Question

You have discovered a new star in the Milky Way: Your new star is red and has 3/5 the tempera-
ture of our Sun. The new star emits a total power that is 100,000 times greater than the power
emitted by our Sun.
(a) Determine the spectral type (i.e. spectral classification) of the new star.
(b) How many times bigger is the radius of the new star compared to the radius of our Sun?

Answer 1

To determine the spectral class of the star we need to know its temperature. We can determine it by using the temperature of the Sun:

$$T_{star}=\frac 3 5 T_{Sun} = \frac 3 5 * 5778K = 3466.8K$$

As the temperature of this star is below 3500K and its color is red, then it belongs to the spectral type M.

We can determine the radius of the new star by using the Stefan-Boltzmann formula:

$$L=\sigma A T^4=4\pi R^2 \sigma T^4;$$

Where L - the luminosity of the star (emission power), A - the total surface of the star, T - the temperature in Kelvin. As the star has a spherical shape, we can transform A into 4*pi*R².

The total emission power of the new star is 100,000 times bigger than the power of the Sun, so we can make an equation:

$$\frac {L_{Star}} {L_{Sun}} =100,000;$$ $$\frac {4\pi R_{star}^2 \sigma T_{Star}^4;} {4\pi R_{Sun}^2 \sigma T_{Sun}^4;} =100,000;$$ $$\frac {R_{star}^2 T_{Star}^4;} {R_{Sun}^2 T_{Sun}^4;} =100,000;$$

T_star is equal to 3/5 of the Sun's temperature, so we can transform the equation above:

$$\frac {R_{star}^2 (\frac 3 5 T_{Sun})^4;} {R_{Sun}^2 T_{Sun}^4;} =100,000;$$ $$\frac {81 R_{star}^2 T_{Sun}^4;} {625 R_{Sun}^2 T_{Sun}^4;} =100,000;$$ $$R_{star} =\sqrt {\frac {625*100,000*R_{Sun}^2} {81}};$$ $$R_{star} =\frac {25*100*\sqrt {10}*R_{Sun}} {9};$$ $$R_{star} =878.41 R_{Sun};$$

Answer: the new star belongs to spectral type M and its radius is approximately 878.41 times bigger than the radius of the Sun.