Europa is one of Jupiter's moons. This moon is in a circular orbit of 6.71x10^8 meters within a period of 3.55 days. What is the mass of Jupiter?
To find the mass of Jupiter, we can use the third law of planetary motion, which relates the period (T) of a moon's orbit to the distance (r) between the moon and its planet, and the mass (M) of the planet. The formula is as follows:
T^2 = (4π^2 / GM) * r^3
Where:
T = Period of the moon's orbit
G = Gravitational constant (6.67430 x 10^-11 m^3 kg^-1 s^-2)
M = Mass of the planet
r = Distance between the moon and the planet
In this case, the period (T) is given as 3.55 days, which we need to convert to seconds:
3.55 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute = 306,720 seconds
The distance (r) is given as 6.71 x 10^8 meters.
Now, we can rearrange the formula to solve for the mass of Jupiter (M):
M = (4π^2 / G) * r^3 / T^2
Substituting the given values:
M = (4 * π^2 / (6.67430 x 10^-11)) * (6.71 x 10^8)^3 / (306,720)^2
Calculating this equation will give us the mass of Jupiter.
To determine the mass of Jupiter, we can use Kepler's third law of planetary motion. This law states that the square of the orbital period of a moon or planet is proportional to the cube of the semi-major axis of its orbit.
First, let's convert the period of Europa's orbit from days to seconds:
3.55 days * 24 hours/day * 60 minutes/hour * 60 seconds/minute = 306,720 seconds
Next, let's calculate the square of Europa's orbital period:
(306,720 seconds)^2 = 9.42 x 10^10 seconds^2
Now, we can use the equation for Kepler's third law:
(T^2 / a^3) = (4π^2 / G * M)
where T is the orbital period, a is the semi-major axis, G is the gravitational constant, and M is the mass of Jupiter.
Rearranging the equation to solve for M:
M = (4π^2 * a^3) / (G * T^2)
Plugging in the given values:
a = 6.71 x 10^8 meters
T = 9.42 x 10^10 seconds
G = 6.67430 x 10^-11 m^3kg^-1s^-2 (gravitational constant)
Calculating the mass of Jupiter:
M = (4π^2 * (6.71 x 10^8)^3) / (6.67430 x 10^-11 * (9.42 x 10^10)^2)
After performing the calculation, the mass of Jupiter is approximately 1.89 x 10^27 kilograms.
Europa is one of Jupiter's moons. This moon is in a circular orbit of 6.71x10^8 meters within a period of 3.55 days. What is the mass of Jupiter?
The time it takes a satellite to orbit a central body, its orbital period, can be calculated from
T = 2(Pi)sqrt[a^3/µ]
where T is the orbital period in seconds, Pi = 3.1416, a = the semi-major axis of an elliptical orbit = (rp+ra)/2 where rp = the perigee (closest) radius and ra = the apogee (farthest) radius from the center of the earth, µ = the central bodies gravitational constant = GM, the gravitational constant times the mass of the central body. In the case of a circular orbit, a = r, the radius of the orbit.
You have T, r, and G = 6.67259x10^-11m^3/kg.sec.^2. Solve for M.