Copy of Black Holes

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Santa Monica College *

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Astronomy

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Apr 26, 2024

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pdf

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Black Holes INTRODUCTION The existence of black holes was originally predicted by Einstein’s Theory of General Relativity over one hundred years ago. But without any observational evidence of their existence, they remained merely a theoretical possibility for many decades. Today, we know without a doubt that black holes are real, physical objects in space. Indeed, we have numerous independent observations that point conclusively to their existence. One of the most compelling sources of evidence for black holes is shown in the animated loop below. Since 1995, astronomers from UCLA have been observing a group of stars near the very center of our Milky Way galaxy - a region where astronomers predict that there is a supermassive black hole. While the black hole itself is invisible, its gravitational effect on nearby stars can be observed. In this lab, we’re going to try to make sense of this data and use it to determine the size of the black hole at the center of our galaxy. UCLA’S GALACTIC CENTER DATA The animation above looks complicated (because it is), but let’s walk through it step by step. Watch this video of UCLA astronomer and Nobel Laureate, Andrea Ghez , describing how the motion of the stars is recorded by taking a series of photographs through a telescope. Notice in the upper right-hand corner of the animation that the date of each observation is given. This data set was captured between 1995 and 2012. The path of the stars over time is added onto the animation so that we can more easily track the orbit of each star.
HOW BIG IS THE BLACK HOLE? When discussing the “size” of a black hole, we can talk about two different properties: 1) the mass (M) of the black hole, as measured in solar masses, or 2) the Schwarzschild Radius (R sh ), which is the area around the black hole from which nothing can escape, not even light. We usually measure R sh in units of kilometers. Fortunately, both properties are related to each other through the equation for the Schwartzschild Radius. R sh = 3·M So, if we can find the mass of the black hole in solar masses, then we can simply multiply by three to find its Schwarzschild Radius in kilometers. CALCULATING MASS To calculate the mass of the black hole, we need to examine the orbital motion of the stars around it. In particular, we need to determine the orbital period (P) and the semi-major axis (a) of the orbits. The orbital period (P) is the time (in years) that it takes an object to complete one orbit. The semi-major axis (a) is one half of the major axis of the elliptical orbit. With that information, we can use the simplified version of Kepler’s 3rd Law to calculate the mass. M = a 3 / P 2 So now the method is clear. If we can measure the orbital period and the semi-major axis for just one star orbiting the black hole, then we can calculate the black hole’s mass and Schwarzschild Radius. If we can perform this measurement and calculation for two stars orbiting the black hole, then we can even double-check our values. That’s what we will do! DATA COLLECTION We will focus our data collection efforts on just two of the stars in this field, SO-2 and SO-102. First, let’s try to determine the orbital period of these two stars.
Use this video of the galactic center to determine how many years it took each of these two stars to complete one full orbit ( please note that the measurements for the star S-102 does not begin until 7 seconds into the video ). Use the pause button and the player bar to move carefully back and forth through the images, trying to obtain the most accurate measurement that you can. Record each star’s orbital period in the table below. Next, we will try to determine the semi-major axis of these two orbits in astronomical units (AU). This still image from the data set shows that the observations are calibrated with an arrow indicating an angular size of 0.1” (that is, a tenth of an arcsecond). At the galactic center, this arrow corresponds to a physical size of 812 AU. Use this information, along with a ruler to measure the semi-major axis of each orbit as accurately as you can. (It may help to print the still image with inverted colors , so you can measure on a flat hard surface.) Remember that the semi-major axis is one half of the major axis of an elliptical orbit. (This step involves a little tricky math. If you need help, watch this video .) Enter your values in the table below . Star Orbital Period (P) in Years Semi-Major Axis (a) in AU SO-2 16 years 726.512 AU SO-102 12 years 747.88 AU CALCULATIONS Use the simplified version of Kepler’s Third Law, along with the data you collected, to calculate the mass of the black hole. Enter your values in the table below . Measurement Black Hole Mass in Solar Masses Schwarzschild Radius in Kilometers Based on SO-2 Data 1,255,543 3,766,629 Based on SO-102 Data 2,423,495 7,270,485 The two stars provide two independent measures of the mass. The difference between these two measurements gives us a sense of how accurate our values are. If you notice a large difference, don’t be discouraged - remember that we did all of this data collection with a ruler and a video! Astronomers have much more sophisticated methods of analyzing the data. Even so, getting two independent measurements that have the same place value is reassuring. Based on your data and analysis, you should be able to confidently answer the question... The black hole at the center of our galaxy has a mass that is equivalent to ( Circle, underline or make bold your answer below ) a. hundreds of suns
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