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The orbital period is the time it takes for a planet to make one complete orbit around a star. According to Newton's Version of Kepler's Third Law, the orbital period is deeply connected with the mass of the star and the distance of the planet from this star. We generally calculate the orbital period using the formula: \[ T^2 = \frac{4\pi^2}{G(M + m)}a^3 \] where

• \(T\) is the orbital period,
• \(G\) is the gravitational constant,
• \(M\) and \(m\) are the masses of the star and the planet respectively,
• \(a\) is the semi-major axis of the planet's orbit.

For small planets like Earth, the mass \(m\) can be ignored as it is insignificant compared to the star. Thus, the orbital period is directly influenced by the star's mass and the semi-major axis.

The mass of the star is a crucial factor in determining the gravitational influence it has on surrounding planets. This influence dictates how fast or slow a planet completes its orbit. In our problem, the star is said to be four times as massive as our Sun, denoted as \(4M_\odot\), where \(M_\odot\) refers to the mass of the Sun. When the star's mass is large, the gravitational attraction it exerts is stronger, and planetary orbits can be faster. That's why, in the scenario given, the star being four times more massive directly affects the equation used to compute the orbital period, reducing it because the star's gravitational pull is stronger, allowing for a quicker trajectory.

The semi-major axis is a measure of the size of an elliptical orbit, represented by the longest diameter stretching across the ellipse. In celestial mechanics, it plays a vital role in determining the planet's orbital period when paired with the star's mass. In this exercise, the semi-major axis \(a\) is given as \(1\, \text{AU}\) (Astronomical Unit), the average distance between Earth and the Sun. This means the calculation assumes the planet is at a distance from its star similar to Earth's distance from our Sun. With this known value, the semi-major axis makes the problem easier to tackle since \(a = 1\) simplifies the computation in our main equation without altering the dimensionality of the result. Having the semi-major axis fixed highlights the relationship between orbital speed and star's mass, showcasing fundamental aspects of Newton's extension to Kepler's Law.

Kepler's Third Law is a cornerstone of classical astronomy and planetary motion, known for stating the relationship between the time a planet takes to orbit its star (orbital period) and its average distance from the star (semi-major axis). Originally expressed as: \[ T^2 \propto a^3 \] Newton's contribution was to refine this understanding by including the mass of the star into the relation, making it applicable in broader contexts. The Newtonian version transforms this into an equation suitable for different celestial planes by considering stellar masses, ensuring its accuracy in predicting real world planetary motions in diverse solar systems. In our scenario, Newton's expanded principle clearly illustrates how varying stellar masses impact planetary orbits, emphasizing the law's applicability beyond our solar system. It showcases the elegance of celestial dynamics, combining simplicity with profound insight.