A star has a power output of 3.2E+26 W. Determine by how much its mass changes per year.

1. Break Down the Problem

To determine the change in mass of a star per year based on its power output, we can use Einstein’s mass-energy equivalence principle, which states:

$E = mc^2$

where:

• $E$ is energy,
• $m$ is mass,
• $c$ is the speed of light in a vacuum ($c \approx 3.0 \times 10^8 \, \text{m/s}$).

Given that power is the rate of energy transfer, we can express the change in mass over a year with the relationship:

$\Delta m = \frac{E}{c^2}$ where $E$ is the total energy produced by the star in one year.

2. Relevant Concepts

We can convert the power output into total energy over one year:

• Power $P = 3.2 \times 10^{26} \, \text{W}$
• Time in one year $t = 365 \times 24 \times 60 \times 60 \, \text{s}$

Using: $E = P \times t$

3. Analysis and Detail

1. Calculate the total energy output in one year:

   $t = 365 \times 24 \times 60 \times 60 = 31,536,000 \, \text{s}$

   $E = P \times t = 3.2 \times 10^{26} \, \text{W} \times 31,536,000 \, \text{s}$

   Calculating $E$: $E = 3.2 \times 10^{26} \times 31,536,000 \approx 1.007 \times 10^{34} \, \text{J}$

2. Determine the change in mass using $E = mc^2$:

   $\Delta m = \frac{E}{c^2} = \frac{1.007 \times 10^{34}}{(3.0 \times 10^8)^2}$

   Calculating $c^2$: $c^2 = (3.0 \times 10^8)^2 = 9.0 \times 10^{16} \, \text{m}^2/\text{s}^2$

   Calculate $\Delta m$: $\Delta m = \frac{1.007 \times 10^{34}}{9.0 \times 10^{16}} \approx 1.119 \times 10^{17} \, \text{kg}$

4. Verify and Summarize

The calculations have been verified and the steps have been followed accurately.

Final Answer

The mass of the star changes by approximately $1.12 \times 10^{17} \, \text{kg}$ per year.