The Triangulum Galaxy is a spiral galaxy located about 2.7 million light-years away from Earth. Its core mass is estimated to be about 1 x 10³⁸ kg. Assuming that the average star in the Triangulum Galaxy has a mass similar to the Sun's mass, which is 1.989 x 10³⁰ kg, estimate the number of stars in the galactic core.

An asteroid with a mass 'm', sharing the orbit with a planet 'X', revolves around the sun. Calculate the orbital radius and velocity of the asteroid given that it has a period of 4.0 Earth years. The Gravitational constant is 6.67 × 10^-11 Nm²/kg².

Astronomers spot a previously unseen planetoid with a diameter on the order of hundreds of meters. It is orbiting the Sun in the same plane as the other planets in the solar system. Given that its period is 560d, determine its mean distance from the Sun. (Assume that the mean distance between the Sun and Earth is 1.50 × 10⁸ km)

Scientists are speculating that there may be another yet undiscovered planet in the solar system called "Planet Nine". By their estimate, it should be at an average distance of 9.0 × 10¹⁰ km from the Sun. Determine the length of the planet's year given that the average distance between the Earth and the Sun is 1.5 × 10⁸ km.

Astronomers discovered a potentially habitable planet with a mass equal to five times that of Earth. It revolves around a star of mass similar to that of our Sun. If its orbital period is 5.0 % more than Earth's, determine the average distance it is from its star. (Assume that the mean distance between the Sun and Earth is 1.50 × 10⁸ km.)

A celestial body located beyond Neptune's orbit takes 1200 years to complete one revolution around the Sun. Given that its closest distance from the Sun is 3 AU, and its furthest distance from the Sun is 223 AU, determine the ratio of its speed at the perihelion (closest distance from the Sun) to the speed at the aphelion (the furthest distance from the Sun).

[1 AU = The mean distance between the Earth and the Sun.]

Astronomers have observed an asteroid orbiting the Sun in an ellipse with an orbital period of 3200 years. It is estimated to be 1.5 astronomical units away from the Sun at its closest approach. Determine its furthest distance from the Sun.

[Note that, 1 AU (astronomical units) = the average distance between the Earth and the Sun.]

A satellite is orbiting a strange space rock at a distance of 3.5 billion km from Earth. The space rock has a mass of $1.5\times 10^{14}kg$ and can be modeled as a sphere with a radius of $2.5\times 10^{3}m$. The satellite orbits at an altitude of 4.0 km above the surface of the space rock. Find the orbital period of the satellite about the space rock.