|
|  |
Introduction
This problem vastly oversimplifies the complexities of astrophysics -- turning a multibody relativity problem into an idealized two-body mechanics problem. Nonetheless, this problem provides both AP Physics B and C students with a valuable opportunity to apply their skills to real-world data sets and to calculate answers that are at least within the realm of physical reason. In addition, this problem provides the physics instructor with an opportunity to highlight the central role of physics in answering questions faced by contemporary astronomy.
Problem Statement
The center of our galaxy, the Milky Way, is normally obscured from our view by the opaque gas and dust of the galactic disk. Its location in the night sky is indicated in the following picture.
Note: Unless otherwise indicated, all images are taken from NASA's Astronomy Picture of the Day Web site: http://antwrp.gsfc.nasa.gov/apod/ap971111.html.
Because the center of our galaxy appears to lie in the constellation of Sagittarius, this object is known as Sagittarius A.
The Milky Way, seen from outside our galaxy, would look similar to spiral galaxy NGC 7331, shown below.
http://antwrp.gsfc.nasa.gov/apod/ap040701.html
The massive structure of the Milky Way is held together by gravity, including mutual attraction between the visible stars and gas, unseen dark matter, and something else -- a HUGE mass located at the galactic center. In this problem, we will use Newtonian mechanics to estimate the mass of this object. To simplify our calculations, we will discount relativistic effects, as well as the influence of dark matter and smaller masses such as "normal" sized stars.
Although the center of the galaxy cannot be seen in visible light, both radio waves and infrared waves (different forms of electromagnetic radiation) readily penetrate the dust and gas, obscuring our view. The following image shows the galactic center in radio wavelengths. This type of image highlights hot gas and dust near the galactic center.
http://antwrp.gsfc.nasa.gov/apod/ap020803.html
Here Sgr A represents Sagittarius A, the massive object that we will be studying.
For our purposes, we will focus on infrared images, which highlight the positions of individual stars. Select the following link to see the movement of stars near the galactic center over a several-year period.
www.mpe.mpg.de/ir/GC/images/movie2003.mpg
The Web site noted above provides the following data, which has been extracted from the image.
Figure 1: Motion of star S0-1 (www.astro.ucla.edu/~ghez/gc_nat.html)
Problem 1: Estimating the Mass of Sgr A Using S0-1
Consider the motion of star S0-1 shown in Figure 1. Although the orbit shown is an ellipse, we will assume a circular orbit for the purposes of this problem (note that part of the distortion is caused by the viewing angle).
(A) The galactic center is located 28,000 light-years from the earth. Given that light travels at 3.0 x 108 m/s, what is the distance to the galactic center in meters?
(B) In Figure 1, distances are given in arcseconds (arcsec). Since a degree is divided into 60 minutes, and a minute is divided into 60 seconds, an arcsec is 1/3600 of a degree.
Estimate the distance represented by 0.1 arcsec on this image.
(C) Assuming a circular orbit, what is the average speed of S0-1? (You will have to estimate distances directly from Figure 1.)
(D) What is the average acceleration of S0-1?
(E) Using Newton's law of gravity, estimate the mass of Sagittarius A. Assume that Sgr A remains motionless.
(F) The mass of the sun is 2.0 x 1030 kg. How many solar masses does Sgr A represent?
Problem 2: Estimating the Mass of Sgr A Using S0-2
We will now estimate the mass of Sgr A using star S0-2. Since this path is far from circular, we will use the conservation of energy, rather than Newton's laws, for this calculation. A more detailed data set for S0-2 is shown in Figure 2.
Figure 2: Data set for S0-2 (www.mpe.mpg.de/ir/GC/index.php).
Note that the units of both axes are arcsec (").
(A) Estimate the speed of S0-2 when it is farthest from Sgr A (do not take into account viewing angle).
(B) Estimate the speed of S0-2 when it is closest to Sgr A.
(C) Use the conservation of energy to estimate the mass of Sgr A.
(D) Based on Figure 2, how many solar masses does Sgr A represent?
Problem 3: Reflections
(A) The accepted value for the mass of Sgr A is 2.6 million solar masses. How do your estimated values for the mass of Sgr A compare with the accepted value? Suggest possible explanations for these differences.
(B) The planet Pluto orbits about 5.6 x 109 km from our sun. Assuming that the orbit of Pluto represents the "edge" of the solar system, we can calculate the density of the solar system as:
.
Based on your estimates, what is the minimum density of Sgr A? How does this compare to the average density of our solar system?
(C) Stars keep themselves from collapsing by turning hydrogen into helium. Once their nuclear fuel is exhausted, they collapse. According to our present understanding, if a star has a mass 10 times greater than our sun, then its final collapse cannot be stopped by any known atomic force -- and the star becomes a black hole. Based on this information, how likely is it that the object in the center of the Milky Way galaxy is a black hole? Are there other possible explanations for the observed mass and density values calculated above?
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.
|
Course Home Pages