Gliese 581 is a star with a mass 31 % of the Sun's mass, and...

Gliese 581 is a star with a mass $31 \%$ of the Sun's mass, and it generates only about $1.3 \%$ as much light as the Sun. a. Use Newton's version of Kepler's third law to show that the nearest planet, which takes 5.37 days to orbit the star, must orbit at a distance of $0.041 \mathrm{AU}$ b. Compare the brightness of light received from Gliese 581 at the distance of $0.041 \mathrm{AU}$ to the brightness of light received at Earth from the Sun. (Use the inverse-square law-Unit 21.2.)

Gliese 581 is a star with a mass $31 \%$ of the Sun's mass, and it generates only about $1.3 \%$ as much light as the Sun.
a. Use Newton's version of Kepler's third law to show that the nearest planet, which takes 5.37 days to orbit the star, must orbit at a distance of $0.041 \mathrm{AU}$
b. Compare the brightness of light received from Gliese 581 at the distance of $0.041 \mathrm{AU}$ to the brightness of light received at Earth from the Sun. (Use the inverse-square law-Unit 21.2.)

Key Concepts

Newton’s version of Kepler’s Third Law

This law relates the orbital period of a body to the distance from the central mass by incorporating the gravitational constant and the mass of the central object. It states that the square of the orbital period (T²) is proportional to the cube of the semi-major axis (a³) divided by the sum of the masses of the two bodies. This relationship enables the calculation of an orbit’s size when the period and the mass of the central body are known.

Inverse-Square Law for Light

The inverse-square law describes how the intensity or brightness of light diminishes as the distance from the source increases. Specifically, the brightness is inversely proportional to the square of the distance from the light source. This concept is essential for comparing the light received from different stars or at different orbital distances, as it quantifies how changes in distance lead to significant variations in observed brightness.

Astronomical Units and Stellar Properties

Astronomical units (AU) are a standard measure of distance used in astrophysics, representing the average distance between the Earth and the Sun. Coupled with an understanding of stellar mass and luminosity, these units help in calculating orbital parameters and in comparing the light output from different stars, which is crucial when determining habitability or the energy received by orbiting planets.

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