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Introduction
It is often difficult for physics students to transfer problem-solving skills learned in one context to another despite similar underlying principles. For example, students who can solve a diffraction grating problem with light (AP Physics B, 1996, question 3) become confused when the same ideas are applied to sound (AP Physics B, 2004, question 4). One way to address this problem is to expose students to the basic ideas of physics in a wide variety of contexts. In this way, irrespective of guise, students can better recognize important principles. This month's problem features a novel type of diffraction grating that is currently being used in infrared astronomy. It is hoped that by connecting student learning to real-world science, both depth of understanding and student interest will be enhanced.
Problem Statement
Astronomers observe the universe at different wavelengths of light for two principal reasons.
1. Certain wavelengths of light are able to penetrate the gas and dust between the Earth and various astronomical objects, thus allowing their observation. For example, the center of our Milky Way galaxy cannot be viewed directly in visible light, but it can be easily seen at X-ray wavelengths.
--> For Milky Way center in visible light see
http://antwrp.gsfc.nasa.gov/apod/ap020923.html
--> For Center of Milky Way galaxy in X-ray light see
http://antwrp.gsfc.nasa.gov/apod/ap030712.html
2. The photons given off by different astronomical events depend upon the event's energy (E = hf). Since most gases in space are relatively cool, they radiate energy at very low frequencies (long wavelengths). The differences in detail that are visible at different wavelengths can be easily seen when comparing the area near Orion. Cool, otherwise hidden, gas clouds are easily visible at infrared light wavelengths.
--> For Orion Nebula in Visible and Infra-Red Light see
www.ipac.caltech.edu/Outreach/Edu/animation.html
Unfortunately, some of the most useful observational wavelengths are absorbed by the water vapor in the Earth's atmosphere. In order to address this problem, NASA, in its SOFIA program, is currently mounting an infrared telescope in a Boeing 747 airliner. The 747 will fly at 41,000 feet in order to keep the scope above much of the water vapor that plagues ground-based infrared astronomy. In addition, the telescope will be cooled to -40° C in order to limit the infrared radiation emitted by the walls of the telescope. Without such cooling, observing in the infrared would be like using an optical telescope in a Wal-Mart parking lot!
--> For Sophia Aircraft and Telescope Image see
www.sofia.usra.edu/Gallery/aircraft/AC/AC19980009.html
More information about the SOFIA program can be found at
www.sofia.usra.edu/Sofia/sofia.html .
The Instrument
Unlike a space based telescope, the observing instruments mounted in SOFIA can be changed readily to meet a variety of observing tasks. One instrument that will be used is the University of Texas' EXES (Echelon X Echelle Spectrograph), which utilizes a unique aluminum reflection diffraction grating to study the gas clouds that are forming around proto-stars. Information on the EXES program is available at http://www.as.utexas.edu/astronomy/research/exes.html. We will be using a modified instrument design in order to simplify calculations and avoid the use of non-AP formulas. For example, the actual EXES instrument uses two diffraction gratings, one to "spread" light sideways and one to "spread" light vertically - the "cross" (i.e., X) Echelle bit).
The Challenge
When gas is spinning around a star, the frequencies of light that the gas emits change slightly due to the Doppler effect (light from gas moving toward us has a slightly smaller wavelength and higher frequency than light from gas that is traveling away from us). The following image shows the wavelength "split" for the H2 gas surrounding a hypothetical star.
www.as.utexas.edu/astronomy/research/exes_sim.html
Note that a star with a continuous distribution of gas orbiting it has a different intensity profile than a star with a discontinuous disk.
(A) Explain qualitatively how this graph is consistent with the basic principles of electromagnetic waves and mechanics. Do not use the term "Doppler effect" in your explanation.
(B) The equation used by astronomers to find the Doppler shift for light is 
, where v is the speed of the object, λ is the "rest" wavelength of the light, and c is the speed of light. Using this equation, find the wavelengths that correspond to the two "peaks" that form when 17.0 µm light is Doppler-shifted by the movement of gas around a protostar. Note: µm (micron or micrometer) = 10-6 m.
(C) The following diagram shows the path of light as it enters the EXES instrument from the SOFIA telescope, diffracts on the grating inside EXES, and reflects onto the CCD detector.
Infrared radiation enters the EXES instrument at the top of the diagram (indicated in yellow -- not the beam's actual color!). The incoming radiation strikes an aluminum parabolic mirror (purple), which makes the wave front parallel. The parallel beam strikes the aluminum reflection grating and is diffracted (blue). Once the beam diffracts, the infrared radiation strikes a second aluminum parabolic mirror (also purple), which makes the radiation parallel again. The effective distance between the diffraction grating and the detector is represented by the blue line. The red line that leads to the detector (not shown) can be of any arbitrary length.
In order to increase the resolution on the detector, EXES uses 1500th-order diffraction lines. The grating has a step spacing of 0.3 inch/grating and is 1.0 meter in length. By taking measurements directly off the schematic diagram, estimate the separation distance of the two wavelengths on the CCD detector.
(D) Is your estimated separation distance reasonable? Explain.
Click here to view the answers and commentary!
David Castro taught AP Physics (B and C), AP Calculus (AB and BC), and AP U.S. and European History in a teaching career spanning 14 years, including 5 years as a master AP Physics teacher. In 1997, he received a Special Recognition Teaching Award, and in 2002 his combined AP Physics and AP Calculus syllabus was published in the AP Physics Teacher's Guide. Active as an AP Physics consultant in the Southwest Region since 1995, his areas of expertise include Pre-AP middle school science, AP Vertical Teams, as well as interdisciplinary physics/calculus. He also serves as a Reader for AP Physics. Mr. Castro recently joined the Charles A. Dana Center at the University of Texas, where he continues to focus on providing support for science educators.
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