Question

The Sun radiates energy into space at the rate of \(3.9 \times 10^{26} \mathrm{~J} / \mathrm{s}\). (a) Calculate the rate of mass loss from the Sun in \(\mathrm{kg} / \mathrm{s}\). (b) How does this mass loss arise? (c) It is estimated that the Sun contains \(9 \times 10^{56}\) free protons. How many protons per second are consumed in nuclear reactions in the Sun?

Short answer

(a) The Sun loses about \(4.33 \times 10^9\) kg of mass per second due to energy-mass conversion in its core. (b) This mass loss arises from nuclear fusion, where hydrogen atoms fuse to form helium, and some mass is converted into energy following Einstein's equation. (c) Approximately \(2.59 \times 10^{36}\) protons are consumed per second in nuclear reactions within the Sun.

Step-by-step solution

Calculate the rate of mass loss from the Sun

To calculate the rate of mass loss from the Sun, we can use the energy-mass conversion formula, according to Einstein's famous equation: \[E = mc^2\] where \(E\) is the energy, \(m\) is the mass, and \(c\) is the speed of light (\(3.0 \times 10^8\) m/s). Given the energy radiated by the sun per second (\(3.9 \times 10^{26}\) Joules/s), we can rearrange the formula to find the mass loss per second: \[\frac{E}{c^2} = \frac{m}{1~\mathrm{s}}\] Now substitute the given values, and solve for \(m\): \[\frac{m}{1~\mathrm{s}} = \frac{3.9 \times 10^{26}~\mathrm{J}}{(3.0 \times 10^8~\mathrm{m/s})^2}\]

Evaluating the expression for mass loss

Now that we have our expression, let's evaluate it to find the numerical value of the rate of mass loss: \[\frac{m}{1~\mathrm{s}} = \frac{3.9 \times 10^{26}~\mathrm{J}}{(9.0 \times 10^{16}~\mathrm{m^2/s^2})}\] \[\frac{m}{1~\mathrm{s}} = 4.33 \times 10^9~\mathrm{kg/s}\] So the Sun loses about \(4.33 \times 10^9\) kg of mass per second.

Discuss the origin of mass loss

This mass loss arises from the nuclear fusion happening in the core of the Sun. The Sun mainly consists of hydrogen, and in its core, hydrogen atoms undergo nuclear fusion to form helium. In this process, some of the mass is converted into energy, following the energy-mass conversion formula (Einstein's equation). As nuclear fusion takes place, some mass is lost, causing the Sun to lose mass continuously.

Calculate the number of protons consumed per second

Given the number of free protons in the Sun (approximately \(9 \times 10^{56}\)), we can estimate the number of protons consumed per second by dividing the mass loss due to fusion per second (Step 1) by the mass of one proton. The mass of one proton is approximately: \(m_p = 1.67 \times 10^{-27}\) kg Now we can find the number of protons consumed per second by dividing the mass loss by the mass of one proton, as follows: \[\frac{4.33 \times 10^9~\mathrm{kg/s}}{1.67 \times 10^{-27}~\mathrm{kg/proton}}\]

Evaluating the expression for protons consumed per second

Evaluating the expression to find the number of protons consumed per second: \[\frac{4.33 \times 10^9~\mathrm{kg/s}}{1.67 \times 10^{-27}~\mathrm{kg/proton}} = 2.59 \times 10^{36} ~\mathrm{protons/s}\] So, approximately \(2.59 \times 10^{36}\) protons are consumed per second in nuclear reactions within the Sun.

Key concepts

Understanding Nuclear Fusion

Nuclear fusion is the process that powers our Sun and stars, providing the energy that sustains life on Earth. It's a kind of nuclear reaction where two light atomic nuclei combine to form a heavier nucleus, releasing a colossal amount of energy as a result. In the core of the Sun, hydrogen atoms fuse to create helium, and this transformation is the cornerstone of the Sun's energy production.

During fusion, a small fraction of the mass of fusing particles is converted to energy. This conversion is described by Einstein's equation, which asserts that mass can be transformed into energy and vice versa. Fusion reactions release energy in several forms, including light and heat, which travel through space and reach Earth, providing us with sunlight and warmth. Understanding this process is crucial to grasp how the Sun functions and affects us here on Earth.

Solar Mass Loss Explained

Solar mass loss is an intriguing phenomenon occurring in the Sun. This mass isn’t actually 'lost' but converted into energy — radiated away from the Sun in the form of light and other types of electromagnetic radiation. According to the mass-energy equivalence principle encapsulated in Einstein's equation, a small amount of mass can create a large amount of energy, and this is what’s happening continuously in the Sun's core.

The process of nuclear fusion, which combines hydrogen atoms into helium, causes a slight discrepancy between the mass of the initial particles and the resulting helium atom. That missing mass turns into energy, radiated as sunlight, and this is what contributes to solar mass loss. Although the amount of mass lost each second might seem large, it's tiny compared to the Sun’s total mass, thus allowing the Sun to shine for billions of years.

Proton Consumption in the Sun

The proton consumption in the Sun is associated with the nuclear fusion processes in its core. Protons, which are positively charged particles within the nucleus of atoms, are the building blocks of the universe. In the sun's extreme core conditions, the abundance of free protons initiates the fusion chain reaction. When protons fuse, they create heavier elements up to iron through a series of interactions known as the proton-proton chain reaction.

This chain of events starts with protons and ends with helium, releasing energy according to the relationship described by Einstein's equation. It's by counting the protons consumed during these reactions that scientists can estimate the Sun's energy output. The calculations showing the amount of mass and corresponding protons consumed per second help us understand the massive scale of solar processes that have been supporting life on Earth for millennia.

Einstein's Famous Equation E=mc^2

Einstein's famous equation E=mc^2 is perhaps the most well-known formula in physics, encapsulating the principle of mass-energy equivalence. This equation tells us that energy (E) and mass (m) are interchangeable; they are different forms of the same thing. The 'c' in the equation represents the speed of light in vacuum (approximately 299,792,458 meters per second), and when squared, it shows that a tiny amount of mass can be converted into a huge amount of energy.

This equation underpins a wide range of phenomena, from the inner workings of atomic bombs to the lifecycle of stars. It explains how the Sun is able to release vast amounts of energy through the process of nuclear fusion, where the mass 'lost' is transformed into energy that travels across the solar system. This formula has implications in both theoretical and practical fields, such as astrophysics and sustainable energy research, making it a cornerstone of modern physics.