The Sun shines by converting mass into energy according to Einstein's well- known relationship \(E=m c^{2}\). Show that if the Sun produces \(3.85 \times 10^{26} \mathrm{J}\) of energy per second, it must convert 4.3 million metric tons \(\left(4.3 \times 10^{9} \mathrm{kg}\right)\) of mass per second into energy.

Einstein's mass-energy equivalence principle is a groundbreaking discovery in physics. This principle states that mass and energy are interchangeable. The famous equation \( E=mc^2 \) encapsulates this concept. Here:

• \( E \) is the energy produced or consumed.
• \( m \) is the mass.
• \( c \) is the speed of light, approximately \( 3.00 \times 10^8 \, \text{m/s} \).

This equation reveals that even a small amount of mass can be converted into a tremendous amount of energy because the speed of light squared (\( c^2 \) ) is a very large number. For example, the calculation in the exercise shows how a certain amount of mass from the Sun is transformed into energy, producing the sunlight that reaches Earth. Mass-energy conversion is a vital principle that highlights the hidden energy potential in matter.
By substituting the given values for mass ( \( 4.3 \times 10^9 \, \text{kg} \) ) and the speed of light ( \( 3.00 \times 10^8 \, \text{m/s} \) ), we can calculate the energy produced using the formula: \[ E = (4.3 \times 10^9) (3.00 \times 10^8)^2 \] .
This will invariably match the given energy value, solidifying the mass-energy conversion process.

Stars, including our Sun, generate energy through nuclear fusion. During fusion, hydrogen nuclei combine to form helium, releasing vast amounts of energy. This process happens under extreme temperatures and pressure found in the star's core. Here, the fusion of four hydrogen nuclei produces one helium nucleus, a positron, a neutrino, and energy. Through fusion, stars convert a tiny fraction of their mass into energy using Einstein's \( E=mc^2 \) relationship.
The Sun converts about \( 4.3 \times 10^9 \, \text{kg} \) of mass into energy every second. This immense energy production keeps our Sun shining and provides the essential light and heat needed for life on Earth. In nuclear fusion reactions, it's essential to understand that the energy produced comes from the mass differences between the initial and final products.
This process not only powers our solar system but also underpins the lifecycle of stars. Understanding nuclear fusion in stars gives us insights into stellar life cycles, producing various elements and the energy dynamics of the universe.

The speed of light ( \( c \) ) is a critical constant in physics, with a value of \( 299,792,458 \, \text{m/s} \), usually approximated as \( 3.00 \times 10^8 \, \text{m/s} \). Labeled as the universe's ultimate speed limit, no object or information can travel faster than light in a vacuum. This speed is essential when dealing with mass-energy conversion. In the \( E=mc^2 \) equation, the speed of light squared ( \( c^2 \) ) significantly amplifies the amount of energy derived from a given mass.
The speed of light plays a pivotal role in many areas of physics, from determining astronomical distances to formulating theories about the universe's structure. In practical terms, light's speed affects how we understand time and space, a cornerstone of Einstein's theory of relativity.
When analyzing energy production in stars, the speed of light ensures the conversion calculations yield accurate and consistent results. It shows the massive energy potential locked away within even a minute mass, further highlighting the profound implications of Einstein's theories on modern science.