Question

Suppose a solar system has a star that is four times as massive as our Sun. If that solar system has a planet the same size as Earth orbiting at a distance of $1\mathrm{AU}$, what is the orbital period of the planet? Explain. (Hint: The calculations for this problem are so simple that you will not need a calculator.

Short answer

The orbital period is 0.5 Earth years.

Step-by-step solution

Understanding Kepler's Third Law

Kepler's Third Law states that the square of the orbital period (T^2) of a planet is directly proportional to the cube of the semi-major axis (a^3) of its orbit, and inversely proportional to the mass of the star it orbits. Mathematically, this is expressed as $$T^2\propto \frac{a^3}{M},$$ where M is the mass of the star. In our scenario, a is 1 AU, the mass of the star is 4 times that of the Sun, and we aim to find T.

Applying Kepler's Third Law

Since the distance a is 1 AU and we know the relationship, we can express T^2 as $$T^2=\frac{1^3}{4},$$ because a = 1 AU and M = 4\ M_\odot.

Solving for T

Next, calculate T by solving the equation for T^2\ = \frac{1}{4}. Taking the square root of both sides, we get $$T=\sqrt{\frac{1}{4}}=\frac{1}{2}\text{ Earth years.}$$ This means the orbital period of the planet is 0.5 Earth years.

Key concepts

Orbital Period Calculation

Orbital period calculation involves determining the time a planet takes to complete one full orbit around its star. Kepler's Third Law is fundamental for these calculations. According to this law, the orbital period (T) is related to both the semi-major axis (a) of the orbit and the mass of the star (M). For simple calculations, particularly when comparing to Earth, we don't always need complex computations or calculators.

Kepler's Third Law is given by the formula: $$T^2=\frac{a^3}{M}$$ where $a$ is in astronomical units (AU) and $M$ is the mass of the star compared to the Sun. In our specific problem, because $a=1\text{ AU}$ and the mass of the star is four times that of the Sun, $$T^2=\frac{1}{4}$$. Taking the square root of both sides, we find the orbital period $T\approx 0.5\text{ Earth years}$. Thus, the planet completes its orbit in half the time Earth does.

Massive Stars

Stars come in different sizes and masses, influencing the characteristics of their solar systems. More massive stars, like in our problem where the star is four times as massive as our Sun, affect planetary orbits significantly.

A massive star's gravity is stronger than a less massive one. This increased gravitational pull means planets around massive stars can move faster and have shorter orbital periods compared to identical planets orbiting less massive stars, even at the same distance.

• In this scenario, a planet similar to Earth, located 1 AU from a four-times massive star, will have a shorter period compared to orbiting the Sun. Thus, the impacts of a star's mass are profound on orbital dynamics and period calculations.

Planetary Orbits

Planetary orbits can vary greatly within a solar system, with each planet's path shaped by the star's mass and the distance from it. In circular or near-circular paths, the orbit is defined primarily by the semi-major axis.

The more massive the star, the more elliptical planetary orbits can become if influenced by gravitational forces or if the star has an irregular mass distribution. In simpler systems like our scenario, we consider the basic principles of orbits as determined by the semi-major axis and the mass of the star.

These basic properties help us understand why the planet in our problem orbits faster; its path around the star is under a more powerful gravitational influence due to the star's mass.

Semi-Major Axis

The semi-major axis is a crucial element in understanding planetary orbits and their characteristics. It is half of the longest diameter of an orbit and mainly describes the size of the orbit. The semi-major axis directly influences the orbital period. In the case of our solar system problem, the semi-major axis of the planet's orbit is 1 AU. This is the same as Earth's distance from the Sun, making calculations straightforward by allowing direct application of Kepler's Third Law.

When the semi-major axis is smaller, closer to the star, planets tend to move faster and complete their orbits in shorter periods. Conversely, a larger semi-major axis means longer distances from the star and longer periods. Understanding the semi-major axis helps us gauge the scale and dynamics of planetary motion, particularly in different star systems. This concept is pivotal in celestial mechanics, offering insight into how planets travel through space.