Question

Estimates show that the total energy output of the sun is \(5 \times 10^{26} \mathrm{~J} / \mathrm{s}\). What is the corresponding mass loss in \(\mathrm{kg} / \mathrm{s}\) of the sun?

Short answer

The sun loses approximately \(5.55 \times 10^9 \, \mathrm{kg/s}\) of mass.

Step-by-step solution

Understanding the Relationship between Energy and Mass

According to Einstein's mass-energy equivalence principle, energy (E) and mass (m) are related by the equation: \( E = mc^2 \), where c is the speed of light in a vacuum, approximately \( 3 \times 10^8 \, \mathrm{m/s} \).

Rearranging the Formula

To find the mass loss \( m \) from the energy \( E = 5 \times 10^{26} \, \mathrm{J/s} \), we rearrange Einstein's equation to:\[ m = \frac{E}{c^2} \]

Substitute the Known Values

Substitute the given energy output and the speed of light into the equation:\[ m = \frac{5 \times 10^{26} \, \mathrm{J/s}}{(3 \times 10^8 \, \mathrm{m/s})^2} \]

Calculate the Mass Loss

Calculate the mass loss by performing the division:\[ m = \frac{5 \times 10^{26}}{9 \times 10^{16}} = \frac{5}{9} \times 10^{10} = 0.555 \times 10^{10} \, \mathrm{kg/s} \]This simplifies to:\[ 5.55 \times 10^9 \, \mathrm{kg/s} \]

Key concepts

Einstein's equation

Albert Einstein's groundbreaking mass-energy equivalence principle is encapsulated in the famous equation, \( E = mc^2 \). This elegant formula demonstrates how energy (\(E\)) and mass (\(m\)) are fundamentally two forms of the same thing. In simpler terms, it tells us that mass can be converted into energy and vice versa. The constant \(c\) represents the speed of light, and squaring it indicates a profound implication that a small amount of mass can be converted into a large amount of energy. This concept is central to nuclear reactions and explains phenomena such as how stars, including our sun, produce energy.

Speed of light

The speed of light, denoted as \(c\), is about \(3 \times 10^8 \, \text{m/s} \). This is not only a constant in Einstein's famous equation but also a fundamental constant of nature. It indicates how fast light travels in a vacuum, and importantly, it is the maximum speed at which all energy, matter, and information can travel in the universe. The large value of \(c^2\) in \(E = mc^2\) illustrates why mass-energy conversion results in the release of immense energy, even when only a small amount of mass is converted.

Sun's energy output

The sun emits an enormous amount of energy, calculated to be \(5 \times 10^{26} \, \text{J/s} \). This signifies the energy that the sun loses every second in the form of light and other radiation. The source of the sun's energy is primarily nuclear fusion occurring in its core. In these reactions, hydrogen nuclei combine to form helium, releasing energy due to the mass-energy equivalence principle. This energy sustains life on Earth, driving our climate and weather.

Mass loss calculation

To calculate the sun's mass loss per second, we apply Einstein's equation. Given the sun's energy output of \(5 \times 10^{26} \, \text{J/s} \), we rearrange the equation \( E = mc^2 \) to \( m = \frac{E}{c^2} \). By inserting the values, we get:

• Energy, \(E = 5 \times 10^{26} \, \text{J/s}\)
• Speed of light, \(c = 3 \times 10^8 \, \text{m/s}\)

Therefore, the mass loss per second \( m = \frac{5 \times 10^{26}}{(3 \times 10^8)^2}\). Simplifying this calculation results in \(5.55 \times 10^9 \, \text{kg/s}\). This calculation shows the actual mass lost by the sun every second converted into energy, driven by the process of nuclear fusion.