Question

The hot glowing surfaces of stars emit energy in the form of electromagnetic radiation. It is a good approximation to assume e = 1 for these surfaces. Find the radii of the following stars (assumed to be spherical):________

(a) Rigel, the bright blue star in the constellation Orion, which radiates energy at a rate of 2.7×1032W and has surface temperature 11,000 K;
(b) Procyon B (visible only using a telescope), which radiates energy at a rate of 2.1×1023W and has surface temperature 10,000 K.
(c) Compare your answers to the radius of the earth, the radius of the sun, and the distance between the earth and the sun. (Rigel is an example of a supergiant star, and Procyon B is an example of a white dwarf star.)

Answer 1

To find the radii of the stars, we can use the Stefan-Boltzmann law, which relates the power radiated by a star to its surface temperature and radius. The radius of Rigel is approximately 2.9 × 10^10 m, while the radius of Procyon B is approximately 6.0 × 10^7 m. These values can be compared to the radii of the Earth, the Sun, and the distance between the Earth and the Sun.

Step-by-step explanation:

To find the radii of the stars, we can use the Stefan-Boltzmann law, which relates the power radiated by a star to its surface temperature and radius. The law states that the power radiated per unit area by a blackbody is given by:

F = σT^4,

where F is the power radiated per unit area, σ is the Stefan-Boltzmann constant (5.67 × 10^-8 W/(m^2K^4)), and T is the temperature of the star's surface.

To find the radii of the stars, we can rearrange the equation to solve for the radius:

R = sqrt(P / (σT^4)),

where R is the radius, P is the power radiated by the star, σ is the Stefan-Boltzmann constant, and T is the temperature of the star's surface.

(a) For Rigel, which radiates energy at a rate of 2.7 × 10^32 W and has a surface temperature of 11,000 K:

R = sqrt(2.7 × 10^32 W / (5.67 × 10^-8 W/(m^2K^4) × (11,000 K)^4)) = 2.9 × 10^10 m.

(b) For Procyon B, which radiates energy at a rate of 2.1 × 10^23 W and has a surface temperature of 10,000 K:

R = sqrt(2.1 × 10^23 W / (5.67 × 10^-8 W/(m^2K^4) × (10,000 K)^4)) = 6.0 × 10^7 m.

(c) To compare the radii of the stars with the radius of the Earth, the radius of the Sun, and the distance between the Earth and the Sun:

The radius of the Earth is approximately 6.4 × 10^6 m.

The radius of the Sun is approximately 6.96 × 10^8 m.

The average distance between the Earth and the Sun (1 astronomical unit) is approximately 1.5 × 10^11 m.

Comparing the radii we calculated for Rigel and Procyon B to these values:

Rigel has a radius of 2.9 × 10^10 m, which is much larger than the radius of the Earth and the distance between the Earth and the Sun, but still smaller than the radius of the Sun.

Procyon B has a radius of 6.0 × 10^7 m, which is larger than the radius of the Earth and the distance between the Earth and the Sun, but much smaller than the radius of the Sun.