The planet discovered orbiting the star 70 Virginis (" \(70 \mathrm{Vir}\) " in Figure 8-17), 59 light-years from Earth, moves in an orbit with semimajor axis \(0.48 \mathrm{AU}\) and eccentricity \(0.40\). The period of the orbit is \(116.7\) days. Find the mass of 70 Virginis. Compare your answer with the mass of the Sun. (Hint: The planet has far less mass than the star.)

The celestial field of astronomy involves studying objects and phenomena beyond Earth's atmosphere. It covers the examination of stars, planets, galaxies, and the broader cosmos. The science relies on various principles to comprehend the workings of the universe, one of which is Kepler's laws of planetary motion. These laws enable astronomers to predict celestial bodies' behavior, such as their orbits around stars.

Using these laws, particularly Kepler's Third Law, astronomers can deduce critical information about planets and stars, such as their mass and the size of their orbits. This is not only fundamental for understanding individual systems but also for appreciating the cosmic scale and organization of the universe. For instance, by calculating the orbital characteristics of a newly discovered planet, astronomers can gain insights into its potential climate and whether it lies within the habitable zone of its star.

The orbital period of a celestial body is the time it takes to complete one full orbit around another body. In the context of planets orbiting a star, it specifies the duration of a planet's year. For example, Earth's orbital period around the Sun is about 365.25 days, which constitutes one Earth year.

This period can vary significantly depending on the distance from the object it is orbiting and the masses involved. Kepler's Third Law provides a direct relationship between the orbital period of a planet and the semi-major axis of its orbit around a star, suggesting that a longer orbital period corresponds to a larger orbit. To solve astronomy problems, scientists often convert the orbital period into years for easier comparison and calculation, as they did with the 116.7-day period of the planet orbiting 70 Virginis.

Semi-major axis refers to half the longest diameter of an elliptical orbit. In simpler terms, it's the average distance between the orbiting body and the focal point of the orbit—the star around which it revolves. In the elliptical orbits typical of planets, the semi-major axis serves a vital role in calculations as it represents a planet's average orbital distance.

Kepler's Third Law proposes a relationship between the semi-major axis and the orbital period of a planet. The semi-major axis is a crucial variable in determining other properties of the orbit, such as its size and shape. For instance, the planet orbiting 70 Virginis, with a semi-major axis of 0.48 AU, is a critical value for computing the star's mass using Kepler's formula.

The eccentricity of an orbit is a measure of its deviation from being circular. It quantifies how elongated an orbit is, with an eccentricity of 0 corresponding to a perfect circle and values approaching 1 indicating increasingly stretched ellipses. Eccentricity is an important aspect of understanding a planet's orbit, as it influences the planet's distance from its star during various points in its orbit.

Orbits with high eccentricity result in a planet experiencing significant changes in temperature and radiation as it moves closer to and further away from the star. The eccentricity value provided in the exercise (0.40) implies that the orbit of the planet around 70 Virginis is quite elliptical, which may have implications for the conditions on the planet's surface.

Solar mass units are a convenient way of expressing mass in astronomy when dealing with masses that are on the order of, or related to, the mass of our Sun. One solar mass unit is equivalent to the mass of the Sun (approximately 1.988 x 10^30 kilograms).

Using solar mass units simplifies calculations and comparisons between different celestial bodies. When Kepler's Third Law is used with a constant that sets the mass in solar mass units, as in the exercise, we get the mass of a star directly in terms of how many times it is more massive than the Sun. This makes it exceptionally straightforward for astronomers to compare the mass of 70 Virginis to our Sun after calculating it from the orbital period and semi-major axis of the planet.