Question

A white dwarf is a star that has exhausted its nuclear fuel and lost its outer mass so that it consists only of its dense, hot inner core. It will cool unless it gains mass from some nearby star. It can form a binary system with such a star and gradually gain mass up to the limit of 1.4 times the mass of the Sun. If the white dwarf were to start to exceed the limit, it would explode into a supernova. How much energy is released by the explosion of a white dwarf at its limiting mass if \(80.0 \%\) of its mass is converted to energy?

Short answer

The explosion releases approximately \(2.00 \times 10^{47}\) Joules of energy.

Step-by-step solution

Understanding the Problem

We need to calculate the energy released by a white dwarf star if it reaches its limiting mass and explodes, converting 80% of its mass into energy. The limiting mass is 1.4 times the mass of the Sun.

Mass of the White Dwarf

The mass of the Sun, denoted by \( M_{\odot} \), is approximately \( 1.9885 \times 10^{30} \text{ kg} \). Therefore, the mass of the white dwarf at its limit is \( 1.4 \times M_{\odot} = 1.4 \times 1.9885 \times 10^{30} \text{ kg} \).

Calculate Limiting Mass

Perform the multiplication to obtain the limiting mass:\[M_{\text{white dwarf}} = 1.4 \times 1.9885 \times 10^{30} = 2.784 \times 10^{30} \text{ kg} \].

Determine Mass Converted to Energy

Since 80% of this mass is converted to energy, we calculate the converted mass as \(0.8 \times 2.784 \times 10^{30}\).

Apply Mass-Energy Equivalence

Using Einstein's mass-energy equivalence equation, \( E = mc^2 \), where \( c \) is the speed of light (\( 3.00 \times 10^8 \text{ m/s} \)), we find the energy released:\[E = (0.8 \times 2.784 \times 10^{30} \text{ kg}) \times (3.00 \times 10^8 \text{ m/s})^2\].

Compute Energy Released

Perform the calculations to determine the energy released:\[E = (2.2272 \times 10^{30} \text{ kg}) \times (9.00 \times 10^{16} \text{ m}^2/\text{s}^2) = 2.00448 \times 10^{47} \text{ J}\].

Key concepts

Stellar Evolution

Stellar evolution is the process by which a star changes over the course of time. Stars are born from clouds of dust and gas known as nebulae. Over millions of years, gravity pulls these particles together, increasing pressure and temperature until nuclear fusion ignites.
Throughout a star's life, it undergoes various phases. Initially, stars spend most of their existence in the main sequence, fusing hydrogen into helium. Once the hydrogen is depleted, they transition into different types depending on their mass.
- **Low to Medium Mass Stars:** These stars, like our Sun, eventually shed their outer layers and form white dwarfs.
- **Massive Stars:** Massive stars may explode as supernovae, leading to either a neutron star or black hole formation. The ultimate stage of a star's life is determined by its initial mass. White dwarfs, neutron stars, and black holes mark the possible endpoints of stellar evolution.

Supernova

A supernova is a colossal and luminous explosion lasting from a few weeks to a few months.
It occurs at the end of a star's life cycle, specifically in massive stars when their nuclear fuel is exhausted. The core collapses under its gravity, causing the outer layers to explode.
There are two main types of supernovae:
- **Type I:** Occurs in binary star systems where a white dwarf gains mass from its companion star. Once it exceeds the Chandrasekhar limit (about 1.4 solar masses), a thermonuclear explosion occurs.
- **Type II:** Results from the collapse of a massive star after nuclear fuel exhaustion.
Supernovae are significant for creating heavier elements and dispersing them into space, contributing to the cosmic chemical enrichment.

Mass-Energy Equivalence

The concept of mass-energy equivalence is encapsulated in Einstein's famous equation, \[ E = mc^2 \]
where:

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

This equation implies a direct relationship between mass and energy, showing that they can be converted into each other. Even a small amount of mass can be converted into a vast amount of energy, as seen in nuclear reactions.
In the context of a white dwarf exploding into a supernova, a significant mass is converted to energy, elucidating the enormous energy release during such cosmic events.

Nuclear Fusion

Nuclear fusion is the process where two light atomic nuclei combine to form a heavier nucleus, releasing energy in the process. Inside stars, this process powers everything from their light to their life cycle.
Here's how it works:

• In hydrogen fusion, four hydrogen nuclei (protons) merge to form one helium nucleus, plus energy.
• Fusion requires high temperatures and pressures to overcome the repulsion between positively charged protons.

Fusion is the sun and stars' main energy source and is responsible for the balance of forces within stars, determining their life stages. Once a star's core ceases fusion, it either expands and cools to form a giant or contracts to a smaller, hotter state like a white dwarf, depending on remaining fusion processes.

Binary Star System

A binary star system consists of two stars orbiting around their common center of mass. These systems are prevalent in our galaxy and exemplify complex gravitational interactions.
Key features of binary systems include:

• Mass exchange can occur between the stars, especially if one star expands beyond its Roche lobe, transferring mass to the companion.
• Binary systems can lead to interesting phenomena, such as the build-up of mass on a white dwarf, potentially causing a supernova explosion.

In the case of a white dwarf companion: If it gathers additional mass from its partner, leading to surpassing the Chandrasekhar limit, a type I supernova can occur, creating a brilliant, explosive end and showcasing the dynamic nature of binary star interactions.