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Chapter 17: Problem 53

If the Crab Nebula has been expanding at an average velocity of \(3,000 \mathrm{km} / \mathrm{s}\) since \(1054 \mathrm{CE}\), what was its average radius in the year \(2013 ?\) (Note: There are approximately \(3 \times 10^{7}\) seconds in a year.)

Short Answer

Expert verified
The average radius in 2013 was \(8.631 \times 10^{13} \text{ km}\).

Step by step solution

01

- Calculate the Time Interval

First, determine the total time during which the Crab Nebula has been expanding. Subtract the initial year from the final year: \[\text{Time Interval} = 2013 - 1054 = 959 \text{ years}\]
02

- Convert Years to Seconds

Convert the time interval from years to seconds using the given approximation that there are \(3 \times 10^7\) seconds in a year: \[\text{Time in Seconds} = 959 \times 3 \times 10^7 = 2.877 \times 10^{10} \text{ seconds}\]
03

- Calculate the Average Radius

Use the average expansion velocity to calculate the average radius of the Crab Nebula in 2013. Multiply the expansion velocity by the total time in seconds: \[\text{Average Radius} = 3000 \frac{\text{km}}{\text{s}} \times 2.877 \times 10^{10} \text{ s} = 8.631 \times 10^{13} \text{ km}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

headline of respective core concept
Astrophysics is the branch of astronomy that deals with the physical properties and processes of celestial objects and phenomena. One interesting example is the study of cosmic objects like nebulae. The Crab Nebula, an expanding supernova remnant, offers a great case study. By measuring its expansion velocity, we can infer crucial insights about its size and growth over time.
headline of respective core concept
To calculate how long something has been expanding, like the Crab Nebula, we need to know the start and end times. For instance, the Crab Nebula exploded as a supernova in 1054 CE. If we want to know its expansion until 2013, we subtract the initial year from the final year:
Time Interval = 2013 - 1054 = 959 years. This gives us the total years it has been expanding.
headline of respective core concept
Often in astrophysics, we need to convert units to make calculations easier. Since we know there are approximately 30 million seconds in a year, to find out how many seconds the nebula has been expanding, we'd convert 959 years:
  • Time in Seconds = 959 years × 3 × 10^7 seconds/year
This calculation gives us 2.877 × 10^{10} seconds.
headline of respective core concept
To find the average radius of an expanding nebula, we multiply its velocity by time. With the Crab Nebula expanding at 3000 km/s, and the total time calculated as 2.877 × 10^{10} seconds:
  • Average Radius = 3000 km/s × 2.877 × 10^{10} seconds
The result is 8.631 × 10^{13} km. This value gives the radius in 2013.

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Most popular questions from this chapter

Go to the Einstein@Home website (http://einsteinathome.org). In this distributed computing project, volunteers use their spare computer processing power to help search for new pulsars. Look over the "News" section on the right. Have any pulsars been found lately? Join the project, create an account, download BOINC, and follow directions to look for pulsars.

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