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Roughly \(90 \%\) of all stars are main-sequence stars. Exceptions include supergiants, giants, and dwarfs. For main-sequence stars (including the sun) there is an important relationship called the mass-luminosity relation between the relative luminosity \({ }^{25} L\) and the mass \(M\) in terms of solar masses. Relative masses and luminosities of several main-sequence stars are reported in the accompanying table. a. Find a power model for the data in this table. (Round the power and the coefficient to one decimal place.) The function you find is known to astronomers as the mass-luminosity relation. b. Kruger 60 is a main-sequence star that is about \(0.11\) solar mass. Use functional notation to express the relative luminosity of Kruger 60 , and then calculate that value. $$ \begin{array}{|l|c|c|} \hline \text { Star } & \begin{array}{c} \text { Solar mass } \\ M \end{array} & \begin{array}{c} \text { Luminosity } \\ L \end{array} \\ \hline \text { Spica } & 7.3 & 1050 \\ \hline \text { Vega } & 3.1 & 55 \\ \hline \text { Altair } & 1 & 1.1 \\ \hline \text { The Sun } & 1 & 1 \\ \hline 61 \text { Cygni A } & 0.17 & 0.002 \\ \hline \end{array} $$ c. Wolf 359 has a relative luminosity of about \(0.0001\). How massive is Wolf 359 ? d. If one star is 3 times as massive as another, how do their luminosities compare?

Step by step solution

01

Understand the Problem

We need to find a power model that describes the relationship between mass and luminosity for main-sequence stars. This model will help determine the relative luminosity of Kruger 60 based on its mass, the mass of Wolf 359 based on its luminosity, and how luminosity changes with mass.

02

Set Up the Power Model

A power model can be written in the form \( L = k \cdot M^n \), where \( L \) is the luminosity, \( M \) is the mass, \( k \) is a constant coefficient, and \( n \) is the power to be determined. We will use the data from the table to find the appropriate values of \( k \) and \( n \).

03

Use Logarithms to Linearize the Data

Transform the data using logarithms: \( \log(L) = \log(k \cdot M^n) = n \cdot \log(M) + \log(k) \). This looks like a linear equation \( y = a \cdot x + b \), where \( y = \log(L) \), \( x = \log(M) \), \( a = n \), and \( b = \log(k) \).

04

Compute the Logs

Calculate \( \log(M) \) and \( \log(L) \) for each star's data: - **Spica**: \( \log(M) = \log(7.3) \approx 0.863 \), \( \log(L) = \log(1050) \approx 3.021 \) - **Vega**: \( \log(M) = \log(3.1) \approx 0.491 \), \( \log(L) = \log(55) \approx 1.740 \) - **Altair**: \( \log(M) = \log(1) = 0 \), \( \log(L) = \log(1.1) \approx 0.041 \) - **The Sun**: \( \log(M) = 0 \), \( \log(L) = 0 \) - **61 Cygni A**: \( \log(M) = \log(0.17) \approx -0.769 \), \( \log(L) = \log(0.002) \approx -2.699 \)

05

Find the Line of Best Fit

Use these logarithms in a linear regression to determine the line of best fit, identifying \( n \) and \( \log(k) \). Solving yields approximately \( n \approx 3.5 \) and \( \log(k) \approx 0 \) (as \( k \approx 1 \)). Thus, the relation is \( L = 1 \cdot M^{3.5} \).

06

Calculate Luminosity for Kruger 60

Using the model \( L = M^{3.5} \), substitute \( M = 0.11 \): \[ L = (0.11)^{3.5} \approx 0.000464 \].

07

Determine Wolf 359's Mass

Given \( L = 0.0001 \), solve for \( M \) using the power relation: \[ 0.0001 = M^{3.5} \Rightarrow M = (0.0001)^{1/3.5} \approx 0.09 \].

08

Compare Luminosities for Different Masses

If one star is 3 times as massive as another, substitute into the power model \( L = M^{3.5} \): \( L_1 = (3M_2)^{3.5} = 3^{3.5} \cdot M_2^{3.5} \), giving \( L_1 \approx 46.77 \cdot L_2 \). So, the more massive star is much more luminous.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

main-sequence stars

Main-sequence stars are a classification of stars within the Hertzsprung-Russell diagram, which is a graphical tool that astronomers use to study the life stages of stars. These stars form a continuous band, known as the main sequence, that stretches diagonally across the diagram from the upper left to the lower right. They are powered by nuclear fusion, primarily converting hydrogen into helium, and this process generates the energy that stars emit as light and heat.

Some familiar examples of main-sequence stars include our very own Sun. Such stars can vary greatly in size, temperature, and luminosity, but they share the common trait of being in a stable phase of hydrogen burning in their cores. This stability can last for millions to billions of years depending on the star's mass.

Main-sequence stars cover a range of spectral types, from hot and blue at the higher mass end to cooler and red at the lower mass end. This class includes stars like supergiants and red dwarfs, with red dwarfs being the most numerous type of star in the universe.

power model

A power model is a type of mathematical relationship used to describe how one variable changes in proportion to another through the use of powers or exponents. When applied to stellar properties, power models help astronomers understand relationships such as the one between a star's mass and its luminosity.

In the case of main-sequence stars, the mass-luminosity relation can be described with the power model equation: \[ L = k \cdot M^n \]where:

• \( L \) is the luminosity of the star.
• \( M \) is the mass of the star in solar masses.
• \( k \) is a constant coefficient.
• \( n \) is the power that indicates how the two variables relate through their logarithmic transformation.

This mathematical representation allows astronomers to predict a star's brightness based on its mass, and vice versa, which is crucial for understanding star behaviors and characteristics.

astronomical calculations

Astronomical calculations are essential for deriving meaningful information about stars from observational data. These calculations often use models, such as the power model, to articulate complex interactions and phenomena in a digestible form.

For example, to determine the luminosity of a star like Kruger 60, astronomers would input its mass into the derived mass-luminosity equation, \[ L = M^{3.5} \] and solve for \( L \). Calculating a star's mass or luminosity can involve taking the logarithms of these quantities to linearize data. Once in a linear form, it can be easier to establish the relationship between variables using techniques like linear regression.

Beyond luminosity and mass, astronomical calculations extend to distances, temperature estimations, and other properties critical to our understanding of celestial mechanics and the life cycles of stars.

linear regression

Linear regression is a statistical method used to model the relationship between two variables by fitting a linear equation to observed data. In the context of astronomy, it is particularly useful when dealing with logarithmic transformations of data, where a nonlinear relationship can be approximated as linear.

For instance, by taking logarithms of the mass and luminosity values of stars, an astronomer can apply a linear regression model to find the best-fitting line through this transformed data. The equation of this line helps deduce the quantitative relationship between star mass and luminosity, revealing key values such as the exponent \( n \) in the power model.

The accuracy and reliability of linear regression make it a valuable tool in the analysis, allowing for predictions and deeper insights into how stars function and evolve. It provides a mathematical foundation for much of the practical work done in astrophysics to study stellar phenomena.