1. At opposition, Nursat(is observed to be in opposition every 374 days.) is observed to subtend an angular radius of 7.1523 arcseconds. Use this and the conversion factors in our Table of Constants to find the radius of Nursat in km.
2. Nursat has a moon, Sunaj, which is observed to orbit the planet with period 0.6947 days, grazing the planet with an orbital radius only 2.606 times the radius of the planet itself. Use this information to find the mass of Nursat. Express your answer as a multiple of the mass of Earth
1. To find the radius of Nursat in kilometers, we can use the formula:
Radius (in km) = (Angular Radius in radians) * (Distance to Nursat)
First, let's convert the angular radius from arcseconds to radians. Recall that 1 arcsecond is equal to 1/3600 degrees, and there are 2*pi radians in a circle. So, we have:
Arcseconds to radians conversion: (7.1523 arcseconds) * (1/3600 degrees/arcsecond) * (pi/180 degrees) = (7.1523 * pi) / (3600 * 180) radians
Next, we need to find the distance to Nursat. Since it is observed to be in opposition every 374 days, we can assume that it is at its closest distance to Earth during opposition. Let's denote this distance as "d". The distance to Nursat can be expressed as:
Distance to Nursat = 2 * Distance to Sun
Since we don't have the actual distance to the Sun, we can assume it is about 149.6 million kilometers (the average distance between the Earth and the Sun). Therefore:
Distance to Nursat = 2 * 149.6 million kilometers = 299.2 million kilometers
Now, we can substitute the values we have into the formula:
Radius (in km) = [(7.1523 * pi) / (3600 * 180)] * 299.2 million kilometers
Calculating this expression will give you the radius of Nursat in kilometers.
2. To find the mass of Nursat, we can apply Kepler's Third Law, which states that the square of the orbital period of a moon is proportional to the cube of its semi-major axis (orbital radius).
Let's denote the period of Sunaj as "T" and the radius of Nursat as "r". According to the given information, the period of Sunaj is 0.6947 days, and its orbital radius is 2.606 times the radius of Nursat.
Using Kepler's Third Law, we have the following relationship:
(T^2 / r^3) = (T^2 / (2.606 * r)^3) = (M / M_earth), where M is the mass of Nursat and M_earth is the mass of Earth.
Since the problem asks for the mass of Nursat in terms of Earth's mass, we want to solve for the ratio (M / M_earth).
Simplifying the equation, we have:
1 / (2.606^3) = M / M_earth
Now, we can substitute the value of the ratio (1 / (2.606^3)) into the equation to find the mass of Nursat relative to Earth's mass.
Keep in mind that the exact value of the mass of Earth (M_earth) is approximately 5.972 × 10^24 kg.