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 $6000\;K$ . Calculate the temperature of the star.
Answer
Verified
116.1k+ views
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 $17000\;E$ .
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-
$\dfrac{E}{{17000E}} = \dfrac{{\sigma T_S^4}}{{\sigma T_R^4}}$
Cancelling $E$ from numerator and denominator we get,
$\dfrac{1}{{17000}} = \dfrac{{{T_S}^4}}{{{T_R}^4}}$
$ \Rightarrow \dfrac{1}{{17000}} = {\left( {\dfrac{{{T_S}}}{{{T_R}}}} \right)^4}$
Rearranging the equation, we can write-
$\dfrac{{{T_S}}}{{{T_R}}} = {\left( {\dfrac{1}{{17000}}} \right)^{\dfrac{1}{4}}}$
$ \Rightarrow \dfrac{{{T_S}}}{{{T_R}}} = \dfrac{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.
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 $17000\;E$ .
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-
$\dfrac{E}{{17000E}} = \dfrac{{\sigma T_S^4}}{{\sigma T_R^4}}$
Cancelling $E$ from numerator and denominator we get,
$\dfrac{1}{{17000}} = \dfrac{{{T_S}^4}}{{{T_R}^4}}$
$ \Rightarrow \dfrac{1}{{17000}} = {\left( {\dfrac{{{T_S}}}{{{T_R}}}} \right)^4}$
Rearranging the equation, we can write-
$\dfrac{{{T_S}}}{{{T_R}}} = {\left( {\dfrac{1}{{17000}}} \right)^{\dfrac{1}{4}}}$
$ \Rightarrow \dfrac{{{T_S}}}{{{T_R}}} = \dfrac{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.
Recently Updated Pages
Uniform Acceleration - Definition, Equation, Examples, and FAQs
Young's Double Slit Experiment Step by Step Derivation
How to find Oxidation Number - Important Concepts for JEE
How Electromagnetic Waves are Formed - Important Concepts for JEE
Electrical Resistance - Important Concepts and Tips for JEE
Average Atomic Mass - Important Concepts and Tips for JEE
Trending doubts
JEE Main 2025: Application Form (Out), Exam Dates (Released), Eligibility & More
JEE Main Chemistry Question Paper with Answer Keys and Solutions
Learn About Angle Of Deviation In Prism: JEE Main Physics 2025
JEE Main 2025: Conversion of Galvanometer Into Ammeter And Voltmeter in Physics
JEE Main Login 2045: Step-by-Step Instructions and Details
Physics Average Value and RMS Value JEE Main 2025
Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs
Dual Nature of Radiation and Matter Class 12 Notes CBSE Physics Chapter 11 (Free PDF Download)
Inductive Effect and Acidic Strength - Types, Relation and Applications for JEE
Degree of Dissociation and Its Formula With Solved Example for JEE
Diffraction of Light - Young’s Single Slit Experiment
JEE Main 2025: Derivation of Equation of Trajectory in Physics