Problem 51 - University Physics with Modern Physics

Question

Magnetar. On December \(27,2004,\) astronomers observed the greatest flash of light ever recorded from outside the solar system. It came from the highly magnetic neutron star SGR \(1806-20\) (a magnetar). During 0.20 \(\mathrm{s}\) , this star released as much energy as our sun does in \(250,000\) years. If \(P\) is the average power output of our sun, what was the average power output (in terms of \(P )\) of this magnetar?

Step-by-Step Solution

Answer

The magnetar's average power output is about 650,000,000,000,000 times the sun's power output (\(6.5 \times 10^{14}P\)).

Step 1: Understand the Problem

We need to determine the average power output of the magnetar in terms of the sun's power output, given that the magnetar released in 0.20 seconds as much energy as the sun does in 250,000 years.

Step 2: Identify Key Information

The magnetar released the equivalent of the sun's energy output over 250,000 years in just 0.20 seconds.

Step 3: Power Formula

Remember that power is defined as energy per unit time: \[P = \frac{E}{t}\] where \(P\) is power, \(E\) is energy, and \(t\) is time.

Step 4: Set Up the Magnetar's Power Output

Let \(P_{s}\) be the average power output of the sun. The total energy the sun releases in 250,000 years is:\[E_{s} = P_{s} \times 250,000 \text{ years}\]

Step 5: Convert Time to Seconds

Convert 250,000 years to seconds. There are 365.25 days in a year (considering leap years), 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute.\[250,000 \times 365.25 \times 24 \times 60 \times 60 \text{ seconds}\]

Step 6: Calculate Magnetar's Energy

Thus, the energy released by the magnetar is equal to the energy released by the sun in that time span:\[E_{m} = P_{s} \times 250,000 \times 365.25 \times 24 \times 60 \times 60 \]

Step 7: Determine Magnetar's Power Output

The average power output \(P_{m}\) of the magnetar is:\[P_{m} = \frac{E_{m}}{0.20} = \frac{P_{s} \times 250,000 \times 365.25 \times 24 \times 60 \times 60}{0.20}\]

Step 8: Simplify the Expression

The average power output of the magnetar in terms of the sun's average power output is then:\[P_{m} = P_{s} \times \frac{250,000 \times 365.25 \times 24 \times 60 \times 60}{0.20}\]

Key Concepts

Neutron Stars

Neutron stars are fascinating celestial objects born from the remnants of massive stars. When a star with a large mass exhausts its nuclear fuel, it may explode in a supernova, leaving behind a core. This core continues to collapse under gravity, forming an incredibly dense object known as a neutron star. Neutron stars contain a unique form of matter, primarily made up of neutrons packed tightly together.

• Neutron stars are about the size of a city, with diameters ranging from 10 to 20 kilometers.
• Despite their tiny size, they can possess the mass equivalent of 1.4 times that of the sun or more.
• The surface gravity of a neutron star is immensely strong due to its dense nature.

Neutron stars can spin rapidly, and some have strong magnetic fields, approximately a trillion (or more) times stronger than Earth's magnetic field. These are called magnetars. The incredible traits of neutron stars make them interesting subjects in the study of the universe.

Astrophysics

Astrophysics is the branch of astronomy that seeks to understand the physical nature of celestial objects and the processes that govern them. It combines principles from physics and astronomy, applying them to the study of stars, planets, and galaxies. By using a multitude of methods and technologies, astrophysicists explore the cosmos to unveil its mysteries.

• Astrophysics helps us understand the life cycles of stars by observing different stages from birth to death.
• It also involves the study of the chemical composition, energy outputs, and dynamics of celestial bodies.
• This field employs data from telescopes that span the electromagnetic spectrum, including radio waves and gamma rays.

Understanding events such as supernovae, black holes, and neutron stars falls under the scope of astrophysics, providing insights into the fundamental laws of nature.

Power Calculation

Calculating power in physics involves understanding the rate at which energy is transferred or converted. The basic formula for power, which is used in various applications, is given by:\[P = \frac{E}{t}\]where:- \( P \) is the power (in watts)- \( E \) is the energy (in joules)- \( t \) is the time (in seconds)This formula is pivotal in many areas, from calculating the energy consumption of household appliances to understanding astronomical events. For example, when comparing the power of the sun and a neutron star like a magnetar, the energy these bodies emit over a specific time helps determine their power output.
It's also important to remember that power is a scalar quantity, meaning it has magnitude but no direction. The concept of power is crucial in both everyday life and scientific research, providing a clearer understanding of energy use and conversion.

Energy Conversion

Energy conversion is a fundamental concept in physics, explaining how energy changes from one form to another. This process is vital for understanding how various systems operate, whether mechanical, electrical, or biological. The law of conservation of energy states that energy cannot be created or destroyed, only transformed.

• In a neutron star, gravitational energy is converted into radiation and kinetic energy.
• Inside engines and batteries, chemical energy is transformed into mechanical and electrical energy, respectively.
• Thermal power plants convert the thermal energy of combusted fuels into electrical energy.

These transformations are crucial for harnessing energy effectively to serve our needs. By studying energy conversion, scientists can improve efficiency and develop new technologies, like using the energy from celestial events to understand better the universe's workings. Neutron stars, with their extreme conditions, offer a natural laboratory for examining energy conversion processes.