[latex]F_d=F_g \ frac{mv^2}{R} =G frac{mM}{R^2} \ v= frac{2pi R}{T} \ frac{m*4pi^2R^2}{rT^2} =G frac{mM}{r^2} \ frac{4pi^2R}{T^2} =G frac{M}{R^2} \ M= frac{4pi^2R^3}{GT^2}[/latex], gdzie M-masa planety, m-masa księżyca, T - okres obiegu, R-odległość dzieląca księżyc i planetę Obliczymy średnią masę Jowisza na podstawie wyznaczenia jego masy z danych dla każdej z planet: Io: [latex]R=421000km=421000000m \ T=1,77dnia=1,77*24*3600s=152928s \ M= frac{4pi^2R^3}{GT^2}= frac{4pi^2*(421000000m)^3}{6,67*10^{-11} frac{m^3}{kg*s^2} (152928s)^2}approx 1,888*10^{27} kg[/latex] Europa: [latex]R=671000km=671000000m \ T=3,55dnia=3,55*24*3600s=306720s \ M= frac{4pi^2R^3}{GT^2}= frac{4pi^2*(671000000m)^3}{6,67*10^{-11} frac{m^3}{kg*s^2} (306720s)^2}approx 1,901*10^{27} kg[/latex] Ganimedes: [latex]R=1070000km=1070000000m \ T=7,16dnia=7,16*24*3600s=618624s \ M= frac{4pi^2R^3}{GT^2}= frac{4pi^2*(1070000000m)^3}{6,67*10^{-11} frac{m^3}{kg*s^2} (618624s)^2}approx 1,895*10^{27}kg[/latex] Kallisto: [latex]R=1883000km=1883000000m \ T=16,7dnia=16,7*24*3600s=1442880 s \ M= frac{4pi^2R^3}{GT^2}= frac{4pi^2*(1883000000m)^3}{6,67*10^{-11} frac{m^3}{kg*s^2} (1442880 s)^2}approx 1,898*10^{27}kg[/latex] [latex]M_{sr}approx1,8955*10^{27}kg[/latex] Wyznaczymy teraz masę każdego z księżyców: [latex]m= ho*V= ho*frac{4}{3}pi r^3[/latex] m-masa planety r-promień planety ρ-gęstość planety Io: [latex] ho=3,53 frac{g}{cm^3}= 3,53 frac{ frac{1}{1000} kg}{ frac{1}{1000000} m^3}=3530 frac{kg}{m^3} \ r= frac{3643km}{2} =1821,5km=1821500m \ m= ho*frac{4}{3}pi r^3=3530 frac{kg}{m^3}*frac{4}{3}pi*(1821500m)^3approx8,936*10^{22}kg[/latex] Europa: [latex] ho=3,01 frac{g}{cm^3}= 3,01 frac{ frac{1}{1000} kg}{ frac{1}{1000000} m^3}=3010 frac{kg}{m^3} \ r= frac{3122km}{2} =1561km=1561000m \ m= ho*frac{4}{3}pi r^3=3010 frac{kg}{m^3}*frac{4}{3}pi*(1561000m)^3approx4,796*10^{22}kg[/latex] Ganimedes: [latex] ho=1,94 frac{g}{cm^3}= 1,94 frac{ frac{1}{1000} kg}{ frac{1}{1000000} m^3}=1940 frac{kg}{m^3} \ r= frac{5262km}{2} =2631km=2631000m \ m= ho*frac{4}{3}pi r^3=1940 frac{kg}{m^3}*frac{4}{3}pi*(2631000m)^3approx1,480*10^{23}kg [/latex] Kallisto: [latex] ho=1,23 frac{g}{cm^3}= 1,23 frac{ frac{1}{1000} kg}{ frac{1}{1000000} m^3}=1230 frac{kg}{m^3} \ r= frac{4821km}{2} =2410,5km=2410500m \ m= ho*frac{4}{3}pi r^3=1230 frac{kg}{m^3}*frac{4}{3}pi*(2410500m)^3approx7,216*10^{22}kg[/latex]
