Question

The luminosity of the Rigel star in the Orion constellation is $17,000$ times that of the sun. The surface temperature of the sun is $6000K$ . Calculate the temperature of the star.

Answer 1

Hint: The luminosity of a star depends on its surface temperature. According to Stefan’s law, the luminosity (the total radiant heat power) of a star is proportional to the fourth power of its temperature. Thus, this proportionality can be used to calculate the temperature of Rigel.

Complete step by step solution:
The luminosity of a body is defined as the total amount of electromagnetic radiation that is emitted by a black body per unit of time. Here the stars Rigel and Sun are assumed to be blackbody and thus their luminosities can be calculated by Stefan’s law (also called the Stefan-Boltzmann law).
It is given in the question that the surface temperature of the sun is, $T_S=6000K$
Let the temperature of Rigel star be $T_R$ .
Assuming that the luminosity of the sun is $E$ , we can write the luminosity of Rigel as $17000E$ .
From the Stefan-Boltzmann law, we know that the power emitted by a black body per unit area is equal to the product of the Stefan-Boltzmann constant$\left(\sigma \right)$ and the fourth power of its thermodynamic temperature.
Thus, it can be written that-
$E=\sigma T^4$
For the sun, we can write-
$E=\sigma {T_S}^4$
And for the Rigel, we can write-
$17000E=\sigma {T_R}^4$
The ratio of both luminosities can be written as-
${\displaystyle \frac{E}{17000E}}={\displaystyle \frac{\sigma T_S^4}{\sigma T_R^4}}$
Cancelling $E$ from numerator and denominator we get,
${\displaystyle \frac{1}{17000}}={\displaystyle \frac{{T_S}^4}{{T_R}^4}}$
$\Rightarrow {\displaystyle \frac{1}{17000}}={\left({\displaystyle \frac{T_S}{T_R}}\right)}^4$
Rearranging the equation, we can write-
${\displaystyle \frac{T_S}{T_R}}={\left({\displaystyle \frac{1}{17000}}\right)}^{{\displaystyle \frac{1}{4}}}$
$\Rightarrow {\displaystyle \frac{T_S}{T_R}}={\displaystyle \frac{1}{11.4185}}$
Upon cross-multiplying and substituting the value of $T_s$ we get,
$T_R=11.4185\times 6000$
$\Rightarrow T_R=68511.5K$

Therefore the temperature of the Rigel star is approximately $68512$Kelvins.

Note: If a calculator is not allowed, the fourth root of the luminosity can be found by using a log table, where we take the log with an appropriate base ( say $10$ ) on both sides. Then the exponent can be removed and the root can be calculated. Since the value $11.4185$ is not a natural number therefore it will be very tough if the root is calculated via the LCM or other method.