A hypothetical moon is orbiting Saturn in a circular orbit that has exactly 3 times the period that an object orbiting in one of one of the gaps in Saturn's rings would have. As measured from the center of Saturn at what fraction [to 3 decimal places] of that moon's orbital radius does the ring appear? [Hint: Use Kepler's 3rd Law]

Kepler's third law says that, for all objects orbiting the same large mass,

P^2/R^3 = constant
P is the period; R is the orbit radius.

If the moon's period is 3 times the ring gap object's period
[R(gap)/R(moon)]^3 = [P(gap)/P(moon)]^2 = 1/9
[R(gap)/R(moon)] = cube root of 1/9

[R(gap)/R(moon)] = cube root of 1/9 = 0.481

To answer this question, we can use Kepler's 3rd Law, which states that the square of the orbital period of a planet or moon is proportional to the cube of its semi-major axis (orbital radius).

Let's denote the orbital period of the moon as Tm, and the semi-major axis (orbital radius) as Rm.

We are given that Tm = 3 * Tg, where Tg is the period of an object orbiting in one of the gaps in Saturn's rings.

According to Kepler's 3rd Law:
(Tm)^2 / (Rm)^3 = (Tg)^2 / (Rg)^3

Since Tm = 3 * Tg, we can rewrite this equation as:
(3 * Tg)^2 / (Rm)^3 = (Tg)^2 / (Rg)^3

Simplifying further:
9 * (Tg)^2 / (Rm)^3 = (Tg)^2 / (Rg)^3

Now, we need to consider the relationship between the radius of the moon's orbit (Rm) and the radius of the gap in Saturn's rings (Rg), where the object is orbiting.

Since the radius of the gap is smaller than the orbital radius of the moon, the fraction (Rg / Rm) will be less than 1. We need to find this fraction.

Let's assume the fraction (Rg / Rm) = x.

Now we can rewrite the equation again using this fraction:
9 * (Tg)^2 / (Rm)^3 = (Tg)^2 / [(Rm * x)^3]

As we can see, the orbital periods (Tg)^2 cancel out, and we can simplify the equation to solve for x:
9 / (Rm)^3 = 1 / [(Rm * x)^3]

To solve for x, we can rearrange the equation:
9 * [(Rm * x)^3] = (Rm)^3

Taking the cube root of both sides:
(Rm * x)^3 = (Rm)^3 / 9

Now, taking the cube root again:
Rm * x = [(Rm)^3 / 9]^(1/3)

Finally, solving for x, the fraction of the moon's orbital radius where the ring appears:
x = [(Rm)^3 / 9]^(1/3) / Rm

x is the fraction we are looking for, which represents the fraction of the moon's orbital radius at which the ring appears when measured from the center of Saturn.

To calculate this fraction, you would need to know the values for the moon's orbital radius (Rm) and the period of an object in one of Saturn's rings (Tg). Once you have these values, simply substitute them into the equation to find x.