Question

Be sure to show all calculations clearly and state your final answers in complete sentences. The Crab Nebula 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 a 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 Nebula 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 estimated age is about 1257 years, while the actual age is 969 years.

Step-by-step solution

Identify Given Information

Identify the given variables: period of pulsar, \( p = 0.0333 \) seconds, and the rate of slowing, \( r = 4.2 \times 10^{-13} \).

Plug Values into Formula

Use the age formula \( \text{age} = \frac{p}{2r} \). Substitute \( p = 0.0333 \) and \( r = 4.2 \times 10^{-13} \).

Perform the Calculation

Calculate the age: \[ \text{age} = \frac{0.0333}{2 \times 4.2 \times 10^{-13}} \]. This equals approximately \( 3.964286 \times 10^{10} \) seconds.

Convert Age to Years

Convert the age from seconds to years by dividing by the number of seconds in a year \( (60 \times 60 \times 24 \times 365.25) \). This gives approximately \( 1.257 \times 10^3 \) years.

Compare with Actual Age

Calculate the actual age: AD 2023 - AD 1054 = 969 years. Compare 1257 years to approximately 969 years.

Key concepts

Crab Nebula

The Crab Nebula is one of the most remarkable objects in our galaxy. It is the remnant of a supernova explosion that was first observed on Earth in July 1054 AD. This spectacular event was recorded by astronomers in several ancient civilizations, including China. Over centuries, the remnants expanded and evolved, creating the Nebula we see today.

Located in the constellation Taurus, the Crab Nebula is roughly 6,500 light-years away from us. It is constantly illuminated by the central pulsar, PSR B0531+21, a rapidly rotating neutron star. This pulsar emits beams of radiation along its magnetic poles. These beams sweep across the sky, much like a lighthouse beam, resulting in detectable pulses from Earth, which we can study to understand the properties of this cosmic wonder.

Pulsar Age Estimation

Estimating the age of pulsars such as the one located in the Crab Nebula involves a fascinating interplay of theoretical astrophysics and observations. The age estimate of a pulsar combines its present period and the rate at which this period is changing.

This is given by the formula: \[ \text{Age} = \frac{p}{2r} \]where:

• \( p \) is the period, the time it takes for the pulsar to complete one full rotation.
• \( r \) is the rate of change of this period over time.

For the Crab Nebula pulsar, its current period is 0.0333 seconds, and it is slowing down at a rate of \( 4.2 \times 10^{-13} \) seconds per second. Plugging these values into the formula, we estimate the age to be about 1,257 years.

This might seem close to the actual age of 969 years, indicating the pulsar's slowdown model is a good approximation for such calculations.

Astrophysics Calculations

Astrophysics calculations help unravel the mysteries of pulsars and the universe. These calculations include using precise measurements, and often complex equations, to answer big questions about the age and evolution of celestial bodies.

In the case of the Crab Nebula pulsar, the calculation involves converting the period in seconds and the slow down rate into a more comprehensible age, expressed in years. Conversion factors like the number of seconds in a year are used to present results in human-understandable terms:
\[ \text{Seconds in a year} = 60 imes 60 imes 24 imes 365.25 \]These formulas allow scientists to make sense of observations and predict the past and future behavior of astronomical phenomena. Thus, they are essential for comparing theoretical models with observations and refining our understanding of the cosmos.

Supernova Remnants

Supernova remnants, like the Crab Nebula, are the afterglow of a star's explosive death. Following the supernova explosion, these remnants are composed of the star’s ejected material. They radiate brightly as the debris races outward through space.

The study of supernova remnants is crucial in astrophysics as these regions are cosmic laboratories for understanding stellar life cycles. They are fertile grounds for studying extreme physical conditions, such as high-energy particles and magnetic fields.

• These remnants also contribute to the enrichment of the interstellar medium with heavy elements, essential for forming new stars and planets.
• Moreover, they provide clues about the original star and the explosion dynamics.

Over time, the Crab Nebula will continue to expand and mix with the surrounding interstellar medium, becoming less visible but nonetheless remaining a significant data source for scientists studying the complex life and death of stars.