Determine the time it takes for a satellite to orbit the Saturn in a circular "near-Saturn" orbit. A "near-Saturn" orbit is at a height above the surface of the Saturn that is very small compared to the radius of the Saturn. [Hint: You may take the acceleration due to gravity as essentially the same as that on the surface.]
mv²/R=GmM/R ²
v= sqrt (GM/R) =…
T=2πR/v= …
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To determine the time it takes for a satellite to orbit Saturn in a circular "near-Saturn" orbit, we can use Kepler's third law of planetary motion. Kepler's third law states that the square of the orbital period of a planet or satellite is directly proportional to the cube of its semi-major axis.
In this scenario, since the satellite is in a circular orbit, the semi-major axis will be equal to the radius of Saturn plus the height above its surface. However, since the height is very small compared to the radius of Saturn, we can approximate the semi-major axis as equal to the radius of Saturn.
The acceleration due to gravity on the surface of Saturn can be considered as the same anywhere in the orbit, as mentioned in the hint. Let's denote this acceleration as "g".
Now, we can calculate the time it takes for the satellite to complete one orbit using the following steps:
Step 1: Determine the radius of Saturn.
You can find the radius of Saturn by looking up the value online or referring to an astronomy resource. The average radius of Saturn is approximately 58,232 kilometers.
Step 2: Calculate the semi-major axis.
Since the height above the surface is very small, we can approximate the semi-major axis as equal to the radius of Saturn.
Semi-major axis (a) = Radius of Saturn = 58,232 kilometers.
Step 3: Calculate the orbital period.
Using Kepler's third law, we express the relationship between the orbital period (T) and the semi-major axis (a).
T^2 = k * a^3,
where k is a constant.
Since we're solving for the orbital period, we can write it as:
T = sqrt(k * a^3),
where sqrt denotes the square root.
Step 4: Calculate the orbital period.
We need to find the value of the constant k. Kepler's third law in its full form includes G, the gravitational constant, but G cancels out when comparing two orbits around the same body. Therefore, we don't need its value for this calculation.
T = sqrt(a^3 / g),
where a = 58,232 km and g is the acceleration due to gravity on the surface of Saturn.
Step 5: Substitute known values and calculate.
To calculate T, we need to know the acceleration due to gravity on the surface of Saturn. Let's assume that it is approximately 10 m/s^2. However, it's always better to check for the most accurate value from reliable sources.
T = sqrt((58,232 km)^3 / (10 m/s^2)),
Convert 58,232 km to meters by multiplying by 1000,
T = sqrt((58,232,000 m)^3 / (10 m/s^2)),
T = sqrt(197,227,085,143,040,000,000 / 10),
T ≈ sqrt(19,722,708,514,304,000,000,000),
T ≈ 4,439,726 seconds.
Therefore, it takes approximately 4,439,726 seconds for a satellite in a circular "near-Saturn" orbit to complete one orbit around Saturn.