An Exploration of Black Holes
Created by Thomas Waters, Acacia Arielle, and Hang Mai.
Our universe is an infinite collection of wondrously beautiful, elegant, and violent objects. A vast and unimaginable web of light and dark that stretches before us in the night sky. However, our perception of this image betrays the true nature of the tiny specks of light we observe. Comets, planets, stars, pulsars, and galaxies sprawl across the observable universe, each more magnificent and menacing than the last. However, one object dwarfs them all in its extremity, the black hole.
Background Image Credit: NASA, ESA
EXPERIMENT
Calculating the Mass of Sagittarius A*
Mathematics and physics are the universal languages of the cosmos. Our goal for this project was to develop an understanding of the concepts that lurk backstage to the fantastic astronomical discoveries that have been made throughout human history.
Using the combined knowledge of our predecessors, we have developed a mathematical model to calculate not only the mass of the supermassive black hole at the center of the Milky Way galaxy, but any system of orbiting bodies. The following details that process:
The galactic center, 27,000 lightyears from Earth.
Credit:
NASA, ESA, and the Hubble Heritage Team (STScI/AURA)
Data Collection
The data collection portion of our experiment centered around footage gathered by the European Southern Observatory using the Adaptive Optics (AO) NAOS-CONICA (NACO) instrument on the 8.2-m VLT YEPUN telescope. We then used an educational software called LoggerPro to manually track the motion of a star, called S2, that orbits closely around Sagittarius A* and to gather position versus time data points.
ESO Footage
This footage shows the motion of several stars clustered within a ten light day radius of the supermassive black hole Sagittarius A*. Our point of focus is the star S2 that shows a stable elliptical orbit about Sagittarius A*. The central point in the video, denoted by the yellow cross, is Sagittarius A*.
Credit: ESO
Credit:
NASA, ESA, N. Smith (University of California, Berkeley), and The Hubble Heritage Team (STScI/AURA)
Using the video analysis feature on Logger Pro, we developed this graph. The x-axis of the graph is the time in seconds and the y-axis represents the x-position (red curve) and the y-position (blue curve) of the star S2 in meters.
Logger Pro Graph
Credit: Thomas Waters
Force
As the data we gathered from the ESO footage was in terms of position and time, we deduced that a logical first step to calculating the mass of Sagittarius A* was to analyze the motion of S2 in terms of gravitational force. Therefore, we set out to derive a mathematical model using Newton's law of universal gravitation and Newton's second and third laws of motion.
Credit: the Physics Classroom
Force Representation of an Orbiting Body
This video is a representation of the gravitational force S2 experiences and how its velocity changes throughout its orbit around Sagittarius A*. At the aphelion (furthest point from Sgr A*), the gravitational force and velocity are at a minimum. At the perihelion (closest point so Sgr A*), the gravitational force and velocity are at a maximum.
Using Newton's third law of motion, we developed this free body diagram that shows the equivalent forces of gravity on S2 by Sagittarius A* and on Sagittarius A* by S2.
Free Body Diagram
Credit: Thomas Waters
Credit:
NASA, ESA, the Hubble Heritage Team (STScI/AURA), A. Nota (ESA/STScI), and the Westerlund 2 Science Team
Mathematical Model: Force
Energy
Since taking the second order derivative of our position vs. time equations to find the acceleration of S2 resulted in the loss of information and a large percent error, we decided to analyze the motion of S2 using energy to improve our accuracy.
Energy Bar Chart
This energy bar chart shows the gravitational potential and kinetic with S2 is at an initial position of the aphelion (furthest point from Sgr A*), and a final position at the perihelion (closest point to Sgr A*). When S2 is at the aphelion, its gravitational potential energy is at a maximum. As S2 approaches the perihelion, most of its gravitational potential energy is converted into kinetic energy.
Credit: Thomas Waters
Mathematical Model: Energy
Credit:
NASA, ESA, SSC, CXC and STScI
Credit:
NASA, ESA and the Hubble Heritage Team (STScI/AURA)-ESA/Hubble Collaboration
Individual Reflection
Thomas Waters:
After developing explanations using motion, graphs, models, etc., what do you understand better? What can you explain now that you could not before this assignment?
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One of the biggest challenges of this project was to develop a mathematical model. At this point, I feel much more confident in experimenting with mathematics. Choosing the correct equations and manipulating them to a point in which our gathered data could be used to solve them was an initially daunting task. However, the mathematics worked out quite nicely in this situation and I feel that in the future, I will be able to manipulate and derive mathematical models much more efficiently.
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Did some representations support your understanding better than others? Why?
As I still struggle with the idea that a tiny star is exerting a gravitational force on a supermassive black hole that is equivalent to the force that keeps it in orbit, I think that examining the motion of S2 was easier to grasp conceptually. The concepts of gravitational potential energy and kinetic energy have always felt more intuitive to me. Also, in many cases, the mathematics of energy calculations are often simpler.
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What do you still wonder? What questions did this bring up or what is left unanswered even after your investigation?
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My biggest unsated curiosity from this project is the effects of Einstein’s theories of special and general relativity. While our energy model got us very close to the actual value of the mass of Sagittarius A*, I would like to examine how much our margin of error would change by accounting for relativity. As S2 is moving over six-million meters per second, I theorize that special relativity will have a measurable effect on our observation of S2. Also, while S2 only comes within approximately 20 billion kilometers of Sgr A*, the gravitational forces generated by Sgr A* are massive. I believe that accounting for these facts could reduce our percent error further.
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What supported your group in working together?
Scheduling was likely the biggest hurdle we faced in this project. Trying to coordinate meetings was a challenge as all of us have our own educational, work, and personal obligations. However, I think that we communicated effectively through email. In addition to this, everyone in our group is genuinely interested in black hole research so completing this project was an exciting experience. This fact was a huge benefit to the group as we did not have to motivate uninterested members to participate.
Hang Mai:
After developing explanations using motion, graphs, models, etc., what do you understand better? What can you explain now that you could not before this assignment?
After this project, I understand more about how strong the gravitational force is inside black holes. Also, I got more knowledge about black holes and especially about Hawking Radiation.
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Did some representations support your understanding better than others? Why?
To derive the equations and find the mass, we used energy and force representation. Using force representation was a little challenge. I would use energy better than force representation. With a background in kinematics, I found that using energy is much easier than force for most of the cases. After finding the mass, it proves that everything happens around the world relate to physics, even black holes.
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What do you still wonder? What questions did this bring up or what is left unanswered even after your investigation?
I did a lot of research about black holes but I still can’t imagine how gravitational force appears in space. I was surprised when I found that information. Also, do black holes create dark matter?
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Acacia Arielle:
After developing explanations using motion, graphs, models, etc., what do you understand better? What can you explain now that you could not before this assignment?
After developing various explanations using motion, graphs, and models, I understand better how a black hole affects other things around it. It helped me see a black hole as a planetary being other than a source of destruction. Stars can orbit black holes like planets orbit stars. Just because a star is orbiting doesn’t mean it’s going to get sucked into the hole too.
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Did some representations support your understanding better than others? Why?
The energy representation helped support my understanding considerable better than the others. Having a slight background in kinematics, delving into the energy of what is happening to the star helped me understand the physics of it. The energy representation helped support my own understanding of kinematics than I thought I did previously.
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What do you still wonder? What questions did this bring up or what is left unanswered even after your investigation?
I still wonder about how a black hole dissolves. We didn’t get to research nearly as much as we wanted to, and we really wanted to find out how something so massive can dissolve into nothing. It’s amazing that something so huge and coherent can have the potential of becoming non-existent.
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What supported your group in working together? For me, what helped to work together was to meet regularly. It’s hard to know exactly what each person is supposed to be doing -- so when we are within near earshot of each other it helps to be able to ask questions straight away without having to wait for an answer. We also relied on each other’s strengths and worked with each other to get the best outcomes.
Credit:
NASA, ESA and the Hubble SM4 ERO Team
References
Boehle, A.; Ghez, A. M.; Schödel, R.; Meyer, L.; Yelda, S.; Albers, S.; Martinez, G. D.; Becklin, E. E.; Do, T.; Lu, J. R.; Matthews, K.;
Morris, M. R.; Sitarski, B.; Witzel, G (2016-07-19). "An Improved Distance and Mass Estimate for Sgr A* from a Multistar Orbit Analysis". The Astrophysical Journal. 830 (1): 17. arXiv:1607.05726. Bibcode:2016ApJ...830...17B. doi:10.3847/0004-637X/830/1/17
ESO. (2002, October 16). Motion of "S2" and Other Stars Around the Central Black Hole. Retrieved from
https://www.eso.org/public/usa/videos/eso0226a/
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​Information@eso.org. (n.d.). Images. Retrieved from https://spacetelescope.org/images/