1. In this collection of problems we will try to use what we have learned to see what information about a hypothetical Solar System planet we can extract from observations. The planet, Nursat, is observed to be in opposition every 374 days. Use this information to find the sidereal period of Nursat. Express your answer in years.

2. Find the radius of Nursat’s orbit, which we will assume is circular, in AU

41.6

41.55 years,

11.996 AU

To find the sidereal period of Nursat, we can use Kepler's Third Law, which states that the square of the sidereal period of a planet is proportional to the cube of its average distance from the Sun.

Step 1: Convert the opposition period to the sidereal period.
The opposition period is the time it takes for a planet to return to the same position relative to the Sun as observed from Earth. Since opposition occurs every 374 days, this is the sidereal period plus the time it takes for Earth to orbit the Sun.

The time it takes for Earth to orbit the Sun is approximately 365.25 days. We can subtract this from the opposition period to get the sidereal period of Nursat.

Sidereal period of Nursat = Opposition period - Earth's orbital period
Sidereal period of Nursat = 374 days - 365.25 days

Step 2: Convert the sidereal period to years.
To express the sidereal period of Nursat in years, we need to divide the obtained value by the number of days in a year.

Sidereal period of Nursat in years = (Sidereal period of Nursat in days) / (Number of days in a year)

Step 3: Calculate the radius of Nursat's orbit in AU.
In a circular orbit, the radius is equal to the average distance from the Sun to the planet. We can use the relationship between the sidereal period and the average distance (radius) to solve for the radius.

Using Kepler's Third Law, we can rearrange the formula: Average distance (radius) = Cube root((Sidereal period of Nursat)^2)

Now we can calculate the radius of Nursat's orbit in AU using the obtained sidereal period from step 1.

Average distance (radius) = Cube root((Sidereal period of Nursat in years)^2)

Remember to express the answer in AU (Astronomical Units).

By following these steps, you should be able to find the sidereal period of Nursat in years and the radius of Nursat's orbit in AU.