Calculating the Mass of Saturn Using Orbital Data

[Fysicsx]

titan, a moon of saturn, has a 16 day orbital period and an orbital radius of 1,222,000 km. what is the mass of saturn?

my attempt:

t = 2*pi*r^(3/2)/sqrt(g*m)
t = 16 days = 1,382,400 s
r = 1,222,000/2 km = 611,000,000 m

1,382,400 = 2*pi*611,000,000^(3/2)/sqrt(6.67e-11*m) => m = 7.065e25 kg

the answer should be 5.6e26 kg, what am i doing wrong?

 

[SammyS]

titan, a moon of saturn, has a 16 day orbital period and an orbital radius of 1,222,000 km. what is the mass of saturn?

my attempt:

t = 2*pi*r^(3/2)/sqrt(g*m)
t = 16 days = 1,382,400 s
r = 1,222,000/2 km = 611,000,000 m

1,382,400 = 2*pi*611,000,000^(3/2)/sqrt(6.67e-11*m) => m = 7.065e25 kg

the answer should be 5.6e26 kg, what am i doing wrong?

Hello Fysicsx. Welcome to PF !

Where does the following equation come from?

t = 2*pi*r^(3/2)/√(g*m)

What is g in this equation?

 

[Fysicsx]

hi sammys,

thanks for the warm welcome

f_g = g*m_1*m_2/r^2 (law of gravitation)
v = sqrt(g*m/r) (circular orbit)

v = 2*pi*r/t
t = 2*pi*r/v = 2*pi*r*sqrt(r/(g*m)) = 2*pi*r^(3/2)/sqrt(g*m) (circular orbit)

g is the gravitational constant 6.67e-11 m^3*kg^-1*s^-2

 

[SammyS]

hi sammys,

thanks for the warm welcome

f_g = g*m_1*m_2/r^2 (law of gravitation)
v = sqrt(g*m/r) (circular orbit)

v = 2*pi*r/t
t = 2*pi*r/v = 2*pi*r*sqrt(r/(g*m)) = 2*pi*r^(3/2)/sqrt(g*m) (circular orbit)

g is the gravitational constant 6.67e-11 m^3*kg^-1*s^-2

You divided r by 2.

Solve this for r, not v : v = sqrt(g*m/r)

 

[Fysicsx]

oops that was careless lol. thanks for your help, sammys.