Question

Be sure to show all calculations clearly and state your final answers in complete sentences. The Crab Pulsar Winds Down. Theoretical models of the slowing of pulsars predict that the age of a pulsar is approximately equal to \(p /(2 r),\) where \(p\) is the pulsar's current period and \(r\) is the rate at which the period is slowing with time. Observations of the pulsar in the Crab Nebula show that it pulses 30 times per second, so \(p=0.0333\) second, but the time interval between pulses is growing longer by \(4.2 \times 10^{-13}\) second with each passing second, so \(r=4.2 \times 10^{-13}\) second per second. Using that information, estimate the age of the Crab pulsar. How does your estimate compare with the true age of the pulsar, which was born in the supernova observed in A.D. \(1054 ?\)

Short answer

The theoretical age estimate for the Crab pulsar is 1,257 years, which exceeds the true age of 969 years by 288 years.

Step-by-step solution

Understand the formula for pulsar age

The age of a pulsar is given by the formula \( \text{Age} = \frac{p}{2r} \), where \( p \) is the period of the pulsar and \( r \) is the rate at which the period changes with time. We have \( p = 0.0333 \) seconds and \( r = 4.2 \times 10^{-13} \) seconds per second.

Substitute the values into the formula

We substitute the given values \( p = 0.0333 \) seconds and \( r = 4.2 \times 10^{-13} \) seconds per second into the formula: \[ \text{Age} = \frac{0.0333}{2 \times 4.2 \times 10^{-13}} \].

Perform the calculation

First, calculate the denominator: \( 2 \times 4.2 \times 10^{-13} = 8.4 \times 10^{-13} \). Then, divide \( 0.0333 \) by \( 8.4 \times 10^{-13} \): \[ \text{Age} = \frac{0.0333}{8.4 \times 10^{-13}} = 3.964 \times 10^{10} \] seconds.

Convert the age from seconds to years

Convert \( 3.964 \times 10^{10} \) seconds into years using the conversion factor of \( 3.154 \times 10^7 \) seconds per year (1 year = 365.25 days): \[ \text{Age} = \frac{3.964 \times 10^{10}}{3.154 \times 10^7} \approx 1,257 \,\text{years} \].

Compare with the true age

Estimate the true age of the pulsar by subtracting the year 1054 from the current year (assume the current year is 2023 for this calculation): \( 2023 - 1054 = 969 \) years. The calculated age \( 1,257 \) years indicates the theoretical age is higher than the actual age by \( 1,257 - 969 = 288 \) years.

Key concepts

Crab Pulsar

The Crab Pulsar is one of the most fascinating celestial objects in our sky. Found in the heart of the Crab Nebula, it is a rapidly spinning neutron star formed from the remnants of a supernova explosion observed on Earth in 1054 A.D.

These neutron stars are extremely dense, compact, and rotate at incredible speeds. The Crab Pulsar, in particular, completes one full rotation every 0.0333 seconds, which means it pulsates 30 times per second. This rapid pulsation is a key characteristic of pulsars, as they emit beams of electromagnetic radiation that appear to pulse due to their rotation.

The study of such objects not only helps us understand the life cycle of stars but also provides insights into the physics of extreme states of matter and magnetism.

Pulsar Period

The pulsar period is the duration of one complete rotation of the pulsar and is a vital measurement in understanding pulsars. For the Crab Pulsar, its period is incredibly short—only 0.0333 seconds. This period is detectable due to the pulsar's regular bursts of radiation as it spins.

Studying the pulsar period is crucial for astronomers as it helps in measuring the stability and changes in a pulsar's rotation over time. By monitoring slight variations in the period, scientists can gather information on the pulsar's energy loss and its interaction with surrounding material.

Period Change Rate

The Period Change Rate ( ") is a measure of how a pulsar's period alters over time. For the Crab Pulsar, this rate is approximately 4.2 × 10^{-13} seconds per second. This signifies that the pulsar's rotation is slowing down, albeit very gradually.

This deceleration is due to the loss of rotational energy, mostly emitted as radiation. Understanding the rate at which this period change occurs is essential for estimating the age of the pulsar using the formula \[ ext{Age} = rac{p}{2r} \]. This precise rate also helps scientists to explore the dynamics of neutron stars and evaluate theoretical models surrounding the nature of pulsars.

Supernova

A supernova is a powerful and luminous explosion that occurs at the end of a star's life cycle. The Crab Pulsar is a result of such an event. When a supernova exploded in 1054 A.D., it was visible from Earth even during the day for several weeks.

Supernovae play a pivotal role in the universe by distributing elements and enriching interstellar space with heavy materials. They also leave behind dense remnants, like neutron stars or black holes. The Crab Pulsar embodies this transformation from a brilliant supernova to a dense, rapidly spinning neutron star, providing a unique opportunity to study the aftermath of such colossal cosmic events.

Cosmic Chronology

Cosmic Chronology refers to the timeline and history of the universe as interpreted through astronomical observations and phenomena. Studying objects like the Crab Pulsar allows astronomers to piece together significant events in the cosmic timeline.

The birth of the Crab Pulsar in a supernova event gives us a reliable date, recorded as 1054 A.D., to anchor other events surrounding this period.

• It helps in understanding the lifecycle of stars from their formation to ultimate fate.
• It serves as a temporal marker to gauge the evolution and transformation of cosmic structures.

By analyzing pulsars and their remnants, we deepen our knowledge about the age and development of various celestial phenomena, offering key insights into the dynamic processes of the universe.