Venus's circular velocity is \(35.03 \mathrm{km} / \mathrm{s}\), and its orbital radius is \(1.082 \times 10^{8} \mathrm{km} .\) Calculate the mass of the Sun.

Orbital velocity refers to the speed at which a planet or object travels around a larger body, like a star or a planet. This velocity is important because it determines whether an object will remain in orbit or escape into space. For Venus, its orbital velocity is given as 35.03 km/s. In order to maintain this speed while orbiting the Sun, the gravitational pull of the Sun balances the centripetal force of Venus in its orbit. This concept can be understood using the formula for circular orbital velocity which is: \ \( v = \sqrt{\frac{GM}{r}} \) \ Here: \ - \(v\) is the orbital velocity\ - \(G\) is the gravitational constant\ - \(M\) is the mass of the Sun\ - \(r\) is the orbital radius of Venus\ This formula helps us determine how fast an object needs to move to stay in a stable orbit at a particular distance from the Sun.

The gravitational constant, denoted as \(G\), is a fundamental constant in physics that appears in Newton's law of universal gravitation. Its value is \(6.67430 \times 10^{-20} \mathrm{km}^3 / \mathrm{kg} / \mathrm{s}^2\) when expressed in kilometers, kilograms, and seconds. This constant helps us understand the strength of the gravitational force between two masses. It's particularly useful in astrophysics for calculating the forces between celestial bodies. In our problem to find the mass of the Sun, \(G\) is used in the formula for orbital velocity to help determine the mass based on Venus's orbital speed and distance from the Sun.

The circular orbit formula, \( v^2 = \frac{GM}{r} \), is crucial in understanding how objects move in space. It relates the orbital velocity \(v\) of an object to its distance \(r\) from the center of mass of the object it is orbiting, the mass \(M\) of the larger object, and the gravitational constant \(G\). By squaring the orbital velocity and rearranging the formula, we can solve for the mass \(M\) of the larger body: \ \( M = \frac{v^2 r}{G} \). \ This rearranged formula allows us to calculate the mass of the Sun given Venus's velocity and orbital radius. The steps to calculate it involve squaring the velocity, multiplying it by the orbital radius, and dividing by the gravitational constant.

Venus orbits the Sun with a circular orbit velocity of 35.03 km/s and an orbital radius of \(1.082 \times 10^{8} \text{km}\). These parameters are essential for calculating the mass of the Sun using the circular orbit formula. Given these values: \(v = 35.03 \text{ km/s}\), \(r = 1.082 \times 10^{8} \text{ km}\), and \(G = 6.67430 \times 10^{-20} \text{ km}^3 / \text{ kg} / \text{s}^2\), we can plug them into the mass formula to find the Sun's mass. First, calculate \(v^2\), then multiply by \(r\), and divide by \(G\). This step-by-step method provides a clear way to understand how Venus's motion and position help us learn about the Sun's mass.