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Chapter 14: Problem 53

Separation of Pluto and Charon. Pluto is located about 4.5 billion km from Earth and has a maximum observable separation from Charon of about 0.9 arcseconds. Access the AIMM (Active Integrated Media Module) called “Small-Angle Toolbox” in Chapter 1 of the Universe Web site or eBook. Use this AIMM and the above data to determine the distance between Pluto and Charon. How does your answer compare with the value given in the text?

Short Answer

Expert verified
The separation distance between Pluto and Charon is approximately 19,600 km.

Step by step solution

01

Convert separation to radians

Firstly, we need to convert the separation of 0.9 arcseconds between Pluto and Charon into radians. Since there are approximately 206265 arcseconds in a radian, we get \(0.9/206265 = 4.36*10^{-6}\) radians.
02

Convert distance to AU

Then, convert the distance to Pluto from kilometers into Astronomical Units (AU). The average distance from the Earth to the Sun is 1 AU, which is about 149.6 million kilometers. So, \(4.5*10^9/149.6*10^6 = 30\) AU.
03

Apply small-angle formula

Now we can apply the small-angle formula \(d = r*θ\), where d is the actual separation distance between Pluto and Charon, r is the radius or distance of Pluto from Earth, and θ is the angle in radians. We substitute r = 30 AU and θ = \(4.36*10^{-6}\) radians into the equation and solve for d, which gives \(d = 30 * 4.36*10^{-6} = 1.31*10^{-4}\) AU.
04

Convert answer back to kilometers

In order to back-convert the computed separation distance into kilometers, we use 1 AU = 149.6 million km. Thus, \(1.31*10^{-4} * 149.6*10^6 = 1.96*10^4\) km.

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Most popular questions from this chapter

(a) Find the semimajor axis of the orbit of an object whose period is \(3 / 2\) of the orbital period of Neptune. How does your result compare to the semimajor axis of Pluto's orbit? (b) A number of Kuiper belt objects called plutinos have been discovered with the same orbital period and hence the same semimajor axis as Pluto. Explain how these objects can avoid colliding with Pluto.

Are there any trans-Neptunian objects that are not members of the Kuiper belt? Are there any members of the Kuiper belt that are not trans-Neptunian objects? Explain.

How do the orientations of Uranus's and Neptune's magnetic axes differ from those of other planets?

If you have access to a large telescope, make arrangements to view Neptune. Like Uranus, Neptune is best seen at or near opposition and can most easily be found using a star chart. Can you see a disk? What is its color?

Use the Stary Night Enthusiast \({ }^{\mathrm{TM}}\) program to observe Pluto and Charon. First select Options \(>\) Viewing Location from the menu. At the top of the Viewing Location dialog box, select position hovering over and Pluto in the drop boxes. Then click on the center of the map of Pluto that appears in the dialog window and click the Set Location button. Use the elevation buttons in the toolbar to increase the distance from the surface of Pluto to about \(35,000 \mathrm{~km}\). Use the Location scroller (hold down the Shift key while holding down the mouse button and moving the mouse, to rotate the view around Pluto. In the toolbar, set the Time Flow Rate to 1 hour, then click on the Run Time Forward button (a triangle that points to the right). (a) Estimate Charon's orbital period. (b) By following a spot on Pluto's surface, estimate Pluto's rotation period. How does it compare to your answer in part (a)? (c) Select Options > Viewing Location from the menu and set the dropdown boxes at the top of the Viewing Location dialog window to read position moving with and Pluto. Then select the Above orbital plane option and click the Set Location button. Open the Find pane and double-click the entry for the Sun to center the Sun in the view. Set the Time Flow Rate to 1 year, and click on the Run Time Forward button to see the apparent motion of the Sun as seen from Pluto. Observe the motion for several centuries of simulated time. Does the Sun always appear to move at the same speed? Use the properties of Pluto's orbit to explain why or why not.
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