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Name:__________________________________________ Date:_________________ PHYS 1 10 L Lab # 10 Dark Matter^1 Instructions: Please read and follow the steps described below and answer all questions. Part #1 Weighing the Milky Way Galaxy Introduction: The Milky Way appears to extend about 10 kiloparsecs (10,000 pc or 50,000 light years) out from the center. Beyond that distance, the density of stars and gas drops off, although a few more distant stars can be found. The total mass of stars and gas (MStars+Gas) in the Milky Way is observed to be about 1 ร 1011 times the mass of the Sun, all contained within a disk of radius 10 kiloparsecs. According to Newtonโs Law of Gravity, the velocity of stars orbiting far from the galaxy should be proportional to the square root of the total mass divided by the distance from the center. ๐ = 2 ร 10 โ^3 โ
In this equation M is the mass of the Galaxy, in units of solar masses, R is the distance from the center of the Galaxy in kiloparsecs, and V is the velocity in kilometers per second. Step 1. Using the above equation, compute the velocity that stars or globular clusters orbiting the galaxy should have at distances of 10, 12, and 16 kiloparsecs from the Galactic center. Enter your values in the table below. Distance (in Kpc) Mass Enclosed Predicted Orbital Velocity (km/s) 10 1 ร 1011 solar masses 200 (^12 1) ร 1011 solar masses 182. 16 1 ร 1011 solar masses 158. (^1) Modified from Mini-Lab Activities for Use in Introductory Astronomy Courses , by Caty Pilachowski, Indiana University Bloomington. Learning Objectives: In this lab assignment you will conduct a series of inquiries regarding the evidence for dark matter in galaxies based on the orbital motion of stars (rotation curve) in spiral galaxies.
The graph given below is a plot of the observed velocity of stars orbiting around the center of the Milky Way, as a function of distance from the Galactic Center. Astronomers call a plot like this a โrotation curve.โ Stars orbit the Galaxy following Newtonโs Law of Gravity, and their orbital speed depends on the total mass contained inside their orbit. The orbital velocities of stars rise quickly from the center as we move out in radius. This is because the center of the Galaxy is dense, so that the mass inside a circle rises quickly with increasing orbital radius. Further out, the density of stars is less, so the mass contained inside a given radius increases more slowly, and the rotation curve flattens out. The wobbles in the curve are due to the spiral arms Step 2. Plot the predicted velocities you calculated in Step 1 (and entered in the table on Page 1) on the observed rotation curve of the Milky Way given below. Question 1: How does the observed orbital speed of distant stars around the center of the Milky Way compare to your prediction based on Newtonโs Law of Gravity? The calculations trend downward linear where the graph shows a more steady rotation speed Question 2: From the graph below, what is the observed orbital speed of stars at a distance of 16 kiloparsecs from the Galactic Center? 235
Gravitational Mass : Below is a rotation curve for NGC 2742 (also known as UGC 4779). The graph plots the radial velocity of stars as a function of distance from the center of the galaxy. The radial velocity of NGC 2742 (or UGC 4779) rises quickly from the center as we move out in radius and reaches a maximum at some distance out from the galactic center. The velocity rises outward because the amount of mass contained is increasing as we move outward. The speed of rotation at a given radius is proportional to the square root of the mass inside that radius and to the square root of the reciprocal of the distance from the center of the galaxy (equation on Page 1). Step 4. At one of the radii given, determine the absolute value (positive value) of the rotational velocity of the galaxy and enter it in the table on Page 5 ( do this for only one value ). Note that the value for a radius of 2 kpc has been included as an example. Step 5. Use the values of the radius and velocity to determine the mass of the galaxy inside the radius you selected. According to Newton's law of gravity ๐ =
๐^2 ๐
where G is the gravitational constant, V is the rotational velocity, and M is the mass contained inside of radius R. Mass is measured in solar masses, radius is measured in kiloparsecs (kpc), and velocity is measured in km/s. Using these units, the gravitational constant has a value of
4 ร 10 โ^6. Record your result in the โGravitational Massโ column in the table. Your number should be large. This is a galaxy we are dealing with and it contains a lot of stars! Use scientific notation! Radius (kpc) Rot. Vel. (km/s) Grav. Mass (solar masses) Luminosity (solar lum.) Luminous Mass (solar masses) Ratio Lum/Grav Mass 2 kpc
100 ๐ ร ๐๐๐^ ๐ ร ๐๐๐^ ๐ ร ๐๐๐^ 0.
4 kpc 125 1.56x10^9 1.5x10^9 3.0x10^9 1. 6 kpc 8 kpc 10 kpc The Luminous Mass - Now that you have found the total mass of the galaxy, we will investigate how much of that mass comes from matter we can see - the stars! The figure on the right is a graph of how many solar luminosities of light NGC 2742 produces as a function of distance from the center of the galaxy. Step 6. At the radius you selected in Step 4, find the value for the luminosity within that radius and record it in the table above. The luminosity is given in solar luminosities. Now that you have measured how much light is coming from NGC 2742, you can estimate the mass of the stars that produced that light. It would be easy if, for every solar luminosity of light we measure, we can assume that one solar mass of stars is producing it. Unfortunately, some light is blocked by dust in the galaxy we are observing, and the galaxy is comprised of a mix of high mass and low mass stars. The luminosity of a star depends on its mass; massive stars produce more light per solar mass, while low mass stars produce less light per solar mass. Much
