The Sun shines by converting mass into energy according to \(E=m c^{2} .\) 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 \(\left(4.3 \times 10^{\circ} \mathrm{kg}\right)\) of mass per second into energy.

Einstein's equation, represented by the famous formula \(E=mc^2\), is a fundamental principle of physics that states energy \(E\) and mass \(m\) are interchangeable. This means mass can be converted into energy and vice versa.
The equation includes:

• \(E\): the amount of energy produced or consumed
• \(m\): the mass involved in the conversion
• \(c\): the speed of light in vacuum, a constant value of approximately \(3 \times 10^8\) meters per second

This insight was groundbreaking because it demonstrated that even small amounts of mass could be converted into tremendous amounts of energy, and it has pivotal applications in various fields.
For instance, in nuclear reactions and processes occurring within stars, including our Sun.

The Sun generates vast amounts of energy by converting mass into energy through nuclear fusion. Inside the Sun's core, hydrogen nuclei fuse to form helium, releasing energy in the process. This energy leaves the Sun in the form of sunlight, which travels to Earth and provides us with light and heat.
The exercise states that the Sun emits \(3.85 \times 10^{26}\) Joules of energy every second. To produce this energy, it must convert a certain amount of its mass into energy. This process is ongoing and crucial to sustaining the Sun's luminosity.
Let's break down how this works:

• Hydrogen atoms collide and fuse under extreme pressure and temperatures.
• This fusion process forms helium and releases energy.
• The energy travels outward from the Sun's core to its surface.
• Finally, this energy is emitted as sunlight.

Mass conversion, as described in the earlier steps, is the process where mass is transformed into energy. Based on Einstein’s equation \(E=mc^2\), the mass that got converted to produce the Sun's energy can be calculated as follows:

• First, identify the energy produced \(E\), which is \(3.85 \times 10^{26}\) Joules.
• Next, use the known speed of light \(c\).
• Insert these values into the equation to find the mass.

By substituting the values, we get:
\[3.85 \times 10^{26} = m (3 \times 10^8)^2\]
Solving for \(m\) gives us:
\[m = \frac{3.85 \times 10^{26}}{9 \times 10^{16}}\]
which simplifies to \(m = 4.28 \times 10^9\) kg. Finally, converting this to metric tons results in \(4.3 \times 10^6\) metric tons.
This stepwise conversion of mass into energy explains how the Sun converts its mass to maintain energy production, ensuring continuous emission of light and heat.

The speed of light, denoted as \(c\), is a critical constant in physics, valued at approximately \(3 \times 10^8\) meters per second. It plays a fundamental role in Einstein's equation for mass-energy equivalence. Here's why it's important:

• It determines the factor by which mass is converted into energy.
• Its squared value \((c^2)\) indicates that even a tiny amount of mass can result in a massive amount of energy.

To put it into perspective, when mass is multiplied by the speed of light squared, the outcome is an immensely large number because of the enormous value of \(c\).
This enormous value is essential for understanding why even small amounts of matter containing a lot of energy can sustain the Sun's energy production over billions of years.