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Chapter 23: Problem 37

The mass of our Galaxy interior to the Sun's orbit is calculated from the radius of the Sun's orbit and its orbital speed. By how much would this estimate be in error if the calculated distance to the galactic center were off by \(10 \%\) ? By how much would this estimate be in error if the calculated orbital yelocity were off by \(10 \%\) ? Explain your reasoning.

Short Answer

Expert verified
The estimated mass of the Galaxy interior to the Sun's orbit would be in error by 30% if the calculated distance to the galactic center were off by 10%, and in error by 20% if the calculated orbital velocity were off by 10%.

Step by step solution

01

Calculation of error in mass due to error in distance

The mass M is proportional to the cube of the distance r. Thus, if the distance is off by \(10 \%\), the mass is off by \(3 \times 10\% = 30\%\), according to the power rule. If the distance was underestimated, then the mass will be underestimated by 30%. If the distance was overestimated, the mass will be overestimated by the same percentage.
02

Calculation of error in mass due to error in velocity

The mass M is proportional to the square of the orbital speed v. Thus, if the velocity is off by \(10 \%\), the mass is off by \(2 \times 10\% = 20\%\). If the speed was underestimated, then the mass will be underestimated by 20%. And if the speed was overestimated, then the mass will be overestimated by the same percentage.
03

Conclusion

So, underestimation or overestimation by \(10\%\) of the distance to the galactic center and the Sun's orbital speed will lead to errors of \(30\%\) and \(20\%\) respectively in the calculated mass of the Galaxy interior to the Sun's orbit.

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Most popular questions from this chapter

How do astronomers determine how fast the Sun moves in its orbit around the Galaxy? How does this speed tell us about the amount of mass inside the Sun's orbit? Does this speed tell us about the amount of mass outside the Sun's orbit?

What are RR Lyrae stars? Why are they useful for determining the distance from our solar system to the center of the Galaxy?

Use the Starry Night Enthusiast \(\mathrm{TM}^{\mathrm{M}}\) program to observe the Milky Way. (a) Display the entire celestial sphere by selecting Favourites \(>\) Guides \(>\) Atlas. Select View \(>\) Stars \(>\) Milky Way to display this galaxy. Select Options \(>\) Stars \(>\) Milky Way, move the Brightness slide-bar to the far right to brighten the Milky Way and click OK. In the View menu, ensure that the Scrollbars are activated and use them to look at different parts of the Milky Way. Can you identify the direction toward the galactic nucleus? In this direction the Milky Way appears broadest. Open the Find pane, enter Sagittarius in the Query box and press Enter to center on this constellation to check your identification. (b) Use this full-sky view to determine the orientation of the plane of the Galaxy with respect to the celestial sphere. Move the vertical scrollbar to its central position to display the Celestial Equator as a horizontal line across the lower part of the view. Move the horizontal scrollbar until the Milky Way is centered upon the view. Estimate the angle between the Milky Way and the celestial equator on the screen. How well aligned is the plane of the Milky Way with the plane of the Earth's equator? (c) A third plane of interest is that of the ecliptic, which is shown as a green line. Use the scrollbars to adjust the view so that the ecliptic appears as a straight line rather than as a curve, thereby ensuring that you are viewing in a direction that lies in the ecliptic plane. Use the horizontal scrollbar to move the view to where you can see where the ecliptic crosses the Milky Way. Estimate the angle between the Milky Way and the ecliptic on the screen. How well aligned is the plane of the Milky Way to the ecliptic, the plane of the Earth's orbit around the Sun? (d) Click on Home in the toolbar to return to your home view, stop Time Flow and set the local time to midnight (12:00:00 A.M.). Select Options > Stars > Milky Way, move the Brightness slide-bar to the far right to brighten the Milky Way and click OK. Adjust the date to January 1, then February 1, and so on. In which month is the galactic nucleus highest in the sky at midnight, so that it is most easily seen from your location?

The disk of the Galaxy is about \(50 \mathrm{kpc}\) in diameter and 600 pc thick. (a) Find the volume of the disk in cubic parsecs. (b) Find the volume (in cubic parsecs) of a sphere \(300 \mathrm{pc}\) in radius centered on the Sun. (c) If supernovae occur randomly throughout the volume of the Galaxy, what is the probability that a given supernova will occur within \(300 \mathrm{pc}\) of the Sun? If there are about three supernovae each century in our Galaxy, how often, on average, should we expect to see one within \(300 \mathrm{pc}\) of the Sun?

Why are infrared telescopes useful for exploring the structure of the Galaxy? Why is it important to make observations at both near-infrared and far- infrared wavelengths?
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