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Chapter 17: Problem 48

The star HIP 72509 has an apparent magnitude of \(+12.1\) and a parallax angle of \(0.222\) arcsecond. (a) Determine its absolute magnitude. (b) Find the approximate ratio of the luminosity of HIP 72509 to the Sun's luminosity.

Short Answer

Expert verified
The absolute magnitude of star HIP 72509 is approximately \(7.98\) and the ratio of the luminosity of HIP 72509 to the luminosity of the Sun is approximately \(0.026\).

Step by step solution

01

Use the distance modulus to calculate distance

The distance modulus equation is \(m - M = 5 \log_{10}(d) - 5\), where \(m\) is the apparent magnitude, \(M\) is the absolute magnitude, and \(d\) is the distance in parsecs. Knowing that the parallax angle in arcseconds is the reciprocal of the distance in parsecs \(p = 1/d\), we can first find that with a parallax \(p = 0.222\) arcsec, the distance to HIP 72509 is \(d = 1/p = 1/0.222 \approx 4.50\) parsecs.
02

Use the distance to find the absolute magnitude

From the previous step, we know the distance and the apparent magnitude, we can rearrange the distance modulus equation to solve for the absolute magnitude: \(M = m - 5 \log_{10}(d) + 5 = 12.1 - 5 \log_{10}(4.50) + 5 \approx 7.98\).
03

Use absolute magnitude to find the luminosity ratio

We can find the ratio of the luminosity of HIP 72509 to the Sun using the equation \(L/L_{\odot} = 10^{0.4(M_{\odot} - M)}\), where \(L\) is the luminosity of the star, \(L_{\odot}\) is the luminosity of the Sun, \(M_{\odot}\) is the absolute magnitude of the Sun, and \(M\) is the absolute magnitude of the star. Using \(M_{\odot} = 4.83\) and \(M = 7.98\), we get \(L/L_{\odot} = 10^{0.4(4.83 - 7.98)} \approx 0.026\), which is the ratio of the luminosity of HIP 72509 to the Sun's luminosity.

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Most popular questions from this chapter

What is the inverse-square law? Use it to explain why an ordinary lightbulb can appear brighter than a star, even though the lightbulb emits far less light energy per second.

Find the average distance from the Sun to Neptune in parsecs. Compared to Neptune, how many times farther away from the Sun is Proxima Centauri?

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Suppose that a dim star were located 2 million AU from the Sun. Find (a) the distance to the star in parsecs and (b) the parallax angle of the star. Would this angle be measurable with present-day techniques?
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