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Orbital Period

Nicolaus Copernicus, a renowned astronomer from the 16th century, made a groundbreaking discovery that revolutionized our understanding of the solar system. He proposed a heliocentric model, which placed the Sun at the center of the system with the planets, including Earth, orbiting around it.

One of Copernicus' significant contributions was his recognition of the relationship between the synodic period (Ps) and the orbital period of planets (T) . The synodic period refers to the time it takes for a planet to return to the same alignment with the Sun and Earth, while the orbital period represents the time it takes for a planet to complete one full orbit around the Sun.

Copernicus observed that the synodic period of a planet, such as Mars or Venus, is longer than its orbital period. This realization led him to infer that the synodic period is influenced by the relative positions and motions of Earth and the observed planet.

The equation can be expressed as follow:

$$T = {1 \over {1 \over Pt} - {1 \over Ps}}.$$

Where Pt is the orbital period of the Earth, i.e. 1 year, or 365.25 days

Taking Jupiter as an example, its synodic period is approximately 399 days (Ps), we can calculate the orbital period of Jupiter:

$$T = {1 \over {1 \over 365.25} - {1 \over 399}}.$$

Using the values for Jupiter and Earth, we find that Jupiter takes approximately 4,318 days (around 11.8 years) to complete one revolution around the Sun.

Distance between the object and the Sun

In the 17th century, the German astronomer Kepler described his eponymous celestial mechanical laws, which can be summarized as follows:

1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3. The square of a planet's orbital period (T) is proportional to the cube of the length of the semi-major axis of its orbit (r). $$ {r^3 \over T^2} = constant.$$

Using Kepler's third law and knowing the orbital periods of Jupiter (11.8 years) and Earth (1 year), along with the average distance between the Sun and Earth (150,000,000 km), we can calculate the average distance between the Sun and Jupiter.

$$a = {150 000 000 * \sqrt[3]{11.8^2/1^2}}.$$

Using the orbital period of Jupiter (11.8 years), the average distance between the Sun and Jupiter is approximately 777,000,000 km.

Based on the equation provided above, can you calculate the mean distance between Venus and the Sun, given that Venus has an orbital period of approximately 225 days?

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