Question

An O star has a temperature of 20,000 K and a radius 10 x that of the Sun. How much more luminous is the O star than the Sun? At what distance would the O star have to be placed to appear to be the same brightness as the Sun. Express your answer in AU and light-years.

Answer 1

Luminosity is the total amount of energy emitted by a star per unit time. So this is basically the power output of the star.

So for a star with Temperature T=20000°K and radius R=10×695510Km=6.95×10⁹ m, the power output is given by the Stefan-Boltzmann law which is expressed below

Where is the Stefan-Boltzmann constant=5.7×10^-8 Wm^{​​​​​-2} K^{​​​​​​-4}

A= surface area of star=4πR² 4× 3.14 ×(6.95×10⁹ )² =6.06×10²⁰ m^{​​​​​​2}

So the luminosity=5.52×10³⁰ Watt.

Now we determine the power per unit area of this star so that we can compare it with the power per unit area of sun to calculate at what distance the star should be to appear as bright as the sun.

The power per unit area of the star=(5.52×10³⁰ )÷(6..06×10²⁰ )..............(power/area of star)

Therefore power per unit area of the star=9.1×10⁹ W/m^{​​​​​​2}

We know the power per unit area (solar insolation) received at earth is 1370Watt/m^{​​​​​​2} .

This is a measure of the brightness of the sun.

now we need to determine how far the star has to be from earth to appear as bright as sun(i e.1370W/m^2)

The solar insolation is a function of inverse of area of the surface on which the insolation is measured. That means it is a function of inverse square of distance of that surface from the sun. By same logic,the farther we go from the star, the dimmer it appears. So we can write the following relation

Where B_{​​​​​​sun} is the brightness of sun, B_{​​​​​​star} is the brightness of the star,r_{​​​​​​1} and r_{​​​​​2} are the distances from earth to the sun and star respectively

We already know

, and

So now let's just plug in the values n

Inlightyears, this distance is about 0.0405 lightyears.

So this star needs to at a distance of 2577.27AU or 0.0405 lightyears to appear as bright as sun.