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Chapter 13: Problem 36

Sirius, the brightest star in the sky, has a parallax of 0.379 arcsec. What is its distance in parsecs? in light-years? How long does it take light to reach Earth?

Short Answer

Expert verified
Sirius is approximately 2.64 parsecs (or 8.6 light-years) away. Light from Sirius takes 8.6 years to reach Earth.

Step by step solution

01

- Understanding Parallax

Parallax is the apparent shift in position of a nearby star against the background of distant objects due to Earth's orbit around the Sun. It is measured in arcseconds.
02

- Calculating Distance in Parsecs

The distance to a star in parsecs can be calculated using the formula: \[ d = \frac{1}{p} \]where \(d\) is the distance in parsecs and \(p\) is the parallax in arcseconds. For Sirius, \(p = 0.379\) arcsec. \[ d = \frac{1}{0.379} \approx 2.64 \text{ parsecs} \]
03

- Converting Parsecs to Light-Years

1 parsec is approximately equal to 3.26 light-years. Therefore, to find the distance in light-years: \[ distance_{ly} = 2.64 \times 3.26 \approx 8.6 \text{ light-years} \]
04

- Calculating Light Travel Time

Light travels at a constant speed of about 299,792 kilometers per second (or about 186,282 miles per second). One light-year is the distance that light travels in one year. Hence, if Sirius is 8.6 light-years away, it takes light 8.6 years to reach Earth from Sirius.

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Key Concepts

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

Parallax
Parallax is a fascinating concept used in astronomy to measure distances to nearby stars. Think of it like observing the position of an object from two different angles. Due to Earth's orbit around the Sun, nearby stars appear to shift against the backdrop of distant stars. This apparent shift is known as parallax and is measured in arcseconds. The smaller the parallax, the farther away the star is. By understanding parallax, astronomers can map out the stars in our galaxy with impressive precision.
Parsec
A parsec is a crucial unit of distance used in astronomy, derived from 'parallax-arcsecond.' It's defined as the distance at which one astronomical unit (AU) subtends an angle of one arcsecond. In simpler terms, if a star has a parallax of one arcsecond, it's one parsec away from Earth. The formula to calculate the distance in parsecs (pc) using parallax (p) is: \( d = \frac{1}{p} \)For example, if a star has a parallax of 0.379 arcseconds, then: \( d = \frac{1}{0.379} \approx 2.64 \) parsecs.So, Sirius, with its parallax of 0.379 arcseconds, is approximately 2.64 parsecs away.
Light-Year
A light-year is another essential astronomical unit used to describe vast distances. It represents how far light travels in one year, which is roughly 9.46 trillion kilometers (about 5.88 trillion miles). Given that one parsec equals approximately 3.26 light-years, converting distances from parsecs to light-years is straightforward. For instance, Sirius is 2.64 parsecs away, so in light-years: \( distance_{ly} = 2.64 \times 3.26 \approx 8.6 \) light-years.This means light from Sirius takes about 8.6 years to reach us here on Earth.
Distance Calculation
Calculating distances in astronomy often relies on the relationship between parallax, parsecs, and light-years. Here's a step-by-step breakdown:

1. Measure the parallax angle of a star in arcseconds.
2. Use the formula \( d = \frac{1}{p} \) to find the distance in parsecs.
3. Convert parsecs to light-years by multiplying by 3.26.

For Sirius, with a parallax of 0.379 arcseconds:
1. \( p = 0.379 \) arcseconds.
2. \( d = \frac{1}{0.379} \approx 2.64 \) parsecs.
3. \( distance_{ly} = 2.64 \times 3.26 \approx 8.6 \) light-years.

Light from Sirius thus takes 8.6 years to reach Earth, offering insight into the vast scales of our universe.

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Most popular questions from this chapter

Citizen science: Go to the website for the Stellar Classification Online Public Exploration (SCOPE) project (http://scope.pari edu/takepart.php. This project uses crowd sourcing to classify stars seen on old photographic plates of photographs taken in the Southern Hemisphere. Create an account, review the science and the FAQ, and then click on "To Take Part" to see some practice examples. Then go to "Classify," choose a photographic plate, and classify a few stars.

Star A and star B are two nearby stars. Star A has a parallactic angle 4 times as large as star \(\mathrm{B}\) 's. Which of the following statements is true? a. Star A is one-quarter as far away as star B. b. Star A is 4 times as far away as star B. c. Star A has moved through space one-quarter as far as star B. d. Star A has moved through space 4 times as far as star \(B\).

A telescope on Mars would be able to measure the distances to more stars than can be measured from Earth because a. the resolution of the telescope would be better. b. Mars has a thin atmosphere. c. it would be closer to the stars. d. the parallax "baseline" would be longer.

Betelgeuse (in Orion) has a parallax of \(0.00763 \pm 0.00164\) arcsec, as measured by the Hipparcos satellite. What is the distance to Betelgeuse, and what is the uncertainty in that measurement?

Capella (in the constellation Auriga) is the sixth brightest star in the sky. When viewed with a high-power telescope, it is clear that Capella is actually two pairs of binary stars: the first pair are G-type giants; the second pair are M-type main sequence stars. What color does Capella appear to be? a. red b. yellow c. blue d. color cannot be determined from this information
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