Two satellites are in circular orbits around Jupiter. One, with orbital radius r, makes one revolution every 17 h. The other satellite has orbital radius 4.2r. How long does the second satellite take to make one revolution around Jupiter?
The orbital period is defined by
T = 2(Pi)sqrt[r^3/µ)] where T = the period in seconds, r = the orbital radius and µ = the planet's gravitational constant.
17hr = 61,200 sec.
61,200 = 2(Pi)sqrt[r^3/µ] from which µ = r^3/94,873,103
At a radius of 4.2r,
T = 2(Pi)sqrt[74.088r^3(94,873,103/r^3] = 23.288 hours.
OOPS - the hand was quicker than the eye.
The orbital period is defined by
T = 2(Pi)sqrt[r^3/µ)] where T = the period in seconds, r = the orbital radius and µ = the planet's gravitational constant.
17hr = 61,200 sec.
61,200 = 2(Pi)sqrt[r^3/µ] from which µ = r^3/94,873,103
At a radius of 4.2r,
T = 2(Pi)sqrt[74.088r^3(94,873,103/r^3] = 146.32
hours.
The same result derives from Kepler's third law, T/To = [R/Ro]^(3/2) where
T/17 = [4.2R/R]^(3/2).
Well, if the first satellite takes 17 hours to make one revolution around Jupiter, then we can calculate the period of the second satellite using the formula for the period of a circular orbit:
T = 2π√(r³/GM)
Here, r is the orbital radius, G is the gravitational constant, and M is the mass of Jupiter.
So, for the second satellite with an orbital radius of 4.2r, let's substitute that value in the formula:
T2 = 2π√((4.2r)³/GM)
Now, we can simplify it:
T2 = 2π√(4.2³r³/GM)
T2 = 2π√(74.088r³/GM)
Hmm, this equation is looking really complicated. I bet even math professors would struggle with this. But don't worry, I'm here to help! Let's simplify it further:
T2 ≈ 2π(8.612r/√GM)
T2 ≈ 17.224π(r/√GM)
Okay, let's substitute the value of T from the first satellite into this equation:
T2 ≈ 17.224π(r/√GM)
T2 ≈ 17.224 × 17
T2 ≈ 292.108 hours
So, the second satellite takes approximately 292.108 hours to make one revolution around Jupiter.
Don't worry, though. While it may seem like it's taking forever, there's plenty of cosmic popcorn and astronaut ice cream to keep you entertained during the wait! Enjoy the show!
To solve this problem, we can use Kepler's third law, which states that the square of the orbital period (T) is directly proportional to the cube of the orbital radius (r) for objects orbiting the same central body.
Let's assume that the orbital period of the first satellite (with radius r) is T1, and the orbital period of the second satellite (with radius 4.2r) is T2.
Using Kepler's third law, we can set up the following equation:
(T2 / T1)^2 = (r2 / r1)^3
Since r2 = 4.2r and r1 = r, we can substitute these values into the equation:
(T2 / T1)^2 = (4.2r / r)^3
Simplifying the right side of the equation:
(T2 / T1)^2 = (4.2)^3
Taking the square root of both sides:
T2 / T1 = sqrt(4.2)^3
T2 / T1 = 4.2^(3/2)
Now we need to solve for T2. To do this, we multiply both sides of the equation by T1:
T2 = T1 * 4.2^(3/2)
We know that the orbital period of the first satellite (T1) is 17 hours, so substituting this value into the equation:
T2 = 17 * 4.2^(3/2)
Calculating the right side of the equation, we find:
T2 ≈ 17 * 12.991
T2 ≈ 220.787 hours
Therefore, the second satellite takes approximately 220.787 hours to make one revolution around Jupiter.
To find the time it takes for the second satellite to make one revolution around Jupiter, we can use Kepler's Third Law. According to Kepler's Third Law, the square of the orbital period of a satellite is proportional to the cube of the orbital radius.
Let's denote the orbital period of the first satellite as T1 and the orbital radius as r. Therefore, we have:
T1^2 ∝ r^3
Similarly, let's denote the orbital period of the second satellite as T2 and the orbital radius as 4.2r. Hence:
T2^2 ∝ (4.2r)^3
Now, comparing the two equations, we can write:
T2^2 = k * (4.2r)^3
where k is the constant of proportionality.
To determine the ratio of the two orbital periods, we can divide the two equations:
T2^2 / T1^2 = (4.2r)^3 / r^3
Simplifying, we get:
(T2 / T1)^2 = 4.2^3
Taking the square root of both sides to solve for T2 / T1, we have:
T2 / T1 = √(4.2^3)
Calculating this value, we find:
T2 / T1 ≈ 2.239
Since we know that T1 is 17 h, we can find T2 by multiplying T1 by the above ratio:
T2 = T1 * (T2 / T1) = 17 h * 2.239
Calculating this value, we can determine that:
T2 ≈ 36.703 h
Therefore, the second satellite takes approximately 36.703 hours to make one revolution around Jupiter.