Question

Be sure to show all calculations clearly and state your final answers in complete sentences. Imagine another solar system, with a star of the same mass as the Sun. Suppose there is a planet in that solar system with a mass twice that of Earth orbiting at a distance of 1 AU from the star. What is the orbital period of this planet? Explain. (Hint: The calculations for this problem are so simple that you will not need a calculator.)

Short answer

The planet's orbital period is 1 year because its orbit is identical to Earth's orbit around the Sun.

Step-by-step solution

Understand Kepler's Third Law

Kepler's Third Law states that the square of the orbital period \( T \) of a planet is directly proportional to the cube of the semi-major axis \( a \) of its orbit around the star. Mathematically, this is written as \( T^2 \propto a^3 \) when the mass of the star remains constant.

Identify Given Values

We know that the mass of the star is the same as our Sun, the semi-major axis \( a \) is 1 AU (which is the Earth-Sun distance), and the planet's mass is not part of the equation since Kepler's Third Law applies to the planet's orbit around the star and not the mass of the planet.

Apply Kepler's Third Law

Given that \( a = 1 \text{ AU} \), we can substitute this into Kepler's Third Law. Since \( T^2 \propto 1^3 \) and using Earth as the baseline (where \( T = 1 \) year for \( a = 1 \text{ AU} \)), we directly find \( T = 1 \text{ year} \).

Interpret the Result

Since the semi-major axis is the same for this planet as it is for Earth, and given that the mass of the star is the same as the Sun, the planet will have an orbital period of 1 year, identical to Earth's orbital period around the Sun.

Key concepts

Orbital Period

The orbital period refers to the time taken for a planet to complete one full revolution around its star. In our example, a planet orbits a star with the same mass as the sun. According to Kepler's Third Law, a planet's orbital period depends solely on the semi-major axis of its orbit when the star's mass remains unchanged.

• This period is expressed in years, assuming the reference distance is in astronomical units (AU), like Earth's orbit around the Sun.
• In this problem's scenario, regardless of the planet's own mass, its orbital period at 1 AU remains unchanged, as only the central star's mass plays a role in determining this period.

Thus, using Kepler's Third Law, we establish that the planet's orbital period is 1 year when it orbits its star at a 1 AU distance. This mirrors Earth's orbit period around the Sun.

Semi-Major Axis

The semi-major axis is the average distance from a planet to the star it orbits, serving as the longest radius of an elliptical orbit. In simpler terms, it helps determine the size of the orbit. Kepler's Third Law highlights that the orbital period is closely linked to the semi-major axis.

• When the semi-major axis increases, the orbital period does too, following the relationship: if the semi-major axis is doubled, the period increases by a factor of about 2.828 (since \(\sqrt{2^3} = 2.828\)).
• However, in the given exercise, the semi-major axis is 1 AU, aligning it with Earth's orbital parameters around the Sun.

In this specific scenario, with a semi-major axis of 1 AU, the planetary orbit mimics Earth's path, leading to an identical 1-year orbital period.

Gravitational Constant

The gravitational constant, denoted as \(G\), is a crucial value in physics that affects the gravitational force between objects. Even though \(G\) is implicit in our example, understanding its role is essential for broader celestial mechanics.

• The constant \(G\) contributes to the formulation of the force between any two masses, helping us understand orbits and the effect of gravity well beyond our solar system.
• In Kepler's Third Law, it combines with the masses involved (the star and planet) to determine the gravitational pull that dictates the orbital motion.

In our current scenario, though the mass of the new planet is twice that of Earth, \(G\) remains unchanged. This consistency ensures that changes in the planet's mass do not alter the orbital time period, as ruled by Kepler's Law, since the only variables that matter here are the semi-major axis and the star's mass.