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Chapter 17: Problem 50

A certain type of variable star is known to have an average absolute magnitude of \(0.0\). Such stars are observed in a particular star cluster to have an average apparent magnitude of \(+14.0\). What is the distance to that star cluster?

Short Answer

Expert verified
The distance to the star cluster is \( d = 10^{3.8} \) parsecs.

Step by step solution

01

Understanding the distance modulus equation

The distance modulus equation is defined as \( m - M = 5 \log(d) - 5 \) where \( m \) is the apparent magnitude, \( M \) is the absolute magnitude, and \( d \) is the distance which we want to find.
02

Plugging in the given values

Now substitute the given values \( m = +14.0 \) and \( M = 0.0 \) into the distance modulus equation, we get \( +14.0 - 0.0 = 5 \log(d) - 5 \).
03

Simplifying the equation

After simplifying we get \( 14 = 5 \log(d) - 5 \). Now let's add 5 to both sides to get rid of the '-5' on the right side of the equation. This gives us \( 19 = 5 \log(d) \).
04

Solving for log(d)

To further isolate \( \log(d) \), we divide both sides by 5, resulting in \( \log(d) = 19/5 = 3.8 \).
05

Solving for d

Now to find \( d \), the distance to the star cluster, we use the property of logarithm that converts log base 10 to a power of 10: \( d = 10^{3.8} \).

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Most popular questions from this chapter

Sketch the light curve of an eclipsing binary consisting of two identical stars in highly elongated orbits oriented so that (a) their major axes are pointed toward the Earth and (b) their major axes are perpendicular to our line of sight.

It is desirable to be able to measure the radial velocity of stars (using the Doppler effect) to an accuracy of \(1 \mathrm{~km} / \mathrm{s}\) or better. One complication is that radial velocities refer to the motion of the star relative to the Sun, while the observations are made using a telescope on the Earth. Is it important to take into account the motion of the Earth around the Sun? Is it important to take into account the Earth's rotational motion? To answer this question, you will have to calculate the Earth's orbital speed and the speed of a point on the Earth's equator (the part of the Earth's surface that moves at the greatest speed because of the planet's rotation). If one or both of these effects are of importance, how do you suppose astronomers compensate for them?

Observe the eclipsing binary Algol ( \(\beta\) Persei), using nearby stars to judge its brightness during the course of an eclipse. Algol has an orbital period of \(2.87\) days, and, with the onset of primary eclipse, its apparent magnitude drops from \(2.1\) to 3.4. It remains this faint for about 2 hours. The entire eclipse, from start to finish, takes about 10 hours. Consult the "Celestial Calendar" section of the current issue of Sky or Telescope for the predicted dates and times of the minima of Algol. Note that the schedule is given in Universal Time (the same as Greenwich Mean Time), so you will have to convert the time to that of your own time zone. Algol is normally the second brightest star in the constellation of Perseus. Because of its position on the celestial sphere (R.A. \(=3^{\mathrm{h}} 08.2^{\mathrm{m}}\), Decl. \(\left.=40^{\circ} 57^{\prime}\right)\), Algol is readily visible from northern latitudes during the fall and winter months.

Kapteyn's star (named after the Dutch astronomer who found it) has a parallax of \(0.255\) arcsec, a proper motion of \(8.67\) arcsec per year, and a radial velocity of \(+246 \mathrm{~km} / \mathrm{s}\). (a) What is the star's tangential velocity? (b) What is the star's actual speed relative to the Sun? (c) Is Kapteyn's star moving toward the Sun or away from the Sun? Explain.

Explain the difference between a star's apparent brightness and its luminosity.
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