Go to the online "Extrasolar Planets Encyclopedia" (http:// exoplanet.eu/catalog a. Find a planet with an orbital period similar to that of Earth. What is the semimajor axis of its orbit? If it is very different from \(1 \mathrm{AU}\), then the mass of the star is different from that of the Sun. Click on the star name in the first column to see the star's mass. What is the orbital eccentricity? b. Click on "Planet" to sort by name, and select a star with multiple planets. Verify that Kepler's third law applies by showing that the value of \(P^{2} / A^{3}\) is about the same for each of the planets of this star. How eccentric are the orbits of the multiple planets?

The orbital period is the time a planet takes to complete one full orbit around its star. For Earth, this is about 365.25 days, which is why we have a leap year every four years. When studying extrasolar planets, one of the first tasks is often to look for planets with orbital periods similar to Earth's. This helps in identifying potential planets that could have Earth-like conditions. Orbital periods can vary greatly depending on the distance from the star and the star's mass. For example, a planet orbiting closer to its star will have a shorter orbital period, while a planet further away will have a longer orbital period.

The semimajor axis is one of the most crucial elements of an orbit, representing half of the longest diameter of an elliptical orbit. For Earth, the semimajor axis is approximately 1 Astronomical Unit (AU), which is the average distance from Earth to the Sun, roughly 93 million miles or 150 million kilometers. In the context of extrasolar planets, the semimajor axis provides insight into the potential habitability of a planet. This is because the distance from the star affects the planet's surface temperature and climate. When identifying planets with similar orbital periods to Earth, we also check if their semimajor axis is near 1 AU. Significant deviations suggest the star's mass is different from the Sun.

Kepler's third law states that the square of a planet’s orbital period (P) is directly proportional to the cube of the semimajor axis (A) of its orbit, expressed as: \( P^2 = A^3 \). This relationship is fundamental in astrophysics for calculating various orbital parameters. In the search for extrasolar planets, verifying Kepler's third law involves calculating \( \frac{P^2}{A^3} \). For planets around the same star, this ratio should be approximately constant. This law helps confirm the consistency and accuracy of the observed data in the exoplanet catalog.

Orbital eccentricity measures how much an orbit deviates from being circular. An eccentricity (e) of 0 describes a perfect circle, whereas values closer to 1 indicate more elongated orbits. For Earth, the eccentricity is very close to 0 (about 0.0167), making its orbit nearly circular. When examining extrasolar planets, identifying their orbital eccentricity helps in understanding the dynamics of their orbits and potential climate conditions. Higher eccentricity can result in more extensive seasonal variations, affecting the planet's habitability.

The Exoplanet Catalog is a database that contains a wealth of information about known extrasolar planets. This includes data on their orbital periods, semimajor axes, eccentricities, and other characteristics. To utilize the catalog, users can search for specific planets or sort information, for instance by a planet's name or period. This catalog is essential for researchers and students to explore and analyze the properties of different exoplanets. Checking the catalog for patterns, like applying Kepler's third law to multiple planets within the same star system, is a practical exercise in understanding planetary dynamics and validating observational data.