Problem 53 - University Physics with Modern Physics

Question

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 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 4.0 \(\times\) 10^{38} times the sun's power \(P\).

Step 1: Identify Variables and Given Data

We know that in 0.20 seconds, the magnetar released as much energy as the sun does in 250,000 years. Let the average power output of the sun be represented by the variable \(P\). The aim is to express the magnetar's average power output in terms of \(P\).

Step 2: Calculate Energy Output of the Sun

The power output of the sun is \(P\), and over a period of 250,000 years, the energy output can be computed using: \[ \text{Energy} = \text{Power} \times \text{Time} = P \times (250,000 \times 365 \times 24 \times 60 \times 60) \] seconds to convert years into seconds.

Step 3: Calculate the Equivalent Energy Output of the Magnetar

The energy output of the magnetar is the same as the energy the sun emits in 250,000 years. Therefore, the energy output of the magnetar in this timeframe remains the same as in Step 2.

Step 4: Convert Magnetar's Energy Output to Power Output

Since the energy output from the magnetar was released in 0.20 seconds, use the formula for power, \( \text{Power} = \frac{\text{Energy}}{\text{Time}} \). Letting \(E\) be the energy from the previous step, the average power output of the magnetar is \( \frac{E}{0.20} \).

Step 5: Express Magnetar's Power Output in Terms of P

Substitute \(E\) with the result from Step 2 into the power formula: \[ \text{Magnetar Power} = \frac{P \times (250,000 \times 365 \times 24 \times 60 \times 60)}{0.20} \]. Thus, expresing the magnetar's power output as a multiple of \(P\): \[ \text{Magnetar Power} = (P \times 250,000 \times 365 \times 24 \times 60 \times 60) \times \frac{1}{0.20} \].

Key Concepts

Neutron Star

Neutron stars are the fascinating remnants of massive stars that have undergone a supernova explosion. When a massive star runs out of nuclear fuel, its core collapses under gravity, causing protons and electrons to combine into neutrons. This process forms an incredibly dense object – a neutron star.
Neutron stars have some peculiar features, such as very strong magnetic fields, rapid rotations, and extreme densities.

• Their density is extraordinary, as a neutron star's mass can be up to twice that of our Sun, but all of it is compressed into a sphere with a radius of about 10 kilometers.
• One peculiar type of neutron star is called a magnetar. Magnetars have an even stronger magnetic field than typical neutron stars, which can be more than a trillion times stronger than Earth's magnetic field.

This immense magnetic power can lead to high-energy radiation bursts, like the one observed from magnetar SGR 1806-20. Understanding neutron stars aids astronomers in exploring the extremes of physics.

Power Output

Power output is the measure of the amount of energy released or used per unit of time. It is commonly expressed in watts, where one watt equals one joule per second.
In the context of stars, the power output is a critical measure because it describes the star's energy-generating capability. For instance:

• The Sun's power output is approximately 3.8 x 10²⁶ watts, a standard reference point when comparing other astronomical bodies.

The average power output of the Sun, represented as \( P \) in the exercise, helps us understand the magnetar's immense energy release when it produced a flash equivalent to the Sun's energy over 250,000 years, but in only 0.20 seconds. This calculated power output of the magnetar illustrates its extraordinary energy-emitting capability and helps astronomers study the differences between typical stars and extreme objects like magnetars.

Energy Calculation

Energy calculations often involve determining how much energy is produced or consumed over a period. Energy is typically measured in joules and is calculated by multiplying power by time.
In the exercise, energy calculation is crucial for understanding the scale of the magnetar's burst.

• First, the energy the Sun releases over 250,000 years is calculated by multiplying its power \( P \), and converting years into seconds, which results in massive energy in joules.
• This same quantity of energy, when divided by the short time span of 0.20 seconds, allows us to compute the magnetar's explosive power output.
• The formula used is \( \text{Energy} = \text{Power} \times \text{Time} \), which emphasizes the dynamic relationship between power and energy.

By comparing the computed energy and time, we can appreciate how different astronomical phenomena employ vast amounts of energy in varied timescales.

Astronomy

Astronomy is the scientific study of celestial objects, space, and the universe as a whole. It encompasses the examination of galaxies, stars, planets, and other worldly phenomena.
By observing astronomical events, such as the magnetar burst from SGR 1806-20, astronomers can gather valuable insights into the universe's physical laws and forces.

• The vastness of the universe means that there are diverse and energetic occurrences beyond our solar system, like the magnetar flare, challenging previous concepts and provoking newer theories.
• Instruments and observational technology in astronomy, like telescopes and space-based observatories, allow for the detection and analysis of such far-reaching bursts of energy.

Engaging with astronomy opens a window into the unknown, keeping the quest to understand cosmic phenomena alive, and consistently reshaping our understanding of the cosmos.