I}{\left( {\frac{{{T_C}}}{{{T_I}}}} \right)^{\frac{2}{3}}}\\{r_C} &= 422 \times {10^3}\;{\rm{km}}{\left( {\frac{{16.7\;{\rm{Ed}}}}{{1.77\;{\rm{Ed}}}}} \right)^{\frac{2}{3}}}\\{r_C} &= 1884 \times {10^3}\;{\rm{km}}\end{aligned}\)

The calculated results and results from the table 5-3 are very close.

Thus, the mean distance for Europa is \({r_E} = 671 \times {10^3}\;{\rm{km}}\), the mean distance for Ganymede is \({r_G} = 1070 \times {10^3}\;{\rm{km}}\), and the mean distance for Callisto is \({r_C} = 1880 \times {10^3}\;{\rm{km}}\).