Saturn’s Moon Janus is observed to orbit the planet with period 0.695 days, grazing the planet with an orbital radius only 2.64 times the radius of the planet itself. Use this information to find the mass of Saturn. Express your answer as a multiple of the mass of Earth M⊕ rounded to two significant digits.
To find the mass of Saturn, we can use Kepler's third law, which states that the square of the orbital period of a planet/moon is proportional to the cube of the semi-major axis of its orbit. We can express this as:
T² = (4π²/GM) * r³
Where T is the orbital period, G is the gravitational constant, M is the mass of Saturn, and r is the orbital radius of the moon.
Given:
T = 0.695 days
r = 2.64 * R (where R is the radius of Saturn)
First, we need to convert the period to seconds:
T = 0.695 days * (24 hours/day) * (3600 seconds/hour) = 60,048 seconds
Next, we can substitute the values into the equation and solve for M:
(60,048 seconds)² = (4π²/GM) * (2.64R)³
Simplifying the equation:
(60,048 seconds)² = (4π²/G) * (2.64³) * R³ * M⁻¹
Let's denote (4π²/G) * (2.64³) by a constant C for simplicity. Therefore,
(60,048 seconds)² = C * R³ * M⁻¹
Now, we can solve for M:
M⁻¹ = [(60,048 seconds)²] / [C * (R³)]
To express the mass of Saturn in terms of the mass of Earth (M⊕), we can divide the calculated value by the mass of Earth:
M/M⊕ = [(60,048 seconds)²] / [C * (R³ * M⊕)]
Finally, we round the calculated M/M⊕ value to two significant digits and that will be the mass of Saturn in terms of Earth's mass.
Please note that to get an accurate calculation, we need the values of the radius of Saturn (R) and the gravitational constant (G), in addition to the conversion of units for the period.