Question

The intensity of light from a star (its brightness) is the power it outputs divided by the surface area over which it’s spread: I = P 4πd2 . Suppose two stars of the same apparent brightness I are also believed to be the same size. The spectrum of one star peaks at 750 nm while the other peaks at 450 nm. Use Wien’s Law and the Stefan-Boltzmann equation to estimate their relative distances from u

Answer 1

`(d_(1))/(d_(2))=0.36`

Step-by-step explanation:

1. We can find the temperature of each star using the Wien's Law. This law is given by:

`\lambda_(max)=(b)/(T)=(2.9x10^(-3)[mK])/(T[K])` (1)

So, the temperature of the first and the second star will be:

`T_(1)=3866.7 K`

`T_(2)=6444.4 K`

Now the relation between the absolute luminosity and apparent brightness is given:

`L=l\cdot 4\pi r^(2)` (2)

Where:

• L is the absolute luminosity
• l is the apparent brightness
• r is the distance from us in light years

Now, we know that two stars have the same apparent brightness, in other words l₁ = l₂

If we use the equation (2) we have:

`(L_(1))/(4\pi r_(1)^2)=(L_(2))/(4\pi r_(2)^2)`

So the relative distance between both stars will be:

`\left((d_(1))/(d_(2))\right)^(2)=(L_(1))/(L_(2))` (3)

The Boltzmann Law says,
`L=A\sigma T^(4)` (4)

• σ is the Boltzmann constant
• A is the area
• T is the temperature
• L is the absolute luminosity

Let's put (4) in (3) for each star.

`\left((d_(1))/(d_(2))\right)^(2)=(A_(1)\sigma T_(1)^(4))/(A_(2)\sigma T_(2)^(4))`

As we know both stars have the same size we can canceled out the areas.

`\left((d_(1))/(d_(2))\right)^(2)=(T_(1)^(4))/(T_(2)^(4))`

`(d_(1))/(d_(2))=\sqrt{(T_(1)^(4))/(T_(2)^(4))}`

`(d_(1))/(d_(2))=\sqrt{(T_(1)^(4))/(T_(2)^(4))}`

`(d_(1))/(d_(2))=0.36`

I hope it helps!