Ch 36: Special Relativity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 36: Special Relativity

Problem 69c

Chapter 36, Problem 69c

The sun radiates energy at the rate 3.8 x 10²⁶ W. The source of this energy is fusion, a nuclear reaction in which mass is transformed into energy. The mass of the sun is 2.0 x 10³⁰ kg. Fusion takes place in the core of a star, where the temperature and pressure are highest. A star like the sun can sustain fusion until it has transformed about 0.10% of its total mass into energy, then fusion ceases and the star slowly dies. Estimate the sun's lifetime, giving your answer in billions of years.

Verified step by step guidance

Step 1: Use Einstein's mass-energy equivalence formula, E = mc², to calculate the total energy that can be produced by the Sun. Here, m is the mass of the Sun that can be converted into energy (0.10% of the Sun's total mass), and c is the speed of light (approximately 3.0 × 10⁸ m/s).

Step 2: Calculate the mass that can be converted into energy by multiplying the Sun's total mass (2.0 × 10³⁰ kg) by 0.10%. This gives m = 0.001 × 2.0 × 10³⁰ kg.

Step 3: Substitute the value of m from Step 2 and c = 3.0 × 10⁸ m/s into the equation E = mc² to find the total energy that can be radiated by the Sun.

Step 4: The Sun radiates energy at a constant rate of 3.8 × 10²⁶ W (watts). To estimate the Sun's lifetime, divide the total energy calculated in Step 3 by the Sun's energy output per second (3.8 × 10²⁶ W). This gives the Sun's lifetime in seconds.

Step 5: Convert the Sun's lifetime from seconds into years by dividing by the number of seconds in a year (approximately 3.15 × 10⁷ seconds per year). Finally, express the result in billions of years.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Nuclear Fusion

Nuclear fusion is a process where two light atomic nuclei combine to form a heavier nucleus, releasing a significant amount of energy. In stars like the sun, hydrogen nuclei fuse to create helium, which powers the star and produces the energy we receive as sunlight. This process occurs under extreme temperature and pressure conditions in the star's core.

Mass-Energy Equivalence

Mass-energy equivalence, expressed by Einstein's equation E=mc², states that mass can be converted into energy and vice versa. In the context of the sun, a small fraction of its mass is converted into energy during fusion, which sustains the sun's luminosity. Understanding this principle is crucial for estimating how long the sun can continue to produce energy before exhausting its fuel.

Stellar Lifespan

The stellar lifespan refers to the duration a star can sustain nuclear fusion before it exhausts its nuclear fuel. For the sun, this is approximately determined by the total mass available for fusion and the rate at which it converts mass into energy. The sun can sustain fusion for about 10 billion years, and its current age is around 4.6 billion years, indicating it has several billion years left before it transitions to the next stage of its lifecycle.

Recommended video:

Guided course

06:36

Star collapses

Related Practice

Textbook Question

Let's examine whether or not the law of conservation of momentum is true in all reference frames if we use the Newtonian definition of momentum: p_x = mu_x. Consider an object A of mass 3m at rest in reference frame S. Object A explodes into two pieces: object B, of mass m, that is shot to the left at a speed of c/2 and object C, of mass 2m, that, to conserve momentum, is shot to the right at a speed of c/4. Suppose this explosion is observed in reference frame S' that is moving to the right at half the speed of light. Use the Lorentz velocity transformation to find the velocities and the Newtonian momenta of B and C in S'.

views

Textbook Question

The sun radiates energy at the rate 3.8 x 10²⁶ W. The source of this energy is fusion, a nuclear reaction in which mass is transformed into energy. The mass of the sun is 2.0 x 10³⁰ kg. What percent is this of the sun's total mass?

views

Textbook Question

Some particle accelerators allow protons (p⁺) and antiprotons (p⁻) to circulate at equal speeds in opposite directions in a device called a storage ring. The particle beams cross each other at various points to cause p⁺ + p⁻ collisions. In one collision, the outcome is p⁺ + p⁻ → e⁺ + e⁻ + γ + γ, where γ represents a high-energy gamma-ray photon. The electron and positron are ejected from the collision at 0.9999995c and the gamma-ray photon wavelengths are found to be 1.0 x 10^-6 nm. What were the proton and antiproton speeds, as a fraction of c, prior to the collision?

views

Textbook Question

An electron moving to the right at 0.90c collides with a positron moving to the left at 0.90c. The two particles annihilate and produce two gamma-ray photons. What is the wavelength of the photons?

views

Textbook Question

The nuclear reaction that powers the sun is the fusion of four protons into a helium nucleus. The process involves several steps, but the net reaction is simply 4p → ⁴He + energy. The mass of a proton, to four significant figures, is 1.673 x 10^-27kg, and the mass of a helium nucleus is known to be 6.644 x 10^-27 kg. What fraction of the initial rest mass energy is this energy?

views

Textbook Question

At what speed, as a fraction of c, is the kinetic energy of a particle twice its Newtonian value?