Time period of any moon of Jupiter is given by
[tex]T = 2\pi \sqrt{\frac{r^3}{GM}}[/tex]
from above formula we can say that mass of Jupiter is given by
[tex]M = \frac{4 \pi^2 r^3}{GT^2}[/tex]
now for part a)
[tex]r = 4.22 * 10^8 m[/tex]
T = 1.77 day = 152928 seconds
now by above formula
[tex]M = \frac{4 \pi^2 r^3}{GT^2}[/tex]
[tex]M = \frac{4 \pi^2 (4.22 * 10^8)^3}{(6.67 * 10^{-11})(152928)^2}[/tex]
[tex]M = 1.9* 10^{27} kg[/tex]
Part B)
[tex]r = 6.71 * 10^8 m[/tex]
T = 3.55 day = 306720 seconds
now by above formula
[tex]M = \frac{4 \pi^2 r^3}{GT^2}[/tex]
[tex]M = \frac{4 \pi^2 (6.71 * 10^8)^3}{(6.67 * 10^{-11})(306720)^2}[/tex]
[tex]M = 1.9* 10^{27} kg[/tex]
Part c)
[tex]r = 10.7 * 10^8 m[/tex]
T = 7.16 day = 618624 seconds
now by above formula
[tex]M = \frac{4 \pi^2 r^3}{GT^2}[/tex]
[tex]M = \frac{4 \pi^2 (10.7 * 10^8)^3}{(6.67 * 10^{-11})(618624)^2}[/tex]
[tex]M = 1.89* 10^{27} kg[/tex]
PART D)
[tex]r = 18.8 * 10^8 m[/tex]
T = 16.7 day = 1442880 seconds
now by above formula
[tex]M = \frac{4 \pi^2 r^3}{GT^2}[/tex]
[tex]M = \frac{4 \pi^2 (18.8 * 10^8)^3}{(6.67 * 10^{-11})(1442880)^2}[/tex]
[tex]M = 1.889* 10^{27} kg[/tex]
