Questions: Part 3. The luminosity and mass for main sequence stars are connected: L ~ M^3.5 Question #6. (2 points) A star with a mass of 2 M⊙ has a luminosity of 11.3136 L⊙ while a star with luminosity of 3,160 L⊙ has an approximate mass of M⊙.

Part 3. The luminosity and mass for main sequence stars are connected:
L ~ M^3.5

Question #6. (2 points) A star with a mass of 2 M⊙ has a luminosity of 11.3136 L⊙ while a star with luminosity of 3,160 L⊙ has an approximate mass of M⊙.

Transcript text: Part 3. The luminosity and mass for main sequence stars are connected: \[ L \sim M^{3.5} \] Question #6. (2 points) A star with a mass of $2 \mathrm{M} \odot$ has a luminosity of 11.3136 L $\odot$ while a star with luminosity of $3,160 \mathrm{~L} \odot$ has an approximate mass of $\mathrm{M} \odot$.

Solution

Solution Steps

Step 1: Understanding the Luminosity-Mass Relationship

The given relationship between luminosity $L$ and mass $M$ for main sequence stars is: \[ L \sim M^{3.5} \]

Step 2: Solving for Mass Given Luminosity

We need to find the mass of a star given its luminosity. The relationship can be written as: \[ L = k M^{3.5} \] where $k$ is a constant. For a star with a mass of $2 \, M_\odot$ and a luminosity of $11.3136 \, L_\odot$, we can determine $k$.

Step 3: Determining the Constant $k$

Using the given values: \[ 11.3136 = k (2)^{3.5} \] \[ 11.3136 = k \cdot 11.3137 \] \[ k \approx 1 \]

Step 4: Using the Constant to Find the Mass

Now, we use the constant $k \approx 1$ to find the mass of a star with a luminosity of $3,160 \, L_\odot$: \[ 3,160 = 1 \cdot M^{3.5} \] \[ M^{3.5} = 3,160 \] \[ M = (3,160)^{1/3.5} \]

Step 5: Calculating the Mass

\[ M \approx (3,160)^{0.2857} \approx 10 \]