91Ó°ÊÓ

Spectroscopic parallax: Stars have an apparent magnitude \(m\), which is the brightness of light reaching Earth. They also have an absolute magnitude \(M\), which is the intrinsic brightness and does not depend on the distance from Earth. The difference \(S=m-M\) is the spectroscopic parallax. Spectroscopic parallax is related to the distance \(D\) from Earth, in parsecs, 35 by $$ S=5 \log D-5 $$ a. The distance to the star Kaus Astralis is \(38.04\) parsecs. What is its spectroscopic parallax? b. The spectroscopic parallax for the star Rasalhague is 1.27. How far away is Rasalhague? c. How is spectroscopic parallax affected when distance is multiplied by 10 ? d. The star Shaula is \(3.78\) times as far away as the star Atria. How does the spectroscopic parallax of Shaula compare to that of Atria?

Step by step solution

01

Calculate Spectroscopic Parallax for Kaus Astralis

Given the distance to Kaus Astralis, \(D = 38.04\) parsecs, we need to find the spectroscopic parallax \(S\) using the formula: \[ S = 5 \log D - 5 \]Substitute \(D = 38.04\) into the formula. First, compute \(\log D\):\[ \log D = \log 38.04 \approx 1.5802 \]Then compute \(S\):\[ S = 5(1.5802) - 5 = 7.901 - 5 = 2.901 \]

02

Determine Distance to Rasalhague

Given the spectroscopic parallax \(S = 1.27\), we need to find the distance \(D\) using the formula: \[ S = 5 \log D - 5 \]Rearrange to find \(D\): \[ 5 \log D = S + 5 \]\[ \log D = \frac{S + 5}{5} = \frac{1.27 + 5}{5} = \frac{6.27}{5} = 1.254 \]Now find \(D\) by taking the antilog:\[ D = 10^{1.254} \approx 17.98 \text{ parsecs} \]

03

Effect of Multiplying Distance by 10 on Spectroscopic Parallax

Consider how \(S = 5 \log D - 5\) changes if \(D\) is multiplied by 10. This modifies \(D\) to \(10D\) and results in:\[ S' = 5 \log(10D) - 5 \]Recognizing the logarithm property, we can express:\[ \log(10D) = \log D + \log 10 = \log D + 1 \]So,\[ S' = 5(\log D + 1) - 5 = 5 \log D + 5 - 5 = 5 \log D \]Thus, when the distance is multiplied by 10, the spectroscopic parallax increases by 5.

04

Compare Spectroscopic Parallax for Shaula and Atria

The distance to Shaula is \(3.78\) times the distance to Atria, i.e., \(D_{\text{Shaula}} = 3.78D_{\text{Atria}}\).Using the formula \[ S = 5 \log D - 5 \], we have:\[ S_{\text{Shaula}} = 5 \log(3.78D_{\text{Atria}}) - 5 \]Using the property of logarithms:\[ \log(3.78D_{\text{Atria}}) = \log 3.78 + \log D_{\text{Atria}} \]Hence:\[ S_{\text{Shaula}} = 5 (\log 3.78 + \log D_{\text{Atria}}) - 5 = S_{\text{Atria}} + 5 \log 3.78 \]Calculate the increment:\[ \log 3.78 \approx 0.5775 \]Thus,\[ S_{\text{Shaula}} = S_{\text{Atria}} + 5 \times 0.5775 = S_{\text{Atria}} + 2.8875 \]Shaula's spectroscopic parallax is greater than Atria’s by approximately 2.89.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Apparent Magnitude

In astronomy, apparent magnitude is a measure of how bright a star or other astronomical object appears from Earth. It's important to remember that this is not an absolute measure of the object's brightness, but rather how bright it seems to us, as viewed from our planet.

• The scale of apparent magnitude is logarithmic. A difference of 1 in magnitude means a difference in brightness by a factor of about 2.512.
• Brighter objects have lower apparent magnitudes. For instance, a star with an apparent magnitude of 1 is brighter than a star with a magnitude of 2.
• Negative magnitudes are possible, and they indicate an object brighter than all stars in the night sky. The Sun, for example, has an apparent magnitude of about -26.74.

This system allows astronomers to compare the brightness of celestial objects easily, but it doesn't provide information about the true luminosity of those objects.

Absolute Magnitude

Absolute magnitude, on the other hand, gives us a way to compare the intrinsic brightness of astronomical objects. It's defined as the apparent magnitude a star would have if it were located exactly 10 parsecs (about 32.6 light-years) from Earth. By standardizing this measurement, astronomers can compare the true brightness of objects regardless of their actual distance from us.

• Like apparent magnitude, the scale is logarithmic. A change of 1 in absolute magnitude reflects a roughly 2.512 times change in brightness.
• Absolute magnitude helps in understanding the luminosity of stars and galaxies, providing insight into their energy output.
• By understanding both apparent and absolute magnitudes, astronomers can calculate the distance to an astronomical object using the formula for spectroscopic parallax.

This concept is especially useful in cosmology and the study of stellar evolution, as it gives a clear picture of the power emitted by stars beyond the influence of distance.

Distance in Parsecs

In the field of astronomy, the parsec is a key unit of length. It's commonly used to describe astronomical distances, offering an alternative to the more familiar light-year. A single parsec corresponds to about 3.26 light-years, which is approximately 3.086 x 10^13 kilometers.

• The term 'parsec' comes from "parallax of one arcsecond." It's the distance at which one astronomical unit subtends an angle of one arcsecond.
• Parsecs are particularly useful in many astronomical calculations, including the determination of a star's distance using spectroscopic parallax.
• The relationship between a star's apparent and absolute magnitudes can be expressed in terms of its distance in parsecs, which is crucial for identifying where heavenly bodies lie in the universe.

By using parsecs, astronomers can carry out calculations more comfortably and understand distances on a cosmic scale, fundamental for exploring the universe's structure and size.

Logarithms in Astronomy

Logarithms play a pivotal role in many areas of astronomy, including magnitude scales and spectroscopic parallax calculations. They simplify the mathematical treatment of the vast ranges in brightness and distance found in the universe.

• The logarithmic scale allows astronomers to manage and process the enormous differences in brightness or luminosity of celestial objects efficiently.
• For spectroscopic parallax, the relationship between apparent magnitude, absolute magnitude, and distance employs logarithms to express these wide-ranging figures in manageable equations.
• The equation for spectroscopic parallax, \( S = 5 \log D - 5 \), demonstrates how logarithms are used to relate star distance with apparent and absolute magnitudes.

By understanding the application of logarithms, students and astronomers can better handle the calculations that arise in their study of stars and the structure of the universe, thus providing deeper insights into their work.